We investigate the quasiparticle density of states in disordered d-wave
superconductors. By constructing a quantum map describing the quasiparticle
dynamics in such a medium, we explore deviations of the density of states from
its universal form ($\propto E$), and show that additional low-energy
quasiparticle states exist provided (i) the range of the impurity potential is
much larger than the Fermi wavelength [allowing to use recently developed
semiclassical methods]; (ii) classical trajectories exist along which the
pair-potential changes sign; and (iii) the diffractive scattering length is
longer than the superconducting coherence length. In the classically chaotic
regime, universal random matrix theory behavior is restored by quantum
dynamical diffraction which shifts the low energy states away from zero energy,
and the quasiparticle density of states exhibits a linear pseudogap below an
energy threshold $E^* \ll \Delta_0$.Comment: 4 pages, 3 figures, RevTe

A. Altland

A. Ghosal

A. Ossipov

A. Yazdani

A.A. Nerseyan

A.G. Yashenkin

B. Huckestein

C. Chamon

C. Pépin

F.M. Izrailev

H. Walter

İ. Adagideli

I.L. Aleiner

J.-X. Zhu

K. Ziegler

M. Bocquet

M.C. Goorden

P.J. Hirschfeld

P.J. Turner

Ph. Jacquod

R. Ketzmerick

S. Fishman

T. Senthil

T.P. Eggarter

W.A. Atkinson

English

arXiv

We investigate the quasiparticle density of states in disordered d-wave superconductors. By constructing a quantum map describing the quasiparticle dynamics in such a medium, we explore deviations of the density of states from its universal form ([proportional]E), and show that additional low-energy quasiparticle states exist provided (i) the range of the impurity potential is much larger than the Fermi wavelength (allowing one to use recently developed semiclassical methods), (ii) classical trajectories exist along which the pair potential changes sign, and (iii) the diffractive scattering length is longer than the superconducting coherence length. In the classically chaotic regime, universal random matrix theory behavior is restored by quantum dynamical diffraction which shifts the low-energy states away from zero energy, and the quasiparticle density of states exhibits a linear pseudogap below an energy threshold E*$\lt$$\lt$Delta0, much smaller than the superconducting gap

Adagideli, Inanc

Jacquod, P.

