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Chemistry » Physical and Theoretical Chemistry » "Superconductivity - Theory and Applications", book edited by Adir Moyses Luiz, ISBN 978-953-307-151-0, Published: July 18, 2011 under CC BY-NC-SA 3.0 license. © The Author(s).

# Effects of Impurities on a Noncentrosymmetric Superconductor: Application to CePt3Si

By Heshmatollah Yavari
DOI: 10.5772/16338

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# Effects of Impurities on a Noncentrosymmetric Superconductor - Application to CePt3Si

Heshmatollah Yavari

## 1. Introduction

In the past two decades, a number of novel superconducting materials have been discovered where order parameter symmetries are different from an s-wave spin singlet, predicted by the Bardeen-Cooper-Schrieffer (BCS) theory of electron-phonon mediated pairing. From the initial discoveries of unconventional superconductivity in heavy-fermion compounds, the list of examples has now grown to include the high-Tc cuprate superconductors, ruthenates, ferromagnetic superconductors, and possibly organic materials.

In most of these materials, there are strong indications that the pairing is caused by the electron correlations, in contrast to conventional superconductors such as Pb, Nb, etc. Nonphononic mechanisms of pairing are believed to favor a nontrivial spin structure and orbital symmetry of the Cooper pairs. For example, the order parameter in the high-Tc superconductors, where the pairing is thought to be caused by the antiferromagnetic correlations, has the d-wave symmetry with lines of zeroes at the Fermi surface. A powerful tool of studying unconventional superconducting states is symmetry analysis, which works even if the pairing mechanism is not known.

In general, the superconducting BCS ground state is formed by Cooper pairs with zero total angular momentum. The electronic states are four-fold degenerate|k, and |k have the same energyε(k). The states with opposite momenta and opposite spins are transformed to one another under time reversal operation Κ|k=|k and states with opposite momenta are transformed to one another under inversion operationI|k=|k. The four degenerate states are a consequence of space and time inversion symmetries. Parity symmetry is irrelevant for spin-singlet pairing, but is essential for spin-triplet pairing. Time reversal symmetry is required for spin-singlet configuration, but is unimportant for spin-triplet state (Anderson, 1959, 1984).

If this degeneracy is lifted, for example, by a magnetic field or magnetic impurities coupling to the electron spins, then superconductivity is weakened or even suppressed. For spin-triplet pairing, Anderson noticed that additionally inversion symmetry is required to obtain the necessary degenerate electron states. Consequently, it became a widespread view that a material lacking an inversion center would be an unlikely candidate for spin-triplet pairing. For example, the absence of superconductivity in the paramagnetic phase of MnSi close to the quantum critical point to itinerant ferromagnetism was interpreted from this point of view (Mathur, 1998; Saxena, 2000). Near this quantum critical point the most natural spin fluctuation mediated Cooper pairing would occur in the spin-triplet channel. However, MnSi has the so-called B20 structure (P21), without an inversion center, inhibiting spin-triplet pairing.

Unusual properties are expected in superconductors whose crystal structure does not possess an inversion center (Edelstein, 1995; Frigeri et al., 2004; Gor’kov & Rashba, 2001; Samokhin et al., 2004; Sergienko& Curnoe, 2004).

Recent discovery of heavy fermion superconductor CePt3Si has opened up a new field of the study of superconductivity (Bauer et al., 2004). This is because this material does not have inversion center, which has stimulated further studies (Akazawa et al., 2004; Yogi et al., 2005). Because of the broken inversion symmetry, Rashba-type spin–orbit coupling (RSOC) is induced (Edelstein, 1995; Rashba, 1960; Rashba & Bychkov, 1984)), and hence different parities, spin-singlet pairing and spin triplet pairing, can be mixed in a superconducting state (Gor’kov & Rashba, 2001).

From a lot of experimental and theoretical studies, it is believed that the most possible candidate of superconducting state in CePt3Si is s+p-wave pairing (Frigeri et al., 2004; Hayashi et al., 2006). This mixing of the pairing channels with different parity may result in unusual properties of experimentally observed quantities such as a very high upper critical field Hc2which exceeds the paramagnetic limit (Bauer et al., 2004; Bauer et al., 2005a, 2005b; Yasuda et al., 2004), and the simultaneous appearance of a coherence peak feature in the NMR relaxation rate T11and low-temperature power-law behavior suggesting line nodes in the quasiparticle gap (Bauer et al., 2005a, 2005b; Yogi et al., 2004). The presence of line nodes in the gap of CePt3Si is also indicated by measurements of the thermal conductivity (Izawa et al., 2005) and the London penetration depth (Bauer et al., 2005; Bonalde et al., 2005).

It is known that the nonmagnetic as well as the magnetic impurities in the conventional and unconventional superconductors already have been proven to be a useful tool in distinguishing between various symmetries of the superconducting state (Blatsky et al., 2006). For example, in the conventional isotropic s-wave superconductor, the single magnetic impurity induced resonance state is located at the gap edge, which is known as Yu-Shiba-Rusinov state (Shiba, 1968). In the case of unconventional superconductor with dx2y2-wave symmetry of the superconducting state, the nonmagnetic impurity-induced bound state appears near the Fermi energy as a hallmark of dx2y2-wave pairing symmetry (Salkalo et al., 1996). The origin of this difference is understood as being due to the nodal structure of two kinds of SC order: in the dx2y2-wave case, the phase of Cooper pairing wave function changes sign across the nodal line, which yields finite density of states (DOS) below the superconducting gap, while in the isotropic s-wave case, the density of states is gapped up to energies of aboutΔ0 and thus the bound state can appear only at the gap edge. In principle the formation of the impurity resonance states can also occur in unconventional superconductors if the nodal line or point does not exist at the Fermi surface of a superconductor, as it occurs for isotropic nodeless p-wave and/or dx+idy-wave superconductors for the large value of the potential strength (Wang Q.H. & Wang,Z.D, 2004).

