Calculation of the effect of random superfluid density on the temperature dependence of the penetration depth
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Calculation of the effect of random superfluid density on thetemperature dependence of the penetration depth.
Thomas M. Lippman
1, 2 and Kathryn A. Moler
1, 2, 3, ∗ Stanford Institute for Materials and Energy Sciences,SLAC National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, CA 94025, USA. Department of Physics, Stanford University,Stanford, California 94305-4045, USA. Department of Applied Physics, Stanford University,Stanford, California 94305-4045, USA. (Dated: December 6, 2018)
Abstract
Microscopic variations in composition or structure can lead to nanoscale inhomogeneity in super-conducting properties such as the magnetic penetration depth, but measurements of these prop-erties are usually made on longer length scales. We solve a generalized London equation with anon-uniform penetration depth, λ ( r ), obtaining an approximate solution for the disorder-averagedMeissner screening. We find that the effective penetration depth is different from the average pen-etration depth and is sensitive to the details of the disorder. These results indicate the need forcaution when interpreting measurements of the penetration depth and its temperature dependencein systems which may be inhomogeneous. PACS numbers: 74.62.En,74.20.-z,74.62.Dh,74.81.-g . INTRODUCTION The penetration depth and its temperature dependence are important characteristics ofany superconductor and are considered key to determining the momentum space structure ofthe order parameter.
The possibility of disorder in exotic superconductors is well known,but analyses performed to date have concentrated on the effect of disorder-induced scatteringon the momentum space structure of the gap.
This paper is motivated by the possibilitythat disorder may lead to nanoscale real space variation and the associated need to modelthe relationship between such spatial variation and properties that are measured on longerlength scales. We address how inhomogeneity in the penetration depth may affect bulkmeasurements of the penetration depth for methods that rely on Meissner screening and canbe analyzed by solutions to London’s equation. In particular, we show that the measuredresult is not simply given by the average value of the penetration depth, but is affected bythe statistical structure of the spatial variations in the penetration depth.Many superconductors are created by chemical doping of a non-superconducting parentcompound. In these systems the inherent randomness of the doping process may give riseto an inhomogeneous superconducting state. The importance of this effect will be deter-mined by the characteristic length over which the dopant atoms affect the superconductivity.Even in the most ordered material, there will be binomial fluctuations in the total number ofdopants in a given region. In general, one does not expect significant spatial variation in ma-terials that are weakly correlated and can be described by a rigid band model. For example,disorder is largely irrelevant in classic metallic superconductors, due to their long coherencelengths and weakly correlated nature. In contrast, the cuprates are doped insulators witha coherence length on the scale of the lattice. They are known to have nanoscale disorderin their superconducting properties, as seen by scanning tunneling microscopy. Similar gapmaps have been observed in the iron pnictide family and in disordered titanium nitridefilms close to the superconductor to insulator transition.
