## 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-

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-* 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

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 * B*20 structure (

*2*P

_{1}), 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 CePt_{3}Si 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 CePt_{3}Si is * s*+

*-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*p

_{3}Si 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 * s*-wave case, the density of states is gapped up to energies of about

*-wave and/or*p

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 shift

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 temperature

In the noncentrosymmetric superconductor with the possible coexistence of * s*-wave and

*-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*p

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 CePt_{3}Si. 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 temperature

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 CePt_{3}Si 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 energy

This system possesses time reversal and inversion symmetry

which removes parity but conserves time-reversal symmetry, i.e.,

In the normal state the eigenvalues of the total Hamiltonan

where

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

where

### 2.1. Impurity averaging in superconducting state

Let us calculate the impurity-averaged Green`s 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).

where

spin state, and the impurity scattering enters the self-energy of the Green`s function of the normal,

here

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

where

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

where

here

The energies of elementary excitations are given by

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

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

with the spin-singlet s-wave component

where m is the bare electron mass.

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

where in this case

and

I consider the superconducting gaps _{3}Si). 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

## 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

The coupling constants

where the pairing interaction is represented as a sum of the k-even, k-odd, and mixed-parity terms:

The odd contribution is

here the amplitudes

Finally, the mixed-parity contribution is

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,

where the angular brackets denote the average over the Fermi surface, assuming the spherical Fermi surface for simplicity,

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

where

The coefficient

For isotropic * s*-wave pairing

*-wave and d-wave states*p

For mixing of * s*-wave state with some higher angular harmonic state, e.g., for example

At

In two particular cases of (i) both nonmagnetic and magnetic scattering in an isotropic s-wave superconductor (

In the strong scattering limit (

From Eq. (29) one finds

One can see that the left hand side of Eq. (38) increases monotonically with both

For strongly anisotropic gap parameter

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

For strongly isotropic case

For the case of

In this case the value of

In the absence of nonmagnetic impurities one obtains

And for the s-wave superconductor with

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 CePt_{3}Si 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 _{3}Si is most likely unconventional. This can be verified using the measurements of the dependence of

It should be mentioned that the lines of gap nodes can exist also for conventional pairing (

In the low

This means that anisotropy of the conventional order parameter increases the rate at which

## 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 tensor

The expression for the response function can be obtained as

where

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

Now we separate out the response function as

where

Doing the summation over Matsubara frequencies for each band one gets

The factor

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 velocity

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

In the pure case there are four relevant energy scales in the low energy sector in the Meissner state: T,

In low temperatures limit the contribution of the fully gap (

In the nonlocal

where

Depending on the effective nonlocal energy scales

For CePt_{3}Si superconductor with

Magnetic penetration depth measurements in CePt3Si did not find a

In the local, clean, and nonlinear limit

Where

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

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

Thus the _{3}Si compound.

Now the effect of impurities when both * s*-wave and

*-wave Cooper pairings coexist is considered.*p

I assume that the superconductivity in CePt_{3}Si is unconventional and is affected only by nonmagnetic impurities. The equation of motion for self-energy can be written as

where the T matrix is given by

here

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

where

and

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,

where in the axial state

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) with

The low temperature dependence of penetration depth in polar and axial states used by Einzel et al., (Einzel et al. 1986) to analyze the _{13} at low temperatures. The axial

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

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

here

In the low temperature limit we can replace the normalized frequency

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

In * p-*wave cuprates, scattering fills in electronic states at the gap nodes, thereby suppressing the penetration depth at low temperatures and changing

## 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

where

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

where

here

with

The Fourier transform of the correlation function is given by

The retarded correlation function is obtained by analytical continuation of the Matsubara correlation function

