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The e?ect of nonmagnetic impurities on the local density of states in s-wave superconductors

P. Miranovi? c, M. Ichioka, and K. Machida

Department of Physics, Okayama University, 700-8530 Okayama, Japan (Dated: February 2, 2008) We study the e?ect of nonmagnetic impurities on the local density of states (LDOS) in s-wave superconductors. The quasiclassical equations of superconductivity are solved selfconsistently to show how LDOS evolves with impurity concentration. The spatially averaged zero-energy LDOS is a linear function of magnetic induction in low ?elds, N (E = 0) = cB/Hc2 , for all impurity concentration. The constant of proportionality c depends weakly on the electron mean free path. We present numerical data for di?erential conductance and spatial pro?le of zero-energy LDOS which can help in estimating the mean free path through the LDOS measurement.

PACS numbers: 74.25.Jb, 74.25.Bt, 74.25.Op

arXiv:cond-mat/0312420v1 [cond-mat.supr-con] 17 Dec 2003

I.

INTRODUCTION

Since Hess and co-workers1 succeeded in measuring the LDOS in the superconducting NbSe2 there have been many reports and theoretical studies on the electronic structure of the superconductor in the mixed state. The novel experimental technique introduced, scanning tunneling spectroscopy, enables one to measure di?erential conductivity (DC) σ (r , V ) at various positions r and bias-voltages V . DC is closely related to LDOS of the superconductor (kB = 1), σ (r , V ) = σN

∞

dE 4T

?∞

N (r , E ) E + eV N0 cosh2 2T

.

(1)

where σN is DC in the normal state, e is electron charge, N (r , E ) is LDOS at the position r and energy E relative to the Fermi level, and N0 is DOS at the Fermi level in the normal state. Only in the limit of zero temperature T → 0 DC and LDOS are proportional: σ (r , V )/σN = N (r , |e|V )/N0 . At ?nite temperature, DC is actually thermally broadened LDOS. It is clear that at low temperatures DC should follow the spatial structure of LDOS. Two prominent features should be mentioned. DC measured at the vortex center revealed a peak at the Fermi level (zero-bias peak) that well exceeds σN . This indicates that vortex core can not be viewed as being “normal” at least in clean superconductors. The zero-bias peak in DC originates from the zero energy peak of LDOS at the vortex center, which is due to the low lying bound states inside the vortex core. The other remarkable feature revealed in Ref. 1 is a star-shaped DC around the vortex core measured at ?xed bias-voltage, with star orientation depending on the bias-voltage value. The six-fold structure of DC in NbSe2 is coming either from the e?ect of the hexagonal vortex lattice, anisotropic s-wave pairing or anisotropic Fermi surface, and most probably it is coming from each e?ect simultaneously. Again, the star-shaped DC originates from the star shaped LDOS in the vortex lattice.

However, the measured DC does not follow the sharp features of the corresponding, theoretically calculated, LDOS even if the experiment is performed at very low temperature.2 The height and width of the zero-bias peak was found to be sample dependent indicating impurities as a plausible explanation for the discrepancy. Indeed, impurities are inevitably present in superconducting samples on which the experiments are performed. Therefore, it is important to quantitatively study how LDOS is changing with impurity concentration. This is the purpose of our paper. There is another one topic that we analyze in this paper: the e?ect of impurities on the speci?c heat ?eld dependence. In s-wave superconductors low energy quasiparticles are trapped inside the vortex core. Therefore, zero-energy LDOS, spatially averaged, is proportional to the number of vortexes: N (E = 0) ∝ N0 ξ 2 B , with B being the magnetic induction and ξ the size of the vortex core. This translates into the linear ?eld dependence of low temperature speci?c heat given by Cs /T = 2π 2 N (E = 0)/3. The nonlinearity in low ?eld Cs (B ) curve should be related to the gap anisotropy. In the case of anisotropic s-wave pairing, addition of nonmagnetic impurities can smear out the gap anisotropy, which can be tracked by examining Cs (B ) curves. This kind of measurement has been performed on Nb1?x Tax Se2 3 and Y(Ni1?x Ptx )2 B2 C.3,4 The intention was to make the gap isotropic by adding impurities, but with the price to have rather dirty s-wave superconductor. The e?ect of impurities on LDOS notwithstanding it would be of value to study how ?eld dependence of spatially averaged LDOS evolves with impurity concentration in the most simple case of s-wave superconductors. So far, the only systematic experimental study of the e?ect of disorder on LDOS is by Renner et al.5 In particular they measured the zero-bias DC at the vortex center in the alloy system Nb1?x Tax Se2 . Substitution of Nb by Ta leads to systematic decrease of the electron mean free path. On the other hand small changes in the electronic spectrum is expected since Nb and Ta are isoelectronic and with similar atomic radii. Zero-energy DC is found to be very sensitive to the impurity concentration. It grad-

