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Modeling nanodevices: from semiconductor heterostructures to Josephson junction arrays

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Modeling nanodevices: from semiconductor heterostructures to Josephson junction arrays

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Content MODELING NANODEVICES: FROM SEMICONDUCTOR HETEROSTRUCTURES TO JOSEPHSON JUNCTION ARRA YS Copyright 2012 by Bruna Pereira Wan derley de Oliveira A Disse rtation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2012 Bruna Pereira Wan derley de Oliveira To my grandfather Eclair, who taught me how to find the stars in the sky, and instilled in me his curiosity and love for science. Pa ra meu avo Eclair, que me ensinou a encontrar estrelas no ceu, e passou para mim sua curiosidade e amor pela ciencia. 11 Acknowledgments This Ph.D. thesis is the culmination of a work done over many years, and I have been fortunate to have received unconditional support from many profe ssors, inst ituti ons, friends, and family. I would like to thank my advisor, Stephan Haas, for all the encouragement and guid ance through the years. I benefited greatly from our interacti ons and I admire his ded ication to the Physics department at USC. Stephan is a passionate teacher and a truly insp irational advisor. I al so would like to give a special thank you to my collaborator and friend To mmaso Roscilde, for the extreme patience and amazing hospitality that he and his lovely wife Lisa offered me in France. To mmaso is a great teacher and I have learned a lot of science from him, but especially he has shown to me, by his acti ons, what it takes to be an exceptional scholar. I hope someday I will learn this lesson. Thank you to the members of my thesis committee, professors Gene Bicke rs, Moh El-Naggar, Wer ner Dappen, and Stephen Cronin, for the guidance and helpful advice. A special thank you to Prof. Dappen, for being a friendly and supportive Department Chair, and to Prof. Cronin, for it was a great pleasure to have had the chance to work with him. My sincere thank you to all professo rs I had as a USC student, especially to Prof. Hubert Saleur, from whom I have learned so much and whom I greatly admire for his kindness and bril liance. Many thanks to the staff at USC Physics and Astronomy, Lisa Moeller, Mary Beth Hicks and Betty Byers, all of whom have helped me so many times. Also thank you to Joseph Vandiver, my laboratory director when I was a TA for Physics and Astronomy. Part of the work in this thesis was done during several visits to the Ecole Normale Superieure de Lyon, in Lyon, France. I am very grateful to the school for hosting me and 111 for providing such a pleasant environment to do research. There I also met some special people: Lucile Sa vary, Stephan Humeniuk, Marianne Corvellec, Amilia Panella, and Sebastien Besse, all very smart ph ysicists and very dear friends. I will always cherish the special moments I have spent with each one of them. Thank you to my colleagues at USC, especially Yaqi Tao , Damian Abasto, Tameem Albash, and Nikolay Bobev. Yaqi is a very sweet person whom I had the honor to know. Damian and his wife Andrea have become very dear friends. Nikolay is a great friend and an amazing scholar whom I deeply admire, but I was also lucky enough to have been his beach volleyball partner. Our second place in the intramurals champ ionship against the fraternity boys and sorority girls was memora ble, and so was our attempt to play in the scalding Golden Sands in Varna. My brief collaboration with Tameem in neuro science was one of the most fun proj ects during my time at USC. He is a great collaborator and friend, and his discipline and dedication inspire me everyday. I was very fortunate to have had the best landlord and landlady in Los Angeles, Karen and Gary Kou snetz. I will never forget all the delicious Thanksgiving dinners and special meals they have prepared for their tenants. Thank you to all members of my American family. I have enjoyed every minute I have known them, and I feel very honored to have become part of such a beautiful fa mily. Thank you so much to my family in Brazil, especially to my grandmother Angelita and my grandfather Eclair, to whom this thesis is dedicate d, and my uncles, aunts, and cousins, who are too many to name, but are all very special. It is always a blast to go back home and visit everyo ne. I am al so thankful for having such special friends in Brazil, especially Gustavo, Eliza, and Paulinha. We have such a beautiful , long-lasting friendship. Thank you for keeping me virtual company whenever I was lonely, and thank you for all the laughs! iv I have had the support of very special people through these years, but I never expected to have such a strong network of people helping me as I had while writing this thesis. Thanks to my aunt and uncle, Fernanda and Paulo, for letting me work on the manuscript in their quiet apartment this last December. Thank you so much to my brother Lucas, who is now a Troj an and helped me at home while I worked on the final prepara tions of this thesis. He has been a great roommate, and more than just sibl ings, we are very close friends. I am so fortunate for having such a wonderful brother. To my parents, Ricardo and Ir ilene, a thank you will never be enough. Mom and dad, you always believed in me. You supported me when I decided to study abroad, even though you knew you would miss me as much as I know you have, because I have missed you too. You have been there for all the phone ca lls, whenever I need a recipe or just to say "hi". I am so thankful for everyth ing. Toby, thank you so much for your love and compani onship. You were with me through the hard and happy moments, and your caring support made the whole differ ence. It has been so fun to share my days with you. Finally, I would also like to acknowledge the inst ituti ons from which I received support during my Ph.D program. Thanks to Wi SE (W omen in Science and Engineering at USC), to the USC Dana and David Dornsife College of Ar ts, Letters and Science, to the USC Graduate School for a Disse rtation Completion Fellowship, and to the French Embassy in Wa shington D.C. for a Chateaubri and Fellowship. v Table of Contents Dedication Acknowledgments List of Figures Abstract Chapter 1: Introduction 1.1 Organization of this diss ertation Chapter 2: Electron-Phonon Interaction in Semiconducting Nanostruc- 11 111 V111 XV 1 2 tures 4 2.1 Introduction . . . . 4 2.2 Model and Method 6 2.3 Numerical Results . 2.4 Conclusion . . . . Chapter 3: Josephson junction arrays 3.1 Introduction .......... . 3.1.1 Josephson Effects ... . 3.2 Classical Josephson junction arra ys: The two dimensional XY Model . . 3. 3 Frustrated Josephson junction array s . . . . . . . . . . . . . . . . . 3.3.1 Phase tran sitions in fully fru strated square Josephson junc tion array s . . . . . . . . . 3.4 Diluted Josephson junction array s 3.5 Quantum Effects ......... . Chapter 4: Phase Transitions in Diluted and Frustrated Josephson Junc- tion Arrays 4.1 Introduction ...... . 4.2 Model ......... . 4. 3 Computational Methods . 4.3.1 Metropolis Updates 4.3.2 Microcanonical Updates: Overrelaxation . 4.3.3 Cluster Updates ............. . 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Helicity Modulus and Choice of Boundary Conditions 8 20 22 22 23 24 29 34 35 36 38 38 40 41 42 43 45 46 46 Vl 4.4.2 Structure Factor . 4.4.3 Phase diagram . 4.5 Conclusions ...... . 53 56 57 Chapter 5: Current-Voltage characteristics offrustrated and diluted Joseph- son junction arrays 58 5.1 Introduction 58 5.2 Model . . . 59 5.3 Results . . . 62 5.4 Conclusion 66 Chapter 6: Conclusions Bibliography Appendix: The Propagation Matrix Method 67 69 76 Vll List of Figures Figure 2.1: (a) Electron transmission probability through a delta poten tial. The solid black line repre sents a repulsive potential with an electron-phonon coupling constant g = 0.117 e V nm. The dashed red line corre sponds to tran smission through an attractive delta potential with g = 0.078 eV nm. In both cases, the electron is allowed to excite two local oscilla tor levels with energies nw = 1.0 eV and 2.0 eV. The blue line is for g = 0. (b) Electron tra nsmission probability through a finite-wid th potential barrier/well of width L = 0.1 nm. The solid black line repre sents a repulsive rectan gular potential of strength V0 = 2.0 eV with g = 0.117 eV nm. The dashed red line cor responds to tran smission through an attractive rectangular potential Vo = - 1.0 e V with g = 0.078 eV nm. The local oscilla tor levels are cho sen to be at nw= 0.03 e V and 0.06 e V, positioned at the centers of the rectangular potential s. . . . . . . . . . . . . 9 Figure 2.2: (a) Symmetric double rectangular potential with barrier width 0.4 nm, well width 0.6 nm, and barrier energy V0 = 1. 0 e V. A local phonon scatterer is located at the center of the double barrier. (b-e) Electron transmission probabiliti es for various numbers of accessible phonon channel s. (b) cor responds to the case without electron-phonon interaction (g = 0), (c) rep resents the tran smission coefficient for one phonon channel , (d) for 2 phonon channel s, and (e) for 40 phonon channel s. In (c-e) the electron-phonon coupling is set to g = 0.0 4 eV nm, and the phonon frequency is nw = 0.01 eV. With increasing number of phonon chan- nels one observes the formation of a band. . . . . . . . . . 11 V111 Figure 2.3 : Electron tran smission resonances for a double rectangular potential barrier with a phonon scatterer located at the cen ter. The parameters are chosen identical to Fig. 2.2, unless otherwi se specified. (a) and (b) correspond to a system with one phonon channel, and (c), (d) to a system with two phonon channel s. In (a) and (c) the electron-phonon scat tering strength is kept constant at g = 0.04 e V nm, and the phonon energy nw is varied. One observes that the entire spectrum shifts to larger values of energy as nw increases. In (b) and (d) nw = 0.01 eV and g is varied. In this case the gaps between the transmission resonance peaks widen with with increasing g. For two phonon channel s, the cen- tral peak does not shift as g is varied. . . . . . . . . . . . . 13 Figure 2.4: Gap (in e V) between bonding and anti bonding peaks for differ ent values of electron-phonon coupling g (in eV·nm), for 1 phonon channel (black circles) and 2 phonon channels (red square s). . . . . . . . . . . . . . . . . . . . . . . . . 14 Figure 2.5: Bound state ener gies in a double rectangular potential bar rier with one phonon scatterer at the center (as depicted in Fig. 2.2(a)). For the case of one phonon channel, the ener gies of the two resulting transmission resonance peaks are plotted as funct ions of the electron-phonon coupling constant g and of the phonon energy nw. Exact numerical results (solid black lines) are compared with the effective theory (dashed red lines) described in the text. In (a) and (c) the electron-phonon coupling is kept constant at g = 0.04 eV nm, and in (b) and (d) the phonon energy is kept con stant at nw = 0.01 eV. In (a) and (b), for the higher bound state energ ies, the effe ctive theory reprodu ces remarkably well the numerical results. In (c), the effe ctive theory and numerical results are off- set by approximately 0.02 eV. In (d), the effective theory matches the numerical results for small values of g. For instance, when g = 0.04 eV nm, the effective theory overestimates the lower bound state energy by 0.01 eV. . . . . . . . . . . . . . . . . . . . . . . . . . 16 IX Figure 2.6: (a) Harmonic oscilla tor potential V(x), with an electron incoming from the left. The effective mass of the electron is 0. 07m (b) Electron energy vs. transmission for the poten tial depicted in (a) . Note resonant tunneling (unit transmis sion) for E0 = 0.14 7 eV, E1 = 0.442 eV and E 2 = 0.765 eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Figure 2.7: (a) Harmonic oscillator potential V(x), with a phonon scat terer site at x = 10 nm. The electron phonon coupling is g = 0.2 eV·nm, and the phonon energy is nw = 0.0 1eV. (b) Red-dashed line: Electron energy vs. transmission for the potential depicted in (a), with electron-phonon interact ion, for one phonon channel availabl e. Note the splitting for the ground state energy Eft = 0.2031 eV, E0 = 0.0 842 eV. The first excited level does not split and the tran smission for the third excited level is suppre ssed. . . . . . . . . . . . 18 Figure 2.8: (a) Potential double barrier with thick ness of 1. 0 nm and separation of 5.0 nm. The height of the barrier is 1. 0 eV, and the electronic effective mass is 0. 07me. (b) Tran s rmsswn vs. energy curves for one excited phonon at the center of the potential well. In the pre sence of a voltage bias across the heterostructure, this curve is shifted towards lower ener gies. (c) Current-voltage curve for the double barrier in (a). The two small peaks at low voltage bias (Vb) corre spond to the two low energy peaks in (b). The calcula tion of current is done by integrating the transmission over an energy window from 0 eV to 50 meV. . . . . . . . . . . 21 Figure 3.1: (a) A positive vortex. (b) A negative vortex. . . . . . . . . 25 Figure 3.2: (a) Configuration of phases near the ground state of XY model. (b) Array above critical temperature T K T· . . . . . . 29 Figure 3.3: A fully fru strated square plaquette with three ferromagnetic coupli ngs (Jij = J) and one antifer romagnetic (effectively �j = -J in Eq. 3.4 ). Note that 2::::: Aij = 21r f = 1r around the plaquette. For the XY model at zero temperature, the fully fru strated plaquette arran ges into the Villain lattice configuration (Fig. 3.4 ). . . . . . . . . . . . . . . . . . . . 31 X Figure 3.4: (a) One plaquette of the Villain 4-sublattice order. (b) Unit cell of Villain 4-sublattice order, with two vortex pairs at zero temperature. . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 3.5: (a) Ground state of a 10% diluted lattice of square Joseph son junct ions. (b) Just below the lattice percolation thresh old, at 35% diluti on. (c) Above the percolation threshold, with 45% of diluted sites. The lines are guides to show the differ ent clusters that form in the latti ce. The fact that the phases align at different angles for each cluster shows that the clusters are non-i nteracting. . . . . . . . . . . . . . . . 35 Figure 4.1: Conjectured phase diagram from Ref. [3] for a bond-diluted and fru strated triangular Josephson junction array. The phases are superconducting (SC), normal (N), and vortex glass (VG). Note that the authors considered that both phase tran sitions occur at a single temperature at zero and low dilu tion, contrary to current knowledge of phase tran sitions in such systems, which occur at close but distinct tempera- tures at zero dilution [68]. . . . . . . . . . . . . . . . . . . 39 Figure 4.2: (a) Ground state of fru strated model. (b) Ground state at 5% diluti on, still displaying 4-sublattice order. (c) Ground state at 30% dilut ion, showing that long-range order is lost. (d) Ground state at 45% dilut ion, above the lattice percola- tion threshold. . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 4.3 : Example of a phase update using the overrelaxation method. The ori ginal phase ¢ is repr � sented by the vector S , the direction of the local field is h and the updated phase a is the vector S '. . . . . . . . . . . . . . . . . . . . . . . . . 44 Figure 4.4: Helicity modulus of fru strated Josephson junction array s at zero diluti on, for system sizes L = 16, 24 and 32. The KT transition temperature TKr is given by the limit in Eq. 4.6. Figure 4.5 : Helicity modulus of site-diluted Josephson junction array s with f = 1/2, for system size L = 24 and different dilu tions d. The KT transition temperature TKr is suppressed 47 as dilution increases. . . . . . . . . . . . . . . . . . . . . 48 Xl Figure 4.6: Helicity modulus for the diluted and fru strated JJA. A "reen trance" is observed for T < 0.18 , due to incommen surabil ity effects in the lattice with periodic boundary condit ion. Each data point is averaged over 100 disorder realizati ons. Here, 15% of sites are dilu ted. . . . . . . . . . . . . . . . 49 Figure 4. 7: Te st sample used to investigate incommensu rability effects in the helicity modulus Y(T). Here the lattice is fully fru s trated, with system size L = 10. The dilution d = 1 0% cor responds to 10 adj acent sites removed from 2 rows and 5 columns of the latti ce, creating a cluster of vaca ncies. Figure 4.8: Right (Lef t): Helicity modulus along the x (y) direction for the array in Fig 4.7. Error bars are smaller than the 50 symbols, and the dashed lines are a guide to the eye. . . . . 51 Figure 4.9: Left (Right) : Supercurrent term in Eq. 4.7 al ong the x (y) direction for the array in Fig. 4. 7. Error bars are smaller than the symbo ls, and the dashed lines are a guide to the eye. We can see that in the y direction the supercurrent is finite at zero temperature. . . . . . . . . . . . . . . . . . . 52 Figure 4.10: Renormali zed phase structure factor as a function of tem perature for 3 system sizes (L = 16, 24 and 32), at zero dilution (left) and 10% dilution (right). The chosen order ing vector Q1 is located at (0, 0) in reciprocal space. The insets show in detail the cro ssing points of S ( Q1, T I J; L) for each dilution d. The error bars are smaller than the sym- bols and the dotted line is just a guide to the eye. . . . . . 