University of Regensburg Publication Server

arXiv:cond-mat/0307159v1  [cond-mat.supr-con]  8 Jul 2003Low-energy quasiparticle excitations in dirty d-wave superconductors and theBogoliubov-de Gennes kicked rotatorI˙. Adagideli1 and Ph. Jacquod21Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands2De´partement de Physique The´orique, Universite´ de Gene`ve, CH-1211 Gene`ve 4, Switzerland(Dated: November 8, 2018)We investigate the quasiparticle density of states in disordered d-wave superconductors. Byconstructing a quantum map describing the quasiparticle dynamics in such a medium, we exploredeviations of the density of states from its universal form (∝ E), and show that additional low-energyquasiparticle states exist provided (i) the range of the impurity potential is much larger than theFermi wavelength [allowing to use recently developed semiclassical methods]; (ii) classical trajectoriesexist along which the pair-potential changes sign; and (iii) the diffractive scattering length is longerthan the superconducting coherence length. In the classically chaotic regime, universal randommatrix theory behavior is restored by quantum dynamical diffraction which shifts the low energystates away from zero energy, and the quasiparticle density of states exhibits a linear pseudogapbelow an energy threshold E∗ ≪ ∆0.PACS numbers: 71.23.-k, 74.72.-h, 05.45.MtIn recent years, considerable attention has been fo-cused on the low-energy properties of the quasiparticlespectrum of disordered cuprate superconductors [1, 2].Because many of the cuprate superconductors are ran-domly chemically doped insulators and disorder is a pair-breaker for d-wave superconductors, the role of nonmag-netic impurities is particularly important for an under-standing of the d-wave superconducting state, its quasi-particle spectral and transport properties. Of special in-terest is the low-energy behavior of the single-particleDensity of States (DoS) ρ(E).In early work [3], the self consistent T -matrix approx-imation was shown to break down for 2-dimensional d-wave superconductors. This led to a series of papers us-ing nonperturbative methods, which predicted (at firstsight) contradictory results: vanishing [3, 4, 5, 6], con-stant [7, 8], and diverging [9, 10] DoS as E → 0. On thenumerical side, several investigations also predicted bothvanishing [11, 12], and diverging [13, 14] DoS at zero en-ergy. It was soon argued, based on numerical analysis,that the reason behind these contradicting predictionsis the fact that the microscopic details of disorder (i.e.details beyond the transport mean free path ℓ, such asthe density of scatterers or the correlation length ζ ofthe impurity potential) as well as the symmetries of theclean Hamiltonian matter both qualitatively and quan-titatively [13, 15] (see also [16]). An important featureshared by the numerics of Refs. [11, 12, 13, 14, 15] is thatthe disorder is introduced via isolated point-like scatter-ers. Long-wavelength disorder, which may arise due tochemical doping away from CuO2 planes, or can be in-duced via ion radiation techniques [17] or via a STMtip [18], is thus ignored. Effects of long-wavelength dis-order are expected to become dominant when the CuO2planes have almost no atomic disorder [19]; they are themain focus of this paper.It has recently been realized, in the context of meso-scopic physics and weak localization, that ζ and ℓ, to-gether with the Fermi wavelength λF , define two classesof complex quantum systems: quantum disordered sys-tems where λF ℓ/ζ2 > 1 and quantum chaotic systemsfor which λF ℓ/ζ2 < 1 [20]. The latter class is character-ized by the emergence of a new diffractive scattering timescale, τE = ν−1 ln[ζ/λF ], defined as the time it takes forthe classically chaotic dynamics (with Lyapunov expo-nent ν) to stretch a wavepacket of minimal initial exten-sion λF to a length ζ. In contrast to quantum disorderedsystems, quantum chaotic systems exhibit nonuniversalproperties due to their short-time classical (i.e. deter-ministic) dynamics. In particular, significant deviationsfrom Random Matrix Theory (RMT) emerge, as was re-cently found by Adagideli et al. [10] in the context ofimpurities in d-wave superconductors. These authorsused a semiclassical approach to calculate the low en-ergy DoS for a collection of extended scatterers (a quan-tum chaotic system) and found an asymptotic behaviorρ(E) ∼ 1/E| lnE2|3 as E → 0. They nevertheless arguedthat the RMT predictions of a linear pseudogap would berestored at lower energy E < E∗, i.e. that the singularityin the DoS would be cut off at an energyE∗ related to τE ,by diffractive (nonclassical) scattering occurring at largertimes τ > τE . The purpose of the present paper is to in-vestigate the modifications