2 ually disappears and for x = 0.2 the zero-energy LDOS in the vortex center is the same as that of normal phase N0 . It was even proposed that DC spectra can serve as a measure of quasiparticle scattering time. In this paper the problem of LDOS in presence of nonmagnetic impurities will be studied within the quasiclassical equations of superconductivity. Quasiclassical approximation is adequate in superconductors where coher?1 ence length ξ is much larger than the atomic length kF . LDOS is studied within the quasiclassical approximation by Ullah et al.6 and Klein7 for the case of isolated vortex in the isotropic s-wave superconductor. The full selfconsistent analysis of LDOS in the case of vortex lattice is performed by Ichioka et al.8 All these studies assume clean superconductor without impurities. Dirtiness of the superconductor is only roughly estimated by Klein.7 As for the single vortex case, the e?ect of impurities was studied in Refs. 9,10. The e?ect of impurities on DOS in extremely high ?eld, where the Landau level quantization of the electronic energies should be taken into account, is studied by Dukan and Teˇ sanovi? c.11 Those phenomena are beyond the scope of this text. As for the study of the extreme case, dirty superconductor, the reader is refered to Refs. 12,13. The paper is organized as follows. In section II the method of solving Eilenberger equations is presented. The readers not interested in technical details may skip that section. In section III spatial and energy dependence of LDOS and DC for various impurity concentration is shown. In section IV the e?ect of impurities on the speci?c heat ?eld dependence is discussed.

[ω ? u (? ? iA)] f ? = Ψ? g + F ? g ? Gf ? .

(3)

These are supplemented by the self-consistency equations for the gap function Ψ and vector-potential A Ψ ln t = 2t

ω>0

f ?

Ψ , ω

(4)

?×?×A =?

2t Im ug , κ2 ω>0

(5)

as well as for the impurity potentials F = 1 f , τ G= 1 g . τ (6)

Born approximation is assumed in treating scattering on impurity. For convenience, equations are written in following dimensionless units: order parameter is measured in units πTc , length in units R0 = v/(2πTc ), v is 2 Fermi velocity, magnetic ?eld in units H0 = Φ0 /2πR0 , where Φ0 is ?ux quantum. Vector-potential is in units 3 A0 = Φ0 /2πR0 , energy in units E0 = (πTc )2 N0 R0 . Scattering time τ is in units 1/(2πTc ). It can be expressed via electron mean free path l: τ = l/0.882ξ0, with ξ0 being BCS coherence length. Eilenberger parameter κ ? is the only material constant that enters the equations κ?2 = 2πN0 π Φ0

2

v4 . (πTc )2

(7)

II.