Materials with intrinsic disorder present two separate challenges. Understanding how therandom doping process gives rise to local superconducting properties, such as the penetrationdepth or local density of states, requires a microscopic model. But even with such a model,we still need to make the connection between the local superconducting properties and bulkmeasurements. The manner in which local superconducting properties relate to the observed2roperties will differ from experiment to experiment. For instance, a measurement of theheat capacity will return the total heat capacity of the macroscopic sample, so the inferredspecific heat capacity will be a volume average over the sample. In contrast, we might expectthe thermal conductivity response to be dominated by a percolation path connecting regionswith small local gap, ∆( r ), or large local density of states.Here we focus on the penetration depth, λ , as measured by screening of the magneticfield, including both resonant cavity frequency shift measurements at radio frequencies andthe local probes of Magnetic Force Microscopy and Scanning SQUID Susceptometry. These methods measure λ ( T ) by detecting the response magnetic field generated by thesuperconductor due to an applied field, and can be analyzed using the London equation.Thus, we can model the effect of inhomogeneity by solving the London equation with λ ( r ) asa random function of position r . Our goal is to find a new equation for the disorder-averagedmagnetic field, as this will determine the measured response. Here, we work in the limit ofsmall fluctuations to find an approximate equation for the disorder-averaged magnetic field,as this will determine the measured response. II. STOCHASTIC LONDON EQUATION
To understand the measured penetration depth when λ ( r ) is a random function of po-sition, we calculate the disorder-averaged magnetic field response to obtain an effectivepenetration depth. For isotropic and local superconductors in three dimensions, the staticmagnetic field h ( r ) is given by the London equation with λ ( r ) a function of position. Thecorrect form of the London equation when the penetration depth is non-uniform is: h + ∇× (cid:2) λ ( r ) ∇× h (cid:3) = 0 , (1)which is derived from the second Ginzburg-Landau equation in the London limit. Weparametrize the penetration depth as the average value plus a fluctuating term: λ ( r ) = Λ [1 + ξ ( r )] , (2)so that h λ ( r ) i = Λ. Then Eq. 1 becomes:( L + M + M ) h = 0 , (3)3here: L ≡ − Λ ∇ I ,M ≡ − ξ ∇ I + 2Λ ∇ ξ × ∇× , and (4) M ≡ − Λ ξ ∇ I + Λ ∇ ξ × ∇× . Here I is the identity tensor, and the “dangling curl” is understood to operate on a vector toits right. The terms are grouped so that M is first-order in ξ , M is second-order in ξ , and L gives the unperturbed London equation. We will work in the limit of small fluctuations, ξ ( r ) ≪
1, so that M + M is a perturbative term in Eq. 3.Our method of solution comes from the similarity of the Helmholtz and London equations.The Helmholtz equation, which governs wave propagation, becomes the London equationwhen the wavevector is purely imaginary. Thus our problem is related to the propagation ofwaves in a random medium, and we can build upon a large and multidisciplinary literaturedevoted to this challenge. The paper by Karal and Keller is particularly relevant,because it retains the vectorial nature of the problem, rather than simplifying to a scalarwave equation.We now derive, from Eq. 3, an approximate equation for the disorder-averaged