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Time reversal symmetry breaking states at the surface of Cuprate superconductors Rodriguez, Jose 2002

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Time Reversal Symmetry Breaking States at the Surface of Cuprate Superconductors A microscopic approach by Jose Rodriguez Engineer, Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico, 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard THE UNWERSITYOF BRITISH COLUMBIA December 10, 2002 © Jose Rodriguez, 2002 In presenting this thesis in partial fulfilment of the requirements for an ad-vanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permis-sion for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia Vancouver, Canada Date /0~Dec Abstract ii Abstract It is known that, in cuprate superconductors, a subdominant order parame-ter can coexist with the dominant d-wave one close to vortices, impurities and boundaries. This subdominant order parameter, together with the dominant one, is responsable for the generation of Time Reversal Symmetry Breaking (TRSB) states at an "ab" oriented surface in cuprate superconductors. Must of the analysis done on this TRSB surface states, have been done within a quasiclassical framework. This thesis analyzes the problem by solving numeri-cally the microscopic Bogoliuvob-de Gennes equations in a lattice. Comparison with the quasiclassical expectations is done and some features are commented. The pairing interactions for the order parameters considered here are: d x 2 _ y 2 , d x 2 _ y 2 + is and d x 2 _ y 2 + id x y . Contents iii Contents Abstract ii Contents iii List of Figures iv Acknowledgements vi 1 Introduction 1 2 Quasiclassical theory 4 2.1 Infinite superconductor 4 2.2 Finite size superconductor 9 3 Microscopic theory 12 3.1 The many body Hamiltonian 12 3.2 Mean field theory and the Bgoliubov-de Gennes equations . . . . 13 3.3 Observables 16 3.3.1 Current 16 3.3.2 Local Density Of States (LDOS) 17 3.3.3 Magnetic flux •-. 19 3.4 The cuprates 20 4 Numerical solutions for the B d G equations 23 4.1 Numerical solutions 23 4.2 Pure d-Wave 26 4.3 d+is 31 4.4 d+id 34 5 Conclusions 39 Bibliography 41 List of Figures iv List of Figures 1.1 Different order parameters allowed by the symmetry in High-Tc superconductors with tetragonal structure 2 2.1 Bound states at the surface of a superconductor. Taken from Lofwander et al. (2002) [17] 5 2.2 Current distribution (left) and magnetic field (right) of a finite d+is island. Taken from Amin et al. (2002) [2] 10 2.3 Current distribution (left) and magnetic field (right) of a finite d+id island. Taken from Amin et al. (2002) [2] 11 3.1 Conduction planes in cuprates 19 3.2 Line integral in the lattice 19 3.3 Cu-0 conducting plane in a cuprate. The picture represents each atom in the lattice by its overlapping orbital. Taken from Poole etal. (1995) [23] 21 4.1 Lattice sites for an infinite wall of superconductor with surfaces parallel to a {1,1,0} plane. The points are the copper atoms. . . 24 4.2 Lattice sites for an finite piece of superconductor with one TRSB edge parallel to a {1,1,0} plane 25 4.3 Differential tunneling current for a d-wave superconducting wall as function of the bias voltage and temperature at site 1. ix = — 1 and Vi = 3 26 4.4 d-wave and s-wave component of the order parameter in a d-wave superconducting wall. Observe the different scaling for the two order parameters, /x = — 1, Vi = 3 and T = 0.05 27 4.5 STM conductance after one iteration in at the surface of a d-wave superconducting wall at site 1. fx = -1, Vi = 3 and T = 0.05. . 28 4.6 Current generated along a d-wave superconducting wall with n = -1, Vi = 3, T = 0.05, 30 sites and 31 cells 28 4.7 Magnitude of the current (parallel to the walls of the supercon-ductor) as a function of position in a d-wave superconducting wall with ix = -1 and Vi = 3 29 4.8 Extended-s component of the order parameter at site 2 vs. cur-rent at the same site in an infinite wall of d-wave superconductor. Each point is taken at a different temperature, starting from the left: 0.125, 0.1, 0.075, 0.065, 0.05 and 0.01. ix = -l and Vi = 3. 30 List of Figures v 4.9 dl/dV against differential potential at two different sites of an infinite wall of d-wave superconductor, fi — — 1, Vi = 3 and T = 0.05 30 4.10 Spontaneous current generated at a finite d-wave island with a subdominant anisotropic s-wave component. The sides of the triangle have 22 sites, n = -1, Vi = 3 and T = 0.025 31 4.11 Magnetic flux generated at a finite d-wave island. The sides of the triangle have 22 sites, n = -1, Vi = 3 and T = 0.025. . . . 32 4.12 Magnitude of the different order parameters along an infinite d+is superconducting wall. \x = -1, Vi = 3, Vo = -1 and T = 0.05. . 32 4.13 Magnitude of the current along an infinite wall of d+is super-conductor as function of position, fi = —1, Vi = 3, Vo = —1. 33 4.14 dl/dV against differential potential at two different sites of a d+is superconducting wall, fi = —1, Vi = 3, Vo = —1 and T = 0.05. . 34 4.15 Distance between peaks on the dl/dV graph (S) against magni-tude of A 0 both at site 1. ii = —1, Vi = 3, Vo = —1- Each point correspond to a different temperature; starting from the left: 0.05, 0.04, 0.03, 0.025, 0.01 and 0.009 ; . . . . 35 4.16 Tunneling density of states at a (110) surface as function of energy (e). Here the result with p = 0.00 correspond to a surface without roughness. Taken from Rainer et al. (1998) [24] 35 4.17 dl/dV against differential potential for site 1 in a d+id supercon-ducting wall, /i = —1, Vi = 3 and V2 = -1 36 4.18 Non selfconsistent dl/dV against differential potential for site 1 in a d+id superconducting wall; the self-consistent solution is presented for comparison, ii = — 1, Vi = 3, V2 = — 1 and T = 0.025 37 4.19 Magnitude of the Current, as a function of position, along a d+id infinite superconducting wall, fj, — — 1, Vi = 3 and V2 = — 1. . . 37 4.20 dl/dV against differential potential at two different sites for an infinite wall of d+id superconductor. \x = -1, Vi = 3, V2 = -1 and T = 0.05 38 Acknowledgements vi Acknowledgements I want to thank my supervisor, Dr. Marcel Franz, for suggesting me to work in this very interesting research topic and for his great guidance through the project. I'm also very grateful to Dr. Franz for being a teacher in what can be called "Career-in-Science Dynamics". I wold like to thank Dr. Alex Zagoskin, for accepting reviewing this script and for all his invaluable comments and suggestions. This thesis got many contributions from every member in my superconduc-tivity group: Tom Davis, Tami Preg-Barnea and Dr. Daniel Sheehy. I thank them all for enriching this thesis with their comments, questions and discussions. I am very grateful to Mariangela for her support during my Master in Sci-ence studies, for reviewing the drafts and for her right advice on computational problems. I thank also Ignacio Olabarrieta for his great help on every computer-related problem I referred to him. Finally, I want to thank my family who has supported me on every decision I have made in my life. Chapter 1. Introduction 1 Chapter 1 Introduction The d-wave nature of the pairing potential in the cuprate superconductors is no longer a question. During the middle 90's a lot of effort was put in dentifying the characteristics of the order parameter in the cuprates. Many experiments (like Angle Resolved Photo Emision) pointed out that it should have a four lobe symmetry, but the question of its real nature, d-wave or anisotropic s-wave, the two possible candidates, (Sigrist (1998) [25]) was still unclear. The strongest evidence for a d-wave pairing came from Josephson tunneling experiments(Wollman et al. (1993) [34], Van Harlingen (1995) [13] and Tsuei (1996) [32]), since they allow the detection of the sign change in the lobes of the order parameter (Sigrist and Rice (1995) [28]). Besides the Josephson tunnel-ing experiments, supporting evidence for a d-wave like order parameter came from the work done by Hu (1994) [14]. He showed, using quasiclassical approxi-mations (chapter 2), that the previously observed Zero Bias Conductance Peak (ZBCP) on Scanning Tunneling Microscopy (STM) conductance studies of "ab" oriented superconducting samples, was a consequence of the d-wave nature of the order parameter. He also proved that this effect could not be explained in terms of an anisotropic s-wave order parameter. As will be shown in chapter 2, a cuprate superconductor with one of its surfaces along a {1,1,0} plane1, creates low energy surface bound states at this face of the sample. These states are responsible for the ZBCP on the STM conductance measurements. In 1995 Sigrist, Bailey and Laughlin [27] used the concept of Time Reversal Symmetry Breaking (TRSB) states to explain the experiments on "ab" oriented grain boundaries done by Kirtley et al. The mechanism trough which this TRSB arise made use of a subdominant order parameter2 that can coexist with the dominant "d" at the surface of the grains. In the same line in 1997, Fogelstorm et al. [9] predicted a ZBCP splitting in STM studies on cuprates with a surface along an {1,1,0} plane; this, as a consequence of the TRSB state of the system. Short after, Covington et al. [6] observed experimentally the splitting of the peak. It was found after, that the ZBCP is energetically unstable and many mech-anisms, besides a subdominant order parameter, could give rise to the ZBCP splitting and, therefore, to a TRSB state (for a review on different mecha-1The cuprate superconductors have a tetragonal structure (chapter 3). 