1. Introduction
In the past two decades, a number of novel superconducting materials have been discovered where order parameter symmetries are different from an
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-
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
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
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
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
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 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 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 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. 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
For mixing of
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 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
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 CePt3Si 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
Now the effect of impurities when both
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
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
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
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
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 below
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 temperature
Acknowledgments
I wish to thank the Office of Graduate Studies and Research Vice President of theUniversity of Isfahan for their support.
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