field h h i .Applying the inverse of L to both sides: (cid:2) L − ( M + M ) (cid:3) h = h , (5)where L h = 0. Solving for h : h = (cid:2) L − ( M + M ) (cid:3) − h , (6)assuming the inverse exists. Averaging both sides: h h i = D(cid:2) L − ( M + M ) (cid:3) − E h , (7)where h comes outside of the average because it is non-random. Solving for h : D(cid:2) L − ( M + M ) (cid:3) − E − h h i = h . (8)Since we assume small fluctuations, we can expand the term inside the average: (cid:10) − L − ( M + M ) + L − M L − M + O ( ξ ) (cid:11) − h h i = h . (9)4veraging and expanding again: (cid:0) − L − h M L − M i + L − h M i (cid:1) h h i = h , (10)since h M i = 0 due to Eq. 2. We then apply L to both sides: (cid:0) L − h M L − M i + h M i (cid:1) h h i = 0 , (11)which yields the average field to second order in ξ . III. RESULTS
We first evaluate the averages in Eq. 11, giving us an equation for h h i in terms of thepenetration depth correlation function, h λ ( r ) λ ( r ′ ) i . We then consider two specific cases forthe correlation function and numerically evaluate the effective penetration depth for a rangeof parameters. A. Evaluating the Averages
We will solve Eq. 11 for a single Fourier mode of h h ( r ) i = h e i k · r , then derive an equationfor k that yields exponentially decaying solutions consistent with Meissner screening.First, we evaluate h M i : h M i = − Λ h ξ ( r ) i ∇ I + Λ h ∇ ξ ( r ) i × ∇× . (12)We introduce the correlation function R ( r , r ′ ) = h ξ ( r ) ξ ( r ′ ) i , which is a function only of | r − r ′ | if ξ is stationary and isotropic. Then we see that h ξ i = R (0) and h ∇ ξ i = ∇ h ξ i = 0,so h M ih h i = Λ k R (0) h e i k · r . (13)We now evaluate the remaining average, (cid:10) M L − M (cid:11) , in three stages to derive Eq. 21.First we expand the differential operations, then evaluate the disorder average. The laststage is to evaluate the integral. We will then combine this integral with Eq. 13 to solveEq. 11.The average to evaluate has the form: (cid:10) M L − M (cid:11) h h i = Z d r ′ D M ( r ) G ( r − r ′ ) M ( r ′ ) E h h ( r ′ ) i . (14)5he Green’s function is the solution to (1 − Λ ∇ ) G ( r , r ′ ) = δ ( r − r ′ ), and is: G ( z ) = 1Λ πz e − z/ Λ . (15)Here, we have defined z = r − r ′ .We now expand the differential operations in Eq. 14. We do this in two segments, firstwith derivatives at r , then with derivatives at r ′ . The first is: M ( r ) G ( r − r ′ ) v ( r ′ ) = (cid:2) − ξ ( r ) ∇ r + 2Λ ∇ r ξ ( r ) × ∇ r × (cid:3) G ( r − r ′ ) v ( r ′ )= 2 ξ v ( r ′ ) (cid:2) δ ( z ) − G ( z ) (cid:3) + 2Λ ∇ ξ × (cid:2) ∇ r G ( r − r ′ ) (cid:3) × v ( r ′ ) . (16)The second, which was represented by v ( r ′ ) above, is: v ( r ′ ) = M ( r ′ ) h h ( r ′ ) i = e i k · r ′ h k ξ ( r ′ ) h + 2 i Λ ∇ r ′ ξ ( r ′ ) × k × h i = 2Λ e i k · r ′ n(cid:2) k ξ ( r ′ ) − i ∇ r ′ ξ ( r ′ ) · k (cid:3) h + i (cid:2) ∇ r ′ ξ ( r ′ ) · h (cid:3) k o . (17)Combining Eqns. 16 and 17, we obtain: M ( r ) G ( r − r ′ ) M ( r ′ ) h h ( r ′ ) i = 2Λ e i k · r ′ ( ξ ( r ) ξ ( r ′ ) (cid:2) δ ( z ) − G ( z ) (cid:3) k h + ξ ( r ) ∇ r ′ ξ (cid:0) h ⊗ k − k ⊗ h (cid:1) i (cid:2) δ ( z ) − G ( z ) (cid:3) − G ( z ) (cid:0) Λ − + z − (cid:1)(cid:20) ξ ( r ′ ) ∇ r ξ (cid:0) h ⊗ ˆ z − ˆ z ⊗ h (cid:1) k + i (cid:0) h ⊗ ˆ z − ˆ z ⊗ h (cid:1)(cid:0) ∇ r ξ ⊗ ∇ r ′ ξ (cid:1) k + i (cid:0) ˆ z ⊗ k − k ⊗ ˆ z (cid:1)(cid:0) ∇ r ξ ⊗ ∇ r ′ ξ (cid:1) h (cid:21)) , (18)where we use ⊗ to indicate the tensor product.To perform the disorder average in the second stage, we need