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High-frequency and high-field magnetic resonance spectroscopy of diamond
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High-frequency and high-field magnetic resonance spectroscopy of diamond
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Content
HIGH-FREQUENCY AND HIGH-FIELD
MAGNETIC RESONANCE SPECTROSCOPY OF DIAMOND
by
Viktor Stepanov
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In partial Fulllment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
May, 2017
c ⃝ Copyright by
Viktor Stepanov
2016
Dedicated to my loving parents, Aleksandr and Galina,
and to the heart of my life, Julia.
Abstract
A nitrogen-vacancy (NV) center is a paramagnetic color center in diamond with
unique electronic, spin, and optical properties including its stable
uorescence (FL)
signalsandlongdecoherencetime. Moreover, itispossibletoinitializethespinstates
of NV centers by applying optical excitation and to readout the states by measuring
the FL intensity. Electron paramagnetic resonance (EPR) of a single NV center is
observed by measuring changes of the FL intensity, a magnetic resonance technique
known as optically detected magnetic resonance (ODMR) spectroscopy. In addition,
NV centers are extremely sensitive to their surrounding electron and nuclear spins.
Sensitivity of a single NV center to a single or a small ensemble of electron or nuclear
spins has been demonstrated using NV-based magnetic resonance (MR) techniques
at low magnetic elds. Despite these fascinating achievements, routine application of
NV-based magnetometry remains challenging.
Long coherence of a NV center is critical for most of NV-based MR techniques
that highly depends on the content of paramagnetic impurities in diamond. Char-
acterization of spin relaxations of NV centers is of particular interest to \road-map"
engineeringofdiamondmaterialsforsensingapplications. Inaddition,similartoEPR
spectroscopy, the spectral resolution of NV-based MR techniques is signicantly im-
provedathighmagneticeld, thushighlyadvantageousindistinguishingtargetspins
from other species (i.e. impurities existing in diamond). However, all NV-based MR
experiments have been performed at low microwave frequencies, while high frequency
iv
(HF) measurements are technologically challenging.
This dissertation is dedicated for investigation of diamond for single NV-based
magnetometry at high magnetic elds. The dissertation is organized as the following:
In Chapter 1, the remarkable mechanical, optical, electrical and magnetic proper-
ties of diamonds are overviewed and the motivation for NV-based magnetometry is
discussed. Investigation of diamond paramagnetic impurities presented in this disser-
tation relays on EPR techniques that are described in Chapter 2. In Chapter 3, the
development of conventional HF EPR spectrometer and low eld ODMR system for
a single NV center are presented in details. HF EPR and low eld ODMR spectro-
scopiesofparamagneticcentersindiamond,whicharestudiedinthisdissertation,are
presented in Chapter. 4. In Chapter 5, a double-electron-electron-resonance (DEER)
basedmethodtodetectconcentrationofparamagneticspinsindiamondisdeveloped.
The method was successfully applied to study and precisely characterize coherence
times of nitrogen impurities in diamond. In addition, method may be combined with
NV-based MR techniques to probe concentrations in a microscopic volume and to
characterize NV relaxations due to paramagnetic impurities in diamond. Finally, the
development of HF ODMR system for a single NV center in diamond is presented
in Chapter 6. Using HF ODMR system, the coherent control of a single NV center
was achieved for the rst time at high magnetic elds ( 4 Tesla), showing that
NV center retains all its unique optical properties up to 12 Tesla and demonstrating
the opportunity for HF NV-based magnetometry. In addition, investigation of T
1
relaxations of NV centers in nanodiamond (ND) was performed in a wide range of
magnetic eld (08 Tesla), showing that NV centers in NDs are highly sensitive to
surface paramagnetic impurities in NDs.
v
Contents
Abstract iv
Contents vi
List of Figures ix
List of Tables xviii
List of Abbreviations xix
List of Physical Constants xxi
List of Units xxii
1 Introduction 1
1.1 Properties of diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Defects in diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Motivation for NV-based nanoscale magnetometry at high magnetic
elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 EPR spectroscopy 10
2.1 Static spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Electron Zeeman interaction . . . . . . . . . . . . . . . . . . . 11
2.1.2 Zero-eld splitting . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Hyperne interaction . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.4 Nuclear Zeeman interaction . . . . . . . . . . . . . . . . . . . 15
2.1.5 Nuclear quadrupole interaction . . . . . . . . . . . . . . . . . 15
2.1.6 Weak dipole-dipole interactions . . . . . . . . . . . . . . . . . 16
2.2 cw-EPR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 cw-EPR signals . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 cw-EPR spectrum. . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Pulsed-EPR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Free induction decay . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Spin echo decay . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vi
3 Instrumentation 36
3.1 High-frequency EPR spectrometer . . . . . . . . . . . . . . . . . . . . 38
3.1.1 High-frequency high-power solid-state transmitter . . . . . . . 38
3.1.2 Quasioptical system. . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.3 Detection system . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.4 Cryogenic-free superconducting magnet and
4
He cryostat . . . 46
3.1.5 Sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Optically detected magnetic resonance system . . . . . . . . . . . . . 50
3.2.1 Spatial resolutions . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Imaging and tracking . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3 Photon statistics measurement . . . . . . . . . . . . . . . . . . 58
4 Magnetic resonance spectroscopy of diamond 61
4.1 High-frequency EPR spectroscopy of diamond . . . . . . . . . . . . . 61
4.1.1 cw-EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Pulsed-EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 ODMR experiments on a single NV center in diamond . . . . . . . . 69
4.2.1 cw-ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Pulsed-ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Measurement of paramagnetic spin concentration in a solid-state
system using double electron-electron resonance 82
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.1 Spin echo measurement . . . . . . . . . . . . . . . . . . . . . . 84
5.2.2 Double electron-electron resonance spectroscopy . . . . . . . . 86
5.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.1 Spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.2 Double electron-electron resonance . . . . . . . . . . . . . . . 91
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4.1 Determination of N spin concentration . . . . . . . . . . . . . 97
5.4.2 T
2
vs N concentration . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 High-frequency and high-eld optically detected magnetic reso-
nance of nitrogen-vacancy centers in diamond 104
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 HF ODMR system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 HF ODMR of a single NV center . . . . . . . . . . . . . . . . . . . . 106
6.4 T
1
eld-dependence of NV centers in nanodiamonds . . . . . . . . . . 111
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Conclusion 115
vii
A Averaging magnetic resonance signals 118
A.1 Averaging over the OU process . . . . . . . . . . . . . . . . . . . . . 118
A.2 Averaging DEER signals over ensemble of dipolar coupled spins . . . 120
B FL and ODMR signals of a single NV center 123
B.1 FL signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.2 ODMR signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
B.3 Autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . . 135
Bibliography 138
viii
List of Figures
1.1 Unitcellofthediamondlatticeshowingthesizeofthecubiccellawiththe
C-Cbond lengthofb
CC
andangle betweenbonds of109.5
◦
. xyz-frameis
used to dene four crystallographic directions in diamond along the C-C
bonds with the use of the Miller indices as [111], [
111], [1
11] and [11
1]. . 2
3.1 Overview of the HF EPR/DEER spectrometer . . . . . . . . . . . . . . . 37
3.2 Schematic overviews of the high-frequency high-power solid-state source
in the HFEPR spectrometer. (a) Conguration of single-frequency mode
usedforcwEPR,SEandDDmeasurements. (b)Congurationofdouble-
frequency mode used for DEER measurements. . . . . . . . . . . . . . . 40
3.3 Overview of the quasioptical system consisting of transmitter and receiver
stages. Thequasiopticalsystemisaperiodicfocusingsystemwiththefocal
lengthof254mmfortheGaussianwaves. HE
11
modeinacorrugatedhorn
connected to the source excites the Gaussian mode with high efficiency
(coupling between HE
11
mode and the Gaussian mode is 99 %). The
periodic focusing system also allows using the same quasioptics for a wide
range of frequencies. The isolation box to reduce background noises is
made of aluminum and the inside is covered by absorbers. . . . . . . . . 42
3.4 Characterization of the quasioptical variable attenuator. (a) Schematic
diagram of the variable attenuator. Transmission of a linearly polarized
microwavewiththeinitialelectriceld(E
i
)throughtherotatingandxed-
angle wire grid polarizers is given by E
f
=E
i
sin
2
. The direction of the
xedwiregridissettobeperpendiculartothepolarizationoftheincident
microwave. The angle between the incident microwave polarization and
theaxisoftherotatingwiregridpolarizer()isvariable. (b)Themeasured
attenuation as a function of . Blue (black in the print version) square
dots with error bars represent the measured attenuation and solid line
indicates the best t to the data using Eq. 3.4. For the measurement ofI
i
and I
f
, a pyroelectric detector (Eltec Instruments) was used. . . . . . . 43
ix
3.5 Circuit diagram of the detection system. Circuit diagrams to produce (a)
IF using subharmonically pumped mixer and (b) I and Q signals from
IF and 3 GHz reference signals. A subharmonically pumped mixer for
EPR detection provides an excellent isolation of the LO to the RF port
(>100 dB). The high-frequency LO consists of a broad-band microwave
synthesizer (2-20 GHz, Micro Lambda Wireless), a directional coupler, an
isolator, a PIN switch, an amplier and a frequency multiplier. The PIN
switch in the LO is for protection of the detection system. The super-
heterodyne detection has 1 GHz bandwidth and its noise temperature
(T
N
) is1200 K. IF power is controlled by a variable attenuator to opti-
mize the power to the IQ mixer. . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Overviewof varioussample holder congurations. Single crystal and thin-
lm samples are positioned directly on an end-plate with conductive sur-
face, and powder and aqueous/frozen solution samples are loaded to a
cylindrical bucket made of Te
on. Depending on the dimensions, samples
are placed either inside or near the bottom end of the waveguide. . . . . 48
3.7 Aqueous sample holder for room temperature HF EPR experiments at
115 GHz. [133] (a) Electric and magnetic eld components of microwave
in the sample holder. The sample is positioned on the aluminum tape.
(b) Schematics for aqueous sample design. The top and bottom caps are
made of Te
on. Doted lines indicate threads and volume for screws that
are used to tighten top and bottom caps. . . . . . . . . . . . . . . . . . 49
3.8 Diagramofthehome-builtODMRsystemforasingleNVcentermeasure-
ments at low magnetic elds. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.9 Numerical calculations of a diffraction limited spot (DLS) of the laser
beam focused by a microscope objective with NA = 1.4. (a) DLS in
a case of aberration free focusing, when the laser is focused within the
immersion oil (no diamond sample) (b) DLS for the laser focused through
oil-diamond interface with the probe depth of 10 m. (c) Lateral and (d)
axial intensity distributions of the DLS calculated in (a) (cyan), (b) (red)
and for a plane wave focused by a thin lens (gray). Dash white line in
(a) and (b) indicates the location of a geometric focus. Colorbar displays
the color code for the laser intensity. Calculation of DLS in (a) and (b)
wereperformedusingelectromagneticwavediffractionofaGaussianbeam
focusedbyamicroscopeobjectivewithahigh-numericalaperturethrough
a planar interface described in [134] with the parameters: (a) n
1
= n
2
=
1.52, NA = 1.4, (b) n
1
= 1.52, n
2
= 2.4, NA = 1.4 and with an input
Gaussianbeamwastethatisequaltotheradiusofobjectivebackaperture.
Planewavediffractionin(c)and(d)wascalculatedaccordingtothescalar
Fresnel-Kirchhoff diffraction theory [135]. . . . . . . . . . . . . . . . . . 54
x
3.10 Imaging and tracking of a single NV center. (a) Laser scanning pattern
usedtoimageadiamondsample. Laserpositionsinxyplanearecontrolled
by FSM in x and y steps at the xed positions of the sample stage. FL
signalsareintegratedover
s
timeconstantateachposition. (b)FLimage
obtained on type-Ib diamond crystal. Image was obtained using 100 by
100 points with
s
= 20 ms integration time. (c) Relative positioning of
the diamond sample with respect to the laser in a tracking loop, where
the focal point of the laser is controlled inxy plane with FSM and sample
is positioned along z axis using piezo translation stage. FL signals are
integrated over
s
time constant at each position. (d) FL signals and x, y,
z positions recording in the continuous tracking mode during 2.5 hours.
Parameters used in tracking:
s
= 20 ms, x = 25 nm, y = 25 nm, z =
40 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Photonstatisticsmeasurement. (a)DiagramoftheHanbury-Brown-Twiss
arrangement of detection of FL signals for second order photon correla-
tionfunctionmeasurement. (b)Experimentallyobtainedhistogramofthe
measured number of events for each detected time delay interval between
photons. The time bin width was set to 128 ps. The data was collected
over 6 hours.
0
was set to 19.3 ns ( 4 meters of BNC cable). (c)
g
(2)
() as obtained from the normalization and background signal correc-
tions (Appendix B.3) of the raw data presented in (b). g
(2)
(0) < 1/2
proves optical insulation of a single NV center (single quantum emitter). 59
4.1 Technicaldetailsofthecw-EPRlineshapedetectioninbulkdiamondsam-
ple. (a)cw-EPRlineshapeandthecorrespondinglineshapesignalrecorded
using lock-in amplier and application of modulation eld. L(B
0
) repre-
sents the absorption lineshape of EPR, L
′
(B
0
) - rst-derivative of L(B
0
),
which the signal detected by the lock-in amplier in cw-EPR measure-
ments, B
m
strength of modulation eld at the sample. (b) Conguration
of sample alignment with respect to external magnetic eld (B
0
). (hkl)
are the Miller indices of the polished face of a diamond sample. . . . . . 62
4.2 cw-EPR spectra of type-Ib diamonds measured at 115 GHz. Upper (or-
ange) and lower (magenta) data show the spectra obtained on the di-
amonds with the polished faces along(100) and (111) crystallographic
planes, respectively. The external magnetic eld was aligned orthogo-
nally to the faces as shown in Fig. 4.1b. Results were obtained withB
m
00.3 mT at 20 kHz frequency, 300 ms integration time of lock-in ampli-
er outputs and 0.13 mT/s sweep rate of magnetic eld. Simulations
(Sim.) of cw-EPR spectra were carried in Matlab using N spin Hamilto-
nian (Eqn. 4.2) and the procedure described in Sect. 2.2.2. . . . . . . . 63
xi
4.3 StructureandenergydiagramofNspinsindiamond. (a)StructureofNin
the diamond lattice. External magnetic eldB
0
is aligned at angle with
the[111]crystallographicaxis. (b)SchematicsofenergylevelsplittingofN
spins according to the dominant spin interactions (Zeeman and hyperne
interactions) for the case of = 0. (c) and (d) B
0
alignment for the case
of diamond samples with the polished face (100) and (111), respectively.
Polished faces are represented by the shaded planes. . . . . . . . . . . . 65
4.4 Spin echo measurements of T
1
and T
2
relaxation times of N spins in bulk
type-Ib diamond. (a) Transient signals recorded by a digital oscilloscope
in pulsed-EPR experiments at 115 GHz. Magnetic eld was applied along
[100] crystallographic direction. The strength of magnetic eld was set
to excite N spins in the central spectral line (Fig. 4.2). Signal was av-
eraged over 16 traces with the following parameters: =2 = 200 ns, =
450 ns, = 1.3 s, repetition rate = 20 ms. Sensitivity range of the fast
oscilloscopeisadjustedtointensityofSEsignalsresultinginsaturatedsig-
nals for microwave pulses. (b) SE intensity as a function of to measure
spin decoherence timeT
2
of central spins in bulk diamond with (111) face
orientation. The inset depicts Hahn echo sequence used to measure SE
decay. Each data point was taken with 32 averages, 20 ms repetition rate
and pulse parameters: =2 = 200 ns, = 350 ns, varied. SE decay
was t with single exponential function to extract T
2
. (c) SE intensity as
function of time interval T to measure spin-lattice relaxation time of cen-
tral spins in bulk diamond with (111) face orientation. The inset depicts
inversion recovery sequence used to measure SE decay. Each data point
was taken with 32 averages, 20 ms repetition rate and pulse parameters:
=2 = 150 ns, = 250 ns, = 1 s, T varied. SE decay was t with
single exponential function to extract T
1
. . . . . . . . . . . . . . . . . . 67
4.5 Structureofanitrogen-vacancycenterinthediamondlattice. N-nitrogen
14 atom, V - vacancy in the lattice, C - carbon 12 atom. . . . . . . . . . 69
4.6 Energy diagram of a nitrogen-vacancycenter in diamond at zero magnetic
eld. jg⟩ and je⟩ are wavefunctions in the electronic ground and excited
state of the NV center, respectively. Spin sub-levels are labeled according
to the spin projection (m
s
) along NV axis, which are split in the ground
and excited state due to the zero-eld splitting (Eqn. 4.3). Green arrows -
transition induced by optical excitation, red arrows - radiative decay from
je⟩ tojg⟩ state, black arrows - non-radiative transitions. . . . . . . . . . 70
xii
4.7 cw-ODMR measurements on a single NV center at zero magnetic eld
(upper panel) and at external magnetic eld of 5 mT. Magnetic eld
was applied along NV axis ([111] crystallographic direction). The insets
show schematic energy diagrams for the spin sub-levels according to the
dominant interaction terms in Eqn. 4.3, i:e: the Zeeman interaction and
zero-eld splitting terms. Fitting of cw-ODMR signals was performed
using Lorentzian lineshape to obtain amplitude (A) and half-width (∆!)
of NV transitions: A = 0.31 0.01 and ∆! = 15.9 1.4 MHz at 2870.0
0.7 MHz, A = 0.21 0.01 and ∆! = 7.0 0.8 MHz at 2734.8 0.5
MHz, A = 0.27 0.01 and ∆! = 10.5 0.5 MHz at 3006.3 0.5 MHz.
Possiblereasonforvariationsinamplitudeandlinewidthofthecw-ODMR
signals is frequency-dependent transmission of the microwave line [166]. 73
4.8 Transient FL signals of NV center. (a) Pulse sequence that was employed
to detect transient FL signals of NV centers. Duration of optical initial-
ization pulse and readout pulses (RO) was set to 5 s. FL was detected
using collection window of 50 ns. t
p
is duration of microwave pulse.
R
is the position of FL collection window with respect to the RO pulse. To
recordtransientFL,FLsignalsaremeasuredwithinthecollectionwindow
of 50 ns as function of
R
. (b) Transient FL signals for three duration of
microwave pulse: t
p
= 0 (blue), t
p
= 37.5 ns (red), t
p
= 75 ns (orange).
Single FL point was measured over 10
6
repetitions of the pulse sequence
at each
R
, to reduce the effect of short-noise, resulting in the overall in-
tegration time of 50 ms. Solid lines were obtained from the simulation
of transient FL signals based on the ve-level model for NV center (Ap-
pendix B.1). A good agreement was found for the optical excitation rate
of 5 MHz and Rabi oscillation frequency of 10.4 MHz. . . . . . . . . . . 75
4.9 Rabi oscillation measurement of a single NV center. (a) Pulse sequence to
detect FL signals in the Rabi oscillation measurement. Optical initializa-
tionpulsewassetto3s,readoutpulse(RO)-300ns,FLreadoutwindow
(Sig) - 500 ns. Position of pulse sequences to detect Max and Min FL sig-
nals are indicated by black arrows. Max-Sig-Min pulse sequence were
repeated over 10
6
times resulting in the integration time of 300 ms for FL
signals. (b) Rabi oscillation signal of a single NV center for j0⟩↔ j1⟩
transition at magnetic eld of 3 mT. The signal was normalized to the
populationinm
s
=0accordingtotheproceduredescribeinAppendixB.2.
The signal was t to the function/ cos(2Ω
R
t
p
) exp[(t
p
=T
R
)
2
], to ob-
tain the Rabi oscillation frequency - Ω
R
= 10:70:1 MHz and damping
time T
R
of 1:40:1s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xiii
4.10 SpinechomeasurementofasingleNVcenter. (a)Pulsesequencetodetect
FL signals in the SE decay measurement. Optical initialization pulse set
to 3 s, readout pulse (RO) - 300 ns, FL readout window (Sig) - 500 ns.
Position of the pulse sequences to detect Max and Min FL signals are
indicated by the black arrows. Max-Sig-Min pulse sequence were repeated
over 10
7
times resulting in the integration time of 300 ms for FL signals.
(b) SE decay signal of a single NV center for j0⟩↔ j1⟩ transition at
magnetic eld of 3 mT. The signal was normalized to the population
in m
s
= 0 according to the procedure describe din Appendix B.2. Good
agreementofthedataandtheSEdecay/ exp[(2=T
2
)
2
]wasfoundwith
T
2
decoherence time for single NV of 2.2 s. . . . . . . . . . . . . . . . . 80
5.1 SE measurements of type-Ib and type-IIa diamond crystals. (a) SE inten-
sity as a function of magnetic elds. The applied pulse sequence is shown
in the inset. In the measurement of the type-Ib diamond, the durations of
/2 and pulses were 150 ns and 250 ns and was 1.5s. The data were
taken with 32 averages with 20 ms of the repetition time. In the measure-
ment of the type-IIa diamond, the durations of the/2 and pulses were
250 ns and 450 ns and was 3s. The data were taken with 256 averages
with 20 ms of the repetition time. The magnetic eld was applied along
the [111] direction for type-Ib crystals and the [100] direction for type-
IIa. (b) SE intensity as a function of to measure spin decoherence time
T
2
. The decays of the SE were tted by a single exponential function to
extract T
2
(solid lines). The data of the type-Ib (type-IIa) diamond was
taken with 128 (256) averages. . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Double electron-electron resonance (DEER) spectroscopy of the type-Ib
and type-IIa diamond crystals. (a) three-pulse DEER sequence used in
the experiment, where t
1
and t
2
denote duration of =2 and pulses for
A spins, respectively. t
p
is duration of pulse for B spins. T is the delay
of t
p
from t
1
. (b)&(c) DEER spectrum of N spins in type-Ib and type-IIa
diamonds, respectively. The DEER signals were normalized by the SE
signals. Experimental parameters were t
1
= 250 ns, t
2
= 450 ns, t
p
= 450
ns, = 2.5 s, T = 2 s in case of type-Ib diamond, and t
1
= 250 ns,
t
2
= 450 ns, t
p
= 450 ns, = 110 s, T = 109.45 s in case of type-IIa
diamond. The data of the type-Ib (type-IIa) diamond was taken with 128
(256) averages. Purple and brown dashed lines represent the best t of
experimental data using Eqn. 5.10. . . . . . . . . . . . . . . . . . . . . . 87
5.3 Analyses of the
ip-
op process of N spins in type-Ib diamond. (a) Anal-
yses of a single exponential SE decay with T
2
= 950 ns (cyan) using
Eqn. (5.1). 4.9, 2.9 and 2 kHz of W
max
were obtained from the ts for
60 (black square), 80 (red diamond) and 100 (pink circle) ppm of N con-
centrations, respectively. (b) Flip-
op rate distribution among N spins
obtained using Eqn. (5.2) for 4.9 (black), 2.9 (red) and 2 kHz (pink) of
W
max
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
xiv
5.4 Schematics for the DEER model. L() is the lineshape function. !
m
is
the center frequency of Group m (m=15). !
A
and !
B
are microwave
frequencies of the probe and pump pulses, respectively. ! and !
j
are the
Larmor frequencies of A and B spins, respectively. and
j
are frequency
offsetsofAandBspinsfromthepumpandprobefrequencies,respectively.
A and B spins were chosen close to the probe and pump frequencies, to
indicate, that spins can be excited by a respective pulse with a small
frequency offsets. However, in general, as in our consideration, they can
be anywhere within the lineshape L. . . . . . . . . . . . . . . . . . . . . 92
5.5 Fit results of DEER spectrum. Left top, middle, bottom panels show
concentration n of N spins, half-width ∆! of inhomogeneous lineshape
and a t error
, respectively, as obtained from the t at a xedt
Ω
values.
The top (bottom) panel on the right shows the result of the t obtained
at t
Ω
= 100 ns (305 ns). The result with t
Ω
= 305 ns is the best t. The
grey shaded area on the left indicates ts with a large
. . . . . . . . . . 98
5.6 (a) Summary of the obtained ∆!. (b) Summary of the obtained t
Ω
. t
2
is
the duration of -pulse used in the present study. . . . . . . . . . . . . . 100
5.7 1/T
2
of N spins as a function of the N concentration. Open squares rep-
resent experimentally obtained data, orange solid line is the best t of
the data to the model of decoherence rate described by Eqn. 5.11. Yellow
region represents the plot of Eqn. 5.11 with the xed
C
in the range of
150250 s and a slope C=0.0139 s
1
ppm
1
as obtained from the best
t of the data. Dashed orange line shows the best t of the data using
Eqn. 5.11 without the nuclear spin decoherence (1=T
13
C
2
= 0). . . . . . . 102
6.1 Overview of the HF ODMR system. The HF source in the HF excitation
component is tunable continuously in the range of 107-120 GHz and 215-
240 GHz. HF microwaves are guided by quasioptics and a corrugated
waveguide. The NV detection system consists of a 532 nm cw diode-
pumpedsolidstatelaser,anAOM,bercouplers,opticallters,aBS,and
APDs. TheexcitationlaserisappliedtoNVcentersthroughamicroscope
objective located at the center of the 12.1 Tesla superconducting magnet
and the FL signals of NV centers are collected by the same objective.
The FL signals are ltered by optical lters in the NV detection system.
For autocorrelation measurements, the FL signals are split into two and
detected by two separate APDs. The microscope system consists of a
microscopeobjective,az-translationstage,andthecorrugatedwaveguide.
The sample stage is supported by the z-translation stage. . . . . . . . . 107
xv
6.2 Fluorescence signals of a single NV center in type-Ib diamond. (a) FL
intensity image of a type-Ib diamond crystal. The scanning area is 5
5 m
2
. Solid circle indicates a single NV center that was used in the
subsequent measurements. (b) Autocorrelation curve observed from the
the single NV center. The observation of g
2
( = 0) < 0:5 conrms the
detection of the single NV center. (c) Magnetic eld dependence of the
singleNVcenterFLintensity. Theeldwasappliedalongthe(111)axisof
the diamond within 8 degrees. The inset shows reduction of FL intensity
at0.05 and 0.1 Tesla due to LAC of the excited and ground states of the
NV center, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 ODMR measurements on a single NV center in type-Ib diamond at 115
GHz. (a) cw-ODMR measurement of the single NV center at 115 GHz.
The ODMR signal of the single NV center was observed at 4.2022 Tesla.
The solid line indicates a t to the Gaussian function. The inset shows
the in-situ EPR measurement on ensemble N spins in the diamond. (b)
Rabi oscillation experiment. The frequency of the observed Rabi oscilla-
tions was 0.8 MHz. The inset shows the applied pulse sequence consist-
ing of the initialization (Init.) and readout (RO) pulses by the 532 nm
laser (Exc.), microwave pulse (MW) of length t
P
, and FL signals (Sig.).
Init.=4 s and RO=Sig.=300 ns were used in the measurement. t
P
was
varied. c) Pulsed-ODMR as a function of magnetic eld. The solid line
indicates a t to the Gaussian function. The inset shows the pulse se-
quence. Init.=4 s, RO=Sig.=300 ns, and t
MW
=500 ns were used in the
measurement. (d) The spin echo measurement to determine the spin de-
coherence time (T
2
) of the single NV center. The solid line indicates a t
to exp[(2=T
2
)
3
] [127]. The inset shows the pulse sequence. Init.=4 s,
RO=Sig.=300 ns, =2=250 ns, and =600 ns were used in the measure-
ment. was varied. The pulse sequence was repeated on the order of 10
6
times to obtain a single point in all measurements. . . . . . . . . . . . . 110
6.4 T
1
measurementsofNVcentersinNDs. (a)T
1
measurementofNVcenters
inaND(ND1). Thesolidlineindicatesattothesingleexponentialdecay
function,exp(T=T
1
). AFLimageofND1isshownintheinset. Theinset
also shows the pulse sequence consisting of the initialization (Init.) and
readout (RO) pulses by the 532 nm laser (Exc.), and FL signals (Sig.).
T was varied. Init.=4 s, RO=3 s, and Sig.=300 ns were used. The
reference (Ref.) of 300 ns was measured 2 s after the signal. The pulse
sequence was repeated on the order of 10
6
times to obtain a single point
in all measurements. In addition, the inset shows T
1
measurement of a
single NV center in the type-Ib bulk diamond obtained using the same
pulsesequenceabove(T
1
= 1:10:5ms). (b)T
1
asafunctionofmagnetic
eld for two NDs (ND1 and ND2). . . . . . . . . . . . . . . . . . . . . . 112
xvi
B.1 Five-level model for NV center to model transient FL signals in pulsed-
ODMR measurements [68]. Ground spins states are labeled as j1⟩
jm
s
= 0⟩,j2⟩jm
s
=1⟩, while excited spin states are labeled asj3⟩
jm
s
= 0⟩, j4⟩ jm
s
=1⟩. Singlet metastable states are considered
as a single level j5⟩. Green solid (dashed) arrows indicate strong spin-
conserving (weak spin-non-conserving) optical transitions from electronic
ground to excited states with the rate r (ϵr). Red solid (dashed) arrows
indicate radiative spin-conserving (weak spin-non-conserving) transitions
from electronic excited to the ground states with the rate
(ϵ
). Black
arrows indicate non-radiative transitions with the shelving rate
S
for
j4⟩!j5⟩ transition and polarization rate
P
forj5⟩!j1⟩ transition. . . 124
B.2 Population dynamics of NV center in pulsed-ODMR measurement. Re-
sultswereobtainedbasedontheve-levelmodelforNVcenterandforthe
pulsesequenceshownonthetoppanel: initializationpulse-2s,rstdark
time interval - 2 s, microwave pulse - 200 ns (Ω = 2.5 MHz), second
dark time interval - 1.8 s, readout pulse - 1 s, third dark time interval
- 1 s.
11
(t) and
22
(t) are populations in the m
s
= 0 and m
s
= 1
ground states. Transient FL is represented by NV populations in the ex-
cited states (
33
(t) +
44
(t)). Simulations were performed for initial NV
spin state
11
(0)= 1=3 and
11
(0) = 2=3 and r = 10 MHz. . . . . . . . 127
B.3 Normalization and background correction of experimentally obtained au-
tocorrelation function g
(2)
(). (a) Raw data of autocorrelation function.
(b)ProleoftheFLspotimagerecordedwithAPD1. (c)ProleoftheFL
spot image recorded with APD2. FL (f
1
, f
2
) and BG (b
1
, b
2
) signals are
obtained from a Gaussian t of the corresponding proles. (d) g
(2)
() of
FL photons as obtained from the C() according to Eqn. B.29. g
(2)
(0)<
1/2 proves FL signals due to a single NV center. . . . . . . . . . . . . . 136
xvii
List of Tables
5.1 Summary of ∆! and n for the studied type-IIa and type-Ib diamonds as
extracted from the analyses of the DEER data. The errors of n and ∆!
were estimated as 95 % condence interval for the t parameters. . . . . 99
xviii
List of Abbreviations
AC Alternating Current
AOM Acousto Optical Modulator
APD Avalanche Photodiode
BNC Bayonet Neill-Concelman
BDPA ,
-BisDiphenylene--PhenylAllyl
BS Beamsplitter
CVD Chemical Vapor Deposition
CW Continuous-Wave
DAQ Data Acquisition Card
DC Direct Current
DEER Double Electron-Electron Resonance
DLS Diffraction Limited Spot
DM Dichroic Mirror
ENDOR Electron Nuclear Double Resonance
EPR Electron Paramagnetic Resonance
FID Free Induction Decay
FL Fluorescence
FSM Fast Steering Mirror
FWHM Full Width at Half Maximum
GPIB General Purpose Interface Bus
HF High-Frequency
HBT Hanbury-Brown-Twiss
HPHT High-Pressure; High-Temperature
I In-phase
IF Intermediate Frequency
IR Infrared
ISC Intersystem Crossing
LAC Level Anti-Crossing
LHe Liquid Helium
LNA Local Area Network
LNA Low-Noise Amplier
LO Local Oscillator
LP Longpass lter
MMF Multi-Mode optical Fiber
xix
N Nitrogen
NA Numerical Aperture
ND NanoDiamond
ND Nuetral Density
NMR Nuclear Magnetic Resonance
NV Nitrogen-Vacancy
ODMR Optically Detected Magnetic Resonance
OU Ornstein-Uhlenbeck
PC Paramagnetic Center
PC Personal Computer
PIN P-I-N
RO Readout
RWA Rotating Wave Approximation
SMF Single-Mode optical Fiber
Q Quadrature
RF Radio Frequency
SE Spin Echo
SMA SubMiniature version A
SPI Surface Paramagnetic Impurity
TCSPC Time-Correlated Single Photon Counting
TLS Two-Level System
TTL Transistor-Transistor Logic
USB Universal Serial Bus
ZPL Zero-Ponon Line
xx
List of Physical Constants
h 6:6260695710
34
kgm
2
s
1
or Js Planck constant
k
B
1:380648810
23
kgm
2
s
2
K
1
or JK
1
Boltzmann constant
0
410
7
NA
2
or TmA
1
Permeability of free space
B
9:2740096810
24
kgm
2
s
2
T
1
or JT
1
Bohr magneton
xxi
List of Units
A Ampere
dB Decibel
g Gram
GHz Gigahertz (=110
9
Hz)
Hz Hertz (=1 cycles
1
)
J Joule (=1 m
2
kg)
K Kelvin
kHz Kilohertz (=110
3
Hz)
ppm Parts per million (=110
6
%)
m Meter
MHz Megahertz (=110
6
Hz)
mA Milliampere (=110
3
s)
ms Millisecond (=110
3
s)
mT Millitesla (=110
3
mT)
mW Milliwatt (=110
3
W)
ns Nanosecond (=110
9
s)
s Second
T Tesla
m Micrometer (=110
6
m)
s Microsecond (=110
6
s)
T Microtesla (=110
6
T)
xxii
Chapter 1
Introduction
Diamond minerals are well known from ancient times. It is believed that diamond
was mined for the rst time in 800 BC in India. Since the discovery of diamond,
diamondhasbeenappreciatedforitshardnessandprimarilyusedasanamulet,which
supposed to give strength and protection to the owner. Following the development
of diamond polishing technique that is associated with Bergham in 1476, the rst
use of diamond as a jewelery is dated back in 1477, when Archduke Maxillian made
an engagement present to Mary of Burgundy in the form of the diamond ring. First
experiments on combustion of diamond were carried out in 1772 [1] and 1797 [2]
revealing that diamond is composed solely from carbon atoms. Since then, many at-
tempt had been made to obtain a synthetic diamond, however, the transformation of
carbon sources to diamond became feasible after a high pressure technology required
to form diamond became available in the early 1950's [3]. First engineered dia-
mond was reported by General Electric in 1955 using high-pressure high-temperature
(HPHT) synthesis [4], followed by the rst successful low pressure synthesis in 1958
using chemical vapor deposition (CVD) [5, 6]. Owning to high stiffness and thermal
conductivities, synthetic diamonds were primarily used for industrial cutting tools.
