# Nonlinear Meissner effect in unconventional superconductors.

**ABSTRACT** We examine the long-wavelength current response in anisotropic superconductors and show how the field-dependence of the Meissner penetration length can be used to detect the structure of the order parameter. Nodes in the excitation gap lead to a nonlinear current-velocity constitutive equation at low temperatures which is distinct for each symmetry class of the order parameter. The effective Meissner penetration length is linear in $H$ and exhibits a characteristic anisotropy for fields in the $ab$-plane that is determined by the positions of the nodes in momentum space. The nonlinear current-velocity relation also leads to an intrinsic magnetic torque for in-plane fields that are not parallel to a nodal or antinodal direction. The torque scales as $H^3$ for $T\rightarrow 0$ and has a characteristic angular dependence. We analyze the effects of thermal excitations, impurity scattering and geometry on the current response of a $d_{x^2-y^2}$ superconductor, and discuss our results in light of recent measurements of the low-temperature penetration length and in-plane magnetization of single-crystals of $YBa_2Cu_3O_{7-\delta}$ and $LuBa_2Cu_3O_{7-\delta}$. Comment: 30 pages, RevTeX file with 16 postscript figures. Submitted to Phys. Rev. B

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**ABSTRACT:**Recent studies of the electromagnetic response of high-Tc superconductors using terahertz, infrared, and optical spectroscopies are reviewed. In combination these experimental techniques provide a comprehensive picture of the low-energy excitations and charge dynamics in this class of materials. These results are discussed with an emphasis on conceptual issues, including evolution of the electronic spectral weight in doped Mott-Hubbard insulators, the d-wave superconducting energy gap and the normal-state pseudogap, anisotropic superfluid response, electronic phase segregation, emergence of coherent electronic state as a function of both temperature and doping, the vortex state, and the energetics of the superconducting transition. Because the theoretical understanding of these issues is still evolving the review is focused on the analysis of the universal trends that are emerging out of a large body of work carried on by many research teams. Where possible data generated by infrared/optical techniques are compared with the data from other spectroscopic and transport methods.Review of Modern Physics 08/2005; 77(2). · 44.98 Impact Factor - R. T. Gordon, N. Ni, C. Martin, M. A. Tanatar, M. D. Vannette, H. Kim, G. Samolyuk, J. Schmalian, S. Nandi, A. Kreyssig, A. I. Goldman, J. Q. Yan, S. L. Bud'ko, P. C. Canfield, R. Prozorov10/2008;
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**ABSTRACT:**Reversible magnetic-torque measurements in untwinned, single-crystal YBa2Cu3O7 show both twofold and fourfold symmetry in the ab plane. Different c-axis anisotropy parameters are found from the London model: γca=mc/ma=7.55+/-0.63 and γcb=mc/mb=8.95+/-0.76 at 90 K in 0.8 T. The ab-anisotropy parameter is γab=ma/mb=1.18+/-0.14. The free energy of the fourfold symmetry has a minimum, and neointrinsic pinning is found in the in-plane irreversible magnetic torque, when the field is applied parallel to the a or b axis. These results are consistent with dx2-y2 symmetry of the bulk order parameter.Physical Review B 11/1997; 56(18):71831-11902. · 3.66 Impact Factor

Page 1

arXiv:cond-mat/9502110v1 27 Feb 1995

The Nonlinear Meissner Effect in Unconventional Superconductors

D. Xu, S. K. Yip and J.A. Sauls

Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208 USA

(draft: July 17, 1994; revised: February 12, 1995)

Abstract

We examine the long-wavelength current response in anisotropic superconductors and show how

the field-dependence of the Meissner penetration length can be used to detect the structure of

the order parameter. Nodes in the excitation gap lead to a nonlinear current-velocity constitutive

equation at low temperatures which is distinct for each symmetry class of the order parameter.

The effective Meissner penetration length is linear in H and exhibits a characteristic anisotropy for

fields in the ab-plane that is determined by the positions of the nodes in momentum space. The

nonlinear current-velocity relation also leads to an intrinsic magnetic torque for in-plane fields that

are not parallel to a nodal or antinodal direction. The torque scales as H3for T → 0 and has a

characteristic angular dependence. We analyze the effects of thermal excitations, impurity scattering

and geometry on the current response of a dx2−y2 superconductor, and discuss our results in light

of recent measurements of the low-temperature penetration length and in-plane magnetization of

single-crystals of Y Ba2Cu3O7−δ and LuBa2Cu3O7−δ.

I. INTRODUCTION

Recent measurements of the Meissner penetration depth [1] and Josephson interference effects in Y Ba2Cu3O7−δ

[2] have been interpreted in support of a spin-singlet order parameter belonging to the one-dimensional, dx2−y2

representation, ∆(? pf) = ∆0(ˆ p2

been proposed by several authors [3–5] based on arguments that the CuO materials are Fermi liquids close to an SDW

instability.

If the cuprates have an order parameter that is unconventional, i.e.

the normal state besides gauge symmetry, then the superconducting state is expected to exhibit a number of novel

properties, including (i) gapless excitations below Tc, (ii) anomalous Josephson effects, (iii) exotic vortex structures

and associated excitations, (iv) new collective modes, (v) sensitivity of superconducting coherence effects to defect

scattering and (vi) multiple superconducting phases. [6,7] Many of these signatures of unconventional pairing have been

observed in superfluid3He, and in heavy fermion superconductors, notably UPt3. [8] The case for an unconventional

order parameter in the cuprates, and particularly a dx2−y2 state, is not settled; there are conflicting interpretations of

closely related experiments, [2,9,10] variation in results that are presumably related to material quality or preparation,

[11,12] and experimental results that are not easily accounted for within the dx2−y2 model. [13,14]

In this paper we examine the long-wavelength current response in superconductors with an unconventional order

parameter, and show how the field-dependence of the Meissner penetration length can be used to detect the structure

of the order parameter. This report extends our earlier work on nonlinear supercurrents, [15,16] and provides the

relevant analysis that could not be included in our short reports. Specifically, we show (i) how the nodes in the

excitation gap, whose multiplicity and position in momentum space depend on the symmetry class of the order

parameter, lead to a nonlinear current-velocity constitutive equation at low temperatures (T ≪ Tc) which is unique

and qualitatively distinct for each symmetry class. The effective Meissner penetration length is linear in H and

exhibits a characteristic anisotropy for fields in the ab-plane. (ii) This anisotropy is determined by the positions of

the nodes in momentum space. For example, in the case of a dx2−y2 state in a tetragonal material the anisotropy is

precisely 1/√2, independent of the detailed shape of the Fermi surface or the gap. (iii) The nonlinear current-velocity

relation leads to an intrinsic magnetic torque for in-plane fields that are not parallel to a nodal or antinodal direction.

