CONDUCTANCE OF SUPERCONDUCTOR-NORMAL METAL CONTACT JUNCTION BEYOND QUASICLASSICAL APPROXIMATION VLADIMIR LUKIC. BSc, University of Belgrade, 1997

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1 CONDUCTANCE OF SUPERCONDUCTOR-NORMAL METAL CONTACT JUNCTION BEYOND QUASICLASSICAL APPROXIMATION BY VLADIMIR LUKIC BSc, University of Belgrade, 1997 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2005 Urbana, Illinois

2 c Copyright by Vladimir Lukic, 2005

3 CONDUCTANCE OF SUPERCONDUCTOR-NORMAL METAL CONTACT JUNCTION BEYOND QUASICLASSICAL APPROXIMATION Vladimir Lukic Department of Physics University of Illinois at Urbana-Champaign, 2005 Anthony J. Leggett, Advisor The subject of this thesis is a study of the superconductor-normal metal (SN) contact junction by systematically treating the corrections of the order /E F in momentum and conductance. We isolated the effects that are already present in the original formulation of Blonder-Tinkham-Klapwijk (BTK) model, but were neglected as the small quantities of the order /E F. The corrections studied are: non-equal momenta of various particles in the system, selfconsistent finite gap onset length scale, non-exact retro-reflection in Andreev process of the particles with finite energy, non-trivial renormalization of the barrier potential due to the non-equal momenta at finite incidence angle, and effects due to an anisotropy of the systems in contact. The main question is what is the interplay of these effects, and can they constructively add to produce the effect of the order 1. The answer required treatment of all the effects from the outset at the same level, and incorporation of these effects in a self-consistent calculation. To achieve that, a new method for self consistent calculation of the behavior of gap at the SN contact is developed, which does not use the quasiclassical approximation, but rather finds solution to the Bogoliubov - De Gennes equations in a simplified, step-wise constant, model of the gap. The conductance is calculated using the same method, thus guaranteeing the same accuracy. A study of self-consistently obtained solution shows that these corrections often have an effect opposite to each other, or have the same target states, which limits the overall effect. As a consequence even for large /E F the overall correction is still relatively small, and the conductance of the system does not differ much from the simple BTK model. We have thus shown the reason for robustness of BTK model, and gained a better view of what might be the cause of larger discrepancies between this simple model and experiment. iii

4 To my family. To Maki. iv

5 Acknowledgments Thanks to my advisor Tony Leggett, for his patience while I was trying to find myself in physics (and world in general), and for the innumerable sound advices he gave me during all these years. Thanks to Jim Eckstein and Laura Greene, who taught me how to perceive physics from the experimental side. Thanks to my friends and collegues - Joseph Jun, Geoffrey Warner, Vivek Aji, Carl Tracy, Argyrios Tsolakidis, Julian Velev. Thanks to to my family - Veljko, Milica and Natasa, for everything. Thanks to Pero, Momir, Zarija, Tijana, Dimitrios, Nemanja, Sale...to all my friends. Most of all, thanks to Maki. I acknowledge financial support from the National Science Foundation under grants NSF DMR , NSF DMR , NSF DMR , NSF DMR COOP, from MRL DOE grant, and from the Department of Physics, University of Illinois. v

6 Table of Contents List of Figures viii Chapters 1 Introduction Superconductor-Normal Metal Contact Junctions and Andreev Reflection The Bogoliubov-De Gennes Equations and the Blonder-Tinkham-Klapwijk Model Bogoliubov - DeGennes Equations The Model of Blonder, Tinkham and Klapwijk The Nature of Gap Edge Conductance Peak, Subgap Conductance and Zero-Bias Properties Corrections to the BTK Conductance Finite Gap Onset Length and Exact Momenta Non-exact Retro-reflection Angle Dependence of Effective Barrier Strength Corrections Not Taken into Account Calculation of the Conductance Particle with Perpendicular Incidence Angle Finite Incidence Angle Self-consistent Gap Calculation Quasi-classical Self-consistent Gap Calculations Improvement on Quasiclassical Approach vi

7 7 Results and Interpretation Effects of Finite Gap Onset Length Effects of Mismatch and Anisotropy on Fermi Surface Effects of Non-exact Retro-reflection Effects of Self-consistence Conclusions Appendix A Quantum Mechanical Ramp Barrier A.1 Exact Solution of the Tunneling Problem A.2 Numerical Results and Comparison of the Solutions B Basic Quasiclassical Equations C Definition and Calculation of Gap References Author s Biography vii

8 List of Figures 1.1 SN (A) and SIN (B) junction. Dashed regions are occupied states.grey block is an interface barrier. Single particle states are not allowed inside the gap Four processes occurring at SN interface: specular reflection (A), Andreev reflection (B), transmission as an electron (C), transmission as a hole (D). Arrows point in a direction of the velocity of the particle, and abbreviations for the directions are: er - a right moving electron, el - a left moving electron, hr - a right moving hole, hl - a left moving hole. Electron trajectories - full line, hole trajectories - dashed line Energy spectrum of BdG equations. Four types of particles with the given energy E are marked by the dots: left moving electron-like (el), right moving hole-like (el), left moving hole-like (hl), right moving electron-like (er) Visualisation of the BTK problem. The properties of N and SC are uniform, and there is a δ-function potential at the boundary The BTK conductance normalized to a high voltage value, for values of Z (top to bottom curve): 0, 0.3, 0.6, 1.0, The BTK conductance normalized to a normal state conductance of a system without barrier for values of Z (top to bottom curve): 0, 0.3, 0.6, 1.0, The conductance contributions from the individual components, Z = 0, E F = 1eV, = 20meV : (upper row) a - transmission without branch crossing, b - transmission with branch crossing, (lower row) c - Andreev reflection, d - specular reflection. The coefficient b and c are zoomed up to a larger scale to stress that they are exactly zero in BTK viii