54 Figure 4.11: Renormali zed chiral structure factor as a function of tem perature for 3 system sizes (L = 16, 24 and 32), at zero dilution (left) and 10% dilution (right) . The ordering vector Q is located at ( 1r, 1r ) in reciprocal space. The insets show in detail the crossing points of Sc( Q, T I J; L) for each dilu tion d. The error bars are smaller than the symbols and the dotted line is just a guide to the eye. . . . . . . . . . . . . 55 xu Figure 4.12: Phase diagram of site-diluted and fully fru strated square JJA. There is a significant separation between the phase tran sitions as more sites are dilute d, creating a phase which has vortex order but no phase coherence, in which phase correlations are short-ra nge. Both tran sitions are suppressed well below the lattice percolation thre shold at around 40%. Figure 5.1: (a) Array of RSJ (Resist ively Shunted Josephson- junct ions) without fru stration and dilut ion. Current is injected from the electrode on top. (b) Same circuit as in (a), but for a diluted array of RSJ in a perpendicular magnetic field iJ, 56 which induces fru stration. . . . . . . . . . . . . . . . . . . 60 Figure 5.2: I-V characteristics of frustrated and diluted array with L = 48. The combined effe cts of fru stration and percolative dis order reduces the critical current to lower values up to the percolation threshold. . . . . . . . . . . . . . . . . . . . . 62 Figure 5.3: dV ldi versus I I L for different diluti ons in the interval d Kr < d < de, for L = 48. The plateaux at high cur rent bias is an indication that the JJA is Ohmic in this limit. At low current bias, the plateau is a finite size effect, as shown in Fig. 5.4. Inset: Detail of dV ldi at low current bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 5.4: Left: Voltage vs current for different system sizes at 20% diluti on. Right : Same plot, in log-log scale. The behavior of the I-V characteristics at low I I L is a finite size effect. Figure 5.5: Curve fits for the values of dilution and L indicated in each plot (Left: L = 32, d = 0.25 (top), d = 0.22 (botto m); Top right: L = 48, d = 0.25; Bottom right: L = 24, d = 0.30.) From these fits we extract the exponent a and the critical 63 current I c. (See Figs. 5.6, 5. 7) . . . . . . . . . . . . . . . . 64 Xlll Figure 5.6: Rough estimate of Ic versus dilution for L = 24, 32 and 48. The values of Ic were found by fitting the data as V (I) = cf +bE>(! - Ic)(I- Ic)a. For the larger system sizes we find a sharp drop at 20% dilut ion. The non-zero Ic at higher diluti ons is a finit e-size effect. . . . . . . . . . . . . . . . 65 Figure 5.7: Exponent a versus dilution for L = 24, 32 and 48. In a rough approximation there is a peak around 20% dilution and a jump to around a � 3 when the system loses phase Figure 1: Figure 2: cohere nce. . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Example of potential barrier with height V0 and width b - a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Example of potential barrier with a phonon sca tterer at x = a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 xiv Abstract In this thesis we study the physical proper ties of two distinct physical systems. First, a generalized propagation matrix method is used to study how scattering off local Einstein phonons affects resonant electron tran smission through quantum wells. In particular, the parity and the number of the phonon mediated satellite resonances are found to depend on the availabl e scattering channel s. For a large number of phonon channel s, the for mation of low-energy impurity bands is observed. Furth ermore, an effective theory is developed which accurately describes the phonon generated sidebands for sufficiently small electron-phonon coupli ng. Finally, the curre nt-voltage characteristics caused by phonon assisted transmission satellites are discussed for a specific double barrier geom etry. In the second part of this thesis we numerically investigate the complex interplay between fru stration and disorder in a magnetically fru strated Josephson junction array on the square lattice with site diluti on, modeled by the fully frustrated classical XY model on the same latt ice. This system has a superconducting ground state featuring a vortex crystal induced by fru stration. In absence of dilution this system is known to exhibit two thermal transitions: a Kosterlitz- Thouless transition at which superconduc tivity and quasi-long-range phase order disa ppear, and a higher-tem perature transition at which the vortex crystal melts-the two critical temperatures delimit a chiral phase. We find that dilution enhances the width of the chiral phase, and that at a critical dilution superconductivity is supp ressed down to zero temperature, while chiral order survives the corresponding ground state of the system becomes therefore a chiral phase glass. At an even higher dilution chiral order disappears in the ground state, leaving the system in a vortex and phase glass state. We reconstruct the complex phase diagram via extensive Monte Carlo simulat ions, and we investigate the main signatures of the various phases in the transport properti es (I-V characteristics). XV Chapter 1 Introduction The theoretical study of collective phenomena in quantum systems has experienced enormous developments in the past century, thanks to the success of a combination of different theoretical tools such as mean-field theory, renormalization group techniq ues, and perturbative approaches. On the other hand, the enormous progre ss in materials science has led to the synthesis of a large fam ily of novel complex material s, such as high-tem perature or disordered superconductor s, fru strated quantum ma gnets, and spin glasses, whose theoretical understanding still remains cha llenging. The electrons in these materials are very strongly interact ing, and the fru strated and/or disordered geo metrical structure leads to a strong competition between different ground states. These ingredients, among others, conspire against the possibility of giving a simple theoreti cal description based on the above-mentioned approaches. Despite several decades of intense research, even the minimal theoretical models elaborated to capture the basic phenomenology of these materials remain poorly understood. The concomitant elab oration of powerful computational al gorithms for the study of classical and quantum many-body systems - along with the increasing computational capability of CPU s - has suggested the pos sibility that numerical methods could come to the rescue of analytical approaches. Unfortunately, numerically exact techniques are in general only limited to small cluste rs, and in only a few cases can the computa tions be pushed to system sizes that permit reliable extrapolati on to the thermodynamic limit. This diss ertation is dedicated to the study of two many-body systems and some of their physical propert ies, namely the transport of electrons through semicon ductor 1 nanostructures and the investigation of transport and phase tran sitions in Josephson junction arra ys. Even though these models display distinct phenomena, they are both systems wherein structural redesign can be performed in order to effect desired proper ties. Since existing analytical tools are not sophisticated enough to provide a precise description of how such systems behave, we resort to using novel numerical tools that have been developed in order to better understand the physics of nanoscale devices. 1.1 Organization of this dissertation Chapter 2 of this thesis is devoted to the study of the tran smission of electrons through multilayer nanostructures and how it is affected by the interaction between these elec trons and the atomic vibrations of the structures. It is well-known that electrons have a finite probability of tunneling through a single potential barrier. If we instead have a double potential barrier, electrons with a given energy can experience several tunneli ngs and reflections at the walls of the barriers in such a way that the incoming electron would have one hundred percent probability of being transmitted through both quantum barri ers. The particular value of energy for which the electron is transmitted corresponds to a bound state of the electron-quantum well system. However, the unity tran smission probability is a very simple ap proxima tion, since we do not consider the presence of local vibrational modes in the quantum well. If the incoming electrons interact with the vibrational degrees of freedom of the device, resulting in the excitation of local vibrational modes, we observed that the elec tro nic transmission probability decreases. On the other hand, in contrast with the case with no coupling to vibrational degrees of freedom, the number of states is now directly 2 proportional to the number of vibrational modes that are excited by the incoming elec tron. The results of this work were pub lished in Physical Review Bin April2009 [24]. The remaining chapters investigate the complex interplay between geometric fru s tration and lattice disorder in Josephson junction array s. These structures consist of superconducting islands connected though oxide layers to form a planar latti ce. Since the size of Cooper pairs is much smaller than that of the island, they can be regarded as point-like composite bosons with charge 2e, moving over the array of superconducting islands via Josephson tunneling. The tunneling of Cooper pairs between neighboring islands favors the establishment of phase coheren ce, with the result that the whole array becomes superconduc ting. A simple superconducting state with uniform phase is also influenced by the appl i cation of a magnetic field transverse to the array. Indeed, due to the connected geometry of the array, the magnetic field induces persistent currents around the plaquettes of the array s. Such circulating currents correspond to the formation of topological defects -i n the form of vorti ces - in the phase configurat ion, as a result of fru stration in the Joseph son couplings induced by the magnetic field. If the magnetic flux threaded through each plaquette is a rational fraction of a superconducting flux quantum, the persistent currents form an ordered structure of vor tices, while the persistent currents can form a quasi-periodic structure in the case of irrational fluxes. Given the high degree to which Josephson junction array s can be tuned in practice, all of our predict ions should be fully accessible to future experiments on ultra-small superconducting islands. In the event that these experimental investigat ions pose new questions and discover new physical behavior, the theoretical tool s and capabi lities that we have developed here will be well suited for providing answers. 3 Chapter 2 Electron-Phonon Interaction in Semiconducting Nanostructures 2.1 Introduction Resonant tunneling through quantum wells has been extensively studied in semicon ductor heterostructures, such as GaAs/ AlxGa1_xAs double barri ers [34, 7, 19, 32, 50]. More recently, ana logous electron transmission processes have also been investigated in the context of molecular juncti ons [67, 4, 65, 10, 31] and mesoscopic rings [84, 13]. Resonant tunneling is a purely quantum effect whereby electrons pass through structures made of potential wells and barri ers with unit or near-unit tran smission probabiliti es if they enter the quantum well at the particular ener gies of the structure 's bound state s. Following the initial experimental observation of satellite peaks of these tran smission resonances [34], a large volume of theoretical work [33, 16, 14, 15, 83, 75, 2, 42, 89, 8, 9, 47, 66, 12] has focused on the effects of phonon scattering on the electronic tunneling. Early on, it was recognized that perturbative treatments tend to miss the essential feed back effects between elastic and inelastic channels which lead to these satellite features in the electronic transmission [33]. In particular, it was found that electron-phonon scat tering processes can cause the formation of polaro nic bound states, leading to phonon assisted resonant tunneling [16, 14, 15, 83]. More recent works have shown that within a tight-binding desc ription these features are further enhanced [75], and phonon bands can form [8, 9]. Furth ermore, theoretical models have been generalized to include the 4 effects of the three-dimensional environment [2, 89], non-equilibrium dynami cs [ 42], and finite temperatures [47]. In this chapter, we exami ne the hierarchy of polaronic resonances in the electron tran smission through quantum well structures. In particular, we focus on even-odd effects with respect to the number of avai lable phonon channels and on the emergence of impurity bands as this number becomes large. We also apply an effective theory which repro duces the dependence of the resonance peaks on the electron-phonon coupling strength and the phonon energy in the limit of suffi ciently small coupli ng. The method we are using is a generalization of the propagation matrix technique [58] which takes into account elastic electron scattering at potential steps as well as scattering off local Einstein phonons. This approach al lows a numerically exact calculation of the electron tran smission through quasi-one-dimensional heterostructures without any perturbative constraints, such as limitati ons to particular parameter regimes, or restrictions to specific energy ranges, such as low-energy resonant states. In addition, our method accounts for the feedback between an ad justable number of phonons and the elastic transmis sion channel , and is therefore suitable to accurately describe the interplay between non perturbative resonances of the many-body system. Before proceeding to the discussion of resonant tunneling through specific semicon ductor double well s, let us bri efly point out some simila rities and diffe rences of this system with electronic transport through molecular junct ions [67, 4, 65, 10, 31]. In the theory of both physical systems, many-body methods are combined with scattering theory to obtain the tunneling density of states and the resulting current-voltage charac teristics for electronic transport through small obj ects with quantized energy level s. In both cases one observes the formation of phonon assisted satellite feat ures as the elec trons scatter off local vibrational modes. However, there are several significant differ ences between these systems, as we will see below. Electron transport through layered 5 semicond uctor structures exhibit resonant tunneling features which coexist with contin uum contribu tions. These are typically absent in molecular tran sistors. Furth ermore, since semiconductor hereostructures are manmade, the specific resonance levels can be controlled by layer thickne ss and composition and are thus tunable. Moreover, the experimental current-voltage curves for semiconductor heterostructures are quite differ ent from molecu les, i.e. they show peak feat ures rather than the steps characteristic for molecular systems [62]. The work desc ribed in this chapter was published in Physical Review B in April 2009. [24] 2.2 Model and Method We wish to determine the tran smission probability of electrons through potential struc tures of arbitrary profile, with the possibility of exciting local Einstein phonon channel s. The basic Hamiltonian for this pro blem, H = L E(k)clck + L V(x)c�Cx + L nwbLbx; + g L J(x- Xi) (bL + bx;) 4ck'(2.1) k X Xi Xi ,k,k1 describes electrons with creation and annihilation operators denoted by c t and c , and a dispersion E(k) = n2k2 /2m, propagating through a potential structure whose real-space profile is given by V (x). In additi on, local Einstein phonon scatterers with creation and annihilation operators bt and b and energy hw are placed at impurity sites x i . The electron-phonon interaction is controlled by the coupling constant g, which has units of energy times length. The particular systems we have in mind are layered semicon ductor 6 heterostructu res, such as GaAs/ AlxGa1_xAs. For these systems, the use of a momen tum independent electron-phonon coupling constant is standard, and can be viewed as a reliable lowest order approach [12]. To find the electronic transmission probability we use the propagation matrix method, which is appl ied in the following way: for each step at position j in the poten tial profile, we construct a propagation matrix P�t e p' and in between neighboring steps at a distance Lj apart, we construct a propagation matrix p�:ee· The elements of Pst e p depend on the boundary condit ions of the electronic wavef unction at the potential step at position j. The matrix p�:ee is diagonal, and its elements depend on the phase picked up by the electron as it propagates through a length Lj between potential steps. The total propagation matrix is given by the product of the individual matrices: (2.2) An example of the propagation matrix method is given in the appendix. For a system without phonon excitat ions, the propagation matrix is simply a 2 x 2 matrix. When the electron excites phonons as it penetrates the structure, the propagation matrix grows as (2n + 2) x (2n + 2), where n is the number of phonon channel s. When several phonons are excited it becomes necessary to find the transmission probability of an electron as a function of energy numerically. The idea is to solve a system of linear equati ons of the form px = a, where x is the vector whose terms cor respond to the tran smission and reflection coef ficients x = (to, ro, ... , tn, rn) and a= (ao, bo, ... ,an, bn), where the coefficients a1 and b1 depend on the initial conditions of the probl em. In our problem there is no reflection as the electron exits the potential profile, so we can set the reflection coef ficients r1 = 0 for all l, therefore reducing the number of equati ons in the system by half: 7 Pn P13 P 1n -1 to P31 P33 P3n -1 (2.3) P n -11 P n -1 2 P n -1n -1 Initially all phonons are in the ground state, and therefore aj = Jjo· All that is left is to deter mine the tj. and we can do so by solving the system above using a Gauss- Jordan elimination. Once these terms are found, we can calculate the transmission probability: � k z (E) 2 T(E) = '2o k o(E) l tz (E) I , (2.4) where k1 is the momentum of the electronic wave function in a channel with l phonon s. With this approach it is possible to plot tran smission probability versus energy. 