Recent progress in CVD syntheses allowed to reduce the cost of synthetic diamond
1
Chapter 1. Introduction 2
1.54A
C C
b
°
−
=
3.57A a
°
=
109.5°
x
y
z
[111]
[111]
[111]
[111]
[100]
[001]
[010]
Figure 1.1: Unit cell of the diamond lattice showing the size of the cubic cella with
the C-C bond length of b
CC
and angle between bonds of 109.5
◦
. xyz-frame is used
to dene four crystallographic directions in diamond along the C-C bonds with the
use of the Miller indices as [111], [
111], [1
11] and [11
1].
and opened possibility for a new range of applications that would utilize not only
superb hardness of a diamond, but also remarkable optical, thermal, chemical and
electronic properties [7, 8]. In addition, CVD method allows fabrication of diamond
with a tailored properties for a specic application leading to various implementa-
tions and proposal of synthetic diamond in the eld of fundamental physics [9{14],
biomedicine [15], high power electronics [16, 17], nanoscale thermometry [18, 19] and
magnetometry [20{34], as well as quantum optics and laser technologies [35, 36].
1.1 Properties of diamond
While a single carbon atom at rest has an electron conguration 1s
2
2s
2
2p
2
, the stable
diamond lattice conguration is adopted through bonding of hybridized sp
3
electron
orbitals of tetrahedral geometry leading to strong covalent bonding of each carbon
atomwithfourneighbors(co-ordinationnumberis4). Theresultingcrystalstructure
Chapter 1. Introduction 3
can be represented by two interpenetrated face-centered cubic Bravais lattices with a
unit cell presented in Fig. 1.1. Diamond unit cell consists of 8 effective carbon atoms
(# of atoms inside the cell + (# of face centered atoms)/6 + (# of corner atoms)/8
) with the size of unit cella =0:357 nm and the length of carbon-carbon (C-C) bond
b
CC
=
p
3a=4 = 0:154 nm [37]. Crystallographic axes in diamond are dened along
four possible orientations of C-C bonds in diamond with the relative angle between
bonds of 109.5
◦
(see Fig. 1.1).
Strong covalent C-C bond gives diamond unique mechanical and wear properties
with diamond being the hardness known substance [37]. Moreover, unique C-C bond
allows for an efficient transfer of lattice vibrations resulting in the highest known
thermal conductivity (2500 W/m K) [38]. These properties were rst utilized for
industrial cutting tool applications, while recently diamond is used for high speed
dry machining of highly abrasive nonmetallic materials for automotive and aerospace
industries. In addition, stiffness of diamond is employed in acoustic applications such
as micromechanical oscillators [39], surface acoustic wave devices [40] and tweeter
domes [41]. Unique thermal properties of diamond are nding uses for thermal man-
agement in high-power electronic, opto-electronic and microwave devices [7, 8], e:g:
multi-kilowatt CO
2
lasers [42], megawatt gyrotrons [43], GaN base high-power RF
devices [44], etc:
Furthermore,strongcovalentbondingresultsinlowchemicalreactivity,highresis-
tivitytooxidationandacids,andlowsusceptibilitytohardradiation. Duetochemical
properties, diamonds are extensively studied for applications in electrochemistry and
electroanalyses as an electrodes offering several advantages over current electrodes [7,
45],i:e:highresistivitytohydro
uoricacidandfouling[46], reducedbackgroundcur-
rents [47], electrochemical sensing in corrosive environments[48],etc. Radiation hard
property combined with a high carrier mobilities in diamond ( 4500 cm
2
V
1
s
1
for
Chapter 1. Introduction 4
electrons and 3800 cm
2
V
1
s
1
for holes [49]) led to the developments in the high
energyresearch. Inparticular,diamondbaseddetectorsfor-,-,
-particles[50,51],
aswellasneutron[52],protonandelectronradiation[53],withcommerciallyavailable
biomedical dosimeters available for X- and
-ray detection in living organisms [54].
While most diamond based detectors are more expensive than other semiconductor
based detectors, they are essential for high precision measurements [8].
Shortnatureofthe C-C bonds givesrisetoa largeenergysplitting betweenbond-
ing and anti-bonding carbon orbitals resulting in the wide bang gap for diamond (
5.47 eV) [55]. Owning to the large band gap, diamond crystals are typically trans-
parent in the wide range of electromagnetic radiation from middle ultraviolet (> 230
nm)extendingtofar-infrared(500m)withatwo-andthree-phononabsorptionin
near-infrared(2.5-6.5m)andbelowthebandgap(<230nm)[56]. Combinedwith
thermal properties, low birefringence CVD diamond are used as an optical windows
in high-power lasers [42]. In addition, using high-quality CVD diamond as intracav-
ity heat spreader allowed two orders of magnitude improvement of power output for
semiconductor disc based laser in the wavelength range of 2-2.5m [57]. Most recent
achievement for optical applications include diamond as an active medium for Ra-
man laser offering two orders of magnitude improvement of gure of merit (thermal
lensing) when comparing to other conventional gain crystals [58].
1.2 Defects in diamond
Early optical spectroscopy investigations of diamonds have revealed nitrogen related
defects that are typically the most abundant defects in natural and synthetic di-
amonds giving rise to a distinct IR absorption pattern in the range of 700 -1600
cm
1
[59]. According to the content of nitrogen atoms (based on IR spectrum), dia-
monds are typically classied into two major groups of type-I (concentration of nitro-
Chapter 1. Introduction 5
genrelateddefects>1ppm)andtype-II(<1ppm)diamonds[60,61],where1ppm
is a sensitivity limit of IR spectroscopy in diamond. Further investigation of type-I
diamonds allowed to distinguish three major forms in which nitrogen atoms prefer-
entially incorporate into diamond lattice, i:e: two neighboring single substitutional
nitrogen atoms (A center with the strongest absorption peak at 1282 cm
1
) [62], four
substitutional nitrogen atoms surrounding single vacancy (B center, 1332 cm
1
) [63],
andanisolatedsinglesubstitutionalnitrogenatom(Ccenter,1344cm
1
)[64]. There-
fore, type-I diamonds that predominantly contain A, B or C centers are classied as
type-IaA, type-IaB or type-Ib, respectively. Most of natural diamonds fall into type-
Ia category with nitrogen aggregates of the form of A or B centers, while synthetic
diamonds typically contain C centers and hence are of type-Ib [59]. Furthermore,
type-II diamonds are subdivided according to the content of boron atoms into type-
IIa with no measurable nitrogen and boron impurities and type-IIb diamonds that
show semi-conductivity due to boron defects, which are acceptor centers with elec-
tronic ground state of 0.37 eV above the valance band of diamond [65]. Unique
spectroscopicpropertiesofdefectsindiamondresultinadistinctcoloringofdiamond
crystals, e:g: type-Ib is of yellow color, while type-IIb appears as blue.
Moreover, many impurities in diamond show paramagnetism due to charge imbal-
ance that results in unpaired electrons for an impurity related defect. In fact, most
of the defects in diamond have been identied using electron paramagnetic resonance
(EPR)spectroscopymakingupalonglistofparamagneticcenters(PCs)(Readerwho
is interested in a comprehensive review of PCs should see Ref. [66]) Among PCs, the
mostintriguingparamagneticcenterindiamondthatboostedtheinterestofstudying
paramagnetism in diamonds over the last decade is nitrogen-vacancy (NV) center.
NV center consists of a single substitutional nitrogen and a neighboring vacancy
predominantly in a negative charge state resulting in spin one (S = 1) for NV. The
Chapter 1. Introduction 6
unique properties of NV center arise from spin-selective relaxations from electronic
excited state [67, 68], short lifetime of the excited states ( 13 ns) [69] and stable
uorescence signals (no blinking or bleaching) allowing for an optical imaging and
optical readout of a single NV center spin [67, 70]. In addition, due to intrinsi-
cally low concentration of paramagnetic centers and weak spin-phonon coupling in
diamond, NV spin has exceptionally weak relaxations comparing to spin systems in
other solids [13, 14, 71{74]. Being an atomic scale system with unique optical and
magneticproperties, NVcenterindiamondhasbeenasubjectofvariousapplications
and proposals in the eld of spintronics [75], quantum information processing [76{84]
as well as nanoscale sensing of temperature [18, 19] and small magnetic elds [20{22,
29{33, 85{90].
1.3 Motivation for NV-based nanoscale
magnetometry at high magnetic elds
Since early foundation of quantum physics, many spectroscopic techniques were de-
veloped to control quantum system and to probe their quantum properties such as
electronic structure, angular, spin, rotational and vibrational degrees of freedom and
their interactions with local environments. Due to sensitivities of available instru-
mentations, the experiments often rely on large ensemble of quantum systems. In
case each quantum system in ensemble is prepared in an identical state, the ensemble
signals would represent an amplied signal of a single quantum system. However,
in practice, it is extremely difficult to prepare quantum system with same physical
properties, same quantum state and especially with same environmental interactions.
Analyses of ensemble experiments is typically accompanied with certain assumptions
about system inhomogeneities or heterogeneities, which should be, in principle, taken
Chapter 1. Introduction 7
into account, but difficult to validate in practice. From fundamental point of view,
studyofasinglequantumsystemwithitslocalenvironmentprovidesagreaterinsight
into system interactions that can be probed by continuous and pulsed spectroscopic
techniquesandcombinedwithensemblemeasurementsmayfacilitatecharacterization
ofcertainsysteminhomogeneitiesandheterogeneities. Studyofasinglequantumsys-
tem can be viewed as a tool to probe physical models for a quantum system allowing
forhomogeneouspropertiestobeextractedandhencemorequalitativeanalysestobe
obtainedofvariationsinsystemstructure, interactionsandrelaxations. Additionally,
thenestructureofthespectroscopicparameterobtainedfromtheensemblemeasure-
ments may be potentially masked by its distributions due system inhomogeneities,
but on the other hand, could be resolved from a single quantum system measure-
ment. Therefore, single quantum system measurement is of particular importance for
fundamental studies and practical applications.
One of the most powerful and versatile analytical tools available today is EPR
spectroscopy that has been widely applied to probe and study local structures and
dynamic properties of various compounds in liquids and solids; for example, struc-
tures and dynamics in biological molecules, magnetic structures and relaxations in
magnetic molecules and quantum coherence in solid-state spin systems. However,
intrinsically low sensitivity of magnetic resonance (MR) techniques precludes mag-
netic resonance analyses of very small volume of samples. For example, sensitivity of
conventional EPR techniques is limited to 10
10
electron spins at room temperature
requiring large amount of sample to be employed. Therefore, EPR studies are typi-
cally limited to large ensemble and are affected by ensemble averaging effects. One
way to improve the sensitivity of EPR techniques, is to detect a dipolar coupling of
a single target spin. This requirement sets several limitations on a potential probe
of dipolar elds, i:e: it should be of a nanoscale size as dipolar eld of a single spin
Chapter 1. Introduction 8
extends over several tens of nanometers, sensitive to small magnetic elds (> 0.1
mT) and should have a readout capability. For investigations of a single biological
molecule at physiological conditions, excellent chemical properties, bio-comparability
and operation at room temperature are desired. Satisfying all of the above makes
NV center in diamond an ideal candidate for a single spin MR detection. In fact,
using optically detected magnetic resonance (ODMR) of a single NV center, recent
experimental demonstrations show NV center sensitivity to a single electron [90] and
nuclear spin [73] inside the diamond. In addition, detection of a single external and
small ensemble of external nuclear spins has been already achieved [30, 91]. Never-
theless,theseareuniqueandrearexamplesofNV-basedmagnetometry,whileroutine
implementation of NV centers for nanoscale MR detection remains challenging.
Most of a single NV ODMR techniques are based on coherent manipulation using
pulsed-EPR spectroscopy and a long coherence is critical for small magnetic eld
sensing. For a strong dipolar couping to an external target spin located on the
surface of the diamond, implementation of shallow NV center is desired with the
depth below diamond surface of 2-10 nm. This can be achieved with the use of NV
engineering techniques developed over the last decade [92{98]. However, when NV
center is close to the surface, its coherent properties become largely affected by the
surface paramagnetic impurities (SPI). Moreover, engineering of shallow NV center
is accompanied with creation of paramagnetic single substitutional nitrogen spins (N
spins,alsoknownasP1centersinEPRwithelectronspinS=1/2)thatarethemajor
source of decoherence of electron spins in bulk diamond. In addition, NV is affected
by nuclear spin decoherence due to carbon isotope
13
C that posses a nuclear spin
(I =1=2). Therefore, surface and intrinsic paramagnetic defects limit the sensitivity
of NV based ODMR nanoscale magnetometry.
Furthermore, NV ODMR techniques relay on independent control of a target spin
Chapter 1. Introduction 9
by microwave excitation requiring a good spectral resolution in ODMR spectrum.
Spectral resolution in ODMR, same as in EPR, depend on g-factors of spin systems
and strength of external magnetic eld. Magnitude of a g-factor as well anisotropy of
g-factororiginatefromaspin-orbitinteractionofanunpairedelectron. Whileg-factor
of a free electron is 2.0023, low spin-orbit coupling for N spins and SPI results in a
similar g-factors of 2.0028 for N spins and 2.0029 for SPI. On the other hand,
g-factor of a commonly used spin-label for biological molecules is 2.002 - 2.004.
Similar g-factors of diamond impurities and a target spin, especially in the case of
a spin-label, makes it impractical to differentiate between ODMR spectrum at low
magnetic elds (< 1 Tesla) and precludes from studying spectrum and dynamics of
a target spin. All of the current applications of NV ODMR magnetomtery has been
demostratedatextremelylowmagneticeldsof0.1Teslamakingthemuniqueand
important demonstrations of NV center sensitivities rather than powerful magnetic
resonance technique. Therefore, development of high eld (HF) ODMR system for
a single NV center, which would allow for a ne spectral resolution of ODMR sig-
nals, would be an essential step towards routine applications of NV-based nanoscale
magnetometry.
Chapter 2
EPR spectroscopy
EPR spectroscopy is a well established technique, which is powerful to probe struc-
ture and electron/nuclear composition of atomic-scale environment of paramagnetic
centers/defects as it is sensitive to electronic properties and magnetic interactions of
unpaired electrons. In combination with pulsed-EPR techniques, EPR spectroscopy
allowsinvestigationoffastspindynamicsanddynamicinteractionstonearbynuclear
and electron spins. For example, EPR spectroscopy has been successfully applied to
investigate local structures and dynamic properties of various biological systems [99{
104], study nature of decoherence in solid state systems [74, 105] that are essential
for quantum memory devices and serve as a key tool to study and control mesoscopic
systems[73, 106, 107]forinvestigationoffundamentalscience[9{11]andapplications
to quantum information processing [78, 108, 109].
Investigationsandapplicationsofdiamondmaterialspresentedinthisdissertation
solely relay on EPR techniques as a main tool to study the paramagnetic defects
in diamond. Therefore, basic concepts of EPR spectroscopy are introduced in this
chapter to support the interpretation of the results presented herein. In particular,
short introduction to spin Hamiltonian formalism of bound electronic states in solids,
spin relaxations, continuous wave (cw-) and pulsed-EPR techniques are given.
10
Chapter 2. EPR spectroscopy 11
2.1 Static spin Hamiltonian
EPR spectroscopy is often used to study electronic state of paramagnetic species due
to electrostatic interactions to lattice ions as well as their magnetic interactions with
the local environments due to electrons and nuclei spins. According to Abragam and
Pryce [110], the energy states of unpaired electrons in solids may be conveniently
described with the static spin Hamiltonian that explicitly depends on spin degrees of
freedom, while spatial degrees of freedom of electronic wavefunctions are accounted
bymagneticconstants,whichareusuallydescribedbysecond-ranktensors. Ageneral
spin Hamiltonian (H
0
) for electron spin S in externally applied magnetic eld (B
0
)
interactingwithnnearbynuclearspinsI andcollectionofremoteelectronandnuclear
spins is given by [110{112]
^
H
0
=
^
H
Ze
+
^
H
ZFS
+
^
H
HF
+
^
H
Zn
+
^
H
Qn
+
^
H
DD
=
B
B
0
↔
g
^
S+
^
S
↔
D
^
S+
n
∑
i=1
(
^
S
↔
A
i
^
I
i
+g
N;i
N
I
i
B+
^
I
i
↔
P
i
^
I
i
)
+
^
H
DD
;
(2.1)
where
^
S and
^
I
i
are electron and i
th
nuclear spin operator, respectively. The terms
in Eqn. 2.1 describe: electron Zeeman interaction (
^
H
Ze
), zero-elds splitting (
^
H
ZFS
),
hyperne interactions between electron and nearby n nuclear spins (
^
H
HF
), nuclear
Zeeman interactions (
^
H
Zn
), nuclear quadrupole interactions (
^
H
Qn
) and weak dipole-
dipole interactions with remote electron and nuclear spins (
^
H
DD
).
2.1.1 Electron Zeeman interaction
Interactionofthemagneticmomentofelectronspinwithexternallyappliedmagnetic
eld B
0
is described by the electron Zeeman term in Eqn. 2.1. In a case of unpaired
electron in an isolated ion, electron magnetic moment arises from orbital and spin
angular degrees of freedom, with a magnetic moment operator ^ :
^ =
B
(
^
L+g
e
^
S); (2.2)
Chapter 2. EPR spectroscopy 12
where
B
is the Bohr magneton, g
e
is the g-factor of a free electron and
^
L is the
orbital angular momentum operator. When an ion is placed into the crystal, the
atomic orbitals interact with electrostatic potential (crystal eld), which originates
fromsurroundingcrystalions. Duetothecrystaleld,theorbitalangularmomentum
of the ground state is quenched (⟨L
z
⟩ = 0) [113] and the magnetic moment of an un-
paired electron is given as ^ =g
e
B
^
S. However, spin-orbit interaction (LS coupling)
partially couples ground and excited states resulting in a nite orbital momentum in
the ground state. To account for this, Russel-Saunders spin-orbit coupling (/
^
L
^
S) is
considered alongside with the magnetic moment interaction with external magnetic
eld (/ ^ B
0
). The total Hamiltonian of electron spin is then described by
^
H =
^
H
Ze
+
^
H
LS
=
B
B
0
(
^
L+g
e
^
S)+
LS
^
L
^
S; (2.3)
where
LS
isaspin-orbitcouplingconstant. Usingasecond-orderperturbationtheory
and treating LJ term as a perturbation, an effective spin Hamiltonian is derived as
^
H =
B
B
0
(g
e
↔
I +2
LS
↔
)
^
S+
2
LS
^
S
↔
^
S+
2
B
B
0
↔
B
0
; (2.4)
with a unit tensor
↔
I and a spin-orbit coupling tensor
↔
given by
↔
=
∑
k̸=0
⟨
0
j
^
Lj
k
⟩⟨
k
j
^
Lj
0
⟩
ϵ
0
ϵ
k
;
where j
0
⟩, ϵ
0
and j
k
⟩, ϵ
k
are wavefunctions and energies of the ground and k
th
excited state, respectively.
The rst term in Eqn. 2.4 is an effective spin Zeeman interaction term that is
reducedtotheformgiveninEqn.2.1withtheuseofaneffectiveg-tensor(
↔
g),dened
as
↔
g =g
e
↔
I +2
LS
↔
;
which describes the effect of spin-orbit interaction on anisotropy and changes of g-
factorfromisotropicg
e
↔
I ofafreeelectronspin. Thesecondtermresultsinazero-eld
Chapter 2. EPR spectroscopy 13
splitting of the spin states due to crystal eld, which is non-zero for a spin system
with S > 1=2 and discussed in the following section. The third term shifts all spin
levels by the same amount and therefore has no effect in the magnetic resonance
experiments. In most cases, effective g-tesnor can be diagonalized and its principle
axes are considered as the molecular axes with all other spin interaction terms in
Eqn. 2.1 referred to this frame.
2.1.2 Zero-eld splitting
InSect.2.1.1,ithasbeenshownthatinthepresenceofspin-orbitinteraction,crystal-
eld lifts the degeneracy of spin sublevels for a spin system with S > 1=2 without
application of magnetic eld. In addition, zero-eld splitting may also originate in a
system of two unpaired electrons localized on a single atom or neighboring atoms. In
this case, Heisenberg exchange interaction may result in the triplet state and due to
close localization of electrons they exhibit strong magnetic dipole-dipole interaction
leading to a splitting of spin sublevels. In both cases, the zero-eld interaction can
be always factorized to the following form
^
H
ZFS
=
^
S
↔
D
^
S; (2.5)
where
↔
D is a traceless tensor. In the frame dened by the principle axes ofD-tensor,
Eqn. 2.5 is written as
^
H
ZFS
=D
x
^
S
2
x
+D
y
^
S
2
y
+D
z
^
S
2
z
=D
(
^
S
2
z
1
3
S(S +1)
)
+E(
^
S
2
x
^
S
2
y
);
(2.6)
wheref
^
Sg
i=x;y;z
are spin operators in the Cartesian coordinate system, D 3D
z
=2
and E (D
x
D
y
)=2. Both D and S strongly re
ect local symmetry of the defect:
cubic symmetry - D = E = 0, axial symmetry - D ̸= 0;E = 0, lower than axial
symmetry - D̸= 0;E ̸= 0. In general, the contribution of the spin-orbit interaction
Chapter 2. EPR spectroscopy 14
can be estimated from anisotropy of g-factor as it depends on both the spin-orbit
coupling constant
LS
and tensor
↔
that also result in zero-eld splitting due to the
crystal eld (Eqn. 2.4).
2.1.3 Hyperne interaction
Similar to an electron spin, a nuclear spin possesses a magnetic moment and the
interaction between electron and nuclear magnetic moments is termed as hyperne
interaction (third term in Eqn. 2.1). Due to a large mass of a nucleus, the nuclear
magnetic elds are short range and have effect on electron spin states in the case of
sufficient electron density at a nucleus. For a nuclear spin I separated by a radius
vector r from an electron, the hyperne Hamiltonian is given by
^
H(r)=
0
4
g
e
g
N
B
N
[
^
S
^
I
r
3
(
^
Sr)(
^
Ir)
r
5
+
8
3
(r)
^
S
^
I
]
; (2.7)
where g
N
and
N
are the nuclear g-factor and nuclear magneton, respectively. The
rst two terms in Eqn. 2.7 are due to the dipole-dipole interaction between two spins,
while the last term takes into account zero distance interaction energy between two
magnetic dipoles (this is the origin of the Fermi contact interaction) [114]. Averag-
ing the Hamiltonian over electron spatial degree of freedom results in the hyperne
interaction Hamiltonian of the form
^
H
HF
=
^
S
↔
A
^
I =
^
S(
↔
A
iso
+
↔
A
D
)
^
I; (2.8)
where
↔
A
iso
is the isotropic hyperne interaction tensor (Fermi contact term)
A
iso
=
2
0
3
g
e
g
N
B
N
j
0
(0)j
2
;
and
↔
A
D
is a traceless tensor with elements
(A
D
)
ij
=
0
4
g
e
g
N
B
N
r
5
⟨
0
j3r
i
r
j
ij
r
2
j
0
⟩:
Chapter 2. EPR spectroscopy 15
A-tensor can be diagonalized in the Cartesian coordinate system to the form
↔
A =
diagfA
xx
; A
yy
; A
zz
g. For axially symmetric hyperne interaction -A
xx
=A
yy
=A
?
,
A
zz
=A
∥
.
2.1.4 Nuclear Zeeman interaction
Interaction of a nuclear magnetic moment with externally applied magnetic eld is
described by the nuclear Zeeman interaction Hamiltonian
^
H
Zn
=g
N
B
B
0
^
I: (2.9)
In most EPR experiments, the nuclear g-tensor is considered as isotropic. For the
case of allowed EPR transitions, the nuclear spin remains unexcited, therefore effect
of the nuclear Zeeman interaction is not observed, as it shifts electron spin states
by the same amount. However, the nuclear Zeeman term may manifest in forbidden
EPR transitions, which could allow to identify nuclei responsible for the hyperne
interaction. Moreover, in the case when the nuclear Zeeman and hyperne couplings
are of the same order, satellite signals from forbidden EPR transitions are observed
on both sides of allowed EPR transition.
2.1.5 Nuclear quadrupole interaction
Nuclei withI > 1/2 posses a nuclear quadrupole moment (Q) originated from a non-
spherical charge distribution. Electric eld gradient due to electrons and nuclei in
the close proximity of the nucleus may align the quadrupole moment along a specic
direction and therefore affect the orientation of the nuclear spin. In this case, nuclear
spin states are described by the nuclear quadrupole interaction term,
^
H
Qn
=
^
I
↔
P
^
I; (2.10)
Chapter 2. EPR spectroscopy 16
where
↔
Pisnuclearquadrupoletensor,whichischosentobetraceless. Intheprincipal
axes system, H
Qn
is usually written in the following form
H
Qn
=
e
2
qQ
4I(2I1)
[
3
^
I
2
z
I(I +1)+(
^
I
2
x
^
I
2
y
)
]
; (2.11)
where e is electron charge, q - electric eld gradient and - asymmetry parameter
denedas = (P
x
P
y
)=P
z
withP
z
e
2
qQ=2I(2I1), whichisthelargestprincipal
value of P-tensor.
2.1.6 Weak dipole-dipole interactions
In a case of well separated unpaired electrons in a crystal, i:e: there is no signicant
overlap of orbitals of unpaired electrons, their magnetic moments may be considered
as isolated point dipoles. The same simplication can be applied for interaction of
electronandnuclearspins, when electrondensityatnucleusisnegligible(j
0
(r
N
)j
2
0). Hence, using the far-eld approximation [114], Hamiltonian for the dipole-dipole
interactions of an unpaired electron with m remote electron and n nuclear spins is
given by [115, 116]
^
H
DD
=
0
4
g
e
B
(
g
e
B
m
∑
j=1
[
^
S
^
S
j
r
3
j
(
^
Sr
j
)(
^
S
j
r
j
)
r
5
j
]
+
+g
N
N
n
∑
i=1
[
^
S
^
I
i
r
3
i
(
^
Sr
i
)(
^
I
i
r
i
)
r
5
i
]
)
;
(2.12)
where r
j
and r
i
are radius vectors connecting an unpaired electron with j
th
electron
and i
th
nuclear spins, respectively. Examples of the dipolar coupled electron spins
considered here include: 1) electron spins (
^
S and
^
S
j
) belong to the same group of
spins, which are dipolar coupled to each other, 2) an electron spin (
^
S) is a central
spin and a collection of remote electron (
^
S
j
) and nuclear (
^
I
i
) spins are bath spins.
Eqn. 2.12 can be further simplied in a weak coupling regime (
^
H
DD
≪
^
H
0
) using
rotating wave approximation (RWA). Dipolar coupled spins are considered to be in
Chapter 2. EPR spectroscopy 17
the weak coupling regime when the spin concentration is low (< 10
20
spin/cm
3
) or
the difference of the Larmor frequencies between the central and the bath spins are
much larger than the strength of the dipolar interaction. For spins in the weak
coupling regime, non-secular terms are omitted under RWA. Therefore, the dipolar
Hamiltonian becomes [117]
^
H
DD
0
4
g
e
B
(
m
∑
j=1
g
e
B
13cos
2
j
r
3
j
(
^
S
z
)
j
+
n
∑
i=1
g
N
N
13cos
2
i
r
3
i
(
^
I
z
)
i
)
^
S
z
;
(2.13)
where
j
(
i
) is polar angle between r
j
(r
i
) and quantization axis of a central spin
(z-axis). Expression in parentheses is an effective magnetic eld (b
DD
) due to elec-
tron and nuclear spin baths. b
DD
strongly depends on relative position of remote
spins and results in energy shifts of central spin states. Given a random distribu-
tion of electron and nuclear spins in a crystal, weak dipole-dipole interactions lead
to the distribution of EPR transition energies around central frequency given by
^
H
0
^
H
DD
Hamiltonian. This effect is termed as inhomogeneous broadening (∆!) of
EPRspectrallines. Inacaseofweakdipolarinteractionswithelectronspinbath, the
inhomogeneous lineshape is expected to be Lorentzian, while for dipolar interactions
with nuclear spin bath the lineshape is expected to be Gaussian [118].
2.2 cw-EPR spectroscopy
In cw-EPR spectroscopy, EPR spectrum is often obtained by measuring absorption
of electromagnetic wave by a spin system as a function of magnetic eld or electro-
magnetic wave frequency. At low magnetic elds, e.g. 0.3 Tesla (X-band), the
choice is to sweep microwave frequency, while at higher magnetic elds (> 1 Tesla)
a common choice is to sweep magnetic eld, in order to avoid frequency dependent
effects of EPR instrumentation. The analyses of cw-EPR spectroscopy based on a
Chapter 2. EPR spectroscopy 18
spin Hamiltonian allows to obtain useful insight into a spin system as described in
Sect. 2.1 . In this section, the quantum-mechanical background of cw-EPR signals is
describedandtheresultsarelinkedtothecommonstrategyforsimulationofcw-EPR
spectrum based on a static spin Hamiltonian.
2.2.1 cw-EPR signals
Here cw-EPR signals are discussed by considering a S=1/2 spin as a two-level system
(TLS), where the unperturbed Hamiltonian of TLS in externally applied magnetic
eldB
0
withanisotropicg-tensorisgivenby
^
H
0
=g
B
B
0
^
S. Inaddition,perturbation
Hamiltonian in the system due to continuous microwave excitation is described by
^
V =
^
Vcos!t =g
B
b
1
^
Scos!t, whereb
1
isthemagneticeldvectorperpendicularto
B
0
and and! is a microwave frequency. Dening the quantization axis (z-axis) along
the direction of external magnetic eld and making RWA, the total Hamiltonian (
^
H
I
)
intherotatingframewith!
0
=g
B
B
0
=~frequency,isgivenby(inunitsoffrequency)
^
H
I
=
^
U
1
(
^
H
0
+
^
V)
^
U!
0
^
S
z
=
Ω
2
(
e
it
j⟩⟨j+e
it
j⟩⟨j
)
;
(2.14)
where
^
U =exp(i!
0
^
S
z
t), Ω⟨j
^
Vj⟩ =g
B
b
1
=2~ and!
0
!. Spin statesj⟩
and j⟩ are eigenvalues of the spin Hamiltonian !
0
^
S
z
, which was used for rotating
frame transformation, and \" and \" stand for spin up and spin down projec-
tions along z-axis, respectively. Using the Liouville-von Neumann equation, unitary
evolution of TLS in the rotating frame is given by [119]
@^
@t
=i[
^
H
I
;^ ]: (2.15)
where ^ is a density matrix operator (^ =
^
U
1
j ⟩⟨ j
^
U withj ⟩ =
∑
i=;
c
i
ji⟩). In
order to account for a realistic situation of a spin system, spin relaxations have to
be introduced. For this purpose, a general dissipation superoperator (
^
^
R) is adopted,
Chapter 2. EPR spectroscopy 19
which represents energy exchange with an arbitrary reservoir, e:g: lattice phonons,
dipolar coupled electron neighboring spins, etc:. The action of the dissipator on the
density matrix of the system is described by [120]
^
^
R^ =
(
^
^ ^
+
1
2
^
+
^
^
1
2
^ ^
+
^
)
+
(
^
+
^ ^
1
2
^
^
+
^
1
2
^ ^
^
+
)
;
(2.16)
where ^
= j⟩⟨j, ^
+
= j⟩⟨j, and
and
are the transition rates from
spin up to spin down states and vice versa, respectively. Using the density matrix in
a thermal equilibrium (^
eq
= exp(~!
0
^
S
z
=k
B
T)=2cosh(~!
0
=2k
B
T), where k
B
is the
Boltzmann constant, T is temperature of a spin system), the matrix representation
of Eqn. 2.16 in the basis ofj⟩,j⟩ states is given by
^
^
R^ =
0
B
@
1
2
1
[
(
)(
eq
eq
)
]
2
2
1
2
1
[
(
)(
eq
eq
)
]
1
C
A
;
(2.17)
where
1
+
and
2
1
=2+
env
with
env
representingadditionaleffective
decay rate due to coupling to a spin environment.
1
is responsible for population
relaxations in the system resulting in relaxation of a longitudinal component of spin
magnetization and therefore is referred as a longitudinal relaxation rate with T
1
1=
1
beingalongitudinalrelaxationtime. Incontrast,
2
isresponsibleforrelaxation
of the transverse component of magnetization and hence is called as a transverse
relaxation rate with T
2
1=
2
being a transverse relaxation time. However, modern
names for T
2
are phase memory time or decoherence time. The latter is adopted
throughout the dissertation. As seen from the denition of
2
above, decoherence in
the system has two contributions - one due to population relaxations and second due
to surrounding spin environment. For a spin system with long T
1
(> 100s), T
2
is
typically dominated by electron or nuclear spin
uctuation processes [74, 105, 121].