that the DoS undergoes as thecorrelation length of the impurity potential increases andτE becomes relevant. We will focus our attention on (i)providing for numerical checks of the theory of Adagideliet al. in the case of long-wavelength disorder [10]; (ii)finding out whether for some E∗ the DoS is suppressedfor E < E∗, in agreement with RMT predictions [4, 5, 6],(this could also reconcile the above mentioned contra-dictory predictions); and (iii) investigate the transitionregion between extended and pointlike disorder.We start by introducing a quantum map model forquasiparticle states in disordered d-wave superconduc-2tors. The main motivation behind this model is to inves-tigate discrepancies (in low-energy DoS) between point-like vs. extended disorder as well as the transition region,i.e. the regime in which the impurity size is intermedi-ate. To the best of our knowledge no disorder modelwhich includes extended impurities as well as pointlikein d-wave superconductors has been studied numericallyso far. This map has two additional advantages: First,as both the density and correlation length of impuritiescan be tuned independently, it is possible to interpolatebetween the two extreme regimes of strong disorder: uni-tary disorder (i.e. disorder due to dilute, pointlike scat-terers [6, 9, 15, 16], viz. quantum disorder) and quasi-classical disorder [10] (i.e. disorder due to extended scat-terers, viz. quantum chaotic). Second, from a numeri-cal point of view, it allows for the investigation of verylarge system sizes, i.e. lattice sizes of up to 256 × 256,which are necessary for both variations of the disordercorrelation length and the numerical extraction of theparametric behavior of the DoS. Our reasons why a dy-namical model is relevant are: (i) In absence of supercon-ductivity, many properties of quasiparticles in disorderedmedia (such as Anderson localization) are correctly de-scribed by 1-D maps [21]. In fact it has been shown byAltland and Zirnbauer that one of those maps, the 1-Dkicked rotator, and quasi-one-dimensional metallic wiresare described by the same effective field theory [22]. Re-cent numerical investigations in D = 2 suggest that thisis also true in higher dimensions [23]. (ii) In presenceof superconductivity, Andreev maps based on the kickedrotator have recently been shown to adequately describequantum dots in contact with a superconductor [24].We first briefly discuss generic properties of quantummaps for uncoupled quasiparticles. The dynamics corre-sponds to a succession of free propagations, interruptedby sudden kicks of period τ0, i.e. instantaneous perturba-tions. Quantum maps are conveniently represented by aunitary, Floquet operator F , giving the time-evolution af-ter p kicks as u(p) = F pu(0), for an initial wavefunctionu(0). The matrix F has eigenvalues exp(−iεm), whichdefine quasi-energies εm ∈ (−π, π) (energies and quasi-energies are expressed in units of ~/τ0). While the en-ergy is not conserved, the periodicity of the kick stillpreserves quasi-energies, much in the same way as a pe-riodic potential breaks translational symmetry, but stillpreserves quasi-momentum. Time evolution of hole ex-citations (being the time-reversed of electronic excita-tions) is given by v(p) = (F ∗)pv(0). Specializing to theD-dimensional kicked rotator, we write the Floquet op-erator as [25]F = exp(−i KI~τ0ΠDj=1 cos rj)exp(i~τ02I~∇2). (1)It describes the free motion of a particle with dimension-less coordinates {rj} (e.g. expressed in units of a latticeconstant), which is interrupted at periodic time intervalsτ0 by a kick of strengthK ·ΠDj=1 cos rj . I is the moment ofinertia of the particle, and K is the kicking strength. ForD = 1 and 2, increasing K makes the classical dynamicsevolve from integrable (K = 0) to fully chaotic [K & 7,with Lyapunov exponent λ ≈ ln(K/2)]. For 0 < K < 7stable and unstable motion coexist (a so-called mixedphase space) [25]. IncreasingK is thus tantamount to in-creasing the amount of disorder, the fully chaotic regimecorresponding to a finite density of impurities.Electron and hole excitations inside a superconductorare however coupled by a nonvanishing pair-potential.Accordingly we extend the kicked rotator of Eq. (1) toa Bogoliubov-de Gennes form. We discuss this construc-tion for the case D = 2. First, we replace the free quasi-particle motion by a coupled electron and hole dynamics,F0 = exp(−iHτ0/~), (2a)H = H σz +∆ σx. (2b)Here, H = −(~2~∇2/2I)−EF , with EF the Fermi energy,σx,z are Pauli matrices acting in particle-hole space, and∆ is the superconducting pair potential. Second, thecoupled quasiparticle motion is followed by a kickFK = exp(−iHK/~), (3a)HK = KIτ0cosx cos y σz . (3b)Exponentiating the Pauli matrices, we end up with theBogoliubov-de