2.3 Numerical Results As a test of the validity of the propagation matrix method, we first examine two cases which have previously been studied in the literature. In Fig. 2.1 , we show the transmis sion through repulsive (solid black line) and attractive (dashed red line) delta potential s, which allow the excitation of two local vibrational modes at nw = 1. 0 eV and 2.0 eV. For comparison, we also show the case (dotted blue line) without coupling to these local Ein- stein modes. In their absence, the transmission increases monotonically with the energy of the incoming electrons. However, in the pre sence of inelastic scattering channel s res onance feat ures in the form of spikes and dips in T( E) occur at ener gies slightly below the local oscilla tor level s. They indicate the formation of bound states [14, 2, 12], mani fested by Fano features which for attractive potentials can completely suppre ss electron 8 1 .0 (a) . - -- !·· \- ... · __ � - �-� -�- __ -:..:. -, _-:._·: ... .- �·:.. :: .. :: .. ::.. 1.0 (b ) 0.2 I ' I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 'I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I � 0 ' 0 .0 10 20 {) 0. 5 . 1.5 . Energy (eV) 0.8 . � 0 . 6 :l ' 8 � E'< 0.4 0.2 2.5 °'8.oo - V = 2.0 eV, g = 0.117 eV nm V = -1.0 eV, g = 0.078 eV nm · V = -1.0 eV, no phonons 0,03 0.06 Energy (eV) 0.09 Figure 2.1: (a) Electron transmission probability through a delta potential. The solid black line repre sents a rep ulsive potential with an electron-phonon coupling constant g = 0.117 e V nm. The dashed red line cor responds to transmission through an attractive delta potential with g = 0.078 eV nm. In both cases, the electron is allowed to excite two local oscillator levels with ener gies nw = 1. 0 eV and 2.0 eV. The blue line is for g = 0. (b) Electron tran smission probability through a finite-width potential barrier/well of width L = 0.1 nm. The solid black line rep resents a repulsive rectangular potential of strength V0 = 2.0 eV with g = 0.117 eV nm. The dashed red line cor responds to transmission through an attractive rectangular potential Vo = -1.0 e V with g = 0.078 eV nm. The local os cillator level s are chosen to be at nw= 0.03 eV and 0.06 eV, positioned at the centers of the rectangular potential s. tran smission right below the resonance energy (red line in Fig. 2.1(a)). These features arise from the strong feedback between inelastic and elastic scattering pro cesses, and are easily missed in perturbative treatments [33]. The parameters in Fig. 2.1 have been chosen identical to previously pub lished data [2, 12] to demonstrate full agreement of methods. 9 As shown in Fig. 2.1(b ), the phenomenon of polaron-type bound state formation persists for finite-width wells and barri ers. [14 ] Here, the location of the Einstein scat terers are chosen at the center of the rectangular potential profiles. Bearing in mind experimentally relevant scales, we consider vibrational energies two orders of magni tude lower than in Fig. 2.1, i.e. at nw = 0.03 eV and 0.06 eV. In analogy to the case of delta potential s, resonances are observed at both energy level s. However, electron tran smission is suppressed with respect to the case of delta potentials because of the finite spatial extent of the wells and bar riers. Next, we turn to the case of electron tran smission through more complex quantum well structures. Focusing on symmetric potential profi les, let us consider rectangular double barriers of length 0.4 nm, separation 0.6 nm, and height Vo = 1 eV. A local Einstein scatterer is placed at the center of the well, as illustrated in Fig. 2.2(a). The vibrational ener gies are nwn = nnw with nw = 0.01 eV and n = 1, 2, 3, .... In the absence of phonon scattering, shown in Fig. 2.2(b ), one observes a bound state at E = 0.358 e V which allows resonant tunneling with unit transmission. In the followi ng, we examine the effects of inelastic scattering on this resonant feature. In the presence of a phonon scatterer with a single avai lable inelastic channel at nw1 = 0.01 eV (Fig. 2.2(c)), the bound state is split into two satellite s, separated by app roxima tely equal energy gaps with respect to the energy of the original bound state. Such "side bands" have been the focus of numerous earlier studies [50, 16, 14, 15, 83, 75, 2, 89, 8]. Note that for the semicond uctor double barrier structures studied here the magnitudes of the energy splits between these phonon assisted satellite peaks are consi derably larger than the weak coupling result, E0 ± 'tlwt. where E0 is the energy of the resonance in the absence of inelastic scattering. This is due to strong renormalization of the bare electron-phonon coupling constant by the confinement of the electron wave function to the small well region, which will be discussed in more detail later on. 10 1.5 (a 1.5 (b) \ (c) \ (d) , (e) \ \ \ - - - Phon on scatterer -- - no phonons \ -- - 1 phonon \ -- - 2 phonons \ , / _-' � .s 1.0 ,//, / / ; i ! ' '-' ';<' > r I I I I I I ' I I I I I I I I 'iii ·.c � 0.5 � 0.5 l I ' ' \ - �� - - -- ,- 0.0 0.0 L__�_L_�_J L_----L-___L�__J '-----'----'--�_JL_----L-___L�__J 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 x (nm) Transmission Figure 2.2: (a) Symmetric double rectangular potential with barrier width 0.4 nm, well width 0.6 nm, and barrier energy V0 = 1. 0 eV. A local phonon scatterer is located at the center of the double barrier. (b-e) Electron tran smission proba bilit ies for various numbers of accessible phonon channel s. (b) cor responds to the case without electron phonon interaction (g = 0), (c) rep resents the transmission coefficient for one phonon channel, (d) for 2 phonon channel s, and (e) for 40 phonon channel s. In (c-e) the electron phonon coupling is set to g = 0.04 eV nm, and the phonon frequency is nw = 0.01 eV. With increasing number of phonon channels one observes the formation of a band. Here, we wish to examine how such phonon assisted satellite features merge into an impurity band with increasing number of availabl e inelastic channel s. The generalized propagation matrix method is particularly suited for this task, as the propagation matrix for the system only increases linearly with the number of added vibrational modes. The pattern which emerges from Fig. 2.2 is that the bound state splits into n + 1 peaks, where n is the number of phonon channels which are excited. For instance, in the case of one excited phonon with energy 0.01 eV and electron-phonon coupling g = 11 0.04eV · nm, we find the peaks to be at positions E1 = 0.2833 eV and E 2 = 0.4340 eV (Fig. 2.2(c)), which differ from the zero-phonon case (Fig. 2.2(b)) by �1 = -0. 0747 eV and � 2 = 0.076 eV. For two phonons (Fig. 2.2(d)), one finds 3 peaks at E1 = 0.22 55 eV, E 2 = 0.3645 eV, and E 3 = 0.4875 eV, giving shifts of �1 = -0.13 25 eV, � 2 = 0.0065 eV and � 3 = 0.1295 eV. This obse rvation points to an interesting even-odd effect, whereby for odd numbers of phonon channel s, there exists a central, non-bonding, peak, whereas for even numbers of phonon channels it is absent. It al so implies that the satellites at En � E0 ± hwn are bonding/anti -bonding pairs. Before investigating this aspect of multi -phonon-assisted resonant tunneling more closely, let us point out that in the limit of many phonon channels (Fig. 2.2(e)) a low-energy band emer ges. Note that the asymmetry in this impurity band is already anticipated in the asymmetry of the satellites for the few-phonon cases [75, 2]. In Fig. 2.3 we exami ne the effects of the electron-phonon coupling and the vibra tional energies on resonant transmission through the same double barrier potential shown in Fig. 2.2(a), i.e. we focus on the low-energy tran smission peaks. The case of one available phonon channel is studied in Figs. 2.3( a) and 2.3(b), and the case of two phonon channel s is illustrated in Figs. 2.3(c) and 2.3(d). Let us first keep the electron-phonon coupling constant fixed at g = 0. 04e V · nm, and vary the energy of the vibrational level s. As observed in Figs. 2.3( a) and 2.3(c), increasing values of nw cause the entire spectrum of transmission resonances to shift to higher ener gies, whereas the gaps between the peaks remain constant. If instead we keep the vibrational energies fixed at nw = 0.01 eV, and vary g, the gaps between peaks are found to increase as g increases (see Fig. 2.4). Note that for the case of even numbers of phonon channels the central non-bonding peak does not shift with increasing g, whereas as the bond ing/antib onding peaks move to higher and lower energies respectively. 12 - 11m · O.OleV (b ) -- - 11m=0.02eV - - 11ro - 0.03 eV -·- 1lm•0.03eV 0.45 Energy( e V) - g=0.02eVnm --- g=0.04eVnm ·-·- g=0.06eVnm - g=0.02eVnm g=0.04eVnm ·-- g=0.06eVnm Figure 2.3 : Electron transmission resonances for a double rectangular potential barrier with a phonon scatterer located at the center. The parameters are chosen identical to Fig. 2.2, unless otherwise specified. (a) and (b) cor respond to a system with one phonon channel, and (c), (d) to a system with two phonon channel s. In (a) and (c) the electron phonon scattering strength is kept constant at g = 0.04 eV nm, and the phonon energy nw is varied. One observes that the entire spectrum shifts to larger values of energy as nw increases. In (b) and (d) nw = 0.01 eV and g is varied. In this case the gaps between the tran smission resonance peaks widen with with increasing g. For two phonon channel s, the central peak does not shift as g is varied. The observation of bound state energy splitting when there are Einstein phonon channels in the system is ana logous to degeneracy breaking in the linear Stark effect. For suffi ciently small electron-phonon coupli ng, we can compute the first order energy shift quantitatively by treating the phonon energy and electron-phonon interaction terms in the Hamiltonian (Eq. 2. 1) as perturbati ons and by using time-independent degenerate perturbati on theory to calculate the resulting energy shif ts. The unperturbed eigenstates are denoted by lx , n ) , where n is the phonon quantum number and x denotes the electron 13 0.4 .---------,--------,-------,------,-------,----------, 0.3 (}· -o 1 phonon a· -o 2 phonons .0 > � 0.2 . . - ----- <l . - - - - · · . - - -- 0.1 . . - · · 0" . . . . . . . . . 0.0 L__ __ ____.L ___ __L_ ___ _J_ ___ ..J.__ __ _ L__ __ __j 0.02 0.04 0.06 g (eV nm) Figure 2.4: Gap (in eV) between bonding and antibonding peaks for different values of electron-phonon coupling g (in eV·nm), for 1 phonon channel (black circl es) and 2 phonon channels (red square s). position in the well region of the potential profile x E [ 0, L ]. In order to make analytical progre ss, the electron wave function is app roximated by the infinite well wavef unction \]1 ( x) � sin( 1rx I L) I VL, where L is the length of the well. The resulting perturbation matrix has elements (x, nll: x; n.wbLbx; + g l: x;,k,k' J (x - xi ) (bL + bx;) ctck'lx , n) , which for the case of one phonon channel yields the 2 x 2 perturbati on matrix p = ( 0 giL ) . giL li.w (2.5) 14 Assuming that the impurity site is located at x0 = L /2, we have sin( 1rx0j L) = 1. The energy shifts are calculated by diagonalization of the perturbati on matrix and are given by (2.6) In practi ce, the well width L can be made rather small, even compared to the scale of molecular junct ions. This can lead to a significant renormalization of the electron phonon coupling constant, g ----'t g / L, which in turn explai ns the the relatively large energy gaps between the phonon assisted satellit es, observed in the propagation matrix results. To further illustrate how the bound state energies depend on the phonon energy and the electron-phonon coupli ng, we plot the lower and higher bound state ener gies as a function of hw (Figs. 2.5(a) and 2.5(c)), and as a function of g (Figs. 2.5(b) and 2.5(d)). The accuracy of the effective theory compared to the numerical results of the full propagation matrix calculation is striking, in particular for predicting the higher bound state energy (Figs. 2.5(a) and 2.5(b)). For the lower bound state, the accuracy increases for smaller g (Fig. 2.5 (d)), although for a fixed value of g = 0.04e V · nm and variable hw the effective theory predicts a lower bound state off- set by about 0.02 eV with respect to the full propagation matrix calculation (Fig. 2.5 (c)). This diff erence is of the order of the phonon energy and one order of magnitude lower than the bound state energy. However we notice that although the curves for the effe ctive theory and numerical results are off, they present the same qualitative behavior. Therefore, one can affirm that the effective theory repro duces the numerical results with striking accuracy for small values of g. The same procedure can be repeated for any number of phonon channels with similar results. 15 (a) t; H o .60 ! 1 0.50 j o .4 0 - - >' (c) �0.34 r � 0.32 "' ] 0.30 ] 0· 28 .00 010 \l 0.05 . 0.15 Phonon Energy (eV) - Numerical results Effecti ve theory (b) 0.55 0.50 0.45 0.40 (d) 0.35 0.30 0.25 0.20 0. 1 5 0.00 0.02 0.04 0.06 0.08 0.18 . 1 ° Electron-phonon coupling g (eV nm) Figure 2.5: Bound state ener gies in a double rectangular potential barrier with one phonon scatterer at the center (as depicted in Fig. 2.2(a)). For the case of one phonon channel, the energies of the two resulting tran smission resonance peaks are plotted as functi ons of the electron-phonon coupling constant g and of the phonon energy liw. Exact numerical results (solid black lines) are compared with the effective the ory (dashed red lines) described in the text. In (a) and (c) the electron-phonon coupling is kept constant at g = 0.04 eV nm, and in (b) and (d) the phonon energy is kept con stant at liw = 0.01 eV. In (a) and (b), for the higher bound state ener gies, the effective theory reprodu ces remarkably well the numerical results. In (c), the effective theory and numerical results are off- set by app roxima tely 0.02 eV. In (d), the effective theory matches the numerical results for small values of g. For instance, when g = 0.04 eV nm, the effective theory overestimates the lower bound state energy by 0.01 eV. To prove the point that any potential shape can be modeled by this propagation matrix method, we repeat the numer ics for the harmonic oscillator potential V (x) = (x- x0 ) 2 / L 2 , which is built with one thou sand steps with L step = 0.01 nm (Fig. 2.6). The resonant tunneling happens at the harmonic oscillator energies En= liw e (n + 1/ 2), 16 1.0 0.8 � � 0. 6 � ] jl 0. 4 e. 0. 2 (a) - - - - o.o o.o I \) 5.0 10.0 15 .0 Position (nm) 1.0 (b) - 0 .8 - - - 0.2 20.0 °·lb .o 0.2 o.4 o.6 o.8 Transmission 1.0 Figure 2.6: (a) Harmonic os cillator potential V(x). with an electron incoming from the left. The effective mass of the electron is 0.07m (b) Electron energy vs. tran smission for the potential depicted in (a). Note resonant tunneling (unit transmission) for E0 = 0.14 7 eV. E1 = 0.442 eV and E 2 = 0.765 eV. with We given by: Therefore. 8 x 10-3eV/m 14 0.07 . 9.11 x 1Q -31 kg = 4.5 x 10 Hz. Eo = 'fiwe/2 = 0.147eV, E1 = 3'fiwe/2 = 0.442eV, E 2 = 5'fiw e/2 = 0.735eV. (2.7) (2.8) (2.9) (2.10) 17 The energy E 2 does not agree exactly with the resonant tunneling energy since the poten tial is finite. If we add a phonon scatterer at the center of the potential (x0 = 10 nm), we get features which are similar to the double square barrier, namely the sidebands due to the electron-phonon coupling (see Fig. 2.7). 1.0 (a) 1.0 (b) 0.8 0.8 - No phonons ',,_ --- 1 phonon ................. ,... ' l ' � � 0.6 R :a >0 .6 � eil 0 � 0.4 p., &] 0.4 0.2 0 ·0 . L- o ��-�� ---:-' 20.o 0 ·0 .o 8 lJ 5.0 10.0 15.0 lJ 0.4 0.6 0. 1.0 Position (mn) Transmission Figure 2.7: (a) Harmonic os cillator potential V(x), with a phonon scatterer site at x = 10 nm. The electron phonon coupling is g = 0.2 eV·nm, and the phonon energy is nw = 0.01eV. (b) Red-dashed line: Electron energy vs. tran smission for the potential depicted in (a), with electron-phonon interacti on, for one phonon channel ava ilable. Note the splitting for the ground state energy E(j = 0.2031 eV, E0 = 0.0842 eV. The first excited level does not split and the tran smission for the third excited level is suppre ssed. The position of the E(j and E0 peaks can be predicted by the same method we pro posed for the double square barrier. Taking the harmonic os cillator wave function for the ground state: (2. 