The torque scales as H3for T ≪ Tcand has a characteristic angular variation with a period of π/2 (for tetragonal

symmetry). The magnitude and angular dependence of this torque are calculated for thick superconducting films or

slabs. (iv) We discuss the effects of thermal excitations, impurity scattering and geometry for observing these features

in a dx2−y2 superconductor. Recent measurements of the low-temperature, zero-field penetration length [1] are used

to determine the relevant material parameters for Y Ba2Cu3O7−δ, which are then used to estimate the magnitudes of

the field-dependence of the penetration depth and the torque anisotropy at low temperatures.

x− ˆ p2

y), which breaks reflection symmetry in the basal plane. Such a pairing state has

one that breaks additional symmetries of

1

Page 2

Our starting point is Fermi-liquid theory applied to anisotropic superconductors; section II includes the relevant

theoretical framework needed to calculate the current response in unconventional superconductors. We derive formulas

relating the equilibrium supercurrent to the magnetic field and discuss the linear response limit in section III. The

nonlinear current-velocity constitutive equation is examined in section IV. A clean superconductor with a line of nodes

in the gap has an anomalous contribution to the current which is a nonanalytic function of the condensate velocity,

? vs, at T = 0. The relation of the anomalous current to the quasiparticle spectrum is discussed, and the contribution

of this current to the Meissner penetration depth is obtained from solutions to the nonlinear London equation.

The effects of impurity scattering and thermally excited quasiparticles on the anisotropy and field-dependence of the

supercurrent are examined in detail; the signatures of the anomalous current survive thermal excitations and impurity

scattering at sufficiently low temperatures and weak (or dilute) impurity scattering. We discuss our results in light of

recent experiments on the low-temperature penetration depth [1] in single crystals of Y Ba2Cu3O6.95. An important

conclusion is that if the linear temperature dependence of the penetration depth reported for Y Ba2Cu3O6.95is due

to the nodes of a dx2−y2 order parameter, then the nonlinear Meissner effect, including the intrinsic anisotropy and

field-dependence, should be observable for T < 1K with a change in λabof approximately 30˚ A over the field range

0 < H < Hc1 ≃ 200G. In section V we discuss the nonlinear current, and associated in-plane magnetic torque,

that develops for surface fields that are not aligned along a nodal or antinodal direction. The torque anisotropy

(or transverse magnetization) is obtained from solutions to the nonlinear London equation at low temperatures.

We also comment on a recent experimental report of a measurement of the in-plane magnetization [17] of a single

crystal of LuBa2Cu3O7−δ. In the rest of the introduction we briefly discuss the symmetry classes and unconventional

order parameters for superconductors with tetragonal symmetry appropriate to the CuO superconductors (see Refs.(

[18,7,19]) for detailed discussions.)

Symmetries of the pairing state

BCS superconductivity is based on a macroscopically occupied equal-time pairing amplitude fαβ(? pf)

?a? pfαa−? pfβ

inversion symmetry (if present) implies that the pairing amplitude decomposes into even-parity (spin-singlet) and

odd-parity (spin-triplet) sectors. Furthermore, the pairing interaction separates into a sum over invariant bilinear

products of basis functions for each irreducible representation of the point group. The resulting ground-state order

parameter, barring the exceptional case of near degeneracy in two different channels, belongs to a single irreducible

representation. For tetragonal symmetry there are four one-dimensional (1D) representations and one two-dimensional

(2D) representation, and each of them occurs in both even- and odd-parity representations.1The residual symmetry

of the order parameter is just that of the basis functions for the 1D representations, but for the 2D representation

there are three possible ground states with different residual symmetry groups. There is no evidence that we are

aware of to support a spin-triplet order parameter in the CuO superconductors; in fact the temperature dependence

of the Knight shift in the cuprates [20,21] is argued to strongly favor a spin-singlet order parameter. [22] Thus, we

limit the discussion to even-parity, spin-singlet states; however, most of the analysis and many of the main results for

the current response are also valid for odd-parity states.

∼

?, for quasiparticle pairs near the Fermi surface with zero total momentum and spin projections α and β.

Fermi statistics requires that the order parameter obey the anti-symmetry condition, fαβ(? pf) = −fβα(−? pf), while

TABLE I. Even Parity Basis functions and Symmetry Classes for D4h

Symmetry Class

A1g

A2g

B1g

B2g

Eg(1,0)

Eg(1,1)

Eg(1,i)

Order Parameter: ∆(? pf)

1

ˆ pxˆ py(ˆ p2

ˆ p2

y

ˆ pxˆ py

ˆ pzˆ px

ˆ pz(ˆ px+ ˆ py)

ˆ pz(ˆ px+ iˆ py)

Residual Symmetry

D4h× T

D4[C4] × Ci× T

D4[D2] × Ci× T

D4[D′

D2[C′

D2[C′′

D4[E] × Ci

Nodes

none

8 lines: |ˆ px| = ±|ˆ py|, ˆ px = 0, ˆ py = 0

4 lines: |ˆ px| = ±|ˆ py|

4 lines: ˆ px = 0, ˆ py = 0

3 lines: ˆ pz = 0, ˆ px = 0

3 lines: ˆ pz = 0, ˆ px+ ˆ py = 0

1 line: ˆ pz = 0

x− ˆ p2

y)

x− ˆ p2

2] × Ci× T

2] × Ci× T

2] × Ci× T

1The principal results and conclusions presented here are not qualitatively modified by a-b anisotropy; the quanitative effects

of a-b anisotropy will be discussed elsewhere.

2

Page 3

Table I summarizes the symmetry classes of the order parameter for spin-singlet pairing. All of the 1D repre-

sentations have residual symmetry groups which include four-fold rotations combined with appropriate elements of

the gauge groups. The states Eg(1,0) and Eg(1,1) have a residual symmetry group that allows only two-fold rota-

tions. The resulting supercurrent, or superfluid density tensor, for such states is in general strongly anisotropic in the

basal plane. The 2D order parameter, Eg(1,i), preserves the four-fold rotational symmetry, but breaks time-reversal

symmetry.

Although the B1g(dx2−y2) and B2g(dxy) order parameters break the C4rotational symmetry of the CuO planes, a

combined C4rotation and gauge transformation by eiπis a symmetry. Since many properties of the superconducting

state depend only on Fermi-surface averages of |∆(? pf)|2, the broken rotational symmetry is not easy to observe. In

particular, the London penetration depth tensor is cylindrically symmetric for any of the 1D pairing states listed in

Table I. Furthermore, all of the unconventional gaps in Table I yield a linear temperature dependence at T ≪ Tcfor

the zero-field penetration depth (in the clean limit).

A distinguishing feature of each phase, which is a consequence of their particular broken symmetries, is that the

nodes of each gap are located in different positions in ? p-space. A point that we make below is that the field-dependence

of the supercurrent may be used to locate the positions of the nodal lines (or points) of an unconventional gap in

momentum space. This gap spectroscopy is possible at low temperatures, T ≪ Tc, and is based on features which are

intrinsic to nearly all unconventional BCS states in tetragonal or orthorhombic structures.