In unconventional superconductors non-magnetic impurities act as pair-breakers, similar to magnetic impurities in s-wave superconductors. A bound state appears near an isolated non-magnetic strong (scattering phase shiftπ2, or unitarity) scatterer, at the energy close to the Fermi level. The broadening of this bound state to an impurity band at finite disorder leads to a finite density of states at zero energy,N(0) , that increases with increasing impurity concentration (Borokowski & Hirschfeld, 1994). The impurity scattering changes the temperature dependence of the physical quantities below T corresponding to the impurity bandwidth: Δλchanges the behavior fromT toT2 the NMR relaxation rate changes fromT3 toT, and specific heat C(T)changes fromT2 toT. In other words, the impurities modify the power laws, especially at low temperatures.

The problem of a magnetic impurity in a superconductor has been extensively studied, but is not completely solved because of the difficulty of treating the dynamical correlations of the coupled impurity-conduction electron system together with pair correlations. Generally, the behavior of the system can be characterized by the ratio of the Kondo energy scale in the normal metal to the superconducting transition temperatureTKTc. ForTKTc1, conduction electrons scatter from classical spins and physics in this regime can be described by the Abrikosov-Gor'kov theory (Abrikosov & Gor'kov, 1961). In the opposite limit,TKTc1 , the impurity spin is screened and conduction electrons undergo only potential scattering. In this regime s-wave superconductors are largely unaffected by the presence of Kondo impurities due to Anderson's theorem. Superconductors with an anisotropic order parameter, e.g. p-wave, d-wave etc., are strongly affected, however and the potential scattering is pair-breaking. The effect of pair breaking is maximal in s-wave superconductors in the intermediate region,TKTc , while in the anisotropic case it is largest forTKTc (Borkowski & Hirschfeld, 1992).

In the noncentrosymmetric superconductor with the possible coexistence of s-wave and p-wave pairing symmetries, it is very interesting to see what the nature of the impurity state is and whether a low energy resonance state can still occur around the impurity through changing the dominant role played by each of the pairing components. Previously, the effect of nonmagnetic impurity scattering has been studied in the noncentrosymmetric superconductors with respect to the suppression of Tc and the behavior of the upper critical field (Frigeri et al., 2004; Mineev& Samokhin, 2007).

This in turn stimulates me to continue studying more properties. My main goal in this chapter is to find how the superconducting critical temperature, magnetic penetration depth, and spin–lattice relaxation rate of a noncentrosymmetric superconductor depend on the magnetic and nonmagnetic impurity concentration and also discuss the application of our results to a model of superconductivity in CePt3Si. I do these by using the Green’s function method when both s-wave and p-wave Cooper pairings coexist.

The chapter is organized as follows. In Sect. 2, the disorder averaged Green’s functions in the superconducting states are calculated and the effect of impurity is treated via the self-energies of the system. In Sect. 3, the equations for the superconducting gap functions renormalized by impurities are used to find the critical temperatureTc.

In Sect. 4, by using linear response theory I calculate the appropriate correlation function to evaluate the magnetic penetration depth. In this system the low temperature behavior of the magnetic penetration depth is consistence with the presence of line nodes in the energy gap.

In Sect. 5, the spin–lattice relaxation rate of nuclear magnetic resonance (NMR) in a superconductor without inversion symmetry in the presence of impurity effect is investigated.

In the last two cases I assume that the superconductivity in CePt3Si is most likely unconventional and our aim is to show how the low temperature power law is affected by nonmagnetic impurities.

Finally sect. 6 contains the discussion and conclusion remarks of my results.

## 2. Impurity scattering in normal and superconducting state

By using a single band model with electron band energyξk measured from the Fermi energy where electrons with momentum k and spin s are created (annihilated) by operatorsCk,s(Ck,s), the Hamiltonian including the pairing interaction can be written as

 H=∑k,sξkCk,s†Ck,s+12∑k,k′∑s,s′Vk,k′Ck,s†C−k,s′†C−k′,s′Ck′,s (1)

This system possesses time reversal and inversion symmetry(ξk=ξk) and the pairing interaction does not depend on the spin and favors either even parity (spin-singlet) or odd parity (spin-triplet) pairing as required. The absence of inversion symmetry is incorporated through the antisymmetric Rashba-type spin-orbit coupling

 Hso=∑k,s,s′αg→k.σ→s,s′Cks†Cks′ (2)

which removes parity but conserves time-reversal symmetry, i.e.,IHsoI1=Hso andTHsoT1=Hso. In Eq. (2),σ denotes the Pauli matrices (this satisfies the above condition IσI1=σ andTσT1=σ), gkis a dimensionless vector [gk=gkto preserve time reversal symmetry], and α(>0)denotes the strength of the spin-orbit coupling. The antisymmetric spin-orbit coupling (ASOC) termαgk.σ is different from zero only for crystals without an inversion center and can be derived microscopically by considering the relativistic corrections to the interaction of the electrons with the ionic potential (Frigeriet al., 2004; Dresselhaus, 1995). For qualitative studies, it is sufficient to deduce the structure of the g-vector from symmetry arguments (Frigeri et al., 2004) and to treat α as a parameter. I setgk2k=1, where ... denotes the average over the Fermi surface. The ASOC term lifts the spin degeneracy by generating two bands with different spin structure.

In the normal state the eigenvalues of the total Hamiltonan (H+Hso) are

 ξk±=εk−μ∓α|gk| (3)

where εk=k22m and and μis the chemical potential.