Chapter 2. EPR spectroscopy 20
Takingintoaccountspinrelaxations,thetotalevolutionequationofdensitymatrix
under microwave excitation is given by [120, 122]
@^
@t
=i[
^
H
I
;^ ]+
^
^
R^ : (2.18)
In cw-EPR experiment, the EPR signals are measured on the time-scale (
cw
) much
longer than relaxations in a spin system (
cw
≫ T
1
;T
2
). Therefore, cw-EPR signals
are calculated using the density matrix at a steady state after all relaxations have
elapsed (@^ =@t = 0). The steady state density matrix elements are obtain from
Eqn. 2.18 using Eqns. 2.14 and 2.17 as
=
eq
∆
2
Ω
2
T
1
T
2
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.19a)
=
eq
+
∆
2
Ω
2
T
1
T
2
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.19b)
e
=e
=
∆
2
Ω(iT
2
T
2
2
)
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.19c)
where ∆
eq
eq
= tanh(~!
0
=2k
B
T) is the thermal magnetization factor and
e
= (e
)
=
e
it
.
Using Eqns. 2.19 the magnetization (M = g
B
Tr(
^
S^ )) components along x-, y-
and z-axes in the rotating frame are found [119, 123]
M
x
=g
B
∆
2
ΩT
2
2
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.20a)
M
y
=g
B
∆
2
ΩT
2
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.20b)
M
z
=g
B
∆
2
1+
2
T
2
2
1+
2
T
2
2
+Ω
2
T
1
T
2
; (2.20c)
which can be also obtained from the stationary classical Bloch equations.
Next, using the fact that the effective microwave absorption rate in a steady state
isequaltotheeffectivepopulationrelaxationrate(j(
^
^
R^ )
j),thecw-EPRabsorption
signals (I
cw
) are obtained using Eqns. 2.19a and 2.19b as
I
cw
()= Ω
2
∆
2
T
2
1+
2
T
2
2
+Ω
2
T
1
T
2
: (2.21)
Chapter 2. EPR spectroscopy 21
As evident form Eqn. 2.21, following the denition of Ω (Ω/b
1
), the cw-EPR line-
shapedependsonthemicrowavepowerandmayresultinbroadening. Toavoidpower
broadening effects, cw-EPR experiment is performed at sufficiently low microwave
power at which Ω
2
T
1
T
2
≪ 1 condition is satised. This leads to
I
cw
(!!
0
)/ Ω
2
∆ L(!!
0
;T
2
); (2.22)
whereL(!!
0
;T
2
)
1
(1=2T
2
)
(!!
0
)
2
+(1=2T
2
)
2
isanintrinsiclinehspahegivenbyLorentzian
function with a full-width at half maximum (FWHM) of 1=2T
2
(in Hz), which is
usually called as a spin packet. The proportionality sign is used in Eqn. 2.22 because
experimentalcw-EPRsignaldependsonexperimentalvariablesincludingthenumber
of spins, temperature, magnetic eld and gain of the EPR spectrometer.
Eqn. 2.22 describes a cw-EPR lineshape that is broadened purely due to decoher-
ence time T
2
. However, as discussed in Sect. 2.1.6, weak dipole-dipole interactions
mayleadtothedistributionofthetransitionfrequency!
0
resultingininhomogeneous
broadening of intrinsic lineshape. The overall lineshape (L
F
) can be expressed as
L
F
(!!
0
;∆!;T
2
) =
1
∫
0
L(!;T
2
)F(!
0
;∆!)d; (2.23)
where F(!!
0
;∆!) is a transition frequency distribution function due to inhomo-
geneous broadening with a characteristic broadening parameter given by ∆!. For
Gaussian or Lorentzian functions, ∆! is typically dened as FWHM. Therefore, cw-
EPR signal of a TLS dipolar coupled to a spin bath is given by
I
cw
(!!
0
)/ Ω
2
∆ L
F
(!!
0
;∆!;T
2
); (2.24)
Analyses of cw-EPR spectrum using Eqn. 2.24 may, in principle, allow forT
2
and ∆!
estimate. However,theanalysesrequirespreciseknowledgeofF(!!
0
;∆!)function.
In case when 2T
2
≪ ∆!, the lineshape can be approximated by inhomogeneous
lineshape (L
F
F). Furthermore, anisotropic spin Hamiltonian terms may also
Chapter 2. EPR spectroscopy 22
contribute to broadening of the lineshape. For more reliable determination of T
2
and ∆!, pulsed-EPR are employed that can facilitate independent measurement of
decoherence and inhomogeneous broadening effects.
2.2.2 cw-EPR spectrum
Procedure to simulation cw-EPR spectrum for a spin system with an arbitrary spin
Hamiltonian is derived from a cw-EPR signals of a TLS given by Eqn. 2.24. First
term in Eqn. 2.24, is a transition probability between two spin states coupled by
microwave excitation. For an arbitrary spin Hamiltonian H
0
given in Eqn. 2.1, the
generalized transition probability between two eigenstatesji⟩ andjj⟩ is described as
Ω
ij
=⟨ij
^
Vjj⟩ =
B
~
⟨ijb
1
↔
g
^
Sjj⟩
=
B
b
1
~
⟨ij(g
xx
^
S
x
+g
xy
^
S
y
+g
xz
^
S
z
)jj⟩;
(2.25)
where z-axis is dened along external magnetic eld and x-axis along microwave
magnetic eld, a typical situation realized in experiment. The second term is a mag-
netizationfactorthattakesintoaccountpopulationdifferencebetweentwomicrowave
coupled spin states. Therefore, in a case H
0
has n eigenstates with corresponding ϵ
n
eigenvalues, the magnetization factor for transition betweenji⟩ andjj⟩ states is given
by
∆
ij
=
e
~ϵ
i
=k
B
T
e
~ϵ
j
=k
B
T
∑
n
k=1
e
~ϵ
k
=k
B
T
: (2.26)
Finally, the third term for the case of inhomogeneously broadened transition between
ji⟩ andjj⟩ states is simplyL
F
(!!
ij
;∆!), where!
ij
ϵ
i
ϵ
j
. Therefore, forn level
system, the cw-EPR spectrum is obtained as
I
cw
/
∑
i;j
Ω
2
ij
∆
ij
L
F
(!!
ij
;∆!;T
2
): (2.27)
For the case of a spin system with random distribution of their principle frames with
respect to magnetic eld, anisotropy of any tensor in Eqn. 2.1 results in orienta-
Chapter 2. EPR spectroscopy 23
tion dependent eigenvalues and eigenfunction of H
0
. Furthermore, for cw-EPR of a
powder sample, where the sample is oriented randomly with respect to the external
magnetic eld (powder spectrum), Eqn. 2.27 is integrated over all possible polar ()
and azimuthal angles (ϕ) resulting in [124]
I
powder
cw
/
2
∫
0
∫
0
∑
i;j
Ω
2
ij
(;ϕ)∆
ij
(;ϕ)L
F
(!!
ij
(;ϕ);∆!;T
2
) sinddϕ: (2.28)
2.3 Pulsed-EPR spectroscopy
To study spin dynamics and spin relaxations, pulsed-EPR techniques are often em-
ployed. While spin relaxations are instantaneously acting on a spin system, in many
cases pulsed-EPR techniques allows to address a particular relaxation mechanism.
This is usually achieved by taking advantage of a time-scale and relaxation mecha-
nism of a particular relaxation. In this section, basic ideas of pulsed-EPR techniques
are described, starting from dynamics of TLS under microwave pulse, following ex-
perimental techniques to probe inhomogeneous broadening of the cw-EPR lineshape
and decoherence times.
2.3.1 Rabi oscillations
For pulsed-EPR experiments it is essential to consider spin dynamics of TLS during
microwave pulses rst. This is easily done for experiments with short microwave
pulses (t
p
), where time-scale of spin relaxations is slower than t
p
(t
p
≪ T
1
;T
2
). The
evolution of TLS wavefunction in the rotating frame with the microwave frequency!
and without spin relaxations is described by the Schr odinger equation [119]
i
@j ⟩
@t
=
^
H
mw
j ⟩; (2.29)
where
^
H
mw
=
^
U
1
(
^
H
0
+
^
V)
^
U !
^
S
z
=
^
S
z
+Ω
^
S
x
with
^
U = exp(i!
^
S
z
t). j ⟩ =
^
U
1
j
0
⟩ withj
0
⟩ being TLS wavefunction of
^
H
0
+
^
V Hamiltonian in the laboratory
Chapter 2. EPR spectroscopy 24
frame (Schr odinger picture). Other parameters are dened in Eqn. 2.14. Hence, the
solution to Eqn 2.29 can be expressed using unitary propagator
^
R as
j (t
p
)⟩ =
^
R(t
p
)j (0)⟩; (2.30)
where
^
R(t
p
) exp
[
i(
^
S
z
+Ω
^
S
x
)t
p
]
and j (0)⟩ is wavefunciton of initial state of
TLS before application of microwave pulse. In the j⟩, j⟩ basis, the matrix repre-
sentation of the propagator is
^
R(t
p
) =
0
B
B
B
@
cos
Ω
R
t
p
2
i
Ω
R
sin
Ω
R
t
p
2
i
Ω
Ω
R
sin
Ω
R
t
p
2
i
Ω
Ω
R
sin
Ω
R
t
p
2
cos
Ω
R
t
p
2
+i
Ω
R
sin
Ω
R
t
p
2
1
C
C
C
A
; (2.31)
where Ω
R
p
2
+Ω
2
. Finally, for TLS intialy in thej⟩ state, the evolution of the
wavefunction is obtained using Eqns. 2.30 and 2.31 as
j (t
p
)⟩ =
(
cos
Ω
R
t
p
2
+i
Ω
R
sin
Ω
R
t
p
2
)
j⟩i
Ω
Ω
R
sin
Ω
R
t
p
2
j⟩: (2.32)
Using the state wavefunction described by Eqn. 2.32, the populations in thej⟩ (p
)
andj⟩ (p
) as function of a microwave duration are obtained
p
(t
p
) =
1
2
[(
1+
2
Ω
2
R
)
+
Ω
2
Ω
2
R
cosΩ
R
t
p
]
; (2.33a)
p
(t
p
) =
1
2
Ω
2
Ω
2
R
[1cosΩ
R
t
p
]: (2.33b)
In the case of =0, Eqns. 2.33 become
p
;
(t
p
)=
1
2
[1cosΩt
p
] (2.34)
Eqn. 2.34 shows that when the resonant excitation is applied to TLS, the spin popu-
lation oscillates between spin states coupled by microwave excitation while the pulse
duration increases. These oscillations are referred as Rabi oscillations. The oscil-
lation frequency Ω is called as Rabi frequency. In a case of off-resonant excitation
Chapter 2. EPR spectroscopy 25
(Eqns. 2.33), amplitude of Rabi oscillations is reduced (less population is transfered
between the spin states) and the population oscillations occur with higher frequency
given by Ω
R
, which is called as generalized Rabi frequency.
Furthermore, using Eqn. 2.32,x-,y- andz-components of the spin magnetization
(M =g
B
⟨ j
^
Sj ⟩) are found
M
x
=
g
B
2
Ω
Ω
2
R
(1cosΩ
R
t
p
); (2.35a)
M
y
=
g
B
2
Ω
Ω
R
sinΩ
R
t
p
; (2.35b)
M
z
=
g
B
2
1
Ω
2
R
(
2
+Ω
2
cosΩ
R
t
p
)
: (2.35c)
When microwave pulse of Ωt
p
= =2 is applied on resonance with TLS, ac-
cording to Eqn. 2.34, the populations in j⟩ and j⟩ are equalized and therefore
M =
g
B
2
[0;1;0] (Eqn. 2.35). This microwave pulse is called as =2 pulse be-
cause it rotates spin magnetization by 90
◦
(
g
B
2
[0;0;1] !
g
B
2
[0;1;0]). Similarly,
microwave pulse is called pulse when Ωt
p
= . This pulse completely trans-
fers population from j⟩ to j⟩ state and therefore rotates magnetization by 180
◦
(
g
B
2
[0;0;1]!
g
B
2
[0;0;1]).
In ensemble EPR experiment, homogeneous on-resonant excitation of spins is
rarely achieved. Due to the distribution of transition frequencies !
0
given by EPR
lineshape,spinswithdifferentfrequencyoffset frommicrowavefrequencyareexcited
to a different extend. Therefore, the effective population transfer in such inhomoge-
neous spin systems can be described by
P
(t
p
) =
1
∫
1
Ω
2
Ω
2
+(!)
2
sin
2
(
√
Ω
2
+(!)
2
t
p
2
)
L(!
0
)d; (2.36)
where integration of p
, given by Eqn. 2.33b, is taken over the EPR lineshape L(
!
0
). As evident from Eqn. 2.36, when microwave is applied at !
0
and the EPR
linewidht is much narrower than p
, P
reduces to the on-resonant case given by
Eqn. 2.34. In most practical situations, width of p
. than the EPR linewidth.
Chapter 2. EPR spectroscopy 26
2.3.2 Free induction decay
Freeinductiondecay(FID)signalsmeasureadecayoftransversecomponentofaspin
magnetization. To maximize the intensity of FID signals in pulsed-EPR,=2 pulse is
often applied to rotate the initial magnetization along the z-axis (e.g. magnetization
due to the thermal spin polarization) into the transverse plane. The FID originates
from the fact that, in addition to an externally applied magnetic eld (B
0
), each
spin is subject to a local magnetic eld due to surrounding spin baths. Given a
randomdistributionofelectron andnuclearspinsinthelattice, eachspinexperiences
aslightlydifferentlocalmagneticeldresultinginthedecayofoverallmagnetization.
Moreover, due to dipole-dipole interactions between the bath spins (
ip-
op process)
or bath spin interaction with the lattice phonons (single
ip process), local magnetic
eld at a particular lattice site is, in general, a time-varying function
e
b(t). On the
other hand,
e
b(t) causes modulation of the Larmor frequency that can be viewed as a
diffusion of transition frequency within a spectral line, hence, this process is termed
as a spectral diffusion.
Considering a single spin described by TLS that is subject to a local mangetic
eld
e
b(t), the evolution equation for initial spin statej
0
⟩ in the rotating frame with
microwavefrequency! afterapplicationof=2pulseonresonancewithTLSfollowed
by a free evolution time (t) is given by [119]
j
t
⟩ =
^
U(t)
^
R
=2
j
0
⟩; (2.37)
where
^
R
=2
is a unitary propagator that describes spin state transformation under
=2 pulse, given by Eqn 2.31 with =0 and Ωt
p
==2.
^
U(t) exp(i
∫
t
0
b(t
′
)
^
S
z
dt
′
)
is a unitary propagator that describes evolution of a spin state during free evolution
time, where b(t) is a coupling constant to a local magnetic eld (
e
b(t)) due to bath
spins, i:e: b(t) g
B
e
b(t)=~. For TLS initially in j⟩ state, a spin state during FID
Chapter 2. EPR spectroscopy 27
experiment is obtained using Eqn. 2.37 as
j
t
⟩ =
1
p
2
exp
(
i
2
∫
t
0
b(t
′
)dt
′
)
j⟩
i
p
2
exp
(
i
2
∫
t
0
b(t
′
)dt
′
)
j⟩: (2.38)
Furthermore, using a spin state given by Eqn. 2.28, transverse components of a single
spin magnetization are obtained
m
x
(t)=m
0
sin
(∫
t
0
b(t
′
)dt
′
)
; (2.39a)
m
y
(t) =m
0
cos
(∫
t
0
b(t
′
)dt
′
)
; (2.39b)
where equilibrium magnetizationm
0
=g
B
∆
eq
=2 was introduced to account for spin
polarization of central spins. Next, to calculate ensemble average FID signals, b(t) is
treatedasastochasticOrnstein-Uhlenbeck(OU)processwithzeromean(⟨b(t)⟩
t
=0)
andcorrelationfunction⟨b(t)b(0)⟩
t
=∆exp(t=
c
), where∆and
c
arevarianceand
correlation time of b(t) process, respectively. In a case of a particular spatial cong-
uration of a spin bath, ∆
g
B
~
√
∑
j
e
b
2
j
, where
e
b
j
= g
0
B
(13cos
2
j
)
j
=(4r
3
j
)
is a magnetic eld produced by j-th bath spin at a central spin connected by radius
vectorr
j
(r
j
;
j
)and
j
isaspinstateofj-thbathspin(
j
=1=2). Correlationtime
in this case represents a characteristic time-scale of magnetic eld
uctuations at a
central spin. Averaging Eqn. 2.39 over the OU process results in (Appendix A.1)
m
x
(t)= 0; (2.40a)
m
y
(t)=m
0
exp
[
∆
2
2
c
(t=
c
+e
t=c
1)
]
: (2.40b)
As expected for a stationary Gaussian process, m
x
is averaged to zero. The obtained
signals in Eqns. 2.40 are FID signals of a single central spin averaged over different
realizations of b(t). This is equivalent to the averaging over an ensemble of central
spins, which are coupled to the identical spin baths (same ∆ and
c
).
Chapter 2. EPR spectroscopy 28
FID signals are typically characterized by a decay rate
*
2
, which is called as a
dephasing rate, or by dephasing time T
*
2
1=
*
2
. Therefore, transverse spin magne-
tization (m
?
(t)=
√
m
2
x
+m
2
y
) using Eqns. 2.40 can be expressed as
m
?
(t)=m
0
exp
[
(
*
2
t)
]
; (2.41)
where
may take values in the range between 1 2, which is obtained from two
limiting cases of Eqns. 2.40. A single exponential decay (
= 1) is obtained for a
spin bath in a motional narrowing regime (
c
≫ t, fast spectral diffusion), while
Gaussian decay (
= 2) is obtained for a spin bath in a quasi-static regime (
c
≪t,
slow spectral diffusion). However, even for a dilute spin systems (typically used in
EPR experiments), FID signals decay on the time-scale smaller than
c
due to static
distributionofLarmorfrequencies. Thereforeaquasi-staticregime(
c
≪t)isfurther
considered, which results in
m
?
(t)=m
0
exp
(
∆
2
t
2
2
)
: (2.42)
According to Eqn. 2.42, in a case of a single spin or ensemble of spins coupled to
identical spin baths, FID signals are governed by a variance of b(t) process resulting
in a Gaussian decay.
For the case of randomly distributed electron spins in a crystal, central spins
are subject to non-identical baths due to difference in spatial congurations of bath
spins. Therefore, to obtain ensemble FID signals (M
?
(t)), Eqn. 2.42 is averaged over
all possible positions and spin states of bath spins (r
j
,
j
,
j
) to give [125, 126]
M
?
(t)=m
0
exp
[
√
2
2
0
2
B
g
2
n
9
p
3~
t
]
; (2.43)
where n is a concentration of bath spins in a crystal. In contrast to Eqn. 2.42 for a
single spin, ensemble FID signal is described by a single exponential decay with the
dephasing rate
*
2
=
√
2
2
0
2
B
g
2
n
9
p
3~
.
Chapter 2. EPR spectroscopy 29
2.3.3 Spin echo decay
As described in Sect. 2.3.2, FID originates from the static Larmor frequency distri-
bution of spins. However, the effect of local static magnetic elds can be reversed
to produce spin echo (SE) signals. SE signals allow to probe the dynamics of a spin
bath through SE decay and are bases for multi-frequency pulsed-EPR techniques,
e:g: double electron-electron resonance (DEER), electron-nuclear double resonance
(ENDOR), DEER-based-NMR,etc. Formation of SE signals is achieved with a Hahn
echo sequence (=2) that consists of =2 and microwave pulses sepa-
rated by a free evolution time with SE signals observed after second free evolution
time, at 2. Following the approach described in Sect. 2.3.2 to obtain spin state in
the rotating frame, a spin state after application of Hahn echo sequence is expressed
as [119]
j
2
⟩ =
^
U
2
()
^
R
^
U
1
()
^
R
=2
j
0
⟩; (2.44)
where
^
R
=2
and
^
R
are unitary operators that describe effect of =2 and mi-
crowave pulses, respectively, which are obtained from Eqn 2.31 with = 0.
^
U
1
()
exp(i
∫
0
b(t)
^
S
z
dt) and
^
U
2
() exp(i
∫
2
b(t)
^
S
z
dt) are unitary propagators that
describe evolution of a spin state during free evolution intervals under a local mag-
netic eld. For TLS initially in j⟩ state, a spin state after Hahn echo sequence is
obtained using Eqn. 2.44 as
j
2
⟩ =
1
p
2
e
iϕ
j⟩
i
p
2
e
iϕ
j⟩; (2.45)
where ϕ
1
2
∫
2
b(t)dt
1
2
∫
0
b(t)dt is a total phase accumulated by a single spin
during Hahn echo sequence for a particular realization of b(t). Usingj
2
⟩ given by
Eqn. 2.45 and taking into account equilibrium spin magnetization (m
0
), transverse
components of a single spin magnetization are obtained
m
x
(2) =m
0
sin
(∫
2
b(t)dt
∫
0
b(t)dt
)
; (2.46a)
Chapter 2. EPR spectroscopy 30
m
y
(2)=m
0
cos
(∫
2
b(t)dt
∫
0
b(t)dt
)
: (2.46b)
Takingintoaccountthatingeneralamagnitudeofb(t)processisdescribedbyasym-
metric distribution with a zero mean value, m
x
(2) signals are averaged to zero and
may be omitted from further consideration. Therefore, the transverse magnetization
of a single spin is found as
m
?
(2) =m
0
cos
(∫
2
b(t)dt
∫
0
b(t)dt
)
: (2.47)
Inacaseofastaticb(t)(b(t) =Const),thetransversemagnetizationat2 (Eqn.2.47)
is aligned in y direction. Therefore, Hahn echo sequence removes the decay of
transverse magnetization due to static magnetic elds (Eqn. 2.42). This allows to
observe a decay due to time-varying component of b(t) by measuring SE signal as
functionofatotalfreeevolutiontime(2). Shapeandacharacteristictime-scale(T
2
)
of SE decay may allow to identify a major source of the decoherence in a solid. In
addition, temperature, concentration or microwave pulse dependence of decoherence
may be employed to reveal different decohernece mechanisms.
Single spin
In the case of a single spin coupled to a spin bath, b(t) can be modeled as the OU
process described in Sect. 2.3.2. Averaging Eqn. 2.47 over all possible realizations of
b(t) results in (Appendix A.1)
m
?
(2)=m
0
exp
[
∆
2
2
c
(2=
c
3+4e
=c
e
2=c
)
]
: (2.48)
For the case of slow and fast spectral diffusions, Eqn. 2.48 is reduced to
m
?
(2) =m
0
exp
[
∆
2
(2)
3
12
c
]
m
0
exp
[
(
2
T
2
)
3
]
; (
c
≫t) (2.49a)
m
?
(2) =m
0
exp
[
∆
2
c
(2)
]
m
0
exp
[
(
2
T
2
)]
; (
c
≪t) (2.49b)
Chapter 2. EPR spectroscopy 31
with T
2
= (12
c
=∆
2
)
1=3
and T
2
= 1=∆
2
c
, respectively. For an intermediate regimes
of the spectral diffusion, SE decay is/ exp(2=T
2
)
with
2 (1;3).
As described in Sect. 2.3.2, the spectral diffusion originates from spin
uctuations
inaspinbath. However,thereareseveralmechanismresponsibleforspin
uctuations,
i:e: the
ip-
op process due to dipolar coupling between bath spins or a single
ip
of a bath spin due spin-phonon interaction. For dilute spin systems with relatively
slowT
1
process (2 ≫T
1
), the dominant spectral diffusion mechanism is the
ip-
op
process in electron spin bath. In this case, SE signal (S
ff
) is given by Eqn. 2.48 and
using decoherence time T
2
is expressed as
S
ff
(2)=m
0
exp
(
2
T
2
)
(2.50)
with
2 [1;3]. Inaddition,considerationofintrinsicspindynamicsduetothedipolar
interactions between bath spins has revealed SE decay of a single spin predominantly
with
2 [2;3] for bath spin concentration 210
18
spins/cm
3
[127].
When the spectral diffusion is dominated by single spin
ips, as expected for
highly dilute spin systems, SE decay (S
1
) is described by a slow spectral diffusion
given in Eqn. 2.49a:
S
1
(2)=m
0
exp
(
2
T
2
)
3
; (2.51)
with T
2
= (12
1
=∆
2
)
1=3
, where
1
is a longitudinal relaxation time of bath spins.
Next, because of slow nuclear dynamics ( 1 ms), SE decay due to nuclear bath
spectral diffusion (S
N
) is also expected to be described by Eqn. 2.49a:
S
N
(2) =m
0
exp
(
2
T
2
)
3
; (2.52)
with T
2
= (12
N
=∆
2
)
1=3
, where
N
is a characteristic time for intrinsic dynamics of
nuclear spin bath due to nuclear
ip-
ops or single nuclear spin
ips.
Finally, T
2
of the central spin, which is completely insulated from the spin envi-
ronment, is affected by T
1
relaxation mechanism that sets the fundamental limit for
Chapter 2. EPR spectroscopy 32
longest T
2
. In this case, SE signal is found from Eqn. 2.18 as
S
T
1
(2)=m
0
exp
(
2
T
2
)
; (2.53)
with T
1
limited decoherence time T
2
=2T
1
.
Ensemble spins
To obtain ensemble SE signals due to the spectral diffusion governed by
uctuations
in electron spin baths, single spin SE signals (Eqn. 2.48) may be averaged over all
possiblespinbathrealizationsthataredescribedbytwoparameters,∆and
c
. While
averaging over ∆ may be carried out similarly to calculation of ensemble FID signals
(Sect. 2.3.2), averaging over
c
requires knowledge of
c
dependence on spatial cong-
uration of bath spins that could be obtained from quantum-mechanical treatment of
intrinsic spin bath dynamics. For this reason, ensemble SE signals are obtained using
different approach as described in [118, 128].
First, it is convenient to adopt a convention of A and B spins, which is widely
used in MR spectroscopy, where A spins are regarded as spin that are excited by
microwave pulses and are identical to central spins considered above, while B spins
are spins that remain unexcited and constitute a spin bath. Next, according to [128],
SE signals are rst calculated for a single A spin dipolar coupled to a single nearby
spin that
uctuates with the rate W and is located at the position described by the
relative radius vector r(r;). In this case, SE signal of a single A spin is given by
v(2) =
⟨
cos
2
2
⟩
L
v
0
(2)+
⟨
sin
2
2
⟩
L
v
(2); (2.54)
where ⟨sin
2
2
⟩
L
∫
1
1
Ω
2
Ω
2
+(!)
2
sin
2
(
√
Ω
2
+(!)
2
t
p
)
L()d denes an average
typingangleofasingleneighboringspinduetopulseinHahnechosequence. Thisis
equivalenttoeffectivepopulationtransferdescribedbyEq.2.36,while⟨cos
2
2
⟩
L
1
⟨sin
2
2
⟩
L
is the probability for spin to remain unexcited. Therefore, due to microwave
Chapter 2. EPR spectroscopy 33
pulse, a single A spin neighbor shows a duality due to quantum nature of a spin
state. It remains unexcited with ⟨cos
2
2
⟩
L
probability resulting in SE signals of A
spin due to B spins given by v
0
(2). On the other hand, it has⟨sin
2
2
⟩
L
probability
to undergo a transition resulting in a sign
ip of dipolar eld at A spin leading to
SE signals of A spin given byv
(2). Thus,v
0
(2) andv
(2) can be dened as SE
signals of a single A spin due to dipolar interaction to nearby B and A spin, which
are expressed as
v
0
(2)=
[
(
coshR +
W
R
sinhR
)
2
+
A
2
4R
2
sinh
2
R
]
exp(2W); (2.55a)
v
(2) =
[
(
coshR +
W
R
sinhR
)
2
A
2
4R
2
sinh
2
R
]
exp(2W); (2.55b)
respectively, with A
0
B
g
2
(1 3cos
2
)=(4~r
3
) and R
2
W
2
A
2
=4. Next,
ensemble SE signals (S) are obtained by considering single A spin coupled to all A
and B electron spins in the sample and averaging SE signals over all possible spatial
congurations, resulting in
S(2) =S
SD
(2) S
ID
(2); (2.56)
where S
SD
(2) is a contribution of spectral diffusion to SE decay of A spins due to
dipolar coupling to
uctuating B spins, while S
ID
(2) is a contribution of instanta-
neous diffusion that originates from dipolar coupling between A spins. S
SD
(2) and
S
ID
(2) are given as
S
SD
(2) = exp
(
n
∫
V
[1v
0
(2;W)]dV
)
; (2.57a)
S
ID
(2)= exp
(
n
⟨
sin
2
2
⟩
L
∫
V
[v
0
(2;W)v
(2;W)]dV
)
; (2.57b)
where n is the total spin concentration (assuming A and B spins belong to the same
spin system). S
ID
is additional SE decay mechanism that is not present in the case
of a single A spin considered above, but is effective in ensembles because multiple
Chapter 2. EPR spectroscopy 34
A spins are excited. For the case of fast
uctuations of A spins (W ≫ 1), effect
of instantaneous diffusion is negligible and SE decay is primarily governed by spec-
tral diffusion mechanism. In addition, instantaneous diffusion may be reduced by
employing short pulse that reduces
⟨
sin
2
2
⟩
L
.
For a spin system with relatively long T
1
and low probability of
ip-
ops, SE
signals due to single T
1
ips of bath spins are obtained from a quasi-static limit
(W ≪ 1) of Eqns. 2.57 with W = 1=T
1
as
S
SD
T
1
(2) = exp
(
n
9
p
3
g
2
2
B
0
~
(2)
2
2T
1
)
; (2.58a)
S
ID
T
1
(2)= exp
(
n
9
p
3
g
2
2
B
0
~
⟨
sin
2
2
⟩
L
(2)
)
: (2.58b)
Eqn. 2.58a that predicts SE decay due to single
ip of B spins was also obtained
using model similar to the OU process described above for a single spin, but with the
assumption of Lorentzian diffusion for b(t) process [125]. Moreover, Eqn. 2.58b can
be independently obtained by averaging a single spin SE signals given by Eqn. 2.47
over all possible positions of bath spins and considering static bath spins partially
excited by pulse in Hahn echo sequence.
For the case of the spectral diffusion due to the
ip-
op process of A and B
spins, Eqns.2.57aremodiedtoaccountforthe
ip-
opdistributionwithinasample
(f(W;W
max
), whereW
max
isW at whichf(W;W
max
) is maximum). Therefore, nal
expressions for SE signals due to
ip-
ops (S
SD
ff
and S
ID
ff
) are obtained as
S
SD
ff
(2)= exp
(
n
∫
1
0
∫
V
[1v
0
(2;W)]dV dW
)
; (2.59a)
S
ID
ff
(2)= exp
(
n
⟨
sin
2
2
⟩
L
∫
1
0
∫
V
[v
0
(2;W)v
(2;W)]dV dW
)
; (2.59b)
where f(W;W
max
) is given by
f(W;W
max
)=
√
3W
max
2W
3
exp
(
3W
max
2W
)
: (2.60)
Chapter 2. EPR spectroscopy 35
To investigate limiting cases of slow and fast spectral diffusion due to the
ip-
op
process, numerical simulations may be employed. Similarly to T
1
dominated spin
ips, fast
ip-
ops of spins average the affect of instantaneous diffusion. Therefore,
signicant contribution of the instantaneous diffusion is expected for relatively slow
dynamics of A spins.
Chapter 3
Instrumentation
Description of the HF EPR spectrometer presented in this Chapter can also be
foundinthearticletitledA high-frequency electron paramagnetic resonance spectrom-
eter for multi-dimensional, multi-frequency, and multi-phase pulsed measurements by
Franklin H. Cho, Viktor Stepanov, and Susumu Takahashi in Review of Scientic In-
struments85, 075110:1-075110:7(2014) (Reprintedwith permission. Copyright 2014,
AIPPublishingLLC)andthebookchaptertitled230/115 GHz electron paramagnetic
resonance/double electron-electron resonance spectroscopy byFranklinH.Cho,Viktor
Stepanov, Chathuranga Abeywardana and Susumu Takahashi in Methods in Enzy-
mology 563, 95-118 (2015) (Reprinted with permission. Copyright 2015, Elsevier).
Design of the aqueous sample holder was published in the book chapter titled High-
frequency electron paramagnetic resonance spectroscopy of nitroxide-functionalized
nanodiamonds in aqueous solution by Rana D. Akiel, Viktor Stepanov and Susumu
Takahashi(accpetedinCellBiochemistryandBiophysics,2016)(Reprintedwithper-
mission. Copyright 2016, Springer)
36
Chapter 3. Instrumentation 37
High-frequency,
high-power
transmitter
Quasioptical
system
Liquid helium cryostat
12.1 T
superconducting
magnet PC control
Lock-in amplifiers
Oscilloscope
Corrugated horns
Pulse generator
IQ mixer
3 GHz
IF
LO Sig
I Q
3 GHz
ref
Ref 1
Ref 2
Microwave signals
(arrowhead indicates direction)
Absorber
Rotating
wiregrid
polarizer
Ellipsoidal
mirror
Fixed
wiregrid
polarizer
Faraday
rotator
Sample
LNA
Corrugated waveguide
Data acquisition via
GPIB, LAN, or USB connection
Pulse triggering via BNC connection
Detection system
Receiver
Figure3.1: OverviewoftheHFEPR/DEERspectrometer. High-powertransmitter
and receiver are custom built by VDI, and the quasioptical system consists of corru-
gated horns, wire grid polarizers, Faraday rotators, corrugated waveguide (Thomas
Keating), and right-angle ellipsoidal mirrors (fabricated by the USC Machine Shop).