II. FERMI-LIQUID THEORY OF SUPERCONDUCTIVITY

Our starting point for calculations of the current response is the Fermi-liquid theory of superconductivity. This

theory is general enough to include real materials effects of Fermi-surface anisotropy, impurity scattering and inelastic

scattering from phonons and quasiparticles, in addition to unconventional pairing. A basic feature of the Fermi-liquid

theory of superconductivity (c.f. Refs.( [23–25]) for a more detailed discussion of the formulation of Fermi-liquid

theory.) is that for low excitation energies (¯ hω,kBT,¯ hqvf,∆) ≪ Ef, the wave nature of the quasiparticle excitations

is unimportant and can be eliminated by integrating the full Matsubara Green’s function over the quasiparticle

momentum (or kinetic energy) in the low-energy band around the Fermi surface,

gαβ(? pf,?R;ǫn) = −

?ωc

−ωc

dξ? p

?β

0

dτ eiǫnτ/¯ h

?

d? re−i? p·? r/¯ h< Tτψα(?R +? r/2,τ)ψ†

β(?R −? r/2,0) >, (1)

where ξ? p = vf(? pf)(|? p| − |? pf|) is the normal-state quasiparticle excitation energy for momentum ? p nearest to the

position ? pf on the Fermi surface and ? vf(? pf) is the quasiparticle velocity at the point ? pf. The resulting quasiclassical

propogator is a function of the momentum direction ? pfon the Fermi surface, the center of mass coordinate?R and the

Matsubara energy ǫn= (2n + 1)πT. The pairing correlations are described by the ξ-integrated anomalous Green’s

functions,

fαβ(? pf,?R;ǫn) = −

?ωc

−ωc

dξ? p

?β

0

dτ eiǫnτ/¯ h

?

d? re−i? p·? r/¯ h< Tτψα(?R +? r/2,τ)ψβ(?R −? r/2,0) > .(2)

The low-energy quasiparticle spectrum, combined with charge conservation and gauge invariance, allows one to formu-

late observables in terms of the quasiclassical Green’s function and material parameters defined on the Fermi surface.

For example, the equilibrium current is given by

?js(?R) = −eNf

?

d? pf? vf(? pf)T

?

n

1

2Tr

?

ˆ τ3ˆ g(? pf,?R;ǫn)

?

, (3)

where Nf is the single-spin density of states at the Fermi level and the integration is over the Fermi surface with a

weight factor of the angle-resolved density of states normalized to unity. We have introduced the 4 × 4 quasiclassical

functions in ‘spin × particle-hole’ space; a convenient representation for the particle-hole and spin structure of the

propagator is,

ˆ g =

?

g(? pf,?R;ǫn) +? g(? pf,?R;ǫn) · ? σ

f(−? pf,?R;ǫn)∗iσy−?f(−? pf,?R;ǫn)∗· iσy? σ

This matrix structure represents the remaining quantum mechanical degrees of freedom; the coherence of particle

and hole states is contained in the off-diagonal elements in eq.(4). The diagonal components are separated into spin-

scalar, g, and spin-vector, ? g, components. The scalar component determines the current response, while the vector

f(? pf,?R;ǫn)iσy+?f(? pf,?R;ǫn) · i? σσy

g(−? pf,?R;−ǫn) −? g(−? pf,?R;−ǫn) · σy? σσy

?

.(4)

3

Page 4

components determine the spin-paramagnetic response. The off-diagonal propagator separates into spin-singlet, f,

and spin-triplet,?f, pairing amplitudes, which are coupled to the diagonal propagators through the quasiclassical

transport equation,

ˆQ[ˆ g, ˆ σ] ≡

?

iǫnˆ τ3− ˆ σ(? pf,?R;ǫn), ˆ g(? pf,?R;ǫn)

?

+ i? vf·?∇ˆ g(? pf,?R;ǫn) = 0,(5)

first derived by Eilenberger [26] by eliminating the high-energy, short-distance structure of the full Green’s function

in Gorkov’s equations. [27] The transport equation is supplemented by the normalization condition,

ˆ g(? pf,?R;ǫn)2= −π2ˆ1, (6)

which eliminates many unphysical solutions from the general set of solutions to the transport equation. [26]

The self energy, ˆ σ, has an expansion (Fig. 1) in terms of ˆ g (solid lines) and renormalized vertices describing the

interactions between quasiparticles, phonons (wiggly lines), impurities and external fields. An essential feature of Fermi

liquid theory is that this expansion is based on a set of small expansion parameters, small ∼ kBTc/Ef,¯ h/pfξ0, ... ≪ 1,

which are the relevant low-energy (e.g. pairing energy) or long-wavelength (e.g. coherence length) scales compared

to the characteristic high-energy (e.g.

Fermi energy) or short-wavelength (e.g.

leading order contributions to the self energy are represented in Fig.1. Diagram (1a) is the zeroth-order in small and

represents the band-structure potential of the quasiparticles. This term is included as Fermi-surface data for ? pf, ? vf

and Nf, which is taken from experiment or defined by a model for the band-structure. Diagram (1b) is first-order in

small and represents Landau’s Fermi-liquid interactions (diagonal in particle-hole space), and the electronic pairing

interactions (off-diagonal in particle-hole space), or mean-field pairing self-energy (also the ‘order parameter’ or ‘gap

function’). Diagram (1b’) represents the leading-order phonon contribution to the electronic self energy (diagonal)

and pairing self-energy (off-diagonal); however, we confine our discussion to electronically driven superconductivity

with a frequency-independent interaction.

Fermi wavelength) scales. [23] The

b

a

small0

small1

b’

d1d2d3

c

...

Fig. 1 Leading order contributions to the quasiclassical self-energy.

In the spin-singlet channel, the order parameter satisfies the gap equation,

∆(? pf,?R) =

?

d? p′

f V (? pf,? p′

f) T

?

ǫn

f(? p′

f,?R;ǫn),(7)

where f(? pf,?R;ǫn) is the spin-singlet pairing amplitude and V (? pf,? p′

this function may be expanded in basis functions for the irreducible representations of the point group,

f) represents the electronic pairing interaction;

V (? pf,? p′

f) =

irrep

?

α

Vα

dα

?

i=1

Yαi(? pf)Yαi(? p′

f),(8)

4

Page 5

where the parameter, Vα, is the pairing interaction in the channel labeled by the αth irreducible representation, and

the corresponding basis functions, {Yαi(? pf)|i = 1,...,dα}, are orthonormal,

Fermi surface average is defined by < A(? pf) >? pf=?d? pf A(? pf).

?

Yαi(? pf)Y∗

βj(? pf)

?