11) 18 Writing a matrix with elements x i) (bL + bx i) cl ck'l x , n), we get for the diagonal elemen ts: nnw J ?j;t (x)?j;(x)dx nnw, for n = 0, 1 and assuming the wavefunction is normali zed. The off- diagonal elements will be: where the term 1'1/J (x o ) 1 2 is simply : The secular equation is therefor e: with solution: (2. 12) = 0, (2. 14) (2. 15) Plu gging in the valu es of g, A0 and nw in this equation we get >. + = 0.064 eV, and ).._ = �0.0 54 eV. Our simulation shows that the new ener gies resulting from the electron-phonon interaction are shifted from the original value (no electron-phonon interaction) by: � + = 0.20 31 � 0.147 = 0.056eV, (2. 16) � - = 0.0842 � 0.147 = 0.063eV, (2. 17) which differ from the effective theory by a value of the order of the phonon energy. 19 Finally, let us turn to the current-voltage charact eristics caused by phonon assi sted tran smission features. As shown in Fig. 2.8(a), application of an external electric field yields a spatial gradient in the potential energy profile. The resulting current flow is deter mined from the transmission functions T(E, V) at given voltage biases V via an integral rV o I(V) = J o T(E, V)dE, (2. 18) where the energy window [0, Vol for currents through semiconductor heterostructures is small compared to molecular junct ions. As a result of thi s, the I (V) dependence shown in Fig. 2.8(c) inherits the peak structure of the individual transmission curve s, some of which are shown in Fig. 2.8(b). This is an important difference from the step-like I(V) curves reported in measurements of molecular juncti ons [65, 10]. 2.4 Conclusion In summary, we have investigated how quantum well electron-phonon resonances are affected by the presence of several inelastic channels in the phonon spectrum. Using a generalized propagation matrix method for multipl e eleastic and inelastic tran smission channel s, we determined the highly non-perturbative effe cts of scattering by Einstein phonons on the electron transmission through potential structures. In particular, we observed a characteristic splitting of the bound state resonances into satellite peaks. The presence or absence of a non-bonding resonance reflects the parity associated with even vs. odd numbers of accessible inelastic channel s. Furth ermore, in the limit of many availabl e channel s, the formation of low-energy impurity bands was observed. The dependence of the resonance satellites on the electron-phonon coupling strength and on the phonon energies could be reproduced using an effe ctive model, which works well 20 1 .5 (a) I I 1.5 (b) (c) - Vb=O.O eV - Vb=0 .2eV 1 .0 I- - 3.6 � 1.0 " i � j � 0.5 1- - ! 0.5 0.0 1-- - � I I '--- I I o .'b. o 0.0 5.0 10.0 0.5 0.03 x (nm) Transmission Figure 2.8: (a) Potential double barrier with thick ness of 1. 0 nm and separation of 5.0 nm. The height of the barrier is 1.0 eV, and the electronic effe ctive mass is 0.07me. (b) Tran smission vs. energy curves for one excited phonon at the center of the potential well. In the presence of a voltage bias across the heterostructure, this curve is shifted towards lower ener gies. (c) Cur rent-voltage curve for the double barrier in (a). The two small peaks at low voltage bias (V b) correspond to the two low energy peaks in (b). The calculation of current is done by integrating the transmission over an energy window from 0 eV to 50 meV. within the limits of perturbati on theory. One promi se of the mul ti-channel propagation matrix method which is developed here lies in the ability to study highly asymmetric quantum systems with strongly interacting itinerant and local features. A further direc tion to pursue is to depart from strictly local oscillato rs, which are nevert heless important for nanoelectro nics, and to consider spatially extended phonon scattering regions. 21 Chapter 3 Josephson junction arrays 3.1 Introduction In the previous chapter we studied some aspects of phonon-assisted one-dimensional electronic transport in semiconductors. Now we turn to the numerical study of the physical properti es of a different system. Namely we investigate phase tran sitions and transport in two-dimensional latti ces of superconducting islands separated by a thin insulating or metallic layer. These arrays of Josephson junct ions, as they are commonly call ed, have been studied extensively in the last 30 years since they serve as test-beds for effe cts of frustration, disorder, and chaos in two dimensional latti ces [6, 20, 76, 56, 38, 30, 39, 88]. More recently, there is a renewed interest in Josephson juncti ons as they can realize artificial two-level systems and are therefore used in the fabrication of superconducting qubits [73]. In our work we investigate the effects of dilution and fru stration on the physical properti es of a Josephson junction array. These systems are interesting because disorder can be introduced in a controlled way. Indeed, since Josephson junction arrays are lithographically fabricated, any geometry is in principle possible, in particular that of a diluted latti ce. This makes our observations a priori accessible to experimental tests. Our results are shown in the next chapters. Here we present a conci se review of the concepts we will explore further on. We start by giving a brief introduction to Josephson juncti ons and the de and ac Josephson eff ects. Then, we discuss some basic Josephson junction array physics. 22 3.1.1 Josephson Eff ects Josephson junctions consist of coupled superconducting materials separated by a thin layer of a metal or insu lator. Even in the absence of an external potential diffe rence, proximity effects will cause a current of Cooper pairs (also known as a supercurrent) to tunnel from one superconductor to the other spontaneously. This supercurrent is proportional to the sine of the phase difference of the mac roscopic Ginzburg-landau wavef unction, \If 1 (2) = I \If 1 (2) I e i 1> 1 c2> , on each side (the superconductors 1 and 2) of the junct ion: (3 .1) Equation 3.1 is known as the de Josephson effect. The ac Josephson effect occurs in the presence of a bias voltage V, where the phase difference /:1¢ = ¢1 - ¢ 2 varies in time as: d/:1¢ 2eV dt n ' (3.2) generating an alternati ng current varying with frequency v = 2e V /h. Note that unlike normal conductors, the voltage is responsible for the frequency of the current, and not for the current itself. These effects are named after the physicist who first predicted such phenomena, B. D. Josephson, in 1962 [49]. From these effe cts we can determine the free energy of one junction [78], F J I s Vdt, or, (3.3) where J is the Josephson coupling constant, given by ni c/2e. 23 3.2 Classical Josephson junction arrays: The two dimensional XY Model Josephson juncti ons can be arranged in a regular two-dimensional proximity-coupled array of mesoscopic superconducting islands. If the islands are sufficiently large and if the temperature of the whole system is well below the superconducting temperature of the islands, then each of them has a well defined superconducting phase ¢, which fluctuates slowly in time. Hence one can treat this phase variabl e as a classical variable, and from the Josephson coupling between two such islands one recovers the ph ysics of the ferromagnetic XY model [43, 44, 45, 61, 27]: H =- L Jij cos(¢i - ¢j), ( i,j ) (3.4) where i, j are nearest neighboring sites on the array and Jij is the coupling constant between two neighboring islands. Since Josephson junction arrays (JJA) can be used to realize magnetic models, at times through this thesis the phases will be referred to as two dimensional vectors §i with components (cos ¢i, sin ¢i), in analogy to spins in a classical two-dimensional XY model. In addition to the phases, another relevant degree of freedom of JJAs is the chirality or vorticity K: K =s ign (�s: X Sj) , �---> J (3.5) here the sum is closed in a plaquette, summing from site i to site j. The vorticity is therefore a discrete degree of freedom assuming values ± 1 associated to each plaquette in the array (Fig. 3.1). Physically, this cor responds to the direction of supercurrent circulation around a plaquette (clockwi se or anti -clockwi se). For the square array where 24 (a) \-----{ .a.i + : � ' ' ' ' ' ' ' ' ' ' : \J )/------- � (b) �-----{ � ! - ! ... /-----� Figure 3.1: (a) A positive vortex. (b) A negative vortex. the indexes 1, 2, 3, 4 cor respond to the 4 sites in a square plaquette, Eq. 3.5 can be written as: K, = sign (sin( ch - (h) + sin( ¢3 - ¢ 2 ) + sin( ¢4 - ¢3) + sin( ¢1 - ¢4)) . (3.6) At zero temperature, if there is no external magnetic field applied to the array, all phases are aligned and the array 's energy is at a minimum (no currents flow through the system). As temperature is increased, phase fluctuati ons become stronger and vortex antivortex excitati ons can be thermally generated. At a certain value of T the vortex- antivortex pairs start to unbind, leading to the loss of quasi-long-range order in the sys- tern. This phase tran sition, driven by the unbinding of vortex-antivortex pairs in the latti ce, was first explained by Berezinskii and independently by Kosterlitz and Thou less in the early 1970's [5, 51, 52]. At the time there was experimental evidence of a phase transition in models with 0(2) symm etries, but it was known that such sys tems do not present spontaneous symmetry breaking as Goldstone modes destroy any long-range order that appears at zero temperature [63]. This topological phase tran si tion, known as a Berezinskii-Kosterlitz- Thouless transition (although in the literature it is more commonly referred to as simply Kosterlitz- Thou less, or KT transition) stems 25 from the unbinding of topological defects (vortex-antivortex pairs) which, like charges of different signs in two dimensions, interact with a loga rithmic potential [64]. Experimentally, T KT can be deter mined by investigating the I-V characteristics of the array. The temperature at which the array 's resistance goes from zero to a finite value corre sponds to the unbinding of vortex-antivortex pairs [7 1]. Numerically, one can sim ulate the I-V characteristics by solving Langevin equati ons for the phase dynamics with a noise term cor responding to the temperature. Alternatively, as proposed by Minnhagen [64], one can determine at which temperature there is a universal jum p in the helicity modulus of the array. This obse rvable is al so known as the superftuid density of the superconducting network, or, for magnetic systems, the spin stiff ness. Physically the helicity modulus is a measure of the array 's resistance to position-dependent twists in the phase variabl e. It is intr insically related to the system's transport propert ies, as it corre sponds to the frac tion of Cooper pairs that carries supercurrent in the array. Math- ematically, this quantity is related to the free energy of the JJA as: (3.7) Here the vector k is the twist vector. The estimator for the helicity modulus of the classical XY model is found by applying a twist to the Hamiltonian (Eq. 3.4). The uniform twist in the phase is given by: ¢i ---r rPi + k · fi . After applying the twist, the Hamiltonian reads: H[ k ] = 2:::: Jijcos(¢i - ¢j + k · (fi - r j)) . (3.8) < i,j > 26 The free energy per volume of the system is: (3.9) where j3 = 1/kBT, and we set kB = 1. Applying Eq. 3.9 in Eq. 3.7, we find the following expre ssion for the helicity modulus: I (T) � ·� (R) - � \ ( � l;; sin(</> ; - <l>; )(t, · ci;))) , (3.10) where (E) = j L Jij cos(¢i- ¢ j)) . \ ( ij } Yet another way to numerically determine T Kr is by investigating the behavior of some observabl es at the phase transition. For instance, the structure factor, a quantity related to the order of the system and measured experimentally by neutron scattering, is defined as the Fourier transform of the correlation funct ion: S( qk ) = L exp( - i qk · fi)r( ri), (3 .11) and in the continuous limit, where we can observe the long- wavelength properti es of the system, Eq. 3.11 takes the form: S(Cj) = j drexp( -iq . f)r( r), (3.12) and the phase correlation function is r( r) = (Si · Sj ) , with r equal to the distance between sites i and j: lf i - fjl = r. The correlation function decays as a power law in rat TKr: 27 (3.13) where Q is the pitch vector. For the two-dimensional XY model (d = 2), renormalization group calculati ons give 71 = 1/4. Applying Eq. 3.13 to Eq. 3.12 we find that the structure factor at Tx r is proportional to the correlation length � as: (3.14) In the next chapter we use this scaling law of the structure factor at the phase tran si tion to find the Txr for a fru strated and diluted JJA. At the phase tran sition, the correla tion length of the infinite system �oo dive rges to infinity exponentially: �oo(T) = � o exp ( J T - a Txr ) . (3.15) However, in our numerical calculati ons we are limited to finite system sizes. Therefore our estimate of Txr requires correcti ons for finit e-size systems. A finite system becomes critical around T*(L). At the phase tran sition, when � saturates at L, �oo(T* (L)) rv L and Eq. 3.14 becomes: (3.16) at a temperature T*(L). From Eq. 3.15 and from �00(T* (L)) = AL, we find that the correct ions to Txr are logarithmic with the square of the system size L: (3.17) with L0 = �0/A. 28 (a) (b) \ \ \ \ \ � � I \ �� I � ' I I ' I ' I ' \ \ \ \ \ ' " I � " " -- -- ...... , I I " ' -- .... \ ------""-, I ' � ' � ' \ ' " ' " - \ ! " '• ' I I � " � \ " " ...... - .... - � , ' I ' " \ \ \ ,.. -"' ....._ j( � ' ' -I I I � - ' ' ' \ \ I ' I I ' " ' ' __ ... __ _ \ ' I I $ I \ \ \ \ \ , I I _, , - - - - " """ .;:: - - I \ \ \ \ \ \ 'I � ;' \ -� ' ..... ;of + ...- ;' ' I ' I I 1 ..... ....-- � " ;' � I I ' I I I I \ I �- I I I I I I I I I I , \ \ I \ \ \ \ \ \ I I \ 'I � I I I '.¥ - - " $ I ' I I \ I \ \ I j \ I I I I , I -I I I \ , I >' I , I I I � -, I " � � $ I " ' \ I ' \ I ' ' ' ' I I ' ' I ' � ' ' ' I , , ' � - \ \ I - ' \ ' I \ \ Figure 3.2: (a) Configuration of phases near the ground state of XY model. (b) Array above critical temperature T K T· 3.3 Frustrated Josephson junction arrays In Josep hson's seminal paper, the effect of an external magnetic field on a supercon- ducting tunnel junction is al so considered [ 49]. The tunneling supercurrent is highly sensitive to the presence of external fields, and to include these effects the phase differ ence !::,.¢ = ¢1 - ¢ 2 has to be written in its gauge-in variant form, r: 2e 1 2 � � 1 = T1 - T 2 - /l 1 A· dl, (3.18) where A is the vector potential of the external magnetic field and the integral is evaluated ac ross the junction. The de Josephson effect is now expr essed as: (3.19) 29 This sensitivity to external magnetic fields make Josephson junctions a very conve nient device for measurement of fields. In fact, such measurement devices, SQUIDs (Superconducting QUantum Interference Devices) are made of a loop of Josephson junct ions. Their critical current is modulated by the strength of the flux of the exter- nal field. When a magnetic field H is applied perpendicularly to an array of Josephson junc tions it enters each plaquette through a flux: 2e f __, __, ID: <I> - A · dl = A·· = 21r� = 21rj n ZJ <I> , i�j 0 (3.20) where Aj is the line integral along the junction between sites i and j, and <!:>0 = h/2e is the superconductor flux quantum. The dimension less value f cor responds to the fru s tration of the array, which has mod (1) value. Ty pical values of f in the literature are commensurate to the latt ice, taking the form f = pjq, where p and q are integers, and q is the size of the unit cell [76]. When f = 0 there is no fru stration, and in a square array, for a value of fru stration f = 1/2 the array is said to be fully frustrated. A convenient choice of gauge makes cor respondence to a square lattice of spins whose plaquettes have three ferromagnetic coupli ngs and one antifer romagnetic coupling (see Fig. 3.3). Using the gauge invariant phase difference (Eq. 3.18 ), the Hami ltonian for an array of Josephson junctions in the presence of an appl ied magnetic field is: H = - 2::: Jij cos((h - ¢j - Aij) . ( i , j ) (3.21) The term Aij from Eq. 3.20 enters the hamiltonian (Eq. 3.21) as an effective cou- piing between the phases, inducing fru stration. For the values of Aij in Fig. 3.3, we recover Eq. 3.4 for the bonds where Aj = 0. For the bond with Aij = 1r, we have an antifer romagnetic XY coupli ng, with Jij = -1 . 