# Effect of spatial variations of superconducting gap on suppression of the transition temperature by impurities

###### Abstract

We calculate correction to the critical temperature of a dirty superconductor, which results from the local variations of the gap function near impurity sites. This correction is of the order of and becomes important for short-coherence length superconductors. It generally reduces a pair-breaking effect. In -wave superconductors small amounts of nonmagnetic impurities can increase the transition temperature.

###### pacs:

PACS numbers: 74.20.Fg, 74.62.Dh, 74.72.-hThe effect of disorder is of considerable significance for the physics of superconductivity. The well known picture for conventional -wave superconductors includes a strong pair-breaking role of magnetic impurities [2] versus a weak effect from time-reversal symmetric perturbations such as nonmagnetic impurities and other lattice defects [3], in agreement with the Anderson’s theorem [4]. Generalization to the case of anisotropic Cooper pairing shows that even nonmagnetic scatters are pair breakers for a momentum dependent gap. Such a behavior was observed for heavy-fermion and copper-oxide superconductors and the -wave superconducting gap of the latter is now well established [5]. The original Abrikosov-Gor’kov (AG) theory was extended to calculate changes of the transition temperature , superfluid density, and residual density of states for arbitrary strength of potential scatters in -wave superconductors [6].

On the experimental side, while some measurements on Zn-substituted Y-123 samples show perfect agreement with the AG theory [7], others observe a substantially slower-than-predicted reduction of with increasing impurity concentration in Y-123 and La-214 systems [8] or with growing ion damage on irradiated samples [9]. Various theoretical explanations were proposed for this discrepancy including strong-coupling effects [10] or an effect from the van Hove singularity [11]. An alternative hypothesis was suggested by Franz and co-workers [12], who presented numerical evidence for the effect of spatial variations of the gap neglected in the AG theory on thermodynamic properties for a model two-dimensional -wave superconductor. First discussions of such effects in conventional superconductors with magnetic impurities date to late 60’s [13, 14, 15]. However, they lack consistency with each other and neglect an important effect of the Friedel oscillations. In the present paper we calculate analytically corrections to the AG result for the transition temperature resulting from local variations of superconducting order parameter induced by nonmagnetic and magnetic impurities for various types of pairing.

Our main results can be summarized as follows. Spatial variations of , which appear only at very short distances of the order of from an impurity site, reduce the pair-breaking effect on . The correction to the AG result is of the order of and becomes important for short-coherence length superconductors such as high- cuprates. In -wave superconductors small amounts of nonmagnetic impurities can increase the transition temperature, if other effects, such as an energy dependent density of states, are small. The effect of spatial inhomogeneity is enhanced in two-dimensional superconductors compared to three-dimensional systems.

In our treatment of impurities in superconductors we go beyond the standard assumption of a spatially constant order parameter [2, 3]. The starting point is the linearized gap equation [16]

(1) |

for a singlet superconducting order parameter (we suppress spin indices) with and related to the relative and the center of mass coordinates, respectively. The finite-temperature normal state Green’s function in an external impurity potential depends on the two momenta. The retarded electron-electron interaction is taken in the form , i.e., we assume pairing according to a nondegenerate one-dimensional irreducible representation of the crystal point group and employ a cut-off , , in sums over Matsubara frequencies. Consequently, the superconducting gap in the mixed momentum-space representation is factorized and Eq. (1) is rewritten for a function of space coordinate as

(2) |

Here, both the gap and the nonuniform part of the kernel depend on impurity positions (). The order parameter is written in the form

(3) |

where , , and the function describes relative variations of the gap in space. Integrating the gap equation (2) over the volume and assuming the dilute limit () we have

(4) |

The terms without , when averaged over impurity positions, give the standard AG result, whereas the remaining members describe corrections due to spatial variations of the gap. In the low impurity concentration limit, the net space variation can be considered as a sum of the variations due to each impurity: , , . Substituting these back into Eq. (4) we notice that all terms with vanish after averaging over impurity positions. The impurity averaged gap equation has the following form

(5) |

where the difference of the di-gamma functions ’s represents the AG result: for nonmagnetic impurities in an unconventional superconductor and for magnetic impurities in an -wave superconductor, and being potential and exchange scattering times, respectively; is the Fermi level density of states. In the dilute limit the correction to the AG result is expressed via single impurity characteristics only. Similar equations have been earlier obtained in [13]. Our consideration departs, however, now from previous works.