? pf

= δαβδij, where the

Impurity scattering

The summation of diagrams (1d) gives the leading-order self energy from a random distribution of impurities in

terms of the impurity t-matrix, [28,29]

ˆ σimp(? pf;ǫn) = nimpˆt(? pf,? pf;ǫn), (9)

ˆt(? pf,? p′

f;ǫn) = ˆ u(? pf,? p′

f) + Nf

?

d? p′′

fˆ u(? pf,? p′′

f) ˆ g(? p′′

f;ǫn)ˆt(? p′′

f,? p′

f;ǫn).(10)

The first term is the matrix element of the impurity potential between quasiparticles at points ? pf and ? p′

Fermi surface, nimp is the impurity concentration, and the intermediate states are defined by the self-consistently

determined quasiclassical propagator.

For a spin-singlet superconductor with non-magnetic impurities, ˆ u(? pf,? p′

that contribute in the transport equation lead to a renormalization of the Matsubara frequency and gap function;

i˜ ǫn = iǫn− σimp(? pf;ǫn) and˜∆(? pf;ǫn) = ∆(? pf) + ∆imp(? pf;ǫn). Thus, the solution to the transport equation and

normalization condition for the propagator becomes,

f on the

f) = u(? pf,? p′

f)ˆ1, and the terms in ˆ σimp

ˆ g(? pf;ǫn) = −π

i˜ ǫn(? pf;ǫn)ˆ τ3−ˆ˜∆(? pf;ǫn)

?

˜ ǫn(? pf;ǫn)2+ |˜∆(? pf;ǫn)|2

.(11)

In the second-order Born approximation for the impurity t-matrix (this is not essential, but simplifies the following

discussion). The impurity renormalization of the off-diagonal self-energy is given by,

˜∆(? pf;ǫn) = ∆(? pf) +

?

d? p′

fw(? pf,? p′

f)

˜∆(? p′

f;ǫn)

?

˜ ǫ2

n+ |˜∆(? p′f;ǫn)|2

,(12)

where w(? pf,? p′

equation for˜∆(? pf;ǫn) has the mean-field order parameter, ∆(? pf), as the driving term. The scattering rate w(? pf,? p′

has the full symmetry of the normal metal; thus, it too can be expanded in basis functions for the irreducible

representations of the point group,

f) = 2πnimpNf|u(? pf,? p′

f)|2is the scattering rate in the Born approximation. Note that the integral

f)

w(? pf,? p′

f) =

irrep

?

α

1

2τα

dα

?

i=1

Yαi(? pf)Y∗

αi(? p′

f),(13)

where 1/2ταis the scattering rate for channel α. The integral equation for the renormalized order parameter separates

into algebraic equations for each representation,

˜∆αi=

?

d? pfY∗

αi(? pf)∆(? pf) +

1

2τα

?

d? pf

Y∗

?

αi(? pf)˜∆(? pf;ǫn)

˜ ǫ2

n+ |˜∆(? pf;ǫn)|2

.(14)

The driving term is non-zero only for the irreducible representation corresponding to ∆(? pf). Thus, the resulting

solution for the impurity renormalized order parameter necessarily has the same orbital symmetry as the mean-field

order parameter; and the magnitude of the impurity renormalization is determined by the scattering probability for

scattering in the same channel as that of ∆(? pf). [30] The argument also holds for the full t-matrix.

For isotropic (‘s-wave’) impurity scattering the renormalized Matsubara frequency and order parameter become

˜ ǫn= ǫn+

1

2τ

?

˜ ǫn

?

˜ ǫ2

n+ |˜∆(? p′

f;ǫn)|2

?

? p′

f

, (15)

5

Page 6

˜∆(? pf;ǫn) = ∆(? pf) +

1

2τ

?

˜∆(? p′

f;ǫn)

?

˜ ǫ2

n+ |˜∆(? p′f;ǫn)|2

?

? p′

f

.(16)

Thus, for an s-wave order parameter these equations give identical renormalization factors for both the Matsubara

frequency and the order parameter, i.e.

˜ ǫn

ǫn

=

˜∆(ǫn)

∆

= Z(ǫn) = 1 +

1

2πτ

1

?ǫ2

n+ ∆2

(s − wave), (17)

in which case the impurity renormalization drops out of the equilibrium propagator and gap equation. [31,32] However,

s-wave superconductors are exceptional; for any unconventional order parameter impurity scattering is pairbreaking.

[29]

Consider an unconventional superconductor with impurities in which the scattering is dominated by the identity

representation. If there is an element of the point group, R, which changes the sign of ∆(? pf), i.e. ∆(? pf)

then from eq.(16) the impurity renormalization of the order parameter vanishes identically:˜∆(? pf;ǫn) = ∆(? pf). The

cancellation between the impurity renormalization factors for the Matsubara frequency and order parameter no longer

occurs, with the consequence that impurity scattering suppresses both Tcand the magnitude of the order parameter.

For isotropic impurity scattering (not restricted to the Born approximation), the renormalization factor for the

Matsubara frequency, ˜ ǫn/ǫn= Z(ǫn), is independent of position on the Fermi surface and given by

R

−→−∆(? pf),

Z(ǫn) = 1 + Γu

Z(ǫn)D(ǫn)

ctn2(δ0) + (Z(ǫn)ǫnD(ǫn))2,(18)

with

D(ǫn) =

?

1

?Z(ǫn)2ǫ2

n+ |∆(? pf)|2

?

? pf

,(19)

where Γu= nimp/πNf and δ0= tan−1(πNfu0) is the s-wave scattering phase shift in the normal state. In the Born

limit, δ0→ πNfu0, while in the strong scattering limit (Nfu0→ ∞) we obtain the unitarity limit, δ0→ π/2.

Given the gap function, ∆(? pf), the impurity renormalization is easily calculated. The magnitude and temperature

dependence of the order parameter are calculated self-consistently from the mean-field gap equation,

∆(? pf) =

?

d? p′

fV (? pf,? p′

f)πT

|ǫn|<ωc

?

ǫn

∆(? p′

n+ |∆(? p′f)|2.

f)

?Z(ǫn)2ǫ2

(20)

The linearized gap equation determines Tc in terms of the pairing interaction, frequency cutoff ωc and impurity

scattering rate. At Tconly the dominant pairing channel α is relevant and the linearized gap equation becomes,

1

Vα

= πTc

|ǫn|<ωc

?

ǫn

1

|ǫn| + Γ,(21)

where Γ = Γusin2δ0is the pair-breaking parameter,

Γ =

?Γusin2δ0=

1

2τ= πnimpNfu2

0

,(Born limit)

(unitarity limit).

Γu=

nimp

πNf

,

(22)

For Γ = 0, this equation determines the clean-limit value of the transition temperature, Tco. Eliminating the pairing

interaction and cutoff gives the well-known Abrikosov-Gorkov formula, [33] except that the pairbreaking parameter

is determined by non-magnetic scattering, [6]

ψ

?1

2+

Γ

2πTc

?

− ψ

?1

2

?

= ln

?Tco

Tc

?

,(23)

where ψ(z) is the digamma function.

