Comment on "Can disorder really enhance superconductivity?"
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Comment on ”Can disorder really enhance superconductivity?”
I. M. SuslovKapitza Institute for Physical Problems,Moscow, Russia
The paper by Mayoh and Garcia-Garcia [arXiv:1412.0029v1] is entitled ”Can disorder re-ally enhance superconductivity?”. In our opinion, the answer given by the authors is notsatisfactory. We present the alternative picture.
The paper by Mayoh and Garcia-Garcia [1] is en-titled ”Can disorder really enhance superconductiv-ity?”. In our opinion, the answer given by the au-thors is not satisfactory and we present the alterna-tive picture.
Mean field solution.
A basis for description of thespatially inhomogeneous superconductivity is givenby the Gor’kov equation for the order parameter∆( r ) ∆( r ) = Z K ( r, r ′ )∆( r ′ ) d d r ′ (1)with the kernel K ( r, r ′ ) satisfying the sum rule [2] Z K ( r, r ′ ) d d r ′ = gν F ( r ) ln 1 . ω T , (2)where g is the Cooper interaction constant, ν F ( r ) isthe local density of states at the Fermi level, ω is acut-off frequency, and d is a dimension of space.The Anderson theorem [3] follows from Eq.1 un-der assumption of a self-averaging order parameter,when ∆( r ) and K ( r, r ′ ) can be independently aver-aged over disorder. Since h ∆( r ) i does not depend on r due to the spatial uniformity in average, the use ofthe sum rule (2) gives h ∆ i = g h ν F i ln 1 . ω T h ∆ i , (3)and the critical temperature T c is given by the BCSformula, which contains the average density of states h ν F i . If the latter is maintained fixed, T c is notchanged by disorder. However, self-averaging doesnot hold in the general case, and deviations fromthe Anderson theorem arise.Equation (1) can be accurately solved for a smallconcentration of the point-like impurities [4]. Thissolution shows possibility of two regimes. The first one corresponds to the moderate variation of ∆( r )in space, so that it remains more or less of the sameorder in the whole space. The corresponding T c isgiven by the formula δT c T c = 1 λL d Z d d r ν ν ( r ) + ν ( r ) ν , (4)where ν ( r ) is a deviation of the local density ofstates ν F ( r ) from its unperturbed value ν , λ = gν is the dimensionless coupling constant, T c is thetransition temperature in the absence of disorder, L d is a volume for one impurity, and integration iscarried out over a vicinity of the single point de-fect. The linear in ν ( r ) term exactly correspondsto the Anderson theorem and relates the change in T c with the change of the average density of states.Generally, ν ( r ) is comparable with ν and alreadyEq.4 predicts a possibility of essential violation ofthe Anderson theorem. It is related with the factthat the initially uniform order parameter is influ-enced by point defects and can increase or decreasein their vicinity.More essential deviations from the Anderson the-orem arouse in the second regime, when the orderparameter is mainly localized at the small numberof ”resonant” impurities producing the quasi-localstates near the Fermi level. The corresponding esti-mate for T c [4] T c ∼ ga − d ∼ λE (5)( a is the lattice spacing, E is a scale of the orderof the Fermi energy or the bandwidth) is valid forsmall g and saturates by a quantity of the order ω ,when g increases.Results (4) and (5), obtained for a small cocen-tration c of impurities, can be qualitatively extrap-olated into the c ∼ T c by a trivial reason,due to increase of the average density of states.(b) If the average density of states remains fixed,then T c can be changed due to deviations from theAnderson theorem. This effect is always positive inthe framework of formula (4).(c) There is a possibility of the catastrophic in-crease of T c due to resonances on the quasi-locallevels, though this regime is affected by fluctuations(see below). Role of multifractality.