In order to express spatial variation of the gap function for a single impurity we return back to Eq. (2). The relation (3) is now replaced by , since vanishes in the thermodynamic limit for one impurity. Neglecting variations of one the right hand side of Eq. (2) we immediately arrive at

(6) |

This is the so called non-self-consistent approximation for the spatial variation of the gap [15, 18] and its validity for short distances from the impurity is justified later. Using symmetry of the kernel we finally get from Eq. (5) our main result:

(7) |

In contrast to previous studies [13], the above equation takes into full account the effect of spatial variations of the gap, including the rapid Friedel oscillations of wave-length in the vicinity of the impurity. Though Eq. (7) contains only linear in corrections, it holds for both magnetic and nonmagnetic impurities and for an arbitrary type of pairing. As is seen from Eq. (7), the qualitative role of the correction is always to reduce the pair-breaking effect. In the case of nonmagnetic impurities in an -wave superconductor, when , the transition temperature increases as a result of spatial variations.

## Nonmagnetic impurities in -wave superconductors.

In order to estimate these effects quantitatively we now turn to the calculation of the relative gap suppression in particular cases, considering first a -function impurity potential in an -wave superconductor. The Nambu Green’s function in superconducting state is expressed via a -matrix as , where satisfies Gor’kov’s equation without impurity potential, but with corresponding variations of the gap:

(8) |

The function has to be found from the gap equation

(9) |

Here , is the quasiparticle dispersion, and are Pauli matrices. The expression for the -matrix is , which we approximate by for .

The short-distance behavior of can be found approximating by . This is exactly the non-self-consistent approximation used above. The real space representation for is

(10) |

where , . Substituting it back into Eq. (9) we find

(11) |

A similar result was obtained for a finite-radius impurity by Fetter [17]. Note that the method used in [13, 15] for a magnetic impurity would give zero instead of (11) and, consequently, no effect on the transition temperature. In the limit one can neglect and find the following analytic asymptotes

(12) |

and for , where is the coherence length. The latter long-distance asymptote is incorrect since no such rapid oscillations are present in the Ginzburg-Landau regime. The correct behavior is recovered in the self-consistent analysis. However, for the purpose of calculating the correction (7) we need to know only the short distance expression (12). The function falls of to a negligible value within a few Fermi wave-lengths . Such rapid relaxation of the gap has also been seen in numerical simulations [18, 19]. Accordingly, the integral in Eq. (7) quickly converges and does not depend on the upper limit. Integral’s value is and, hence, the transition temperature of an -wave superconductor with nonmagnetic impurities is

(13) |

where is the zero-temperature gap of a pure system and is the normal state scattering rate. The result (13) does not contradict Anderson’s theorem, which is valid only for a spatially homogeneous superconducting state [4].

In order to prove the validity of the non-self-consistent approximation we return back to Eq. (8) and write its solution to the first order in :

(14) |

Substituting this into the gap equation and keeping only terms of first order in impurity potential we find after Fourier transformation

(15) |

The difference with the non-self-consistent solution appears because of the term in the curl brackets with the sum over momenta given near by

(16) |

The short distance behavior of corresponds to the large- asymptote for . In this case we neglected Matsubara frequencies in the above equation and found the remaining integral to be . The self-consistent correction in Eq. (15) is and is small for . In the corresponding space region the non-self-consistent solution (12) is a good approximation for the gap variations. Numerical data show that this may be an accurate approximation even for a strong impurity potential [18].