6

Page 7

Finally, the linearized gap equation is used to eliminate the pairing interaction and cutoff in favor of Tcin the full gap

equation (20). For pairing in a 1D representation, V (? pf,? p′

where ∆ is obtained from eq.(20). Multiplying eq.(20) by e(? pf)∗, integrating over the Fermi surface, and adding and

subtracting the RHS of the linearized gap equation (with Tc→ T) to eliminate V gives,

?

∞

?

which is solved self-consistently with eq.(18) to give Z(ǫn) and ∆ as a function of T/Tcand Γ.

f) = V e(? pf)e(? p′

f)∗, the order parameter is ∆(? pf) = ∆e(? pf),

ln(T/Tc) + ψ(1

2+

Γ

2πT) − ψ(1

2+

Γ

2πTc)

?

=

2πT

n=0

?

|e(? pf)|2

n+ ∆2|e(? pf)|2

?Z(ǫn)2ǫ2

?

? pf

−

1

ǫn+ Γ

, (24)

Gauge-invariant coupling to the condensate flow field

The self-energy term representing the diamagnetic coupling of quasiparticles to a static magnetic field (Fig. 1c) is

determined by local gauge invariance. Under a gauge transformation of the Fermion fields, ψ(? r) → ψ(? r)e−iΛ(? r)/2, the

quasiclassical propagator transforms as

ˆ g

Λ

−→ˆ g′=ˆU(Λ)†ˆ gˆU(Λ), (25)

whereˆU(Λ) = exp(+i

the transport equation (5) givesˆU(Λ)† ˆQ[ˆ g, ˆ σ]ˆU(Λ) =ˆQ[ˆ g′, ˆ σ′+ ˆ σ∇]. Thus, the form of the transport equation is

invariant, but the local gauge field generates an additional self-energy, ˆ σ∇= −iˆU(Λ)†? vf(? pf)·?∇ˆU(Λ). This property

of the transport equation is used to eliminate the phase degree of freedom of the order parameter in favor of a spatially

varying flow field. We parametrize the spatial variations in terms of a physical gauge, the phase χ(?R), and the local

amplitude, ∆0(? pf;?R) = |∆(?R)|e(? pf)

ˆ∆(? pf,?R) =ˆU[χ(?R)]ˆ∆0(? pf,?R)ˆU†[χ(?R)].

2Λ(?R)ˆ τ3), as does the self-energy and order parameter. [23] Applying this transformation to

(26)

Thus, the transport equation becomes

?

iǫnˆ τ3−ˆ∆0− ˆ σv− ˆ σ

′, ˆ g′?

+ i? vf(? pf) ·?∇ˆ g′= 0, (27)

where ˆ σv=1

invariance, and can be represented in terms of the gauge-invariant condensate flow field,

2? vf(? pf) ·?∇χ ˆ τ3. The diamagnetic coupling to a magnetic field,?b =?∇ ×?A, is then determined by gauge

? vs=1

2(?∇χ +2e

c

?A),(28)

and the self-energy,

ˆ σv= ? vf(? pf) ·? vs(?R) ˆ τ3.(29)

In the Meissner geometry (?H parallel to the interface) the driving term associated with the applied surface field is

of order,

????

σv

πTc

????∼

e

cvfA

πTc

∼H

Hc, (30)

where Hc∼ φ0/(ξλ) is the thermodynamic critical field. This term should be compared to the gradient term arising

from spatial variations of the screening current,

?????

? vf·?∇g

πTc

?????∼

?????

? vf·?∇(σv/∆)

πTc

?????∼ξ

λ

H

Hc. (31)

In the strong type II limit the velocity field is effectively uniform on the scale of the coherence length so we are

generally justified in dropping the gradient term in eq. (27).

7

Page 8

Linear response

The linear response limit is simply obtained from a perturbation expansion of the propagator, transport equation

and normalization condition, and is expected to be valid for low magnetic fields, |σv/πTc| ∼ H/Hc≪ 1. Assume an

expansion of the form, ˆ g = ˆ g0+ ˆ g1+..., where ˆ g0is the zero-field solution to the transport equation given by eq.(11),

and ˆ g1is the first-order correction to the propagator, formally of order |ˆ g1| ∼ O|(σv/∆)ˆ g0|. The linearized transport

equation and normalization condition,

?

i˜ ǫnˆ τ3−ˆ˜∆, ˆ g1

?

− [ˆ σv, ˆ g0] = 0,

{ˆ g0, ˆ g1} = 0, (32)

are inverted with the aid of eqs. (11) and (6) to give

ˆ g1= πˆ σv

|˜∆|2− i˜ ǫnˆ τ3ˆ˜∆

n+ |˜∆(? pf,ǫn)|2]3/2.

[˜ ǫ2

(33)

The resulting supercurrent calculated from eq. (3) can be written in terms of the superfluid density tensor,

?↔

ρs

?

ij= 2Nf

?

d? pfΦ(? pf)vi

f(? pf)vj

f(? pf),(34)

where

Φ(? pf) = πT

?

ǫn

|˜∆(? pf,ǫn)|2

n+ |˜∆(? pf,ǫn)|2]3/2,

[˜ ǫ2

(35)

which reduces to the angle-dependent Yosida function in the clean limit.

Fermi-liquid effects

Fermi-liquid effects arise from the leading order electronic self-energy (diagram 1b). For the diamagnetic response

the most important Fermi-liquid effect is the screening correction to the diamagnetic current; the relevant self energy

is, ˆ σb(? pf,?R) = σb(? pf,?R) ˆ τ3, with

σb(? pf,?R) =

?

d? p′

fAcur(? pf,? p′

f)T

?

ǫn

g(? p′

f,?R;ǫn), (36)

where Acur(? pf,? p′

Fermi liquid effects can contribute substantial temperature-dependent corrections to the penetration depth as the

gap opens and the number of thermal quasiparticles drops rapidly. We include formulas for the Fermi-liquid correction

to the supercurrent for a model uniaxial Fermi surface. The position on the Fermi surface can be parametrized by

the direction of the Fermi wavevector, ˆ p, and the Fermi velocity is given by,

f) is the dimensionless quasiparticle interaction.

? vf= v||

f(ˆ pxˆ x + ˆ pyˆ y) + v⊥

fˆ pzˆ z .(37)

Similarly, the quasiparticle interaction is parametrized by two Landau parameters corresponding to current flow in

the basal plane and along the ˆ z-axis,

Acur(? pf,? p′

f) = A||?ˆ pxˆ p′

x+ ˆ pyˆ p′

y

?+ A⊥(ˆ pzˆ p′

z) .(38)

The linear response result for the supercurrent is easily obtained from eq.(33) with the replacement ˆ σv → ˆ σsc =

? vf(? pf) ·? vs+ σb(? pf). The resulting current is given by

?

with the screening field satisfying the self-consistency equation,

?js(?R) = −2eNf

d? pf? vf(? pf)Φ(? pf)σsc(? pf),(39)

8

Page 9

σsc(? pf) = ? vf(? pf) ·? vs+

?

d? p′

fAcur(? pf,? p′

f)Φ(? p′

f)σsc(? p′

f).(40)

Equations (34,35,38,40,18,24) are the basic equations used to calculate the temperature-dependent penetration depth

for unconventional superconductors.