various derivatives of thecorrelation function R ( z ): (cid:10) ξ ( r ) ∇ r ′ ξ (cid:11) = ∇ r ′ R (cid:0) | r − r ′ | (cid:1) = − ˆ z ˙ R ( z ) , (cid:10) ξ ( r ′ ) ∇ r ξ (cid:11) = ∇ r R (cid:0) | r − r ′ | (cid:1) = ˆ z ˙ R ( z ) , and (cid:10) ∇ r ξ ∇ r ′ ξ (cid:11) = ∇∇ ′ R ( | r − r ′ | ) = − (cid:20) ˙ Rz I + ˆ z ⊗ ˆ z (cid:16) ¨ R − ˙ Rz (cid:17)(cid:21) , where the overdot indicates differentiation with respect to z, and I is the identity tensor.Then averaging Eq. 18 gives: (cid:10) M ( r ) G ( r − r ′ ) M ( r ′ ) (cid:11) h h ( r ′ ) i = 2Λ e i k · r ′ n(cid:2) A ( z ) + 2 k R ( z ) δ ( z ) (cid:3) h − (cid:2) B ( z ) + 2 i ˙ R ( z ) δ ( z ) (cid:3)(cid:0) k ⊗ h − h ⊗ k (cid:1) ˆ z − C ( z ) h (cid:0) ˆ z ⊗ ˆ z (cid:1)o , (19)6ith the scalars A, B , and C given by: A ( z ) = 2 k h ˙ R ( z )Λ G ( z ) (cid:0) Λ − + z − (cid:1) − R ( z ) G ( z ) i ,B ( z ) = 2 i h ¨ R ( z )Λ G ( z ) (cid:0) Λ − + z − (cid:1) − ˙ R ( z ) G ( z ) i , and C ( z ) = 2Λ k ˙ R ( z ) G ( z ) (cid:0) Λ − + z − (cid:1) . The final stage in evaluating Eq. 14 is to perform the integral over r ′ . We first changevariables from r ′ to z , then integrate over the orientation of z . Using the relations Z dˆ z e − i k · z = 4 π sin( kz ) kz ≡ F ( k, z ) , Z dˆ z ˆ z e − i k · z = ˆ k iz ∂ k F, and Z dˆ z ˆ z ⊗ ˆ z e − i k · z = − z (cid:20) ∂ k Fk I + ˆ k ⊗ ˆ k (cid:18) ∂ k F − ∂ k Fk (cid:19)(cid:21) , (20)we find that Eq. 14 evaluates to: Z d r ′ D M ( r ) G ( r − r ′ ) M ( r ′ ) E h h ( r ′ ) i = 4Λ k e i k · r n [ X + R (0)] h + Y ˆ k ( h · ˆ k ) o . (21)The functions X and Y are given by: X ( k ) = Z ∞ d z G ( z ) (cid:26) Λ (cid:0) Λ − + z − (cid:1)h ˙ R (cid:0) z F + k − ∂ k F (cid:1) − ¨ Rzk − ∂ k F i + ˙ Rzk − ∂ k F − Rz F (cid:27) ,Y ( k ) = Z ∞ d z G ( z ) (cid:26) Λ (cid:0) Λ − + z − (cid:1)h ¨ Rzk − ∂ k F + ˙ R (cid:0) ∂ k F − k − ∂ k F (cid:1)i − ˙ Rzk − ∂ k F (cid:27) . (22)We require the average magnetic field to have ∇· h = 0, which means that k · h = 0. Wenow collect our results from Eqns. 21 and 13, and insert them into Eq. 11: h e i k · r h k (cid:0) − R (0) − X (cid:1)i = 0 . (23)We are interested in solutions consistent with Meissner screening, so we require that k bepositive and purely imaginary. Then the field decays on a length scale λ eff = ik , which weidentify as the experimentally measured penetration depth. To calculate λ eff , we will solvethe equation: λ Λ = 1 − R (0) − X. (24)7nserting k = iλ eff into our equation for X , we get: X = 4 π Z ∞ d z G ( z )sinh (cid:18) zλ eff (cid:19) (cid:26) Λ (cid:0) Λ − + z − (cid:1)h ˙ Rλ eff z − (cid:0) z + λ (cid:1) − ¨ Rλ i + ˙ Rλ − Rzλ eff (cid:27) + 4 π Z ∞ d z G ( z )cosh (cid:18) zλ eff (cid:19) (cid:26) Λ (cid:0) Λ − + z − (cid:1)h − ˙ Rλ + ¨ Rzλ i − ˙ Rzλ (cid:27) . (25)Valid solutions for λ eff will require the integral for X to converge and Eq. 24 to havesolutions. B. Correlation Function
A full solution of the disorder-averaged magnetic field, h h i , requires knowledge of the cor-relation function R ( z ) and hence requires not only a detailed knowledge of the composition,structure, and disorder of the sample, but also a microscopic model to locally determinethe superconducting properties from that structure. Without guidance from microscopiccalculations, we will use the Mat´ern one-parameter family of correlation functions to tunethe smoothness, as well as the magnitude and correlation length, of the penetration depthfluctuations. Handcock and Wallis parametrize the Mat´ern class of covariance functionsas: R ( z ) = R (0)2 ν − Γ( ν ) (cid:16) √ ν zl (cid:17) ν K ν (cid:16) √ ν zl (cid:17) , (26)where K ν is a modified Bessel function of the second kind and Γ( z ) is the Gamma function.The intercept at zero separation is the normalized variance of the penetration depth, R (0) = σ λ / h λ i = ( h λ i − h λ i ) / h λ i , and quantifies the magnitude of the inhomogeneity in λ ( r ).The correlation length, l , controls the size of the fluctuations in λ ( r ). The parameter ν controls the smoothness of λ ( r ). Larger ν gives a smoother random field, since it is ⌈ ν ⌉ − ⌈·⌉ is the ceiling function. Two members of the family deserve specific mention. When ν = 1 /
2, Eq. 26 reducesto the exponential correlation function, R ( z ) = R (0) exp (cid:0) − z √ /l (cid:1) , which is the correlationfunction of a Markov process in one dimension. The integrals for X in Eq. 25 divergewhen R ( z ) ∝ e − z , making the case ν = 1 / ν → ∞ , R ( z ) → R (0) exp( − z /l ), which is labeled the squared exponential correlation function, to preventconfusion with the Gaussian probability distribution. This correlation function gives thesmoothest possible λ ( r ) that can be described within the Mat´ern covariance family.8 . Squared Exponential Correlations We now consider the case of squared exponential correlations, R ( z ) = R (0) e − z /l . InFig. 1 we plot four realizations of a normally distributed penetration depth with squaredexponential correlations, illustrating the effect of the two parameters l and R (0) on λ ( r ).Evaluating Eq. 25 gives: X = − R (0) Z ∞ d z e − z/ Λ e − z /l sinh (cid:18) zλ eff (cid:19) λ eff Λ h(cid:0) Λ l (cid:1) + 2 z Λ l i (cid:0) λ l (cid:1) + 2 R (0) Z ∞ d z e − z/ Λ e − z / l cosh (cid:18) zλ eff (cid:19) λ Λ h z l (cid:0) Λ l (cid:1) + 2 z Λ l i . (27)All of these integrals converge, so we evaluate X as: X = R (0) 2 λ l + R (0) λ eff √ π l Λ ((cid:16) l − l Λ λ eff + 4Λ λ (cid:17) exp " l (cid:18)
1Λ + 1 λ eff (cid:19) erfc (cid:20) l (cid:18)
1Λ + 1 λ eff (cid:19)(cid:21) − (cid:16) l + 2 l Λ λ eff + 4Λ λ (cid:17) exp " l (cid:18) − λ eff (cid:19) erfc (cid:20) l (cid:18) − λ eff (cid:19)(cid:21)) . (28)After inserting Eq. 28 into Eq. 24, we solve for λ eff over three decades in the corre-lation length, l , and in the disorder variance, R (0) (Fig. 2). At large correlation lengththe effective penetration depth is larger than the average value, representing suppressed Meissner screening. Conversely, at small correlation length the effective penetration depthis smaller than the average, indicating enhanced screening. The separatrix, where λ eff = Λfor all values of R (0), occurs near l = 1 . R (0) →
1. This is truefor both linear and logarithmic spacing around the separatrix. In other words, neither | λ eff ( l s + ∆ l ) − h λ i| = | λ eff ( l s − ∆ l ) − h λ i| nor | λ eff ( al s ) − h λ i| = | λ eff ( l s /a ) − h λ i| are true,where l s denotes the separatrix, and a is an arbitrary positive real number. As expected, λ eff → Λ as R (0) → . Yet even at small disorder, λ eff has variations on the one percentscale, shown by the contours in Fig. 2. As we will discuss below, sub-percent variations of λ eff could be significant in the context of a typical measurement of ∆ λ ( T ).The trends in λ eff can also be seen in Fig. 3, where we plot λ eff / Λ vs. R (0) at fixedcorrelation length. All three curves taper to λ eff = Λ as the magnitude of disorder decreases.At large correlation length, in this case l = 10Λ, λ eff increases by ten percent when R (0) =9 .