30 I A ;,� O 1 A-=0 ! iA -=7r ,, l A,; � 0 I ? , , Figure 3.3: A fully fru strated square plaquette with three ferromagnetic couplings (Jij = J) and one antifer romagnetic (eff ectively Jij = - J in Eq. 3.4). Note that 2:::: Aij = 21r f = 1r around the plaquette. For the XY model at zero temperature, the fully fru strated plaquette arran ges into the Villain lattice configuration (Fig. 3.4). In the special case of the fully fru strated square array there is an extra symmetry that the system acquires in the ground state. Namely, fru stration forces the phases to arrange in vortex pairs down to low temperatu res, and at T = 0 the vorti ces are positioned in a checkerboard pattern, showing a Z 2 symmetry analogous to a two dimensional antifer romagnetic Ising latti ce. In 1977, Villain [79] determined analytically the ground state of a square fully fru strated XY spin latt ice. He found that three of the phases around a plaquette in the ground state can be written as a function of a fourth phase ¢. For the plaquette in Fig. 3.4: cp- T7rj 4 cp + T7rj4 cp- T7r/2, (3.22) (3.23) (3.24) (3.25) 31 (a) + - 2 Figure 3.4: (a) One plaquette of the Villain 4-sublattice order. (b) Unit cell of Villain 4-sublattice order, with two vortex pairs at zero temperature. with T = ± 1. This solution is now known as the Villain 4-sublattice order 1 and it corre sponds to the exact ground state of the fru strated XY model. As a consequence of this extra symmetry, one expects the fully fru strated JJA to have another phase tran sition, in addition to the KT tran sition: At a finite temperature Tc, the array undergoes a breaking of the Z 2 (Ising) symmetry. There are several observabl es from which we can get an estimate of Tc, such as the net magnetization and the specific heat. In the next chapter our approach to find the Ising transition temperature of fru strated and diluted JJA is ana logous to the method used for the KT tran sition. Namely we calculate numerically the chiral structure factor Sc(ij), defined as the Fourier transform of a chiral correlation funct ion: (3.26) 1 Some authors [70] refer to this ground state configuration as simply the Villain lattice, which can be somewhat misleading since this nomenclature is related to the order that the degrees of freedom acquire in the ground state, and not to the lattice geometry. 32 and in the continuous limit: Sc(i/) = j drexp(-iq · r)x(r) , (3.27) where the two-point chiral correlation function is x(r) = \K,i • K,j J, with K, given by Eq. 3.5, and r is the distance between two plaquettes i and j. At the critical point x(r) also decays as ljrd -2+ !J. Coincidentally, 77 = 1/ 4 al so for the two-dimensional Ising model. Similarly to the analysis we carried out for the phase structure factor, we have that at the Ising transition Sc(iiJ is proportional to the correlation length: (3.28) Considering finit e-size eff ects, just as Eq. 3. 16, we get Sc(Q) ex L 2- !J at T c *(L). Finite size correct ions in Tc are found by exami ning the divergence of the correlation length � at the phase transition: (3.29) In the two-dimensional Ising model, v = 1. The finite-size correcti ons in r; ( L), where �co(T ;( L)) = AL are : (3.30) The following subsection is dedicated to a short review of phase tran sitions in this model. 33 3.3.1 Phase transitions in fully frustrated square Josephson junc- tion arrays In 19 82, Voss et al. [80] pub lished their results on the Kosterlitz- Tho uless transition in Josephson junction arrays under a magnetic field, where they observed a suppre ssion in TKT· Following this result, the first numerical effort to find the suppre ssion of TKT in fru strated Josephson junction arrays was carried out by Teitel et al. [77]. Their results showed that TKT for the fru strated XY model is about one half of TKT for the unfru s trated case. In order to find this result, they ran a Monte Carlo simulation to compute the helicity modulus of the fru strated array. The Ising transition was deter mined by looking at the divergence of the specific heat at Tc. They found Tc to be very close to TKr, but their calculat ions for small system sizes were not precise enough to determine whether both phase tran sitions occur at the same temperature or whether Tc was slightly above TKT· In fact, this question remained unanswered for many years. Early on, numerical simulations led some authors to argue that the fru strated XY model was in fact a realiza tion of a coupled XY-Ising Hamiltonian with a unique transition at a critical temperature Tc, belonging to an XY-I sing universality class [40, 41, 54]. Meanwhile, others argued that these were two separate transiti ons, Kosterlitz- Thou less and Ising transitions, which occur at temperatures very close to each other, but essentially distinct [55, 69, 70]. The ambiguity existed beca use early numerical simulations were unable to distinguish with accuracy the values of TK T and Tc. Only recently, after the advent of more reliable simulations on faster computer s, there is consensus that the two tran sitions are in fact distinct, with Tc > TKr, meaning that as the system goes from a superconducting to a normal phase, vortex order remains up to a slightly higher temperature [68]. The current best estimates for the transition temperatures are Tc = 0.45324(1) and TKT = 0.442(2), from Ref. [68]. 34 / / / / / / / / / / / I I / / / I I (a) / / / / / / / I / / / / / / / / / / / (c) / / / / / / / / / I I / / I I / / / / I / / / I / / / / I / / / / I � " I I I !. ••••• .! L� J ·--· r--� 1 / 1 1/ 1 r 1 " I I I I I I ......• ..... .. ... ... . .. : / / / / / I I I r••• :...--- / ...--- : : / / I I 1.--i ·· · .. - .. I / : I I 1/ / / I / / I I . ----- -· I I I I 1 / / I / I / I / I I I I / I I . .. .. . r-----� !. •• - -- . . I I --- _....,.. I / I I I : / / -- . . r · · " ·-- I -- .. I I - I . - . . I I I ...._ -- -· 1- 1 I / I - I I ---. I I / / I I I I I I / I I (b) I I I I I I I I I Figure 3.5: (a) Ground state of a 10% diluted lattice of square Josephson junct ions. (b) Just below the lattice percolation threshold, at 35% diluti on. (c) Above the percolation threshold, with 45% of diluted sites. The lines are guides to show the different clusters that form in the latt ice. The fact that the phases align at different angles for each cluster shows that the clusters are non-interac ting. 3.4 Diluted Josephson junction arrays Another aspect of Josephson junction array s that has been studied extensively since the 1980's is the effect of disorder on phase tran sitions and dynamical propert ies. Disorder can be realized in several ways: By assuming random values of the coupling constants Jij or flux Aij, one can recover the properti es of glassy systems (gauge glass) [1]. Such a form of disorder can be implemented by slightly disloca ting islands on the grid (posi tional disorder) [18, 21, 30, 29, 86]. A different form of disorder, which does not lead 35 to random gauge fields, consists of randomly diluting superconducting islands or indi vidual juncti ons [85, 87, 88]. In this section and the next chapters we will focus on site-diluted Josephson junction arra ys. When studying the physical properti es of a diluted Josephson junction array a ques tion one may ask is: How does percolative disorder affect the phase tran sition? One of the first experiments to try to answer this question was carried out by Harris et al . [46], who studied site dilution on a square array of Nb superconducting islands in zero magnetic field and found that random percolative disorder suppre sses the transition tem perature down to the percolation thre shold of the latti ce, when the transition ceases to exist. Also, they observed that the transition appeared to belong to the Kosterlitz Tho uless universality class independently of the strength of the percolative disorder. Subsequent experiments and numerical simulations by other research groups supported this evidence [25, 88]. From Fig. 3.5 we see that dilution destroys quasi-long-range order in the ground state above the percolation threshold, since the system forms several non-interacting cluste rs. In the next chapter we will study phase tran sitions in Josephson junction array s which are fru strated and diluted. Namely, we investigate how site dilution affects the Ising and Kosterlitz-Thouless transition temperatures as it app roaches the lattice perco lation threshold. 3.5 Quantum Effects The previous sections were focused on the classical limit of proximity coupled Joseph son junction array s; we assumed mesoscopic superconducting islands for which capac itive effects could be ignored, and the superconducting phase was the only degree of freedom of the system (for f = 0). However, if the superconducting grains are made 36 small enough, the number of Cooper pairs in a site i of the array is a relevant degree of freedom and the energy to add a Cooper pair to an island becomes an important energy scale of the system. In this case, capacitive effects must be included in the Hamiltonian as an on-site charging energy [78]: H = -EJ L cos(¢i - ¢j) + � c L n 7. «� i (3.31) By this point, we are already familiar with the first term in Eq. 3.31, which corre sponds to the coupling energy E1 between neighboring islands. The second term, the charging energy, is the energy related to adding or removing Cooper pairs from a super conducting site. Writing the phases ¢i and the number of Cooper pairs in a given island ni as operator s, we find that they are conjugate variabl es that obey the commutation relation [¢i, nj] = ibij · The quantum operator form of Eq. 3.31 is equivalent to the two-dimensional quantum rotor model Hamiltonian [27, 72]: (3.32) At zero temperature, in the limit where quantum fluctuati ons drive the phase tran- sitions [74], the physical properti es of the system are characteri zed by the interplay between E1 and E0. There are two limits of interest for the charging and coupling energ ies. If E c » E 1, Cooper pairs are confined to the islands, the Josephson current Is ceases to flow, transfor ming the array into a Mott insu lator. In the opposite limit, E1 » Ec, phases are well-defined and Cooper pairs can freely move from island to island, making the array superconducting [27]. 37 Chapter 4 Phase Transitions in Diluted and Frustrated Josephson Junction Arrays 4.1 Introduction In the previous chapter we have shown the different models of classical Josephson junc tion arrays that have been extensively studied in the literature. However, combinations of the effects of fru stration and disorder have not been explored as often. The first work of this nature was done by Zeng et al. [88], in which they studied the effect of per colative disorder in unfrustrated and fru strated array s. Their Monte Carlo simulation data for the unfrustrated square array s with site dilution were shown to match experi mental results by Harris et al. [ 46] , but their results for the fru strated and diluted JJA were largely inconclusive. At the time, the prevailing idea regarding phase tran sitions in fru strated array s was that they belonged to an XY-Ising universality class with a single transition temperature at Tc. The Monte Carlo calculati ons carried out in [88] could not confirm this as their results appeared to show only a KT-like transition and no Ising-like tran sitions for the diluted and fru strated JJA. The authors admitted that their data might not have completely equilibrated since the combination of these two effects seemed to increase considerably the relaxation times. Therefore they claimed that Monte Carlo simulations longer than the ones that they reported in their work would be necessary in order to obtain more accurate information about this model. 38 The question on the effect of percolative disorder in fru strated systems remained open for several years. In 1998. Benakli et al. [3] revived the discussion by suggesting a conjectured phase diagram of bond-diluted and frustrated Josephson junction array s. Their motivati on was the study of a two-dimensional vortex glass phase which appears at zero temperature in disordered systems. The choice to use diluted and fru strated JJA was appropriate since these systems can be constructed and investigated in a control- !able fa shion. They propo sed that as the fru strated array is diluted. the KT and Ising tran sitions are suppressed at different rate s. Since dilution affects phase coherence more than it affects vortex order. the KT transition is completely suppressed at dilution dKr while vortex order still exists up to a dilution dv L · Both dKr and dv L are well-below the lattice percolation threshold d P and a vortex -glass phase at zero temperature emer ges at dilut ions d in the interval dv L < d < d P . All their calculati ons were done at zero temperature. and by extending the dilution dependence of the phase tran sitions to finite temperature s. they conjectured the phase diagram in Fig. 4.1. T N VG d Figure 4.1: Conjectured phase diagram from Ref. [3] for a bond-diluted and fru strated triangular Josephson junction array. The phases are superconducting (SC). normal (N). and vortex glass (VG). Note that the authors considered that both phase tran sitions occur at a single temperature at zero and low diluti on. contrary to current knowledge of phase tran sitions in such systems. which occur at close but distinct temperatures at zero dilu tion [68]. 39 In our work, we addre ss quantitatively the phase diagram of fru strated and diluted classical Josephson junction array s. In particular, we were motivated to answer the questions raised by the studies prev iously cited, namely whether the percolative dis order changes the nature of the phase transition in fru strated arrays [88] and whether we can numerically deter mine the critical valu es of dilution dKT and dvL for the fru s trated and diluted JJA at finite temperature. We conduct a careful Monte Carlo study of site-diluted and fully fru strated Josephson junction arrays at various temperatures and dilut ions. From these data, we then construct a phase diagram confirming Fig. 4.1, showing that indeed the character of the Kosterlitz- Thouless and Ising tran sitions are unchanged as d increases, and that phase coherence and vortex order dis appear below the lattice percolation thre shold at d P = 0.407. In the next sections we explain the model and computational methods, then, from a study of the observab les discussed in the pre vious chapter we extract the KT and Ising transition temperatures for the site-diluted and fully fru strated Josephson junction array. 4.2 Model The model we study consists of a square proximity-coupled Josephson junction array with site dilution in the pre sence of a perpendicular magnetic field. The cor responding hamiltonian is: H = - L Jij cos(¢i - ¢j - Aij). < ij > (4. 1) Since we consider site dilut ion, Jij will be equal to zero if either site i or site j (or both) are vacant. Otherwise, Jij is equal to the coupling constant J, which is the same for all junct ions. To account for the external magnetic flux, we choose the Aij such that 40 the lattice is fully fru strated (! = 1/2 and ID:: Aij = 1r). Sites are removed randomly with probability d. At low dilut ion, the ground state will resemble the Villain 4-su perlattice order at regions far from vacan cies, but near vacanci es the phases will arran ge into a low-energy configuration which depends on their positions relative to the vacancy. If the array has a high percentage of diluted sites, however, the ground state resembles a glass, where phases inside a cluster will be correlated but the correlation between clusters is very small, meaning that the phases are ordered locally but not globally (see Fig. 4.2). 4.3 Computational Methods We compute all observabl es for arrays with free boundary condit ions and sizes L = 16, 24 and 32. Even though our array is fully fru strated, the parity of the system size does not matter because of the choice in boundary condit ions. To simulate disorder by site dilut ion, we prepare between 200 to 400 disorder realizati ons with sites removed randomly with probability d. All obs ervables are then aver aged over disorder realiza tions. In order to find the ground state of the diluted and fru strated array s, the lattice is annealed by means of a Monte Carlo method. Annealing is achieved by starting the sys tem in a high-tem perature disordered state and slowly cooling down to the ground state. We implement the standard Metropolis update for annealing. However, when used by itself this method is not very effi cient due to critical slowing down at the phase tran si tion [44]. Therefore, we add to our Monte Carlo simulation an overrelaxation method: a microcanonical update which flips the component of each spin perpendicular to its local field, therefore not changing the total energy of the array. We al so implement a Wolff cluster algorithm [82], which is found to speed up the simulation consi derably despite the fact that it is known that the Wolff cluster algorithm does not work for fru strated 41 (a) ' f ' I ' I ' I ' I ' I � I ' I ' ' I ' I ' I ' ' ' ' I ' (b) - � - � - � - � - � - ' - ' - ' � � - � - ' � # � , - I ' I ' I ' I ' f ' f ' I ' I ' I ' I ' I ' f ' ' ' f ' I ' I \ - � # - # - � � - � - ' - ' - ' ' _, - ' _ , - ' - ' - � - ' - ' f ' \ ! \ ' I ' f ' f ' f ' f ! ' I ' I ' ! ' I ' I ' I ' I - � - � - - - � # � - � - � - � ' - ' _, _, - ' _, - � - ! ' \ ' \ ! f ' \ ! \ ! \ ' \ ' \ I ' I ' I ' f ' I ' I ' I ' \ ' � - � # � - � -- # � - ' - � ' - ' _, - ' - ' - ' - ' � ' ! \ ' I ! \ ' I ' I ' I ' I ' I ' ! ' I ' I ' I ' I ' I ' I ' - � - � ' - - ' - � - � -, - � ' _, _, - ' - ' -! - ' � � ' f ' I ! I ' I ' I ' I ' f ' f \ ' I ' I \ I \ I ' f ' I ' I ' - ' - - - ... - ' - ' - ' - ' - � ' - ' - ' -- -- - " - ' - ! ! - ' I � I ' I ' I ' I ' I ' I ' I I ' I ' I ' ' I ' I ' I I ' - ' - ' - ' - , - , - ' - ' - ' ' _, - ' - # - # -, _, - ' - ' I ' I ' I ' I ' I ' I � I ' I ' - - ! ' ! ' I ' I ' ! ' I ' - � - ... - � - ... - ... - ' - ' - ' _, -' - ' - ' _, _, - ' - (c) ' ' ' ' ' I I ' I ' ' I I ' � ' ' I I (d) ' , __ ' \ ' ' - \- - � ' \ I ' I ' - ' ' - - - -- ' - -- ' '-- � - ' - ' \ I ' I ' ! \ ' I I I ' I - ' ' ' ' ' I ' ' - - ' - - - ' - - ' ' I ! I I ' ' ' \ ! ! ' I ' I ' ! I ! ' \ - ' ' ' I ' - ' ' - ' ' \ I I I ' \ \ - I ' ! I I ' I ' ' I ' ' ' ' - # ' I - '- ' - I- ' - - - - - I ' ' - _, ' - \ - ' ' \ - ' ' ' - � ' - ' , __ ' ' I ' ,- ' ' ' - - - ! I ! I , ' ' ' ' ' \ I ' ' ' ' ' � ... - - - - - -- .. ' - -- I ! ' ' ' ' .. I ! ' ... ... - ' ' ' \ � ' ' Figure 4.2: (a) Ground state of fru strated model. (b) Ground state at 5% dilut ion, still displaying 4-sublattice order. (c) Ground state at 30% dilut ion, showing that long-range order is lost. (d) Ground state at 45% diluti on, above the lattice percolation threshold. systems (see discussion later). The Monte Carlo simulation consisted of 40,000 Monte Carlo steps (MCS) per disorder realizati on, 10% of which were needed for equilibrati on. 4.3.1 Metropolis Updates The annealing part of the simulation is carried out during the Metrop oli s updates. A Metro polis update consists in choosing a site at random and chan ging, also randomly, 42 the phase associated to it. In order for this update to be accepted by the algorithm, the conditions below have to be obe yed: • If the move lowers the total energy of the array, it is accepted with probability 1. • If the move increases the total energy of the system, it is accepted with probability p = e -6-EfT . During this step in the simulation, each site in the array has a finite probability to be chosen at least once. Allowing a move to increase the overall energy of the array prevents the simulation from becoming trapped in a local energy minimum. This method is effi cient at temperatures far above and below the phase tran sition. However, in the proximity of a transition it can get signifi cantly slower, since the number of correlated spins becomes of the order of the system size (�00 (T*(L)) "' L). Therefore, the system becomes more resistant to flipping only one spin at a time, since most spins are correlated, and the time to complete the step grown as ��· In our simulation, for each MCS we implemented the Metro polis update twice. 