It is obvious from here that the time reversal symmetry is lost and the shape of the Fermi surfaces does not obey the mirror symmetry.

Due to the big difference between the Fermi momenta we neglected the pairing of electronic states from different bands. The structure of theory is now very similar to the theory of ferromagnetic superconductors with triplet pairing (Mineev, 2004).

Effects of disorder are described by potential scattering of the quasiparticles, which in real-space representation is given by

 Himp=∑i∫ψs†(r→)Uimpψs(r→)dr→ (4)

whereUimp=Un+Um,Un is the potential of a non-magnetic impurity, which we consider rather short-ranged such that s-wave scattering is dominant andUm=J(r)S.σ is the potential interaction between the local spin on the impurity site and conduction electrons, here Jis the exchange coupling and S is the spin operator.

### 2.1. Impurity averaging in superconducting state

Let us calculate the impurity-averaged Greens functions in the superconducting state. The Gor’kov equations with self-energy contributions are formally analogous to those obtained for system with inversion symmetry (Abrikosov et al., 1975).

 (iωn−ξk±−∑˜G(iωn))ℑ˜±(k,iωn)+(Δk+∑˜F(iωn))F˜±†(k,ωn)=σ^0 (5)
 (iωn+ξ−k±+∑˜G†(iωn))F˜±†(k,ωn)+(Δk†+∑˜F†(iωn))ℑ˜±(k,ωn)=0 (6)

where ωn=(2n+1)πTare the Matsubara Fermionic frequencies, σ^0is the unit matrix in the

spin state, and the impurity scattering enters the self-energy of the Greens function of the normal,˜G , and the anomalous type,˜F , their mathematical expressions read

 ∑˜G(iωn)=(nn|Un|2+nm|Um|2)∫dk′→(2π)3ℑ˜(k→′,iωn) (7)
 ∑˜F(iωn)=(nn|Un|2+nm|Um|2)∫dk′→(2π)3F↼(k→′,iωn) (8)

here nnand nmare the concentrations of nonmagnetic and magnetic impurities, respectively.

The equations for each band are only coupled through the order parameters given by the self-consistency equations

 Δk±=−T∑k′,υ,nV±υ(k→,k→′)Fυ(k→′,ωn) (9)

where

 υ=± (10)
.

Solving the Gor’kov equations one obtains the following expressions for the disorder-averaged Green’s functions

 ℑ˜±(k→,ωn)=(ℑ±(k→,iωn)F±(k→,iωn)F±†(k→,iωn)−ℑ±(−k→,−iωn)) (11)

where

 ℑ±(k→,ωn)=iωn+ξ−k±(iωn−∑imp+v→s.k→F−ξk±)(iωn−∑imp+v→s.k→F+ξ−k±)−Δk±Δk±† (12)
 F±(k→,ωn)=−Δk±(iωn−∑imp+v→s.k→F−ξk±)(iωn−∑imp+v→s.k→F+ξ−k±)−Δk±Δk±† (13)

hereimp=imp(n)+imp(m) is the self energy due to non magnetic and magnetic impurities.

The energies of elementary excitations are given by

 Ek±=ξk±−ξ−k±2±(ξk±+ξ−k±2)2+Δk±Δk±† (14)

The presence of the antisymmetric spin-orbit coupling would suppress spin-triplet pairing. However, it has been shown by Frigeri et al., (Frigeri et al., 2004) that the antisymmetric spin-orbit coupling is not destructive to the special spin-triplet state with the d vector parallel to gk(dkgk). Therefore, by choosinggk=321kF(ky,kx,0), one adopts the p-wave pairing state with parallel d vectordk=Δ(k˜y,k˜x,0). Here the unit vectork˜=(k˜x,k˜y,k˜z)=(cosφsinθ,sinφsinθ,cosφ).

By considering this parity-mixed pairing state the order parameter defined in (5) and (6) can be expressed as

 Δ(r→,k→)=[Δ0(r→)σ^0+d→(k→).σ→]iσ^y=[Δ0(r→)σ^0+Δ(r→)(−kyσ^x+kxσ^y)] (15)

with the spin-singlet s-wave component Δ0(r) and the dvectordk(r)=Δ(r)(ky,kx,0), here, the vectorrindicates the real-space coordinates. While this spin-triplet part alone has point nodes (axial state with two point nodes), the pairing state of Eq. (14) can possess line nodes in a gap as a result of the combination with the s-wave component (Hayashi et al., 2006; Sergienko 2004). In the presence of uniform supercurrent the gap function has the rdependence as

 Δ(r→,k→)=Δkei2mv→s.r→ (16)

where m is the bare electron mass.

The particular form of order parameter prevents the existence of interband terms in the Gor’kov equations

 (iωn−ξk,±−∑G(iωn))ℑ±(k,iωn)+(Δk,±+∑F(iωn))F±†(k,ωn)=1 (17)
 (iωn+ξ−k,±+∑G†(iωn))F±†(k,ωn)+(Δk,±†+∑F†(iωn))ℑ±(k,ωn)=0 (18)

where in this case

 ∑˜G(iωn)=(nn|Un|2+nm|Um|2)∫dk′→(2π)3[ℑ+(k→′,iωn)+ℑ−(k→′,iωn)] (19)
 ∑˜F(iωn)=(nn|Un|2+nm|Um|2)∫dk′→(2π)3[F+(k→′,iωn)+F−(k→′,iωn)] (20)

and

 Δ±=Δ0±d|gk| (21)

I consider the superconducting gaps |Δ0+Δsinθ| and |Δ0Δsinθ| on the Fermi surfaces I and II, respectively (such as superconductor CePt3Si). Such a gap structure can lead to line nodes on either Fermi surface I or II (Hayashi et al., 2006). These nodes are the result of the superposition of spin-singlet and spin-triplet contributions (each separately would not produce line nodes). On the Fermi surface I, the gap is |Δ0+Δsinθ|and is nodeless, (not that we choose Δ0>0 andΔ>0). On the other hand, the form of the gap on the Fermi surface II is|Δ0Δsinθ|, where line nodes can appear forΔ0<Δ (Hayashi et al., 2006).