9 2.6 Contribution to the conductance from the individual components, Z = 2, E F = 1eV, = 10meV : (upper row) a - transmission without branch crossing, b - transmission with branch crossing, (lower row) c - Andreev reflection, d - specular reflection Temperature dependence of s-wave BCS gap, and the smearing factor f/ V Temperature dependence of the BTK conductance given for Z=0 (left) and Z=0.5 (right). Curves from the bottom correspond to T=0, 0.2Tc, 0.4Tc, 0.6Tc, 0.8 Tc, Tc. Each curve is offset by +1 from the previous one. We use µ = 1eV, = 10meV The plots of a zero bias conductance as a function of temperature, normalized to a high voltage value, for different values of Z - from top: Z=0, 0.3, 0.6, 0.9, 1.2, SN junction with a displaced barrier. Three processes represent possible trajectories after AR at the SC interface. The barrier position is a vertical line with T, R, interface is at the gap onset Point contact conductance between Au and c-axis of CeCoIn 5, from [14] A comparison of the BTK conductance (full line) to the similar calculation with gap onset length ξ, and µ = 1eV, = 10meV, Z = 0 (left) and Z = (right). Note that y-axis doesn t start at zero Andreev reflection for a particle above Fermi surface Particles on the outside of the space limited with lines AB and CD cannot AR (momenta k 1 and k 2 ). Particle k 3 is allowed to AR. Left hand side is a case k F SC > k F N, right side k F SC < k F N The effect of limited tunneling due to the non-exact retro-reflection in a system with µ = 1eV, = 10meV, T = T c /2 = 33K in a dirty limit (lower curve) compared to the finite temperature BTK calculation (upper curve) Limit on retro-reflection as given by (4.4) - k max = 2k F A contact of two metals with different Fermi wavevectors. Tunneling to (and from) regions above the line AB and below CD is forbidden. Note that k is conserved ix

10 4.8 Angle dependence of Z eff for Z 0 = 0, m 1 /m 2 = 5 and values of E F 1 /E F 2, from the bottom curve: 2, 1, 1/2, 1/5, 1/8, 1/ Z eff as a function of incident angle, for a system with m 1 /m 2 = (1 + 4 cos θ) and ratio E F 1 /E F 2 (from the left): 20, 10, 5, 3, 2.5, Z eff as a function of incident angle, for a system with m 1 /m 2 = (1 + 4 cos θ) and ratio E F 1 /E F 2 (from the left): 2, 1, 1/2, 1/5, 1/10, 1/ Effect of the proper inclusion of the angle dependent Z, for a system with Z 0 = 0.1, r k = 2/3. Dots: calculation with correction taken into account; full line - BTK calculated with corresponding Z eff = Schematics of approximation of a real potential by a piecewise constant model potential One segment with corresponding particle amplitudes Components of the conductance for a system with step-function gap and exact momenta retained throughout the calculation, with Z = 0, µ = 1eV, = 0.2eV (upper row): a-transmission without branch crossing, b - transmission with branch crossing, (lower row) c - Andreev reflection, d- specular reflection. Note a different scale in parts c and d. Compare with Fig.2.5 to see an effect of exact momenta. Vertical axes - current, normalized to the incoming particle; horizontal axes - energy, in units 0.1meV Distribution of the current in space for each component er, el (upper row), hr, hl (lower row), normalized to the incoming er current. Z = 0.7, µ = 1eV, = 0.1eV. An electron is incoming from the right, position of the barrier is at the mark 50. Length ξ is 10 divisions on x axis Schematics of the change of incident angle for a sequence of segments, due to the increase of gap at the barrier, as calculated in (4.12) Boundary conditions for a SN contact: an electron incoming from the left (A) and a hole outgoing to the right (B). Full line - electron, dotted line - hole. Arrows point in the direction of propagation. N metal is on the left side of the interface in both figures x

11 6.2 Processes (A) and (B) from the Fig.6.1 drawn to include AR along the trajectory (left). Contributions to the gap from two trajectories at every point in space (right) The unnormalized contribution to the pair correlation function (6.5) from the particles of one energy and along one incident angle for Nb, Z=0, T=0 (E (left), E 3 (right)). Units on x-axis are 1/10ξ, and contact is at the mark 100, N is to the left, SC to the right of it The unnormalized contribution to the pair correlation function from particles at an incident angle θ = 0 (left) and integrated over all angles (i.e. after complete first iteration) for Nb, Z=0 (E (left), E 3 (right). Units on x-axis are 1/10ξ, and contact is at the mark The calculated gap in each iteration (top to bottom) (right) and the unnormalized pair correlation function after three iteration loops (left) for Nb, Z=0. Units on x-axis are 1/10ξ, and contact is at the mark Self-consistent gap and normalized pair correlation function for T = 0.95T c, all parameters are the same as in other figures Self-consistent gap and normalized pair correlation function for barrier parameter Z = 4.0, T = 0, all parameters the same as in other figures Self consistent gap after 4 iterations for E F = 1eV and = 0.1eV (left) and = 0.2eV (right) at T=0, Z= Evolution of the conductance curve for m SC /m N = (1+4 cos θ) and E F SC /E F N 1/5, 1/2 (upper row), 2, 3 (lower row). E F = 1eV, = 10meV, Z 0 = 0. Full line is BTK curve, fitted to the high energy values, dotted line is this calculation. Vertical axis is a conductance, normalized to a perfect contact, horizontal - energy, in units 1/ A conductance curve for m SC /m N = (1+4 cos θ) and E F SC /E F N - 2 (left)and 3 (right). Z 0 = 0, E F = 1eV, = 100meV (upper row) and = 200meV (lower row). Full line is BTK curve, fitted to the high energy values, dotted line is this calculation. Vertical axis is a conductance, normalized to the perfect contact, horizontal - energy, in units 1/ xi