Recently there have beenclaims [5, 6, 7] that a great increase of T c is pos-sible in the vicinity of the Anderson transition dueto multifractality of wave functions. The analysis of[5, 6] is based on equation∆( ǫ ) = λ ω Z − ω I ( ǫ, ǫ ′ )∆( ǫ ′ ) p ǫ ′ + ∆( ǫ ′ ) tanh p ǫ ′ + ∆( ǫ ′ )2 T ! dǫ ′ (6)with the kernel of the form I ( ǫ, ǫ ′ ) = (cid:12)(cid:12)(cid:12)(cid:12) E ǫ − ǫ ′ (cid:12)(cid:12)(cid:12)(cid:12) γ , (7)motivated by multifractal properties of wave func-tions. T c is determined by equation (6) linearized in∆: accepting ∆ to be a function ǫ/T and dimension-alizing the integral, one has for infinite ω [5, 6] T c ∼ E λ /γ . (8)The singular limit of small γ was analyzed in [1] andlead to result T c ∼ E (cid:20)(cid:18) E ω (cid:19) γ + γλ (cid:21) − /γ (9)which reproduces (8) in the limit ω → ∞ , the BCSresult in the limit γ → ∼ ω for large λ . The claim of [1]that ω is always less than E is incorrect: E canbe small in semimetals and narrow band materials.The linearized form of equation (6) is not equiva-lent to (1) (see discussion in [6]) and based on two as-sumptions: (i) truncation of the Hamiltonian in theBCS spirit, and (ii) averaging of the kernel indepen-dently of ∆ . Both approximations are uncontrol- It means a self-averaging assumption, but in the modifiedform: instead of the usual equation h ∆ i = h K ih ∆ i one uses h S ∆ i = h SKS − ih S ∆ i where S is a certain operator. It canbe interpreted in the variational spirit, considering S as a kindof the trial function. lable. The partial justification of (i) was suggestedin [1]: truncation of the Hamiltonian is rigorous fora pure superconductor and should be valid approx-imately in the case of weak spatial inhomogeneity.One can partially excuse assumption (ii), using someeffective exponent in the capacity of γ . Indeed, thekernel I ( ǫ, ǫ ′ ) is determined by a matrix element R d d r | ψ ( ǫ, r ) | | ψ ( ǫ ′ , r ) | whose estimation gives dif-ferent values of γ for the ”usual” and ”typical” aver-aging [6]; this uncertainty is aggravated, if averagingis made with the weight ∆( r ).The first argument explains why the strong local-ization regime (corresponding to (5)) does not con-tain in (6) for I ( ǫ, ǫ ′ ) = const . The second argu-ment shows that a difference between (5) and (8)arises on the level, which is not controllable in theapproach based on Eq.6. In fact, the difference be-tween (5) and (8) is not quantitative, but qualitative.Result (5) is not restricted by a vicinity of the mo-bility edge and remains approximately the same forany energy inside the band; correspondingly, it hasno relation to multifractality. Nevertheless, the realphysical mechanism is the same for results (5) and(8) and related with resonances at quasi-local levels[4]. Indeed, if the local density of states ν ( ǫ, r ) is con-sidered as a smooth function of ǫ , then its variationis finite: it corresponds to shifts of the whole bandby a value W or − W for the Anderson model withdistribution of site energies in the interval ( − W, W ).Unbounded fluctuations of ν ( ǫ, r ), arising in the con-text of multifractality, are necessarily related with apartial discretization of the spectrum due to a pres-ence of quasi-local levels. If the usual value of γ isexploited in (7), then the T c value given by (5) isgreater than (8); it means that the Cooper instabil-ity occurs at configurations, which are governed byindividual peaks and not fractal clasters [4].Weak multifractality considered in [1] is practi-cally actual only for the 2 D case in the regime ofweak disorder. However, this regime is describedby formula (4), which can be obtained in this casemaking two iterations of the Gor’kov equation andexploiting the sum rule (2). In this case one canintegrate along all the system and use its whole vol-ume in the capacity of L d : dividing of disorder intoseparate ”impurities” is not necessary. There is noneed to use the approximate equation (6), when theaccurate result is available. By the way, in the frame-work of (4) the order parameter is proportional to ν F ( r ) and the logarithmically normal distributionfor ∆( r ) [1] follows trivially in the weak multifrac-2ality regime.It is clear from this consideration, that multifrac-tality is not a direct cause for the increase of T c andresults of kind (8) are related with the more univer-sal mechanism. Role of fluctuations.
