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Inhomogeneous order-parameter suppression in disordered d-wave superconductors

Gap inhomogeneities and the density of states in disordered d-wave superconductors
W. A. Atkinson1 , P. J. Hirschfeld1 , A. H. MacDonald2
2

Department of Physics, University of Florida, PO Box 118440, Gainesville FL 32611 Department of Physics, Indiana University, Swain Hall W. 117, Bloomington IN 47405 (February 1, 2008)

1

arXiv:cond-mat/0002333v2 [cond-mat.supr-con] 22 Jul 2000

We report on a numerical study of disorder e?ects in 2D d-wave BCS superconductors. We compare exact numerical solutions of the Bogoliubov-deGennes (BdG) equations for the density of states ρ(E) with the standard T-matrix approximation. Local suppression of the order parameter near impurity sites, which occurs in self-consistent solutions of the BdG equations, leads to apparent power law behavior ρ(E) ? |E|α with non-universal α over an energy scale comparable to the single impurity resonance energy ?0 . We show that the novel e?ects arise from static spatial correlations between the order parameter and the impurity distribution. 74.25.Bt,74.25.Jb,74.40.+k

In spite of strong electronic correlations in the normal state, the superconducting state of high Tc materials seems to be accurately described by a conventional BCS-like phenomenology. The debate over the k-space structure of the BCS order parameter ?k has now been resolved in favour of pairing states with d-wave symmetry. Since this symmmetry implies that ?k changes sign under rotation by π/2, there are necessarily points on the 2D Fermi surface at which ?k vanishes. The unique low-energy properties of high Tc superconductors are determined by the quasiparticle excitations in the vicinity of these nodal points. For conventional s-wave superconductors, the density of states (DOS) ρ(E) has a well de?ned gap and is largely una?ected by non-magnetic disorder. In contrast, ρ(E) ∝ |E| in clean d-wave superconductors, and can be substantially altered by disorder. Much of the current understanding of disorder e?ects comes from perturbative theories, such as the widely-used self-consistent T-matrix approximation (SCTMA). In particular, the SCTMA is exact in the limit of a single impurity, and has been used in studies of the local DOS near an isolated scatterer [1]. For su?ciently strong scatterers, an isolated impurity introduces a pair of resonances at energies ±?0 [2,3], where ?0 < ? is a function of of the impurity potential u0 and of the band asymmetry. Analytic expressions for ?0 (u0 ) have been given for a symmetric band [2], and in this special instance the unitary limit ?0 → 0 coincides with u0 → ∞. For a realistic (asymmetric) band, the relationship is more complex [3]. For a ?nite concentration of impurities ni , the SCTMA predicts that the impurity resonances broaden, with tails which overlap at the Fermi energy, leading to a ?nite residual DOS ρ(0) [3–8]. The region over which ρ(E) ≈ ρ(0), the “impurity band”, and has a width comparable to the scattering rate γ at E = 0. In the Born limit, the ±?0 resonances are widely separated in energy, and the overlap of their tails is exponentially small. In the strong-scattering limit, however, the overlap is substan√ tial and γ ? Γ?d , where ?d is the magnitude of the d-wave gap, Γ = ni /πN0 is the scattering rate in the 1

normal state, and N0 is the 2D normal-state DOS at the Fermi level. Several recent experiments [9–12] have studied quasiparticles in the impurity band in Zn-doped high Tc materials. Of particular note are attempts to verify a provocative prediction [10,11] that transport coe?cients on energy scales ω, T < γ have a universal value, independent of Γ [13,14]. The SCTMA which forms the basis of this world-view has several limitations. It is an e?ective medium theory, in which one solves for the eigenstates of an isolated impurity in the presence of a homogeneous mean-?eld representing all other impurities. This approach ignores multiple impurity scattering processes which are responsible for localization physics in metals and may lead to novel e?ects in 2D d-wave superconductors [13,15,16]. Another limitation of most SCTMA calculations is the use of a δ-function potential as an impurity model. While this simpli?es the calculation substantially, numerical results hint that the detailed structure of the impurity potential may be important [8]. A related issue, which will be discussed at length in this Letter, is inhomogeneous order-parameter suppression. It is well-known that d-wave superconductivity is destroyed locally near a strong scatterer, and in the single-impurity limit [19,20], the additional scattering was found to renormalise ?0 but, surprisingly, to leave most other details of the scattered eigenstates unchanged. Here we show, using exact numerical solutions of the Bogoliubov-deGennes (BdG) equations, that additional novel physics does arise in the many-impurity case. The main results of this work are summarised in Fig. 1. For a homogeneous order parameter, ρ(E) saturates at a constant value as E → 0 (in agreement with SCTMA), down to a mesoscopic energy scale ? 1/ρL2 where level repulsion across the Fermi surface induces a gap. This small gap may be a precursor to a regime associated with strong localization in the thermodynamic limit as discussed by Senthil et al. [16]. These authors predicted asymptotic power-law behavior in ρ(E) over an exponen2 tially small energy scale E2 ? 1/ρ(0)ξL , where ρ(0) is the residual density of states in the impurity band plateau