12 7.3 A comparison of the self-consistent calculation of the gap for a system with m SC /m N = (1 + 4 cos θ) and E F SC /E F N (dots) and 3 (full line), after four iterations. Vertical axis is gap in ev, horizontal is distance in units 1/10ξ - contact at Evolution of the subgap structure with incident angle, Z=1.3, E f = 1eV, = 1meV : full line - BTK, dots - this calculation, normalized so that conductance at high voltage without barrier is σ 0 = 1. Consequence of this normalization is that subgap conductance σ SN (E) = 2 means that particle at that energy does not feel the presence of the barrier The schematics of the model of slowly varying gap. N - normal side, SC - superconducting side (with gap ). Region R is either superconducting (with gap < SC ), or normal ( = 0) The schematics of the condition (7.6): (A) - side view (distance vs. energy), (B) - view from the above. A thick vertical line is a surface barrier, a thin line is the position where Andreev reflection occurs Conductance at very large incident angle θ > 88 o, Z = 1.3, E f = 1eV, = 1meV. Full lines are BTK formula. Dots are calculation without (left) and with (right) k = k f approximation. The normalization is the same as above A.1 Ramp potential in standard quantum-mechanical problem. We set V 0 = 0.01eV. 89 A.2 The reflection coefficient of a ramp potential for U = 0.01eV, and values of L = 0, 10, 20, 40, 80, 160, 320, 640A (descending curves) A.3 A numerical approximation to the real potential form Fig. A.1 (steps), and the ramp potential (straight line), for n = 10 steps A.4 Fitting the conductance on the energy scale of a gap. Full line - the exact solution of the ramp potential problem, dots - the numerical solution. U=0.01eV, L=10A (upper curve) and L=320A (lower curve) A.5 Fitting the conductance on a very small scale - energy axis is offset so that gap energy is at zero. Note that R-axis shows details smaller than Fig. A.4. Full line - the exact solution, dots - the numerical solution. U=0.01eV, L=320A 93 xii

13 Chapter 1 Introduction Transport phenomena in a point contact junction between superconductor and normal metal are dominated by Andreev reflection, a process that transforms an electron into a hole that retraces the path of an incoming particle. Theoretical description of the conductance of this system is given by Blonder, Tinkham and Klapwijk, in a theory that is usually referred to as BTK. Their approach is very simple, requires a single fitting parameter, yet it is extremely successful. There are many corrections to this equation, but they usually do not produce significant departures from BTK, and are therefore neglected. In cases when BTK description is not satisfactory, one may try adding additonal fitting parameters, such as quasiparticle lifetime. Addition of an extra fitting parameter adds new fitting curves to the BTK family, but it is not always physically justified. One would naturally like to avoid addition of a new free variable to the data fit. There is a class of corrections to BTK that does not introduce a new physical effect to the problem. One can study the physics already there with a higher accuracy. These corrections do not introduce a new fitting parameter, but they do change family of functions described by it. In essence, they provide a better fitting functions than BTK for same problem. The purpose of this work is study of such one-parameter corrections. The basic problem is that we know of several corrections to BTK of the order /E F, and none of them has significant effect...but how do they act together? Do they add independently, do they mutually enhance, or suppress? In particular, for a system with /E F 1

14 of the order several percent, can these corrections add to produce a result of the order 1? To answer, we will have to treat all corrections on the same level from the beginning and account for them self-consistently. The answer turns out not to be spectacular - the corrections tend to cancel each other out, but we can answer how does it happen, and the reasons for it are interesting. To some extent, we managed to explain why is the BTK so robust and effective. The motivation for this topic was real life problems of the experimental groups in Urbana. Point contact junctions conductance data often look surprisingly similar to BTK, but they are not exactly the same - and the discrepancy cannot be accounted for easily. Depending on the geometry of the device, ordinary BTK can be twisted into an unrecognizable form. I cannot say that I managed to describe said experiments well, but it was a great inspiration for the work, and a pointer to what type of result is interesting to find. The outline of the thesis is as follows: in the rest of Chapter 1 we will discuss phenomenological aspects of Andreev reflection without a quantitative approach. Chapter 2 is a review of BTK, which we will use as a standard for comparison of all results. Chapter 3 is a study of the origin of various aspects of BTK. Chapter 4 is a discussion of the corrections introduced, and isolated effects of each correction separately. Chapter 5 is a discussion of the algorithm used to calculate the conductance with a given gap profile. Chapter 6 is a self-consistent calculation of the gap profile using a similar algorithm. Chapter 7 gives results for the combination of effects and a discussion of their influence on each other, and Chapter 8 is a brief conclusion. 1.1 Superconductor-Normal Metal Contact Junctions and Andreev Reflection Contact between superconductor and normal metal can be in two different regimes - a tunneling junction (also called SIN junction) and a contact junction (SN junction). The difference between them stems from the properties of the interface. A clean interface (one with no potential barrier or impurities), which is considerably harder to make, results in a contact 2