2 A subdominant order parameter is an "extra" order parameter that is almost completely suppresed by the dominant d-wave one in the bulk of the sample. Nevertheless, this subdom-inant order parameter can get enhanced close to vortices, impurities and, as will be shown later on this script, close to boundaries Chapter 1. Introduction 2 nisms see Lofwander et al. (2001) [17]). Nevertheless, the splitting through a subdominant order parameter is the most supported theory (Kos (2001) [33], Sigrist (2000) [26]) and will be the TRSB mechanism used in this thesis. Almost all the theoretical work in TRSB surface states has been done within the framework of the quasiclassical theory (chapter 2). Also, in these studies, almost all the attention has been given to pure d-wave superconductors and superconductors with an s-wave like subdominant order parameter (this super-conductors are also called d+is 3 , the subdominant order parameter has a phase difference equal to e1*^ 2 with respect to the dominant one). In this thesis, a microscopic theory is used for analyzing TRSB surface states in a superconductor. A microscopic approach is used because, as said by Tanuma et al. (1997), quasiclassical calculations "ignored several distinctive features characteristic to High-Tc materials: i) short coherence length, which invalidates the quasiclassical approximation and ii) strong correlation". The formalism that will be used, the Bogoliubov-de Gennes (BdG) equations, account for the short coherence length displayed by High-Tc superconductors. Nevertheless, the strong correlation effects are ignored by this microscopic theory due to the mean field approximation in which it is based. Since the real symmetry of the subdominant order parameter is still un-known, in this thesis the three possible scenarios are considered (see figure 1.1): d (no subdominant order parameter), d+is and d+id (in this last one the sub-dominant order parameter has a d-wave symmetry and its lobes make an angle of 45° with respect to the dominant d-wave one). The d+id superconductors in a TRSB state, have been studied few within the quasiclassical framework (Matsumoto and Shiba [19], [20], [21]; Rainer et al. [24]) and, as far as we know, never with a microscopic theory4. One of the objectives of this work is to produce some clues towards the determination of the subdominant order parameter symmetry. Figure 1.1: Different order parameters allowed by the symmetry in High-Tc superconductors with tetragonal structure. Besides the intrinsic importance of every scientific piece of knowledge, study-ing TRSB systems is important from a technological point of view. Amin et 3Through this thesis: d x 2 _ y 2 = d, d x 2 _ y 2 + is = d + is and d x 2 _ y 2 + i d x y = d + id. 4 The superconductors in a TRSB state that have been studied microscopically are pure d-wave (Tanuma et al. [30]) and d+is (Zhu et al. [36]) Chapter 1. Introduction 3 al. [2] point out in May of this year that, small islands of superconductors in a TRSB state might be able to work as q-bits. This possibility is due to the fact that the TRSB state of the system is double degenerate. The difference between the two states is only the direction of the current along the surface. If the magnetic field produced by this currents is measurable, then, this two level system can be used as a q-bit. Must be said that all the systems analyzed in this thesis have the TRSB surface parallel to a {1,1,0} plane. For a microscopical study on the effect of different surface orientations in pure d-wave superconductors see Tanuma et al. (1998) [30]. Also, an analysis of roughness of the surface and the effect of impurities in the superconductor is off the scope of the present work. It has been found quasiclassically that the roughness of the surface has a significant effect on TRSB surfaces [24]. The possibility of a microscopic simulation of a rough TRSB surface is discussed in the conclusion. The thesis is structured as follows. Chapter 2 presents a review of some calculations and predictions for TRSB superconducting surfaces within a qua-siclassical formalism. In the first part we study the simplified picture5 of an infinite superconductor coated by a thin metal layer, in other words, a quantum well is analyzed. A pure d-wave superconductor is assumed and the origin of the ZBCP is investigated. Then a subdominant order parameter is added in the calculation. As a consequence, it is shown that the ZBCP is splitted. Two subdominant order parameters are considered, "d" and "s" and it is shown that they give rise to different Local Densities Of States (LDOS). The second part reviews the results obtained by Amin et al. [2] for two finite size islands of d+is and d+id superconductor. In Chapter 3 a short review on the microscopic BdG formalism is given. Starting from the many-electron Hamiltonian, the self consistent Bogoliubov-de Gennes equations are developed for the case of a tight binding crystalline lattice. After that, observables and interesting quantities are written in terms of the excitation energies En and the quasiparticle amplitudes u and v. Then the application of this model to cuprate superconductors is discussed. Chapter 4 contains the results from the numerical simulations. First, some important points on the numerical algorithms are made. Then, three kinds of pairing interactions are analyzed: d, d+is and d+id. For each of them two sit-uations are considered: an infinite wall of material; and a finite island (without much success for the last one, as will be pointed out). Finally, in Chapter 5, some comments on the previous chapter are carried further; agreements and disagreements between the microscopic and quasiclas-sical results are pointed out, and conclusions are made. 5 For a complete quasiclassical analysis of TRSB states at a surface of a superconductor the reader is refereed to the series of papers by Matsumoto and Shiba [19], [20], [21] Chapter 2. Quasiclassical theory 4 Chapter 2 Quasiclassical theory This chapter presents an overview on quasiclassical results and estimations for a d-wave superconductor with a surface along a {1,1,0} plane. Later, in chapter four and five, the quasiclassical theory presented here will be compared with the microscopical one. In the first part of the chapter, we follow T. Lofwander et al.[17] in an exposition of the physics produced at the edge of a semi infinite d-wave su-perconductor, with its surface along a {1,1,0} plane. After that, the chapter finishes with a review of the work done by Amin et al. [2] on finite TRSB superconductors (islands). 2.1 Infinite superconductor First, it will be seen how surface bound states can arise at the edge of a very long superconductor. The physical picture is sketched in figure 2.1. A semi infinite superconductor x — L > 0 is covered by a thin normal-metal layer of thickness L and the rest of the space is nothing but vacuum. The situation is that of a quantum well1. At a first glance, electrons in the metal with energy within the gap of the superconductor will not be able to tunnel into the superconductor, nether as cooper pairs nor excitations 2 ; therefore, they are expected to form bound states in the well. The quasiclassical prediction for the energy of these bound states is given by the Bohr-Sommerfeld quantization condition (the phase acquired by the particle along the classical trajectory must be equal to an integer times 27r). Assuming plane wave solutions on the metal this condition is written as: - (7i + 72) ¥ (Xi ~ X2)+v 0 ^ = 2mr (2.1) a b where all the terms will be explained next: a ) Phase due to Andreev reflections. In 1964, Andreev [3] studied the reflection of particles in a superconductor-normal metal interface, moti-vated by the anomalously large thermal resistance in superconductors. His 1 The dominant order parameter in a superconductor decrease its bulk vlue as it approach the surface of the superconductor (as will be seen on chapter 4). The physical meaning of the metal layer in our simplified model corresponds to the transition region of the dominant order parameter. 