4.3.2 Microcanonical Updates: Overrelaxation Overrelaxation is a local algorithm that helps reducing critical slowing down. We choose at random a site i in the latti ce. In the classical XY model, the local field on this site will be given by the sum of its neighboring spins: (4.2) where the j's are the nearest neighbors of i. Once we know the local field in i, the spin s i will be updated according to: 43 _, _, 2h-(h· . §. ) S f s z z z i = - i + h 2 ' t which corre sponds to flipping the component of §i perpendicular to the local field. (4.3) Figure 4.3: Example of a phase update using the overrelaxation method. The ori ginal phase ¢ is rep resented by the vector S , the direction of the local field is h and the updated phase o: is the vector S '. In terms of the phases, this equation can be simply written as (see Fig. 4.3): (4.4) There is no acceptance ratio associated with a phase change in this method-every phase chosen at random will be cha nged. This is possible because this move does not alter the total energy of the array. In our simulation, microcanonical updates were imple mented 8 times consecutively for each Monte Carlo step. 44 4.3.3 Cluster Updates The Wolff cluster update [81, 82] for the XY model flips the spin component al ong the fi direction for a cluster of spins which is dynamically built around a spin. The size of the cluster is of the order of the correlation length, which avoids critical slowing down. A Wolff cluster algorithm consists of the following steps: • Choose a random direction fi; • Choose a random spin § i , which will be our seed; • Calculate fi · § i ; • Find all nearest neighbors § j of §i that have not been added to the cluster yet; • Add § j to cluster with probability: Since f3 = 1/T is always positive, if sign( S i · fi) -1- sign( S j · fi), the probability of adding § j to the cluster is zero. Else, if sign( S i · fi) = sign( S j · fi), § j is added with probability Padd (Si, S j) = 1- exp { - 2f3 [(Si · fi) ( S j · fi) ]}. If the spin § j is added to the cluster, we repeat the procedure for all its neighbors that have not been added to the cluster yet, and so on. Once the cluster stops growing we perform a rotation of the phases in the cluster, by reflecting all spins about the plane along to the direction fi. In summary, each Monte Carlo step is completed after the simulation performs 2 Metro poli s updates for the entire latti ce, 8 overrelaxation steps and 2 Wolff cluster updates. In the case of fru strated systems, the Wolff cluster update is not used because clusters are either as big as the system size or very small. It has been shown numerically that 45 there is no gain in using Wolff cluster updates. However, with dilution the size of the cluster is limited, being confined to a blob connected by weak links. 4.4 Results 4.4.1 Helicity Modulus and Choice of Boundary Conditions As seen in Chapter 3, the helicity modulus of Josephson junction array s can be used to deter mine the critical temperature of the Kosterlitz- Thouless tran sition. We can replicate the results in [77] for the helicity modulus and using the finite size correction in [11] (also from Eq. 3.17 ): (4.6) In the case of no fru stration and finite dilut ion, we can al so observe the suppre ssion of !(T) in Fig. 4.5, agreeing with the calculati ons of helicity modulus in [88, 57]. We also find that, as expected, there is a finite critical temperature up to the lattice percolation thre shold of 40% diluti on, confirming the experimental results in [46]. Since we can reproduce the known results for a fru strated JJA and a diluted lat tice, now we combine these effe cts and calculate the helicity modulus of a site-diluted array of Josephson junctions under a perpendicular magnetic field. Our result for fully fru strated arrays with L = 16, 24 and 32 at 15% dilution is shown in Fig. 4.6. We obtained similar curves for all values of dilution and all system sizes we investigated. At high temperatures we observe the expected behavior: as the system cools, the helic ity modulus goes from zero to a finite value at around Tc = 0.25. Then, as the system cools below Tc something unexpected hap pens: Instead of slowly increasing towards a positive value, the helicity modulus peaks and falls to negative values. In principle, this odd feature could be a sign of reentrant behavior, meaning that at low temperatures 46 0.8�--�----�--�----�----�--�----�--�----�--� Til Figure 4.4: Helicity modulus of fru strated Josephson junction array s at zero diluti on, for system sizes L = 16, 24 and 32. The KT transition temperature TKr is given by the limit in Eq. 4.6. the system would be return ing to the normal state from a superconducting state. This kind of reentrance in the helicity modulus is observed in phase tran sitions in quantum Josephson junction array s, as found by Capriotti et al. [17]. However, there was no prev ious indication of reentrant behavior in classical JJA in the literature and instead it seemed that the numerical calculat ions were the culprits for the observed "reentranc e". Thus, we ran a few tests to investigate what was behind the unexpected drop in helicity modulus at low temperatures for this particular system. The tests we ran were done by diluting specific sites, creating clusters of vacanci es in the middle of a sample, with periodic boundary condit ions. One of the array s we investigated is pictured in Fig. 4.7, where we removed 10 sites from the latti ce, with L = 10. 47 1.0 f- I I I I - TIJ Figure 4.5 : Helicity modulus of site-diluted Josephson junction arrays with f = 1/2, for system size L = 24 and different dilut ions d. The KT transition temperature TKr is suppressed as dilution increases. Let us discuss the particular case of Fig. 4.7 to explain the "reentrance" in Fig. 4.6. Removing 10 sites in 2 rows and 5 columns of the array is the same as removing 6 bonds in each of the 2 rows and 3 bonds in the 5 col umns. We anneal the lattice to the ground state and save the values of the x and y contributions of Y(T), Y xx(T) and Yyy(T), at each temperature step. We then average the results over 8 different initial configurat ions. The helicity modulus in the x and y directions are shown in Fig. 4.8. The direction in which an even number of bonds was removed (x direction) the helicity modulus has the expected behavior and no reentrance is observed. In the y direction, however, we removed an odd number of bonds and we see a drop in Yyy(T) at low T. If we examine the equation for the helicity modulus for a diluted and fru strated JJA along a direction k (k = x or y): 48 I -0.6 - j j 0.0 0.2 0.4 Temperature I o---o L = 16 o---o L = 24 o---o L = 32 j 0.6 Figure 4.6: Helicity modulus for the diluted and fru strated JJA. A "reentrance" is observed for T < 0. 18, due to incommensu rability effects in the lattice with periodic boundary conditi on. Each data point is averaged over 100 disorder realizati ons. Here, 15% of sites are diluted. (4.7) we find that the numerator of the second term on the right-hand side (cor responding to current fluctuati ons) is finite at zero temperature if the number of bonds removed is odd along a row or column of the array. To verify this, we can do a simple calculati on. In the ground state, the phases are approximately org anized in the Villain 4-sublattice order configuration. The current fluctuati ons term from Eq. 4. 7 in the k direction (k = x or k = y) is: 49 / / / - / '-... / " I " I " "- I \ / - / -- / - / / \ I I \ I / / / \ I \ I \ \ / '-... / - / � / / " I \ \ I I I " / / � / / � / \ I " \ I \ '.. Figure 4.7: Te st sample used to investigate incommensu rability effects in the helicity modulus Y(T). Here the lattice is fully frustrate d. with system size L = 10. The dilution d = 10% cor responds to 10 adj acent sites removed from 2 rows and 5 columns of the latti ce. creating a cluster of vacan cies. (4.8) Using Eqs. 3.22 and calculating the supercurrent term (Eq. 4.8) al ong the x direction in the first row of Fig. 4.7. Ii1L 12) � ((5sin(¢- ¢ - ?r/4) - 5sin(¢- ¢ - ?r/4) + 5sin(¢ + 7r/4 - ¢) - 5sin(¢ + 7r/4 - ¢))2) = 0. Now we do the same calculation for the fifth row of Fig. 4.7: Ii5) � ((2sin(¢- ¢ - ?r/4) - 2sin(¢- ¢ - ?r/4) + 2sin(¢ + 7r/4 - ¢) - 2sin(¢ + ?r/4 - ¢))2) = 0, (4.9) (4. 10) 50 0.6, o 1 0.5 - \ •• ' 0 ' 9 ' " � 0.4 - � �� 0. 3 - 0.2 - 0.1 - ' ' Q ' ' Q ' ' ' � ' ' � ' ' 0 ' ' ' � ' I ' o, I R - ,r � 0.0 f- / - m ' ' ' ' ' Q ' - 0.5 I - "' � �, - � �0 � - - ' ' ' ' 0 -1 5 r- - - - I I I "'""'+<>- 0.0 �� c:'-::- ��c-�� 0.0 0.2 0.4 0.6 0.8 - 2.0 �� :'-::--- � �� 1 -=-- �� 1 0.0 0.2 0.4 0.6 0.8 T!J T!J Figure 4.8: Right (Lef t): Helicity modulus along the x (y) direction for the array in Fig 4.7. Error bars are smaller than the symbols, and the dashed lines are a guide to the eye. which is expected from Fig. 4.8. Let us now look at the y direction. The supercurrent term along the third column, 1�3), is: 1�3) � ((3sin(¢- ¢ + ?r/4) - 4sin(¢- ¢ + 7r/4) + 4sin(¢- 7r/4 - ¢) - 3 sin(¢- ?r/4 - ¢))2) = 2. (4. 11) This implies that the term ( ( � Jij sin( Ti - ¢ j) ( eAk · e;j )) ) does not vani sh as the temperature app roaches zero, what can be verified in the plot of the supercurrent term in Eq. 4. 7 as a function of temperature (Fig. 4.9). Therefore, in this limit the second term on the right-hand side of Eq. 4.7 is: 51 "' � �" - 3 0 ,- , - ,1 -- , - ,1 -- , - ,1 -- , - ,1- p - .' e( 25 - 20 - 15 - 10 - 5 - • ' ' ' ' � ' ' ' ' ' 0 ' ' ¢ ' ' ¢ ' ' ¢ ' ' 0 ' ' 0 ' ¢ ,/ // - l - - - J / 1 0 �� - -� 1 0.0 0.2 0.4 T!J 0.6 0.8 3 0 ,-,-,1--,-,-1,--,-1,-,1-, O L_��--L_�I�_L_I�� 0.0 0.2 0.4 0.6 0.8 T!J Figure 4.9: Left (Right) : Supercurrent term in Eq. 4.7 along the x (y) direction for the array in Fig. 4.7. Error bars are smaller than the symbols, and the dashed lines are a guide to the eye. We can see that in the y direction the supercurrent is finite at zero temperature. (4. 12) This is clearly a numerical probl em: the periodic boundary conditions which we assume ca use the appearance of incommens urability effe cts if an odd number of bonds is removed along a row or column of the array. Therefore, when adding dilution to the fru strated model we cannot retrieve TKr from the helicity modulus data due to these incommens urability eff ects. From now on in this chapter, we will assume op en boundary conditions along all boundar ies of the system to avoid any pro blems that may arise when using PBC. Plus, this assumption br ings our model much closer to reality, since this boundary condition mim ics the condit ions found in experimental samples. However, 52 the helicity modulus can only be defined with closed boundary condit ions, becoming then necessary to look for another variable from which we can retrieve the Kosterlitz Tho uless transition temperatures reliably. 4.4.2 Structure Factor As we mentioned in Chapter 3, the structure factor is a measure of the correlations of the latti ce. The Bragg peaks in the reciprocal space of the lattice become sharper and higher as the system approaches its ground state. For the Villain 4-sublattice order, the Bragg peaks appear at the ordering vectors Q1 = (0, 0) , Q 2 = (0, 1r ) , Q 3 = (1r, 0 ) , and Q4 = ( 1r, 1r ), and equivalent points. For the two-dimensional Ising antifer romagnet, there is a peak at Q = ( 1r, 1r ) , and equivalent points. To find the Kosterlitz-Thouless transition temperature TKr, we rewrite Eq. 3.12 as: (4. 13) Since Eq. 4.13 is non-zero in the ground state only at the four ordering vectors Q1, Q 2 , Q 3 , and Q4, we can fix our value of qat one of these vectors in reciprocal space. For each temperature step of our Monte Carlo run we save the value of S ( Qi), for all dis order realizat ions. We repeat the procedure for 3 different system sizes, L = 16, 24 and 32. Averaging over disorder realizati ons we obtain curves S( Qi, T; L) vs T. We know from the previous chapter that at the phase tran sition, S( Qi, TKT) oc e-ry. Renormal izing the structure factor by Cf 4 and plotting it for different system sizes as a function of temperature, we find that the temperature at which the curves cross corre sponds to TKT· However, since our system sizes are finite, we need to take into account finite size effects in the prediction of TKT· The temperatures cor responding to the crossing 53 0.8 � 0.6 ....J 3 �- E-< ;:... c; '[;{ 0.4 0.2 0.35 030 025 020 0.15 0.10 lll . . \ · o \ if . . - 0.8 <il :\ - ··i. :·· .. - \ . e· .. . - l · . . G 04) 045 (;} .. . � L = 16 13 . .. . -a L = 24 � .. · -¢ L = 32 1.0 0. 0 .__ ...__ _.__ .....__ __.__ _._ --L..: "-= """"- '""'"' -- ........... 0.0 0.2 0.4 0.6 0.8 1.0 T/J Figure 4.10: Renormalized phase structure factor as a function of temperature for 3 system sizes (L = 16, 24 and 32), at zero dilution (left) and 10% dilution (right). The chosen ordering vector Q1 is located at (0, 0) in reciprocal space. The insets show in detail the crossing points of S ( Q1, T I J; L) for each dilution d. The error bars are smaller than the symbols and the dotted line is just a guide to the eye. points for the renormalized structure factor S ( Q i , T; L) I L 7 I 4 al so depend on the system size, and correcti ons in ln £ 2 are needed to find the true transition temperature. Fig. 4.10 shows the renormalized structure factor as a function of temperature for differ ent system sizes at different values of diluti on. For the chiral structure factor, we rewrite Eq. 3.27 as: (4. 14) Here the chirality Kn of the n-plaquette is defined in a slightly differ ent way from Eq. 3.5 in order to account for dilut ion: 54 0.5 04 5 050 1.0 030 035 04J o---o L = 16 [3 - ··· Q L = 24 o- - -- � L = 32 Figure 4.11: Renormalized chiral structure factor as a function of temperature for 3 system sizes (L = 16, 24 and 32), at zero dilution (left) and 10% dilution (right). The ordering vector Q is located at (1r, 1r ) in reciprocal space. The insets show in detail the crossing points of Sc( Q, T / J; L) for each dilution d. The error bars are smaller than the symbols and the dotted line is just a guide to the eye. Kn = -] � dijkl sin( ¢i - ¢j - Aij), i ->j (4. 15) where 5ijkl = 1 if all sites i, j, k, l around a plaquette are occupied or dijkl = 0 if at least one of the 4 sites is diluted. The procedure to deter mine Tc from the chiral structure factor fol lows the same steps as the method to find TKT· In Fig. 4.11 we show the chiral structure factor for 3 system sizes at differ ent values of diluti on. 55 4.4.3 Phase diagram Now that we found Tc and TKT for different values of dilut ion, we can plot the phase diagram, represented by the critical temperatures versus dilution (Fig. 4.12). 0. 5 ,--- -,---- ---.-- , - .--- -. ,- .---- ---. ,,.--- -,-- - .---- , -.-- -----r ,- .---- ---., , - --,--- - .---- , ---.-- -----r , - .------ , Q) 0.3 t !3 � [) P.. a Q) E--< 0.21- 0.1 t- 8.oo I I I 0.05 0.10 0.15 I I I I I I l \ 0.20 0.25 Dilution I 0.30 I G---o KT tra nsition I G- --o Ising tra nsition I I 0.35 0.40 - - - - 0.45 Figure 4. 12: Phase diagram of site-diluted and fully fru strated square JJA. There is a significant separation between the phase tran sitions as more sites are dilu ted, creating a phase which has vortex order but no phase coherence, in which phase correlations are short-range. Both tran sitions are suppressed well below the lattice percolation threshold at around 40%. Note that Fig. 4. 