15 Figure 1.1: SN (A) and SIN (B) junction. Dashed regions are occupied states.grey block is an interface barrier. Single particle states are not allowed inside the gap. regime - particles ballistically propagate between two metals. Interface with a potential barrier - an insulating layer (therefore SIN)- produces junction in a tunneling regime. An insulating barrier is usually consequence of a naturally occuring oxide at a metal surface, but it can also be artificially produced. In a tunneling junction (Fig.1.1 (B)), normal metal (N) and superconductor (SC) are largely independent of each other, their wave function overlap is exponentially small and can be treated as a perturbation in a standard method called the tunneling Hamiltonian approach [1]. In this regime the internal states of SC and N are to a large extent unaffected by a system on the other side of a junction (though there are situations where it can be very important, e.g. [2]), and we usually observe how does the given state influence a particle tunneling from the other side. We find, e.g., that a particle with energy E smaller then a SC gap cannot penetrate the SC side, since there are no available final states for it. Until we give a particle sitting at the Fermi surface enough energy (as an electric potential offset V ) to reach the gap, there is no charge transfer between two systems (in case of s-wave SC), and conductance σ = di/dv = 0. Only for V > does transport occur, and we find that conductance measurements effectively map the density of states of a SC system. 3

16 In a contact junction (Fig.1.1 (A)), on the other hand, two systems are in a direct physical contact, and overlap of the wave functions is large. One cannot apply perturbation theory, but rather has to match the wave functions and solve the problem of two systems in contact simultaneously. Influence of two systems on each other is much larger than in the SIN case, and the state of a system near the contact is significantly different from that of an isolated sample. The mutual effect of SC and N system in SN junction is called the proximity effect. To study the transport of SN contact properly, one takes into account not only the influence of a, say, SC state on a particle incoming from a N side, but also the other way around - effect of the particles from N on the ground state of SC system. It is this problem that we shall study in the present work. The dominant effect in a subgap transport (energies E < ) in an SN contact junction and the main cause of proximity effect is Andreev reflection. Andreev reflection (AR) is a process in which the incoming electron gets reflected as a hole, nearly retracing the trajectory of the electron. This type of reflection is also called retroreflection. Opposite conversion, hole into electron, is also possible, but to be definite we shall discuss electron to hole AR only. Minor differences between two processes will be addressed later. AR is a process typical of superconducting state, and most prominent in area around an SN contact. Its overall effect is the transfer of a pair of electrons from N to SC side. Since only pairs exist in SC at energies E < and T = 0, this is the only transfer mechanism in that energy range. An incoming electron with momentum k and E < can be transfered to SC, only if it finds a single electron of the opposite momentum k and forms a Cooper pair. Since free electrons are not available in SC at that energy, the pairing electron must come from the N side, leaving behind a hole with the momentum k. This effect was first studied by Andreev [3] in order to explain the anomalous heat conduction properties at SN contact. Saint-James [4, 5] was the first to study its influence on transport of charge in SN junction, independently of Andreev s work. Sometimes the Andreev effect used in this context is called Andreev-Saint-James effect [6]. 4

17 If there is a finite barrier at SN interface, a particle can also get specularly reflected with E <. The particles with energy E > can be transmitted as electron- or hole-like quasiparticles into SC. While details of these processes will be given later in Sec.2.2, we can now note how these processes change perpendicular (v ) and parallel (v ) components of the velocity (relative to the interface) for two systems in contact with the identical Fermi surfaces, the same Fermi energy E F and the same effective mass m. These four processes exhaust all possibilities. In Fig.1.2 we see that these changes are: specular reflection : v v, v v Andreev reflection : v v, v v transmission as an electron : v v, v v transmission as a hole : v v, v v Process of transmission as a hole might look counterintuitive, but one should bear in mind that the parallel component of the momentum has to be conserved, and it is conserved in all four processes listed here - by virtue of the fact that the momentum of a hole is opposite to its direction of propagation. While the presence of a gap is crucial for the AR, it should be noted that it works only for a SC gap. A particle entering the region with semiconducting gap will be only specularly reflected. The reason for this is in the very nature of a SC state. An electron entering a semiconductor will have its wave function matched to that of a corresponding state in the gap - which is just an exponentially decaying electron wave function. The hole wavefunction does not play a role, since neither N nor semiconducting state mix electrons and holes. The eigenstates of the SC are the coherent mixtures of electron and hole parts. Thus, an electron entering a SC will have its wave function matched to a decaying part that has both electron and hole in it. But, then, the hole part on the SC side has to be matched too, and the only way to do it is to produce a hole wave function on the N metal side. That is the essence of AR. The transfer of a pair into the SC by AR has the spectacular consequences for the low-energy electrical transport properties of an SN contact. If an electron along the given 5

18 Figure 1.2: Four processes occurring at SN interface: specular reflection (A), Andreev reflection (B), transmission as an electron (C), transmission as a hole (D). Arrows point in a direction of the velocity of the particle, and abbreviations for the directions are: er - a right moving electron, el - a left moving electron, hr - a right moving hole, hl - a left moving hole. Electron trajectories - full line, hole trajectories - dashed line. 6