METHOD OF SOLUTION

It is related to GL parameter κ via κ2 = (7ζ (3)/18)κ2 in 3D case and κ ?2 = 7ζ (3) 2 κ . 8 (8)

There are various methods to solve Eilenberger equations for the vortex lattice. The main problem in the numerical procedure is that the initial conditions for the di?erential equations are unknown. One way to overcome that problem is to use special, divergent, gauge in which Green’s functions are periodic, and solve equations in Fourier space (periodic boundary condition).14 The other method is based on the fact that during the integration process Green’s functions exponentially grow (explode). Fortunately, the exponentially growing unphysical solutions can be manipulated to form the physical one. This is the essence of the so-called “explosion method”.8,15,16 Here we will use a di?erent approach. It is interesting that if one parameterizes quasiclassical Eqs. in the form of Riccati’s di?erential equation, then during the numerical integration the physical solution is stabilized regardless of the initial condition. Here we will give more details. For the s-wave superconductor in presence of nonmagnetic impurities Eilenberger equations are [ω + u (? + iA)] f = Ψg + F g ? Gf, (2)

in 2D case. Here ζ is Riemmann’s zeta function. ω = t(2n +1) is Matsubara frequency with integer n, t = T /Tc is reduced temperature, u is unit vector directed along Fermi velocity. Eilenberger Green’s functions f , f ? and g are normalized so that g = 1 ? f f ? . Fermi surface is assumed to be isotropic and two-dimensional.17 Average over the isotropic cylindrical Fermi surface reduces to · · · = (1/2π ) · · · d?, average over polar angle ?. The quantity of our interest, LDOS as a function of position r and quasiparticle excitation energy E , is de?ned as N (r , E ) = N0 Re g (r , u, ω ?→ δ ? iE ) , (9)

where Eilenberger function g describes normal excitations and δ is a small number. It is very convenient to introduce auxiliary functions a and b through the following transformation18,19 f= 2a , 1 + ab f? = 2b , 1 + ab g= 1 ? ab . 1 + ab (10)

3 Equations for auxiliary functions a and b are decoupled and have the form of Riccati’s di?erential equation a2 Ψ+F ? (Ψ? + F ? ) , 2 2 (11)

? 2

A.

Iteration procedure

u?a = ? (ω + G + iuA) a +

Iterative procedure for solving Eq. (15) is the following. We start from some potentials Ψ(r ), A′ (r ), F and G. It is usual to start from the Abrikosov solution for Ψ(r ), A′ (r ) = 0, and local values of impurity potentials: F = 1 τ Ψ(r ) ω2 + |Ψ(r )|2 , G= 1 τ ω ω2 + |Ψ(r )|2 .

Ψ +F b u?b = (ω + G + iuA) b ? + (Ψ + F ) . (12) 2 2 Auxiliary functions a(r , u, ω ) and b(r , u, ω ) are not independent. Once we solve the Eq. (11), function b can be readily calculated: b(r , u, ω ) = ?a? (?r , u, ω ). (13)

?

In the coordinate system (ρ, η ), where Fermi velocity direction u coincides with ρ-axis, ρ = x cos φ + y sin φ, η = y cos φ ? x sin φ, Eq. (11) reduces to ?a Ψ′ a2 Ψ′? = ? (ω + G + iuA) a + ? , ?ρ 2 2 (15)

(14)

(18) After solving the Eq. (15) the new values of potentials are obtained from the self-consistency eqs. (4), (5), and (6), and the new potentials we plug again into the Eq. (15) and solve it. This iterative procedure is repeated until the selfconsistency is achieved. The maximum frequency ωcut = t(2Ncut +1) should be chosen so the result does not depend on the number of Matsubara frequencies. On the other hand the number of iteration cycles needed to stabilize pair potential increase with the Ncut . We followed Klein15 and choose ωcut = 20πTc (in physical units) as appropriate for various temperatures. This gives the number of Matsubara frequencies Ncut ≈ Int 10 t . (19)