2 All the energies on this thesis are measured from the Fermi energy Chapter 2. Quasiclassical theory 5 I N S k i 9 L x Figure 2.1: Bound states at the surface of a superconductor. Taken from Lofwander et al. (2002) [17]. calculations had a surprising result; incoming electrons (holes) from the normal metal do not suffer the expected specular reflection on the bound-ary; instead, the incoming electron (hole) with velocity Vg is "reflected" as a hole (electron) with velocity -Vg (see figure 2.1). Andreev also no-ticed that the reflected particle has an extra phase that depends on its energy and the phase of the order parameter as seen by the particle upon reflection. A more explicit treatment is given next. The Bogoliubov-de Gennes equations in a d-wave superconductor are: In a superconductor (or a metal) only the electrons close to the the Fermi surface are involved in the properties of the material. This motivates the separation of the fast oscillations by means of: (2.3) (2-2) (2-4) Chapter 2. Quasiclassical theory 6 After substituting eq. 2.4 in the Bogoliubov-de Gennes equations, the wave function inside the integral can be expanded in series. Only terms of first order on (fc/^o)-1 are kept since for a superconductor (fc/£o) - 1 ~ Tc/Tf >C 1 [5]. Then the BdG equations can be written as: — ihvfn • Vu(f) + A(n, r)v(r) — Eu(r) ihvfn-Vv(r) + A*(n,f)u(f) - Ev(r) (2.5) These are the quasiclassical Andreev equations. Here n is a dimensionless unitary vector in the kf direction; and Vf is the Fermi velocity. In order to get insight on the "reflection" of normal electrons (holes) at the interface with a superconductor, the Andreev equations will be solved for an infinite junction; that is, a semi-infinite superconductor for x > 0 and a normal metal for x < 0. Let's assume that the mean free path of the particles is infinite and that the order parameter is given by: 0 x<0 . Ae* x > 0 { °> where A is a real number and x is the phase of the order parameter that, in general, depends on the incidence angle of the particle. Must be said that this choice of the order parameter is not self-consistent and it neglects any proximity effect 3 . Also, this order parameter does not agree with the quasiclassical assumption (Cofc/)-1 <C 1. Nevertheless, it can be shown that the results obtained with this oversimplified pair-ing potential agree qualitatively with other quassiclasical self-consistent calculations (for a complete self-consistent treatment see Matsumoto and Shiba. [19] [20] [21]). Using the gap from eq. 2.6 into Andreev equations (eq. 2.5), it can be seen that the solutions for the normal metal are: E = ±hvfk-f^- (2.8) where k is parallel (or antiparallel) to kf. The + sign in the energy corresponds to the electron-like solution and the - to the hole-like one (9%). 3In a metal-superconductor junction "... Cooper pairs form a superconducting metal in close proximity diffuse into the normal metal" (Tinkham, 1996, [31]) overlapping the wave functions. It can be said that the interface becomes "smooth". Chapter 2. Quasiclassical theory 7 In this last equation it is seen that the group velocity (ifg = jrVkTE; where E = ±(kr — kf)hvf and kr = kf + k) is opposite for the electron-like or hole-like part of the solution. Since we are interested in energies within the gap of the superconductor, the solutions on the superconducting part wil l be given by: U 0 \ e - f r e i k f T + j g l ^ V 0 \ p r - r p i k r f v/A2 - E2 \voe '*/ \ u oe % x ' E-hvfM V o = A e * U o where f points along kf. Let's assume an electron is coming from —oo to the boundary. The solu-tions on each side of the material will be given by: incoming Andreev reflected electron hole * w = >lQ e i(*/+*V + s Q e ^ - ^ V (2.10) * S = j D ( v o e - x ) e ~ f r V f c " / r " < 2 - U ) where the exponentially decaying wave function is chosen for the super-conductor (the exponentially increasing one gives no physical solution). It must be noticed that a specularly reflected electron can't be part of the solution since it is energy is not equal to that of the incoming electron (see eq. 2.8). Applying continuity conditions on the wave function at the boundary, it is found that: u 0 ' \ u o / The outgoing hole carries an extra phase given by: B = e-^+x* ; e** = ; 7 = arccos f (2.12) where the relation for 7 can be obtained substituting eq. 2.9 (with A' = 0) on the Andreev equations. The (7 + x) term is the extra phase obtained by a particle upon reflection from the surface of a superconductor. If the incoming particle is a hole, it can be shown by the same procedure that x'~* ~X- It worth notice that on each reflection a charge of 2e is transferred to or from the condensate. Chapter 2. Quasiclassical theory 8 b ) Phase acquired by the particle along the path. This is written as: _ ( ! _ - _ ] ) + cos 8 where /?o is the phase acquired by the particle on the refection at the normal metal-vacuum interface. Note the (—) sign before the absolute value of the hole wave vector; this sign is due to the fact that, even though the hole group velocity after the reflection is opposite to the electron one, the total wave vector kh = kf — k still points along the electron one (since k <C kf), giving rise to a negative sign on kh, • dr. Now the =F sign on the Bohr-Sommerfeld quantization condition (eq. 2.1) will be explained. When the electron has its parallel wave vector component along y (see fig.2.1), the \ contribution from the Andreev reflections to eq. 2.1 is given by —(xi ~ X2); the phase of the order parameter "seen" by the electron is xi and the one "seen" by the hole is X2- If the situation on fig.2.1 is inverted (that is, the electron travels down and the hole up), the order parameter "seen" by the electron is X2 and the one "seen" by the hole is xi; then the x contribution will be given by +(xi - X2)-For a d-wave superconductor with an order parameter oriented as shown figure 2.1 4 , the Bohr-Sommerfeld condition can be written as: -(7i +72)+ ir + P = 2im (2.14) where it was used the fact that the order parameter changes sign after a normal reflection (the order parameter is d-wave), that is, Ax = X i ~ X2 = X — X — 7 r = — TT. One of the main predictions of the quasiclassical theory for the problem can be noticed now. If eqs. 2.8, 2.12 and 2.13 are substituted in eq. 2.14, it can be observed that the obtained equation has the root E=0. This means that, if we compute the Local Density Of States (LDOS) using eq. 2.14, some states will be found at E=0, further more, a peak on the LDOS will be formed at this point (mid-gap states). In a similar way Hu [14] explained the zero bias conductance peak observed in some High-Tc superconductors. The previous phenomenon contrast with what would have been predicted for a superconductor where the order parameter has s-wave symmetry [15]. In this case it can be shown [17] that the E=0 root is not allowed and the LDOS has to go to zero at this point. Equation 2.14 also shows that the d-wave symmetry of the order parameter removes the =r- dependence on the Bohr-Sommerfeld condition; producing a degenerancy on the sign of k% (the component of the electron wave vector along y). It will be seen in chapter 4 that close to the TRSB surface a subdominant order parameter can coexist with the dominant d-wave analyzed so far. Now, 4This orientation of the order parameter correspond to the case of a cuprate superconductor with a surface parallel to a /1,1,0/ plane, as will be seen on section /refcuprates Chapter 2. Quasiclassical theory 9 the equation 2.1 will be studied for this situation. Let's assume that the gap in the superconductor (at the boundary) is given by A<j(0) + iAs. Then the absolute value of the phase seen by the particle on each Andreev reflection is X r e i = arctan A<j/As. The sign of this phase depends on the type of incident particle (electron-like or hole-like) and on the sign of ky. Substituting Xrei m eq. 2.1 gives: - 2 7 + 2Xrei + v + 0(E) = 2wr (2.15) where the + sign corresponds to a negative ky and the — sign corresponds to a positive ky. This subdominant order parameter removes the +" degenerancy that was bring by the d-wave order parameter (see 2.14). This means that, if a subdominant order parameter is present, the energy or the bounded particles will depend on the sign of ky. Using the solutions from the Andreev equations and appropriate boundary conditions, Yang and Hu [35] showed that the quasiparticle energy is given by: E = -AsSign(ky) (2.16) Two interesting things come out from eq. 2.16: first, the zero energy peak on the local density of states splits, one peak is located at A„ and the other at —A3; and second, the time reversal symmetry of the system will break for these states. At half filling band and low temperatures almost all the +ky states will be filled producing a net current along the edge of the superconductor. This current is the signature of the Time Reversal Symmetry Breaking (TRSB) effect. For a d+id superconductor, Yang and Hu showed in the same article that the energies of the quasiparticles are given by: E = -Ad±^^Lsign{ky) (2.17) 0 *2 + *?) Since this relation is dispersive they expect a broadening of the zero energy peak on the local density of states at the edge of a d+id superconductor but no further calculation is reported. Ranier et al. [24] calculated the LDOS at the the edge of a d+id superconductor and found a double peak, like in the d+is case, but more states are found between the peaks; a broadening as expected from eq. 2.17 (see figure 4.16). 