19 trajectory is transmitted with the energy E >, it carries a charge e to the other side. Along the same trajectory an electron with E < is AR - thus transferring a charge 2e to a SC. The conductance below the gap is increased by a factor 2! This is in great contrast to a tunneling junction, where there is no charge transfer at all below the gap. There are no single particle states available at E <, but the pair states are available. Since AR is a two-particle tunneling process, it is greatly suppressed in a SIN junction, since it is a higher-order process compared to single particle tunneling. Note that this is in contrast to an SIS junction - a pair tunneling in that case is a process of a same order as single particle tunneling (which is the basis of the Josephson effect). An SIS junction has correlated systems on both sides of the barrier, and tunneling of one electron in a pair automatically ensures tunneling of the other. It is not so in the SIN case - electrons on the N side are uncorrelated, and they have to tunnel separately, making it a higher order process. The barrier plays little role in an SN contact, since by definition it is orders of magnitude smaller. It is this factor of 2 that plays a central role in the study of SN junctions. The method we employ is simply shooting the electrons toward the barrier one by one at various angles and energies, and calculating the probabilities for each of the processes listed above. While below the gap the matter is simplified by the fact that no transmission is allowed (to the extent that we can get BTK results without actually performing a microscopic calculation), above the gap we have to take into account properly the combination of all possible processes. The resulting solutions for the conductance are drastically different in the two energy ranges. an If for a certain system every trajectory that is transmitted in an NN junction gets Andreev reflected in SN contact - the overall conductance will have an increase by a factor 2 after the SC transition. Our main task will be tracing the trajectories that are not Andreev reflected, and in particular studying the factors not included in the original BTK that can change the conditions for Andreev reflection. For that we will need a microscopic theory, which we shall study in the following section. 7

20 Chapter 2 The Bogoliubov-De Gennes Equations and the Blonder-Tinkham-Klapwijk Model This section reviews briefly the Bogoliubov - DeGennes (BdG) equations, used to describe a superconducting system near the boundaries and the inhomogeneities, and the theory by Blonder, Tinkham and Klapwijk (BTK) that uses these equations to describe conductance of a superconducting-normal metal (SN) contact. We will need only the simplest solution of the BdG, that for an isotropic system. 2.1 Bogoliubov - DeGennes Equations The Bogoliubov-De Gennes equations [7] are the mean-field equations for a superconducting system. They are obtained as the equations of motion for the mean-field approximation to BCS Hamiltonian. In their final form, they are coupled system of the two second-order differential equations and the two self-consistence conditions: ) E n f n (r, t) = ( h2 2 µ(r) + U(r) f 2m r2 n (r, t) + (r)g n (r, t) ) E n g n (r, t) = ( h2 2 µ(r) + U(r) g 2m r2 n (r, t) + (r)f n (r, t) (r) = V Σ n f n (r, t)gn(r, t) (1 2n(E n )) (2.1) U(r) = Σ n U(r) f n (r, t) 2 n(e n ) + g n (r, t) 2 (1 n(e n ))) 8

21 where n = (1 + exp( β(e µ)) is the Fermi occupation factor, β = 1/k B T na inverse temperature, µ a chemical potential (which, in principle, can be position dependent), U(r) a single particle potential calculated in the normal metal, - BCS gap, and u and v - the wave functions for an electron and a hole. As written this system is impossible to solve in the general form. For a homogeneous, clean system it yields the usual BCS value of the gap and the wavefunctions: E 2 = 2 + ɛ 2 q f q = e iet qr u 0 g q = e iet qr v 0 (2.2) u 2 0 = 1 ( ) E E v0 2 = 1 ( ) E E BdG equations have the solutions for both positive and negative E, connected by (f, g) T (E) = ( g, f )( E). Since these are not independent, we will always deal with the positive energy solutions only, and therefore all the sums run over the positive values of energy unless explicitly stated otherwise. For a given energy E we have four solutions propagating along the direction r. The momentum q corresponding to these solutions is given by: q ± = 2mSC h 2 µ ± E 2 2 (2.3) The particles and their momenta are given in Fig.2.1. The names electron-like and hole-like are referring to the corresponding solutions in 0 limit, whereas in the SC state they are really mixture of an electron and a hole component - a property explicitly captured in the spinor representation (an electron component in the first row, a hole in second): ψ er = u 0 e iq+r ; ψ el = u 0 e iq+ r ψ hr = v 0 v 0 u 0 e iq r ; ψ hl = v 0 v 0 u 0 e iq r Later, we will be interested in solving (2.1) in a more complicated case. (2.4) 9

22 Figure 2.1: Energy spectrum of BdG equations. Four types of particles with the given energy E are marked by the dots: left moving electron-like (el), right moving hole-like (el), left moving hole-like (hl), right moving electronlike (er). 2.2 The Model of Blonder, Tinkham and Klapwijk Blonder, Tinkham and Klapwijk [8, 9, 10] used the BdG equations to describe the nature of the excess current observed in some SN contact junctions. The idea that this is caused by the Andreev reflection has been around for a while [11], but BTK put it in its most useful and most often quoted form. A geometry of the problem is simple. BTK models a SC-N interface as a flat surface with a normal metal (N) on one (say, left) side, and a superconductor (SC) on the other. The SC side is characterized by the mass m SC, the Fermi energy E F SC and the order parameter. The order parameter is assumed to be constant everywhere on the SC side, up to the interface. The N side is characterized by the mass m N and the Fermi energy E F N. Fermi energy is measured from the bottom of the conduction band. Both sides have perfectly quadratic dispersion relations, and no band effects beyond the effective mass; they are treated as free electrons, apart from the SC gap. The interface potential is modeled by a delta-function potential H. The particles are moving in the direction perpendicular to the contact, which is chosen as a z-axis. 10