EPR/DEER signals are rst down converted to an intermediate frequency (IF) of 3
GHz by the receiver, amplied by a low-noise amplier (LNA, noise gure 0.5 dB,
Miteq), then by a second amplier (AML Communications). A 3-GHz reference is
produced using references from transmitter and receiver synthesizers (Ref. 1 and 2)
and a mixer (Marki Microwave) to down convert the IF signals to I and Q compo-
nentsofd.c. signalsusinganIQmixer(MarkiMicrowave). A12.1-Tsuperconducting
magnet (Cryogenic Limited) is cooled by a closed-cycle pulse tube cryocooler system
(Cryomech), and a 4He cryostat (Janis Research) is utilized for low temperature
measurements. Data acquisition and triggering of electronics are controlled by a
computer.
Chapter 3. Instrumentation 38
3.1 High-frequency EPR spectrometer
An overview of the HF EPR spectrometer at USC is shown in Fig. 3.1. Outputs
of a high-frequency high-power (the peak power of 100/700 mW at 230/115 GHz
respectively) solid-state source for cw and pulsed excitations are rst propagated in
a quasioptical transmitter stage consisting of an isolator, a circulator and quasiopti-
cal mirrors, then propagated through a corrugated waveguide to couple to a sample
located at the center of 12.1 Tesla cryogenic-free superconducting magnet. The spec-
trometer employs the induction mode detection for EPR detection [129] where the
cross-polarizedcomponentofEPRsignalsisseparatedfromtheinputusingacircula-
tor based on a wire-grid polarizer. EPR signals are then detected bya mixer detector
in a quasioptical receiver stage. Detected EPR signals are down-converted by mixing
with a local oscillator (LO) in the receiver stage to produce intermediate frequency
(IF) signals at 3 GHz. In the detection system, IF signals are further mixed with a 3
GHzreferencesynthesizedfromthesourceandtheLOtoobtaind.c. in-phase(I)and
quadrature (Q) signals. In pulsed-EPR measurements, I and Q components of tran-
sient EPR signals are sampled by a 2 GSamples/s digital oscilloscope (Agilent). In
cw-EPR measurements, intensities of I and Q signals at a eld-modulation frequency
(20100 kHz) are measured using lock-in ampliers (Stanford Research Systems). A
pulse generator (SpinCore), magnet power supplies, data acquisition from the digital
oscilloscope and lock-in ampliers are controlled by National Instruments LabVIEW
VIs in a control computer.
3.1.1 High-frequency high-power solid-state transmitter
Figure 3.2a shows a circuit diagram of the high-frequency high-power solid-state
sourcecustom-builtbyVirginiaDiodes,Inc(VDI).Theoutputfrequencyofthesource
istunableinbetween215240and107120GHz. Thehigh-frequencysourceconsists
Chapter 3. Instrumentation 39
of two microwave synthesizers (810 GHz and 911 GHz, Micro Lambda Wireless),
isolators (Ditom Microwave), fast p-i-n (PIN) switches (American Microwave), di-
rectional couplers (ATM), power splitters and combiners (Narda Microwave East),
variable phase shifters (ARRA), preampliers and frequency multipliers. As shown
in Fig. 3.2a, the single-frequency conguration, e:g: for cw-EPR, SE and DD mea-
surements,utilizesonesynthesizer. Outputofthesynthesizerissplitintotwooutputs
using the power splitter where one of them is fed to a variable phase shifter in order
tooutputsubsequentpulseswithtworelativephases,e:g:XandYphases. Thephase
is continuously controllable with the accuracy of2.4 degrees in 215240 GHz and
1.2degreesin107120GHz. Next,outputsareconnectedtothedirectionalcoupler
to provide a reference for the detection system (Reference 1 in Fig. 3.2a), then to the
isolator to prevent from microwave re
ections and standing waves. When relative
phase control between microwave pulses is not require, output from the synthesizer
is directly connected to the directional coupler in upper circuit arm in Fig. 3.2a. In
pulsed-EPR operation, cw outputs of the synthesizer are gated using the PIN switch
where the timing of the switching is controlled by transistor-transistor logic (TTL)
signals from the pulse generator (see Fig. 3.1). Typical rise and fall times of the
PIN switch are 12 ns, which makes it possible to produce as short as 20 ns pulse.
After the switches, the outputs are subsequently transmitted into an amplier, then
to the frequency multiplier chain using a power combiner. The frequency multiplier
chain is made up of an active frequency tripler and three passive frequency doublers,
therefore base synthesizer frequencies are multiplied by 24 and 12 for the nal mi-
crowave output in the range of 215240 and 107120 GHz, respectively. The output
powerofthesourceis30100mWin215240GHz(200700mWin107120GHz)
where the peak power is 100 mW at 230 GHz (700 mW at 115 GHz). For DEER
measurements two microwave synthesizers are employed with the transmitter con-
Chapter 3. Instrumentation 40
Microwave signals via
SMA connection
Mirocwave signals via
rectangular waveguide
connection
TTL trigger signal from
pulse generator via BNC
connection
(a)
(b)
High-power
frequency multiplier
chain
Final output
to corrugated horn
10 MHz
ref
9-11 GHz
synthesizer
8-10 GHz
synthesizer
Directional
couplers
Isolators
PIN
switches
X3 X2 X2 X2
Ref 1 to
Detection system
108-120 GHz
output
216-240 GHz
output
TTL trigger
TTL trigger
Amplifier
Active frequency tripler
Passive frequency doublers
Power
combiner
10 MHz
ref
9-11 GHz
synthesizer
(terminated)
8-10 GHz
synthesizer
Directional
couplers
Isolators
PIN
switches
Ref 1 to
Detection system
TTL trigger
φ
Power
splitter
Power
combiner
Variable phase
shifter
TTL trigger
High-power
frequency multiplier
chain Final output
to corrugated horn
X3 X2 X2 X2
108-120 GHz
output
216-240 GHz
output
Amplifier
Active frequency tripler
Passive frequency doublers
Figure3.2: Schematicoverviewsofthehigh-frequencyhigh-powersolid-statesource
in the HFEPR spectrometer. (a) Conguration of single-frequency mode used for cw
EPR, SE and DD measurements. (b) Conguration of double-frequency mode used
for DEER measurements.
Chapter 3. Instrumentation 41
gured to the double-frequency mode (Fig. 3.2b). As described in Fig. 3.2b, each
synthesizer chain consists of the directional coupler, the isolator, the PIN switch, the
pre-amplier and the power splitter, and the microwaves from the both chains are
sequentially transmitted into the pair of the frequency multiplier stages. In the same
mannerasthesingle-frequencymode,thedouble-frequencymodeprovidestheoutput
frequencyrangeof215240GHzand107120GHzandthepeakoutputpowerof100
mW and 700 mW at 230 GHz and 115 GHz, respectively. Therefore, the transmitter
system provides a high-power and extremely wide range of the tunable frequency for
EPR and DEER spectroscopy.
3.1.2 Quasioptical system
HF excitations and EPR/DEER signals are efficiently guided by the quasioptical sys-
tem. As shown in Fig. 3.3, the quasioptical system consists of corrugated horns, wire
grid polarizers, Faraday rotators, corrugated waveguide (Thomas Keating), right-
angle ellipsoidal mirrors and transmitter/receiver stages (fabricated by the USC ma-
chineshop). Thecorrugatedhorninthetransmitterstagejointsasingle-moderectan-
gular waveguide (WR-3.4 and WR-8.0 for 215240 GHz and 107120 GHz, respec-
tively) to a corrugated waveguide (HE
11
) and converts the transmitter outputs to
linearly polarized Gaussian waves (represented by red horizontal double-sided arrows
in Fig. 3.1). The Gaussian waves are guided by the right-angle ellipsoidal mirrors
with a focal length f=254 mm, which focus the Gaussian waves periodically (period
= 508 mm) to cancel the frequency dependence of the quasioptics [130, 131].
The intensity of HF excitation waves is controlled by a quasioptical variable at-
tenuator based on a combination of rotating and xed-angle wire grid polarizers (see
Fig. 3.4a). The dynamic range of the variable attenuator is > 30 dB as shown in
Fig. 3.4b. In addition, the system utilizes a quasioptical isolator made up of a com-
Chapter 3. Instrumentation 42
EPR
signals
230/115 GHz
pulses
Corrugated waveguide
(To sample)
Transmitter stage
Receiver stage
Isolation box
Monitor port
Ellipsoid
mirror
To mixer
Source
outputs
Faraday
rotator
Rotating
wire-grid
polarizer
Wire-grid
polarizer
Corrugated
horns
Circulator
(wire-grid)
Figure 3.3: Overview of the quasioptical system consisting of transmitter and re-
ceiver stages. The quasioptical system is a periodic focusing system with the focal
lengthof254mmfortheGaussianwaves. HE
11
modeinacorrugatedhornconnected
to the source excites the Gaussian mode with high efficiency (coupling between HE
11
modeandtheGaussianmodeis99%). Theperiodicfocusingsystemalsoallowsus-
ing the same quasioptics for a wide range of frequencies. The isolation box to reduce
background noises is made of aluminum and the inside is covered by absorbers.
bination of the xed-angle wire grid polarizer and a Faraday rotator to suppress
standing waves in the transmitter stage (as shown in Fig. 3.1, re
ections represented
by a purple dashed double-sided arrow are directed to an absorber). The Gaussian
waves are coupled to the corrugated waveguide to excite a sample located at the bot-
tom end of the waveguide. The coupling efficiency between the Gaussian mode and
HE
11
modeinthecorrugatedwaveguideis99. FordetectionofEPR/DEERsignals,
the system employs the induction-mode detection scheme [129], where a wire grid
polarizer separates the circularly polarized EPR/DEER signals (represented by blue
circular arrows in Fig. 3.1) from the linearly polarized excitations (typical isolation
of more than 30 dB), then guides the induction signals (represented by blue vertical
double-sided arrows in Fig. 3.1) to the receiver. The quasioptics in the receiver stage
Chapter 3. Instrumentation 43
(a) (b)
θ (°)
Attenuation (dB)
E
f
= E
i
sin
2
(θ)
θ
0 20 40 60 80 100 -20
0
-5
-10
-15
-20
-25
-30
-35
Meas.
Fit
Incident microwave
electric field (E
i
)
Rotating wiregrid
polarizer
E
i
sin(θ)
Fixed
wiregrid
Figure 3.4: Characterization of the quasioptical variable attenuator. (a) Schematic
diagram of the variable attenuator. Transmission of a linearly polarized microwave
with the initial electric eld (E
i
) through the rotating and xed-angle wire grid po-
larizers is given by E
f
= E
i
sin
2
. The direction of the xed wire grid is set to be
perpendicular to the polarization of the incident microwave. The angle between the
incident microwave polarization and the axis of the rotating wire grid polarizer () is
variable. (b) The measured attenuation as a function of . Blue (black in the print
version)squaredotswitherrorbarsrepresentthemeasuredattenuationandsolidline
indicates the best t to the data using Eq. 3.4. For the measurement of I
i
and I
f
, a
pyroelectric detector (Eltec Instruments) was used.
are also designed as a periodic focusing system using f=254 mm right-angle ellip-
soidalmirrorsandasimilarisolatortothetransmitterstageforreductionofstanding
waves (as shown in Fig. 3.1, re
ections represented by a purple dashed double-sided
arrow are directed to an absorber). In addition, quasioptics in the receiver stage
are enclosed by an isolation box to isolate it from background noises, e:g: scattered
excitation waves from the quasioptics. The quasioptics in the transmitter and re-
ceiver stages are mounted on separate breadboards to adjust their coupling to the
corrugated waveguide independently and to maximize the coupling efficacy.
Figure 3.4 shows the performance of the quasioptical variable attenuator. As
shown in Fig. 3.4a, a linearly polarized microwave rst goes through the rotating
wire grid polarizer. Because the electric eld component parallel to the wire grid
Chapter 3. Instrumentation 44
polarizer is re
ected, the transmission after the rotating wire grid is given by
E
t
=E
i
sin; (3.1)
where E
i
and E
t
are strengths of the incident and transmitted electric eld and is
the angle between the polarization of E
i
and the direction of the rotating wire grid.
After going through the rotating wire grid, the microwave passes through the xed
wire grid where the direction of the wire grid is perpendicular to the polarization of
E
i
(see Fig. 3.4a). Therefore, the nal transmission after the xed wire grid (E
f
) is
given by
E
f
=E
i
sin
2
; (3.2)
and the transmitted power (I
f
/E
2
f
) of the incident microwave (I
i
/E
2
i
) including
possibleleakagethroughthewiregridpolarizersandbackgroundsinthemeasurement
(C) is written by
I
f
=I
i
(sin
4
+C): (3.3)
Thus, the attenuation in dB (Attn) is given by
Attn = log
10
(
I
f
I
i
)
= log
10
(sin
4
+C): (3.4)
Figure 3.4b shows measurements of the attenuation by the variable attenuator.
AsshowninFig.3.4b,theexperimentaldataiswellexplainedbythemodeldescribed
in Eq. 3.4, concluding the dynamic range of the attenuator to be32 dB.
3.1.3 Detection system
For detection of EPR signals, we built a phase-sensitive superheterodyne detection
system (see Fig. 3.5). In the detection system, EPR signals from the quasioptical
system are rst down-converted to 3 GHz IF signals by a subharmonically pumped
mixer (VDI) and the high-frequency LO (VDI). The high-frequency LO consists of
Chapter 3. Instrumentation 45
SMA connection
Rectangular waveguide
connection
10 MHz
ref
2-20 GHz
synthesizer
Isolator PIN
switch
Single pole
double throw
switch
X3
X2 X2
X2
TTL trigger signal from
pulse generator via BNC
connection
TTL trigger
Amplifier
Active frequency
doubler
Directional
coupler
Ref 2 to
Detection system
Passive frequency
doubler
X3
Passive frequency
tripler
Passive frequency
tripler
Subharmonically
-pumped mixers
Passive frequency
doubler
IF
(3GHz)
IF
(3 GHz)
Sig
(115 GHz)
Sig
(230 GHz)
(a)
(b)
Ref 2 from
receiver synthesizer
Ref 1 from
transmitter synthesizer
Mixer X2 X2 X2 X3 φ
Variable phase
shifter
Frequency multiplier chain
(x12 for 115 GHz or x24 for 230 GHz)
Amplifier
High-pass filters
250 MHz for 115 GHz
or 125 MHz (for 230 GHz
For 115 GHz
For 230 GHz
IQ mixer
3 GHz
ref
IF (3 GHz)
I
Q
Figure 3.5: Circuit diagram of the detection system. Circuit diagrams to produce
(a) IF using subharmonically pumped mixer and (b) I and Q signals from IF and 3
GHz reference signals. A subharmonically pumped mixer for EPR detection provides
an excellent isolation of the LO to the RF port (>100 dB). The high-frequency LO
consists of a broad-band microwave synthesizer (2-20 GHz, Micro Lambda Wireless),
a directional coupler, an isolator, a PIN switch, an amplier and a frequency mul-
tiplier. The PIN switch in the LO is for protection of the detection system. The
superheterodyne detection has1 GHz bandwidth and its noise temperature (T
N
) is
1200 K. IF power is controlled by a variable attenuator to optimize the power to
the IQ mixer.
Chapter 3. Instrumentation 46
a microwave synthesizer (220 GHz, Micro Lambda Wireless), a directional coupler
(ATM), a PIN switch (American Microwave), an isolator (Ditom Microwave) and
frequency multipliers (see Fig. 3.5a). The IF signals are immediately amplied by
a low-noise amplier (LNA, noise gure=0.5 dB, MITEQ), then by a second ampli-
er (AML Communications). As shown in Fig. 3.5b, a 3 GHz reference is produced
by a reference circuit including a mixer (Marki Microwave), ampliers, a 24/12
frequency multiplier (Mini Circuits), a phase shifter and variable attenuator (ATM).
In the reference circuit, a 125/250 MHz reference synthesized by mixing Reference
1 from the source and Reference 2 from the LO is fed into the 24/12 frequency
multiplier to produce the 3 GHz reference signals for 230/115 GHz operation, respec-
tively. Finally the IF signals at 3 GHz are down-converted to I and Q d.c. signals
by mixing with the 3 GHz reference with a IQ mixer (Marki Microwave). In cw-EPR
measurements, the power of the IF signals is adjusted by a variable attenuator (GT
Microwave) to optimize the mixer response, then intensities of the I and Q signals
are measured by lock-in ampliers. In pulsed-EPR measurements, transient I and Q
signals are sampled simultaneously by a fast digital oscilloscope.
3.1.4 Cryogenic-free superconducting magnet and
4
He
cryostat
The HF EPR/DEER spectrometer employs a 12.1 T cryogenic-free superconducting
magnet (Cryogenic Limited). The magnet system consists of two superconducting
solenoidcoils;themaincoilissweepablefrom0to12Tandthesweepcoilissweepable
from 0.1 to 0.1 T. The solenoid coils are controlled by separate magnet power
supplies. The magnet is cooled by a single cryocooler (Cryomech), which operates at
2.8K with no load and is surrounded by a radiation shield at 40K. The cryocooler is
specied to operate in the stray eld of the magnet and provide in excess of 20,000
Chapter 3. Instrumentation 47
hours continuous operation. The magnet has a 89-mm room temperature bore for
the access of a sample and a variable temperature control system.
Sample temperature is controlled by a
4
He cryostat (Janis) inserted in the mag-
net bore (see Fig. 3.1). The cryostat has a 62 mm diameter for sample access and
an optical window at the bottom for excitation and detection of visible and infrared
light. Liquid Helium (LHe) is stored in a LHe reservoir integrated in the cryostat
and the
ow rate of LHe is adjusted by a rotary vane pump and a pressure regula-
tor. By controlling the temperature of the
owing helium vapor in which samples are
immersed, boththesampleandholderaresimultaneouslycooledtoadesiredtemper-
ature, therebyeliminatingtheneedforthermalanchoringandsamplemountheating.
The LHe
ow rate and a heater current in the system are balanced to provide
owing
vapor and sample temperatures over the range of 300-1.4 K. The cryostat can also be
operated with liquid nitrogen to perform experiments from room temperature to78
K. A temperature controller (Lake Shore) together with application of two tempera-
ture sensors at the cryostat and the sample provides precise and stable temperature
control.
3.1.5 Sample holder
In the present HFEPR setup, we employ sample holders without a cavity to mea-
sure various forms of samples, e:g: single crystals, thin lms, powders and frozen
solutions [132]. Single crystals and thin lms are directly placed on a conductive
end-plate and are positioned inside or near the end of the corrugated waveguide. For
powder or frozen solution samples, we use a "bucket" with a volume of 12527 mi-
croliters to hold the samples, e:g: the dimensions of 12 (527) microliter buckets are
2.0 (9.4) mm of the diameter and 3.8 (7.6) mm of the height. The bucket is made
of Te
on. The samples in the bucket are placed on the end-plate and are positioned
Chapter 3. Instrumentation 48
Corrugated
waveguide
Modulation coil
made of
thin copper wire
Sample
mount
made of
G10
Conductive
end-plate
(sliver-coated
mirror)
Single crystal
and thin-film samples
“Bucket” for powder
and aqueous/frozen
solution samples
Figure 3.6: Overview of various sample holder congurations. Single crystal and
thin-lmsamplesarepositioneddirectlyonanend-platewithconductivesurface,and
powder and aqueous/frozen solution samples are loaded to a cylindrical bucket made
of Te
on. Depending on the dimensions, samples are placed either inside or near the
bottom end of the waveguide.
Chapter 3. Instrumentation 49
D
h
d
Aluminum
tape
Sample
volume
Bottom
cap
Top cap
Corrugated
waveguide
E
1
B
1
λ/2
(a) (b)
Thin aqueous
sample
0
E,B
Sample
mount
made of
G10
Modulation coil
made of
thin copper wire
Figure 3.7: Aqueous sample holder for room temperature HF EPR experiments at
115GHz.[133](a)Electricandmagneticeldcomponentsofmicrowaveinthesample
holder. The sample is positioned on the aluminum tape. (b) Schematics for aqueous
sample design. The top and bottom caps are made of Te
on. Doted lines indicate
threads and volume for screws that are used to tighten top and bottom caps.
inside or near the end of the waveguide.
HE EPR experiments on aqueous samples at room temperature are often chal-
lenging due to large amount of water absorption of HF microwave. In particular, the
absorption is signicant at 0.1-1 THz. (32). For example, 1 mm thickness of water
absorbs 10% of microwave power at 10 GHz while at 115 GHz it absorbs 99.98%.
As demonstrated in Ref. (33), one way to overcome the water absorption of HF mi-
crowave is to employ a thin layer of the aqueous sample where the thickness of the
layer (h) is much smaller than the microwave wavelength (). As shown in Fig. 3.7a,
the magnetic component of the microwave (B
1
) is maximum at a conducting surface
(i:e: aluminum tape) while the electric component of the microwave (E
1
) is mini-
mum. This allows to mask thin aqueous samples from the electric eld, resulting in
reduction of the microwave absorption.
ToimplementthisideaforourHFEPRsystemat115GHz, wedesignedasample
Chapter 3. Instrumentation 50
holder as shown in Fig. 3.7b [133]. The sample holder made of Te
on is designed to
hold an aqueous sample in a cylindrical well (Fig. 3.7b). In the present design, the
sampleholderisusedwithnomicrowavecavityandaqueoussamplesolutionisplaced
onaconductingsurfacewhichisthenodeofthemicrowaveelectriccomponentaswell
as the antinode of the magnetic component (Fig. 3.7b). To minimize the microwave
absorptionsandtomaximizethesample volume, thewellheight(h)waschosentobe
100m and the well diameter (D) is 5 mm, whereD is similar to the aperture of the
corrugatedwaveguide. Thesamplewellissealedbyanaluminumtape. Moreover,the
sample well is located at a distance (d) from the corrugated waveguide that satises
the conditiondn
T
+hn
W
= 2, wheren
T
- refraction index of Te
on (n
T
= 1.44),n
W
- refraction index of water (n
W
= 1.33) and - microwave wavelength at 115 GHz.
3.2 Optically detected magnetic resonance
system
A home-built ODMR system for room temperature experiments on NV centers in
diamond at low magnetic elds is presented in Fig. 3.8. The optical system operates
asascanningconfocalmicroscopewithasingleNVcentersensitivity. Forexcitationof
NV center, a diode-pump solid-state laser (CrystaLaser CL-532-100-S) with a single-
mode radiation at 532 nm of 100 mW power is employed. For pulsed operations
of the laser, the laser beam is rst propagated through acousto-optical modulator
(AOM, Isomet) with the on and off states controlled by RF signals generated by
AOM driver. To maximize the response of the AOM, the laser beam is focused to
the AOM by the bi-convex lens L1 ( with the focal distance f
1
= 50 mm) resulting
in 20 ns rise time of optical power allowing for optical pulses with duration > 30
ns. Moreover, the laser beam is incident on the AOM at the Bragg angle producing
Chapter 3. Instrumentation 51
APD
APD
AOM
LP L5
L1 L2
L3
L4
Laser line lters
ND lters
BS
DM
FSM
Piezo positioner
sample
Laser
MW AMP
MMF
MMF
PC
Objective
TCSPC
Figure 3.8: Diagram of the home-built ODMR system for a single NV center mea-
surements at low magnetic elds.
several diffraction beams with the maximized intensity into the 1
st
order and with>
30dBopticalpowerinsulationbetween on andoff states. Next,the1
st
orderbeamis
collimated by the bi-convex lens L2 (f
2
= 50 mm) and transmitted through the laser
line lters (Omega Optical) to attenuate fundamental mode emission of the laser at
1064 nm. The optical power in the system is controlled by a combination of neutral
density (ND) lters (Newport). The laser excitation is directed towards the sample
using silver coated mirrors (Thorlabs), dichoric mirror (DM, Omega Filters) and fast
steering mirror (FSM, Newport). Combination of FSM and system of two bi-convex
lenses, L3 (f
3
= 60 mm) and L4 (f
4
= 250 mm), allows for a laser beam projection
at the back aperture of the objective at a variable incidence angle without loss of the
opticalpower. TheincidenceangleiscontrolledbyrotatingFSMovertwoorthogonal
Chapter 3. Instrumentation 52
axes with the application of DC voltage to the FSM driver using analog outputs of
the data acquisition card (DAQ card, National Instruments, PCI-6321e) installed in
thecomputer. FullrangeofallowedFSManglescorrespondstotheimagingof50x
50m
2
samplearea. Moreover, systemoftwolensesexpandsthelaserbeamtollup
thebackapertureoftheobjective. Finally,thelaserbeamisfocusedtothesampleby
the oil-immersion objective (Nikon, NA = 1.4, magnication 60x, plan-apochromat
innity corrected system).
FL signals produced by excited NV centers (wavelength600 - 800 nm) are col-
lected with the same objective. Collimated FL signals follow the laser beam pathway
and are separated from the laser beam at the DM. After DM, FL signals rst pass
through the long-pass lter (LP, Omega Filter, cut-off wavelength 600 nm) to lter
out residual transmitted and scattered laser light and unwanted background signals,
andarefocusedbythebi-convexlensL5(f
5
=200mm)intomulti-modeber(MMF,
Thorlabs, 100 m in diameter). Aperture of the MMF functions as a pin-hole in a
conventionalconfocal microscopedening the spatialresolutionof anopticalimaging
along the optical axis (axial resolution). For single photon counting of FL signals,
MMFisconnectedtotheavalanchephotodiode(APD,PerkinElmer,70%quantum
efficiency for FL photons, 10 MHz maximum counting rate), which outputs a single
TTL pulse (TTL high > 2.5 V for15 ns duration) at each detected single photon.
TTL signals from the APD are detected using digital counter in the DAQ card (250
MHz internal clock). In pulsed-ODRM experiments, counter is gated to record tran-
sient FL signals. To detect reference signals in ODMR measurements, APD signal
is shared among several counters, which are gated separately. To avoid dark time
effects of the APD (dark time 35 ns) in the photon statistic measurements (auto-
correlation measurements), detection system is set into the Hanbury-Brown-Twiss
(HBT) conguration, where FL signals are split by the beamsplitter (BS, Thorlabs)
Chapter 3. Instrumentation 53
and are detected by two independent APDs with the TTL outputs connected to the
Time-Correlated Single Photon Counting (TCSPC, PicoQuant) module.
A diamond sample is typically mounted on the sample stage made of an acrylic
plate, which is xed on the three axis piezo stage for precise and controllable sample
positioning with respect to the objective. The piezo stage allows sample positioning
along each axis independently using a manual mechanical adjustment over 4 mm
range with 50 m resolution or piezo-electric adjustment over 20 m range with 20
nm resolution using USB controlled voltage driver. For microwave application to NV
center in diamond, a thin gold wire (25 m in diameter) is located on the surface
of the diamond and connected to the microwave circuit with the microwave source
(Rohde & Schwarz), microwave amplier (Mini-Circuit) and 60 dB termination. To
avoid overlap of the excitation and FL signals with a gold wire, we typically study
NV centers, which are located 10 m away from the wire. With this conditions,
0.2-0.3 mT microwave elds at NV centers are usually achieved with 16 Watt
microwave input into the gold wire. To apply external magnetic eld, a permanent
magnet is mounted on a three-axis translation stage below the sample allowing to
control magnetic eld at NV centers in the range of 0-150 mT. The magnetic eld is
usually calibrated using ODMR signals of NV centers.
3.2.1 Spatial resolutions
To facilitate ODMR experiment on a single NV center, diamond sample with low
concentrations of NV centers (< 1 center perm
3
) are typically studied. In addition,
good spatial resolutions of optical imaging are required. This is achieved by focusing
laserbeamintoadiffractionlimitedspot(DLS)withamicroscopeobjectiveofahigh-
numerical aperture. In microscopy, DLS is typically described by spatial distribution
of the laser intensity. The axial and lateral resolutions are then dened as the half
Chapter 3. Instrumentation 54
0
0.5
Normalized Intensity
1
0
1
0.5
Normalized Intensity
0
z (µm)
5 10 15 20 25
0 -0.5 0.5
x (µm)
(c)
(d)
plane wave
no sp.aberrations
with sp.aberrations
10
0
20
30
x (µm)
5 -5 0
z (µm)
(a)
x (µm)
5 -5 0
10
0
20
30
z (µm)
(b)
0.02 0 0.1 0.4 1 0.02 0 0.1 0.4 1
no sp.aberrations with sp.aberrations
plane wave (Airy pattern)
no sp.aberrations
with sp.aberrations
Figure 3.9: Numerical calculations of a diffraction limited spot (DLS) of the laser
beam focused by a microscope objective with NA = 1.4. (a) DLS in a case of aber-
ration free focusing, when the laser is focused within the immersion oil (no diamond
sample) (b) DLS for the laser focused through oil-diamond interface with the probe
depthof10m. (c)Lateraland(d)axialintensitydistributionsoftheDLScalculated
in(a)(cyan),(b)(red)andforaplanewavefocusedbyathinlens(gray). Dashwhite
line in (a) and (b) indicates the location of a geometric focus. Colorbar displays the
color code for the laser intensity. Calculation of DLS in (a) and (b) were performed
using electromagnetic wave diffraction of a Gaussian beam focused by a microscope
objectivewithahigh-numericalaperturethroughaplanarinterfacedescribedin[134]
with the parameters: (a) n
1
=n
2
= 1.52, NA = 1.4, (b) n
1
= 1.52, n
2
= 2.4, NA =
1.4 and with an input Gaussian beam waste that is equal to the radius of objective
back aperture. Plane wave diffraction in (c) and (d) was calculated according to the
scalar Fresnel-Kirchhoff diffraction theory [135].
Chapter 3. Instrumentation 55
size of DLS along optical (z axis) and lateral (x axis) axes, respectively.
For the case of aberration free focusing (laser beam is focused in the immersion
oil in the absence of a diamond sample), the DLS of our ODMR system is shown in
Fig. 3.9a with the lateral and axial laser intensity distributions that are almost iden-
ticaltotheintensitydistributionsofaplanewavefocusedbyathinlens(seeFig. 3.9c
and Fig. 3.9d, respectively). Therefore, lateral (v) and axial (u) resolutions can be
approximated by the well known results for a focused plane wave of a wavelength
as v = 0:61=NA and u = 2=NA
2
, respectively, resulting in v 230 nm and u
540 nm for our microscope objective (NA = 1.4).
However, when the laser beam is focused inside the diamond, the DLS is affected
by spherical aberrations due to mismatch of refraction indexes at the oil-diamond
interface (refration index of immersion oil and diamond are n
O
=1:52 andn
D
=2:4,
respectively). As shown in Fig. 3.9b, in the case of laser focused 10 m below the
diamond surface, spherical aberrations result in the broadening of the DLS predom-
inantly along the z axis and 5 times reduced maximum laser intensity. Further-
more, while the lateral size of DLS is almost unaffected by the spherical aberrations
(Fig.3.9),theaxialsizeextendstoapproximately15mwiththemaximumintensity
of DLS shifted from the geometric focus by 8 m.
To reduce the effect of the oil-diamond interface on laser DLS,i:e: elongation inz
axis and reduction of maximum laser intensity, we typically perform experiments on
NVcentersthatare0-2mbelowthesurface. Inaddition,weemployaberbased
confocal detection (Fig. 3.8) allowing to perform an optical sectioning of FL signals,
i:e: FL signals are collected from a smaller volume than the excitation volume given
by DLS. This technique improves axial resolution to 500 nm [136].
Chapter 3. Instrumentation 56
x
y
δx
δy
δz
z
z - fixed δx
δy
-0.5 0
x (µm)
0.5
y (µm)
0.5
-0.5
0
0
10
20
30
(a)
(c)
FL (counts/ms)
0
20
40
position (µm)
0
1
2
3
# of tracking loops
5000 10000
5000 10000
x
y
x
y
z
τ
s
FL
τ
s
FL
(x
0
,y
0
,z
0
)
(b)
(d)
Figure3.10: ImagingandtrackingofasingleNVcenter. (a)Laserscanningpattern
used to image a diamond sample. Laser positions in xy plane are controlled by FSM
inxandystepsatthexedpositionsofthesamplestage. FLsignalsareintegrated
over
s
time constant at each position. (b) FL image obtained on type-Ib diamond
crystal. Image was obtained using 100 by 100 points with
s
= 20 ms integration
time. (c) Relative positioning of the diamond sample with respect to the laser in a
tracking loop, where the focal point of the laser is controlled in xy plane with FSM
and sample is positioned along z axis using piezo translation stage. FL signals are
integrated over
s
time constant at each position. (d) FL signals and x, y, z positions
recording in the continuous tracking mode during 2.5 hours. Parameters used in
tracking:
s
= 20 ms, x = 25 nm, y = 25 nm, z = 40 nm.
Chapter 3. Instrumentation 57
3.2.2 Imaging and tracking
Imaging of a diamond sample is performed at a xed position of the sample stage
(piezo driven translational stage is xed) using a raster scan, where the laser beam
position in the focal plane of the objective is controlled using the FSM (Fig. 3.8)
following a rectangular scanning pattern shown in Fig. 3.10a. During the scan, the
beam position is incremented by x or y with a time constant
s
(typically for 15
ms), which is used to collect FL signals from a given sample volume. Recorded FL
signals are then mapped to obtain FL image of a scanned sample region (Fig. 3.10a),
where NV centers are identied by bright diffraction-limited FL spots. Scanning area
is controlled by the step size (x andy) and number of scanning points. Example of
FL image obtained on a single NV center is shown in Fig. 3.10b.