Equation (5) is a result ofthe mean field theory. The corresponding configura-tion of the order parameter is a uniform background∆ with abrupt peaks at resonant impurities, whoseconcentration is of the order T c /E . The order pa-rameter can be considered as positive , and so itsphase is the same in the whole volume. When wecome to a fluctuational description, the modulus ofthe order parameter remains practically unchanged,while the essential phase fluctuations arise. If theuniform background is neglected, then the system isdivided into practically independent superconduct-ing ”drops”, whose phases are fluctuating freely anddestroy the macroscopical coherence of the super-conducting state. If the uniform contribution ∆ is taken into account, the Josephson coupling be-tween drops arises and their phases become corre-lated. The accurate fluctuational analysis of sucha system is nontrivial, but the general character ofresults is the same as for the granular superconduc-tors [8]. If the ratio T c /E is not too small, thenthe resonant impurities are close to each other andtheir Josephson interaction is strong enough for sta-bilization of the mean-field solution at practicallythe same T c value. Contrary, if T c /E is sufficientlysmall, then the Josephson coupling between drops isweak and fluctuations destroy superconductivity attemperatures close to the mean-field T c value. How-ever, decreasing of temperature stimulates the grow-ing of tails of the localized solutions [9]; the Joseph-son coupling between drops increases and stabilizessuperconductivity before T c is reached. Hence, fluc-tuations suppress T c in comparison with its mean-field value, but do not eliminate enhancement of T c completely.Analogous arguments can be given for configura-tions corresponding to (8), where a fraction of thesuperconducting phase is estimated as ( T c /E ) γ [6].However, such configurations are not actual, sincethe Cooper instablity corresponds to (5).The paper [1] suggests another way to deal withfluctuations. Firstly, solution of (6) for T = 0 is In the absence of magnetic effects, the kernel K ( r, r ′ ) ispositive, and the Cooper instability corresponds to the node-less eigenfunction. found, giving the spatially inhomogeneous order pa-rameter ∆ ( r ). Secondly, the field T c ( r ) of ”local T c ” values is introduced, such as T c ( r ) ∝ ∆ ( r ). Finally, the global T c is defined as a percolationthreshold in the field T c ( r ). Such percolation pic-ture has a sense for certain conditions [10], but it isnot the case for weak spatial inhomogeneity.Indeed, the Gor’kov equation (1) defines the spa-tially inhomogeneous configuration ∆( r ), which ap-pears at a certain temperature. This temperature,by definition, is a final result for T c : there is no needto consider ”local T c ” or ”percolation”. Of course,one should attend for insignificance of fluctuations,but this condition is rather weak. The Ginzburgnumber is incredibly small for a pure superconduc-tor; it increases gradually with increasing of spatialinhomogeneity, but it is possible to reach rather largevalues of ratio ∆ max / ∆ min before this number willapproach unity and fluctuations will become essen-tial. Surely, no percolation is necessary for weakspatial inhomogeneity. The percolation picture be-comes reasonable for the Ginzburg number of theorder of unity, when the mean-field estimate of T c ispoorly defined and the use of percolation allows torefine it. Role of interaction.
The BCS constant λ corre-sponds to some effective interaction. In a more de-tailed description it is combined from the electron-phonon coupling and the Coulomb pseudopotential.The latter is known to increase in the presence ofdisorder and it is the main cause for T c degradation[11, 12]. This effect was not considered explicitly in[1, 4, 5, 6], but is surely essential in discussion of theexperimental situation. It is the main reason whyenhancement of T c is a rare thing in reality.In paper [7] the result analogous to (8) is obtainedin the framework of the Finkelstein renormalizationgroup approach. However, it is completely differentfrom [5, 6] in the initial assumptions and the dis-cussed physical mechanism. The Cooper constant g is kept fixed in [5, 6], but an attempt is made to ad-vance beyond applicability of the Anderson theorem.Contrary, the authors of [7] consider renormalizationof g by disorder, while self-averaging is accepted forgranted. By the latter reason, the strongly localizedregime was not accessible in this approach, while theobtained effect is lesser than (5). The questions alsoarise, how results of [7] agree with the usual picture In fact, this relation is violated due to the presence ofscale T c .