Fortunately it is not necessary to solve Eq. (15) for all ω . For high frequencies the solution can be well approximated by: a≈ 1 2 1 uΠ (uΠ)2 ? + ω′ ω ′2 ω ′3 (Ψ + F ) , (20)

where Ψ′ = Ψ+ F . Integrating along the direction ρ from ρ′ ? ρ∞ to the desired point ρ′ , the physical solution a+ is stabilized. Note that integrating in the opposite direction, toward decreasing ρ, one will get solution a? = ?1/a+ . How long integration path ρ∞ should be taken depends on ω (is it real or complex) and on impurity concentration. Vector-potential is written as A(r) = B×r + A′ (r ), 2 (16)

where ω ′ = ω + 1/τ and Π = ? + iA. For all n > Ncut /2 we use the equation (20). Solution is quasi-periodic. Translation by Rnm = nr1 + mr2 will amount in phase factor exp(iχ(r , Rnm )) a(r + Rnm , v , ω ) = a(r , v , ω )eiχ(r,Rnm ) , (21) where r1 and r2 are primitive vectors of vortex lattice, n, m are integers, and χ=π mx y (n + m cos β ) ? + nm + n ? m . a0 a0 sin β (22)

where B is magnetic induction, and A′ is periodic with ? · A′ = 0. Therefore, the selfconsistent equation for vector-potential can be written as ?2 A′ = 2t Im ug . κ ?2 ω>0 (17)

It can be easily solved in the Fourier space. Auxiliary function a has the same symmetry properties as Eilenberger function f , which are described in Ref. 8. The equilibrium vortex lattice structure is assumed to be hexagonal. Therefore, it is su?cient to solve equation (15) in the whole vortex lattice cell and only for velocity directions 0 < ? < π/6. With the help of symmetry properties, a(r , u, ω ) can be obtained for all velocity directions.

The angle between primitive vectors is denoted as β (β = π/3 in our case). Once the selfconsistent potentials Ψ(r ) and A(r ) are calculated, the Eilenberger Eqs. are solved again but this time for ω = δ ? iE where δ is small number and E is quasiparticle energy. Note that in the presence of impurities the Eq. (15) has to be solved selfconsistently with respect to impurity potentials F and G. As for the choice of δ one should be very careful. It was already noted20,21 that density of states N (E = 0) is very sensitive to the absolute value of δ . Finite δ has roughly the e?ect of impurities and suppresses the peak in DOS at the vortex center. For small values of δ , N (E = 0) is spiked at the vortex centers and very ?ne mesh is needed to evaluate average LDOS. We ?nd that δ = 0.001 su?ce for our calculation.

4

III. LOCAL DENSITY OF STATES AND DIFFERENTIAL CONDUCTANCE

1

The physics of the vortex core in the clean limit is very di?erent from the physics of the vortex core in dirty superconductors. Properties of the vortex core are governed by Andreev bound states in the clean limit, while in the dirty limit properties of the core are governed by normal electrons.22 To understand the role of impurities we brie?y explain the formation of bound states. Andreev scattering from the pair potential (order parameter) inside the vortex core converse electron-like excitation into hole-like excitation and vice versa. States inside the core are coherent superposition of particle and hole states. At certain energies the coherent superposition of particle and hole states is constructive and the bound state is formed. The lowest bound state has the energy E = ?/kF ξ . In the quasiclassical limit kF ξ ? 1, the lowest bound state energy is pushed to zero. Zero-energy bound state inside the vortex core will manifest as a peak in zero-energy LDOS at the vortex center. Scattering on impurities will randomize the motion of electron, and the coherency is lost. Thus, the impurities will smear out the sharp structure of LDOS. To illustrate this we focus on the spatial structure of zero-energy LDOS N (r , E = 0). In Fig. 1 spatial variation of LDOS along the line connecting two nearest neighbor vortexes is shown. Data for a clean superconductor (ξ0 /? = 0.0), for a relatively large mean free path ξ0 /? = 0.1 and for impure case ξ0 /? = 4.0 are presented. Distance between vortexes is normalized so that 0 and 1 on the abscissa are position of vortexes. To remind the reader again, in the clean limit the height and width of the LDOS peak depend on the small parameter δ , which measures how far we are from the pole of the Green function g . In this sense height and width of the peak in the clean limit are arbitrary. At the vortex core zero energy DOS (ZEDOS) in the clean limit highly exceeds the normal state value N0 . This was in the beginning at odds with generally accepted naive picture of vortex core as being “normal”. Analyzing the ZEDOS in the impure case, it is clear that coherency is crucial in forming the main peak at the vortex. In the dirty limit ξ0 /? → ∞ ZEDOS within the vortex core approaches normal state value N0 , and only in this limit one can view vortex core as being “normal”. Even a small impurity concentration has a great impact on ZEDOS pro?le. The comparison of the ideal case of a clean superconductor ξ0 /? = 0 with rather pure superconductor ξ0 /? = 0.1 reveals a change of the vortex core size by a factor 2. The change of the vortex core size is compensated by the reduction of the peak height, so the ZEDOS averaged over vortex lattice cell is approximately the same in all cases. It is instructive to see how the spatial structure of ZEDOS within the vortex lattice evolves by adding impurities. In the clean limit ZEDOS around the single vortex is cylindrically symmetric. As soon as vortex lattice is formed cylindrically symmetric ZEDOS transforms into