For a gap |∆(? pf)| with four-fold rotational symmetry about the ˆ z-axis the resulting supercurrent is given by a

diagonal superfluid density tensor with in-plane (ρ||

s) and z-axis (ρ⊥

s) superfluid densities given by

ρ||,⊥

s

= 2Nf(v||,⊥

f

)2

φ||,⊥

3A||,⊥φ||,⊥,

1 −1

(41)

with

φ||=3

2

?

dΩ

4π

?ˆ p2

x+ ˆ p2

y

?Φ(? pf),φ⊥= 3

?

dΩ

4πˆ p2

zΦ(? pf). (42)

Since φ||,⊥∼ ∆2

but may give substantial corrections to ρ||,⊥

0∼ (1 − T/Tc) near Tc, the Fermi liquid renormalizations of the penetration depth drop out near Tc,

s

(T) at low-temperatures. [34,35]

III. ZERO-FIELD PENETRATION DEPTH OF A DX2−Y2 SUPERCONDUCTOR

Consider the model of the CuO superconductors based on a dx2−y2 order parameter. The general form of the order

parameter is,

∆(? pf) = ∆(ˆ p2

x− ˆ p2

y) ∗ I(? pf), (43)

where I(? pf) is invariant under the full point group, and ˆ px,ydefine the direction of the Fermi wavevector in the basal

plane of the Fermi surface. Note that the nodes are required by the broken reflection symmetries. Two parameters

determine the excitation spectrum in the clean limit; (i) the maximum value of |∆(? pf)| (= ∆0), and (ii) the angular

slope of the gap near the node,

µ ≡

1

∆0

d|∆(ϑ)|

dϑ

???

ϑ=ϑnode,(44)

where ϑ is the angle measured relative to one of the nodes (see Fig. 2).

px

py

ϑ

Fig. 2Gap function for a dx2−y2 superconductor.

9

Page 10

A simple two-parameter model for |∆(? pf)|, which is useful for numerical calculations, is

|∆(ϑ)| =

?

∆0

µ∆0ϑ

;

1

µ≤ ϑ ≤π

0 ≤ ϑ <1

4

µ.;

(45)

The maximum gap, for a fixed µ, is obtained from a self-consistent solution to the gap equation. Figure 3 shows

solutions of eq.(24) for ∆0(T) as a function of temperature and impurity scattering. We obtain a gap ratio of

∆0/Tc= 1.9 at T = 0 for µ = 2.7 and Γ = 0. Note that the leading temperature-dependent correction to the gap

parameter for T ≪ Tcis δ∆0(T) ∼ T3, in the clean limit (appendix A).

T/Tco

∆o/Tco

0.000.20 0.400.600.801.00

0.00

0.50

1.00

1.50

2.00

0.0000.1000.200

1.896

1.898

1.900

1.902

1.904

Γ/πTco=0

Γ/πTco=0.001

Fig. 3

of ∆0(T) at low-temperature.

Maximum gap for a dx2−y2 superconductor as a function of temperature. The inset shows the T3deviation

The maximum gap is relatively insensitive to the angular slope of the gap near the nodes, except for small µ,

in which case the nodal region occupies a significant fraction of phase space. This behavior can be qualitatively

understood by noting that the condensation energy at zero temperature is effectively determined by the strength of

the pairing interaction, and therefore Tc. For an anisotropic gap ∆0 is enhanced to compensate for the regions of

small gap. If we assume that the Fermi-surface average of |∆(ϑ)|2is constant (fixed by Tc), then the maximum gap

is given by ∆0(µ) ≃ 1.8Tc/(1 − 8/3πµ), which is qualitatively the behavior obtained from the numerical solution to

the gap equation shown in Fig. 4. Note that the ususal one-parameter dx2−y2 gap, ∆ = ∆0(ˆ p2

µ = 2.

x− ˆ p2

y), corresponds to

µ

∆o/Tco

0.00 2.004.00 6.008.0010.0012.00

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

2.50

T/Tco=0.01

Fig. 4Maximum gap for a dx2−y2 superconductor as a function of the angular slope parameter, µ.

10

Page 11

The suppression of Tcby impurity scattering is determined by the pair-breaking parameter Γ = Γusin2δ0. Equation

(23) for Tc implies that unconventional pairing can be sensitive to impurity scattering, e.g.

transition is completely suppressed for (Γ/πTco)crit=

path of lcrit= vf/2Γ ≃ 3.6ξ0, where ξ0= vf/2πTc0is the coherence length in the clean limit. The relatively small

coherence length in the CuO superconductors is then an advantage for an unconventional order parameter. For weak

pair-breaking, Γ/2πTc≪ 1, the suppression of Tcis given by, ∆Tc/Tc0= −πΓ/8Tc0. Thus, for CuO superconductors

with Tc0= 100K and a suppression of less than 0.5K we have Γ < 1.3K. For an in-plane coherence length of ξ0= 14˚ A

this corresponds to an impurity mean-free path, l > 3,450˚ A.2

The magnitude of the gap parameter is also suppressed by impurity scattering. Figure 5 shows the suppression of

∆0(0) as a function of Γ. Note that ∆0(0), in contrast to Tc, is more strongly suppressed in the unitarity limit than

in the Born limit for the same suppression of Tc.

the superconducting

1

2e−γ≃ 0.28, which corresponds to an impurity mean-free

Γ/πTco

∆o/Tco

0.000.050.10 0.15 0.20

0.80

1.00

1.20

1.40

1.60

1.80

2.00

Born limit

Unitary limit

Tc/Tco

0.00 0.10 0.200.30

0.00

0.20

0.40

0.60

0.80

1.00

Fig. 5

parameter. The inset shows the impurity pair-breaking effect on Tc.

Maximum gap at T = 0.01Tc for a dx2−y2 superconductor as a function of the impurity pair-breaking

In the clean limit the angle-resolved density of states, obtained from eq. (11), is given by the familiar BCS form,

N(? pf,E) =

|E|

?E2− |∆(? pf)|2Θ(E2− |∆(? pf)|2).(46)

For low energies, |E| ≪ ∆0, the total density of states is dominated by the low-lying states near the nodes, and is

linear in |E|. For the model gap function in eq.(45) the total density of states is

?

These low-lying excitations are responsible for the linear temperature dependence of the penetration depth for a

superconductor with a line of nodes (or point nodes in 2D). Not surprisingly, the density of low-lying states near

the nodes is determined by the angular slope of the gap; therefore, µ also determines the coefficient of the linear

temperature dependence of the penetration depth (in the clean limit).