Gennes-Floquet (BdGF) operatorF = FK F0, (4a)F0 = cos√(H2 +∆2)(τ0/~)2 I+i sin√(H2 +∆2)(τ0/~)2√H2 +∆2[Hσz +∆σx], (4b)FK = cos[KI~τ0cosx cos y]I+ i sin[KI~τ0cosx cos y]σz , (4c)with I, the identity matrix in particle-hole space. For∆ = 0, Eq. (4) describes uncoupled electron and holeexcitations in a disordered 2D metal. Once this metalbecomes superconducting, ∆ couples these excitationsduring their free propagation, while it is neglected dur-ing the instantaneous kick. As in the case of a BdGeigenproblem, the 2M quasienergies [with average spac-ing δ ≡ 〈εm+1 − εm〉 = π/M ] of the BdGF equationFφm = exp(−iεm)φm, come in pairs with opposite signεm = −ε2M−m+1, similarly to the spectral properties ofa BdG Hamiltonian. These considerations establish thecorrespondence between the map of Eq. (4) and quasi-particles in a dirty superconductor.We next quantize the phase space on a 4-torus{x, y; px, py}, with dimensionless momentum px,y =3-3 -2 -1 0 1 2 3E/∆001234ρ(E) δes-wave d-waveFIG. 1: Density of states for the d-wave (right) and extendeds-wave (left) Bogoliubov-de Gennes kicked rotator defined byEqs. (2-4), and parameters Lx × Ly = 128 × 128, ∆0 = 0.4,EF = 2pi2/5, and K = 0. (solid lines), 2. (dashed lines) and8. (dotted-dashed lines).−i~eff∂/∂(x, y) ∈ (0, 2π) [25]. The effective Planck con-stant ~eff ≡ ~τ0/I0 takes on values ~eff = 2π/M , withinteger M = Lx · Ly, in term of the real-space linearsystem sizes Lx,y (also expressed in units of a latticespacing), and the impurities have a spatial extensionζ = O(Lx, Ly). The BdGF operator is then a 2M × 2Munitary matrix, and we consider the two cases of d-wave[∆(px, py) = ∆0(p2x−p2y)/(p2x+p2y)] and extended s-wave[∆(px, py) = ∆0|p2x − p2y|/(p2x + p2y)] pair potentials, forwhich F0 is diagonal in momentum representation. Not-ing that FK is diagonal in real space representation, werewrite F asF~p~p′ = ([UFKU†] F0)~p~p′ , (5)where U = UI, and U is the unitary matrix of the 2DFourier transform between real space and momentum co-ordinates, U~p~p′ =M−1/2 exp[(2πi/M) ~p · ~p ′]. We numer-ically extract the quasienergy DoS from the eigenvaluessin εm of the hermitean matrix12i (F − F†), which wediagonalize using the Lanczos algorithm [26].In Fig. 1 we show the quasienergy DoS for d-wave andextended s-wave pair potentials away from half filling(EF = 2π2/5 < π2/2), as the kicking strength increases.In the clean case (K = 0) the two DoS are the same.The gap singularity at E/∆0 = 1 gets washed out as Kincreases in both cases, however, a peak emerges in the d-wave DoS around E = 0, while ρ(E) = 0 in the extendeds-wave case. This is in agreement with Ref. [10, 27], i.e.the existence of low-energy states requires a change inthe sign of the pair potential. In the extended s-wavecase, the low energy peak is shifted by an energy corre-sponding to the gap averaged over all momenta mixed bythe impurity potential.We focus on the d-wave symmetry from now on. A0 0.05 0.1 0.15 0.2E/∆000.51ρ(E) δ10-3 10-2 10-1 100E/∆010-210-1100ρ(E) δFIG. 2: Main plot: Low energy DoS for the d-waveBogoliubov-de Gennes kicked rotator of Eqs. (2-4),Lx × Ly = 256 × 256, ∆0 = 0.4, EF = 2pi2/5, andK = 8. Inset: Asymptotic of the density of states atlow excitation energy for the same set of parameters, forlong-wavelength disorder (quantum chaotic; black circles)and diffractive disorder as defined in Eq. (6) with NH = 27(quantum disorder; empty diamonds). The solid and dashedlines indicate a ∝ E and ∝ E−1 behavior respectively.closer look at the DoS in the fully chaotic regime withK = 8. is provided in Fig. 2. It indicates that the char-acteristic semiclassical singularity exhibited by the DoSas E → 0 is cut off at an energy E∗ ≪ ∆0, where a sharpdrop occurs and ρ(E) → 0. RMT predicts such a dropto occur over an energy scale given by the Thouless en-ergy [2], which in our case is however significantly largerthan ∆0 [28]. We thus attribute this drop to the emer-gence of diffractive scattering at times larger than τE [20]as follows. According to Ref. [10], the DoS correspond-ing to low energy semiclassical states can be estimatedfrom a mapping onto a tight-binding chain with randomhoppings, for which the eigenfunctions are localized withan energy-dependent localization length ξ(E) [29]. Atlow energies, ξ exceeds the diffractive scattering lengthvF τE , (vF is the Fermi velocity) in which case hoppingsbetween otherwise uncoupled tight-binding chains (cor-responding to different classical trajectories) have to betaken into account. The emergence of these processes sig-nals the breakdown of semiclassics and the restoration ofRMT. One thus expects the vanishing of the DoS belowa threshold energy given by the condition ξ(E∗) ≈ vF τE .Since ξ(E) is bounded by the superconducting coherencelength, ξ(E) & ~vF /∆0 [10], the observation of the semi-classical peak in the DoS requires a long enough diffrac-tive scattering length vF τE > ~vF /∆0. While prelimi-nary results corroborate this argument, a detailed inves-tigation of E∗ will be presented elsewhere [30].In the inset to Fig. 2 we show the asymptotic behav-ior of the DoS on a log-log scale. Once abstraction ismade of the drop in the DoS below E∗, the semiclassicaldata exhibit a singular behavior slightly below ∝ E−14-0.5 -0.25 0 0.25 0.5E/∆000.250.50.751ρ(E) δFIG. 3: Low energy DoS for the d-wave Bogoliubov-deGennes kicked rotator with decreasing disorder range asdefined in Eqs. (2), (4) and (6), with Lx × Ly = 256 × 256,∆0 = 0.4, EF = 2pi2/5, K = 8., and NH = 1, 3, 5, 7, 17, and27 (from top to bottom).(black circles) which is in qualitative agreement with theprediction ρ(E) ∝ E−1| lnE|−3 of Ref. [10].Having established the validity of semiclassical predic-tions in the quantum chaotic regime, we next decreasethe range of the disorder and enter the quantum disor-dered regime. We accomplish this via the inclusion ofhigher harmonics to the kicking potential, and replaceEq. (3) byHK = KIN2Hτ0NH∑l,m=1cos[lx] cos[my] σz. (6)The typical impurity size decreases as ζ ∝ N−1H . Fig. 3shows the disappearance of the low-energy peak in theDoS as NH increases. For the set of parameter con-sidered, once NH ≃ 27 is reached, the DoS vanishes atE = 0. Note that the resolution used in Fig. 3 does notallow to see the opening of the RMT gap below E∗. Amore precise look at the DoS for NH = 27 is provided onthe inset to Fig. 2 (empty diamonds). The data clearlyindicate the expected RMT linear suppression of the DoS.Our results thus clarify the competition between RMT[2] and semiclassics [10]. The next step is to investigatethe transport properties and to explore the parametricdependence of E∗. Work along those lines is in progress[30].This work was supported by the Dutch Science Foun-dation NWO/FOM and the Swiss National Science Foun-dation. We thank I. Gornyi, A. Yashenkin, M. Vojta, N.Trivedi, and P. M. Goldbart for interesting discussionsand comments.[1] P. J. Hirschfeld and W. A. Atkinson, J. Low Temp. Phys.126, 888 (2002).[2] A. Altland, B. D. Simons, and M. R. Zirnbauer Phys.Rep. 359, 283 (2002).[3] A. A. Nerseyan, A. M. Tsvelik, and F. Wenger, Nucl.Phys. B 438, 561 (1995).[4] T. Senthil, M. P. A. Fisher, L. Balents, and C. Nayak,Phys. Rev. Lett. 81, 4704 (1998); T. Senthil and M. P.A. Fisher, Phys. Rev. B 60, 6893 (1999).[5] M. Bocquet, D. Serbon, and M. R. Zirnbauer, Nucl. Phys.B 578, 628 (2000).[6] A. Altland, Phys. Rev. B 65, 104525 (2002).[7] B. Huckestein and A. Altland, cond-mat/0007413.[8] K. Ziegler, M. H. Hettler, and P. J. Hirschfeld, Phys.Rev. B 57, 10825 (1998)[9] C. Pe´pin and P. A. Lee, Phys. Rev. B 63, 054502 (2001).[10] I˙. Adagideli, D. E. Sheehy, and P. M. Goldbart, Phys.Rev. B 66, 140512(R) (2002).[11] W. A. Atkinson, P. J. Hirschfeld, A. H. MacDonald, andK. Ziegler Phys. Rev. Lett. 85, 3922 (2000)[12] A. Ghosal, M. Randeria, and N. Trivedi, Phys, Rev. B63, 020505 (2000).[13] W. A. Atkinson, P. J. Hirschfeld, and A. H. MacDonald,Phys. Rev. Lett. 85, 3926 (2000).[14] J.-X. Zhu, D. N. Sheng, and C. S. Ting, Phys. Rev. Lett.85, 4944 (2000).[15] A. G. Yashenkin, W. A. Atkinson, I. V. Gornyi, P. J.Hirschfeld, and D. V. Khveshchenko, Phys. Rev. Lett.86, 5982 (2001).[16] C. Chamon and C. Mudry, Phys. Rev. B 63, 100503(2001).[17] H. Walter et al., Phys. Rev. Lett. 80, 3598 (1998).[18] A. Yazdani, C. M. Howald, C. P. Lutz, A. Kapitulnik,and D. M. Eigler; Phys. Rev. Lett. 83, 176 (1999).[19] P. J. Turner et al., Phys. Rev. Lett. 90, 237005 (2003).[20] I.L. Aleiner and A.I. Larkin, Phys. Rev. B 54, 14423(1996).[21] S. Fishman, D. R. Grempel, and R. E. Prange, Phys.Rev. Lett. 49, 509 (1984).[22] A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. 77,4536 (1996).[23] A. Ossipov, T. Kottos, and T. Geisel, Phys. Rev. E 65,055209(R) (2002).[24] Ph. Jacquod, H. Schomerus, and C. W. J. Beenakker,Phys. Rev. Lett. 90, 207004 (2003); M. C. Goorden, Ph.Jacquod, and C. W. J. Beenakker, cond-mat/0306731.[25] F. M. Izrailev, Phys. Rep. 196, 299 (1990).[26] This diagonalization is performed with only O(M2 lnM)operations, if the multiplication with U in Eq. (5) is per-formed with the Fast-Fourier-Transform algorithm, see:R. Ketzmerick, K. Kruse, and T. Geisel, Physica D 131,247 (1999).[27] I˙. Adagideli, P. M. Goldbart, A. Shnirman, and A. Yaz-dani, Phys. Rev. Lett. 83, 5571 (1999).[28] Quantizing the kicked rotator on a torus corresponds to aballistic chaotic system [25] so that the Thouless energyis of the order of the size of the band.[29] T. P. Eggarter and R. Riedinger, Phys. Rev. B 18, 569(1978).[30] Ph. Jacquod and I˙. Adagideli, in preparation.