N(E=0)/N0

0.5

ξ0/l=0.0 ξ0/l=0.1 ξ0/l=4.0

0 0

0.5

1

Nearest neighbour direction

FIG. 1: Spatial variation of zero energy DOS along the nearest neighbor vortex direction. Full line corresponds to the clean limit and dashed lines correspond to the superconductors with ξ0 /? = 0.1 and ξ0 /? = 4.0. The calculation is performed at approximately the same relative ?eld B = 0.1Hc2 .

FIG. 2: ZEDOS within the vortex lattice for superconductors with ξ0 /? = 0.0, ξ0 /? = 0.1 and ξ0 /? = 4.0 (in order from left to right). Only data points N (E = 0)/N0 < 1 are presented. Small parameter δ = 0.03 is used for clean limit data to clarify the spatial distribution.

the star-shaped structure within the hexagonal vortex lattice.8 This is presented in Fig. 2. The e?ect of vortex lattice notwithstanding, the other e?ects such as the anisotropy of the pairing function23 and the anisotropy of the Fermi surface in hexagonal crystal can also contribute to the speci?c star-shaped structure of ZEDOS. By reducing the mean free path, star-shaped structure gradually disappears and is completely absent in the dirty limit even at relatively high ?elds. This indicates that periodicity of the order parameter is not the key element to explain the structure of N (r , 0) in Fig. 2. Only coherent superposition of electron and hole states in the periodic vortex lattice can account for the star-shaped ZEDOS. In Fig. 3a) LDOS at the vortex center is plotted as a function of quasiparticle excitation energy E (in units πTc ) for the clean case. LDOS oscillates with energy, the result previously reported in

5

2 2

N(E,r=0)

l/ξ0=10

l/ξ0=2

1

1 0 2

0

N(E,r=0)

l/ξ0=1

l/ξ0=0.5

?3

?1

1

3 ?3

?1

1

3

1

E

E

FIG. 3: a) LDOS N (E, r = 0)/N0 at the vortex center as a function of excitation energy E (in units πTc ). b) DC σ (E )/σN at the vortex center at T = 0.1Tc . Both data are for superconductor in the clean limit.

0

?3

?1

E

1

3 ?3

?1

E

1

3

Ref. 24. This phenomenon has the same origin as oscillation of DOS in superconducting-normal proximity systems:25,26,27 interference of quasiparticles re?ected at the superconducting-normal barrier. The mixed state can be viewed as periodically arranged in?nite number of “normal”-superconducting boundaries. Here the vortex cores play the role of normal region in the sense that gap drops to zero at the vortex axes. In Fig. 3b) DC at T = 0.1Tc, calculated according to Eq. (1), is presented. At this temperature DC is thermally broadened LDOS, but the oscillating pattern is still visible. The coherency of quasiparticles is essential both for zero-bias peak and oscillation of LDOS with energy at the vortex axis. In Fig. 4 LDOS at the vortex center as a function of energy is plotted for various values of impurity concentration. Oscillation amplitude is very sensitive to the presence of impurities and is almost lost even in very clean samples with ξ0 /? = 0.1. Proliferating impurity concentration will manifest as a ?attening of LDOS at the vortex center: disappearance of zero-energy peak of LDOS, as well as disappearance of deep minima for E < Ψ(B = 0). In the dirty limit ξ0 /? → ∞, LDOS at the vortex center is equal to N0 for all quasiparticle energies N (r = 0, E ) = N0 .12,13

FIG. 4: LDOS at the vortex center as a function of energy plotted for various values of mean free path ?. Normalized DC σ/σN is almost indistinguishable from the normalized LDOS N (E, r = 0)/N0 at this temperature T = 0.1Tc .