2.2 F i n i t e s ize s u p e r c o n d u c t o r Experimental work and technological applications have to deal with finite size superconductors. This motivated Amin and coworkers [2] to study such systems. The theoretical treatment of those 'more realistic' systems is harder; in their work, Amin et al. used a modified Eilenberger formalism to deal with the complex boundary conditions of the problem. A summary of their numerical results on triangular shaped pieces of superconductor is shown next. Chapter 2. Quasiclassical theory 10 Figure 2.2 shows the spontaneous current distribution generated on a d+is piece of superconductor and the magnetic field produced by it. The orientation of the dominant order parameter is shown within the picture and the length of the longer edge is 30£o- It can be noticed that a wide spread peak of magnetic field is generated along the pair breaking edge; the magnitude of the magnetic field at this peak is of the order of 10 - 4 - 10~5 G . The width of this peak was found to extend over a distance of the order of the penetration depth; therefore the Meissner effect provides a mechanism for returning the current. The flux generated by this magnetic field can be large enough to be measurable. Figure 2.2: Current distribution (left) and magnetic field (right) of a finite d+is island. Taken from Amin et al. (2002) [2] The current and the magnetic field generated by a triangle of d+id super-conductor are shown in figure 2.3. Now the current does not travel counter clock wise all along the piece; two vortices are formed at the corners of the island and their currents oppose to the one along the edge. Since the phase of the order pa-rameter does not wind after going around the vortex then no flux quantization is observed. The magnitude of the magnetic field at the peaks of the vortices is of order of 10 - 4 — 10 - 5 G and magnitude of the magnetic field in the peak between them is one order of magnitude less. This last peak is localized within a short distance (<C A L ) from the border. This is due to the superscreening effect, which is expected to show up at the edge of the superconductor [1]. The flux generated by material is strongly peaked only at the vortices, therefore is small and difficult to measure. Chapter 2. Quasiclassical theory 11 Figure 2.3: Current distribution (left) island. Taken from Amin and magnetic field (right) of a finite d+id et al. (2002) [2] Chapter 3. Microscopic theory 12 Chapter 3 Microscopic theory The previous chapter presented a quasiclassical approach to the physics pro-duced at a Time Reversal Symmetry Breaking (TRSB) surface in a supercon-ductor. The purpose of this chapter is to present a microscopic model for High-Tc superconductors. The equations developed here will be solved numerically in the next chapter for the case of a TRSB surface in a superconductor. The present chapter is structured as follows. First, the microscopic Bogoliu-vob-de Gennes (BdG) equations in a crystalline lattice are reviewed. After that, different observables are written in terms of the quasiparticles amplitudes u and v. Then, in the last section, the connection between the microscopic model and the High-Tc cuprates is analyzed. 3.1 The many body Hamiltonian For materials where the lattice constant is "big" enough, so just the outermost orbitals of the atoms in the lattice overlap, the tight binding approximation can be applied (see Ashcroft and Mermin, 1976, chapter 10 [4]). Any formalism that uses this approximation must include electron-electron interactions if it aims to reproduce any High-Tc superconductivity behavior. The tight binding many body problem can be written in terms of annihilation-creation operators for fermions. Landau and Lifshitz, in the third volume of their famous "Course on Theoretical Physics" [16], show that this Hamiltonian is written as: i,S,(T i,<7 + 2 X] ^ri,k,l,mC'l(TlCl<a.2Cm>a2Ci^1 (3.1) Here the operators C\ and Ci create and destroy, respectively, an electron in the lattice site i; that is, they create and destroy an electron with a determined Wannier function [4]. In eq. 3.1 the sum over i is done over all the lattice sites; the sum over S is done over the nearest neighbors sites, and the sum over <jj is done over the spin degrees of freedom. Chapter 3. Microscopic theory 13 The "t" in the above Hamiltonian is called "hopping amplitude" and can be written in terms of the Wannier functions of nearest neighbor sites (<j)i) as: t = J 4>*Hs<j>i+sdr (3.2). where Hs is the one electron tight binding Hamiltonian. The hopping amplitude in eq. 3.2 is independent of 6. That is, the overlap integral is the same in every direction of the lattice and, therefore, this symmetry must be reflected in the overlapping orbitals. This is the case for the cuprate superconductors as will be shown in section 3.4. The many electron Hamiltonian in eq. 3.1 includes hopping terms just be-tween nearest neighbor atoms. The hopping terms to farther neighbors have been neglected compared with "t". This can be thought as an "extreme" tight binding approximation, since the next degree of approximation would be a chain of independent atoms. In the second sum in eq. 3.1, "/J," is the chemical potential of the system. If /i equals the energy of one atomic orbital for a single atom, it can be shown that the energy band originated from our tight binding model will be half filled [4j. From now on the zero energy will be defined at this level so if the system is doped with holes, then the chemical potential n will take a negative value. The last term on eq. 3.1, the electron-electron interaction, is very general. It can express Coulomb repulsion 1 or the interactions that are responsible for superconductivity. These latter interactions can be of two kinds: on-site interactions, or interactions between electrons on neighbor atoms. These two types of interactions can be introduced in the Hamiltonian (eq. 3.1) as: i,d,<r i,o + V0 ^ rii-fJlii + ~Y «»+(5,o-l«i,<7-2 (3.3) where the following definition have been used: n,^ = C\aCi^. is the number operator, its eigenvalue equals to the number of electrons in site i with spin a (1 or 0) and its derivation is not much different from the number operator for the harmonic oscillator that can be found in any quantum mechanics book. 3.2 Mean field theory and the Bgoliubov-de Gennes equations The many body Hamiltonian in eq. 3.3 is difficult to solve. Nevertheless it can be diagonalized using mean field theory in the interaction terms. The following 1 Coulomb interactions will not be included in the Hamiltonian since it can be shown that, within the mean field approximation that will be done later, this term would just add a constant to \i [7] . Chapter 3. Microscopic theory 14 definition will be made: &oi = V0(CnCu,) Ali,s = V1{Ci+SttCi<l) (3.4) in this definition () express thermal average. The physical meaning of this thermal averages will become clear later when they will be written in terms of annihilation-creation operators for excitations2. It must be said that, after this mean field approach, the quasiparticles do not "see" each other any more. They interact only through an average field. This unable the theory to describe any strong correlation effect. Using the definitions on eq. 3.4, the Hamiltonian in eq. 3.3 is written as: i,<5,<7 i i,S The Bogoliubov transformation diagonalizes the above Hamiltonian and is defined as [7]: CiA = __tT».tu».< ~ l l M (3-6) n n here the u and v amplitudes are the same as in the BCS wave function. The operator 7* acting on the ground state wave function, creates an excitation in the energy state n with spin a [7]. The anticommutation relations for this operators are: {7n,CTi»7m,0-2} = ^nm<5<7i<T2 {7n,<ri,7m,a2} = 0 ( 3 - 8 ) Since the operators 7 annihilate and create excitations, it is expected that the thermal average of the number operator for excitations (7 ]^<T7n,<r) is a fermi function. This is the case, more precisely [7]: (7^,0-17m,<7 2) = ^n,m<5(7i,(T2 eEn/kBT + I (7n,<717m,<r2) = 0 (3.9) where En is the energy of the excitation. 2 A is the Ginsburg-Landau order parameter. See de Gennes, 1999 [7] Chapter 3. Microscopic theory 15 If the Bogoliubov transformation and the thermal averages for 7 are used on the definitions in equation 3.4, it is found that A 0 and Ai can be expressed as: A o i = V o ^ u « X , i ( ( 7 n , t 7 i 4 ) - ( 7 i 4 7 n , t » n = Vo ^ U n , i K , i t a n h ^ 2kBT It Aii,j = y 5]un,i+«<i((7n,t7i,t> - <7i,4.7n4» + U n , x<, i + 4((7n,t7i , t > - ( 7 i 4 7 n , i ) ) = y $3[un,i+a<i + « n , i < i + a ] t a n h (3-10) The Bogoliubov transformation diagonalize the Hamiltonian on equation 3.5. Then, it must be possible to write this Hamiltonian as: H = £ g r o u n d s t a t e + ^ -En7n,<r7n,<7 (3.11) n,cr The commutator [Ci^,H] can be calculated using the Hamiltonian on eq. 3.5 or the the one on eq. 3.11 (and using the anticommutation relations for 7, eq. 3.8). If the coefficients for 7* and 7„ are compared in the two commutation relations [Ci,t, H], the next set of equations is obtained. - Un,i+S ~ Vun,i + A 0 iW„,j + ^2 &li,6Vn,i+d = EnUn,i S S t^Vnii+s+/lVn,i +&0iun,i + 'z2&litSUn>i+s = EnVn,i (3.12) ,5 <5 these are the Bogoliubov-de Gennes (BdG) equations. Together with the self-consistent relations in eq. 3.10, they determine the quasiparticle amplitudes it and v, for a given problem. This equations are usually written in a more compact way as: ( i - ' ) ( : ; ) - M : ) <-3> where the matrices | and A are given by: (£un)i,l = ~t Un,j+5 - IMn,j s (Av„)i,i = An,6Vn,i+5 + A 0 i W „ , i (3.14) Chapter 3. Microscopic theory 16 In the next chapter the BdG equations will be solved numerically for super-conductors with different order parameters (A). For example, if the interactions considered are between quasiparticles at nearest neighbors and next-nearest neighbors, then, the order parameters that would be used in the BdG equations are only Ai (nearest neighbors) and A 2 (next-nearest neighbors). 