## 3. Effects of impurities on the transition temperature of a noncentrosymmetrical superconductor

In the case of large SO band splitting, the order parameter has only intraband components and the gap equation (Eq. (9)) becomes

 Δk±=−T∑n,ν∫d3k′(2π)3V±υ(k→,k→′)Δk±(iωn−∑imp−ξk±)(iωn−∑imp+ξ−k±)−Δk±Δk±† (22)

The coupling constants Vλλ(k,k)′ I have used in previous considerations can be expressed through the real physical interactions between the electrons naturally introduced in the initial spinor basis where BCS type Hamiltonian has the following form

 Hint=14Ω∑kk′q∑αβμδ[Vαβμδs(k→,k→′)+Vαβμδt(k→,k→′)+Vαβμδm(k→,k→′)]×ck→+q→,λ†c−k→,λ†c−k→′,λck→+q→,λ′ (23)

where the pairing interaction is represented as a sum of the k-even, k-odd, and mixed-parity terms:V=Vs+Vt+Vm. The even contribution is

 Vαβμδs(k→,k→′)=Vs(k→,k→′)(iσ2)αβ(iσ2)μδ† (24)

The odd contribution is

 Vαβμδt(k→,k→′)=Vijt(k→,k→′)(iσiσ2)αβ(iσjσ2)μδ† (25)

here the amplitudes Vs(k,k)and Vijt(k,k) are even and odd with respect to their arguments correspondingly.

Finally, the mixed-parity contribution is

 Vαβμδm(k→,k→′)=Vim(k→,k→′)(iσiσ2)αβ(iσ2)μδ†+Vim(k→′,k→)(iσ2)αβ(iσiσ2)μδ† (26)

The first term on the right-hand side of Eq. (25) is odd in k and even in k′, while the second term is even in k and odd in k′.

The pairing interaction leading to the gap function [Eq. (14)] is characterized by three coupling constants, Vs, Vt, andVm. Here,Vs , and Vt result from the pairing interaction within each spin channel (s: singlet,t: triplet). Vmis the scattering of Cooper pairs between those two parity channels, present in systems without inversion symmetry. The linearized gap equations acquire simple algebraic form

 Δ0=VsπT∑n〈Ε+〉+VmπT∑n〈sinθΕ−〉 (27)
 Δ=VtπT∑n〈sinθΕ−〉+VmπT∑n〈Ε+〉 (28)

where the angular brackets denote the average over the Fermi surface, assuming the spherical Fermi surface for simplicity, Ε±=ΕI±ΕII2, ΕI,II=Δ0±ΔsinθΒI,II, and

 ΒI,II=[(ωn+i∑imp)2+|Δ0±Δsinθ|2]12 (29)

From Eqs. (26) and (27) one obtains then the following expression for the critical temperature

 lnTc0Tc=(1−Χ)[Ψ(12+14πTτm)−Ψ(12)]+Χ{Ψ[12+14πTc(1τn+1τm)−Ψ(12)]} (30)

where

 1τn=2πnnN0|Un|2,1τm=2πnmN0|Um|2 (31)

Ψ(x)is the digamma function, N0=(N++N)/2, N±are the densities of state (DOS) of the two bands at the Fermi level, and Tc0 is the critical temperature of the clean superconductor.

The coefficientΧ=1Δ(p)FS2Δ2(p)FSquantifies the degree of anisotropy of the order parameter on the Fermi surface (FS), where the angular brackets...FSstand for a FS average.

For isotropic s-wave pairing Δ(p)FS2=Δ2(p)FS(Χ=0)and for any pairing state with angular momentuml>1, e.g., for p-wave and d-wave states(l=1,2),(Χ=1,1τm=0) Eq. (29) reduces to the well-known expressions (Abrikosov, 1993; Abrikosov, A. A. & Gor’kov, 1959).

 lnTc0Tc=Ψ(12+14πTcτm)−Ψ(12) (32)
 lnTc0Tc=Ψ(12+14πTcτn)−Ψ(12) (33)

For mixing of s-wave state with some higher angular harmonic state, e.g., for example s+pands+d, (0<Χ<1,1τm=0), Eq. (29) becomes

 lnTc0Tc=Χ[Ψ(12+14πTcτn)−Ψ(12)] (34)

At τnTc01and τmTc01 (weak scattering) one has from Eq. (29):

 Tc0−Tc≈π4[Χ21τn+1−Χ2τm] (35)

In two particular cases of (i) both nonmagnetic and magnetic scattering in an isotropic s-wave superconductor (Χ=0) and (ii) nonmagnetic scattering only in a superconductor with arbitrary anisotropy ofΔ(p) (1τm=0,0<Χ<1), the Eq. (34) reduces to well-known expressions

 Tc0−Tc≈π4τm (36)
 Tc0−Tc≈πΧ8τn (37)

In the strong scattering limit (τnTc1,τmTc1), by using

 Ψ(12+14πTcτ)−Ψ(12)≈ln(γπτTc)+2π23(τT)2+O(τT)3 (38)

From Eq. (29) one finds

 (1τm)1−Χ(1τn+1τm)Χ=πγTc02Χ−1 (39)

One can see that the left hand side of Eq. (38) increases monotonically with both1τnand 1τmfor any value ofΧ, with the exception of the case Χ=0which does not depend on magnetic impurities.

