Effects of impurities on Hc2(T) in superconductors without inversion symmetry
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Effects of impurities on H c ( T ) in superconductors without inversion symmetry K. V. Samokhin
Department of Physics, Brock University, St.Catharines, Ontario L2S 3A1, Canada (Dated: November 29, 2018)We calculate the upper critical field, H c ( T ), due to the orbital pair breaking in disorderedsuperconductors without inversion symmetry. Differences from the usual centrosymmetric case arehighlighted. The linearized gap equations in magnetic field, with the singlet and triplet pairingchannels mixed by impurity scattering, are solved exactly for a cubic crystal. PACS numbers: 74.20.-z, 74.25.Op
I. INTRODUCTION
Recently, superconductivity has been discovered in a number of compounds lacking inversion symmetry, such asCePt Si (Ref. 1), UIr (Ref. 2), CeRhSi (Ref. 3), CeIrSi (Ref. 4), Li (Pd − x Pt x )B (Ref. 5), and many others. Muchof the theoretical work in the field has focussed on searching for the features which are specific to noncentrosymmetricsystems. These include the magnetoelectric effect, a large residual spin susceptibility and reduced paramagneticlimiting, and various novel nonuniform superconducting states. In this paper we study the effects of the absence of inversion symmetry on the upper critical field, H c ( T ), atarbitrary temperature. We assume the pairing to be of the Bardeen-Cooper-Schrieffer (BCS) type, and includeonly the orbital pair breaking. The main qualitative difference from the centrosymmetric case is that the spin-orbit (SO) coupling of electrons with the crystal lattice changes the nature of single-electron states, lifting spindegeneracy of the energy bands. Then even scalar impurities can mix the singlet and triplet channels in the Cooperpair propagator, thus making the theory considerably more complicated. The derivation of the H c equations forarbitrary noncentrosymmetric crystal symmetry is presented in Sec. II below, with some of the technical detailsrelegated to Appendices A and B. In Sec. III, we apply the general equations to a cubic superconductor with thepoint group G = O . Assuming that both the band structure and the SO coupling are fully isotropic, we are able toexactly solve the coupled equations for the singlet and triplet channels, obtain the H c equation in a closed form, andderive analytical expressions for the upper critical field in the “dirty” limit. This isotropic model clearly shows thedeviations from the usual, i.e. centrosymmetric BCS, case, for which the upper critical field was calculated in theclassic papers by Helfand, Werthamer, and Hohenberg in 1960s (Refs. 17,18). Sec. IV contains a discussion of ourresults.The magnetic phase diagram of noncentrosymmetric superconductors has been discussed previously in several works.The upper critical field for a clean three-dimensional Rashba superconductor was calculated in Ref. 15, while theeffects of disorder in the Ginzburg-Landau regime were studied in Ref. 19. Two-dimensional case, in which only theparamagnetic pair breaking is present, was considered in Ref. 20. Recently, H c at all temperatures was calculatedin Ref. 21, neglecting the impurity-induced triplet channel in the pair propagator in the limit when the SO bandsplitting is small compared with the Fermi energy. In this paper, we relax this last condition and include both thesinglet and triplet channels.Throughout the paper we use the units in which ~ = k B = 1. II. DERIVATION OF H c EQUATIONS: GENERAL CASE