To perform ODMR experiments on NV centers, the laser beam is positioned at
the maximum of FL spot. However, due to mechanical instabilities in the system,
relativepositionofNVcenterandthelaserbeammaydriftonthetimescaleofODMR
measurements. The drifts may exceed the size of DLS leading to the reduction of FL
signals. Moreover, FL reduction is more pronounced when high-power microwave
is applied to the sample in ODMR experiments owning to additional thermal drifts
of a sample. Therefore, for optimum NV centerlaser beam coupling, maximum
FL signals of NV center are continuously re-tracked using a tracking loop shown in
Fig. 3.10c. Tracking loop consists of three xy scans, where laser beam is positioned
using FSM and z axis is scanned by moving a sample along z axis using piezo driven
translation stage (Fig. 3.8). In each xy scan, the laser beam is incremented over 9
positions around the central point (x
0
, y
0
) with the step size x and y in x and
y directions, respectively, with FL signals integrated over
s
at each position. The
optimum position (x, y, z) is then found based on the maximum FL signals with
the precision in x, y and z positions given by x/2, y/2 and z/2, respectively. (x,
Chapter 3. Instrumentation 58
y, z) position is used for a permanent localization of the NV center until re-track
is required or used as a central point for a next tracking loop in the continuous
tracking. As shown in Fig. 3.10d, continuous tracking preserves stable FL signals
with 3 % root-mean-square
uctuations, while drifts in positions may exceed fewm.
3.2.3 Photon statistics measurement
To verify if FL signals originate from a single NV center, photon statistics measure-
ment is preformed. In particular, photon statistics can be revealed by a second order
photoncorrelationfunction(g
(2)
(),where isatimeintervalbetweenphotons)[137].
Given a light intensity I(), g
(2)
is expressed as
g
(2)
()=
⟨I()I(0)⟩
⟨I()⟩⟨I(0)⟩
; (3.5)
which describes relative probability of detecting two photons with a time interval
. Photon statistics can be revealed by examination of g
(2)
at zero time interval (
= 0) [138]. For the case of a classical light source, g
(2)
(0) > 1, where equality is
satised for a perfectly coherent light (Poissonian photon statistics) meaning that
a time interval between two successive photons is random, while g
(2)
(0) > 1 is for
a thermal light (super-Poissonian photon statistics) re
ecting higher probability to
detect photons with a short time interval (bunching effect). On the other hand,
single quantum emitter (sub-Poissonian photon statistics) has a zero probability to
emit two photons at a time (antibunhing effect) and therefore g
(2)
(0) = 0, whereas
g
(2)
(0) =11=n for n independent single quantum emitters. Therefore, observation
of g
(2)
(0) < 1 veries quantum property of light, i:e: FL signals, while g
(2)
(0) < 1=2
proves the detection of a single quantum emitter.
For experimental measurement of g
(2)
(), the FL signals are detected in the
Hanbury-Brown-Twiss (HBT) conguration [139] (Fig. 3.11a) to avoid APD dead
time effects. TTL signals of APD1 and APD2 are connected to the \start" and
Chapter 3. Instrumentation 59
APD1
APD2
τ
τ+τ
0
photon 1
photon 2
TCSPC
module
start
stop
TTL1
TTL2
(a)
(b)
250
time delay τ+τ
0
(ns)
500
0
# of events
100
200
0 -20 0
time delay τ (ns)
g
(2)
(τ)
0
1
2
(c)
single quantum
emitter
τ
0
Delay generator
20
Figure 3.11: Photon statistics measurement. (a) Diagram of the Hanbury-Brown-
Twiss arrangement of detection of FL signals for second order photon correlation
functionmeasurement. (b)Experimentallyobtainedhistogramofthemeasurednum-
ber of events for each detected time delay interval between photons. The time bin
width was set to 128 ps. The data was collected over 6 hours.
0
was set to 19.3 ns
( 4 meters of BNC cable). (c)g
(2)
() as obtained from the normalization and back-
ground signal corrections (Appendix B.3) of the raw data presented in (b). g
(2)
(0)<
1/2 proves optical insulation of a single NV center (single quantum emitter).
Chapter 3. Instrumentation 60
\stop"channelsoftheTSCPCmodule, respectively. Inasingleeventcountingmode,
TCSPC measures a time delay between rst and second photons, in the case they
are detected at the \start" and \stop" channels, respectively, otherwise the measure-
ments is restarted. To record theg
(2)
() for negative values of the time delay (second
photon is detected by APD2 earlier than a rst photon by APD1), additional time
delay
0
is introduced for TTL2 signals using BNC cable ( 5 ns per 1 m of ca-
ble). In the experiment, the accumulated number of events for each measured are
histrogrammed (Fig. 3.11b) and g
(2)
() (Fig. 3.11c) is found after normalization and
background corrections of the raw data (Appendix B.3).
Chapter 4
Magnetic resonance spectroscopy
of diamond
4.1 High-frequency EPR spectroscopy of
diamond
4.1.1 cw-EPR
In cw-EPR measurements, we employ a magnetic eld modulation technique where
theappliedmagneticeldismodulatedwithB
m
cos!
m
t(B
m
and!
m
aremodulation
eld strength and frequency, respectively) and the intensities of I and Q signals at
!
m
are recorded as a function of external magnetic eld (B
0
) using lock-in ampliers
that are set for operation at !
m
. In the limit of a weak modulation eld, an input
signal (S
in
(B)) to the lock-in amplier can be expressed as
S
in
(B)S(B
0
)+
@S
@B
B
0
B
m
cos!
m
t:
In a lock-in amplier, the input signals are mixed with the reference signal (cos!
m
t)
followed by the frequency lter to output a DC signal. The output signal (S
in
(B))
61
Chapter 4. Magnetic resonance spectroscopy of diamond 62
B
0
L (B
0
)
∝ L' (B
0
)
L' (B
0
)
B
m
(a)
(b)
B
0
(hkl)
Figure 4.1: Technical details of the cw-EPR lineshape detection in bulk diamond
sample. (a) cw-EPR lineshape and the corresponding lineshape signal recorded using
lock-inamplierandapplicationofmodulationeld. L(B
0
)representstheabsorption
lineshape of EPR,L
′
(B
0
) - rst-derivative ofL(B
0
), which the signal detected by the
lock-in amplier in cw-EPR measurements, B
m
strength of modulation eld at the
sample. (b) Conguration of sample alignment with respect to external magnetic
eld (B
0
). (hkl) are the Miller indices of the polished face of a diamond sample.
from a lock-in amplier is then given by
S
out
(B
0
) =
1
2
@S
@B
B
0
B
m
: (4.1)
Therefore, when the absorption lineshape (L(B
0
)) is scanned, the recorded spectrum
is proportional to the rst derivative of the lineshape (L
′
(B
0
), see Fig. 4.1a).
To study a bulk diamond, a diamond sample with a polished face of (hkl) orien-
tation is placed orthogonally to the B
0
(see Fig. 4.1b), so thatB
0
is aligned with the
[hkl] crystallographic direction. A typical cw-EPR spectra obtained on the synthetic
type-IbdiamondcrystalsareshowninFig4.2forthecaseoffaceorientationsof(100)
and (111). The observed cw-EPR spectra from both diamonds are well explained by
absorption signals of N spins (P1 centers). Whilst many paramagnetic impurities
have been identied in natural diamonds (Chapter 1), the most abundant impurities
typically found in synthetic diamonds by EPR spectroscopy are N spins.
Chapter 4. Magnetic resonance spectroscopy of diamond 63
4.100
Magnetic field (Tesla)
4.104 4.108
Exp.
Sim.
Exp.
Sim.
B
0
// [111]
B
0
// [100]
Intensity (a.u.)
∗ ∗
∗ ∗
Figure 4.2: cw-EPR spectra of type-Ib diamonds measured at 115 GHz. Upper
(orange) and lower (magenta) data show the spectra obtained on the diamonds with
the polished faces along(100) and (111) crystallographic planes, respectively. The
external magnetic eld was aligned orthogonally to the faces as shown in Fig. 4.1b.
Results were obtained with B
m
00.3 mT at 20 kHz frequency, 300 ms integration
time of lock-in amplier outputs and 0.13 mT/s sweep rate of magnetic eld. Sim-
ulations (Sim.) of cw-EPR spectra were carried in Matlab using N spin Hamiltonian
(Eqn. 4.2) and the procedure described in Sect. 2.2.2.
Chapter 4. Magnetic resonance spectroscopy of diamond 64
N has ve valance electrons and therefore posses an unpaired electron in diamond
that stably resides along one of the crystallographic axis (Fig. 4.3a) due to the Jahn-
Teller distortion [64, 140]. In addition, an unpaired electron is hyperne coupled to
14
N nuclear spin (I = 1). The spin-Hamiltonian of the N spin is well-established and
is given by (in units of frequency)
^
H
N
()=
B
h
B
0
()
↔
g
^
S+
^
S
↔
A
^
Ig
N
B
B
0
()
^
I+
^
I
↔
P
^
I; (4.2)
where
↔
g isanisotropicg-tensor(
↔
g =g
↔
I)withg =2:0024[64],
↔
Aisaxiallysymmetric
hyperne tensor with A
x
= A
y
= 81.3 MHz and A
z
= 114 MHz [141, 142], g
N
=
0.4037 [143],
↔
P is axially symmetric traceless nuclear quadrupole tensor with P
x
=
Py =P
z
=2 = 1.99 MHz [144] and is a polar angle between N spin axis and B
0
.
As shown in Fig. 4.3b, for the case of = 0, according to the dominant spin in-
teractions (i:e: Zeeman electron and hyperne interactions), the energy levels of the
spin states are split into 6 non-degenerate levels giving rise to 3 allowed transitions
(∆m
S
=1 with ∆m
I
=0) with the transition frequencies given by =
0
+A
z
m
I
,
where
0
is the Zeeman interaction frequency (
0
g
B
B
0
=h). On the other hand,
when ̸= 0, the hyperne splitting is reduced due to anisotropy of
↔
A, taking values
in the range [A
x
;A
z
] depending on , resulting in angle-dependence of the transition
frequencies that explains the dependence of cw-EPR spectra on sample orientation
(Fig. 4.2). In the caseB
0
is aligned with [100] direction, all four principle axis ([111],
[11
1], [1
11] and [
111]) are equivalent with respect to B
0
making up the equal angles
( = arccos(1=
p
3) = 54:74
◦
) with the direction of magnetic eld (Fig. 4.3c). There-
fore, the hyperne splitting experienced by all N spins is same, giving rise to cw-EPR
spectrum that consists of three hyperne lines of equal intensities (Fig. 4.2). In con-
trast, when B
0
is along the [111] direction, the hyperne splitting for N spins with
the [111] orientation is maximum (A
z
), while other three orientations ([11
1], [1
11]
and [
111]) are equivalent making up 109.5
◦
with B
0
(Fig. 4.3d) and experience a re-
Chapter 4. Magnetic resonance spectroscopy of diamond 65
x
y
z
[111]
C
N
[111]
[111]
[111]
B
0
θ
[111]
C
N
[111]
[111]
[111]
B
0
[111]
N
[111]
[111]
[111]
C
B
0
(a) (b)
1/2
-1/2
m
S
m
I
1
-1
0
-1
1
0
(c) (d)
ν
0
ν
0
+ A
z
ν
0
- A
z
ν
0
Figure4.3: StructureandenergydiagramofNspinsindiamond. (a)StructureofN
inthediamondlattice. ExternalmagneticeldB
0
isalignedatangle withthe[111]
crystallographic axis. (b) Schematics of energy level splitting of N spins according to
the dominant spin interactions (Zeeman and hyperne interactions) for the case of
= 0. (c) and (d)B
0
alignmentfor the case of diamond samples with the polished face
(100) and (111), respectively. Polished faces are represented by the shaded planes.
Chapter 4. Magnetic resonance spectroscopy of diamond 66
duced hyperne splitting, resulting in two sets of the three hyperne lines overlapped
at the central line (isotropic g-tensor). Therefore, cw-EPR spectrum from diamond
with (111) face orientation consists of 5 peaks with intensities that scale as 1:3:4:3:1
(Fig. 4.2) representing distribution of spin populations among spectral lines. The
simulation of cw-EPR spectrum (Sect. 2.2.2) using the full spin Hamiltonian for N
spins (Eqn 4.2), reveals two additional peaks, which are also observed experimentally
(labeled by asterisks in Fig 4.2). These peaks correspond to twoforbidden EPR tran-
sitions that become pronounced due to mixing of nuclear spin states by quadrupole
interactions, when ̸=0. As shown in Fig. 4.2, the simulated cw-EPR spectra for N
spins are in excellent agreement with the cw-EPR experiments on type-Ib diamonds.
4.1.2 Pulsed-EPR
A typical signal trace (
√
(I
2
+Q
2
)) recorded by a fast oscilloscope (Sect. 3.1) in
pulsed-EPR measurements on N spins in diamond is presented in Fig. 4.4a, showing
microwavepulsesignals(=2and)andsignalsduetoexcitedNspins(FIDandSE).
Shortest in the experiments is chosen as when no signicant signal distortions
are observed. To maximize SE signals, transmitter and receiver stages (Sect. 3.1)
are positioned for optimization of microwave coupling to the sample and detection
of EPR signals. For enhancement of SNR in the measurements, a full area of SE
(schematically indicated by the shaded region in Fig. 4.4a) is recorded over repeated
cycles with the repetition rate longer than 5=T
1
.
The intensity of SE signals constitutes transverse magnetization, therefore, to
measure decoherence time T
2
, the decay of transverse magnetization is measured
using the Hahn echo technique, where the intensity of SE signals is recorded as a
function of 2. As shown in Fig. 4.4b, a typical T
2
time for N spins in type-Ib dia-
monds is on the order of severals as extracted from SE decay. The measured values
Chapter 4. Magnetic resonance spectroscopy of diamond 67
5
T (ms) 2τ (μs)
Echo intensity (a.u.)
10 15 0 10 20
Echo intensity (a.u.)
Time (μs)
0 1 2 3 4
Signal intensity (a.u.)
Exp.
Fit
Exp.
Fit
π/2 π FID FID Spin echo
π/2 π
τ τ
π
T
π/2 π
τ τ
T
2
= 1.38±0.01 μs
T
1
= 2.9±0.1 ms
(a)
(b) (c)
Figure4.4: SpinechomeasurementsofT
1
andT
2
relaxationtimesofNspinsinbulk
type-Ib diamond. (a) Transient signals recorded by a digital oscilloscope in pulsed-
EPRexperimentsat115GHz. Magneticeldwasappliedalong[100]crystallographic
direction. ThestrengthofmagneticeldwassettoexciteNspinsinthecentralspec-
tral line (Fig. 4.2). Signal was averaged over 16 traces with the following parameters:
=2 = 200 ns, = 450 ns, = 1.3 s, repetition rate = 20 ms. Sensitivity range
of the fast oscilloscope is adjusted to intensity of SE signals resulting in saturated
signals for microwave pulses. (b) SE intensity as a function of to measure spin de-
coherence time T
2
of central spins in bulk diamond with (111) face orientation. The
inset depicts Hahn echo sequence used to measure SE decay. Each data point was
taken with 32 averages, 20 ms repetition rate and pulse parameters: =2 = 200 ns,
= 350 ns, varied. SE decay was t with single exponential function to extract
T
2
. (c) SE intensity as function of time interval T to measure spin-lattice relaxation
time of central spins in bulk diamond with (111) face orientation. The inset depicts
inversion recovery sequence used to measure SE decay. Each data point was taken
with 32 averages, 20 ms repetition rate and pulse parameters: =2 = 150 ns, = 250
ns, = 1s,T varied. SE decay was t with single exponential function to extract
T
1
.
Chapter 4. Magnetic resonance spectroscopy of diamond 68
of T
2
for N spins in diamond is much shorter than 2 T
1
(see below). Therefore, 1=T
2
decoherence rate is largely affected by decoherence due to surrounding spin baths
(Sect. 2.2.1). In fact, temperature dependence of T
2
for an ensemble of N spins in
type-Ib diamond has revealed two major decoherence sources in diamond [74]. The
rst source is due to the
ip-
op process in the bath of N spins that dominates T
2
in diamond at room temperature, while the second source was attributed to decoher-
ence due to
13
C nuclear spins (I = 1/2) with the decoherence rate of 0.004 s
1
(T
2
250 s) that is still faster than T
1
limited decoherence (. 0.001 s
1
). In
addition, N spin concentration dependence ofT
2
has been observed [145], supporting
the decoherence due to the
ip-
op of nitrogen spins as the main decoherence source
in type-Ib diamonds. This question is addressed in Chapter 5, where DEER based
method to probe spin concentrations in solid-state is developed, allowing to study T
2
dependence on concentrations of N spins.
To detect the decay of the longitudinal magnetization, the inversion recovery
sequence (T =2, inset in Fig. 4.4c) is applied, where the rst
pulse inverts spin magnetization (M
0
) and the degree of the longitudinal relaxation
is probed with the Hahn echo pulse sequence after a time interval T during which
the magnetization is relaxing back to the equilibrium value (M
0
). To measure T
1
relaxation times, intensity of SE signals is measured as a function of T with the
xed . As shown in Fig. 4.4c, a typical T
1
time for N spins in type-Ib diamonds
is on the order of several milliseconds. The spin-phonon induced tunneling has been
suggested as the main source of T
1
relaxations for N spins in diamond, where the
spin-phonon interactions of N spin with local phonons excites N spin reorientations
between principle axis resulting in spin demagnetization [74, 146].
Chapter 4. Magnetic resonance spectroscopy of diamond 69
x
y
z
[111]
V
N
[111]
[111]
[111]
C
S = 1
Figure 4.5: Structure of a nitrogen-vacancy center in the diamond lattice. N -
nitrogen 14 atom, V - vacancy in the lattice, C - carbon 12 atom.
4.2 ODMR experiments on a single NV center in
diamond
ODMR system was build to image, coherently control and readout a spin state of
a single NV center in diamond. NV center is an impurity defect existing in di-
amond formed by a substitutional nitrogen atom and neighboring vacancy defect
(Fig. 4.5) [147, 148]. In the neutral charge state (NV
0
) NV posses an unpaired elec-
tron (S = 1=2) [149, 150], however, a negatively charged NV (NV
; S=1) is more
often found in diamond [66]. Unique optical properties of NV centers arise for NV
duetospinselectiverelaxationsfromtheelectronicexcitedstate[68, 70, 151]. Inthis
work, only NV centers in the charged state are considered.
As shown in Fig. 4.6, NV center has a triplet ground electronic state (jg⟩) with
a zero-eld splitting of 2.87 GHz between the m
s
= 0 and m
s
=1 spin states that
results from the dipole-dipole interactions between the unpaired electrons [152{155].
Moreover, the electron spin is hyperne coupled to the
14
N nuclear spin (I = 1) [66].
The resulting spin-Hamiltonian for NV center in the ground state is [66, 147, 156,
Chapter 4. Magnetic resonance spectroscopy of diamond 70
(
1
A)
0
±1
0
±1
m
s
Optical
excitation
(532 nm)
Fluorescence
(~ 600 - 800 nm)
2.87 GHz
ISC
1.42 GHz
non-radiative
decay
e
g
(
3
A
2
)
(
3
E)
Figure 4.6: Energy diagram of a nitrogen-vacancy center in diamond at zero mag-
netic eld. jg⟩ and je⟩ are wavefunctions in the electronic ground and excited state
of the NV center, respectively. Spin sub-levels are labeled according to the spin pro-
jection (m
s
) along NV axis, which are split in the ground and excited state due to
the zero-eld splitting (Eqn. 4.3). Green arrows - transition induced by optical exci-
tation, red arrows - radiative decay fromje⟩ tojg⟩ state, black arrows - non-radiative
transitions.
157]
^
H
NV
=
g
B
h
B
0
^
S+D[
^
S
2
z
1
3
S(S +1)]
+A
∥
^
S
z
^
I
z
+A
?
(
^
S
x
^
I
x
+
^
S
y
^
I
y
)+P[
^
I
2
z
1
3
I(I +1)];
(4.3)
where g = 2:0028, D = 2:87 GHz, A
∥
= 2:3 MHz, A
?
= 2:1 MHz, P 5 MHz.
The rst two terms in Eqn. 4.3 describe isotropic Zeeman and zero-eld interactions
for NV electron spin, respectively. The next two terms describe axially symmetric
hyperne interaction between NV electron spin and
14
N nuclear spin, while the last
term is due to quadrupole interaction for
14
N nuclear spin. The nuclear Zeeman
interaction is omitted in the present consideration. Moreover, electronic excited state
(je⟩) of NV center is also a triplet state [12]. However, due to the larger distribution
Chapter 4. Magnetic resonance spectroscopy of diamond 71
of electronic wavefunction in theje⟩ state, the dipolar interaction between unpaired
electrons is reduced in the excited state resulting in the zero-eld splitting of 1.42
GHz [69, 158]. In addition, the hyperne coupling in the excited state is isotropic
with the hyperne constant of 40 MHz [69].
Owning to the allowed electric dipole transition (
3
A
2
↔
3
E) [159], NV center can
be promoted to the electronic excited state using a standard green laser excitation
(532 nm). When excited, NV is rst thermalized to the lowest vibrational state in
the je⟩ state by fast vibrational relaxations with the lattice phonons, followed by
spin selective relaxations to the ground state (Fig. 4.6) [70, 160, 161]. In the case NV
centerisinthem
s
=0excitedstate,NVcenterprimarilyundergoesradiativedecayto
the phonon sideband by emitting FL photons that spectrally extend from 600-800nm
with the zero-phonon line (ZPL) at 637 nm [149, 162]. On the other hand, due to
intersystem crossing (ISC) that occurs primarily for them
s
=1 excited states [160,
163], NV from the m
s
=1 excited state is shelved into the metastable singlet state
(
1
A), from where a non-radiative decay brings NV center into the m
s
= 0 ground
state [68, 160]. Therefore, after several optical cycles, NV in the m
s
= 1 state
is preferably transfered into the m
s
= 0 state allowing for optical spin polarization
into the m
s
= 0 state [68, 163]. In addition, due to ISC for the m
s
= 1 excited
state, NV in them
s
=1 emits fewer FL photons. This enables optical spin readout
of NV spin state based on the intensity of FL signals [67, 68, 163]. Due to weak
transitions that do not preserve the spin state (Appendix B.1), the experimentally
achieved polarization is 70-90 % [70, 160, 164], which gives the ODMR contrast of
30 % [68, 151, 165].
Chapter 4. Magnetic resonance spectroscopy of diamond 72
4.2.1 cw-ODMR
In cw-ODMR measurements, continuous optical and microwave excitations are ap-
plied to a single NV center (Sect. 3.2). When the microwave frequency is away from
resonant spin excitation of NV center, NV is polarized in the m
s
= 0 state and un-
dergoes optical cycling between the m
s
= 0 ground and excited states. As a result,
the maximum steady state FL signals are observed. However, when the microwave
is resonant with one of the allowed spin transitions of NV center, a new steady
state is reached with a reduced population in the m
s
= 0 state and consequently
reduced FL signals. To detect ODMR signals, FL signals are scanned as a function
of the microwave frequency. In order to reduce unwanted effects, such as noise and
background from drifts of the NV position due to heating, NV center is continu-
ously tracked (Sect. 3.2) and two FL signals with (FL
mw
) and without (FL
no mw
)
microwave are measured from two subsequent tracking loops, then the normalized
signal (1FL
mw
=FL
no mw
) is obtained.
cw-ODMR measurements on a single NV center are shown in Fig. 4.7. At no
externalmagneticeld,them
s
=1statesaredegeneratedaccordingtothezero-eld
splitting in Eqn. 4.3 (dominant term that denes the central transition frequency).
As a result, ODMR signals due to two microwave induced transitions (j0⟩↔ j1⟩
and j0⟩↔ j+1⟩) are overlapped and appear at 2.87 GHz frequency (Fig. 4.7a). In
a case of nite magnetic eld along NV axis ([111] crystallographic direction), the
degeneracy of states is lifted and two ODMR signals are observed (Fig. 4.7b). The
shift in central frequencies is described by the Zeeman interaction term in Eqn. 4.3
and is given by 2g
B
B
0
=~, which can be used to measure the magnetic eld at NV
center. Application of magnetic eld separates the spin transitions and allows to
address them individually by selective microwave excitation. In this case, resonant
spin states (e.g. j0⟩ andj1⟩) can be treated as a two-level system.
Chapter 4. Magnetic resonance spectroscopy of diamond 73
Normalized FL contrast
Microwave frequency (MHz)
2870
0.75
1
0.75
1
Exp.
Fit
Exp.
Fit
3080 2660
B
0
= 0 mT
B
0
≈ 5 mT
0 1 ↔ − 0 1 ↔ +
0 1 ↔ ±
0
1 ±
1 −
1 +
0
Figure 4.7: cw-ODMR measurements on a single NV center at zero magnetic eld
(upper panel) and at external magnetic eld of 5 mT. Magnetic eld was ap-
plied along NV axis ([111] crystallographic direction). The insets show schematic
energy diagrams for the spin sub-levels according to the dominant interaction terms
in Eqn. 4.3, i:e: the Zeeman interaction and zero-eld splitting terms. Fitting of cw-
ODMR signals was performed using Lorentzian lineshape to obtain amplitude (A)
and half-width (∆!) of NV transitions: A = 0.31 0.01 and ∆! = 15.9 1.4
MHz at 2870.0 0.7 MHz, A = 0.21 0.01 and ∆! = 7.0 0.8 MHz at 2734.8
0.5 MHz, A = 0.27 0.01 and ∆! = 10.5 0.5 MHz at 3006.3 0.5 MHz.
Possible reason for variations in amplitude and linewidth of the cw-ODMR signals is
frequency-dependent transmission of the microwave line [166].
Chapter 4. Magnetic resonance spectroscopy of diamond 74
Moreover, optical and microwave excitations have opposite affect on NV popu-
lation in the m
s
= 0 state in cw-ODMR measurements, i:e: the optical excitation
increases the spin polarization between the m
s
= 0 and 1 states while the mi-
crowaveexcitationdecreases thepolarizationduetothepowersaturation. Therefore,
cw-ODMR lineshape is typically power broadened and is highly sensitive to the op-
tical and microwave power during the measurement [166]. While the optical power
is very stable over the measurement time (< 1%
uctuations), the microwave power
applied at the NV center may be considerably different at the frequencies of 2735,
2870 and 3006 MHz due to frequency-dependence of the microwave transmission, ex-
plaining the observed difference in the amplitude and width of the cw-ODMR signals
(Fig. 4.7).
4.2.2 Pulsed-ODMR
Inpulsed-ODMRmeasurements,pulsedopticalandmicrowaveexcitationsareapplied
to facilitate control of the NV spin state. Resonant microwave pulses are used to
manipulate NV spin state, while optical pulses are employed for preparation of NV
center into the m
s
= 0 state and readout a spin state of NV center.
Spin readout
In contrast to cw-ODMR that detects changes of FL intensity in the steady state,
spin readout in pulsed-ODMR measurements relies on the detection of changes in
transient FL signals. Fig. 4.8 shows transient FL measurements at zero magnetic
eld with different spin states of NV center. After optical initialization pulse of 5 s
followedbythedarktimeintervalof5s,theNVcenterwaspolarizedintothem
s
=0
state. The dark time was chosen in access to the lifetime of the metastable singlet
state (Fig. 4.7) that is 300 ns, to allow for major population decay tom
s
= 0 from
Chapter 4. Magnetic resonance spectroscopy of diamond 75
initialization RO
FL
Microwave
Laser
t
p
τ
R
(ns)
0 400 800 1200 1600
Normalized FL
0.5
1
0
t
p
= 0
t
p
= 37.5 ns
t
p
= 75 ns
spin state readout window
τ
R
(a)
(b)
Figure4.8: TransientFLsignalsofNVcenter. (a)Pulsesequencethatwasemployed
to detect transient FL signals of NV centers. Duration of optical initialization pulse
and readout pulses (RO) was set to 5 s. FL was detected using collection window
of50ns. t
p
isdurationofmicrowavepulse.
R
isthepositionofFLcollectionwindow
withrespecttotheROpulse. TorecordtransientFL,FLsignalsaremeasuredwithin
the collection window of 50 ns as function of
R
. (b) Transient FL signals for three
duration of microwave pulse: t
p
= 0 (blue), t
p
= 37.5 ns (red), t
p
= 75 ns (orange).
Single FL point was measured over 10
6
repetitions of the pulse sequence at each
R
,
to reduce the effect of short-noise, resulting in the overall integration time of 50 ms.
Solid lines were obtained from the simulation of transient FL signals based on the
ve-level model for NV center (Appendix B.1). A good agreement was found for the
optical excitation rate of 5 MHz and Rabi oscillation frequency of 10.4 MHz.
Chapter 4. Magnetic resonance spectroscopy of diamond 76
the singlet state during the dark time interval. Then, a resonant microwave pulse
of a xed duration (
p
) was applied to prepare NV center in a superposition of the
m
s
= 0 and m
s
= 1 ground states. Finally, transient FL signals were measured
within short FL collection window as a function of
R
during optical readout pulse
(RO). To reduce shot-noise, FL signals were collected over 10
6
repetitions.
As shown in Fig. 4.8b, the maximum transient FL signals are observed from the
m
s
= 0 spin state (no microwave pulse). In a case of a short resonant microwave
pulse, a superposition of the m
s
= 0 and m
s
=1 states is prepared and therefore
populationinthem
s
= 0stateis reduced leadingtothereductionofFLsignals. Fur-
thermore, transient FL signals are partially recovered for the twice longer microwave
pulse, indicating increase of the population in the m
s
= 0 state. This behavior
is expected for microwave induced Rabi oscillations with the characteristic pulse
shorter than the microwave pulse (Sect. 2.3.1), i:e the microwave pulse induces spin
rotation on the Bloch sphere by more than 180
◦
. The observed transient FL signals
are in a good agreement with the simulation described in Appendix B.1 (solid lines
in Fig. 4.8b), where FL signals were calculated based on the ve-level model for NV
center. This conrms successful readout of the spin state through the transient FL
measurement. In practice, to readout the spin state, the transient FL signals are
integrated over 300 ns interval (see Fig. 4.8b). In addition, the optical initialization
pulse of 23 s is typically used because the FL signals approach a steady state
FL level within the optical pulse of 2 s (Fig. 4.8b).
Rabi oscillations
A pulse sequence to observe coherent oscillations of NV populations between mi-
crowavecoupledspinstatesisillustratedinFig. 4.9a, whichissimilartothetransient
FL measurements with minor modications. NV is rst polarized into the m
s
= 0
Chapter 4. Magnetic resonance spectroscopy of diamond 77
stateduringopticalinitializationpulseof3sanddarktimeintervalof5s,followed
by microwave pulse (t
p
) and optical readout pulse (RO) that are separated by 10 s.
The FL signals are counted over 500 ns time interval (Sig), which is centered with
the RO pulse of 300 ns. Time count intervals of 100 ns before and after RO pulse
are introduced to prevent effects of timing errors on the FL counting, such as due
to response of electronics, falling and rising edges of trigger pulses, etc: In this case,
the effective integration time of FL signals per pulse sequence is governed by RO
pulse and is 300 ns, as there are no photons in the system prior the laser pulse.
To perform Rabi oscillation measurement, FL signals are measured as a function of
microwave pulse duration (Sig(t
p
)).
In addition, two reference FL signals are measured with the same pulse sequence,
but without the microwave and with the microwave pulse, which are called as
Max and Min, respectively. In this case, Max signal refers to FL signals due to
maximum population in the m
s
= 0 state, while Min signal refers to FL signals with
minimum population in the m
s
= 0 state after application of pulse. Max and
Min signals are measured using the corresponding pulse sequence applied before and
after the pulse sequence for Sig, respectively. FL signals are accumulated over 10
6
10
7
repetitions and the normalized signal N(t
p
) (Sig(t
p
) - Min)/(Max - Min) is
recorded. Signal, normalized to the population in the m
s
=0 state, is then obtained
from the normalization procedure described in Appendix B.2.
A typical result of the Rabi oscillation measurement on a single NV center in
type-Ib diamond is shown in Fig 4.9b, which was measured for the j0⟩ ↔ j1⟩
transition at magnetic eld of 3 mT. As shown in Fig. 4.9b, clear population
oscillations between j0⟩ and j1⟩ are observed. The signal is well explained by
1=2+1=2cos(2Ω
R
t
p
) exp[(t
p
=T
R
)
2
], with the Rabi oscillation frequency of 11
MHz and Rabi oscillations damping time (T
R
) of 1.4 s. The result of the Rabi
Chapter 4. Magnetic resonance spectroscopy of diamond 78
initialization RO
0
1
p (m
s
= 0)
Exp.
Fit
Microwave frequency (MHz)
500
0.5
1000 1500 0
FL
Microwave
Laser
Sig
t
p
Ω
R
= 10.7 ± 0.1 MHz
T
R
= 1.4 ± 0.1 μs
0 1 ↔ −
(a)
(b)
Max Min
Figure 4.9: Rabi oscillation measurement of a single NV center. (a) Pulse sequence
to detect FL signals in the Rabi oscillation measurement. Optical initialization pulse
was set to 3 s, readout pulse (RO) - 300 ns, FL readout window (Sig) - 500 ns.
Position of pulse sequences to detect Max and Min FL signals are indicated by black
arrows. Max-Sig-Min pulse sequence were repeated over 10
6
times resulting in the
integration time of 300 ms for FL signals. (b) Rabi oscillation signal of a single NV
center forj0⟩↔j1⟩ transition at magnetic eld of 3 mT. The signal was normal-
izedtothepopulationinm
s
=0accordingtotheproceduredescribeinAppendixB.2.
The signal was t to the function/ cos(2Ω
R
t
p
) exp[(t
p
=T
R
)
2
], to obtain the Rabi
oscillation frequency - Ω
R
= 10:70:1 MHz and damping time T
R
of 1:40:1s.