For strongly anisotropic gap parameter(Χ1), Eq. (38) reduces to

 1τn+1τm=πγTc0 (40)

i.e., the contribution of magnetic and nonmagnetic impurities to pairing breaking is about the same.

For strongly isotropic case(Χ1), one has

 1τm=π2γTc0 (41)
andTc is determined primarily by magnetic impurities.

For the case of s+pwave pairing in the absence of magnetic impurities, one has

 (1τn)Χ=πγ2Χ−1Tc0 (42)

In this case the value of Tc asymptotically goes to zero asτn1 increase, whereas Tcof a d or p wave superconductor with Χ=1vanishes at a critical value1τnc=πTc0γ.

In the absence of nonmagnetic impurities one obtains

 (1τm)=πγ2Χ−1Tc0 (43)

And for the s-wave superconductor with Χ=0 one has1τmc=π2γTc0.

Application of these results to real noncentrosymmetric materials is complicated by the lack of definite information about the superconducting gap symmetry and the distribution of the pairing strength between the bands.

As far as the pairing symmetry is concerned, there is strong experimental evidence that the superconducting order parameter in CePt3Si has lines of gap nodes (Yasuda et al., 2004; Izawa et al., 2005; Bonalde et al., 2005). The lines of nodes are required by symmetry for all nontrivial one-dimensional representations of C4v(A2,B1 , andB2), so that the superconductivity in CePt3Si is most likely unconventional. This can be verified using the measurements of the dependence of Tcon the impurity concentration: For all types of unconventional pairing, the suppression of the critical temperature is described by the universal Abrikosov-Gor’kov function, see Eq. (32).

It should be mentioned that the lines of gap nodes can exist also for conventional pairing (A1representation), in which case they are purely accidental. While the accidental nodes would be consistent with the power-law behavior of physical properties observed experimentally, the impurity effect on Tc in this case is qualitatively different from the unconventional case. In this case in the absence of magnetic impurities one obtains the following equation for the critical temperature:

 lnTc0Tc=Χ[Ψ(12+14πTcτn)−Ψ(12)] (44)

In the low(τnTc1) and dirty(τnTc01) limit of impurity concentration one has

 Tc0−Tc≈Χπ8τnτTc0≫1 (45)
 Tc=Tc0(πτnTc0eC)Χ1−ΧτTc0≪1 (46)

This means that anisotropy of the conventional order parameter increases the rate at which Tcis suppressed by impurities. Unlike the unconventional case, however, the superconductivity is never completely destroyed, even at strong disorder.

## 4. Low temperature magnetic penetration depth of a superconductor without inversion symmetry

To determine the penetration depth or superfluid density in asuperconductor without inversion symmetry one calculates the electromagnetic response tensorK(q,vs,T), relating the current densityJ to an applied vector potential A

 J→(q)=−K(q→,v→s,T)A→(q→) (47)

The expression for the response function can be obtained as

 K(q→,v→s,T,ωm)=ne2mc(1+2πm〈T∑n,kk^∥2ℑ±(k→+,ωn)ℑ±(k→−,ωn,ωm)〉) (48)

wherek±=k±q2, k^2is the direction of the supercurrent and ........represents a Fermi surface average.

By using the expression of Green`s function into Eq. (47) one obtains

 K(q→,v→s,T,ωn)=ne2mc(1+2πTm∑n∫d2k(2π)2k^∥2(iωn−∑imp+v→s.k→F)2+ξk+,±ξk−,±+Δk+Δk−[(iωn−∑imp+v→s.(k→+q→2))2−Ek+,±2][(iωn−∑imp+v→s.(k→−q→2))2−Ek−,±2]) (49)

Now we separate out the response function as

 K(q→,v→s,T)=K(0,0,0)+δK(q→,v→s,T) (50)

where K(0,0,0)=c4πλ2(0)(λ(0)=(mc24πne2)12is the zero temperature London penetration depth).

Doing the summation over Matsubara frequencies for each band one gets

 δK(q→,v→s,T)=−2n+e2mc〈k^∥2∫0∞dωRe[f(ω−v→s.k→F)−f(−ω−v→s.k→F)Δk,+2](ω−i∑imp)2−Δk,+2[Δk,+2+(q→.k→F2m)2−(2αgk)2−(ω−i∑imp)2]〉−2n−e2mc〈k^∥2∫0∞dωRe[f(ω−v→s.k→F)−f(−ω−v→s.k→F)Δk,−2](ω−i∑imp)2−Δk,−2[Δk,−2+(q→.k→F2m)2−(2αgk)2−(ω−i∑imp)2]〉=−n+e2mc{1−2〈k^∥2sinh−1(q→.k→F2mΔk,+)(q→.k→F2mΔk,+)1+(q→.k→F2mΔk,+)2+(2αgkΔk,+)2〉+〈k^∥2ln[Δk,+2+(q→.k→F2m)2−(1+(q→.k→F2mΔk,+)2−(2αgkΔk,+)2)∑imp−(q→.k→F2mΔk,+)1+(q→.k→F2mΔk,+)2−(4mΔk,+αgkq→.k→F)2∑imp2−Δk,+2Δk,+2+(q→.k→F2m)2+(1+(q→.k→F2mΔk,+)2−(2αgkΔk,+)2)∑imp−(q→.k→F2mΔk,+)1+(q→.k→F2mΔk,+)2−(4mΔk,+αgkq→.k→F)2∑imp2−Δk,+2]〉+2〈k^∥2∫dωRe[f(ω−v→s.k→F)+f(ω+v→s.k→F)](ω−i∑imp)2−Δk,+2[Δk,+2+(q→.k→F2m)2−(2αgk)2−(ω−i∑imp)2]〉} (51)
 −n−e2mc{1−2〈k^∥2sinh−1(q→.k→F2mΔk,−)(q→.k→F2mΔk,−)1+(q→.k→F2mΔk,−)2+(2αgkΔk,−)2〉+〈k^∥2ln[Δk,−2+(q→.k→F2m)2−(1+(q→.k→F2mΔk,−)2−(2αgkΔk,−)2)∑imp−(q→.k→F2mΔk,−)1+(q→.k→F2mΔk,−)2−(4mΔk,−αgkq→.k→F)2∑imp2−Δk,−2Δk,−2+(q→.k→F2m)2+(1+(q→.k→F2mΔk,−)2−(2αgkΔk,−)2)∑imp−(q→.k→F2mΔk,−)1+(q→.k→F2mΔk,−)2−(4mΔk,−αgkq→.k→F)2∑imp2−Δk,−2]〉+2〈k^∥2∫dωRe[f(ω−v→s.k→F)+f(ω+v→s.k→F)](ω−i∑imp)2−Δk,−2[Δk,−2+(q→.k→F2m)2−(2αgk)2−(ω−i∑imp)2]〉} (52)