12 resembles the conjectured phase diagram in Fig. 4.1, except that at zero dilution the two tran sitions are already well-s eparated. Then, as percolation disorder is increased, this separation between Tc and TK T is accentua ted. There appears a phase between the TKT and Tc lines which possesses vortex order but no phase coher ence. At zero temperature, phase coherence ceases to exist above dKr, but below de the system might still present some non-zero conductivity. We confirm the results in [3] that 56 both Ising and KT tran sitions are suppressed at higher dilu tions. Our results also show that the diluted array is not superconducting for d > 0.22 at zero temperature. In the next chapter we study the I-V curves of the system in the vortex lattice (VL) phase at zero temperature. 4.5 Conclusions In this chapter, we calculated numerically the phase diagram of site-diluted and fru s trated Josephson junction array s. Even though the phase diagram for this model has been conjectured in [3], for the first time a quantitative analy sis is provided. We used a combined Metropo lis, overrelaxation and cluster algorithm in our Monte Carlo sim ulat ion, from which we could calculate the array 's observabl es, such as the structure factor, which we used to find the critical temperatures of the Kosterliz- Thouless tran si tion, TKr, and of the Ising tran sition, Tc. In particular, a finit e-size scaling of the phase and chiral structure factor gives us an estimate for the value of the KT and Ising tran si tion temperatures at various values of dilut ion. The phase diagram that we deter mined shows the emergence of a non-superconducting vort ex-crystal phase between TKr and Tc. The vortex lattice phase that sets in as the system loses phase coherence can serve as a test-bed for the study of two-dimensional metal s, since in this region the system could in principle still have a finite conductivity in spite of the loss of superconductivity, and therefore display metallic behavior. This is a question we addre ss in the next chapter, when we calculate the I-V characteristics of fru strated and diluted Josephson junction array s. 57 Chapter 5 Current-Voltage characteristics of frustrated and diluted Josephson junction arrays 5.1 Introduction Most experimental works to investigate phase tran sitions in Josephson junction array s do so by studying their dynamical behavior, such as the array 's I-V characteristics. As we briefly mentioned in Chapter 3, finding the array 's transition to a superconducting state is possible upon measuring the temperature at which the linear resi stivity vanishes. Fisher et al. [28] found that in two dimensions the Kosterlitz- Thou less transition is char acterized by power-law I-v characteristics. Namely, at TKT, v rv J 3 . Below TK T the linear resi stivity vanishes but non-linear dissipation created by the thermal excita tions (vortex-antivortex pairs) appears in the I-V characteristics as V rv I8 ( T ) . Some of the most recent developments for our system of interest are the experimental works by Yun et al. [85, 87, 86], and the numerical works of Granato et al. [35, 36, 37] and Lv et al. [60, 59]. In this chapter we find the I-V characteristics of a fru strated and diluted JJA at zero temperature. We focus on the region of the phase diagram (Fig. 4.12) in the interval dKT < d < de, where the system has no phase coherence but still presents vortex order. Our motivati on to investigate the conductivity of this phase at zero temperature comes 58 from an interest in finding out whether fru stration and dilution turn the array into a two dimensional metal at zero temperature. If this is the case, this would be a remarkable result, since a system of composite bosons (Cooper pairs) might remain conducting but non-superconducting at T = 0. 5.2 Model We study the I-V characteristics of a square array of shunted Josephson junction s, in the presence of a magnetic field perpendicular to the array which induces fru stration. Site dilution is realized by randomly removing superconducting islands. Two non-adjacent sides of the array are connected to electrodes from which an external current I ext is injected through the system (see Fig. 5.1) . The boundary conditions are open along the interface with the electrodes and periodic in the perpendicular direct ion. We assume weak coupling such that we can ignore the magnetic field induced by the external cur- ren ts. We consider Josephson junctions which are resi stively shunted, as it is commonly the case in experiments (the shunt being rep resented by quasi- particle current through the junction or through the substrate ). From Josephson equati ons for the current we obta in: (5.1) where the first term in the right-hand side is the supercurrent through the junction, R is the shunt resistance, and the voltage in the second term is given by the ac Josephson effect: (5.2) 59 Figure 5.1: (a) Array of RSJ (Resistively Shunted Josephson-juncti ons) without fru stra tion and dilut ion. Current is injected from the electrode on top. (b) Same circuit as in (a), but for a diluted array of RSJ in a perpendicular magnetic field iJ, which induces fru stration. Adapting Eq. 5.1 for a fru strated array, we get for the current through a site i: Ii = L (I0 sin(¢i - ¢j - Aij) + � j/Rij ) , j (5.3) where the sum is over the j neighbor of site i, and J i = I ext if site i is in contact with an electrode, or zero otherwi se. We can rewrite this equation in terms of gij = n/(2e R ij): � g · . (X - ), - ) = I -- � J c _ sin( .-f.,· - k - A -· ) D t} 'Yt o/J t D t} 'Yt 'Y1 t1 . j j (5.4) The left-hand side of Eq. 5.4 can be written as a sparse square matrix of order L 2 . The non-zero matrix elements are : 60 Mij Mij L 9ij, for i = j, j -gij, for i -1- j. (5.5) (5.6) We can also rep resent the time derivati ves of the phases as a vector of N = L 2 ele- .... ments eft= (¢; 1, ¢; 2 , ... ,cft p). Writing a vector cwith elements ci = Ii - L j Jf j sin(¢i - ¢j - Aj), where the index i corre sponds to the site i in the array, Eq. 5.4 is simplified to: (5.7) .... The solution is given by simply inverting the matrix M and finding ¢. However, N N as it turns out, M is a singular matrix by construction, since L Mij = L Mij = 0, j= l i= l for all i, j. To avoid this probl em, we remove one row and one column of M, which corre sponds to grounding one of the superconducting islands so that its phase does not .... change in time [22]. We now have a new matrix M' and vectors eft', ?!. The solution of the linear system now requires inverting a matrix of order ( L 2 - 1): (5.8) which gives one set of (L 2 - 1) coupled ODEs. The voltage acro ss the array is then calculated as V = (V(t))t/ I0 R0, where ( ... )tis a time average over a time interval of 4000bt to 1200 0Jt, depending on the system size. 61 0.5 ,-- ---,- - --, l - ---.- - -. 1 - ---,--- - -, l - ---.- - .-- 1 -----. - -.- - 1 - ---. -- ,----,1 .,. 0.4 - 0.3 - 0.2 - 0. 1- 0 0 d = 0.15 0 ·{] d= 0.20 q. . . . ., d= 0.22 ... " d=0.25 • .. d = 0.30 ... ... d = 0.40 0.5 .; - - - 0.6 Figure 5.2: I-V characteristics of fru strated and diluted array with L = 48. The com bined effects of fru stration and percolative disorder reduces the critical current to lower values up to the percolation threshold. 5.3 Results We calculate V(I) vs I by solving Eq. 5.8 for 60 to 80 different disorder realizati ons, with a time step varying between c5t = 0.01 and c5t = 0.05 for 5000 to 15000 time steps, depending on the system size. In the time average we discard the first 20% time steps which cor respond to the equilibration phase. The current bias is varied in steps of c5 I = 0.01. We looked at system sizes L = 16, 24,32 and 48. The I-V curves for the fru strated and diluted array are depicted in Fig. 5.2. In Fig. 5.2, the effect of disorder is to reduce the critical current Ic, below which the voltage drop acro ss the array is zero. If instead we plot dV I di vs I, as in Fig. 5.3, we observe two plateaux which could suggest that the array is Ohmic at low bias, and at high I I L, where the slope of the I-V characteristics dV I di � 1. 2 is independent of the diluti on. However, the plateau at low I I L is a finite size effect, as suggested by Fig. 5.4. 62 1.2 0.0120 0.0100 1.0 0.0080 0.0050 00040 ooceo - d=0.20 - d=0.22 - d=0.25 � 0.02 0()4 � 0.6 0.4 0.2 0.0 0.0 Figure 5.3: dV I di versus I I L for different diluti ons in the interval dKr < d < de, for L = 48. The plateaux at high current bias is an indication that the JJA is Ohmic in this limit. At low current bias, the plateau is a finite size effect, as shown in Fig. 5.4. Inset: Detail of dV I di at low current bias. 0.4 0.3 .....l > 0.2 0. 1 G···-0 L= 48 G··· - a L= 16 G·-.O L=24 A· ·A L = 32 10 -5 g_ OL1 _ .L__ .L.....L ....L.L .Ll..L O. L1 _ .L__ .L.....L ....L.L ..Ll..l..J IlL Figure 5.4: Left: Vol tage vs current for different system sizes at 20% diluti on. Right: Same plot, in log-log scale. The behavior of the I-V characteristics at low I I Lis a finite size effect. 63 0.1 "' > 001 0.1 ..,! > 001 0.001 �---Q L=3 2 .d= 0.25 - C1Jl"'ii'e Fit - OlrveF it 03-···-o L=3 2 .d= 0.22 0.1 IlL O.Dl O.Dl 0.1 "' G-···G L=48. d=0. 25 - Curve Fit - Curve Fit t;>---o L=2 4.d=03 0 0.1 IlL / > O.Dl 0.001 0� 0 �01--�--��0�1--��LU IlL Figure 5.5: Curve fits for the valu es of dilution and L indicated in each plot (Left: L = 32, d = 0.25 (top ), d = 0.22 (bot tom); Top right: L = 48, d = 0.25; Bottom right: L = 24, d = 0.30.) From these fits we extract the exponent a and the critical current Ic. (See Figs. 5.6, 5.7) We can extract the critical currents Ic for each dilution by fitting the curve as a power-law in the small current bias interval 0.0 :<::: I :<::: 0.2. We fit the data using the function V (I) = ci + b9(I - I c) (I - Ic)a, where the term ci is size dependent (we assume that finite system-size effects are strong in the small current bias limit, thus the linear term). The fit gives us Fig. 5.6 for the critical current versus site dilution for different system sizes, and Fig. 5. 7 shows the dependence of a upon dilut ion. 64 0.14 I I I I <>···<> L= 24 0.12 1- [] - - o L=3 2 - <> · <> L=48 0.10 1- l - 0.08 1- · . ... - � ::so I � 4 0.06 1- - ·. . . . . · · . . . .. ! ·. .. > · < .. · 0.04 1- - . . . . I · · .. . - · · · - : : .. / 0.02 .. \ / 1- _/ - 0.0 8 I I I .1 0 0.15 0.20 0.25 0.30 0.35 d Figure 5.6: Rough estimate of Ic versus dilution for L = 24, 32 and 48. The values of Ic were found by fitting the data as V( J ) = ci + b8(I - Ic)(I - Ic)a. For the larger system sizes we find a sharp drop at 20% dilut ion. The non-zero Ic at higher dilut ions is a finite-size effect. 8.0 7.0 6.0 � ::s "' 5.0 4.0 3.0 2. 8 .10 0.15 0.20 0.25 d <>···<> L=24 o ···o L=3 2 <> <> L=48 0.30 0.35 Figure 5.7: Exponent a versus dilution for L = 24,32 and 48. In a rough approximation there is a peak around 20% dilution and a jump to around a � 3 when the system loses phase coheren ce. 65 5.4 Conclusion We have calculated the I-V characteristics of fru strated and diluted JJA at zero temper ature. We found a strong finite-size effect at low current bias, and Ohmi c conductivity at high current bias. The array is Ohmic in this limit because the critical current is low ered by fru stration and current flows through the shunts instead of the junct ions. At low current bias our results are still inconclusive due to finite size effec ts. In fact, if the helicity modulus for this system is calculated, it will show non-zero current fluctua tions at zero temperature, since we are limited to work with only a few hundred supercon ducting islands in the array. Even with this limitati on, we can plot curve fits to the I-V characteristics for different diluti ons and system sizes, which indicate a phase transition at around 20% dilut ion, and therefore agreeing with our results for the structure factor of the array in the previous chapter. Figs. 5.6 and 5.7 show that as the system size is increased the jump at 20% dilution becomes more prono unced. Our results seem to indicate a saturation of the exponent a rv 3 after the tran sition. 66 Chapter 6 Conclusions In this diss ertation we performed extensive numerical simulati ons to study some of the physical proper ties of two kinds of nanodevices, namely, one-dimensional semiconduc tor heterostructu res, and proximity-coupled Josephson junction arra ys. These systems have been investigated extensively in the last few decades, and even as new devices and new technology are developed at an astonishing pace, their usefulness in novel applica tions in physics and engineering is continuously reaffirmed. As mentioned in Chapter 2, one-dimensional potential wells are now widely used to model molecular junct ions, which have shown great promise in their applicati ons in nanobiotechnology. Also, the study of strong coupling between phonon channels and electronic degrees of freedom has become a maj or issue in quantum informat ion, since the effect of decoherence which arises from the coupling of qubits to a thermal bath needs to be well-understood if a prospective quantum computer is to properly preserve information in quantum bits [53]. Quantum computation has also been a fundamental motivati on in recent works on Josephson junct ions. In fact, nowadays there is much speculation on a recent claim that an alleged 128-q ubit chip, an intricate device made of Josephson junct ions, is able to perform quantum annealing [48]. In spite of this still controversial claim, it has been proven that quantum Josephson junctions are useful realizati ons of quasi-two-level sys tems, as they have been largely used as superconducting qubits [23]. However, not only quantum Josephson junctions have recently received special attention for their applicability. Mesoscopic Josephson junction array s, such as the mod els we studied in this diss ertation, are extremely versatile two-dimensional systems to 67 realize a wide variety of phenomena. A paper published in Nature Physics in December 201 1 features the experimental study of superconductor-metal phase tran sitions in two dimensional systems at T = 0 [26]. The authors utilize a regular Josephson junction array whose superconducting islands are made of individual grains, each with a well defined phase (h Several array s are made, and in each of them the islands are spaced by a fixed length which varies from array to array. The islands are coupled by a term that decays exponentially with distance. This model is analogous to an engineered diluted array whose clusters are weakly coupl ed. In fact, the Monte Carlo code we developed to study phase tran sitions in randomly diluted array s can be slightly modified to model this experiment. In principl e, by doing so we could verify numerically their claim for the existence of two superconductor- metal quantum phase tran sitions as the temperature is extrapolated to zero. This indicates that our research and the focus of this thesis are in sync with most recent developments of appl ied physics. 68 Bibliography [1] V. Alba, A. Peli ssetto, and E. Vicari . 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Ph ys., 75(3):1829, Oct 1994. 75 Appendix The Propagation Matrix Method As an illustration of how to construct the generalized propagation matrix in practi ce, let us consider the example of a rectangular potential barrier given by: 0, x <a V(x) = V0, a <:: x <:: b 0, X> b V(x) V o a b X Figure I: Example of potential barrier with height Vo and width b - a, The wavef unctions are written as super positions of plane waves, 1/J n(x <a) 1/J n(X >b ) 1/Jn(a <:: X <:: b) a e i kn x + b e - i kn x k = VE - nw n n , n , (I) (2) (3) (4) 76 where n repre sents the available phonon channel s. It is assumed as an initial condition that the incident electrons enter the potential structure from the left, i.e. a0 = 1, an fO = 0. To determine the tran smission coefficients for the elastic (tn=o) and inelastic (tnfo) channels we use the propagation matrix method, matching the conditi ons 1/Jn, j = 1/Jn, j+l and d?/Jn,j/dx = d?/J n,J + ddx at each boundary. In the absence of phonon scattering (n = 0), the propagation matrix p of the system is obtained by multiplication of the step matrices (at x = a and x =b ) and the free propagation matrix for a -s: x -s: b, which are given by A a 1 Pstep = 2 A b 1 Pstep = 2 And in the interval a -s: x -s: b: with L = b- a. ( ( 1 + � 1 - � ) k o k o 1 - K{) 1 + K{) k o k o 1 + � 1 - � ) ti;Q ti;Q 1 _ ko 1 + ko K{) K{) (5) (6) (7) (8) In order to find the tran smission probability, we need to solve the matrix equation pt = a where the coefficients of the wave functions (7) and (8) are the elements of a and t, respectively: 77 ( P11 P 1 2 ) ( to ) ( a o ) P2 1 P22 0 b o (9) For this system without phonon channel s, the transmission probability T( E) is sim- ply: (10) If instead phonon scatterer centers are pre sent, we do not have the above condition on the derivati ve, but rather we integrate Schrodinger's equation around x = 0 (from -E to +E). If for instance, we add a phonon scatterer at x = a in the same potential barrier, we have for the step and free matrices are : V(x) Phonon scatte rer V o ·• a b X Figure 2: Example of potential barrier with a phonon scatterer at x = a. A a 1 Pstep = 2 1 + il'£o ko 1- i.® ko igm k1n2 -igm k1h2 1- i i'£Q ko 1 + i.® ko igm k1n 2 -igm k1h 2 igm igm kofi2 koh2 -igm -igm koh2 koh2 1+ � k1 1- � k1 1- i l'£ 1 k1 1 + i l'£ 1 k1 (11) 78 A b 1 Pstep = 2 A L PJree = 1 + � x:o 1 - � x:o 0 0 e "' o L 0 0 0 1 - }£Q_ x:o 1 + � x:o 0 0 0 e -x:o L 0 0 0 0 0 0 1 + � 1 - � X:j X:j (12) 1 - l£l 1 + l£l X:j X:j 0 0 0 0 (13) e "' 1 L 0 0 e -x: �L Once again we determine T(E) by solving pt = a. However, now we have two tran smiss ion coef ficients t0 and tb one for each phonon channel, and the tran smis sion probability is given by: (14) 79