Chapter 4. Magnetic resonance spectroscopy of diamond 79
measurement is used to determine durations of microwave pulses, such as =2 and
pulses. The damping of Rabi oscillations is due to coupling of a single NV center to
its local spin environment. In the case of type-Ib diamond, NV is largely affected by
the bath of nearby N spins as well as weakly coupled bath of
13
C nuclear spins [74].
Both baths induce a local magnetic eld at NV center. The local magnetic eld at
the NV shifts the Larmor frequency of NV center resulting in a small frequency offset
with respect to the microwave frequency. In the case of repeated readout of the NV
spin state, the measured signal is an average over different realizations of the spin
baths,i:e: distribution of the local magnetic eld, leading to the damping of the Rabi
oscillations. FurtherdiscussiononthedecayofRabioscillationsofasingleNVcenter
is given in Appendix B.2
Spin Echo decay measurement
TodetermineT
2
decoherencetimeofasingleNVcenter,weemployamicrowavepulse
sequence that consists of the conventional Hahn echo sequence (=2) and
additional=2 pulse (Fig. 4.10a). Because FL signals are sensitive to the populations
in the m
s
= 0 state, the additional =2 pulse is applied to convert the coherent
state of NV center after the Hahn echo sequence into a population difference. The
pulse sequence for the optical spin initialization and spin readout is identical to the
Rabi oscillation measurements, as well as the detection of the reference - Max and
Min signals. During the measurement, FL signals are typically collected over 10
7
repetitions of the pulse sequence at a xed free evolution time.
The result of SE decay measurement for a single NV center in type-Ib diamond is
shown in Fig. 4.10b, which was measured for thej0⟩↔j1⟩ transition at magnetic
eldof3mT.They-axisofFig.4.10bisthenormalizedsignal(AppendixB.2). The
decay was t with the decay function / exp[(2=T
2
)
2
], to obtain T
2
decohernece
Chapter 4. Magnetic resonance spectroscopy of diamond 80
initialization RO
0.5
2τ (μs)
10
Exp.
Fit
1
0 2 4 6 8
p (m
s
= 0)
FL
Microwave
Laser
Sig
π τ τ
π
2
π
2
T
2
= 2.2 ± 0.1 μs
0 1 ↔ −
(a)
(b)
Max Min
Figure 4.10: Spin echo measurement of a single NV center. (a) Pulse sequence to
detect FL signals in the SE decay measurement. Optical initialization pulse set to
3 s, readout pulse (RO) - 300 ns, FL readout window (Sig) - 500 ns. Position of
the pulse sequences to detect Max and Min FL signals are indicated by the black
arrows. Max-Sig-Min pulse sequence were repeated over 10
7
times resulting in the
integration time of 300 ms for FL signals. (b) SE decay signal of a single NV center
for j0⟩↔ j1⟩ transition at magnetic eld of 3 mT. The signal was normalized
to the population in m
s
= 0 according to the procedure describe din Appendix B.2.
Good agreement of the data and the SE decay/ exp[(2=T
2
)
2
] was found with T
2
decoherence time for single NV of 2.2 s.
Chapter 4. Magnetic resonance spectroscopy of diamond 81
time of 2 s. The observed shape of SE decay is predicted by the OU process
(Sect. 2.3.3).
When the SE decay is measured for several NV centers in the same diamond
crystal, variations in T
2
are observed. The T
2
values in type-Ib diamonds typically
range from several hundreds of ns to several s. The main source of decoherence for
NV centers in type-Ib diamonds is bath of N spins [74], which can be modeled by
the OU process in the case of a single NV center [91, 127, 167{169]. According to
the OU process (Sect. 2.3), SE decay depends on the effective dipolar coupling (∆)
between central spin and the bath spins, and depends on the correlation time (
c
)
of the bath spin
uctuations that arise from the dipolar induced
ip-
ops between
the bath spins. Both ∆ and
c
strongly depend on the the spatial conguration of
the bath. Given random probability to occupy lattice cites for N in the diamond,
different NV centers are exposed to unique spin baths, and therefore unique ∆ and
c
, explaining the variations in T
2
between single NV centers.
Chapter 5
Measurement of paramagnetic spin
concentration in a solid-state
system using double
electron-electron resonance
Materials presented in this Chapter can be also found in the article titled Determi-
nation of nitrogen spin concentration in diamond using double electron-electron res-
onance by Viktor Stepanov and Susumu Takahashi in Physical Review B 94, 024421
(2016) (Reprinted with permission. Copyright 2016, American Physical Society).
5.1 Introduction
FortheapplicationsofNVcenterinnanoscaleMRsensingandfundamentalsciences,
long coherence of a NV center is critical. Coherence of a NV center highly depends
on contents of paramagnetic impurities in diamond. In particular, nitrogen related
impurities including well-known N spins are often abundant in many diamond crys-
tals. Forexample, type-Ibandtype-IIadiamondstypicallycontainnitrogenimpurity
concentration in the range of 10100 parts-per-million (ppm) and tens of parts-per-
82
Chapter 5. Spin concentration DEER-based measurement in diamond 83
billion (ppb), respectively. Coherence in such diamond crystals are largely affected
by the concentration of nitrogen impurities [127, 145].
Moreover,interesttofabricateensemblesofNVcenters(NVconcentration1100
ppm) have been rapidly growing for applications of NV-based quantum devices [170{
172], showing that precise determination of the concentration of NV centers and N
spins in diamond is highly useful. Unfortunately, currently available techniques have
several limitations. For example, infrared absorption spectroscopy is a commonly-
used technique to determine N spin concentration, however the sensitivity is often
nothighenoughtomeasuretype-IIadiamond[173]. LineshapeanalysisofEPRspec-
troscopyhasbeenappliedtodeterminetheconcentrationofparamagneticimpurities.
Although high precision of the spin concentration determination ( 3 %) has been
achieved using X-band EPR spectroscopy [174], the method remains challenging for
wide applications as it highly depends on the choice of the reference sample [175],
positionofthesamplesinthecavity[176,177],spinrelaxations[102]andrequirespre-
ciseknowledgeofeffectofsamplesonmicrowaveelds[178,179],lling[175,179]and
quality factors of the cavity [175, 178], microwave and modulation eld distributions
over the sample volume [180{182].
This Chapter presents a HF DEER technique to determine the concentration of
paramagnetic impurities in solid-state systems with high precision and no reference
sample. DEER spectroscopy is known to be a powerful technique to probe the mag-
neticdipoleinteractionbetweenparamagneticspins. Fortheinvestigation,weemploy
a home-built HF EPR/DEER spectrometer described in Sect. 3.1, which enables to
perform high spectral resolution EPR/DEER spectroscopy with different groups of
spins. Inthepresentdemonstration,115GHzEPR/DEERspectroscopyisperformed
at room temperature. First, we measure HF EPR spectrum of paramagnetic spins
in diamond which allows us to identify a type of impurities. The EPR spectrum
Chapter 5. Spin concentration DEER-based measurement in diamond 84
analysis conrms that a majority of paramagnetic spins in both type-Ib and type-IIa
diamonds are N spins. Then we perform pulsed EPR experiment to determine spin
decoherence time (T
2
) in the diamond crystals and DEER spectroscopy to determine
the concentration of N spins in the range of 0.1100 ppm. Finally, we investigate the
relationship between the concentration of N spins and their spin decoherence time
(T
2
).
5.2 Experiment
Fortheinvestigation, weemployedseveralsyntheticdiamondcrystalsincludingtype-
Ib and type-IIa crystals from DiAmante Industries, LLC [183], Element 6 [184] and
Sumitomo Electric [185]. The EPR/DEER measurements were performed using a
home-built 115 GHz EPR/DEER spectrometer at room temperature (Sect 3.1). The
systemhasawide-bandDEERcapability(13GHz)whichisrequiredforthepresent
study.
5.2.1 Spin echo measurement
Figure 5.1 shows 115 GHz EPR measurements of type-Ib and type-IIa diamond crys-
tals performed by monitoring the SE intensity as a function of magnetic elds. The
type-Ib diamond crystal has a polished face normal to the [111] crystallographic axis
while the type-IIa diamond crystal has a polished face normal to the [100] axis. In
bothmeasurements,themagneticeldwasappliedperpendiculartothepolishedsur-
face. As shown in Fig. 5.1a, the EPR spectrum of the type-Ib diamond sample shows
vepronouncedpeaksrepresentingNspins(Sect.4.1). Thesevepeaksoriginate(la-
beledas1,2,3,4,and5)fromthefourprincipleaxesofNspins, i.e.,[111],[11
1],[1
11]
and [
111], and the hyperne interaction to
14
N nuclear spin [66, 74]. The intensity of
Chapter 5. Spin concentration DEER-based measurement in diamond 85
4.100 4.104 4.108
Type-Ib
B
0
// [111]
Type-IIa
B
0
// [100]
(a)
Magnetic field (Tesla)
Echo intensity (a.u.)
1 2 3 4 5
1,2 3 4,5
π/2 π
τ τ
Type-Ib
T
2
= 2.00±0.01 μs
Type-IIa
T
2
= 146±2 μs
(b)
10 100 1000
2τ (μs)
Echo intensity (a.u.)
π/2 π
τ τ
Figure 5.1: SE measurements of type-Ib and type-IIa diamond crystals. (a) SE
intensity as a function of magnetic elds. The applied pulse sequence is shown in the
inset. In the measurement of the type-Ib diamond, the durations of/2 and pulses
were 150 ns and 250 ns and was 1.5 s. The data were taken with 32 averages
with 20 ms of the repetition time. In the measurement of the type-IIa diamond, the
durations of the /2 and pulses were 250 ns and 450 ns and was 3 s. The data
were taken with 256 averages with 20 ms of the repetition time. The magnetic eld
was applied along the [111] direction for type-Ib crystals and the [100] direction for
type-IIa. (b) SE intensity as a function of to measure spin decoherence time T
2
.
The decays of the SE were tted by a single exponential function to extract T
2
(solid
lines). Thedataofthetype-Ib(type-IIa)diamondwastakenwith128(256)averages.
Chapter 5. Spin concentration DEER-based measurement in diamond 86
the EPR signals represents the population of each group, with the population ratio
corresponding to 1 : 3 : 4 : 3 : 1 for Group 15, respectively. In addition, we mea-
sured the SE intensity of the N spins as a function of magnetic elds in the type-IIa
diamond. As shown in Fig. 5.1a, the width of the observed signals were signicantly
narrower than those of the type-Ib crystal. Next, Fig. 5.1b shows spin decoherence
time (T
2
) measurements of the type-Ib and type-IIa samples. We observed that the
SE decayed exponentially as a function of 2 in both cases. As indicated in Fig. 5.1b,
T
2
for the type-IIa diamond was nearly two orders of the magnitude longer than that
of the type-Ib diamond while both samples have similar spin-lattice relaxation times
(T
1
) of several ms (data not shown). We also found that T
2
values of all groups were
very similar.
5.2.2 Double electron-electron resonance spectroscopy
Next, we performed DEER spectroscopy to probe the magnetic dipole interaction
between N spins. For DEER spectroscopy of the type-Ib diamond, the N spins at B
0
= 4.099 Tesla (Group 1), whose axis is along [111] and whose nuclear spin state is
jm
I
= 1⟩, wereusedasprobespins(Aspins). Bspins(otherNspinsinGroup25in
Fig. 5.1a)wereused as pump spins. Then weappliedthe three-pulseDEER sequence
to probe the magnetic dipolar coupling between N spins in diamond [186]. As shown
in the inset of Fig. 5.2a, the applied DEER sequence consisting of the SE sequence
for A spins at the frequency of
A
= 115 GHz and a single pulse for B spins at the
frequency of
B
. In the DEER spectroscopy, changes in the SE signal occur when
the effective magnetic dipolar elds at A spins are altered by B spins that are
ipped
by the pulse. As shown in Fig. 5.2b, four DEER signals of N spins were clearly
observed as reductions of the SE intensity of A spins. The signals were centered at
114.971, 114.886, 114.801 and 114.772 GHz, corresponding to B spins in Group 2,
Chapter 5. Spin concentration DEER-based measurement in diamond 87
T
π/2 π
τ τ
ν
A
ν
B
t
P
t
1
t
2
0
0.5
1.0
Exp.
Fit
0
0.5
1.0
Exp.
Frequency of pump pulse (GHz)
114.7 114.8 114.9 115.0
114.8 114.9 115.0
Frequency of pump pulse (GHz)
ν
A
ν
A
Fit
Normalized echo intensity Normalized echo intensity
Type-Ib
Type-IIa
(b)
(c)
(a)
Figure5.2: Doubleelectron-electronresonance(DEER)spectroscopyofthetype-Ib
and type-IIa diamond crystals. (a) three-pulse DEER sequence used in the experi-
ment, where t
1
and t
2
denote duration of =2 and pulses for A spins, respectively.
t
p
isdurationof pulseforBspins. T isthedelayoft
p
fromt
1
. (b)&(c)DEERspec-
trum of N spins in type-Ib and type-IIa diamonds, respectively. The DEER signals
were normalized by the SE signals. Experimental parameters were t
1
= 250 ns, t
2
=
450ns,t
p
=450ns, =2.5s,T =2sincaseoftype-Ibdiamond, andt
1
=250ns,
t
2
= 450 ns,t
p
= 450 ns, = 110s,T = 109.45s in case of type-IIa diamond. The
data of the type-Ib (type-IIa) diamond was taken with 128 (256) averages. Purple
and brown dashed lines represent the best t of experimental data using Eqn. 5.10.
Chapter 5. Spin concentration DEER-based measurement in diamond 88
3, 4, and 5, respectively. Thus, the result conrms direct observation of the dipolar
couplingbetweenNspinsinthetype-Ibdiamond. Similarly,weperformedtheDEER
measurement with the type-IIa diamond, and, as shown in Fig. 5.2c, observed the
DEER signals.
5.3 Model
5.3.1 Spin echo
There exist several processes which can contribute to the SE decay, including the
spin
ip-
ops of N spin bath, the instantaneous diffusion,
13
C nuclear spins and the
single spin
ips (T
1
process). As reported previously, the spin
ip-
op (also known
as the spectral diffusion) is one of the major decoherence sources in type-Ib diamond
crystals [74, 145]. The spin
ip-
op process causes dipolar-eld
uctuations at the
sites of the excited spins and the decoherence rate of this process linearly depends on
the concentration of surrounding non-excited N spin bath [127, 145]. On the other
hand, in the case of type-IIa, it has been shown that the nuclear spin decoherence is
pronounced [13, 73]. In addition, the SE decay may be speeded up by the process
of instantaneous diffusion that manifests itself upon application of pulse due to
dipole-dipole interactions between the excited spins. In the case of the instantaneous
diffusion process, the SE decay depends on the concentration of the excited spins,
therefore the contribution of the instantaneous diffusion will be different between
spin groups with different concentrations of N spins,i.e. group 1 and 3 in Fig. 5.1a.
However our observation of similar T
2
times between different groups indicates that
the instantaneous diffusion is insignicant in our experiments. The spectral diffusion
due to T
1
process is also negligible in the present case because of the observed long
T
1
.
Chapter 5. Spin concentration DEER-based measurement in diamond 89
Next, we discuss the SE decay to estimate the spin
ip-
op rate with the use of
a model for the dipolar-coupled electron spins developed in Ref. [128]. According to
Ref. [128], the SE decay due to the spectral diffusion is described by the following
expression,
SE(2)= exp
0
@
n
1
∫
0
f(W;W
max
)
∫
V
[
1v
0
(2;W)
]
dV dW
1
A
; (5.1)
where W is the rate of the spin
ip-
ops of bath spins. v
0
represents SE signals of a
single excited spin dipolar-coupled to a non-excited bath spin with the relativeradius
vector (⃗ r(r;)) given by,
v
0
(2;W)=
[
(
coshR +
W
R
sinhR
)
2
+
A
2
4R
2
sinh
2
R
]
exp(2W);
where A
0
2
B
g
1
g
2
(13cos
2
)=(4~r
3
) and R
2
W
2
1
4
A
2
.
0
is the vacuum
permeability,
B
is the Bohr magneton, ~ is the reduced Planck constant, g
1
and
g
2
are g-factors of the excited and bath spins, respectively. The integration over the
samplevolumeV inEqn.(5.1)takesintoaccountallpossiblerand. Theintegration
over W accounts for a distribution of the
ip-
op rate within the sample where the
distribution function f(W;W
max
) is given by [128],
f(W;W
max
)=
√
3W
max
2W
3
exp
(
3W
max
2W
)
: (5.2)
where f(W;W
max
) is maximum at the
ip-
op rate of W =W
max
.
Usingthemodelabove,weestimateanaverage
ip-
oprateofNspinsindiamond.
We rst consider a single exponential SE decay with T
2
= 950 ns (the shortest T
2
observed in our experiments). We performed a t using Eqn. (5.1) with a xed N
concentration to extractW
max
. As shown in Fig. 5.3a, the SE model (Eqn. (5.1)) ts
well with a single exponential decay with T
2
= 950 ns and the t results give4.9,
2.9,and2kHzofW
max
for60,80and100ppmoftheconcentrations,respectively.
The
ip-
op distribution function (Eqn. 5.2) for the obtained W
max
are plotted in
Chapter 5. Spin concentration DEER-based measurement in diamond 90
2 4 6 8
0
1
n = 60 ppm
n = 80 ppm
n = 100 ppm
Exponential (T
2
= 950ns)
2τ (μs)
Normalized echo intensity
1E-3 0.01 0.1 1
0.5
1.0
W (MHz)
Population (10
-4
)
0
1E-4
f (W, W
max
= 4.9 kHz)
f (W, W
max
= 2.9 kHz)
f (W, W
max
= 2 kHz)
(a)
(b)
Figure 5.3: Analyses of the
ip-
op process of N spins in type-Ib diamond. (a)
Analyses of a single exponential SE decay with T
2
= 950 ns (cyan) using Eqn. (5.1).
4.9, 2.9 and 2 kHz ofW
max
were obtained from the ts for 60 (black square), 80 (red
diamond) and 100 (pink circle) ppm of N concentrations, respectively. (b) Flip-
op
rate distribution among N spins obtained using Eqn. (5.2) for 4.9 (black), 2.9 (red)
and 2 kHz (pink) of W
max
.
Chapter 5. Spin concentration DEER-based measurement in diamond 91
Fig. 5.3b. As shown in Fig. 5.3b, a major population of the
ip-
op rate ranges from
1 kHz to1 MHz. In addition, an average
ip-
op rate is given by,
⟨W⟩
80%
=
[
√
6W
max
b
exp
(
3W
max
2b
)
3W
max
erfc
(
√
3W
max
2b
)]
b=50Wmax
7:1W
max
;
where the upper limit of the integration was set at 50 W
max
(corresponding to 80%
of the cumulative percentage) to avoid the divergence of the integral to evaluate the
⟨W⟩. Using values for ⟨W⟩
80%
, the average
ip-
op events 2⟨W⟩
80%
during the
DEER sequence (2 = 3 s for the sample with the shortest T
2
) were estimated as
0.1,0.06and0.04for60, 80and100ppm, respectively. Moreover,forlongerT
2
times,
the
ip-
op probability is expected to be even lower. With the given small
ip-
op
probability on the time scale of the DEER experiment, we consider the N spins to be
in the static regime to model the DEER signal.
5.3.2 Double electron-electron resonance
In this section, we model DEER signals for ensemble N spins. The DEER signal is
produced by probe N spins (A spins) interacting with resonant N spins to the pump
pulse (B spins) and the rest of spins in diamond (C spins). C spins include both
non-resonant N spins and nuclear spins. The center frequencies of EPR transitions
of N spins are given by the Hamiltonian of N spins (Eqn. 4.2) Moreover, all EPR
transitions have equal linewidths due to randomly distributed N and nuclear spins
in the diamond lattice, giving rise to inhomogeneously broadened spectral lines (i.e.
Group15inFig.5.4). Asshownpreviously[118,125],theinhomogeneouslineshape
due to dipolar interactions between electron spins is expected to be Lorentzian while
the lineshape due to electron-nuclear dipolar interactions is expected to be Gaussian.
We here describe each spectral line by Lorentzian lineshape with a half-width of ∆!
Chapter 5. Spin concentration DEER-based measurement in diamond 92
ω
1
ω
2
ω
3
ω
4
ω
5
ω
Β
ω
Α
ω
j
ω
δ
j
δ
Frequency
L (ξ)
Figure 5.4: Schematics for the DEER model. L() is the lineshape function. !
m
is the center frequency of Group m (m=15). !
A
and!
B
are microwave frequencies
of the probe and pump pulses, respectively. ! and !
j
are the Larmor frequencies of
A and B spins, respectively. and
j
are frequency offsets of A and B spins from the
pump and probe frequencies, respectively. A and B spins were chosen close to the
probe and pump frequencies, to indicate, that spins can be excited by a respective
pulse with a small frequency offsets. However, in general, as in our consideration,
they can be anywhere within the lineshape L.
(the analysis of the type-IIa samples with the Gaussian lineshape was also tested (see
Sect. 5.4.1). Thus the total lineshape is given byL() =
1
∑
m
f
m
∆!
∆!
2
+(!m)
2
, where
f
m
and!
m
being fraction of spins and transition frequency of Groupm, respectively.
Here, we focus on the case when magnetic eld is applied along the [111] direction
and a DEER lineshape is shown in Fig. 5.4. We start by considering a single A spin
with the Larmor frequency ! (see Fig. 5.4) as a two-level system (TLS) represented
by Hamiltonian in units of frequency,
^
H
0
=!
^
S
z
. During the application of the probe
pulse with microwave frequency!
A
, applied at the center frequency of Group 1 (!
1
),
the total Hamiltonian is given by
^
H =
^
H
0
+
^
H
MW
= !
^
S
z
+ 2Ω
^
S
x
cos!
A
t, where
Ω = g
B
b
1
=2~ and b
1
is the strength of the microwave eld. The frequency offset
Chapter 5. Spin concentration DEER-based measurement in diamond 93
() in Fig. 5.4, dened as !!
A
, is due to local magnetic elds from B and C
spins, i.e. =g
B
(b
B
+b
C
)=~, where b
B
(t) =
∑
j
b
j
(t) and b
C
(t) =
∑
k
c
k
(t), j and
k are indexes of B and C spins, and b
j
(t) and c
k
(t) are magnetic elds produced by
j-th B and k-th C spins at a single A spin, respectively. Due to the low probability
of the
ip-
op as discussed in Sect. 5.3.1, is considered to be time-independent.
Moreover, to calculate DEER signals below, we assumejg
B
b
B
=~j≪jj for A spins
contributing to SE signals in DEER experiment because of the low concentration (<
10
19
spins/cm
3
)andpartialexcitationofNspins. jg
B
b
B
=~j≪jjisalsocommonly
employed in dilute spin systems (< 10
20
spins/cm
3
) [118]. The above assumptions
ensure constant during DEER sequence.
First,wecalculateSEsignalproducedbyasingleAspinduringthepulsesequence
(t
1
t
2
). The spin state by the end of the sequence (j
2
⟩) is given by,
j
2
⟩ =
^
U
2
()
^
R(t
2
)
^
U
1
()
^
R(t
1
)j
0
⟩; (5.3)
wherej
0
⟩ is the initial state.
^
R(t
i
) exp
[
i
(
^
S
z
+Ω
^
S
x
)
t
i
]
is a propagator that
describes evolutionof TLS under the microwaveexcitation in the rotating frame with
the microwave frequency (!
A
). In a matrix representation in the basis ofj+⟩ andj⟩
states,
^
R(t
i
) is given by,
^
R(t
i
) =
0
B
@
c
i
i
Ω
A
s
i
i
Ω
Ω
A
s
i
i
Ω
Ω
A
s
i
c
i
+i
Ω
A
s
i
1
C
A
;
where Ω
A
p
2
+Ω
2
, c
i
cosΩ
A
t
i
=2 and s
i
sinΩ
A
t
i
=2. U
i
is a free evolution
propagator dened as
^
U
i
() =
0
B
@
e
i(φ
i
+ϕ
i
)=2
0
0 e
i(φ
i
+ϕ
i
)=2
1
C
A
;
with φ
1
g
B
~
∫
0
b
B
(t)dt, φ
2
g
B
~
∫
2
b
B
(t)dt, ϕ
1
g
B
~
∫
0
b
C
(t)dt and ϕ
2
g
B
~
∫
2
b
C
(t)dt
Chapter 5. Spin concentration DEER-based measurement in diamond 94
Using Eqn. (5.3), the magnetic eld component in the rotating frame alongy-axis
of a single A spin with the initial statej
0
⟩ =j⟩, is calculated as
⟨
^
S
y
⟩
s
=⟨
2
j
^
S
y
j
2
⟩
=
[
Ω
Ω
A
c
1
s
1
c
2
2
2
Ω
Ω
3
A
(c
1
s
1
s
2
2
+2s
2
1
c
2
s
2
)
]
cos2
+
[
3
Ω
Ω
4
A
s
2
1
c
2
2
Ω
Ω
2
A
(2c
1
s
1
c
2
s
2
+s
2
1
c
2
2
)
]
sin2
+
Ω
Ω
3
A
c
2
s
2
[
2
+Ω
2
(c
2
1
s
2
1
)
]
cos
Ω
Ω
4
A
s
2
2
[
2
+Ω
2
(c
2
1
s
2
1
)
]
sin
+
[
Ω
3
Ω
3
A
c
1
s
1
s
2
2
]
cos(φ
1
φ
2
)+
[
Ω
3
Ω
4
A
s
2
1
s
2
2
]
sin(φ
1
φ
2
):
After omitting the FID signals that are averaged out on the time scale of T
2
[128],
the
⟨
^
S
y
⟩
s
is reduced to
⟨
^
S
y
⟩
s
[
Ω
3
Ω
3
A
c
1
s
1
s
2
2
]
cos(φ
1
φ
2
)+
[
Ω
3
Ω
4
A
s
2
1
s
2
2
]
sin(φ
1
φ
2
): (5.4)
Similarly,
⟨
^
S
x
⟩
s
in the rotating frame is found as
⟨
^
S
x
⟩
s
[
Ω
3
Ω
3
A
c
1
s
1
s
2
2
]
sin(φ
1
φ
2
)+
[
Ω
3
Ω
4
A
s
2
1
s
2
2
]
cos(φ
1
φ
2
): (5.5)
Next, the SE signal of a single A spin in the DEER measurement is calculated.
Whenthepumppulsewiththefrequency(!
B
)excitesBspins,thephaseaccumulated
by the A spin during 2 is expressed as
φφ
1
φ
2
=
g
B
~
∑
j
(
b
j
(Tt
p
=2)+
tp
∫
0
b
MW
j
(t)dt
+b
MW
j
(t
p
)[(Tt
p
=2)]
)
;
(5.6)
where b
j
0
B
g
B
(3cos
2
j
1)
j
=(4r
3
j
) is a magnetic eld produced by the j-th
B spin at the A spin before the pump pulse is applied.
j
is the spin state of thej-th
B spin (
j
=1=2). ⃗ r
j
(r
j
;
j
) is the radius vector of the dipole interaction between
Chapter 5. Spin concentration DEER-based measurement in diamond 95
thej-th B spin and the A spin. b
MW
j
=b
j
[
2
j
+Ω
2
(c
2
j
s
2
j
)
]
=Ω
2
B;j
with
j
!
B
!
j
(!
j
is the Larmor frequency of the j-th B spin. See Fig. 5.4), Ω
B;j
√
2
j
+Ω
2
,
c
j
cosΩ
B;j
t=2 and s
j
sinΩ
B;j
t=2. It is important to note that Eqn. (5.6) takes
into account off-resonant excitation of the B spins which is represented by (
j
,r
j
,
j
)
and
j
. Moreover, Eqn. (5.6) can be further simplied in the present case (t
p
≪ 2
and T ) to give
φ
0
4
2
B
g
A
g
B
(2T)
~
∑
j
Ω
2
2
j
+Ω
2
sin
2
(
√
2
j
+Ω
2
t
p
2
)
(3cos
2
j
1)
j
r
3
j
:
To obtain ensemble SE signal,
⟨
^
S
y
⟩
s
and
⟨
^
S
x
⟩
s
are averaged over B spins (r
j
,
j
,
j
,
j
), as described in Appendix A.2, to give
⟨
⟨
^
S
y
⟩
s
⟩
B
[
Ω
3
Ω
3
A
c
1
s
1
s
2
2
]
exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
; (5.7)
and
⟨
⟨
^
S
x
⟩
s
⟩
B
[
Ω
3
Ω
4
A
s
2
1
s
2
2
]
exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
; (5.8)
where⟨sin
2
2
⟩
L
∫
+1
1
Ω
2
(!
B
)
2
+Ω
2
sin
2
(
√
(!
B
)
2
+Ω
2
tp
2
)
L()d.
To calculate DEER signal components in the rotating frame (I
x
andI
y
) produced
by an ensemble of A spins, the DEER signals are rst obtained for a single A spin
with the j ⟩ = j+⟩ initial spin state, similarly to above calculations, and averaged
overj+⟩ andj⟩ spin states with the use of thermal populations in each state, result-
ing in the thermal magnetization factor (∆ tanh(~!
A
=2k
B
T
0
) where T
0
is sample
temperature) for Eqns. (7) and (8). Next, the signals are averaged over the lineshape
(L) to give
I
y
=∆
⟨
Ω
3
Ω
3
A
c
1
s
1
s
2
2
⟩
L
exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
;
and
I
x
= ∆
⟨
Ω
3
Ω
4
A
s
2
1
s
2
2
⟩
L
exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
:
Chapter 5. Spin concentration DEER-based measurement in diamond 96
where ⟨:::⟩
L
represents averaging over the inhomogeneous lineshape L. The latter
being averaged out to zero when the probe frequency is centered with Group 1, thus
the DEER intensity (I
Ω
) is given by
I
Ω
√
I
2
x
+I
2
y
=jI
y
j
=∆
⟨
Ω
3
Ω
3
A
c
1
s
1
s
2
2
⟩
L
exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
exp
(
2
T
2
)
;
(5.9)
wheretheSEdecay(exp(2=T
2
))wasadded. Inthecasewhentheexcitationband-
width is larger than the inhomogeneous line (≪ Ω, then⟨sin
2
2
⟩
L
= 1), Eqn. (5.9)
reduces to the result obtained previously [186, 187],
I
DEER
(n) exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
)
:
Furthermore, the obtained⟨sin
2
2
⟩
L
function in Eqn. (5.9) has been previously con-
sidered in the context of instantaneous diffusion [128, 188] and DEER background
signals in stabilized radical systems [189]. The SE intensity was also calculated pre-
viously without fully taking into account the off-resonant excitation [118, 128]. In
general,theoff-resonantexcitationnotonlyreducesthetippingangle,butalsoresults
in the nite spin projection along the microwave eld that was not considered in the
previous models, however, in the present case, this contribution is critical.
In the present experiment, the microwave power is distributed across the sample,
thereforeEqn.(5.9)hastobefurtheraveragedtoaccountfordistributionofΩ. Using
the normalization signal (N
Ω
= ∆⟨
Ω
3
Ω
3
A
c
1
s
1
s
2
2
⟩
L
exp(2=T
2
)), which is the SE signal
with no pump pulse applied (⟨sin
2
2
⟩
L
= 0 in Eqn. (5.9)), the analytical expression
of the DEER spectrum (I
DEER
=⟨I
Ω
⟩
Ω
=⟨N
Ω
⟩
Ω
) is derived as,
I
DEER
(!
B
;[!
A
;t
1
;t
2
;t
p
;T;ff
m
g;f!
m
g];[Ω;∆!;n])
=
1
⟨⟨S
A
⟩
L
⟩
Ω
⟨
⟨S
A
⟩
L
exp
(
2
0
2
B
g
A
g
B
Tn
9
p
3~
⟨S
B
⟩
L
)⟩
Ω
;
(5.10)
where
S
A
=
Ω
3
Ω
3
A
cos(Ω
A
t
1
=2)sin(Ω
A
t
1
=2)sin
2
(Ω
A
t
2
=2)
Chapter 5. Spin concentration DEER-based measurement in diamond 97
and
S
B
=
Ω
2
Ω
2
B
sin
2
(Ω
B
t
p
=2):
1=⟨⟨S
A
⟩
L
⟩
Ω
is the normalization factor. ⟨:::⟩
Ω
denote averaging over the distribution
of the Rabi frequency Ω. Among the arguments, in the DEER measurement, !
B
is
variable, and !
A
, t
1
, t
2
, t
p
, T, ff
m
g and f!
m
g are xed values. Fitting parameters
(Ω, ∆! andn) are determined from the analysis of the DEER spectrum as described
in Sect. 5.4.1.
5.4 Discussion
5.4.1 Determination of N spin concentration
Inthissection,wepresenttheanalysisofDEERspectrumtoobtaintheconcentration
ofNspins. TheanalysiswasperformedbyttingEqn.(5.10)totheDEERsignals. In
thecaseofthetype-Ibdiamond(Fig.5.2b),theDEERpulseparameters(t
1
=250ns,
t
2
= 450 ns,t
p
= 450 ns,T = 2s,!
A
= 115 GHz) and the experimentally obtained
f!
m
g(114.7714,114.8008,114.8865and114.9724GHz)wereused. Inaddition,dueto
themagneticeldalignmentalongthe[111]crystallographicdirection, thefractionof
spins in each spectral lineff
m
g was set tof1=12;3=12;4=12;3=12;1=12g. To account
for the microwave eld distribution, we used a sinusoidal function, Ω = Ω
0
(1 +
cos(2x=
D
))=2, wherex is a distance of N spin from the surface of the diamond,
D
is the wavelength of the microwave in diamond (
D
= 1.08 mm at 115 GHz) and Ω
0
is the maximum Rabi frequency in the diamond expressed in units of MHz, which
was dened through the shortest duration of pulse (t
Ω
) in diamond as Ω
0
=1=2t
Ω
.