The factor αgkcharacterizes and quantifies the absence of an inversion center in a crystal lattice.

This is the main result of my work i.e. nonlocality, nonlineary, impurity and nonsentrosymmetry are involved in the response function. The first two terms in Eq. (50) represent the nonlocal correction to the London penetration depth and the third represents the nonlocal and impure renormalization of the response while the forth combined nonlocal, nonlinear, and impure corrections to the temperature dependence.

I consider a system in which a uniform supercurrent flows with the velocityvs, so all quasiparticles Matsubara energies modified by the semiclassical Doppler shiftvs.kF.

The specular boundary scattering in terms of response function can be written as (Kosztin & Leggett, 1997)

 Δλspec(T)λ0=2π∫0∞dq˜δK(q→,v→s,T)(q˜2+1)2 (53)

In the pure case there are four relevant energy scales in the low energy sector in the Meissner state: T, Enonlin, Enonloc, andαgk. The first two are experimentally controlled parameters while the last two are intrinsic one.

In low temperatures limit the contribution of the fully gap (|Δ0+Δsinθ|) Fermi surface I decrease and the effect of the gap |Δ0Δsinθ|Fermi surface II is enhanced. I consider geometry where the magnetic field is parallel to c axis and thusvs and the penetration direction q˜are in the ab plane, and in general, vsmakes an angle φ with the axis. There are two effective nonlinear energy scales Enonlin+=vskFuφl1 andEnonlin=vskFuφl2.where uφl=|cosφ+lsinφ| andl1,l2=±1.

In the nonlocal(q0), linear (vs0)limit, i.e., in the range of temperature where EnonlinTEnonlocone gets

 δK(q,0,T)={−c4πλ02(2ln2)TΔ0α′wθl≪T−c4πλ02∑luθl2(π4α′wθlΔ0+32ξ(3)T3Δ0α′2wθl2)α′wθl≫T (54)

wherewθl=|sinθlcosθ|, uθl=|cosθ+lsinθ|, andα=qvF222αgk.

Depending on the effective nonlocal energy scales (Enonloc+=vFuθl1λ0,Enonloc=vFuθl2λ0,l1,l2=±1)one obtains

 Δλspec(T)λ0∝{TEnonloc+,Enonloc−≪TTEnonloc−≪T≪Enonloc+T2Enonloc+,Enonloc−≫T (55)

For CePt3Si superconductor withTc=0.75K, the linear temperature dependence would crossover to a quadratic dependence belowTnonloc0.015K.

Magnetic penetration depth measurements in CePt3Si did not find a T2law as expected for line nodes. I argue that it may be due to the fact that such measurements were performed above 0.015K. On the other hand, it is note that CePt3Si is an extreme type-II superconductor with the Ginzburg-Landau parameter,K=140 , and the nonlocal effect can be safely neglected, and because this system is a clean superconductor, neglect the impurity effect can be neglected (Bauer et al., 2004; Bauer et al., 2005).

In the local, clean, and nonlinear limit(q0,vs0) the penetration depth is given by

 λspec(loc)=(c4πδK(q→0,v→s,T))12 (56)

Where

 δK(q→,v→s,T)==−n+e2mc{1−2〈k^∥2sinh−1(q→.k→F2mΔk,+)(q→.k→F2mΔk,+)1+(2αkFΔk,+)2〉+2〈k^∥2∫dωRe[f(ω−v→s.k→F)+f(ω+v→s.k→F)]ω2−Δk,+2[Δk,+22−(2αgk)2−ω2]〉}−n−e2mc{1−2〈k^∥2sinh−1(q→.k→F2mΔk,−)(q→.k→F2mΔk,−)1+(2αkFΔk,−)2〉+2〈k^∥2∫dωRe[f(ω−v→s.k→F)+f(ω+v→s.k→F)]ω2−Δk,−2[Δk,−2−(2αgk)2−ω2]〉} (57)

Thus by considering only the second term in the right hand side of Eq. (55) into Eq. (51) one gets

 δλspecλ0≈{12ln2∑l=±1uθl2TΔ0Enonlin+,Enonlin−≪T12ln2uθl22TΔ0+uθl1322[vskF2Δ0+2αgkΔ0]Enonlin−≪T≪Enonlin+12∑l=±uθl3[vskF2Δ0+2αgkΔ0]2+o(Te−(vskF+4αgk)2T)Enonlin+,Enonlin−≫T (58)

The linear temperature dependence of penetration depth is in agreement with Bonalde et al's result (Bonalde et al., 2005).

Thus the T behavior at low temperatures of the penetration depth in Eq. (56) is due to nonlineary indicating the existence of line nodes in the gap parameter in CePt3Si compound.