0.5
ρ(E)

0.4
SCTMA SCTMA+OP uncorr.

0.4

0.2

0.3

ρ(E)

0.0 ?0.5

0.2
BdG+OP BdG

0.0 E

0.5

0.1

vF = 5v?

0.0 ?1.0

?0.5

0.0 E

0.5

1.0

FIG. 1. Density of states for ni = 0.04 and u0 = 10. Numerical solutions of the BdG equations are shown with (BdG+OP) and without (BdG) self-consistent calculation of ?ij . Inset: T-matrix calculations of ρ(E) with (SCTMA+OP) and without (SCTMA) o?-diagonal scattering. Also shown is a model with uncorrelated impurity and o?-diagonal potentials (uncorr.).

and ξL ? (vF /γ) exp(vF /v? + v? /vF ) is the quasiparticle localization length. (vF is the Fermi velocity and v? is the gradient of ?k along the Fermi surface at the gap node). The actual value of the DOS at E = 0 is consistent with zero but not determined in our work; theoretically this point is still controversial. [18] Figure 1 shows that when the order parameter is determined self-consistently from the BdG equations ρ(E) is quite di?erent. At low energies the DOS can be ?t to a power law ρ(E) ? |E|α with nonuniversal α (Fig. 2). The power-law is the result of spatial correlations between the order-parameter and the impurity potential, and is therefore fundamentally di?erent from those of Nersesyan et al. [15] and Senthil et al [16], where asymptotic powerlaws were found, with α = 1/7 and α = 1 respectively. Unlike the DOS, the dimensionless conductance and inverse participation ratio (Fig. 3) are not changed significantly by self-consistency. Finally, we remark that the energy scale for the low-energy regime is >?0 , which is ? orders of magnitude larger than E2 for realistic parameters. We employ a one-band lattice model with nearest neighbour hopping amplitude t and a nearest neighbour attractive interaction V . Substitutional impurities are represented by a change in the on-site atomic energy. The Hamiltonian is H = ?t ? c? cjσ ? iσ [? ? Ui ]c? ciσ iσ (1)

are nearest neighbours, Ui is the impurity potential which takes the value u0 at a fraction ni of the sites and is zero elsewhere, and ?ij = ?V cj↓ ci↑ is the mean-?eld order parameter, determined self-consistently by diagonalizing Eq. (1). Throughout this work, energies are measured in units of t, where t is of order 100 meV for high Tc materials. In non-self-consistent calculations the OP has the familiar k-space form ?k = ?d [cos(kx ) ? cos(ky )], 1 where ?d = 2 ± (?i i±x ? ?i i±y ) is independent of i. Unless otherwise stated, V = ?2.3 and ? = 1.2 which yields ?d = 0.4 (corresponding to vF /v? ≈ 5) in the absence of disorder. Self-consistent solutions show that ?ij is suppressed within a few lattice constants of each strongly-scattering impurity. Throughout this Letter, curves marked BdG and BdG+OP refer to the neglect or inclusion of self-consistency in ?ij . The DOS is ρ(E) = L?2 α δ(E ? Eα ), where L is the linear system size and Eα are the discrete eigenenergies of H. Our numerical calculations were performed on periodically continued systems with L ≤ 45. Typical DOS curves were obtained for L = 25 by averaging ρ(E) over ? 50 ? 500 impurity con?gurations and ? 50 ? 100 kvectors in the supercell Brillouin-zone. For system sizes L ≥ 35, computational constraints restricted us to real periodic and anti-periodic boundary conditions. An important motivation for the present study is the need for a test of the reliability of SCTMA predictions. Thus, we use the same disordered lattice model for the SCTMA as for the BdG calculations. We would also like to model order-parameter suppression within a selfconsistent T-matrix approximation (SCTMA+OP), [17] and follow the ansatz of [19] that the o?-diagonal potential is δ?ij = ??ij [δi,0 + δj,0 ]. This term appears in the o?-diagonal block of the e?ective potential. In both cases, the T-matrix is a 2×2 matrix in particle-hole space which satis?es Tij (E) = Uij +
R,R′