Low-energy quasiparticle excitations in dirty d-wave superconductors and the Bogoliubov-de Gennes kicked rotator

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-wave superconductors and the Bogoliubov–de Gennes kicked rotator

We investigate the quasiparticle density of states in disordered d-wave superconductors. By constructing a quantum map describing the quasiparticle dynamics in such a medium, we explore deviations of the density of states from its universal form (∝E), and show that additional low-energy quasiparticle states exist provided (i) the range of the impurity potential is much larger than the Fermi wavelength (allowing one to use recently developed semiclassical methods), (ii) classical trajectories exist along which the pair potential changes sign, and (iii) the diffractive scattering length is longer than the superconducting coherence length. In the classically chaotic regime, universal random matrix theory behavior is restored by quantum dynamical diffraction which shifts the low-energy states away from zero energy, and the quasiparticle density of states exhibits a linear pseudogap below an energy threshold E*≪Δ0, much smaller than the superconducting gap

Adagideli, İ.

Jacquod, Philippe

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Low-energy quasiparticle excitations in dirt <i>d</i>-wave superconductors and the Bogoliubov–de Gennes kicked rotator

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Low-energy quasiparticle excitations in dirty d-wave superconductors and
  the Bogoliubov-de Gennes kicked rotator

Low-energy quasiparticle excitations in dirty d-wave superconductors and the Bogoliubov-de Gennes kicked rotator

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