A T linear dependence of the penetration depth in the low temperature region is expected for clean, local and nonlinear superconductors with line nodes in the gap function.

Now the effect of impurities when both s-wave and p-wave Cooper pairings coexist is considered.

I assume that the superconductivity in CePt3Si is unconventional and is affected only by nonmagnetic impurities. The equation of motion for self-energy can be written as

 ∑imp(n)(p→,iωn)=nnT(p→,p→′,iωn) (59)

where the T matrix is given by

 T(p→,p→′,iωn)=uσ31−uσ3ℑ(p→,p→′,iωn) (60)

here σ3is the third Pauli-spin operator.

By using the expression of the Green’s function in Eq. (58) one can write

 T(p→,p→′,iωn)=πN0u02I1+(πN0u0I)2 (61)

where

 I=∫02πdΩ4π−(ω+i∑imp(n))[(ω+i∑imp(n))2+Δ+Δ−]12 (62)

and u0 is a single s-wave matrix element of scattering potentialu.Small u0puts us in the limit where the Born approximation is valid, where largeu0(u0), puts us in the unitarity limit.

Theoretically it is known that the nodal gap structure is very sensitive to the impurities. If the spin-singlet and triplet components are mixed, the latter might be suppressed by the impurity scattering and the system would behave like a BCS superconductor. For p-wave gap function the polar and axial states have angular structures, Δk(T)=Δ0(T)cosθkand Δk(T)=Δ0(T)sinθkrespectively. The electromagnetic response now depends on the mutual orientation of the vector potential A and I^(unit vector of gap symmetry), which itself may be oriented by surfaces, fields and superflow. A detailed experimental and theoretical study for the axial and polar states was presented in Ref. (Einzel, 1986). In the clean limit and in the absence of Fermi-Liquid effects the following low-temperature asymptotic were obtained for axial and polar states

 Δλ(T)∥,⊥λ(0)=a∥,⊥(kBTΔ0)n∥,⊥ (63)

where in the axial state n=2(4)anda=π2(7π415), and in the polar state n=3(1) anda=27πξ(3)4(3πln22), for the orientations().

The influence of nonmagnetic impurities on the penetration depth of a p-wave superconductor was discussed in detail in Ref (Gross et al., 1986). At very low temperatures, the main contribution will originated from the eigenvalue with the lower temperature exponent n, i.e., for the axial state (point nodes) withT2 low, and for the polar state (line nodes) the dominating contribution with a linearT. The quadratic dependence in axial state may arise from nonlocality.

The low temperature dependence of penetration depth in polar and axial states used by Einzel et al., (Einzel et al. 1986) to analyze the λ(T)T2behavior of Ube13 at low temperatures. The axialAI^ case seems to be the proper state to analyze the experiment because it was favored by orientation effects and was the only one withT2 dependence. Meanwhile, it has turned out thatT2 behavior is introduced immediately by T-matrix impurity scattering and also by weak scattering in the polar case. The axial sate., and according to the Andersons theorem the s-wave value of the London penetration depth are not at all affected by small concentration of nonmagnetic impurities.

Thus, for the polar state, Eq. (60) can be written as

 I=∫02πdθ2π−(ω+i∑imp(n))[(ω+i∑imp(n))2+Δ02cos2θ]12 (64)

Doing the angular integration in Eq. (62) and using Eqs. (57) and (59) one obtains

 ∑imp(n)=−[2ω˜N(0)nnu02/ω˜2+Δ02]K(Δ02ω˜2+Δ02)1+[4N(0)2u02ω˜2/(ω˜2+Δ02)]K2(Δ02ω˜2+Δ02) (65)

hereK is the elliptic integral andω˜=ω+iimp(n). We note that in the impurity dominated gapless regime, the normalized frequencyω˜ takes the limiting formω˜ω+iγ, whereγ is a constant depending on impurity concentration and scattering strength.

In the low temperature limit we can replace the normalized frequency ω˜everywhere by its low frequency limiting form and after integration over frequency one gets

 δK(q→,v→s,T)=−N1e2mc{4γπ2T23〈k^∥2Δk2(Δk2+γ2)52〉} (66)

As in the case of d-wave order parameter, from Eqs. (64) and (51) one finds

 δλ(T)λ(0)=γ4πΔ0ln(4Δ0γ)+π24γΔ0T2 (67)

In p-wave cuprates, scattering fills in electronic states at the gap nodes, thereby suppressing the penetration depth at low temperatures and changing T-linear to T2behavior.

## 5. Effect of impurities on the low temperature NMR relaxation rate of a noncentrosymmetric superconductor

I consider the NMR spin-lattice relaxation due to the interaction between the nuclear spin magnetic moment γnI (γnis the nuclear gyro magnetic ratio) and the hyperfine field h, created at the nucleus by the conduction electrons. Thus the system Hamiltonian is

 H=H0+Hso+Hn+Hint (68)

where H0 and Hso are defined by Eqs. (1) and (2), Hn=γnIHis the Zeeman coupling of the nuclear spin with the external fieldH, and Hint=γnIh is the hyperfine interaction.

The spin-lattice relaxation rate due to the hyperfine contact interaction of the nucleus with the band electron is given by

 R=1T1T=−J22πlimImΚ+−R(ω)ωω→0 (69)

where ω is the NMR frequency, J=8π3γnγe(γeis the electron geomagnetic ratio) is the hyperfine coupling constant, andΚ+R(τ), the Fourier transform of the retarded correlation function of the electron spin densities at the nuclear site, in the Matsubara formalism is given by (in our unitskB==1)

 Κ+−R(τ)=−〈TτS+(τ)S−(0)〉 (70)

here Tτ is the time order operator, τis the imaginary time,S±(τ)=eHτS±eHτ , and

 S+(r)=ψ↑†(r→)ψ↓(r→),S−(r)=ψ↓†(r→)ψ↑(r→) (71)

with ψσ(r)andψσ(r) being the electron field operators.