Although the small pair size in the cuprates leads to relatively weak suppression of Tcfrom impurity scattering,

the density of states at low-energy, |E| ≪ ∆0, and therefore the leading temperature dependence of the penetration

depth, are more sensitive to impurity scattering, particularly in the strong scattering limit. Figure 6 shows the density

of states as a function of the scattering phase shift for fixed Γu = 0.1∆0, corresponding to a rather high impurity

¯ N(E) = d? pfN(? pf,E) ≃

2|E|

µ∆0,

|E| < ∆0. (47)

2For non-magnetic, s-wave impurities in 2D, the impurity resistivity is given by ρ−1

breaking parameter that enters the Abrikosov-Gorkov formula. Thus, we can express Tc/Tc0 as a function of ρimp, independent

of the scattering phase shift, δ0. However, inelastic scattering, which is important at T ∼ Tc, destroys this simple result. [36,37]

imp= e2Nfv2

f/4Γ, where Γ is the same pair-

11

Page 12

concentration. The modification of the density of states at low-energy is negligible in the Born limit (δ0= π/20) and

remains essentially linear even for intermediate phase shifts (δ0= π/4). However, as the strength of the scattering

increases a finite density of states at E = 0 develops, becoming of order¯ N(0) ≃ 0.4 in the unitarity limit. In addition,

¯ N(E) deviates from linearity below E ≃ 0.4∆0. This is the cross-over energy scale, below which impurity scattering

strongly modifies the low-energy spectrum. The cross-over energy, ε∗, can be calculated from the lowest energy scale

for the renormalized Matsubara frequency ˜ ǫnas T → 0. From eq. (18) the cross-over scale at T = 0 is given by

1

σΓu

2τ =

=

D(ε∗)

(1 − σ) + σ(ε∗D(ε∗))2,(48)

where σ = sin2δ0, and D(ε∗) can be calculated to leading logarithmic accuracy (ln(∆0/ε∗) ≫ 1) for the dx2−y2 state,

4

πµ∆0

In the unitarity limit this scale can be a sizable fraction of ∆0 even in the dilute limit, ε∗∼√Γu∆0, but in the

Born limit the cross-over scale is exponentially small, ε∗∼ 2∆0exp(−µπτ∆0/2). For more detailed discussions of

the density of states of d-wave superconductors see Refs. ( [38,39]).

D(ε∗) ≃

ln(2∆0/ε∗).(49)

E/∆0

N(E)/Nf

Γu=0.1∆0

0.00 0.501.00 1.502.00

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

δ0=π/20

δ0=π/4

δ0=π/3

δ0=π/2

Fig. 6

the finite density of states and the suppression of the maximum gap in the unitarity limit.

Density of states vs. scattering phase shift. The impurity concentration is fixed with Γu/∆0 = 0.1. Note

Penetration depth for the dx2−y2 gap

The temperature dependence of the penetration depth for a dx2−y2 gap function is obtained from eq.(34) (neglecting

Fermi liquid corrections). We assume a cylindrical Fermi surface and a gap function parametrized by µ and ∆0as in

eq. (45). The in-plane penetration depth becomes

1

λ2

?

=

4πe2Nf(v?

f)2

c2

?

dϑ

2ππT

?

ǫn

|∆(ϑ)|2

n+ |∆(ϑ)|2]3/2,

[Z(ǫn)2ǫ2

(50)

where the integration is over the Fermi circle in the basal plane, and Z(ǫn) is the impurity renormalization factor for

s-wave scattering centers. In the clean limit the leading correction to the penetration depth at low-temperatures is

linear in T, typical of a gap with a line of nodes, [35,40–42]

δλ?(T)

λ?(0)

=

?

2ln2

d|∆(ϑ)|/dϑ|node

?

T + ...;T ≪ Tc,(51)

where d|∆(ϑ)|/dϑ|nodeis the angular slope of |∆(ϑ)| at the node. For the model gap in eq. (45), d|∆(ϑ)|/dϑ|node=

µ∆0.

12

Page 13

The report by Hardy, et al. [1] of a linear temperature dependence to δλ||(T) for single crystals of YBa2Cu3O6.95

provided the first substantial evidence for a superconducting state with a line of nodes in the excitation gap. Figure

7 shows the low-temperature data for δλ||(T) reported in Ref.( [43]). The solid line is a calculation of the penetration

depth for a dx2−y2 gap with the angular slope adjusted to fit the data for T < 20K. For the absolute penetration

depth we assume λ||(0) = 1,400˚ A. [44] The slope of the penetration depth at low temperature is 4.3˚ A/K, and the fit

to eq.(51) gives µ = 2.7, ∆0(0)/Tc= 1.9, and d|∆(ϑ)|/dϑ|node= µ∆0(0) ≃ 5.1Tc. For comparison, the one-parameter

dx2−y2 model, ∆ = ∆0cos(2φ) with ∆0calculated self-consistently, gives d|∆(ϑ)|/dϑ|node= µ∆0(0) = 4.3Tc.

T(K)

λ(T)-λ(5K) (Å)

0.00 5.00 10.0015.0020.00 25.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

0.01.0 2.03.04.05.0

-21

-18

-15

-12

-9

-6

-3

0

Fig. 7

[1]) the solid line is the best fit with µ ≃ 2.7, ∆0/Tc ≃ 1.9. The inset shows the effect of a tiny gap at the nodal

positions on the temperature dependence of the penetration depth. The axes are the same as those of the main

graph. The middle (top) curve in the inset corresponds to a tiny gap of ∆min/∆0 = 1% (2%).

Penetration depth as a function of temperature. The (*) are experimental data obtained from Ref. (

Also shown in Fig.

anisotropic conventional superconductors, [45] or if the unconventional order parameter contains a small compo-

nent of another representation. For example, an order parameter of the form dx2−y2 + i√εdxy has a gap function,

|∆(? pf)| = ∆0[cos2(2φ)+εsin2(2φ)]1/2, which is strongly anisotropic for small ε with a tiny gap of order ε∆0near the

nodes of the dx2−y2 component. [46] The data of Ref. ( [1]) implies that ε < 2%.

Precision measurements of the penetration depth in thin films show a T2behavior at low temperatures rather than

the linear temperature dependence characteristic of a dx2−y2 order parameter. [47–52,12] This difference may be due

to scattering by a higher concentration of defects present in the films. Hirschfeld and Goldenfeld [42] argue that the T2

dependence in films and the T dependence of δλ||(T) reported for single crystals can be understood within the same

dx2−y2 model for the pairing state provided the films are relatively dirty compared to the single crystals. However,

impurity scattering or defect scattering is pair-breaking in unconventional superconductors, so in order to explain the

weak or negligible suppression of Tcin films (compared to Tcin the single crystals of Ref.( [1]), the authors of Ref.

( [42]) argue that the scattering responsible for the T2dependence of δλ||(T) results from a dilute concentration of

strong scattering centers with phase shifts near the unitarity limit. A small concentration of unitarity scatterers leads

to strong modification of the density of states in the small phase space region near the nodes. Since δλ||is determined

by these low-energy excitations, the temperature dependence of δλ||(T) for T ≪ Tcmay be strongly modified even

when the suppression of Tcby a dilute concentration of scatterers is negligible.