Abstract (if available)

Abstract In this thesis we study the physical properties of two distinct physical systems. First, a generalized propagation matrix method is used to study how scattering off local Einstein phonons affects resonant electron transmission through quantum wells. In particular, the parity and the number of the phonon mediated satellite resonances are found to depend on the available scattering channels. For a large number of phonon channels, the formation of low-energy impurity bands is observed. Furthermore, an effective theory is developed which accurately describes the phonon generated sidebands for sufficiently small electron-phonon coupling. Finally, the current-voltage characteristics caused by phonon assisted transmission satellites are discussed for a specific double barrier geometry. In the second part of this thesis we numerically investigate the complex interplay between frustration and disorder in a magnetically frustrated Josephson junction array on the square lattice with site dilution, modeled by the fully frustrated classical XY model on the same lattice. This system has a superconducting ground state featuring a vortex crystal induced by frustration. In absence of dilution this system is known to exhibit two thermal transitions: a Kosterlitz- Thouless transition at which superconductivity and quasi-long-range phase order disappear, and a higher-temperature transition at which the vortex crystal melts - the two critical temperatures delimit a chiral phase. We find that dilution enhances the width of the chiral phase, and that at a critical dilution superconductivity is suppressed down to zero temperature, while chiral order survives - the corresponding ground state of the system becomes therefore a chiral phase glass. At an even higher dilution chiral order disappears in the ground state, leaving the system in a vortex and phase glass state. We reconstruct the complex phase diagram via extensive Monte Carlo simulations, and we investigate the main signatures of the various phases in the transport properties (I-V characteristics).

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Asset Metadata

Creator de Oliveira, Bruna Pereira Wanderley (author)

Core Title Modeling nanodevices: from semiconductor heterostructures to Josephson junction arrays

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 05/01/2012

Defense Date 03/05/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag electronic transport,Josephson junction arrays,OAI-PMH Harvest,phase transitions

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan (committee chair), Bickers, Nelson Eugene (committee member), Cronin, Stephen B. (committee member), Daeppen, Werner (committee member), Däppen, Werner (committee member), El-Naggar, Mohamed Y. (committee member), Roscilde, Tommaso (committee member)

Creator Email bdeolive@gmail.com,deolivei@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-19647

Unique identifier UC11289350

Identifier usctheses-c3-19647 (legacy record id)

Legacy Identifier etd-deOliveira-685.pdf

Dmrecord 19647

Document Type Dissertation

Rights de Oliveira, Bruna Pereira Wanderley

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