E E tanh 2T 2T

∞

+2

?∞

E 2 N (E )dE . E 4T 3 cosh2 2T

(23)

One can utilize this expression only if the energy dependent, spatially averaged, LDOS N (E ) is provided. However, in the limit T → 0 the ?rst integral is zero. For small T the function to be integrated in the second integral is nonzero only in the small vicinity of E = 0. Therefore we can replace N (E ) by N (E = 0) E 2 N (E = 0)dE 2π 2 N (E = 0) . = 3 E 4T 3 2 cosh ?∞ 2T (24) In the normal phase Cn /T = 2π 2 N0 /3 which gives us the well known result Cs =2 lim T →0 T lim N (E = 0) Cs = . Cn N0 (25)

∞

T →0

IV.

LOCAL DENSITY OF STATES AND SPECIFIC HEAT

The low energy quasiparticle excitations play the important role in the low temperature thermodynamics. Speci?c heat Cs (T ) of a superconductor is given by

∞

Cs =2 T

dE

?N (E ) ?T

ln 2 cosh

E 2T

?

?∞

If the low energy quasiparticles are localized in the vortex cores, which is true for s-wave superconductors at least in the limit of very small ?elds, then N (E = 0) ? ρ2 /Scell. Here Scell = Φ0 /B is vortex lattice cell area and ρ is the size of the vortex core. If we further assume that ρ2 ? Φ0 /Hc2 then we arrive at the following scaling relationship N (E = 0) ? B/Hc2 , for s-wave superconductors. However there is a number of reports of nonlinear ?eld dependence of γs (H ) in s-wave superconductors. One of the o?ered explanations is that vortex core size ρ itself is ?eld dependent which in turn lead

6

1

1

<N(E=0,B)>/N0

<N(E=0)>/N0

2 0.5

0.5

ξ0/l=0.1 ξ0/l=0.5 ξ0/l=1.0 ξ0/l=2.0 ξ0/l=4.0 ξ0/l=6.0

ξ(B)/ξ(0)

1.6 1.2 0.8 0

B/Hc2

1 1

0 0

0 0

0.5

B/Hc2

0.5

1

B/Hc2

FIG. 5: Field dependence of spatially averaged zero-energy LDOS in the clean limit. Straight line is guide for eye. In the inset is shown normalized core size ξ (B )/ξ (0) as a function of B/Hc2 . FIG. 6: Field dependence of spatially averaged zero-energy LDOS for various mean free path.