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

where

where

In low temperatures limit the contribution of the fully gap (

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 for

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

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

Thus from Eq. (75) one has

Therefore, line nodes on the Fermi surface II lead to the low-temperature

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

In the limit,

where

where

In the unitary limit

Thus the power-low temperature dependence of

## 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 (

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

*-wave Cooper pairings coexist, has been considered.*p

For superconductor CePt_{3}Si, 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 below

Finally, I have calculated the nuclear spin-lattice relaxation of CePt_{3}Si superconductor. In the clean limit the line nodes which can occur due to the superposition of the two spin channels lead to the low temperature

## Acknowledgments

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

## References

- 1.
Abrikosov A. A. 1993 Influence of the Gap Anisotropy on Superconducting Properties . - 2.
(1959).On the theory of superconducting alloys.Abrikosov A. A. Gor’kov L. P. - 3.
Abrikosov A. A. Gor’kov L. P. 1961 Contribution to the theory of superconducting alloys with paramagnetic impurities . - 4.
Abrikosov A. A. Gor’kov L. P. Dzyaloshnskii I. E. 1975 Methods of Quantum Field Theory in Statistical Physics , Dover Publications Inc,0-48663-228-8 York. - 5.
Akazawa T. Hidaka H. Fujiwara T. Kobayashi T. C. Yamamoto E. Haga Y. Settai R. Onuki Y. 2004 Pressure-induced Superconductivity in UIr without inversion symmetry. - 6.
Anderson P. W. 1959 Theory of dirty superconductors . - 7.
Anderson P. W. 1984 Structure of triplet superconducting energy gap. - 8.
Bauer E. Bonalde I. Sigrist M. 2005 Superconductivity and normal state properties of non-centrosymmetric CePt3Si:a status report. - 9.
Bauer E. Hilscher G. Michor H. Paul Ch. Scheidt E. W. Gribanov A. Seropegin Yu. Noël H. Sigrist M. Rogl P. 2004 Heavy Fermion Superconductivity and Magnetic Order in Noncentrosymmetric CePt _{3}Si. - 10.
Bauer E. Hilscher G. Michor H. Sieberer M. Scheidt E. W. Gribanov A. Seropegin Yu. Rogl P. Amato A. Song W. Y. Park J. G. Adroja D. T. Nicklas M. Sparn G. Yogi M. Kitaoka Y. 2005 Unconventional superconductivity and magnetism in CePt_{3}Si_{1-x}Ge_{x}. - 11.
Blatsky A. V. Vekhter I. Zhu J. X. 2006 Impurity-induced states in conventional and unconventional superconductors. - 12.
Bonalde I. Brämer-Escamilla W. Bauer E. 2005 Evidence for Line Nodes in the Superconducting Energy Gap of Noncentrosymmetric CePt _{3}Si from Magnetic Penetration Depth Measurements. - 13.
Borkowski L. S. Hirschfeld P. J. 1992 Kondo effect in gapless superconductors . - 14.
Borkowski L. S. Hirschfeld P. J. 1994 Distinguishing d-wave superconductors from highly anisotropic s-wave superconductors . - 15.
Dresselhaus G. 1955 Spin-Orbit Coupling Effects in Zinc Blende Structures.Phys. Rev. 100, 580. - 16.
Einzel D. Hirschfeld P. J. Gross F. Chandrasekhar B. S. Andres K. Ott H. R. Beuers J. Fisk Z. Smith J. L. 1986 Magnetic Field Penetration Depth in the Heavy-Electron Superconductor UBe_{13}. - 17.