UiR G(R ? R′ , E)TR′ j (E),

(2a)

where G has the Fourier transform, Gk (E) = Eτ0 ? ?k τ3 ? ?k τ1 ? Σk (E)
?1

,

(2b)

i,j

σ

i,σ

i,j

{?ij c? c? i↑ j↓

+ h.c.},

where the angle-brackets indicate that site indices i and j 2

?k is the tight-binding dispersion, τi are the Pauli matrices, and Σk (E) = ni Tkk (E). Equation (2a) is solved in real-space to take advantage of the short range of the e?ective potential U . Finally, ρ(E) = ?π ?1 L?2 Im k Tr Gk (E), where the trace is over particle-hole indices. Except for the mesoscopic gap discussed previously, the BdG curve in Fig. 1 is quantitatively similar to the SCTMA (inset). In contrast, the BdG+OP curve vanishes smoothly as E → 0, indicating that qualitatively new physics has been introduced by the inclusion of order parameter suppression. To emphasize that this result is unexpected, we point to the popular “Swiss cheese” model, in which it is assumed that pair-breaking causes a pocket of normal metal of radius ξ0 (the coherence

1

1

0.008

ρ(E)

a(L)

(b)
0.1

0.012 0.01 0.008
0.004

a< E=1.25

0.8

10%

0

a(E)

0.01

0.1

1 0.1

0

0.001 0.002 0.003

4%
0.6

E
1

0.006 0.004

1/L

2

ρ(E)

(c)
α

L=20

ρ(0)

2%
0.4

0.5 0 ?0.2

0.002

0

0 0.2 0.4 0.6

vF = 1.5v?
0 ?1 0 1

25 30 35 40 45 2

2%
0.2

1/u0
0.8

E

(d)
α

vF=5v?
0 ?1 ?0.5 0

(a)
0.5

0.4 0

u0=10 u0=5 u0=3.3

1

0

0.04

E

ni

0.08

FIG. 2. Dependence of ρ(E) on ni and u0 : (a) BdG+OP (solid) and SCTMA (dotted) for u0 = 5 (top three curves) and u0 = 10 (bottom). Also shown (dot-dash) is a ?t ρ(E) = A|E|α for ni = 0.04. (b) Logarithmic plot showing power law behavior for ni = 0.04, u0 = 5. (c) Dependence of power law on u0 for ni = 0.02. For comparison, SCTMA of ρ(0) vs. u0 is shown. Unitary limit is u0 ≈ 5. (d) Scaling of power law with ni . Note that when ni = 0, α = 1.

FIG. 3. Scaling of the inverse participation ratio for BdG+OP (solid curves) calculations. A BdG calculation (dashed) is also shown for L = 20. Parameters are V = ?4.47 (?d = 1.34), ni = 0.06, and ? = 1.0 with between 14 (L=45) and 500 (L=20) impurity con?gurations and real boundary-conditions. Inset: Scaling inside and outside the impurity band. a< is the average of a(E) for |E| < 0.8.

length) to form around each impurity. Within this model, 2 ρ(0) is enhanced relative to the SCTMA by ? ni N0 ξ0 . We emphasize that the correct spatial correlations between an impurity con?guration and its self-consistent o?-diagonal potentials must be preserved for the BdG+OP results to arise. This point is illustrated with a simple numerical calculation (inset of Fig. 1), in which ?ij is found self-consistently for random impurity distributions which are di?erent from those appearing in the diagonal block of H. The system therefore has two distinct types of impurity, one of which is purely o?diagonal, with uncorrelated distributions. It is striking that there is no hint of the correct low-energy behaviour in this calculation. It is instructive to compare the BdG+OP result with the SCTMA+OP (inset, Fig. 1), since they include o?diagonal scattering from the order parameter at di?erent levels of approximation. Furthermore, the SCTMA+OP does preserve the correlation between impurity location and order-parameter suppression. It has been used succesfully to describe shifts in ?0 due to o?-diagonal scattering [19,20] but, here, fails to reproduce the correct low energy DOS in the bulk disordered case. Instead, ρ(E) is quantitatively similar to the SCTMA, which is a direct result of the relative smallness of the o?-diagonal potential ?d /u0 ≈ 0.04. In Fig. 2 we illustrate the dependence of the low en3