to the nonlinear ?eld dependence of zero-energy DOS. The shrinking of the vortex core with increasing ?eld is detected in NbSe2 28 and YBa2 Cu3 O6.60 .29 This is further supported by numerical calculations in dirty13,28 and clean8 limit. Such an explanation brings out another puzzle. Experimental study on in?uence of nonmagnetic impurities on the γ (H ) in Y( Ni1?x Pt x )2 B2 C and Nb1?x Tax Se2 revealed that linear γ (H ) is achieved only in dirty samples.3 This result suggest that the vortex core size in the dirty superconductors is ?eld independent. Numerical calculation by Golubov and Hartman,13 as well as Sonnier et al.28 shows quite contrary, that even in the dirty limit ρ should shrink with increasing ?eld. Here we emphasize the necessity to perform the calculation at low temperature in order to analyze the speci?c heat data through the ZEDOS. In Ref. 8 calculation performed at T = 0.5Tc revealed that N (E = 0) ? ξ 2 (B )B , where ξ (B ) is independently calculated vortex core radius. At lower temperatures, due to the Kramer-Pesch e?ect, the core radius is smaller and it might have di?erent ?eld dependence. The result for the ?eld dependence of ZEDOS in the clean limit, for T = 0.1Tc , is shown in Fig 5. In the inset we plot the ?eld dependence of the core radius at the same temperature. We de?ne the core radius ξ as 1/ξ = (? |Ψ(r)|/?r)/|ΨN N | where |ΨN N | is the maximum value of the order parameter along the nearest neighbor direction, and derivative is taken along the same direction. Compared to the previously reported result at higher temperature T = 0.5Tc, where ξ (B ) decreases with ?eld,8 at T = 0.1Tc , vortex core radius is rather constant at low ?elds. As a consequence, zero-energy LDOS is also linear function of magnetic induction. In the clean limit ZEDOS in between vortexes is negligible in ?elds as large as B = 0.4Hc2 . In other words,

the main contributions to ZEDOS is coming from the vortex cores. On the other hand, in the dirty limit, ZEDOS is not con?ned to the vortex cores, but it is spread throughout the vortex lattice cell. It is large even in between vortexes. Thus, the scaling relation N (E = 0) ? ξ 2 (B )B is of no use in the dirty limit. This is the reason why we do not attempt to correlate vortex core size ξ (B ) and ?eld dependence of LDOS in the impure case. However, N (E = 0, B ) is a linear function of magnetic induction at low ?elds for any impurity concentration: N (E = 0, B )/N0 = c(τ )B/Hc2 . Constant of proportionally c(τ ) weakly depends on the electron mean free path and saturates to c ≈ 0.8 in the dirty limit. Numerical calculation of N (E = 0, B )/N0 as a function of mean-free path value is presented in Fig. 6. We note the concave curves for dirtier cases. This behaviors coincide with the analysis near Hc2 by Kita.30 In Fig. 7 is shown the ?eld dependence of the core radius as calculated from the pair potential pro?le Ψ(r ). For a ?xed relative ?eld B/Hc2 core radius ξ is a nomonotonic function of mean free path, ?rst sharply increases and then slowly decreases with increasing of ratio ξ0 /?. In the dirty limit vortex core shrinks with increasing ?eld, which is consistent with the previous calculations,13,28 in sharp contrast with vortex core enlargement with increasing ?eld in the clean limit. The experimental data, however, revealed that constant c = 1 in the dirty limit.3 It also shows that scaling N (E = 0, B )/N0 = c(τ )B/Hc2 for all ?eld values, which is a remarkable feature that still lacks the explanation. Worth is mentioning that in Ref. 4 speci?c heat is a nonlinear function of ?eld in samples Y( Ni1?x Pt x )2 B2 C for all 0 < x < 1. In these materials, we need to consider also the e?ect of gap anisotropy.

7

ξ0/l=0.0 ξ0/l=0.1 ξ0/l=0.5 ξ0/l=1.0 ξ0/l=2.0 ξ0/l=4.0 ξ0/l=6.0

1.2

1

0.8

0.6

that coherency is crucial in forming the spatial structure of LDOS. As soon as impurities are introduced into the superconductor, scattered electrons loose the information on their initial state, the coherency is lost and sharp LDOS structure is ?attened. It is calculated how DC spectra evolve with electron mean-free path. Although the impurities have a great impact on LDOS, spatially averaged LDOS shows weak dependence on relative ?eld B/Hc2 . We hope that present calculation can be helpful to roughly estimate the electron mean free path through the LDOS measurement.

ξ [in units 0.882ξ0]

0.4 0

0.2

0.4

0.6

0.8

1

B/Hc2

FIG. 7: Field dependence of vortex core size for various mean free path. Acknowledgments V. SUMMARY

In this paper we examined the e?ect of impurities on LDOS in isotropic s-wave superconductors. We showed

We acknowledge the useful communication with T. Dahm at the initial stage of this work.

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