02. The effect at small correlation is more modest, but still reaches nearly ten percent bythe time R (0) = 0 . l = 0 . R (0) and l will have a tem-perature dependence of their own, which will create a temperature-induced change in λ eff .This change contributes to any measurement of λ ( T ), but is not related to the gap structurein momentum space, because it arises from the spatial arrangement of the superconductingstate. If we neglected the spatial variation of λ we would erroneously attribute the entiretemperature dependence to the order parameter. D. General Mat´ern Correlations
To understand the impact of the smoothness of λ ( r ) on the measured penetration depth, λ eff , we now consider the general case of Mat´ern covariance. Recall that the parameter ν controls the smoothness of the penetration depth. With the correlation function defined byEq. 26, we evaluate Eq. 25: X = − R (0)2 ν − Γ( ν ) Z ∞ d z e − z/ Λ sinh (cid:18) zλ eff (cid:19) λ eff Λ l " l (cid:16) √ ν zl (cid:17) ν K ν (cid:16) √ ν zl (cid:17) + 4 νl (cid:0) Λ + Λ z + λ (cid:1) (cid:16) √ ν zl (cid:17) ν − K ν − (cid:16) √ ν zl (cid:17) + 16 ν λ Λ(Λ + z ) (cid:16) √ ν zl (cid:17) ν − K ν − (cid:16) √ ν zl (cid:17) + R (0)2 ν − Γ( ν ) Z ∞ d z e − z/ Λ cosh (cid:18) zλ eff (cid:19) νλ Λ l " l z (cid:16) √ ν zl (cid:17) ν − K ν − (cid:16) √ ν zl (cid:17) + 4 ν Λ(Λ + z ) z (cid:16) √ ν zl (cid:17) ν − K ν − (cid:16) √ ν zl (cid:17) . (29)These integrals can be evaluated using equation 6.621.3 in Gradshteyn and Ryzhik: Z ∞ x µ − e − αx K ν ( βx )d x = √ π (2 β ) ν ( α + β ) µ + ν Γ( µ + ν )Γ( µ − ν )Γ( µ + ) F (cid:18) µ + ν, ν + ; µ + ; α − βα + β (cid:19) , (30)which requires Re µ > | Re ν | and Re ( α + β ) >
0. The function F ( a, b ; c ; z ) is Gauss’hypergeometric function. Using the integral in Eq. 30 to evaluate Eq. 29, we find the10onstraints ν >
32 and λ eff Λ > ll + 2Λ √ ν . (31)The full solution for X is then: X = R (0) √ π Γ( ν ) (cid:18) νl (cid:19) ν λ eff Λ ( λ + Λ
2Λ Γ(2 ν − ν + ) h a − (2 ν − F (cid:3) ) − b − (2 ν − F ♣ ) i + Γ(2 ν )Γ( ν + ) (cid:20) λ eff + Λ2Λ a − ν F (cid:3) ) + λ eff − Λ2Λ b − ν F ♣ ) (cid:21) + 1Λ Γ(2 ν + 1)Γ( ν + ) h a − (2 ν +1) F (cid:3) ) − b − (2 ν +1) F ♣ ) i + λ Λ4 Γ(2 ν − ν − h a − (2 ν − F (cid:3) ) − b − (2 ν − F ♣ ) i + λ eff ν − ν + ) h(cid:0) λ eff + Λ (cid:1) a − (2 ν − F (cid:3) ) − (cid:0) λ eff − Λ (cid:1) b − (2 ν − F ♣ ) i + λ eff ν − ν + ) h a − (2 ν − F (cid:3) ) + b − (2 ν − F ♣ ) i) , (32)where we have introduced the variables a = 1Λ + 1 λ eff + 2 √ νl ,b = 1Λ − λ eff + 2 √ νl , (cid:3) = l ( λ eff + Λ) − λ eff Λ √ νl ( λ eff + Λ) + 2 λ eff Λ √ ν , ♣ = l (Λ − λ eff ) + 2 λ eff Λ √ νl (Λ − λ eff ) − λ eff Λ √ ν , and functions F · ) = F (2 ν − , ν − ; ν + ; · ) ,F · ) = F (2 ν, ν − ; ν + ; · ) ,F · ) = F (2 ν + 1 , ν + ; ν + ; · ) ,F · ) = F (2 ν − , ν − ; ν − ; · ) ,F · ) = F (2 ν − , ν − ; ν + ; · ) , and F · ) = F (2 ν − , ν − ; ν + ; · ) . Inserting this expression for X into Eq. 24, we can solve for λ eff after choosing a valuefor the smoothness parameter ν . In Fig. 4, we have chosen ν = 2, close to the lower bound11f required for convergence of X . The results are almost identical to the case of squaredexponential correlations (Fig. 2); evidently λ eff is not much affected by changes in thesmoothness of λ ( r ) for the Mat´ern family of correlation functions. The qualitative featuresof interest to us are still present: there are regions of enhanced screening and regions ofsuppressed screening, the effect grows on increasing the variance of λ ( r ), and changes in λ eff at the one percent level persist down to small disorder. Quantitatively, the results in Figures4 and 2 differ by five percent in the region near l = 1 and R (0) = 1, where the difference islargest. IV. DISCUSSION