ergy DOS on both ni and u0 . In Fig. 2(a) a series of curves shows how the low-energy regime scales towards zero-width as ni → 0 for the near-unitary scattering potential u0 = 5 [21]. That the size of the regime should scale faster than γ with ni is consistent with our earlier assertion that the novel behavior stems from a multipleimpurity e?ect. The details of the dilute impurity limit depend on the particular value of u0 however: at 2% impurity concentration, the low-energy regime is significantly larger for u0 = 10, which lies farther from unitarity, than for u0 = 5. For ?xed ni , we ?nd that the low-energy regime is >?0 . ? In Fig. 2(b), a logarithmic plot of ρ(E) reveals that the low energy DOS has an apparent power-law dependence on E, with non-universal exponent 0 < α < 1 [Fig. 2(c),(d)]. At low impurity concentrations, α is a strong function of both ni and u0 , with α a minimum for unitary scatterers. For larger ni , α appears to saturate at a value which is independent of u0 . We assert that this power law is fundamentally di?erent from those reported elsewhere [15,16], since it is only observed when o?-diagonal scattering is present and, as we will see next, is unrelated to strong quasiparticle localization. We have studied the scaling of both the inverse participation ratio a(E) and the Thouless number g(E) with system size. The inverse participation ratio is de?ned in the usual way, a(E) = x4 (E)/x2 (E)2 , where xm (E) = [ρ(E)L2 ]?1 n,r [|un (r)|m + |vn (r)|m ]δ(E ? En ) and un (r) and vn (r) are the particle and hole eigenfunctions, and is plotted in Fig. 3 for several system sizes. a(E) is a direct measure of the spatial extent of the wavefunction; it scales as 1/L2 for extended states and is constant for states with localization length ξL < L.

The scaling and energy dependence of a(E) is in general agreement with ref. [22] where calculations were made with similar parameters—there is a crossover in scaling between E < γ and E > γ consistent with a shorter localization length in the impurity band. Unlike ref. [22], we do not ?nd saturation in a< [the average of a(E < γ)] for L ≤ 45. For larger disorder concentrations (not shown) where ξL < 45 sites, we ?nd that a(E) saturates at E = 0 ?rst indicating that ξL (E) is an increasing function of E in the impurity band. The most important result for this work is that a(E), and therefore ξL , is not significantly di?erent in the BdG and BdG+OP calculations despite a substantial di?erence in ρ(E). In the calculation which is shown, a(E) is actually decreased slightly by self-consistency, corresponding to a slight increase in the localisation length. We have also studied the Thouless number, de?ned as g(E) = n [En (π) ? En (0)]δ(E ? En ), where the argument of En refers to the application of periodic or antiperiodic boundary conditions in the x-direction. As L is increased, g(E) is expected to cross over from a constant in the di?usive regime to exponential scaling indicative of strong localization. For L ≤ 45 we ?nd no signi?cant scaling of g(E) with L, consistent with what is found for a(E). Most signi?cantly, we ?nd that g(E) is nearly identical in the BdG and BdG+OP calculations, even for low-energy states where substantial changes in ρ(E) occur. This behaviour is reminiscent of what is seen in Hartree-Fock studies of interacting electrons in disordered conductors [24]. There, the Coulomb interaction enforces spatial correlations between the disorder and charge distributions and leads to the formation of a gap in ρ(E) [23], yet leaves the dimensionless conductance (a two-particle property related to the Thouless number) unchanged. The power-law DOS we observe here may have a similar origin; it is certainly clear that ρ(E) depends crucially on the spatial correlations between the impurity potential and d-wave order parameter. In the current work, however, it is the pairing interaction which is relevant and the BdG equations provide a mean-?eld description of the pairing interaction which is analogous to the Hartree-Fock description of the Coulomb interaction. We speculate that the short-ranged pairing interaction produces spatial correlations between distant impurities via the overlap of the long range tails [2] of the single impurity resonances. In this work we have shown that spatial correlations between order parameter and impurity distributions in d-wave superconductors lead to apparent power-laws in ρ(E) at low energies. These results are potentially relevant to quasi-2D superconductors like BSCCO-2212. Unfortunately, most disorder studies have been performed on the anisotropic 3D YBCO system, where correlation e?ects are expected to be less pronounced. Indeed there is considerable evidence, particularly in Zn-substituted YBCO, that disorder does indeed induce a ?nite DOS at the Fermi level [9,10,12] and somewhat weaker evi4