3.3 Observables Equations 3.12 with the self-consistent conditions 3.10 determine the quasipar-ticle amplitudes u and v. The purpose of this section is writing observable quantities (current, magnetic flux, and LDOS) as function of u and v. 3.3.1 Current The classical current is defined as: ; = f p (3.i5) where the polarization vector P is defined as: P = £V(%* (3.16) i here the sum is over all particles in the system, r is their position and qi is the charge of the ith particle. The quantum mechanical analogue for the equations 3.15 and 3.16 (in the Heisenberg picture) is [18]: Jw = *j:[ff,Pw..] ( 3 1 7 ) where it has been assumed that the particles are electrons (with charge e). Note that within the tight binding approximation, the current flows between nearest neighbors only. If the commutator on eq. 3.17 is calculated with the Hamiltonian 3.3, it can be shown that the current between nearest neighbors is given by: j i , * = -*Y __ (c\+5,„ci,° ~ 4,*ci+s,*) (3-18) Using the Bogoliubov transformation and obtaining the thermal average, the current is expressed in terms of the u and v amplitudes: ji<s = real 2iet " f t " E ( ( l - t a n h ^ ) < i + ^ + (l + tanh V n , i + * < i ) (3.19) Chapter 3. Microscopic theory 17 This equation give the currents flowing through each bond of the crystalline lattice. It will be seen on section 3.4, that the problem to be solved will take place in a two dimensional square lattice (see figure 3.1). There fore, each site in the lattice has four currents associated. In order to report currents in a simpler way the next definitions will be used: ii,x Ji, — x Jxi -jyi = hAZJhzl (3.20) All the currents reported on next chapter are those which components are calculated using equations 3.20. 3.3.2 Local Density Of States (LDOS) During an Scanning Tunneling Microscopy (STM) study a metallic tip is held very close to the surface of a conducting sample in a vacuum chamber. A voltage between the tip and the sample allows to control the chemical potential difference between them. If a difference of chemical potentials is present, the quantum tunneling current in one direction (eg. from the normal metal tip to the sample) will be bigger than the one in the opposite direction (eg. from the sample to the normal metal tip). Assuming that the sample is at zero potential, this two currents can be written as [23]: Jn^3 = e JPN(E)^S(B)Dn{E - eV)f(E - eV) dE (3.21) Ds(E)f(E) dE (3.22) In these equations Dn and Ds are the LDOS in the normal metal tip and in the sample; f(E) is the Fermi distribution function and e is the electron charge. In equations 3.21 and 3.22, the term Pi(E)-vj(E) 1 S the probability for a quantum mechanical transition from the state i(E), in one material, to the state j(E) in the other material and is given, to first order, by the Fermi golden rule for time dependent perturbation theory as: Pi(E)^(E) = y|(i(£)|i?p|i(£))|2^(^)(l -f(E)) = T\HPiiB)ilE)\2Dj(E){l-f{E)) (3.23) where Hp is the perturbation in the Hamiltonian. Adding the equations 3.21, 3.22 and multiplying by the effective area of the normal metal tip (A); the total current is: Chapter 3. Microscopic theory 18 I = — \HP j{EF)i{BF)\2Dn(EF) J D.(E) (f(E - eV) - f(E)) dE (3.24) In this last equation it has been assumed that the matrix element Hp ji and Dn, are independent of energy. This is a good approximation for a small range of energies around the Fermi energy, that is, in the range of energies that contribute to the integral in equation 3.24. If the equation 3.24 is differentiated with respect to V: dl j D.{E)f'(E - eV) dE (3.25) Then, as the temperature is lowered: f'(E - eV) —> S(E - eV) and dl/dV —> D8{eV) (3.26) Therefore an STM scan at low temperatures allows to measure the LDOS. It is desired to have a model for dl/dV in terms of the quasiparticle ampli-tudes u and v since: 1) at low temperatures, dl/dV is approximately equal to the LDOS, and 2) it is needed for comparison with experimental data. This quantity was modeled by Gygi and Shiilter [12] in the following way: the current is related to the one-particle spectral functions in the tip of the microscope and in the superconductor: It will be assumed that the presence of the superconductor surface (a {0,0,1} plane) does not affect the u and v amplitudes; and that the tip of the probe is a point like normal metal. Then the spectral functions are: As(i,E) = 27rJ2[\un,i\26(E-En) + \vn,i\26(E + En)] (3.27) n AN(i,E) = 2nJ2HE-ek) (3.28) k The tunneling current is then: I{i,V)<x j ^As{i,E)AN{i,E + eV) tanh E + eV 2kBT tanh E ' 2kBT_ dE (3.29) and its derivative: dI(i,V) av oc cosh 2 E-eV 2kBT cosh 2 E+eV 2kBT (3.30) V Chapter 3. Microscopic theory 19 3.3.3 Magnetic flux The structure for cuprate superconductors will be explained in the next section. There, it will be seen that the conduction in these materials takes place along planes of atoms like those shown in figure 3.1. The conduction, or "hopping", of electrons between planes is so small that will be neglected. Electrons can hope just between atoms at nearest neighbors (this is the approximation that has been used on equations 3.12) therefore the currents can flow just between nearest neighbor atoms, along the bonds of the square lattice planes in figure 3.1. The unit cell of this 2D square lattice will be called plaquette. Figure 3.1: Conduction planes in cuprates. Assuming that there is an infinite set of conducting planes, .one over the other, a simple expression for the total flux passing through a plaquette will be obtained. 4 'I '••Xr-.-------*. a. Figure 3.2: Line integral in the lattice Figure 3.2 shows an horizontal view of the infinite stack of planes (it is required to be infinite just along I). The circles in the graph represent the lattice bonds between atoms entering in the sheet. Integrating the Maxwell equation V x B = (4n/c)J along the shown trajectory: B -.df = MOJA (3-31) where J A is the current crossing the surface delimited by the trajectory, that Chapter 3. Microscopic theory 20 is, the current passing through the bond "A". The integrals along the curved trajectories cancel each other and the integrals on the lattice planes are equal to zero. The magnetic field along the vertical trajectories will be substituted by a constant effective field parallel to the integration path. Then the integral in eq. 3.31 is: In the previous equation do is the spacing between the planes, and B\ and B2 are the effective magnetic field at the right and left part of the trajectories respectively. If the last equation is integrated over one unit cell: The direction of the current in this equation is from site i to site i + 8, and the flux 4>i is the flux passing through the left plaquette when positioning in site i and looking towards site i + S. This last equation relates the magnetic flux on adjacent plaquettes to the current passing between them and it allows calculating the flux through each of the plaquettes up to an additive constant. 3.4 The cuprates The BdG equations (on eq. 3.12) have been widely used for describing cuprate superconductors (Franz et al [10]; Feder et al. [8]; Soininen et al[29]). This section presents the answer to the question "how to apply this model to high temperature superconductors (cuprates)?" but no justification for the applica-tion is given. The discussion of this last topic is out of the scope of the present thesis and has been done elsewhere (for a review see Micnas (1990) [22]). Many high temperature superconductors (or High -Tc superconductors) have a tetragonal structure with copper-oxygen planes lying parallel to the {0,0,1} planes. The conduction of quasiparticles takes place along this copper-oxygen planes. This makes the problem almost two-dimensional since the tunneling between planes is very small3. Figure 3.3 present a diagram for one of this copper-oxygen planes. It can be seen that the quasiparticles, in order to hope between copper atoms, have to hope first into an oxygen atom. In the BdG equations the presence of the oxygen bonds will be reflected in an effective t (which represents the process of first hopping to oxygen and then to copper). The hopping amplitude, t, changes from one superconductor to another (~ 0.25eV to 0.5eV). The dependence on this parameter will be avoided by writing all the energies for the problem in units of t (eg. / i = —0.9t = —0.9). Nevertheless for a given problem t is a constant, that is, it does not depends on which neighbor the quasiparticle is hopping into, since the lattice has a square 3 For a complete introduction to the structure of High-Tc superconductors see C P . Poole et al.[23] (1995) {Bx(x,y) - B2(x,y))d0 = HQJA (3.32) M0jt,<5 (3.33) Chapter 3. Microscopic theory 21 Copper Oxygen / "P.: Q) (2) P, -Py <X:><r>^£> C±X=> C Z X E y " ' tf p«, UNIT y A CELL A tf tf £ l «5k • • 0-<±X±> CIXZ> G X D <G><3> <4v±;> tf tf tf Figure 3.3: Cu-0 conducting plane in a cuprate. The picture represents each atom in the lattice by its overlapping orbital. Taken from Poole et al. (1995) [23]. symmetry (figure 3.3). This approximation neglects any orthorhombic effect (that can exist in YBCO for example) The interaction terms (or coupling terms) in the BdG equations are of two kinds: one express interactions of the particles in the site (Ao) and the other express interaction between particles at nearest neighbor atoms (Ai). The on site interactions are also called s-wave pairing, this name is due to the direction-independent nature of Ao-The nearest neighbor interaction can have two symmetries 4 : s-wave like or d-wave like. The first one occurs when the interaction of the particles at neighbor sites is, let's say, attractive, independent of the direction in the lattice. In the second one the interaction between particles at neighbor sites depends on the position of these, that is, the interaction between particles at the right and left neighbor sites can be attractive while the interaction with the particles located at the upper and lower neighbor atoms can be repulsive. Both symmetries come embedded in Ai since the BdG equations allow both. Once the selfconsistent result for Ai is obtained, the two symmetries, s-wave like and d-wave like, can be analyzed defining the quantities: Adi = ^ ( A i M - Aii,j + A u , - i - Aii.