The measured λ ( T ) in a non-uniform superconductor will be determined by both themomentum space gap structure and the real space variations of the penetration depth. Wecalculate the influence of spatial fluctuations in the penetration depth by solving the stochas-tic London equation in the limit of small fluctuations. This gives an equation (Eq. 24) forthe disorder-averaged magnetic field in terms of the penetration depth correlation function.We then solve this equation for two example correlation functions to find λ eff , the decaylength of the disorder-averaged field, which we identify as the penetration depth measuredexperimentally. We find that λ eff can be either smaller or larger than the average penetrationdepth, depending on the correlation length of λ . More importantly, the variance and correla-tion length of λ will likely change with temperature, endowing the experimentally measuredpenetration depth with temperature dependence that is unrelated to the superconductingorder parameter.This work shows that there can be a disorder-induced change of the penetration depththat is not caused by the structure of the superconducting gap in momentum space. Rather,it reflects the real space variations of the order parameter. An interpretation that assumeda spatially uniform penetration depth would infer a larger modulation of ∆( k ) than trulyexists. Because ∆( k ) is the starting point for investigations of the mechanism of the su-perconductor, this omission could lead us astray when we seek to determine the underlyingmechanism.How significant is the effect of disorder-induced change in the penetration depth, giventhat λ eff / h λ i approaches 1 over a large segment of the R (0)- l plane? Modern measurements12an routinely resolve sub-nanometer changes in the penetration depth; in cuprates andpnictides the penetration depth is approximately 200 nm, and a 1-nm change in λ yields ∆ λλ of 0.5% – making even small changes in λ eff / h λ i potentially significant.Two issues are worth emphasizing. First, we have made no assumption about the dis-tribution of λ ( r ), i.e., whether it is normally distributed or follows a different probabilitydistribution. However, the calculation presented here only extends to second order, andany non-normality only enters at third order and above. Second, λ eff has a complicated de-pendence on the correlation function R ( z ), and we know neither its functional form nor itstemperature dependence. Hence we cannot make any tidy prediction for the low-temperaturebehavior of λ ( T ); there is no power-law to be had.Even without perfect knowledge of R ( z ), it may be possible learn more about λ eff bytaking advantage of the general constraints that apply to all correlation functions. Inparticular, the strong similarities between the two cases presented here (Figs. 2 and 4) leadus to expect qualitatively similar behavior in λ eff for most possible correlation functions.To make a stronger statement about λ ( T ), we need to determine the local supercon-ducting properties of a given chemically doped and intrinsically disordered material, whichnaturally depends on the microscopic details of the superconducting mechanism. Althoughit should be possible to extract a local penetration depth or superfluid density from nu-merical methods such as solving the Bogoliubov-de Gennes equations on a lattice, to thebest