Therefore,⟨:::⟩
Ω
in Eqn. (5.10) is equivalent to the averaging over the sample height
h (the dimension of the diamond sample along the magnetic eld and h = 2 mm in
the present case).
Chapter 5. Spin concentration DEER-based measurement in diamond 98
0
30
60
0 200 400 600
0
3
6
0 200 400 600
0.1
0.2
concentration
half-width
fit error
n (ppm) Δω (MHz) error γ
t
Ω
(ns)
Best fit
115.0 114.9 114.7 114.8
Pump frequency (GHz)
I
Exp.
I
DEER
(t
Ω
= 305 ns)
I
Exp.
I
DEER
(t
Ω
= 100 ns)
0
0.5
1
0
0.5
1
115.0 114.9 114.7 114.8
Normalized echo intensity
(b)
(a)
Figure 5.5: Fit results of DEER spectrum. Left top, middle, bottom panels show
concentrationn of N spins, half-width ∆! of inhomogeneous lineshape and a t error
, respectively, as obtained from the t at a xed t
Ω
values. The top (bottom) panel
on the right shows the result of the t obtained at t
Ω
= 100 ns (305 ns). The result
with t
Ω
= 305 ns is the best t. The grey shaded area on the left indicates ts with
a large
.
Chapter 5. Spin concentration DEER-based measurement in diamond 99
Table 5.1: Summary of ∆! and n for the studied type-IIa and type-Ib diamonds
as extracted from the analyses of the DEER data. The errors of n and ∆! were
estimated as 95 % condence interval for the t parameters.
n (ppm) ∆! (MHz) t
Ω
5 (ns)
0.0950.012 0.340.20 285
0.1390.011 0.490.14 300
0.220.02 0.540.16 395
0.260.03 0.400.16 460
22.40.4 2.360.12 110
38.20.8 2.960.13 305
50.72.1 2.180.21 400
86.10.8 3.930.26 370
With the parameters dened above, we performed the t of the experimental
DEER spectrum I
Exp
(!
B
) using a least squares minimization procedure with a xed
value of t
Ω
and tting parameters of ∆! and n. The results of this procedure are
showninFig.5.5where∆!,nandaterror(
)denedasasumofsquaredresiduals
were plotted as a function of t
Ω
(t
Ω
= 20600 ns). We performed the t in the wide
range of t
Ω
with a step size of 5 ns. As seen in Fig. 5.5, the result of the t highly
depends on t
Ω
and the t error becomes smaller with t
Ω
& 220 ns. The minimum
error value was obtained at t
Ω
of 305 ns. The values of ∆! and n for the best t
(dashed violet line in Fig. 5.5) were obtained as 2.960.13 MHz and 38.20.8 ppm,
respectively, where the error was calculated as 95 % condence interval for the t
parameter. Similarly, in the case of type-IIa diamond (Fig. 5.2b), the t parameters
were obtained as t
Ω
= 300 ns, ∆! = 0.490.14 MHz and n = 0.140.01 ppm. The
t results for all studied diamonds are summarized in Table 1. The concentration
for the shortest measured T
2
was found as 86.10.8 ppm, which is within the static
model (Sect. 5.3.1).
In Fig. 5.6a, we present the concentration dependence of the inhomogeneous
linewidth (∆!). ∆! at the high concentrations (10100 ppm) depends strongly on
Chapter 5. Spin concentration DEER-based measurement in diamond 100
0.01 0.1 1 10 100
0
2
4
Δω (MHz)
Nitrogen concentration (ppm)
(a)
(b)
0
250
500
Duration time (ns)
t
Ω
t
2
0.01 0.1 1 10 100
Nitrogen concentration (ppm)
Figure 5.6: (a) Summary of the obtained ∆!. (b) Summary of the obtainedt
Ω
. t
2
is the duration of -pulse used in the present study.
Chapter 5. Spin concentration DEER-based measurement in diamond 101
the concentration of N spins, suggesting that the linewidth is governed by the dipolar
coupling between N spins. In contrast, at the low concentrations (< 1 ppm), the
linewidth is almost independent of the concentration, suggesting that the broadening
is dominated by other impurities, most probably
13
C nuclear spins. We also analyzed
the DEER spectra of the type-IIa diamond crystals with the Gaussian lineshape. We
foundthatthetresultswiththeLorentzianlineshapearebetterandthediscrepancy
inthedeterminedconcentrationsiswithintheerror. Ontheotherhand,theobtained
t
Ω
is independent of the concentration (n). Thet
Ω
values are also consistent with the
experiment as shown in Fig. 5.6b, where the lengths of the microwave pulses were
chosen to maximize the SE signals (the durations of the experimental -pulse were
150-450 ns as shown in Fig. 5.6b). Possible reasons for the variations are different
sizes of the diamond crystals and imperfect sample positioning [190].
5.4.2 T
2
vs N concentration
Finally, we discuss the relationship between T
2
and the concentration of N spins. As
shown in Fig. 5.7, 1=T
2
increases while the N concentration increases in both type-
Ib and type-IIa diamond, however, the concentration dependence of the 1=T
2
values
are less pronounced in the type-IIa diamond. To analyze the observed concentration
dependence of 1=T
2
, we considered the two decoherence processes including the spin
ip-
op process of N spins (1=T
N
2
), where the contribution from the N spin is consid-
ered to be proportional to the N concentration (1=T
N
2
n), and the
13
C decoherence
(1=T
13
C
2
). Thus, the decoherence rate (T
2
) is considered by,
1
T
2
=
1
T
N
2
+
1
T
13
C
2
=Cn+
1
T
13
C
2
; (5.11)
where C is a proportional constant. As shown in Fig. 5.7, the data is well explained
with Eqn. 5.11. From the t using Eqn. 5.11, C was found to be 0.01390.0005
s
1
ppm
1
as well as T
13
C
2
to be 19010 s. The N spin concentration dependence
Chapter 5. Spin concentration DEER-based measurement in diamond 102
0.01 0.1 1 10 100 1000
1E-3
0.01
0.1
1
10
Nitrogen concentration (ppm)
1/T
2
(μs
-1
)
Exp.
Fit
Figure 5.7: 1/T
2
of N spins as a function of the N concentration. Open squares
represent experimentally obtained data, orange solid line is the best t of the data to
the model of decoherence rate described by Eqn. 5.11. Yellow region represents the
plot of Eqn. 5.11 with the xed
C
in the range of 150250s and a slopeC=0.0139
s
1
ppm
1
as obtained from the best t of the data. Dashed orange line shows the
best t of the data using Eqn. 5.11 without the nuclear spin decoherence (1=T
13
C
2
=
0).
in T
2
was observed in type-Ib and natural type-Ia diamond crystals although the
previous study did not reveal the nuclear spin decoherence [145]. The obtained T
13
C
2
value is in a good agreement with the decoherence time due to
13
C nuclear spins [13,
74].
5.5 Summary
Insummary,wedemonstratedthecapabilityof115GHzDEERspectroscopyatroom
temperature to determine a wide range of N spin concentrations. Using the pulsed
115 GHz EPR spectroscopy, we rst determined T
2
in type-Ib and type-IIa diamond
crystals and performed DEER spectroscopy to probe the magnetic dipole interaction
Chapter 5. Spin concentration DEER-based measurement in diamond 103
between N spins. From the analyses of the SE decay and the DEER spectra, we
determinedconcentrationsofNspinsintherangeof0.1100ppmwithnoreference
sample. OurDEERanalysistoextractthespinconcentrationisstronglysupportedby
theextractedNconcentrationdependenceoftheinhomogeneouslinewidthandbythe
agreement of the estimated microwave power with our experimental values. Finally,
we showed that the measurement of the N spin concentrations allows us to determine
contributions of N spins and
13
C nuclear spins to T
2
quantitatively. The present
method is applicable to determine the concentration of NV ensembles and various
other spin systems in solid. In addition, by combining nanoscale magnetic resonance
techniques based on NV centers, this method may pave the way to determine spin
concentrations within a microscopic volumes.
Chapter 6
High-frequency and high-eld
optically detected magnetic
resonance of nitrogen-vacancy
centers in diamond
Materials presented in this Chapter can be also found in the article titled High-
frequency and high-eld optically detected magnetic resonance of nitrogen-vacancy
centers in diamond byViktorStepanov,FranklinH.Cho,ChathurangaAbeywardana
andSusumuTakahashiinAppliedPhysicsLetters106,063111(2015)(Reprintedwith
permission. Copyright 2015, AIP Publishing LLC).
6.1 Introduction
Key motivations for the NV magnetic sensing include the detection of a single spins
and improvement of sensitivity of EPR and NMR spectroscopies to the level of a
single spin. For example, detection of a single or a small ensemble of electron and
nuclear spins surrounding a NV center has been demonstrated using DEER and EN-
DOR spectroscopy of a single NV center at low magnetic elds [23, 25, 29, 91, 168].
104
Chapter 6. HF ODMR 105
Inaddition, spinrelaxometrybasedonthelongitudinalrelaxationtime(T
1
)measure-
ment of a single NV center has been employed to detect several electron spins [27,
28].
In this Chapter, the development of a HF ODMR system to investigate NV cen-
tersindiamondispresented. SimilartoNMR spectroscopy, thespectralresolutionof
ODMR and NV-based MR techniques including DEER and ENDOR is signicantly
improved at HF, thus highly advantageous in distinguishing target spins from other
species (i.e., impurities existing in diamond) for NV-based MR spectroscopy. In ad-
dition, HF MR spectroscopy can produce extremely high spin polarization at low
temperatures, which improves the signal-to-noise ratio of NV-based MR measure-
ments on ensembles of target electron spins and increases spin coherence of target
spins signicantly [74, 105, 191{193]. The HF ODMR employs a confocal FL imag-
ing system, a HF excitation component and a 12.1 Tesla superconducting magnet.
The HF excitation component is a part of the HF EPR spectrometer operating in
the frequency range of 107-120 GHz and 215-240 GHz [190], therefore the system is
capable of performing in-situ HF ODMR and HF EPR experiments. We also present
HF ODMR measurements of a single NV center in diamond. First, a single NV
center is identied by FL imaging, autocorrelation, and eld-dependent FL measure-
ments. Then, we perform cw-ODMR at the external magnetic eld of 4.2 Tesla and
microwave frequency of 115 GHz. We also demonstrate the coherent manipulation of
a single NV center spin at 4.2 Tesla by performing Rabi oscillations, pulsed-ODMR,
and SE measurements. Finally, we perform measurements of the longitudinal relax-
ation time (T
1
) of NV centers in nanodiamonds (NDs) using the HF ODMR system.
The experiment shows that T
1
in NDs is shorter than T
1
in bulk diamonds, and is
nearly eld-independent in the range of 08 Tesla. The observation agrees with T
1
process due to paramagnetic spins surrounding NV centers in NDs.
Chapter 6. HF ODMR 106
6.2 HF ODMR system
Figure 6.1 shows an overview of the HF ODMR system consisting of a NV detection
system, a HF excitation component, a microscope system, and a sample stage. A
cw 532 nm laser in the NV detection system is employed for optical excitation of
NV centers. The cw excitation laser rst couples to an AOM for pulsed operations,
then is transmitted through a single-mode optical ber (SMF) to the bottom of a
12.1 Tesla superconducting magnet where the excitation laser couples to the micro-
scope system. As shown in Fig. 6.1, the microscope system consists of a microscope
objective (Zeiss) for both the optical excitation and collection of NV's FL signals,
a z-translation stage (Attocube), and a corrugated waveguide (Thomas-Keating) for
the HF microwave excitation. The HF microwave excitation component is a part of
our HF EPR spectrometer [187, 190]. The output frequency of the HF microwave
source is continuously tunable in the range of 107-120 GHz and 215-240 GHz. The
sample stage is supported by the z-translation stage. The z-direction of the sample
position is adjusted by the z-translation stage, and the xy-direction of the laser exci-
tation volume is controlled by a combination of a sterring mirror and a pair of lenses.
The HF ODMR employs a confocal microscope system to detect FL signals of NV
centers. The FL signals are collected by the same objective, then are transmitted to
the detection system by a 50 m multi-mode optical ber (MMF) cable that enables
confocalFLimaging. FinallytheFLsignalsarelteredbyopticalltersanddetected
by APDs in the NV detection system.
6.3 HF ODMR of a single NV center
Using the HF ODMR system, we performed FL measurements on a single crystal
of type-Ib diamond (1.5 1.5 1.1 mm
3
, Sumitomo Electric Industries). First,
Chapter 6. HF ODMR 107
12.1 T
magnet
HF excitation component Optical table
Laser
APD 1
AOM
Optical fiber
FSM
Lenses
APD 2
Filters BS
HF source
Quasioptics
Corrugated
waveguide
MMF
SMF
Sample stage
FL
NV
Diamond
Coverslip
TCSPC
HF excitation
Z stage
Microscope system
Laser FL
Aluminum frame
B
0
Figure6.1: OverviewoftheHFODMRsystem. TheHFsourceintheHFexcitation
component is tunable continuously in the range of 107-120 GHz and 215-240 GHz.
HF microwaves are guided by quasioptics and a corrugated waveguide. The NV
detection system consists of a 532 nm cw diode-pumped solid state laser, an AOM,
ber couplers, optical lters, a BS, and APDs. The excitation laser is applied to
NV centers through a microscope objective located at the center of the 12.1 Tesla
superconducting magnet and the FL signals of NV centers are collected by the same
objective. The FL signals are ltered by optical lters in the NV detection system.
For autocorrelation measurements, the FL signals are split into two and detected by
two separate APDs. The microscope system consists of a microscope objective, a
z-translation stage, and the corrugated waveguide. The sample stage is supported by
the z-translation stage.
Chapter 6. HF ODMR 108
(a)
(c)
NV
(b)
50 100
X (µm)
Y (µm)
5 0
5
0
0
1
2
3
g
2
(τ)
Delay time τ (ns)
0 2 4 6 8 10
12
16
20
FL intensity (arb. units)
Magnetic field (Tesla)
FL changes at level anti-crossing
0.05 0.10 0.15
NV
20 0 10
FL intensity (counts/ms)
Single quantum emitter
Figure 6.2: Fluorescence signals of a single NV center in type-Ib diamond. (a) FL
intensity image of a type-Ib diamond crystal. The scanning area is 5 5m
2
. Solid
circle indicates a single NV center that was used in the subsequent measurements.
(b) Autocorrelation curve observed from the the single NV center. The observation
of g
2
( = 0) < 0:5 conrms the detection of the single NV center. (c) Magnetic
eld dependence of the single NV center FL intensity. The eld was applied along
the (111) axis of the diamond within 8 degrees. The inset shows reduction of FL
intensity at0.05 and 0.1 Tesla due to LAC of the excited and ground states of the
NV center, respectively.
Chapter 6. HF ODMR 109
as shown in Fig. 6.2a, FL imaging was carried out to map FL signals from the di-
amond crystal. After choosing a well-isolated FL peak (Fig. 6.2a), we performed
anti-bunching measurement and observed the autocorrelation signal which veries
that the FL signal was due to a single quantum emitter (Fig. 6.2b). Then, the FL
intensity was monitored continuously as we applied the external magnetic eld from
0 to 10 Tesla. As shown in the inset of Fig. 6.2c, we observed two dips of the FL
intensity at 0.05 and 0.1 Tesla, originating from the level anti-crossing (LAC) in
NV'sopticallyexcitedandgroundstates, respectively[12]. Thereforetheobservation
of the LAC and autocorrelation signal conrmed the detection of a single NV center.
In addition, as shown in Fig. 6.2c, we found that the FL intensity is stable in high
magnetic elds up to 10 Tesla.
Next, we demonstrated ODMR measurements of the single NV center at the
microwave frequency of 115 GHz. First, we performed cw-ODMR spectroscopy of
the single NV center with applications of cw microwave and laser excitations. As
shown in Fig. 6.3a, the observed FL signals as a function of magnetic eld showed
a dip of the FL intensity at 4.2022 Tesla, which corresponds to the m
s
= 0 $ 1
transitionoftheNVcenter. Theg-factoroftheNVcenterwasestimatedtobe2.0027-
2.0041 by taking into account for the strength of magnetic eld at the resonance
and uncertainty of the orientation of magnetic eld (< 8 degrees). The calibration
of magnetic eld was done by in-situ ensemble EPR measurement of N spins in
diamond (see the inset of Fig. 6.3a) [74]. Then, we carried out pulsed experiments
at 115 GHz and 4.2022 Tesla. In the pulsed experiments, the NV center was rst
prepared in the m
s
= 0 spin sublevel with the application of the laser initialization
pulse, then the microwave excitation pulse sequence was applied. The nal state of
the NV center was determined by measuring its FL intensity with the application of
the laser readout pulse. In all pulsed measurements, the FL intensity was normalized
Chapter 6. HF ODMR 110
Pulse length t
P
(ns)
2000 4000
Exc.
MW
Init. RO
Sig.
FL
t
P
f
Rabi
= 0.8 MHz
Exp.
Fit
FL intensity (arb. units)
↔ 0 -1
FL intensity (arb. units)
Magnetic field (Tesla)
(a)
200 400 600
FL intensity (arb. units)
2τ (ns)
(d)
Exc.
MW
Init. RO
Sig.
FL
π/2 π/2 π τ τ
Exp.
Fit
T
2
= 325±20 ns
4.201 4.202 4.203
Exp.
Fit
4.11 4.12
Intensity
↔ 0 -1
(c)
↔ 0 -1
(b)
Magnetic field (Tesla)
FL intensity (arb. units)
4.201 4.202 4.203
Exc.
MW
Init. RO
Sig.
FL
t
MW
Exp.
Fit
↔ 0 -1
Figure6.3: ODMRmeasurements on a single NV centerin type-Ib diamond at 115
GHz. (a) cw-ODMR measurement of the single NV center at 115 GHz. The ODMR
signal of the single NV center was observed at 4.2022 Tesla. The solid line indicates
a t to the Gaussian function. The inset shows the in-situ EPR measurement on
ensemble N spins in the diamond. (b) Rabi oscillation experiment. The frequency
of the observed Rabi oscillations was 0.8 MHz. The inset shows the applied pulse
sequence consisting of the initialization (Init.) and readout (RO) pulses by the 532
nmlaser(Exc.),microwavepulse(MW)oflengtht
P
,andFLsignals(Sig.). Init.=4s
andRO=Sig.=300nswereusedinthemeasurement. t
P
wasvaried. c)Pulsed-ODMR
asa functionof magnetic eld. Thesolid lineindicates a tto the Gaussianfunction.
The inset shows the pulse sequence. Init.=4 s, RO=Sig.=300 ns, and t
MW
=500 ns
were used in the measurement. (d) The spin echo measurement to determine the
spin decoherence time (T
2
) of the single NV center. The solid line indicates a t to
exp[(2=T
2
)
3
] [127]. The inset shows the pulse sequence. Init.=4s, RO=Sig.=300
ns, =2=250 ns, and =600 ns were used in the measurement. was varied. The
pulse sequence was repeated on the order of 10
6
times to obtain a single point in all
measurements.
Chapter 6. HF ODMR 111
by the reference signal, which is the FL intensity measured without the microwave
pulse sequence in order to cancel noises associated with laser intensity
uctuations
and thermal/mechanical instability of the setup. In addition, each data point was
obtained by repeating the pulse sequence and averaging FL signals on the order of
10
6
times. As shown in Fig. 6.3b, Rabi oscillations of the same single NV center
wereobservedbyvaryingthedurationofthesinglemicrowavepulse(t
P
). Figure 6.3c
shows pulsed-ODMR signals of the single NV center as a function of magnetic eld.
The observed full-width at half-maximum was 0.29 mT (8 MHz) which is typical for
NVcentersintype-Ibdiamondcrystals[34]. Next,wemeasuredthespindecoherence
time (T
2
) of the single NV center at 4.2022 Tesla. As shown in the inset of Fig. 6.3d,
the applied microwave sequence consists of the SE sequence and an additional =2
pulsethatconvertstheresultantcoherenceoftheNVcenterintothem
s
=0state[67].
By tting the observed FL decay to exp[(2=T
2
)
3
] [127],T
2
of the single NV center
was determined as 32520 ns (see Fig. 6.3d).
6.4 T
1
eld-dependence of NV centers in
nanodiamonds
Finally, using our HF ODMR system, we studied the eld dependence of T
1
of NV
centers in NDs. The samples we studied here were NV-enhanced type-Ib NDs with
theaveragediameterof35nm(AcademiaSinica)[194]. TheNDsweresparselyplaced
on a coverslip, and identied by performing FL images (i.e., the inset of Fig. 6.4a for
ND1) and by observing the LAC signals from the eld-dependent FL measurements.
Then T
1
relaxation measurements were performed as shown in Fig. 6.4a. The pulse
sequence employed for T
1
measurements is shown in the inset of Fig. 6.4a. The NV
centers in NDs were rst prepared in them
s
= 0 spin sublevel with the application of
Chapter 6. HF ODMR 112
Exc. RO
Sig.
FL
T
Ref.
0 100 200 300
FL intensity (arb. units)
T
1
= 72±14 µs
Exp.
Fit
T (µs)
0 4
0
4
ND1
25 0
X (µm)
Y (µm)
Init.
T
1
(µs)
0 2 4 6 8
0
200
400
Magnetic field (Tesla)
ND1
ND2
Bulk
0 2 4
T
1
=1.1±0.5 ms
T (ms)
(a)
(b)
Figure 6.4: T
1
measurements of NV centers in NDs. (a) T
1
measurement of NV
centers in a ND (ND1). The solid line indicates a t to the single exponential decay
function,exp(T=T
1
). AFLimageofND1isshownintheinset. Theinsetalsoshows
the pulse sequence consisting of the initialization (Init.) and readout (RO) pulses by
the 532 nm laser (Exc.), and FL signals (Sig.). T was varied. Init.=4 s, RO=3 s,
and Sig.=300 ns were used. The reference (Ref.) of 300 ns was measured 2 s after
the signal. The pulse sequence was repeated on the order of 10
6
times to obtain a
single point in all measurements. In addition, the inset shows T
1
measurement of a
singleNVcenterinthetype-Ibbulkdiamondobtainedusingthesamepulsesequence
above (T
1
= 1:10:5 ms). (b) T
1
as a function of magnetic eld for two NDs (ND1
and ND2).
Chapter 6. HF ODMR 113
the laser initialization pulse, then a dark time interval (T) was applied with no laser
excitation. The nal state of the NV centers was determined by measuring the FL
intensity of the NV centers with the application of the laser readout pulse. The FL
signal with the laser excitation was used as a reference signal for the normalization.
In order to detect the T
1
relaxations from the m
s
= 0 state to the thermal equi-
librium state, the normalized FL signal was measured as a function of T. As shown
in Fig. 6.4a, an exponential decay of the FL intensity was clearly observed. From
a single exponential t of the FL signal, we obtained T
1
= 7214 s. The mea-
sured T
1
is 10100 times shorter than typical T
1
in bulk diamonds (see the inset of
Fig. 6.4a) [27, 74, 77]. The shorter T
1
is often observed in NV centers in NDs and
near the diamond surface, and it has been reported that transverse magnetic eld
uctuations from paramagnetic impurities located on or nearby the diamond surface
provide an additional contribution to T
1
of NV centers [27, 28, 195{197]. Because
the intensity of the
uctuations depends on the difference of the Larmor frequencies
between the NV centers and the impurities [27], the contribution toT
1
will be nearly
independent of the strength of magnetic eld for the impurities observed in NDs hav-
ing g-values2 [23, 198]. Moreover, the T
1
measurements were performed at several
magnetic elds for two different NDs. As shown in Fig. 6.4b, we observed similar
values of T
1
for both NDs in the range of 08 Tesla, which supports the T
1
process
due to paramagnetic spins [27, 28, 195, 197].
6.5 Summary
In summary, we presented the development of the HF ODMR system, which enables
ustoperformtheODMRmeasurementsathighfrequenciesandhighmagneticelds.
UsingtheHFODMRsystem,wedemonstratedcw-andpulsed-ODMRmeasurements
of a single NV center in a type-Ib diamond crystal at the microwave frequency of 115
Chapter 6. HF ODMR 114
GHz and the magnetic eld of 4.2 Tesla and studied T
1
of NV centers in NDs as a
function of magnetic eld.
Chapter 7
Conclusion
Instrumentation development
The home-built HF EPR/DEER spectrometer at USC described in Chapter 3 op-
erates at two microwave frequencies - 115 GHz and 230 GHz with high power mi-
crowave output and capabilities to perform multi-dimensional, multi-frequency and
multi-phase pulsed measurements. It has been already applied to perform unique
HF EPR studies in solid-state systems. In particular, HF EPR spectroscopy of NDs
allowed to clearly resolve surface paramagnetic spins and to study eld dependence
of their absorption lineshape to determine g-strain and concentration of the surface
paramagnetic impurities [199]. In addition, using HF EPR spectrometer, tempera-
ture and magnetic eld dependence measurements of T
1
and T
2
relaxations in NDs
wereperformed [199]. Furthermore, HF EPR spectrometer enabled coherence studies
in Mn
3
single molecule magnet (SMM), which is a candidate for quantum memory
devices,qunatumcomputingandmolecularspintronics,revealingthedipolardecoher-
enceasamainsourceofdecoherenceinMn
3
SMMandallowedtoextenddecoherence
by factor of four at 9 Tesla [200].
Careful design of the aqueous sample holder for HF EPR system (Sect. 3.1.5)
115
Chapter 7. Conclusion 116
almost completely eliminated absorption of HF microwave excitation and paved the
way for room temperature HF EPR spectroscopy in solutions. This is of a particular
importance in applications to study biological samples in their native environment.
ApplicationofHFEPRandaqueoussampleholdertonitroxide-functionalizedNDsin
solutionallowedforreliableinvestigationofthedynamicsofnitroxideradicalsgrafted
on the surface of the NDs by spectrally separating EPR signals of paramagnetic
impurities in NDs and nitroxide radicals at HF [133].
Finally, developedlowmagneticeldODMRsystemforasingleNVcenterindia-
mond(Sect.3.2)wassuccessfullyusedtoperformasingleNV-basedMRspectroscopy
on a small ensemble of nearby N spins using DEER technique [201].
Measurement of spin concentration in diamond
In this dissertation, the method to determine spin concentrations in a solid-state
system using DEER spectroscopy was developed (Chapter 5). The method was ap-
plied to probe concentrations of N spins in diamond and precisely characterize the
dependence of decoherence rate in diamond on N spin concentration. We show that
decoherence rate due to N spins is proportional to their concentration. In addition,
we were able to reveal
13
C nuclear spin decoherence in type-IIa diamonds.
In the context of future optimization of diamond materials for sensing and quan-
tum applications, spin concentration measurement is a unique capability. The break-
through feature of our method is that no reference sample is required. This opens up
an opportunity to combine our method with NV-based MR techniques to probe spin
concentrations in the microscopic volumes and precisely characterizeT
1
andT
2
relax-
ations of NV centers, which is of a particular importance to engineering of diamond
materials with desirable NV concentrations and NV spin relaxations.
Chapter 7. Conclusion 117
HF ODMR on a single NV center in diamond
Finally, development of HF ODMR system for a single NV center in diamond was
presented in Chapter 6. Using HF ODMR system, the coherent control of a single
NV center in diamond was achieved at high magnetic eld ( 4 Tesla) for the rst
time, showing that NV center retains all its unique optical properties up to 12 Tesla.
ThisexperimentaldemonstrationlaysthegroundworktowardsmorerobustNV-based
magnetometry, indicating the opportunity for HF single NV-based MR spectroscopy
with the ne spectral resolution.
As an application of HF ODMR, we performed microwave free measurements of
T
1
relaxations of NV centers in ND. Our eld-dependence study of T
1
in the wide
range of magnetic elds (0-8 Tesla) shows T
1
in NDs is dominated by magnetic noise
due to surface paramagnetic spins.
Appendix A
Averaging magnetic resonance
signals
A.1 Averaging over the OU process
Inthissection,averagingofthepulsedmagneticresonancesignalsovertheOUprocess
is presented. Using Eqn. 2.39b and Eqn. 2.47, average FID and SE signals can
expressed as
S(t) =
⟨
cos
(
t
∫
0
(t
′
)b(t
′
)dt
′
)⟩
; (A.1)
where amplitude factors were omitted. ⟨:::⟩ denotes averaging over all possible re-
alizations of b(t), which is considered as the OU process with the zero mean value
(⟨b(t)⟩
t
= 0) and the correlation function ⟨b(t)b(0)⟩
t
= ∆exp(t=
c
), where ∆ and
c
are the variance and correlation time of b(t) process, respectively. (x) accounts
for the direction of spin evolution under b(t) in a particular experiment, dened as
(x) = 1 in the FID measurements and (x) = 2(xt=2)1 in the SE measure-
ments, where (x) is the Heaviside function and t is a total free evolution time (in
the case of SE measurements, pulse is aplied at t=2). Next, taking into account
118
Appendix A. Averaging magnetic resonance signals 119
that b(t) is a symmetrically distributed variable, Eqn. A.1 becomes
S(t) =
⟨
exp
(
i
t
∫
0
(t
′
)b(t
′
)dt
′
)⟩
: (A.2)
Eqn. A.2 is in the form of the characteristic function of a random variable. For an
arbitrary random process x with the distribution function f(x), the characteristic
function is dened as ⟨e
ikx
⟩. When all moments (⟨x
n
⟩ =
∫
+1
1
x
n
f(x)dx) of the
random process x exist, characteristic function can be expressed using cumulant ex-
pansion: ⟨e
ikx
⟩ = exp
[
∑
1
n=1
(ik)
n
n!
c
n
(x)
]
[202]. In the case of the Gaussian variable,
only the rst two cumulants are non-zero that are related to the moments asc
1
=⟨x⟩
and c
2
=⟨x
2
⟩⟨x⟩
2
. Therefore, the characteristic function of the Gaussian variable
x is given by ⟨e
ikx
⟩ = exp
[
ik⟨x⟩
k
2
2
(⟨x
2
⟩⟨x⟩
2
)
]
[203]. Applying this property
to Eqn. A.2, signal S becomes
S(t) =exp
(
1
2
t
∫
0
dt
1
t
∫
0
dt
2
(t
1
)(t
2
)⟨b(t
1
)b(t
2
)⟩
)
=exp
(
∆
2
2
t
∫
0
dt
1
t
∫
0
dt
2
exp
[
jt
2
t
1
j
c
]
(t
1
)(t
2
)
)
;
(A.3)
where the correlation function of b(t) was used in the last step. After change of
variables (t
2
t
1
= and t
1
=s), Eqn. A.3 is transformed to
S(t)= exp
(
∆
2
t
∫
0
d exp
(
c
)
t
∫
0
( +s)(s)ds
)
: (A.4)
Next, using for the FID and SE measurements, FID signal is expressed as
FID(t)= exp
(
∆
2
t
∫
0
(t) exp
(
c
)
d
)
; (A.5)
while SE signal is expressed as
SE(t) =exp
(
∆
2
[
t=2
∫
0
(t3) exp
(
c
)
d +
t
∫
t=2
(t) exp
(
c
)
d
])
:
(A.6)
Appendix A. Averaging magnetic resonance signals 120
After evaluation of the integrals in Eqn. A.5 and Eqn. A.6, the nal expressions for
the signals are found as [203, 204]
FID(t) =exp
(
∆
2
2
c
[
t
c
+exp
(
t
c
)
1
])
(A.7)
and
SE(t)= exp
(
∆
2
2
c
[
t
c
3+4exp
(
t
2
c
)
exp
(
t
c
)])
: (A.8)
Eqn. A.7 is identical to the FID signal in the main text given by Eqn. 2.40b, while
the SE signal given by Eqn. 2.48 is obtained from Eqn. A.8 by setting t =2.
A.2 Averaging DEER signals over ensemble of
dipolar coupled spins
TheproceduretoaverageDEERsignalofasingleAspinoverdifferentcongurations
of B spins is presented in this section. According to the calculation in Sec. 5.3.2,
averaging over B spins involves averaging of cosϕ and sinϕ functions (see Eqn. 5.4
and Eqn. 5.5) over all B spins \degrees of freedom" (r;;;) with ϕ given by
φ = 2
∑
j
D(
j
)R(r
j
)T(
j
)
j
;
where summation runs over all B spins coupled to a single A spin, =
0
2
B
g
A
g
B
T=(4~), R(r
j
) = 1=r
3
j
, T(
j
) = (3cos
2
j
1), D(
j
) =
Ω
2
(
j
!
B
)
2
+Ω
2
sin
2
(
√
(
j
!