3f the Coulomb pseudopotential enhancement.In conclusion, disorder can enhance T c : (a) dueto increase of the average density of states; (b) dueto deviations from the Anderson theorem; (c) dueto resonances on the quasi-local levels. The latterregime is affected by fluctuations, and in some casescan be essentially suppressed. Practically, enhance-ment of T c by disorder is a rare thing due to theincrease of the Coulomb pseudopotential.The author is indebted to M.V.Sadovskii for thediscussion of paper [1]. References [1] J. Mayoh, A. M. Garcia-Garcia,arXiv:1412.0029v1.[2] P. G. Gennes, Rev. Mod. Phys., , 225 (1964).[3] P. W. Anderson, J. Phys. Chem. Solids , 26(1959).A. A. Abrikosov, L. P. Gor’kov, Zh. Eksp. Teor.Fiz. , 1158 (1958).[4] I. M. Suslov, Zh. Eksp. Teor. Fiz. , 1184(2013) [J. Exp. Theor. Phys , 1042 (2013)].[5] M. V. Feigelman, L. B. Ioffe, V. E. Kravtsov,E. A. Yuzbashyan, Phys. Rev. Lett. , 027001(2007).[6] M. V. Feigelman, L. B. Ioffe, V. E. Kravtsov,E. Cuevas, Annals of Physics , 1368 (2010).[7] I. S. Burmistrov, I.V.Gornyi, A.D.Mirlin, Phys.Rev. Lett. , 017002 (2012).[8] B. M¨ u hlschlegel, D. J. Scalapino, R. Denton,Phys. Rev. B 6, 1767 (1972).G. Deutscher, Y. Imry, L. Gunter, Phys. Rev.B 10, 4598 (1974).[9] I. M. Suslov, Zh. Eksp. Teor. Fiz. , 949 (1989)[Sov. Phys. JETP , 546 (1989)].[10] L. B. Ioffe, A. I. Larkin, Zh. Eksp. Teor. Fiz. ,707 (1981) [Sov. Phys. JETP , 378 (1981)].[11] L. N. Bulaevskii, M. V. Sadovskii, J. Low-Temp. Phys. , 89 (1985).[12] M. V. Sadovskii, Phys. Reports , 225(1997). [13] I. M. Suslov, arXiv:1501.05148.[14] J. Mayoh, A. M. Garcia-Garcia,arXiv:1502.06282. Reply to arXiv:1502.06282
As a reaction to the preceding overview [13], Mayohand Garcia-Garcia have submitted the comment [14].This comment has a form of the personal attack andcontains the whole series of untrue statements. Wegive brief remarks, in order to reveal these statements.
1. Our mean field results for a superconductorwith a small concentration of the point-like impu-rities [4] are based on the accurate solution of theGor’kov equation. These results still persists, evenif somebody does not like them. Contrary, the ap-proach of [1] involves uncontrollable approximations[13].2. It is repeatedly stated in [14] that we pre-dict T c ∼ K for the superconducting transitiontemperature, which ”has never, and very likely willnever, been found numerically or experimentally”.In fact, it was clearly indicated (see the text afterEq.2 in [4]) that T c is bounded by the quantity ω /π ( ω is the cut-off frequency), which for the phononmechanism corresponds to values already observedfor oxide superconductors ( T c ∼ K , ω ∼ K ).A possibility to describe the latter in terms of ourmodel was discussed in Sec.7 of [4].3. The authors of [14] write on ”unjustified useof the mean field approximation”. In fact, insuffi-ciency of the mean field approach was indicated in[4, 13] and the role of fluctuations was extensivelydiscussed. We cannot help, if our arguments are ig-nored.4. According to [14], our formula (4) in [13] is re-lated with ”poorly defined” quantities and unclearconditions of applicability. In fact, all explanationswere given in [4, 13] and we can repeat those of themthat are questioned in [14]: L d is a volume for oneimpurity, i.e. the whole volume divided for a num-ber of impurities; the quantity ν ( r ) is the difference ν F ( r ) − ν , where ν F ( r ) is the local density of states,defined in a standard manner (see Eq.8 in [4]), and ν is its unperturbed value. For a small amplitudeof disorder ( | ν ( r ) | ≪ ν ) the indicated formula can4e obtained by two iterations of the Gor’kov equa-tion [13]. It remains valid for a small concentrationof strong impurities, when | ν ( r ) | ∼ ν [4]. It isnot restricted by any assumption on the correlationbetween impurities.5. According to [14], ”Suslov claims . . . that wedo not discuss phase-fluctuations”. There is no suchstatement in [13]; our comments on the role of inter-action are brief and contain no criticism of [1].6. According to [14], there is a wrong statement in[13]: ”Suslov also states that we claim E is alwayslarger than ǫ D ”. One can compare it with the text in[1]: ”We believe that this is necessary since ǫ D ≪ E so it is inconsistent to take the Debye energy to in-finity while keeping E finite”. This argument wasused in [1] to reject the result T c ∼ E λ /γ appear-ing in preceding publications: ”This is an expressionthat, we are at pains to stress, is not recovered inour formalism” [14]. In fact, this result follows fromEq.22 of [1] in the limit ǫ D → ∞ , independently ofthe desire of authors.7. According to [14], ”Suslov claims that our re-sults are only valid in small and strictly two dimen-sional systems”, while they are valid also for thinfilms and two-dimensional systems with spin-orbitinteractions. In fact, it is written in [13] that ”weakmultifractality considered in [1] is practically actualonly for the 2 D case in the regime of weak disorder”,and there is no rejection of the indicated additionalapplications.8. Our main criticism of [1] refers to the use of thepercolation picture [10] beyond the limits of its appli-cability. Indeed, the case of the ”moderate spatialinhomogeneity” is described by the Gor’kov equa-tion, which has solution ∆( r ) arising at a certaintemperature. This temperature, by definition, is afinal result for T c : there is no need to consider ”lo-cal T c ” or ”percolation”. Such situation persists, tillfluctuations are insignificant and the mean field esti-mation of T c is well-defined. The percolation picturebecomes reasonable for ”very strong spatial inhomo-geneity”, when the Ginzburg number becomes of theorder of unity; in this case the mean-field estimate of T c is poorly defined and the use of percolation allowsto improve it.The authors of [1] try to disprove our argumentsbasing on the difference between ”weak inhomogene-ity” and ”weak multifractality”. This attempt is notsuccessful due to the facts: (a) The suggested in [1] partial justification ofEq.6 in [13] refers to ”weak inhomogeneity” and notto ”weak multifractality”.(b) It is clear from [13] that for applicability ofpercolation, the ratio ∆ max / ∆ min should be greaterthan some large parameter; here ∆ max and ∆ min aretypical (not exclusive) values. The distribution for∆( r ) is presented in Fig.4 of [1]. This distributioncan be cut-off on both sides, since its tails corre-spond to local perturbations, which have no conse-quences for global superconductivity. After that theratio ∆ max / ∆ min is typically of the order of unity:it corresponds to ”moderate inhomogeneity” and ap-plicability of the Gor’kov equation.(c) The situation can be discussed constructively.Practically, ”weak multifractality” corresponds tothe weakly disordered 2D case, which is describedby formula (4) in [13]. The corresponding order pa-rameter is either slightly perturbed (for weak im-purities), or its perturbations are local (for a smallconcentration of strong impurities). In both cases,there are no problems with fluctuations, and hencethere is no place for percolation.The authors of [14] write correctly that ”accord-ing to Suslov our percolation analysis is intended todescribe fluctuations”: it is right, in spite of theirobjections. They are wrong in ascribing to us anidea that ”the value of T c resulting from percolationis similar to that obtained by averaging over T c ( r )”.9. It is stated in [14] that ”in weakly coupled su-perconductor λ ≪ T c /E is always small and theglobal critical temperature must be necessary zero”.This statement reveals a complete misunderstandingof our arguments. We say that, for not very small ra-tio T c /E , the average distance a ( E /T c ) / betweenresonant impurities becomes comparable with a andthe background value ∆ of the order parameter be-comes comparable with its resonant peaks. Then our”strongly localized regime” becomes ”moderately lo-calized” and has no problems with fluctuations. Inany case, we do not restrict ourselves by weak cou-pling, and in the worst situation T c falls to T c0