of our knowledge this has never been attempted. Several groups have calculated the disorder-averaged superfluid stiffness using this approach, for both s-wave and d-wave models. The full temperature dependence of the disorder-averaged superfluid density canalso be calculated, but is incomplete, for we have shown that the real space inhomogeneityof the superconducting state also contributes to the temperature dependence.The larger message is that some measured properties of disordered superconductors willnot be determined by their disorder averages alone; inhomogeneities can affect the measuredproperties in an experiment-dependent manner. For example, the heat capacity will begiven by the disorder average because it is additive, but we have seen that the penetrationdepth is non-trivially affected by the disorder. Nonetheless, these two experiments are bothtraditionally interpreted as measuring the same thing – the magnitude of the single-particlegap, ∆( k ).These results give a specific example of the potential impact of spatial variation on mea-13urements of the penetration depth. With a full consideration of the impact of spatial vari-ation on different measured quantities, as well as a complete understanding of how randomchemical doping gives rise to a non-uniform superconducting state, we will be able to inte-grate a complete account of the effects of disorder into our understanding of unconventionalsuperconductivity. ACKNOWLEDGMENTS
We thank John Kirtley, Steve Kivelson, Jim Sethna, and J¨org Schmalian for helpfuldiscussions. We would also like to thank John Kirtley for checking some of these calculations.This work is supported by the Department of Energy, Office of Basic Energy Sciences,Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515 ∗ [email protected] W. N. Hardy, D. A. Bonn, D. C. Morgan, R. Liang, and K. Zhang,Physical Review Letters , 3999 (1993). R. Prozorov and R. W. Giannetta, Superconductor Science and Technology , R41 (2006). W. N. Hardy, S. Kamal, and D. A. Bonn, in
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1, controls the width the penetrationdepth distribution, and the correlation length establishes the characteristic length over which λ ( r )changes. (0) = σ λ / h λ i l h λ i . . . . . . . . λ eff h λ i FIG. 2. The effective penetration depth is a strong function of the parameters that characterize thedistribution of local penetration depths. Here we show the value of λ eff as the correlation length, l , and variance, R (0), run across three orders of magnitude. This figure considers the case ofsquared exponential correlations in the penetration depth; a different case is shown in Fig. 4. Themost important features of this color plot are the range of λ eff / h λ i and the appearance of valuesboth above and below 1. The calculation is valid when R (0) ≪
1, but we show the region with R (0) > . l or R (0) will contributeto λ ( T ). This temperature dependence is not accounted for by the superconducting gap. −4 −3 −2 −1 l/ h λ i R (0) = σ λ / h λ i λ eff h λ i FIG. 3. The screening can either be enhanced ( λ eff < h λ i ) or suppressed ( λ eff > h λ i ), dependingon the correlation length. The curves for l = 0 . h λ i and l = 0 . h λ i overlap. (0) = σ λ / h λ i l h λ i ν = 2 . . . . . . . . λ e ff h λ i FIG. 4. The effective penetration depth for Mat´ern correlations when ν = 2 (shown here) hasstrong similarities to Fig. 2, which represents the limiting case where ν → ∞ . These similaritiesimply that the smoothness of the random penetration depth does not strongly affect λ eff ..