Edelstein V. M. 1995 Magnetoelectric Effect in Polar Superconductors. - 18.
Frigeri P. A. Agterberg D. F. Sigrist M. 2004 Spin susceptibility in superconductors without inversion symmetry . - 19.
Frigeri P. A. Agterberg D. F. Koga A. Sigrist M. 2004 Superconductivity without Inversion Symmetry: MnSi versus CePt _{3}Si. - 20.
Gor’kov L. P. Rashba E. I. 2001 Superconducting 2D System with Lifted Spin degeneracy: Mixed Singlet-Triplet State. - 21.
Gross F. Chandrasekhar B. S. einzel D. Andres K. Herschfeld P. J. Ott H. R. Beuers J. Fisk Z. Smith J. L. 1986 Anomalous temperature dependence of the magnetic field penetration depth in superconducting UBe ._{13} - 22.
Hayashi N. Wakabayashi K. Frigeri P. A. Sigrist M. 2006 . - 23.
Hayashi N. Wakabayashi K. Frigeri P. A. Sigrist M. 2006 . - 24.
Izawa K. Kasahara Y. Matsuda Y. Behnia K. Yasuda T. Settai R. Onuki Y. 2005 Line Nodes in the Superconducting Gap Function of Noncentrosymmetric CePt _{3}Si. - 25.
Kosztin I. Leggett A. J. 1997 Nonlocal Effects on the Magnetic Penetration Depth in d-Wave Superconductors . - 26.
Mathur N. D. Grosche F. M. Julian S. R. Walker I. R. Freye D. M. Haselwimmer R. K. W. Lonzarich G. G. 1998 Magnetically mediated superconductivity in heavy rmion compounds. - 27.
Mineev V. P. Samokhin K. V. 2007 Effects of impurities on superconductivity in noncentrosymmetric compounds . - 28.
Mineev V. P. Champel T. 2004 Theory of superconductivity in ferromagnetic superconductors with triplet pairing . Phys. - 29.
Rashba E. I. 1960 Properties of Semiconductors with an Extremum Loop I. Cyclotron and Combinatorial Resonance in a Magnetic Field Perpendicular to the Plane of the Loop. - 30.
Rashba E. I. Bychkov Yu. A. 1984 Oscillatory effects and the magnetic susceptibility of carriers in inversion layers . - 31.
Salkalo M. I. Balatsky A. V. Scalapino D. J. 1996 Theory of Scanning Tunneling Microscopy Probe of Impurity States in a D-Wave Superconductor. - 32.
Samokhin K. V. Zijlstra E. S. Bose S. K. 2004 CePt ._{3}Si: An unconventional superconductor without inversion center - 33.
Saxena S. S. Agarwal P. Ahilan K. Grosche F. M. Haselwimmer R. K. W. Steiner M. J. Pugh E. Walker I. R. Julian S. R. Monthoux P. Lonzarich G. G. Huxley A. Sheikin I. Braithwaite D. Flouquet J. 2000 Superconductivity on the border of itinerant-electron ferromagnetism in UGe_{2} - 34.
Sergienko I. A. Curnoe S. H. 2004 Order parameter in superconductors with nondegenerate bands . - 35.
Sergienko I. A. 2004 Mixed-parity superconductivity in centrosymmetric crystals , - 36.
Shiba,H. 1968 Classical Spins in Superconductors. - 37.
Wang Q. H. Wang Z. D. 2004 Impurity and interface bound states in d_{x}^{2}_{-y}^{2}+id_{xy}and p_{x}+ - 38.
Yasuda T. Shishido H. Ueda T. Hashimoto S. Settai R. Takeuchi T. Matsuda T. D. Haga Y. Onuki Y. 2004 Superconducting Property in CePt ._{3}Si under Pressure - 39.
Yogi M. Kitaoka Y. Hashimoto S. Yasuda T. Settai R. Matsuda T. D. Haga Y. Onuki Y. Rogl P. Bauer E. 2004 Evidence for a Novel State of Superconductivity in Noncentrosymmetric CePt _{3}Si: A^{195}Pt-NMR Study. - 40.
Yogi M. Kitaoka Y. Hashimoto S. Yasuda T. Settai R. Matsuda T. D. Haga Y. Onuki Y. Rogl P. Bauer E. 2005 Novel superconductivity in noncentrosymmetric heavy-fermion compound CePt_{3}Si: a^{195}Pt-NMR study.