dence that it scales with disorder in accordance with the SCTMA [9]. Our work should therefore provide a strong motivation to study Zn doping, and other types of planar disorder, in the quasi-2D BSCCO-2212 system at low temperatures. This work is supported by NSF grants NSF-DMR9974396 and nsf-dmr9714055. The authors would like to thank M.P.A. Fisher, K. Muttalib, S. Vishveshwara and K. Ziegler for helpful discussions.

[1] For a recent review see M. E. Flatt? and J. M. Byers, e Solid State Phys. 52, 137 (1999). [2] A. V. Balatsky, M. I. Salkola, and A. Rosengren, Phys. Rev. B 51 15 547 (1995); A. V. Balatsky and M. I. Salkola, Phys. Rev. Lett. 76, 2386 (1996). [3] R. Fehrenbacher, Phys. Rev. Lett. 77, 1849 (1996); R. Fehrenbacher and M. R. Norman, Phys. Rev. B 50 R3495 (1994); R. Fehrenbacher, Phys. Rev. B 54, 6632 (1996). [4] L.P. Gor’kov and P. A. Kaugin, JETP Lett. 41, 253 (1985). [5] S. Schmitt-Rink, K. Miyake, and C. M. Varma, Phys. Rev. Lett. 57, 2575 (1986). [6] P. J. Hirschfeld, D. Vollhardt, and P. W¨l?e, Solid State o Commun. 59, 111 (1986). [7] R. Joynt, J. Low Temp. Phys. 109, 811 (1997). [8] T. Xiang and J. M. Wheatley, Phys. Rev. B 51, 11 721 (1995). [9] K. Ishida et al., J. Phys. Soc. Jpn. 62, 2803 (1993). [10] A. Hosseini et al., (unpublished); Kuan Zhang et al., Phys. Rev. Lett. 73, 2484 (1994); S. Kamal, Ruixing Liang, A. Hosseini, D. A. Bonn, and W. N. Hardy, Phys. Rev. B 58, R8933 (1998). [11] Louis Taillefer et al., Phys. Rev. Lett. 79, 483 (1997). [12] D. L. Sisson, et al., cond-mat/9904131. [13] P. A. Lee, Phys. Rev. Lett. 71, 1887 (1993). [14] P. J. Hirschfeld, W. O. Putikka, and D. J. Scalapino, Phys. Rev. Lett. 71, 3705 (1993); Phys. Rev. B 50, 10 250 (1994). [15] A. A. Nersesyan, A. M. Tsvelik, and F. Wenger, Nucl. Phys. B 438, 561 (1995); Phys. Rev. Lett. 72, 2628 (1994). [16] T. Senthil, Matthew P. A. Fisher, Leon Balents, and Chetan Nayak, Phys. Rev. Lett. 4704, 1998; T. Senthil and Matthew P. A. Fisher, cond-mat/9810238. [17] Matthias H. Hettler and P. J. Hirschfeld, Phys. Rev. B 59, 9606 (1999). [18] K. Ziegler, M.H. Hettler, and P.J. Hirschfeld, Phys. Rev. B 57, 10825 (1998). ˙ [19] Alexander Shnirman, Inanc Adagideli, Paul M. Goldbart, and Ali Yazdani, Phys. Rev. B 60, 7517 (1999). [20] W. A. Atkinson, P. J. Hirschfeld, and A. H. MacDonald, cond-mat/9912168. [21] For ? = 1.2, the band is asymmetric and the unitary limit corresponds to u0 ≈ 6 [20]. Thus, u0 = 5 is a stronger scatterer of low energy quasiparticles than u0 = 10 See

[7,3] for further discussion. [22] M. Franz, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 54, R6897 (1996). [23] A.L. Efros and B.I. Shklovskii, J. Phys. C 8, L49 (1975). [24] S.-R. Eric Yang and A.H. MacDonald, Phys. Rev. Lett. 70, 4110 (1993).

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