-y) (3.34) Aai = J ( A l i i S + A i i , e - l - A l j , _ * + Ai J,_j) (3.35) 4 For a discussion on allowed symmetries of the order parameter A on cuprates see M. Sigrist [25] (1998) Chapter 3. Microscopic theory 22 In the bulk of a d-wave superconductor, where the d-wave like behavior is known to dominate over the s-like, |Arf| = A (A is a constant) and |A S | « 0. The absolute values of this two quantities are called extended-d (|A l^) and extended-s (|AS|) components of the order parameter. The numerical solutions that will be shown, were initialized in such a way that the d-wave like behavior is preferred over the s-wave one. Nevertheless it will be seen that, close to the edge of the superconductor, the s-wave like part of the pairing potential is enhanced with respect to the value at the bulk of the sample. The different symmetries of the order parameter are used to classify the superconductors. A pure d-wave superconductor has interactions just between particles located at the nearest neighbor atoms; the order parameter of this superconductors (Ai) has a dominant d-wave like behavior. In a d+is super-conductor, besides the dominant d-wave like behavior of Ai , an on site interac-tion (Ao) is present and has an extra phase equal to e,7r/2 with respect to the dominant order parameter. The d+id superconductors are d-wave superconductors where the particles interact also with others located at next nearest neighbor atoms. This order parameter, A2, has a e1*/2 phase with respect to the dominant Ai 5 . 5The different symmetries of the order parameter are usually presented in the crystal momentum space, the description shown here is done in the real lattice. Chapter 4. Numerical solutions for theBdG equations 23 Chapter 4 Numerical solutions for the BdG equations In this chapter the Bogoliubov-de Gennes (BdG) equations are applied to the problem of a Time Reversal Symmetry Breaking (TRSB) superconducting sur-face and numerical solutions are obtained. Two geometries for the materials are considered, an infinite superconducting wall and a finite piece of superconduc-tor (island). Different pairing interactions are consider (d, d+is, and d+id) and observables are calculated. Also, the agreement with the quasiclassical results (chapter 2) is commented. 4.1 Numerical solutions This section explains the procedure followed for solving the self-consistent BdG equations. Once the parameters on the BdG equations have been chosen (/i1, Tem-perature, Vi, Vo, etc.), the quasiparticle amplitudes can be determined self-consistently by numerical iterations. That is, a convenient initial condition for the order parameter (Ai) is assumed, then, the matrix on the left hand of equation 3.13 can be diagonalized numerically, and the eigenvalues (En) and eigenvectors (u and v) are found. After that, new order parameters are found by using the self consistent conditions on equation 3.10. This cycle is repeated until the desired precision is achieved. In order to produce a dominant d-wave behavior in the material, the order parameters are initialized with the values they would have in the bulk of a d-wave superconductor. There, it is known that any subdominant order parameter has to be very small and the dominant order parameter must be d-wave like. Then, the initial conditions for the order parameters are A 0 = 0, Ai it± — Ax = 1 + iO.001, Aii,j) = Au-y = -1 + zO.001 and A 2 i j = 0. It can be checked, with eq. 3.35, that this initial conditions produce a big extended-d component and a very small extended-s. The small complex component in the initial conditions for Aj is a pertur-bation. The homogeneous state, as initial condition, can be a higher-energy instable solution for the problem; then, the purpose of the small extra compo-nent on Ai is "shaking" the system so it can evolve in a lower energy state. 1In an antiferromagnetic insulator n = 0. Since High-Tc superconductors are believed to be hole doped antiferromagnetic insulators then, in the BdG equations, fi < 0 Chapter 4. Numerical solutions for theBdG equations 24 The numerical process was implemented in FORTRAN and the BdG matrix diagonalization was done using the ZHEEV routine from the LAPACK library. For each superconductor (d, d+is and d+id) two kinds of situations are considered: an infinite wall (or slab) of superconductor, and a finite triangle (island). The figure 4.1 presents a scheme for the infinite wall of superconductor. Planes like the one shown, are stacked one over the other and the lattice extends to infinity in the direction of the arrows. The solution for this wall, takes advantage from the translational invariance of the material along the wall. Using Born Von Karman boundary conditions along the arrows direction 2,the u and v amplitudes on the BdG equations can be written as: un(x + ja, y - ja) = e'l27T:'/Nun(x, y) vn(x + ja, y - ja) = ei2*j/Nvn(x, y) (4.1) where a is the lattice constant, N is the number of "cells" in the arrows direction and; = 0,1,2 N - 1. , -Line 1 Line 2 1 2 3 4 5 6 1/ i/ 9 10 Figure 4.1: Lattice sites for an infinite wall of superconductor with surfaces parallel to a {1,1,0} plane. The points are the copper atoms. This symmetry allows to write the BdG equations as: 2 Free boundary conditions were used at the surface of the superconductor Chapter 4. Numerical solutions for theBdG equations 25 For sites along line 1: 6 +V4li,J(e- ,2*i/N(4,{ + 8-z,s) + l)«n,i+« = Enun>i 6 (4-2) <s For sites along line 2: + e'2nj/N) ^2 un,i+6 ~ + A 0 U „ , i s +yA 1 J,i(e' 2^' v(4,j + S-t,s) + l)vnii+s = Enun,i s (4-3) r ( l + e-%2nj/N) ^ vn,i+6 + A*«n.t + A o U n . i (5 tS(e-**ilN$i,s + L*,s) + l)u„,,-+* = (5 Despite of how they look, this set of equations simplify the script of the program and allow to find the solution for bigger systems (since just the sites along an horizontal line (see figure 4.1) are included in the calculation). Typical size of these systems were 100 lattice sites and 101 j values, that is 10100 eigenvalues were computed. This resolution allows to compute dl/dV along a horizontal line through the wall. The scheme for the finite island simulations is presented in figure 4.2. In this kind of simulations every point in the lattice enter in the simulation. The lattices simulated had less than 465 sites (the same amount of eigenvalues) and therefore not enough resolution is obtained for the calculation of dl/dV. Free boundary conditions were used in every surface of the island. Figure 4.2: Lattice sites for an finite piece of superconductor with one TRSB edge parallel to a {1,1,0} plane. Chapter 4. Numerical solutions for theBdG equations 26 Convergence for the simulations was checked periodically, and accuracy up to 8 digits was obtained. The results for each kind of superconductor are presented next. 4.2 P u r e d-Wave This section presents the numerical results for a d-wave superconductor, that is, a superconductor where the only interaction present is, lets say, attractive between particles located at the upper and lower nearest neighbor atoms; and repulsive for particles located at the left and right nearest neighbors. The figure 4.3 presents the plot for dl/dV as a function of the bias potential. This graph shows that, for low temperatures where the LDOS sa dl/dV, the Zero Bias Conductance Peak (ZBCP) is split. Figure 4.3: Differential tunneling current for a d-wave superconducting wall as function of the bias voltage and temperature at site 1. /x — — 1 and Vi = 3. This phenomena is on disagreement with the quasiclassical predictions, where a subdominant order parameter is needed for splitting the ZBCP on the Local Density of States (LDOS). This effect has been reported by Tanuma et al. [30] and by Zhu et al. [36]. In section 3.4, it was said that a very small s-wave com-ponent (called extended-s) of the order parameter can coexist with the dominant d-wave in the bulk of the superconductor. Close to the edge, the presence of the boundary reduces the symmetry of the superconductor, making the d-wave component of the order parameter incompatible with the new symmetry. Nev-ertheless, the s-wave component is not affected by this symmetry reduction and Chapter 4. Numerical solutions for theBdG equations 27 gets enhanced once the d-wave component is "out of the competition" (see fig-ure 4.4). In general, it can be said that whenever the pairing potential Ai is changing in space, an s-wave like component is generated. This effect has been predicted in vortices (Franz and Tesanovic [11]; Soininen et al. [29]) and close to non-magnetic impurities (Franz et al. [10]) In a quasiclassical calculation this extended-s component needs to be put in by hand, as every other pairing interaction. In contrast, the BdG equations do not assume any symmetry for the nearest neighbors interactions, this is why both components arise. 