The Fourier transform of the correlation function is given by

 Κ+−R(iωn)=∫0βdτeiωnτΚ+−R(τ) (72)

The retarded correlation function is obtained by analytical continuation of the Matsubara correlation functionΚ+R(ω)=Κ+R(iωn)|iωnω+iδ.

From Eqs. (66)- (70), one gets

 1T1T=−J22πlim1ωω→0×Im{T∑p,p′,ωn[Tr(ℑ±(p→,iωn+iΩm)ℑ±(p→′,iωn))−Tr(F±(p→,iωn+iΩm)F±(p→′,iωn))]iΩn→ω+iδ} (73)

where Ωm=2mπT are the bosonic Matsubara frequencies. By using Eqs. (11) and (12) into Eq. (71), the final result for the relaxation rate is

 1T1T=J2∫0∞dω(−∂f∂ω){N+(ω)N−(ω)+M+(ω)M−(ω)} (74)

where f(ω)=(eωT+1)1 is the Fermi Function., Nσ(ω)and Mσ(ω) defined by the retarded Green’s factions as

 Nσ(ω)=−∑p,ν=±ImℑνR(p→,ω) (75)
 Mσ(ω)=−∑p,ν=±ImFνR(p→,ω) (76)

In low temperatures limit the contribution of the fully gap (|Δ0+Δsinθ|) Fermi surface I decrease and the effect of the gap |Δ0Δsinθ| Fermi surface II is enhanced.

As I mentioned above, the experimental data for CePt3Si at low temperature seem to point to the presence of lines of the gap nodes in gap parameter (In our gap model forΔ0<Δ, |Δ0Δsinθ|has line nodes). Symmetry imposed gap nodes exist only for the order parameters which transform according to one of the nonunity representations of the point group. For all such order parametersMσ=0.Thus, Eq. (72) can be written as

 1T1T=J24T∫0∞dωcosh2(ω2T){N+(ω)N−(ω)} (77)

In the clean limit the density of state can be calculated from BCS expression

 Nσ(ω)=N0Re〈ωω2−Δ±2〉 (78)

For the gap parameter with line nodes from Eq. (76) one gets

 N(ω)=N0π2ωΔ0 (79)

Thus from Eq. (75) one has

 1T1=J2π2N02T32Δ02 (80)

Therefore, line nodes on the Fermi surface II lead to the low-temperature T3 law in T11which is in qualitative agreement with the experimental results.

In the dirty limit the density of state can be written as

 Nimp(ω)=∫dΩNBCS(ω,θ)1+u02NBCS2(ω,θ) (81)

In the limit, Γ<<Δ0where Γ=nimpπN0(NV) (NVis the electron density) the density of state is

 Nimp(ω)≈N(0)+ac2ω (82)

where c=cotgδ0 (δ0is the s-wave scattering phase shift), ais a constant, and N(0) the zero energy(ω=0) quasi-particle density of state is given by

 N(0)=No(ς1+14ς2+12ς)12 (83)

where

 ς=ΓΔ (84)
.

In the unitary limit(u0),c=0(δ0=π/2), from Eqs. (75) and (80) one obtains

 1T1=J2N(0)2T (85)

Thus the power-low temperature dependence of T11 is affected by impurities and it changes to linear temperature dependence characteristic of the normal state Koringa relation again is in agreement with the experimental results.

## 6. Conclusion

In this chapter I have studied theoretically the effect of both magnetic and nonmagnetic impurities on the superconducting properties of a non-centrosymmetric superconductor and also I have discussed the application of my results to a model of superconductivity in CePt3Si.

First, the critical temperature is obtained for a superconductor with an arbitrary of impurity concentration (magnetic and nonmagnetic) and an arbitrary degree of anisotropy of the superconducting order parameter, ranging from isotropic s wave to p wave and mixed (s+p) wave as particular cases.

The critical temperature is found to be suppressed by disorder, both for conventional and unconventional pairings, in the latter case according to the universal Abrikosov-Gor’kov function.

In the case of nonsentrosymmetrical superconductor CePt3Si with conventional pairing (A1representation with purely accidental line nodes), I have found that the anisotropy of the conventional order parameter increases the rate at which Tc is suppressed by impurities. Unlike the unconventional case, however, the superconductivity is never completely destroyed, even at strong disorder.

In section 4, I have calculated the appropriate correlation function to evaluate the magnetic penetration depth. Besides nonlineary and nonlocality, the effect of impurities in the magnetic penetration depth when both s-wave and p-wave Cooper pairings coexist, has been considered.

For superconductor CePt3Si, I have shown that such a model with different symmetries describes the data rather well. In this system the low temperature behavior of the magnetic penetration depth is consistence with the presence of line nodes in the energy gap and a quadratic dependence due to nonlocality may accrue belowTnonloc=0.015K. In a dirty superconductor the quadratic temperature dependence of the magnetic penetration depth may come from either impurity scattering or nonlocality, but the nonlocality and nodal behavior may be hidden by the impurity effects.

Finally, I have calculated the nuclear spin-lattice relaxation of CePt3Si superconductor. In the clean limit the line nodes which can occur due to the superposition of the two spin channels lead to the low temperatureT3 law inT11. In a dirty superconductor the linear temperature dependence of the spin-lattice relaxation rate characteristic of the normal state Koringa relation.

## Acknowledgements

I wish to thank the Office of Graduate Studies and Research Vice President of theUniversity of Isfahan for their support.

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