23 They then consider an incoming particle from the N side. Upon incidence on the interface, it undergoes the reflection (either as a particle or a hole) or the transmission (also as a particle or a hole). The transmission is such that it does conserve the current, thus a right-going particle on the N side will produce only a right going particles on the SC side. We solve the BdG equations (2.1) separately on the SC and the N side, with the appropriate parameters on each side and match the boundary conditions. Bogoliubov quasiparticles on the SC side have weight u 0 in the particle channel and v 0 in the hole channel, while particles on the N side have only one component (either particle or hole). The definition of the particle momenta in the problem is: k + 2mN = h 2 EF N + E k 2mN = h 2 EF N E q + 2mSC = h 2 E F SC + E 2 2 (2.5) q 2mSC = h 2 E F SC E 2 2 where k + is momentum of an electron on the N side, k momentum of a hole on the N side, q + momentum of an electron-like quasiparticle on the SC side, and q momentum of a hole-like quasiparticle on the SC side. We solve the BdG equations separately in two regions, and match the boundary conditions (see Fig.2.2): 1 e ik+ z 0 + C 1 e ik+ z 0 + D 0 e ik z 0 = A u v 0 e iq+ z 0 + B v 0 u 0 e iq z 0 (2.6) where on the left hand side (LHS) we have an incoming particle with the amplitude 1 (and thus the probability equal to 1), and a reflected electron and a hole with the probabilities C and D, both moving toward the left. On the right hand side (RHS) we have an outgoing electron- and a hole-like quasiparticles with the amplitudes A and B, both moving to the 11

24 Figure 2.2: Visualisation of the BTK problem. The properties of N and SC are uniform, and there is a δ-function potential at the boundary. right. For the derivatives, we have: h 2 2m N ik+ 1 0 h 2 2m SC e ik+ z 0 ik + C v e ik+ z 0 + ik D 0 e ik z 0 1 = (2.7) iq+ A u 0 e iq+ z 0 iq B v 0 e iq z 0 + H A u 0 e iq+ z 0 + B v 0 e iq z 0 v 0 where z 0 is the position of the barrier, in the original problem z 0 = 0, but this more general form will be useful for the comparison with the later results. The system (2.6), (2.7) is a system of four equations with four unknown variables - A, B, C, D. Their physical meaning is that they are the amplitudes for: - a right going electron-like quasiparticle on the SC side: the amplitude for the transmission without branch crossing (A) - a right going hole-like quasiparticle on the SC side: the amplitude for the transmission with branch crossing (B). 12 u 0 u 0

25 - a left going electron on the N side: the amplitude for the specular reflection (C) - a left going hole on the N side: the amplitude for the Andreev reflection (AR) (D) Branch crossing is the name we use for a tunneling process where the particle crosses from an electron-like to a hole-like branch of the energy spectrum. In that sense the Andreev reflection is also a branch crossing process, but we shall exclusively use that name for a transmitted particle. The major simplification that BTK use to solve the system (2.6, 2.7) is setting k + = k = k F N and q + = q = q F SC (though we will write these terms explicitely in the formulas ( ), in order to facilitate comparison with the corrections of the following chapters). This induces the error of the order δk/k F = E 2 2 /2E F, thus of the order /E F. Using v SC F = hk F SC /m SC and v F N = hk F N /m N, we define Z 0 = H/ h v F N v F SC, and Z 2 Z 2 = Z0 2 + (1 r v ) 2 /4r v EF N m SC r v = v F N /v F SC = (2.8) E F SC m N By using Z instead of Z 0, we can set E F N = E F SC and m N = m SC. If the contact is perfect (H = Z 0 = 0) the effect of difference of the masses m SC and m N and the Fermi energies E F SC and E F N is absorbed into the renormalized barrier strength Z eff through a single parameter r v. Note that Z eff is insensitive to the exchange m SC m N, so it retains the properties of a real barrier. To get the actual transmission coefficients for every branch we have to take into account the difference in momenta and weight of a hole and a particle part of the wavefunction. Thus we get: a = ( A 2 ( u 2 0 v 2 0 )) q + SC k + N c = C 2 b = B 2 ( u 2 0 v 2 0 d = D 2 k N k + N ) q SC k + N (2.9) The solution of the system (2.6, 2.7) is very different in two regions E < and E >, 13

26 as expected: 0 ; E < a(e) = (u 2 0 v2 0 )u2 0 (1+Z2 ) ; E γ 2 1 d = 4Z2 (1+Z 2 )( 2 E 2 ) ; E < E c(e) = 2 +( 2 E 2 )(1+2Z 2 ) 2 ; E (u 2 0 v2 0 )Z2 (1+Z 2 ) γ 2 0 ; E < b(e) = (2.10) (u 2 0 v2 0 )v2 0 Z2 ; E γ 2 2 ; E < E d(e) = 2 +( 2 E 2 )(1+2Z 2 ) 2 ; E u 2 0 v2 0 γ 2 where we defined γ = u (u 2 0 v 2 0)Z 2. Knowing A, B, C and D, we can calculate the differential conductance of SC-N junction as: σ SC N (E) = 2 D 2 k N k + +( A 2 (u 2 0 v0)) 2 q+ SC N k + + B 2 (u 2 0 v0)) 2 q SC N k + N = 2 d(e)+a(e)+b(e) (2.11) Since the total current through the junction has to be conserved (and equal to the current carried by the incoming particle), the following condition is satisfied: C 2 + D 2 k N k + N + ( A 2 (u 2 0 v 2 0)) q+ SC k + N + B 2 (u 2 0 v 2 0)) q SC k + N and we have the alternative expression for the conductance: = a + b + c + d = 1 (2.12) σ SC N (E) = 1 + D 2 k N k + N C 2 = 1 + d(e) c(e) (2.13) To get these formulas normalized to the high-voltage value (i.e. to the normal state conductance σ N N ), we need to divide them by the same formulas in the limit E. It s not hard to see that the normalization coefficient is just (1 + Z 2 ). The essence of these formulas is following: as we increase the voltage between two systems by an infinitesimal amount δv, a new particle at the energy E = µ + V + δv becomes available for the tunneling. It carries a charge and participates in the total current. The differential conductance σ = di/dv is equal to the current carried by that particle. In an NN junction we get σ N (V ), which is constant on the energy scale we are interested in (of the order ). In the superconducting state a particle can undergo the Andreev reflection, which effectively carries over 2 electrons to a SC. If a particle with energy E gets completely AR, we get σ SC (E)/σ N = 2. If it has a finite chance of specular reflection off the barrier, we have σ SC (E)/σ N < 2. 14