0.4 0.3 I T> V 0.25 g 0.2 0.15 — Extended-d 20 30 60 70 0.15 I T 3 0.05 X x(a) Figure 4.4: d-wave and s-wave component of the order parameter in a d-wave superconducting wall. Observe the different scaling for the two order parameters, /J, = -1, Vi = 3 and T = 0.05. The extended-s component of the order parameter is responsible for the ZBCP splitting. This was checked by solving the BdG equations without any self-consistent condition, that is, the algorithm made just one iteration. In this none-selfconsistent solution the extended-d component of the order parameter is a constant and extends all the way to the edge of the sample (a very similar situation to the one considered on section 2.1); and the extended-s component is zero everywhere. The figure 4.5 presents the Local Density of states at the face of the superconducting wall for this non self-consistent situation. No splitting of the ZBCP is shown since no extended-s component is present. The splitting of the ZBCP is the signature for the Time Reversal Symmetry Breaking State of the System. The expected spontaneous current generation along the faces of the superconducting wall are shown in figure 4.6 and 4.7. This current is consistent with the magnitude of the extended-s component of Chapter 4. Numerical solutions for theBdG equations 28 0.015 eV(t) Figure 4.5: STM conductance after one iteration in at the surface of a d-wave superconducting wall at site 1. LI = — 1, Vi = 3 and T = 0.05. Figure 4.6: Current generated along a d-wave superconducting wall with LI = -1, Vi = 3, T = 0.05, 30 sites and 31 cells. Chapter 4. Numerical solutions for theBdG equations 29 the order parameter; figure 4.8, shows a plot of magnitude of the extended-s component at site 2 as a function of the magnitude of the current at the same site. The relation is almost linear, a least squares fit produce: A8 2 - 2.25041 {current) + 0.00206 with a correlation of 0.9987. As the temperature increase, the current goes to zero at the same rate as the extended-s component. 61 1 1 . 1 1 1 1 1 i n — - T=0.05 -4H _6I i I i I i I i I i u 0 20 40 60 80 100 X(a) Figure 4.7: Magnitude of the current (parallel to the walls of the superconduc-tor) as a function of position in a d-wave superconducting wall with fi = — 1 and Vi = 3. When T=0.05 in figure 4.7, 91% of the current is contained within a distance of 4a/\/2 from the face of the wall and the maximum magnetic field generated by this current is of the order of 0.1G to 0.01G. It must be noticed in figure 4.3 that for T = 0.1 there is only one peak centered at eV= 0 even though there are still currents in the system (figure 4.7). Remember that LDOS « dl/dV is only valid for low enough temperatures, the two peaks are still present on the LDOS but they can't be resolved after the integral that gives rise to dl/dV. Figure 4.9 shows the STM conductance for two different sites (see figure 4.1). At site number 6 the split peaks have almost disappear (16% of the value at site 1). That is, at a distance bigger than 6a/y/2 from the boundary, no appreciable ZBCP split is expected to be observed. Before finishing this section the self-consistent solutions for the BdG equa-tions in a finite d-wave superconductor will be discussed. The geometry of the problem is shown in figure 4.2. Chapter 4. Numerical solutions for theBdG equations 30 Current Figure 4.8: Extended-s component of the order parameter at site 2 vs. current at the same site in an infinite wall of d-wave superconductor. Each point is taken at a different temperature, starting from the left: 0.125, 0.1, 0.075, 0.065, 0.05 and 0.01. n = -l and Vi = 3. Figure 4.9: dl/dV against differential potential at two different sites of an infi-nite wall of d-wave superconductor, fj, = -1, Vi = 3 and T = 0.05. Chapter 4. Numerical solutions for theBdG equations 31 In order to get enough accuracy in this finite island problem, 2000 iterations had to be performed. This is too big compared with other simulations using BdG equations (in which convergence up to six or seven digits is obtained after 60 iterations) and might be an indication of something going wrong. More about this will be said in the next section and in the last chapter, now the obtained results will be presented. The current and the magnetic field obtained are shown in figures 4.10 and 4.11 respectively. The current flows down the TRSB edge and then returns through the bulk of the superconductor. The current (and therefore the mag-netic field) at this edge is three orders of magnitude smaller than the one found for the infinite superconducting wall. > v k k \ \ k k V V V V > 1 r \ k V k V k , 1 «. - V > ^ V > k > * \ V k i / ' - ' — \ Figure 4.10: Spontaneous current generated at a finite d-wave island with a sub-dominant anisotropic s-wave component. The sides of the triangle have 22 sites, n = -1, Vi = 3 and T = 0.025. The magnetic flux, figure 4.11, is enhanced close to the 45° edges. The magnitude of the magnetic field at this regions goes up to 10 _ 3G to 10 _ 4G. This increase of the magnetic field at the acute edges of the island resembles the behavior for a d+id superconducting island found by Amin and coworkers [2] (see figure 2.3). In this regions of high magnetic field the order parameter never goes to zero therefore no vortex is present. 4.3 d+is In this section the results for a d+is superconductor are presented. As it was said on section 3.4, the interactions present in this kind of superconductors are on site repulsion (Vb < 0) and attraction between particles at nearest neighbors atoms(Vi > 0). The characteristics of these superconductors are very similar to the ones an-alyzed on the last section. Figure 4.12 show the behavior of the two differen Chapter 4. Numerical solutions for theBdG equations 32 Figure 4.12: Magnitude of the different order parameters along an infinite d+is superconducting wall, fi — -1, Vi = 3, Vo = -1 and T = 0.05. Chapter 4. Numerical solutions for theBdG equations 33 torder parameters as a function of the position for an infinite wall of supercon-ductor. It is seen that the on-site interaction follows the same behavior as the extended-s component of Ai . As expected, Ao and A s break the time reversal symmetry close to the edge. The current generated (see figure 4.13) is 89% contained within a distance of 3a/v/2 from the edge. The magnetic field produced by this current is of the same order as in the d-wave case (0.1G to 0.01(7). s g o V -4 — • T=0.05 T=0.075 T=0.1 JL 20 40 60 80 100 X(a) Figure 4.13: Magnitude of the current along an infinite wall of d+is supercon-ductor as function of position, /i = -1, Vi = 3, Vo = —1. Comparing figure 4.7 and figure 4.13 is noticed that at T = 0.1 a sizable current is still present for the d-wave superconductor while in the d+is it has been decreased two orders of magnitude. This tell us that the presence of Ao has the effect of reducing Tc23-Since the system is in a TRSB state, besides the currents, a split in the ZBCP is expected for low temperatures. Figure 4.14 shows the STM conductance at two different sites. At the first site the ZBCP is appreciably split but at the site number 6 the split peaks have almost disappeared (15% of the value at site 1). That is, at a distance bigger than 6a/\/2 from the boundary, no appreciable ZBCP split is expected to be observed. The simplified quasiclassical estimation of the LDOS in chapter 2 predicted a linear relation between the distance between peaks (5) and the magnitude of the subdominant order parameter (Ao). The relation for these two quantities obtained by numerical solution of the BdG eqs. is shown on figure 4.15. The relation appears to be linear for higher temperatures and the slope decrease at 3Tc2 is the transition temperature for the subdominant order parameter Chapter 4. Numerical solutions for theBdG equations 34 low temperatures. Nevertheless the slope of the graph, at any point, is at least one order of magnitude less than the expected quasiclassical estimation. The solutions for the BdG equations on a finite triangular island are not reported for a d+is and a d+id superconductor. No convergence was obtained for the order parameters (they keep on decreasing after to 8000 iterations) and the conservation of charge was violated. Nevertheless must be noticed that on the first iterations a big current appear along the TRSB edge of the island but this current died as more and more iterations were carried out. Possible interpretations of this problem are exposed on the next chapter. 