## Magnetic impurities.

We consider a classical spin, which interacts with conduction electrons via an exchange Hamiltonian , where are spin Pauli matrices. In order to express appropriately corresponding effects, one should include spin indices in Eq. (1). Invariance of the singlet superconducting order parameter with respect to spin rotations [20] suggests, however, that only rotationally-invariant part of the impurity -matrix produces a non-zero effect. Therefore, we can average with respect to the direction of spin, though it does not imply rapid fluctuations of . The averaged -matrix is now diagonal in spin indices and is approximated as for [15]. Assuming a particle-hole symmetry we have . Following exactly the same procedure as described above we find for short distances

(17) |

For pair-breaking magnetic impurities there is an additional non-oscillatory suppression of the superconducting order parameter close to the impurity. Note that Eq. (17) [as well as Eq. (12)] fails at very short distances , where an exact band structure becomes important (see, e.g., [19]). The correction (7) can be nevertheless evaluated using the expression (17) with the lower limit cut-off . The result is

(18) |

where . The qualitative difference in a way how the gap function relaxes near magnetic and nonmagnetic impurities has no effect on the integral in (7), which is always of the order of due to rapid relaxation. The difference appears, however, in the dependence of the transition temperature correction on the strength of perturbation , showing an additional smallness of the effect for magnetic impurities.

## Nonmagnetic impurities in unconventional superconductors.

The real space formalism used above becomes inapplicable in this case, since local variations of the gap function depend on the exact nature of the equilibrium gap and are generally highly anisotropic. Our goal is to show that the order of magnitude of the correction to the AG result remains unchanged compared to the case of nonmagnetic impurities in conventional superconductors. For that we rewrite all equations in the momentum representation, including and the gap equation (9) as

We further neglect in the arguments of and average over the Fermi surface; this procedure changes the numerical factor but not the order of magnitude. After that the integrand becomes the same as for the Fourier transform of (12). Therefore, the correction in Eq. (7) will be of the same order of magnitude and finally we have

(19) |

with . We now compare this result with the numerical data of Franz et al. [12]. For these authors found a 14% correction to the AG result, which corresponds to . For their result is a 50% correction to the slope of at or . Experiment on Zn-substituted Y-123 superconductor () found a 40% discrepancy with the AG theory [8], implying . Somewhat larger values for the factor found experimentally and numerically are related to the two-dimensional (2D) nature of the copper oxide materials (see below). 3–5; in the BCS theory

## Two-dimensional superconductors.

We consider the case of an -wave gap. The effect of nonmagnetic impurities can be calculated as in the 3D case with the only change that the Green’s functions have to be taken in the 2D form: . Using the asymptotic form of the Hankel function of the first kind at we have . Substituting into Eq. (10) we obtain for

(20) |

The characteristic feature of the reduced dimensionality is that the order parameter relaxes more slowly to its bulk value. Because of this the integral in Eq. (7) formally diverges for the above and becomes dependent on the upper limit. This yields an additional logarithmic factor 2–4 to the correction in Eq. (13) enhancing effect of spatial variations near impurities on the mean-field in 2D. The previous arguments suggest that this result must hold for any symmetry of the gap.

In conclusion, we have calculated a reduction of the pair-breaking effect on the superconducting transition temperature due to local variations of the order parameter near impurities. Such an effect is important in short-coherence-length superconductors and is further enhanced for quasi two-dimensional systems. In singlet superconductors, spatial variations of the order parameter near magnetic impurities are less important for than near nonmagnetic. Our theory explains numerical data for a model -wave superconductor [12] and shows the significance of such effects in high- cuprates. We also predict an enhancement of the critical temperature of an -wave superconductor with nonmagnetic impurities. This prediction can be tested numerically extending previous model simulations [12, 18] and may have an experimental relevance to NdCeCuO and other electron-doped cuprates, which are believed to be layered -wave superconductors.

This work was supported by the National Science and Engineering Research Council of Canada. We are grateful to V. I. Rupasov for useful discussions.

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- [*] On leave of absence from L. D. Landau Institute for Theoretical Physics, Moscow, Russia.
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