27 Figure 2.3: The BTK conductance normalized to a high voltage value, for values of Z (top to bottom curve): 0, 0.3, 0.6, 1.0, 2.0. Figure 2.4: The BTK conductance normalized to a normal state conductance of a system without barrier for values of Z (top to bottom curve): 0, 0.3, 0.6, 1.0,

28 This is an one-dimensional problem, but the same rationale applies to a three dimensional system. In that case for every given energy there will be number of particles hitting the barrier at various angles. In three dimension the ratio σ SN /σ NN = 2 means that every single trajectory that was transmitted in a normal state, got reflected in an SN junction. Saying that certain effect reduces the SN conductance, means that some of the trajectories are disallowed to undergo AR. The plots of the conductance are shown in Fig.2.3 for several values of Z. The curves are normalized to a conductance at a high value of voltage, equivalent to a normal state conductance σ N = σ SC (V ). The figure 2.4 has different normalization: the curves are normalized to a conductance of system without a barrier. This normalization has a nice property that the value σ(e) = 2 means that a particle incoming with energy E is not affected by a presence of the barrier. Two normalizations are different by the factor (1+Z 2 ). The contributions to a total conductance from the individual components is given in Fig.2.5 for Z = 0, and in Fig.2.6 for the case Z = 2.0. We see several interesting features. In the BTK problem there is no specular reflection and no branch crossing transmission for the system with the clean contact (Z = 0) (Fig.2.5 b, c)). The branch crossing processes have a very small probability, negligible everywhere except at the energies E + (Fig.2.6, c). For a system with a strong barrier specular reflection dominates everywhere except close to the gap, where the AR peak occurs. To get the current we simply integrate the conductance: I NS = 1 R 0 (1 + d(e) c(e))de (2.14) and at a finite temperature we get: I NS = 1 R 0 (1 + d(e) c(e)) (f(e V ) f(e)) de (2.15) Where R 0 is given by the normal state resistivity, and is as such a fitting parameter. However, this expression will not be used extensively. Often defined quantity is the excess current I exc = (I NS I NN ) E. 16

29 Figure 2.5: The conductance contributions from the individual components, Z = 0, E F = 1eV, = 20meV : (upper row) a - transmission without branch crossing, b - transmission with branch crossing, (lower row) c - Andreev reflection, d - specular reflection. The coefficient b and c are zoomed up to a larger scale to stress that they are exactly zero in BTK. 17

30 Figure 2.6: Contribution to the conductance from the individual components, Z = 2, E F = 1eV, = 10meV : (upper row) a - transmission without branch crossing, b - transmission with branch crossing, (lower row) c - Andreev reflection, d - specular reflection. 18

31 Figure 2.7: Temperature dependence of s-wave BCS gap, and the smearing factor f/ V. Finally, by differentiating 2.15 we get the finite temperature conductance as (using f E = f V ): σ SC N σ N N = Clearly, + ( (1 + d(e) c(e)) f ) E de = E ev ( σ(e) f ) E E ev f de = 1. In practice the limits of integration in (2.16) are ±20T. E (2.16) There are two effects of the finite temperature - smearing the features by mixing the contributions from different momenta with weight f/ E, and change of a gap magnitude with temperature, given by the gap self-consistency equation (2.1) (we assume all features of SC are of the BCS kind). Fig.2.7 illustrates these two factors. The results of calculation (2.16) are given in Fig.2.8. We see that finite temperature smears the sharp features prominent at T = 0. For that reason we shall restrict ourselves mostly to a study of the case T = 0, where any new features should be clearly visible. For a study of finite temperature effects, more useful plot is one of the zero-bias conductance as a function of temperature. We simply perform calculation (2.16) for V = 0 over a relevant range of temperatures. The results are given in Fig.2.9. Again, same factors as above determine the shape of these curves. System with different Z produce strikingly diverse plots, and this plot is most convenient way to determine Z eff for the contact junction. 19

32 Figure 2.8: Temperature dependence of the BTK conductance given for Z=0 (left) and Z=0.5 (right). Curves from the bottom correspond to T=0, 0.2Tc, 0.4Tc, 0.6Tc, 0.8 Tc, Tc. Each curve is offset by +1 from the previous one. We use µ = 1eV, = 10meV. Figure 2.9: The plots of a zero bias conductance as a function of temperature, normalized to a high voltage value, for different values of Z - from top: Z=0, 0.3, 0.6, 0.9, 1.2,