B
)
2
+Ω
2
t
p
2
)
with
j
dened here as a Larmor frequency
of j-th B spin (identical to !
j
in Sec. 5.3.2) and all other parameters dened in
Sect. 5.3.2. Then, the average functions to be found (⟨C⟩
B
and⟨S⟩
B
) are expressed
as
⟨C⟩
B
=
⟨
cos
(
2
∑
j
D(
j
)R(r
j
)T(
j
)
j
)⟩
j
;r
j
;
j
;
j
; (A.9a)
Appendix A. Averaging magnetic resonance signals 121
⟨S⟩
B
=
⟨
sin
(
2
∑
j
D(
j
)R(r
j
)T(
j
)
j
)⟩
j
;r
j
;
j
;
j
; (A.9b)
where ⟨:::⟩ denotes averaging over variables specied by the subscripts. Rewriting
the cosine and sine functions in Eqns. A.9 as the real and imaginary parts of the
exponential results in
⟨C⟩
B
= Re
[
∏
j
⟨
exp
[
2D(
j
)R(r
j
)T(
j
)
j
]
⟩
j
;r
j
;
j
;
j
]
; (A.10a)
⟨S⟩
B
= Im
[
∏
j
⟨
exp
[
2D(
j
)R(r
j
)T(
j
)
j
]
⟩
j
;r
j
;
j
;
j
]
; (A.10b)
wherethesummationintheargumentsofthecosineandsinefunctionswasfactorized
to the product of exponentials. First averaging of Eqns. A.10 over
j
(
j
= 1=2
with equal probabilities) gives
⟨C⟩
B
=
∏
j
⟨
cos
[
D(
j
)R(r
j
)T(
j
)
]
⟩
j
;r
j
;
j
; (A.11a)
⟨S⟩
B
=0: (A.11b)
⟨S⟩
B
is averaged out due to the fact that sine is an odd function. Form of Eqn. A.11a
implies that averaging over j-th B spin is independent from the rest of B spins. In
addition, each B spin has an equal probability to occupy any lattice site and may
experience any Larmor frequency () with a probability given by the inhomogeneous
lineshapeL(). Therefore, averaging over ,r, is identical for all B spins and⟨C⟩
B
can be expressed as
⟨C⟩
B
=
⟨
cos
[
D()R(r)T()
]
⟩
N
;r;
; (A.12)
whereN isthenumberofBspins. InthelimitofN !1,usinglim
y!1
(1+x)
y
=e
yx
forjxj< 1, Eqn. A.12 is reduced to
⟨C⟩
B
=e
nI
; (A.13)
Appendix A. Averaging magnetic resonance signals 122
where n =N=V is the volume concentration of B spins and
I =V
⟨
1cos
[
D()R(r)T()
]⟩
;r;
:
To proceed further, it is convenient to dene the volume integral (I
V
) as I =
⟨
I
V
⟩
,
which is then given by
I
V
=2
∫
0
1
∫
0
r
2
sin[1cos(D()R(r)T())]drd: (A.14)
After variable substitution (u =1=r
3
), Eqn. A.14 becomes
I
V
=
∫
0
sind
1
∫
1
[1cos(D()T()u)]
u
2
du:
Next, evaluating integration over u results in
I
V
=
2
D()
3
∫
0
j3cos
2
1jsind =
8
2
D()
9
p
3
and therefore
I =
8
2
9
p
3
⟨
D()
⟩
: (A.15)
Averaging over is performed over the distribution of Larmor frequencies, i:e: inho-
mogeneous lineshape L(), and hence
⟨
D()
⟩
term is identical to ⟨sin
2
2
⟩
L
dened
in the main text as
⟨
sin
2
2
⟩
L
+1
∫
1
Ω
2
(!
B
)
2
+Ω
2
sin
2
(
√
(!
B
)
2
+Ω
2
t
p
2
)
L()d:
Finally, using the denition for and Eqn. A.15,⟨C⟩
B
is obtained from Eqn. A.13 as
⟨C⟩
B
= exp
(
2
0
2
B
g
A
g
B
T
9
p
3~
n
⟨
sin
2
2
⟩
L
)
;
which is the decay function used in Eqn. 5.7 and Eqn. 5.8 of the main text.
Appendix B
FL and ODMR signals of a single
NV center
B.1 FL signals
To calculate transient FL signals in pulsed-ODMR measurements, ve-level model
for NV center is adapted (Fig. B.1) [68]. As shown in Fig. B.1, when optical exci-
tation is applied, NV is promoted to the excited state predominantly through the
spin-conserving transitions j1⟩! j3⟩ and j2⟩! j4⟩ with the transition rate r that
depends on the optical power density at NV center. Finite probability to change
a spin state during optical excitation is accounted by ϵ resulting in ϵr rate for the
spin-non-conserving transitions under optical excitation (j1⟩! j4⟩ and j2⟩! j3⟩).
The optical NV transitions are single sided due to fast vibrational relaxations in
the excited electronic states that result in the states, which are non-resonant with
the optical excitation (no stimulated emission). Next, population from the excited
states radiatively decay to the ground states with the spontaneous decay rate
for
the spin-conserving transitions j3⟩! j1⟩ and j4⟩! j2⟩ and, similarly to the opti-
cal transitions, small fraction of population may undergo spin-non-conserving decay
123
Appendix B. FL and ODMR signals of a single NV center 124
±1
0
±1
0
3
E
1
2
3
4
5
3
2
A
1 1
1
, A E
S
Γ
γ r
εγ
r ε
P
Γ
FigureB.1: Five-level model for NV center to model transient FL signals in pulsed-
ODMR measurements [68]. Ground spins states are labeled asj1⟩jm
s
=0⟩,j2⟩
jm
s
=1⟩, while excited spin states are labeled asj3⟩jm
s
=0⟩,j4⟩jm
s
=1⟩.
Singlet metastable states are considered as a single level j5⟩. Green solid (dashed)
arrows indicate strong spin-conserving (weak spin-non-conserving) optical transitions
from electronic ground to excited states with the rate r (ϵr). Red solid (dashed)
arrowsindicateradiativespin-conserving(weakspin-non-conserving)transitionsfrom
electronicexcitedtothegroundstateswiththerate
(ϵ
). Blackarrowsindicatenon-
radiativetransitionswiththeshelvingrate
S
forj4⟩!j5⟩transitionandpolarization
rate
P
forj5⟩!j1⟩ transition.
(j3⟩!j2⟩ and j4⟩!j1⟩) with the rate given by ϵ
. Additionally, population from
thej4⟩ state (m
s
=1 excited states) is partially shelved to thej5⟩ state (metastable
singlet states
1
A
1
and
1
E) with the rate
S
from where it deshelves to thej1⟩ state
with
P
rate. Since optical coherences are damped on the timescale much shorter
than a time used to optically initialize or optically readout NV center (s), optical
excitation can be described simply by the rate equations for diagonal elements of the
Appendix B. FL and ODMR signals of a single NV center 125
density matrix (populations):
@
d
(t)
@t
=
^
E
d
(t); (B.1)
where
d
diag[^ ] = [
11
;
22
;
33
;
44
;
55
] and
^
E is given according to the Fig. B.1
as
^
E =
0
B
B
B
B
B
B
B
B
B
B
@
r(1+ϵ) 0 ϵ
P
0 r(1+ϵ) ϵ 0
r ϵr (1+ϵ) 0 0
ϵr r 0 [(1+ϵ)+
p
] 0
0 0 0
S
P
1
C
C
C
C
C
C
C
C
C
C
A
:
The values of the transition rates in
^
E are [68]
rdepends on the optical power at NV center
ϵr =1:9510
2
r
=77 MHz
ϵ = 1:5 MHz
S
= 30 MHz
P
= 3:3 MHz
Furthermore,inthepulsed-ODMRmeasurements,microwavepulsesareappliedsepa-
ratelyfromtheopticalpulseswhenentireNVpopulationisinthegroundspinstates.
To model the effect of the microwave pulses, microwave excitation is considered to
couple thej1⟩ andj2⟩ states resulting in Rabi oscillations. Following similar consid-
erationgiveninSect. 2.3.1 forTLS,theeffectoftheresonantmicrowavepulseisthen
described by
11
(t) =
11
(0)cos
2
=2+
22
(0)sin
2
=2; (B.2a)
22
(t) =
22
(0)sin
2
=2
22
(0)cos
2
=2; (B.2b)
for the case of an arbitrary initial populations in the j1⟩ and j2⟩ states (
11
(0) and
22
(0)), where = Ωt
p
(Ω is the Rabi frequency, t
p
is the duration of the microwave
Appendix B. FL and ODMR signals of a single NV center 126
pulse). TodescribepopulationdynamicsofNVcenterinthepulsed-ODMRmeasure-
ments, Eqn. B.1 is used during optical pulses and dark time intervals (with r = 0,
i:e: laser is off), while during microwave pulse, Eqns. B.2 are employed.
Fig. B.2 shows population dynamics in the ground state of NV center during the
pulsesequencethatissimilartothesequenceusedinthepulsed-ODMRmeasurements
(Sect.4.2). FLsignalsarerepresentedbythetotalpopulationintheexcitedstates,i:e:
33
+
44
. Thermal equilibrium state for NV center (
11
(0) = 1=3 and
11
(0) = 2=3)
wasusedastheinitialstate. Insimulation,initializationpulsewasappliedfor2s(
3 s in our experiments), optical readout for 1 s (300 ns in experiments) to display
population dynamics and transient FL on a longer time scale. As shown in Fig. B.2,
during initialization pulse (02 s), the population in the m
s
=1 state is mainly
transfered to the m
s
= 0 state through the singlet state. NV dynamics equilibrate
with the optical pulse at around 2 s, i:e: steady state under optical excitation is
reached. In the case of FL signals, the fast rise in the beginning is due to population
transfer from the ground states to the excited states, followed by the decrease due to
partialshelvingofthem
s
=1excitedstatetothemetastablestateandlaterincrease
duetopolarizationeffect(populationfromthesingletstateispartiallydeshelvedinto
them
s
=0groundstate, whichisthebrightFLstateofNVcenter). Duringthedark
time interval (24 s), NV population in the ground states is rst increased by the
radiative decay of the excited states (lifetime 12 ns) with the further population
increase in the m
s
= 0 ground state due to complete deshelving of the singlet state
at the
P
rate. The steady state during the dark interval is reached on the order
of 1 s completing NV polarization process with 90% spin polarization in the
m
s
= 0groundstate(duetolongT
1
ofNVcenter,spin-latticerelaxationhasnoeffect
in the pulsed-ODMR measurements). Next, microwave pulse is applied (at 4 s)
resulting in the inversion of populations between the m
s
= 0 and m
s
=1 ground
Appendix B. FL and ODMR signals of a single NV center 127
1
mw π pulse
no mw
initialization RO
Microwave
Laser
π
time (μs)
FL
0 2 1 3 4 5 6 7 8
0
0.5
1
0
0.5
0.1
0.05
0
mw π pulse
no mw
mw π pulse
no mw
11
ˆ σ
22
ˆ σ
Figure B.2: Population dynamics of NV center in pulsed-ODMR measurement.
Results were obtained based on the ve-level model for NV center and for the pulse
sequence shown on the top panel: initialization pulse - 2s, rst dark time interval -
2s, microwave pulse - 200 ns (Ω = 2.5 MHz), second dark time interval - 1.8 s,
readout pulse - 1s, third dark time interval - 1s.
11
(t) and
22
(t) are populations
in the m
s
= 0 and m
s
= 1 ground states. Transient FL is represented by NV
populations in the excited states (
33
(t) +
44
(t)). Simulations were performed for
initial NV spin state
11
(0) = 1=3 and
11
(0) =2=3 and r = 10 MHz.
Appendix B. FL and ODMR signals of a single NV center 128
states. While there are no changes in NV populations during second dark time (after
microwave pulse to 6 s), it is employed in the experiments to separate microwave
andopticalpulses. Finally, opticalreadoutisapplied(67s)toobtaintransientFL
signals. As evident from Fig. B.2, transient FL signals strongly depend on the NV
spin state. In the case of no microwave pulse applied, 90% of the NV population
is them
s
= 0 ground state before optical readout pulse and the maximum FL signals
are observed. In the case of application of microwave pulse, 90% of the NV
population is in the m
s
= 1 ground state resulting in the minimum FL. In both
cases, FL signals mimic total NV population in the ground states (
11
(t)+
22
(t))
explaining the dependence of transient FL on the NV spin state before the readout
pulse.
The above procedure to simulate population dynamics of NV center in pulsed-
ODMR measurements was used to analyze transient FL signals of a single NV center
(Sect. 4.2.2). In the experiment, FL traces were measured for three initial spin states
ofNVcenter-withoutthemicrowavepulseandformicrowavepulsesof37.5nsand75
nsduration. ToexplaintheobservedFL,opticalexcitationrater andRabifrequency
ΩwereconsideredasfreeparametersandtransientFLsignalswerecalculatedforthree
durations of the microwave pulses used in the experiment. Good agreement of the
experiment and simulation was found for r =5 MHz and Ω = 10:4 MHz (Fig. 4.8).
B.2 ODMR signals
In all pulsed-ODMR measurements on a single NV center described in this disserta-
tion, the nite magnetic eld was applied along NV axis resulting in the splitting of
NV spin sublevels (Sect. 4.2.1). In this case, microwave pulses selectively excite one
of the allowed transitions of NV center and the resonant spin states (j0⟩ andj1⟩ or
j0⟩ andj+1⟩) may be treated as a two-level system. In the remainder of this Section,
Appendix B. FL and ODMR signals of a single NV center 129
j0⟩↔j1⟩ NV transition is considered, while description ofj0⟩↔j+1⟩ transition is
identical. Therefore, when magnetic eld B
0
is applied along the NV axis, the NV
Hamiltonian given by Eqn. 4.3 can be modied to the following form:
^
H
NV
(t)= [(t)+!
0
]
^
S
z
+A
∥
^
I
z
^
S
z
+A
?
[
^
I
x
^
S
x
+
^
I
y
^
S
y
]; (B.3)
where!
0
=Dg
B
B
0
=~and(t) =g
B
b(t)=~withb(t)beinga
uctuatingmagnetic
eld at NV center due to surrounding electron and nuclear spin baths. NV electron
spin operators
^
S
x
,
^
S
y
and
^
S
z
correspond to the spin operators of the spin 1/2 system.
NVspinstates,j0⟩andj1⟩, canbearbitraryassignedtothespin1/2states,j⟩and
j⟩. The following denition is further adapted: j0⟩j⟩ and j1⟩j⟩. In this
case, NV is polarized into thej⟩ state after application of the optical initialization
pulse. Spin parameters in the Hamiltonian are dened in units of circular frequency,
i:e: D = 22:87 GHz, A
∥
= 22:3 MHz, A
?
= 22:1 MHz. In addition, nuclear
quadrupole interaction term was omitted as it has no effect on the dynamics of the
NV electron spin.
Furthermore,accordingtotheSect.2.3.1,theNVHamiltonian(
^
H)intherotating
frame at the Larmor frequency !
0
is obtained using Eqn. B.3:
^
H(t) =
[
(t)+A
∥
^
I
z
]
^
S
z
; (B.4)
where the non-secular terms were omitted due to RWA.
During the application of the microwave pulses with the microwave frequency
set to the central frequency of the NV transition, the perturbation Hamiltonian in
the rotating frame at !
0
frequency is given by ΩS
x
(Sect. 2.3.1), where the Rabi
frequency is dened as Ω g
B
b
1
=
p
2~. Ω here is
p
2 faster than for a spin 1/2
system (Chapter 2) to account for NV center electron spin (S = 1). The rotating
frame NV Hamiltonian under the microwave excitation (
^
H
mw
) is then given by
^
H
mw
(t) =
[
(t)+A
∥
^
I
z
]
^
S
z
+Ω
^
S
x
: (B.5)
Appendix B. FL and ODMR signals of a single NV center 130
Rabi oscillations
In the simplest case, when (t);A
parallel
≪ Ω, the NV Hamiltonian under microwave
excitation (Eqn. B.5) is approximated as
^
H
mw
= Ω
^
S
x
: (B.6)
In this case, for NV initially in thej⟩ state, the population oscillations betweenj⟩
andj⟩ states due to resonant microwave pulse oft
p
duration are given by Eqn. 2.34:
p
;
(t
p
)=
1
2
[1cosΩt
p
]: (B.7)
The decay of Rabi oscillations that is observed in the experiments originate from
the rst two terms in Eqn. B.5. To describe the decay, several models have been
developed. When a rst term in Eqn. B.5 is considered to be static ((t) = const)
and introduced into Eqn. B.6, it results in the Rabi oscillations with a static offset
from the microwave frequency described by Eqns. 2.33. Averaging over the local
magnetic elds with a Gaussian distribution leads to the decay of Rabi oscillations
accordingtothepowerlawdecay[205,206]. Ontheotherhand,when(t)ismodeled
by the OU process, the decay is predicted to be governed by an exponential or a
power law decay [207]. Finally, when the full NV Hamiltonian is considered in the
rotating frame (Eqn. B.5) with static (t) ((t) = const), the decay is described
by a Gaussian envelope in combination with a power law decay [208]. However, in
practice, we typically observe the decay of Rabi oscillations, which is well explained
by an exponential or a Gaussian function. Therefore, to account for the decay of
Rabi oscillations in the experiments, we consider the decay envelope to be given by
exp[(t
p
=T
R
)
n
], where n = 1 for an exponential decay, n = 2 for a Gaussian decay
andT
R
is a characteristic decay time of Rabi oscillations. The populations in thej⟩
andj⟩ states after application of the microwave pulse t
p
are then given by
p
;
(t
p
)=
1
2
(
1cos(Ωt
p
)exp
[
(
t
p
T
R
)
n
])
: (B.8)
Appendix B. FL and ODMR signals of a single NV center 131
In the experiment, a normalized signal N(t
p
) = (Sig(t
p
) - Min)/(Max - Min) is
recorded,whereMaxsignalstandsforFLsignalsofthem
s
=0state,Min-FLsignals
after application of microwave pulse and Sig - FL signals of a superposition of NV
spin states prepared by the microwave pulse t
p
. To calculate analytical expression
(R(t
p
)) for normalized signalN(t
p
), and are dened to be the number of FL pho-
tonscollectedfromasingleNVcenterwhenitisinthej⟩andj⟩state, respectively.
Then, using Eqn. B.8, the FL signals (R
Max
, R
Min
, R
Sig
(t
p
)) are expressed as
R
Max
=; (B.9a)
R
Min
=p
(t
)+p
(t
) =
+
2
+
2
cos(Ωt
)D
R
(t
); (B.9b)
R
Sig
(t
p
) =p
(t
p
)+p
(t
p
) =
+
2
+
2
cos(Ωt
p
)D
R
(t
p
); (B.9c)
where D
R
(t) exp[(t=T
R
)
n
]. Next, R(t
p
;Ω;T
R
) is obtained as
R(t
p
;Ω;T
R
;n) =
(R
Sig
(t
p
)R
Min
)
(R
Max
R
Min
)
=
cos(Ωt
p
)D
R
(t
p
)cos(Ωt
)D
R
(t
)
1cos(Ωt
)D
R
(t
)
; (B.10)
where cos(Ωt
)̸=1 since the experimental t
may not be the true pulse.
Ω, T
R
and n are determined from the t of R(t
p
;Ω;T
R
;n) to the N(t
p
). The
normalized signal to the population in them
s
= 0 state (p(m
s
=0)) is obtained from
the transformation of N(t
p
) based on the Eqn. B.10:
p(m
s
=0;t
p
) =
1
2
+
1
2
[
N(t
p
)[1cos(Ωt
)D
R
(t
)]+cos(Ωt
)D
R
(t
)
]
; (B.11)
while the model function (p
model
(m
s
= 0;t
p
)) is equivalent to the p
in Eqn. B.8:
p
model
(m
s
= 0;t
p
)=
1
2
+
1
2
cos(Ωt
p
)D
R
(t
p
): (B.12)
FID
FID signals of a single NV center are observed using Ramsey pulse sequence (=2
=2), where NV FL signals are measured as function of a free evolution time
Appendix B. FL and ODMR signals of a single NV center 132
. To calculate NV population dynamics in the FID experiment, the evolution of
the NV density matrix ^ is considered in the rotating frame with !
0
frequency. The
NV dynamics are then governed by the Hamiltonian given in Eqn. B.4 during free
evolutioninterval,andbytheHamiltoniangiveninEqn.B.6duringapplicationofthe
microwavepulses. Moreover,sinceNVelectronspinisalwaysinitializedintoj⟩state
before the application of Ramsey sequence, the initial density matrix of NV electron
spin is ^
e
(0)=j⟩⟨j, while in the case of NV nuclear spin, the initial density matrix
is given by the maximally mixed equilibrium state ^
n
(0) = 1=3
∑
j
jj⟩⟨jj (j = 0, -1,
+1). Therefore, the initial density matrix of NV center is ^ (0) = ^
e
(0)
^
n
(0). The
evolution of NV density matrix in the Ramsey experiment is then described by [119]
^ ()=
^
R(
2
)
^
U()
^
R(
2
) ^ (0)
^
R
y
(
2
)
^
U
y
()
^
R
y
(
2
); (B.13)
where
^
R(
2
)= exp(i
^
H
mw
t
=2
) = exp(i
2
^
S
x
)and
^
U() exp
[
i(()+A
∥
^
I
z
)
^
S
z
]
with()
∫
0
(t
′
)dt
′
. Using the identity operator (
^
I =
^
R
y
(
2
)
^
R(
2
)), the Eqn. B.13
is transformed to
^ ()=
^
E()
[
j⟩⟨j
^
n
(0)
]
^
E
y
(); (B.14)
where
^
E() exp
[
i(()+A
∥
^
I
z
)
^
S
y
]
. Next, the reduced density matrix for NV
electron spin (^
e
()) is obtained by carrying out the trace over the nuclear spin
states:
^
e
() = Tr
n
{
^
E()
[
j⟩⟨j
^
n
(0)
]
^
E
y
()
}
=
1
3
1
∑
k;j=1
⟨kj
^
E()jj⟩
j⟩⟨j
⟨jj
^
E
y
()jk⟩
=
1
3
1
∑
k=1
exp
[
i(()+kA
∥
)
^
S
y
]
j⟩⟨jexp
[
i(()+kA
∥
)
^
S
y
]
=
^
I
2
+
1
3
1
∑
k=1
[
^
S
z
cos[()+kA
∥
]
^
S
x
sin[()+kA
∥
]
]
:
(B.15)
Appendix B. FL and ODMR signals of a single NV center 133
Using ^
e
, NV population in the bright FL state (p
()) in the FID experiment is
obtained as
p
()= Tr
e
f^
e
()j⟩⟨jg
=
1
2
1
6
1
∑
k=1
cos[()+kA
∥
]:
(B.16)
In the case (t) is model by the OU process, averaging of Eqn. B.16 over () is
carried out according to Appendix A.1, resulting in
p
() =
1
2
1
6
(
1+2cosA
∥
)
exp
[
∆
2
2
c
(=
c
+e
=c
1)
]
; (B.17)
or in the quasi-static regime ( ≪
c
)
p
() =
1
2
1
6
(
1+2cosA
∥
)
exp
[
∆
2
2
=2
]
;
which coincides with [168, 208]. Eqn. B.17 was calculated assuming perfect rotation
of the NV electron spin by the microwave pulses. However, due to the decay of Rabi
oscillations, the population transfer by the microwave pulses is reduced, which can
be accounted with the use of Eqn. B.8. Then, p
becomes
p
(;T
*
2
;m;Ω;T
R
;n) =
1
2
+
1
2
G()D
F
()cos(Ωt
)D
R
(t
); (B.18)
where G() 1=3
(
1+2cosA
∥
)
, D
F
() exp((=T
*
2
)
m
). D
F
is the decay func-
tion in Eqn B.17 in the form according to Sect. 2.3.2, where T
*
2
is the characteristic
decay time of the FID signal and m may the take values in range [1-2]. Set of
parameters Ω;T
R
;n are found from the analyses of Rabi oscillations and therefore
cos(Ωt
)D
R
(t
) is the constant term in Eqn. B.18, whileT
*
2
;m are unknown param-
eters that are obtained from the analysis of the FID signal.
Analytic expression (F()) for the normalized FL signal N() in the FID mea-
surement is calculated similarly to R(t
p
) (Eqn. B.10) using Eqn. B.18:
F(;T
*
2
;m)=
G()D
F
()cos(Ωt
)D(t
)cos(Ωt
)D
R
(t
)
1cos(Ωt
)D
R
(t
)
: (B.19)
Appendix B. FL and ODMR signals of a single NV center 134
T
*
2
andm are found from the t of Eqn. B.19 to theN(), and the normalized signal
tothepopulationinthem
s
= 0state(p(m
s
=0))isobtainedfromthetransformation
of N() based on the Eqn. B.19:
p(m
s
= 0;)=
1
2
+
1
2
[
N()[1cos(Ωt
)D
R
(t
)]+cos(Ωt
)D
R
(t
)
]
; (B.20)
while the model function (p
model
(m
s
= 0;)) to p(m
s
= 0;) is given by Eqn. B.18.
SE
SinceT
1
of NV nuclear spin is much longer than the microwave pulse sequence in the
SEmeasurement,theeffectofNVnuclearspinonNVelectronspinisstaticandhence
averaged out in the SE measurement. The dynamics of NV electron spin are then
identical to the treatment of SE signals of the TLS given in Sect. 2.3.3. In addition,
the main source of NV decoherence in type-Ib diamonds is N spin bath, which can
be well described by the OU process. Using the results of Sect. 2.3.3, the population
dynamics of NV center in the SE measurement are found as
p
;
(2) =
1
2
1
2
exp
[
∆
2
2
c
(2=
c
3+4e
=c
e
2=c
)
]
: (B.21)
Taking into account the decay of Rabi oscillations, p
;
becomes
p
;
(2;T
2
;
;Ω;T
R
;n) =
1
2
1
2
D
S
(2)cos(Ωt
2
)D
R
(t
2
); (B.22)
where D
S
(2) exp((2=T
2
)
). D
S
is the decay function in Eqn B.21 in the form
according to Sect. 2.3.2, where
may take the values in range [1-3].
Analytic expression (S(2)) for the normalized FL signal N(2) in the SE mea-
surement is calculated similarly to R(t
p
) (Eqn. B.10) and F() (Eqn. B.19) using
Eqn. B.22:
S(2;T
2
;
)=
D
S
(2)cos(Ωt
2
)D
R
(t
2
)cos(Ωt
)D
R
(t
)
1cos(Ωt
)D
R
(t
)
: (B.23)
Appendix B. FL and ODMR signals of a single NV center 135
T
2
and
arethenobtainedfromthetofEqn. B.23toN(2). Thenormalizedsignal
to the population in the m
s
= 0 state (p(m
s
= 0)) is found from the transformation
of N(2) according to the Eqn. B.23:
p(m
s
= 0;2)=
1
2
+
1
2
[
N(2)[1cos(Ωt
)D
R
(t
)]+cos(Ωt
)D
R
(t
)
]
: (B.24)
The model function (p
model
(m
s
= 0;2)) to p(m
s
= 0;2) is given by Eqn. B.22.
B.3 Autocorrelation function
As described in Section 3.2.3, the outcome of the photon statistics measurement is a
histogram of number of events (C()) as a function of the measured time intervals
() between photons (Fig. B.3a). In this section, the procedure to obtain the second
order correlation function of FL photons (g
(2)
()) from C() is discussed.
The experimentally measured second order correlation function (G
(2)
()) is found
from the normalization ofC() by number of events at a longer time intervals, where
no correlations between photons is observed (C
1
in Fig. B.3a), so thatG
(2)
(1)! 1.
Therefore, G
(2)
() is expressed as
G
(2)
() =
C(
0
)
C
1
; (B.25)
where shift along axis was preformed due to experimentally introduced
0
delay.
On the other hand, following the denition of the second order correlation function,
G
(2)
() is given by
G
(2)
()=
⟨s
1
s
2
⟩
⟨s
1
⟩⟨s
2
⟩
; (B.26)
where s
1
=f
1
+b
1
, and s
2
=f
2
+b
2
. f
1
and b
1
, f
2
and b
2
are
uorescence (FL) and
background(BG)signalsattherstandsecondAPDinHBTarrangement(Fig.3.11),
respectively. The BG signals are due to scattered/transmitted laser light as well as
dark counts of the APDs ( 200 counts/s). FL and BG signals are obtained from the
Appendix B. FL and ODMR signals of a single NV center 136
500
time delay τ+τ
0
(ns)
1000
0
0
0
time delay τ (ns)
(a)
C (τ)
g
(2)
(τ)
(d)
-40 40
1
2
3
100
200
0
FL (counts/ms)
0
0.5
1
1.5
(c)
0 1
x (μm)
FL (counts/ms)
0
5
10
(b)
0 1
x (μm)
f
1
b
1
f
2
b
2
APD2 APD1
C
∞ τ
0
Figure B.3: Normalization and background correction of experimentally obtained
autocorrelationfunctiong
(2)
(). (a)Rawdataofautocorrelationfunction. (b)Prole
of the FL spot image recorded with APD1. (c) Prole of the FL spot image recorded
with APD2. FL (f
1
, f
2
) and BG (b
1
, b
2
) signals are obtained from a Gaussian t
of the corresponding proles. (d) g
(2)
() of FL photons as obtained from the C()
according to Eqn. B.29. g
(2)
(0)< 1/2 proves FL signals due to a single NV center.
Appendix B. FL and ODMR signals of a single NV center 137
proles of NV FL spot imaged by the APDs (Fig. B.3b and Fig. B.3c). It should be
noted, thatinordertoavoidthe\pile-up"effectduetoelectronicsofTCSPC,signals
at the APD2 are attenuated resulting in a smaller count rate than at APD1. Based
on the denition of s
1
and s
2
, the numerator in Eqn. B.26 is given by
⟨s
1
s
2
⟩ =⟨f
1
f
2
⟩+⟨f
1
⟩⟨b
2
⟩+⟨f
2
⟩⟨b
1
⟩+⟨b
1
b
2
⟩; (B.27)
where the rst term is the correlation term between FL photons, the second and
third are correlation terms between FL and BG signals, and the last term stands for
correlation between BG signals. The form of the second and third terms indicates
that FL and BG signals are, in principle, uncorrelated. The BG signals are also
uncorrelated as they are described by a Poissonian photon statistics (⟨b
1
b
2
⟩=⟨b
1
⟩⟨b
2
⟩
= 1 for all ) and therefore fourth term in Eqn. B.27 can be replaced by⟨b
1
⟩⟨b
2
⟩.
Finally, combining Eqn. B.26 and Eqn. B.27, g
(2)
(g
(2)
()
⟨f
1
f
2
⟩
⟨f
1
⟩⟨f
2
⟩
) is found as
g
(2)
()=
G
(2)
()[1p
1
p
2
]
p
1
p
2
; (B.28)
or using Eqn. B.25,
g
(2)
()=
C(
0
)=C
1
[1p
1
p
2
]
p
1
p
2
; (B.29)
where p
1
f
1
=s
1
and p
2
f
2
=s
2
. Fig. B.3d shows the g
(2)
() that was obtained for
C() in Fig. B.3a according to the Eqn B.29.
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Abstract (if available)
Abstract
A nitrogen-vacancy (NV) center is a paramagnetic color center in diamond with unique electronic, spin, and optical properties including its stable fluorescence (FL) signals and long decoherence time. Moreover, it is possible to initialize the spin states of NV centers by applying optical excitation and to readout the states by measuring the FL intensity. Electron paramagnetic resonance (EPR) of a single NV center is observed by measuring changes of the FL intensity, a magnetic resonance technique known as optically detected magnetic resonance (ODMR) spectroscopy. In addition, NV centers are extremely sensitive to their surrounding electron and nuclear spins. Sensitivity of a single NV center to a single or a small ensemble of electron or nuclear spins has been demonstrated using NV-based magnetic resonance (MR) techniques at low magnetic fields. Despite these fascinating achievements, routine application of NV-based magnetometry remains challenging. ❧ Long coherence of a NV center is critical for most of NV-based MR techniques that highly depends on the content of paramagnetic impurities in diamond. Characterization of spin relaxations of NV centers is of particular interest to “road-map” engineering of diamond materials for sensing applications. In addition, similar to EPR spectroscopy, the spectral resolution of NV-based MR techniques is significantly improved at high magnetic field, thus highly advantageous in distinguishing target spins from other species (i.e. impurities existing in diamond). However, all NV-based MR experiments have been performed at low microwave frequencies, while high frequency (HF) measurements are technologically challenging. ❧ This dissertation is dedicated for investigation of diamond for single NV-based magnetometry at high magnetic fields. The dissertation is organized as the following: In Chapter 1, the remarkable mechanical, optical, electrical and magnetic properties of diamonds are overviewed and the motivation for NV-based magnetometry is discussed. Investigation of diamond paramagnetic impurities presented in this dissertation relays on EPR techniques that are described in Chapter 2. In Chapter 3, the development of conventional HF EPR spectrometer and low field ODMR system for a single NV center are presented in details. HF EPR and low field ODMR spectroscopies of paramagnetic centers in diamond, which are studied in this dissertation, are presented in Chapter 4. In Chapter 5, a double-electron-electron-resonance (DEER) based method to detect concentration of paramagnetic spins in diamond is developed. The method was successfully applied to study and precisely characterize coherence times of nitrogen impurities in diamond. In addition, method may be combined with NV-based MR techniques to probe concentrations in a microscopic volume and to characterize NV relaxations due to paramagnetic impurities in diamond. Finally, the development of HF ODMR system for a single NV center in diamond is presented in Chapter 6. Using HF ODMR system, the coherent control of a single NV center was achieved for the first time at high magnetic fields (~ 4 Tesla), showing that NV center retains all its unique optical properties up to 12 Tesla and demonstrating the opportunity for HF NV-based magnetometry. In addition, investigation of T₁ relaxations of NV centers in nanodiamond (ND) was performed in a wide range of magnetic field (0 - 8 Tesla), showing that NV centers in NDs are highly sensitive to surface paramagnetic impurities in NDs.
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Asset Metadata
Creator
Stepanov, Viktor
(author)
Core Title
High-frequency and high-field magnetic resonance spectroscopy of diamond
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Chemistry
Publication Date
03/08/2017
Defense Date
10/13/2016
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
diamond,electron paramagnetic resonance,magnetometry,nitrogen-vacancy center,OAI-PMH Harvest,optically detected magnetic resonance,single spin
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Takahashi, Susumu (
committee chair
), Bradforth, Stephen E. (
committee member
), Venuti, Lorenzo Campos (
committee member
)
Creator Email
stepanov@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-347990
Unique identifier
UC11255804
Identifier
etd-StepanovVi-5128.pdf (filename),usctheses-c40-347990 (legacy record id)
Legacy Identifier
etd-StepanovVi-5128.pdf
Dmrecord
347990
Document Type
Dissertation
Rights
Stepanov, Viktor
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
diamond
electron paramagnetic resonance
magnetometry
nitrogen-vacancy center
optically detected magnetic resonance
single spin