In the Born limit (weak scattering) the density of states, even near the nodes is nearly unchanged. Thus, a much

higher concentration of defects is needed to generate δλ||(T) ∼ T2for T<

suppression of Tc. The sensitivity of the low-energy excitation spectrum to the scattering strength is reflected in the

temperature dependence of the penetration depth shown in Fig. 8. The concentration of s-wave scatterers is fixed

and the curves show the evolution from a linear T dependence in the Born limit (δ0= π/20), and intermediate phase

shifts, to the T2dependence in the unitarity limit (δ0= π/2). Note that the cross-over from δλ||∼ T to δλ||∼ T2is

abrupt, occuring very near the unitarity limit for dilute concentrations. A sharp cross-over in the excitation spectrum

as a function of phase shift has also been noted by Preosti, et al. [38]. Thus, it is worth emphasizing that for dilute

point impurities it is not merely ‘strong scattering’ that is required to obtain δλ||∼ T2with minimal reduction in Tc,

but scattering with δ0→ π/2.

7 is the effect of a tiny gap at the nodal positions. Tiny gaps can arise in strongly

∼0.2Tc, which is accompanied by a sizeable

13

Page 14

T(K)

δλ(Å)

Γu=0.01Tco

0.05.010.015.020.025.0

0.0

20.0

40.0

60.0

80.0

100.0

120.0

expt. data

δ0=π/20

δ0=π/4

δ0=3π/8

δ0=4π/9

δ0=π/2

Fig. 8

unitarity limit. The concentration is fixed with Γu = 0.01Tc.

Penetration depth as a function of temperature for phase shifts ranging from the Born limit to the

In the unitarity limit even dilute concentrations of impurities can strongly modify the low-energy spectrum. In

the unitarity limit the cross-over energy scale is ε∗∼√Γu∆0. Thus, δλ||(T) deviates from the linear T dependence

for T < ε∗. The sensitivity of δλ||(T) at T ≪ Tc for unitarity scattering (see Fig. 9) places a strong constraint

on the concentration of scatterers that can be present in clean single crystals that shows δλ||(T) ∼ T down to low

temperatures. For the data of Ref. ( [1]) we find Γu/∆0<

∼0.0001 given that δλ||(T) ∼ T down to T ≃ 1K.

T(K)

∆λ(Å)

0.005.0010.00 15.0020.0025.00

0.00

20.00

40.00

60.00

80.00

100.00

120.00

expt. data

Γ/Tc0=0

Γ/Tc0=0.0002

Γ/Tc0=0.001

Γ/Tc0=0.004

Fig. 9

scattering limit.

Penetration depth as a function of temperature and impurity concentration (∼ Γu) in the unitarity

IV. NONLINEAR CURRENT RESPONSE

In the Meissner geometry the screening current is proportional to the applied surface field, js∼ cH/λ. As H is

increased nonlinear field corrections to the constitutive equation for the supercurrent may become significant. In

conventional type II superconductors nonlinear corrections to the current-velocity relation arise from the thermal

population of quasiparticles, and vortex nucleation generally occurs before these nonlinear effects become important.

In unconventional superconductors with nodes in the excitation gap the nonlinear field correction to the supercurrent

is substantially larger than in conventional superconductors with other similar material properties. The origin of the

anomalous nonlinear Meissner effect is the contribution to the screening current associated with the quasiparticle

14

Page 15

states near the nodal lines. As a result the nonlinear Meissner effect may be used to detect the nodal structure of the

gap of an unconventional superconductor. [15]

In the limit |? vf·?∇|∆||/πTc∼ ξ/λ ≪ 1 the current can be expressed as a local function of the condensate velocity,

? vs(?R). In the presence of a condensate flow the local solution to the transport equation, ˆ g(? pf,?R;ǫn), is given by eq.

(11), but evaluated with i˜ ǫn→ i˜ ǫn− σv, where σv(? pf,?R) = ? vf·? vs(?R). The current response obtained from eq. (3) is

?

ǫn

?js= −2eNf

d? pf? vf(? pf)πT

?

σv(? pf) − i˜ ǫn

?

(˜ ǫn+ iσv(? pf))2+ |˜∆(? pf;ǫn)|2

.(52)

One point to note in calculating the density of states, or current, at finite flow with impurity scattering is that the

impurity-renormalization of ∆(? pf) need not vanish, even for s-wave impurity scattering and ?∆(? pf)? = 0. The reason

is clear from eq. (12) for˜∆(? pf;ǫn); the kernel no longer vanishes by symmetry with the replacement, ˜ ǫn→ ˜ ǫn+i? vf·? vs

for a general flow field. However, for special directions of ? vs, e.g.

∆(? pf) vanishes,˜∆ = ∆(? pf).

Equation (52) for the current can be transformed by contour integration and analytic continuation to the real axis

to give

? vsparallel to a node, the impurity correction to

?js= −2eNf

?

σv>0

d? pf? vf(? pf)

?+∞

−∞

dE f(E) [N+(? pf,E) − N−(? pf,E)] , (53)

where N±(? pf,E) is the density of states for quasiparticles that are co-moving (+? vf· ? vs > 0) and counter-moving

(−? vf· ? vs< 0) relative to the condensate flow. The integral is taken over the half space σv = ? vf· ? vs> 0 with the

counter-moving excitations included by inversion symmetry: Ciσv = −σv and |∆(Ci? pf)| = |∆(? pf)|. This result is

general enough to cover nonlinear field corrections to the current for superconductors with an unconventional order

parameter and pair-breaking effects from impurity scattering.

Fig. 10

Fermi-surface where |? vf ·? vs| < |∆(? pf)|.

Density of states for co-moving (+? vf·? vs) and counter-moving (−? vf·? vs) excitations at a point ? pf on the

The difference in the nonlinear current-velocity relation for conventional and unconventional order parameters

appears in the contributions to the current from the co-moving and counter-moving excitation spectrum at T = 0.

The spectrum is shown Fig. 10 in the clean limit for a specific direction ? pf in which ? vf· ? vs < |∆(? pf)|. At zero

temperature only the co-moving and counter-moving quasiparticle states with E < 0 contribute to the current.

Nonlinear current: conventional gap

For a conventional superconductor with an isotropic gap at T = 0 the current is easily calculated from the difference

in the number of co-moving versus counter-moving quasiparticles that make up the condensate,

?js= −2eNf

?

σv>0

d? pf? vf(? pf) [2? vf·? vs] = −eρ? vs

,vs< vf/∆0, (54)

with ρ = Nfv2

to the bulk critical velocity, vc = vf/∆0. At vs = vc the edge of the spectrum for the upper branch (E > 0 for

ffor a cylindrical Fermi surface. The main point is that the current is linear in ? vs for velocities up

15

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- Available from J. A. Sauls · May 23, 2014
- Available from J. A. Sauls · May 23, 2014
- Available from ArXiv
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