4.4 d+id In this kind of superconductors the particles interact with others located at the nearest neighbor atoms (Ai) and with the ones located at second nearest neighbors atoms (A2). Close to the face of the infinite superconducting wall, the quasiclassical the-ory would predict an STM conductance like the one shown in figure 4.16. It must be remembered that, within the quasiclassical approximation, the LDOS of a d+id superconductor is not expected to decrease to zero between the peaks since equation 2.17 is a dispersive relation. The numerical solution for the BdG equations in a d+id superconductor produces an STM conductance shown in figure 4.17. In this graph the tun-neling conductance decrease considerably between the peaks. Actually as the Chapter 4. Numerical solutions for theBdG equations 35 0.07 0.065 h 0.06 h 0.055 0.1 0.12 0.14 0.16 |A0| (t) 0.18 Figure 4.15: Distance between peaks on the dl/dV graph (6) against magnitude of Ao both at site 1. fx — -1, Vi = 3, Vo = -1. Each point correspond to a different temperature; starting from the left: 0.05, 0.04, 0.03, 0.025, 0.01 and 0.009. Figure 4.16: Tunneling density of states at a (110) surface as function of energy (e). Here the result with p = 0.00 correspond to a surface without roughness. Taken from Rainer et al. (1998) [24]. Chapter 4. Numerical solutions for theBdG equations 36 temperature is lowered (and LDOS « dl/dV) the STM conductance value at E = 0 does not converge to some value as it would have been expected from a non-zero LDOS at that point. eV(t) Figure 4.17: dl/dV against differential potential for site 1 in a d+id supercon-ducting wall. \x = —1, Vi = 3 and V 2 — —1. This further splitting of the peaks is due, again, to the presence of the extended-s component in both pairing potentials. In a microscopical theory both components, "d" and "s", are inseparable and figure 4.17 shows that the "s" part wins the competition close to the edge. That the extended-s component is responsible for the almost zero STM con-ductance at E = 0 was checked in the following way. The initial conditions for Ai were set as a pure d-wave order parameter all the way to the face of the wall. A2 was initialized as zero everywhere inside the wall except very close to the faces of the wall where a pure imaginary d-wave order parameter was assumed. After one iteration the non self-consistent STM conductance in figure 4.18 was obtained. This unreal situation corresponds to the simplified model of chapter 2. Except for the small peak at E = 0, the similarities with the quasiclassical results (figure 4.16) are evident. The splitting of the LDOS tell about the system being in a TRSB state. The expected current along the faces of the infinite wall is shown in figure 4.19. When T = 0.05, 91% of the current is contained within a distance of 4a/\/2 from the face of the wall. The magnetic field produced by this current has a maximum value of 10 _ 2G to 10 _ 3G. Figure 4.20 shows the derivative of the tunneling current for two different sites. At site number 6 the split peaks have almost disappear (16% of the value Chapter 4. Numerical solutions for theBdG equations 37 0.008 h ^ 0.006 0.004 0.002 — Non self consistent solution — Self consistent solution Figure 4.18: Non selfconsistent dl/dV against differential potential for site 1 in a d+id superconducting wall; the self-consistent solution is presented for comparison, LI — -1, Vi = 3, V 2 = -1 and T = 0.025. Figure 4.19: Magnitude of the Current, as a function of position, along a d+id infinite superconducting wall, LI = -1, Vi = 3 and V2 = — 1. Chapter 4. Numerical solutions for theBdG equations 38 at site 1). That is, at a distance bigger than 6 a / f r o m the boundary, no appreciable ZBCP split is expected to be seen. Comparing figures 4.13 and 4.19 one notices that the d+id superconductors has a Tc2 higher than the one for a d+is superconductor. 0.006 0.005 0.004 \ -0.003 0.002 r -0.001 Figure 4.20: dl/dV against differential potential at two different sites for an infinite wall of d+id superconductor, /x = -1, Vi = 3, V 2 = -1 and T = 0.05. Chapter 5. Conclusions 39 Chapter 5 Conclusions In the last chapter it was noticed that the presence of the extended-s component in the dominant order parameter is responsible for some features not account in the quasiclassical theory. In a d-wave superconductor the extended-s component of the order parame-ter is responsible for the ZBCP split (see figure4.3). In order to observe this ef-fect within the framework of the quasiclassical theory a subdominant anisotropic s-wave order parameter would have to be included in the calculation. This extended-s component is presumably responsible for the deviation found in the quasiclassical expectation of the distance between peaks in the LDOS for a d+is superconductor (see figure4.15). Nevertheless the microscopical calcula-tion produce a relation appears to be linear 1 as expected from the chapter 2 calculations. The effect of A8 on the LDOS in a d+id superconductor is also important. Quasiclassical calculations expect to have a finite LDOS at E = 0 but figure 4.17 shows that the extended-s components of both order parameters overcome the effect of the d-wave component of the subdominant order parameter. In con-clusion, it can be said that every time a quasiclassical calculation is performed, it is recommended to include a subdominant anisotropic s-wave component in the calculation. A. comparison with the quasiclassical calculations on a finite piece of super-conductor (done by Amin et al. (2002) [2]) could not be done in this thesis. The problem was the non-convergence of the quasiparticle amplitudes (u and v) in the d+is and d+id numerical algorithms. After a few iterations of the algorithms it was noticed a big current flowing along the TRSB edge of the is-land but as more and more iterations were performed this current decreased its amplitude all the time. After 8000 iterations no convergence could be achieved. The reasons for this non-convergent behavior are not clear. The problem seems to be that the current generated at the TRSB edge of the island is not finding a stable way to return trough the superconductor. This could be due to a physical impediment of the material to return the currents (due to its small size). Other possible explanation can relay on the initial conditions for the order parameter along the island . Amin et al. (2002) [2] expect a non-trivial behavior for the order parameter (as its expected by looking at the current and the magnetic field reported in their paper, figures 2.3 and 2.2). This strange behavior would have to be introduced approximately by hand when the order At least for high enough temperatures. Chapter 5. Conclusions 40 parameter is initialized. It can be happening that the used initial conditions2 set the system in a point where an energy barrier does not let it evolve into the right solution. Several islands with an initial vortex-like behavior were simulated without success. The result reported for the d-wave finite island shouldn't be trusted. The magnetic field generated along the TRSB edge is two orders of magnitude less than the one generated in the superconducting wall case. Also, 800 iterations had to be performed in order to obtain reasonable accuracy. This number is too big .when compared with the 70 iterations needed for convergence in the infinite wall algorithm. Even though this results shouldn't be trusted, they are reported because they display a very similar behavior as the one expected by Amin and coworkers (see figure 2.3). These results can give a clue for guessing what is happening with the order-parameter in the finite island. Maybe the "vortex-like" behavior of the currents at the corners of the island plays an important role in the mechanism for returning the current trough the superconductor. The problem of returning the current is not present in the the supercon-ducting wall due to its infinite nature. For this situation, stable currents and magnetic fields were obtained. The magnitude of these magnetic fields was found to be three orders of magnitude bigger than those found by Amin et al. (see section 2.2) in a d+is superconductor. Also, the magnetic field produced close to the surface of the d+id superconducting wall was found to be four orders of magnitude bigger than the one calculated by Amin and coworkers. Actually the current generated along the TRSB surface was the same order as the critical current at low temperatures. The magnetic flux was found to be localized within less than 4a from the surface of the wall. The distance from the edge of the superconductor in which the STM con-ductance reaches the bulk value was found to be less than 6a. Roughness in the surface is expected to play an important roll in STM measurements (impeding to observe any peak split, see Ranier et al. (1998) [24] for a quasiclassical treat-ment). This issue was out of the scope of the present thesis. A microscopic simulation with roughness at the faces of the material, presents some problems. Translational invariance symmetry is broken by the random roughness of the sample, therefore whole finite samples would have to be simulated bringing back the problems mentioned in the previous; paragraphs. No relevant difference was found between superconductors with different order parameters. The magnetic fields and the transition temperatures were, in general, higher in the d-wave and d+id. case. Contrary to what was expected quasiclassically, the LDOS close to the edge of the cut superconductor in the d+is and d+id case is very similar. 2 Several initial conditions were tried without obtaining convergence Bibliography 41 Bibliography [1] M.H.S. Amin, A.N. Omelyanchouk, S.N. Rashkeev, M. Coury, and A.M. Zagoskin. Quasiclassical theory of spontaneous currents at surfaces and interfaces of d-wave superconductors. Physica B, 318:162, 2002. [2] M.H.S. Amin, S.N. Rashkeev, M. Coury, A.N. Omelyanchouck, and A.M. Zagoskin. d+is vs d+id' time reversal symmetry breaking states in finite size systems. Preprint cond-mat/0205495, 2002. [3] A.F. Andreev. The thermal conductivity of the intermediate state in su-perconductors. JETP, 19:1228, 1964. [4] N.W. Ashcroft and N.D. Mermin. Solid State Physics. Harcourt College Publishers, 1976. [5] Chr. Bruder. 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