33 Chapter 3 The Nature of Gap Edge Conductance Peak, Subgap Conductance and Zero-Bias Properties An obvious question to ask is why is there a conductance maximum at the gap edge? One is tempted to argue that it is a density of states effect, as in the tunneling Hamiltonian calculations. But BTK is explicitly an one particle calculation, and there is no density of states factor appearing anywhere in the formulas. While there is no doubt that the origin of the effect in BTK and tunneling calculation is the same (by the fact that BTK with large Z reproduces the tunneling formalism calculations result of SIN junction [8]), a question remains how does it appear in the BTK framework. The answer, as we will now see, is related to the other properties of the subgap conductance in BTK, and in particular to the apparent paradox of suppression of the normalized gap conductance for a finite barrier strength. The nature of this paradox is following. In a BTK-type SN system without the barrier every electron with E < is Andreev reflected. Since that process transfers two electrons from N to SC, the normalized conductance for that (and every other) trajectory is σ SS /σ SN = 2. In an NN junction with a barrier, let s say that the fraction T of all electrons gets transmitted, and the fraction R reflected (since we shall study only the zero temperature case, we can use this notation without the possibilily of confusion). T and R are reflection 21

34 and transmission coefficients, and T + R = 1. In terms of the BTK parameter Z, they are: R = Z2 1 + Z ; T = 1 (3.1) Z 2 The reflected particles do not participate in the conductance, and we are really interested only in probability of an electron going through the barrier. We have σ nn P NN (er er) T. Since the SC transition does not change the barrier properties, one would naively expect that in SN junction, exactly the same fraction R of electrons gets reflected, and since for E < there is no transmission, the fraction T gets AR. Each of the AR electrons contributes two times the amount of charge transfer compared to the NN case, so we expect that σ SN 2T, and the normalized conductance is σ SN /σ NN = 2, regardless of the barrier strength. It is not so. Normalized subgap conductance decreases with the increasing barrier strength, as we can see in Figs. 2.3, 2.4 and 2.8. The microscopic BTK calculation doesn t give us much insight into why this happens. The reason for this, and the nature of the gap edge peak will be demonstrated more clearly in somewhat unphysical situation presented below. BTK is a special case of this, more general, argument. But first we have to observe φ - the change of a phase of a hole wavefunction compared to that of an incident electron after AR. To isolate this effect let us study a particle with energy E <, and let us observe penetration of a wave function inside the gapped region at z = 0 (so that we have no accumulation of phase difference due to the distance traveled). Then the wave functions at this point are: ψ er (z = 0 ) = e i φ/2 1 ; ψ hl (z = 0 ) = e i φ/2 0 ; ψ er (z = 0 + ) = u i 0 1 v i (3.2) Here we explicitly write the amplitudes as complex numbers of the norm 1, since we already know that under these conditions an electron is completely converted into a hole. Here u i 22

35 and v i are given by (2.2). On the SC side, for E <, u i ad v i are complex conjugates, and we included that into the ansatz of the phase of wave function (though that ansatz is in no way crucial for the final result). To calculate the difference of phases of ψ er (z = 0 ) and ψ hl (z = 0 ) we seek the difference of phases of an electron and a hole component of the wave function. We get: φ = arg(u i ) arg(v i ) = arg (u i /v i ) = arg(e/ + i 2 E 2 / ) φ = arccos(e/ ) (3.3) We see that AR itself creates the initial phase difference between an electron and a hole. At E = 0 that phase difference is π/2, and at E = it is zero. For AR of the particles above the gap there is no phase change. For the transfer of the incoming hole into the outgoing electron, the same result is still valid. This is because the matching function on a SC side is (v, u) T, so the argument is exactly the same. Going back to a SN system with the barrier, let us displace the barrier from the SN interface by a distance d, so that we have situation given in Fig.3.1. For an electron incident from the N side the following processes can occur: specular reflection back to N at the barrier - probability 1 T Fig.3.1, P1: transmission at barrier (probability T ), AR, transmission (T ) - total probability for the process T 2 Fig.3.1, P2: transmission at barrier (T ), AR, specular reflection at the barrier (1 T ), AR, transmission (T ) - total probability T 2 (1 T ) Fig.3.1, P3: transmission (T ), AR, specular reflection (1 T ), AR, specular reflection (1 T ), AR, transmission (1 T ) - total probability T 2 (1 T ) 2 process with N + 1 AR - probability T 2 (1 T ) N. Processes with the odd number of ARs, result in transfer of a hole back to the N side, and are thus akin to a simple AR (total charge transfer 2e). Processes with the even number of AR, result in transfer of an electron back to the N, and have an overall effect of specular 23

36 Figure 3.1: SN junction with a displaced barrier. Three processes represent possible trajectories after AR at the SC interface. The barrier position is a vertical line with T, R, interface is at the gap onset. reflection. It is only processes that of the former kind that contribute the conductance. Thus, of T electrons that get through the barrier, not all of them are reflected back as holes! Obviously, the conductance will not have zero voltage value equal to 2, as naively expected. We shall be able to quantify this result. Just for a purpose of making the intention clear, let us observe what would the result be if these processes could be considered separately. Then for probability that incident electron resulted in outgoing hole we would have: P incorrect (er hl) = T 2 + T 2 (1 T ) 2 + T 2 (1 T ) 4 T 2 n=0 (1 T )2n = T 2 /(1 (1 T ) 2 ) = T/(2 T ) (3.4) This is not the right thing to do, since wave functions of the outgoing holes should be added first, and then squared - i.e. in the process of reflection we should be operating with the wave function amplitudes, not with the probabilities. Recalling that in simple scattering model with same masses on two sides of the barrier reflection and transmission coefficients are given by the squares of the amplitude of reflected and transmitted wave, we define 24