Let us consider a noncentrosymmetric superconductor with the Hamiltonian given by H = H + H imp + H int . Thefirst term, H = X k [ ǫ ( k ) δ αβ + γ ( k ) σ αβ ] a † k α a k β , (1)describes non-interacting electrons in the crystal lattice potential, where α, β = ↑ , ↓ are spin indices, ǫ ( k ) is thequasiparticle energy counted from ǫ F , and ˆ σ are the Pauli matrices. In Eq. (1) and everywhere below, summation overrepeated spin indices is implied, while summation over space and band indices is always shown explicitly. The secondterm in Eq. (1), with γ ( k ) = − γ ( − k ), describes a Rashba-type (or antisymmetric) SO coupling of electrons withthe crystal lattice. In addition, there might be a usual (symmetric) SO coupling, present even in centrosymmetriccrystals. If the latter is included, then α, β in Eq. (1) should be interpreted as pseudospin projections. Diagonalizationof H yields two non-degenerate bands labelled by the helicity λ = ± : ξ λ ( k ) = ǫ ( k ) + λ | γ ( k ) | . (2)The Fermi velocities in the two bands are given by v λ ( k ) = ∂ξ λ /∂ k . The Fermi-level densities of states are definedin the usual way by N λ = V − P k δ [ ξ λ ( k )] ( V is the system volume), and the difference between N + and N − ischaracterized by a parameter δ = N + − N − N + + N − . (3)If the SO coupling is small compared with the Fermi energy, then δ ∼ O ( E SO /ǫ F ), where E SO = 2 max k | γ ( k ) | is ameasure of the SO band splitting.Scattering of electrons at isotropic scalar impurities is introduced according to H imp = Z d r U ( r ) ψ † α ( r ) ψ α ( r ) . (4)The random potential U ( r ) has zero mean and is characterized by the correlator h U ( r ) U ( r ′ ) i = n imp U δ ( r − r ′ ),where n imp is the impurity concentration and U has the meaning of the strength of an individual point-like impurity.The field operators are given by ψ α ( r ) = V − / P k e i kr a k α .Neglecting the paramagnetic pair breaking, which is a good assumption in many bulk noncentrosymmetric materials,the effect of a uniform external magnetic field H is described by the Peierls substitution: ˆ h = ǫ ( K ) + γ ( K ) ˆ σ + U ( r ) , (5)where K = − i ∇ + ( e/c ) A ( r ), and e is the absolute value of the electron charge.We describe the pairing interaction by a BCS-like Hamiltonian: H int = − V Z d r ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) , (6)where V > η ( r ), see Ref. 24. The critical temperature at a given field, or inversely the upper critical field, H c ( T ), at a given temperature, is found from the condition that the linearized gap equation (cid:20) V − T X n ′ ˆ X ( ω n ) (cid:21) η ( r ) = 0 (7)has a nontrivial solution. Here ω n = (2 n + 1) πT is the fermionic Matsubara frequency, the prime in the second termmeans that the summation is limited to | ω n | ≤ ω c , where ω c is the BCS frequency cutoff, and the operator ˆ X ( ω n ) isdefined by the following kernel: X ( r , r ′ ; ω n ) = 12 (cid:10) tr ˆ g † ˆ G ( r , r ′ ; ω n )ˆ g ˆ G T ( r , r ′ ; − ω n ) (cid:11) imp , (8)where ˆ g = i ˆ σ . The angular brackets denote the impurity averaging, and ˆ G ( r , r ′ ; ω n ) is the Matsubara Green’sfunctions of electrons in the normal state, which satisfies the equation( iω n − ˆ h ) ˆ G ( r , r ′ ; ω n ) = δ ( r − r ′ ) , (9)where the single-particle Hamiltonian ˆ h is given by expression (5).At zero field, Eq. (9) yields the following expression for the average Green’s function:ˆ G ( k , ω n ) = X λ = ± ˆΠ λ ( k ) G λ ( k , ω n ) , (10)where ˆΠ λ ( k ) = 1 + λ ˆ γ ( k ) ˆ σ δαβ γ g + βα γδµ ν g + ... ρ σ + g g + FIG. 1: Impurity ladder diagrams in the Cooper channel. Lines with arrows correspond to the average Green’s functions ofelectrons, ˆ g = i ˆ σ , and the impurity (dashed) lines are defined in the text, see Eq. (15). are the band projection operators (ˆ γ = γ / | γ | ), and G λ ( k , ω n ) = 1 iω n − ξ λ ( k ) + i Γ sign ω n , (12)are the electron Green’s functions in the band representation. Here ξ λ ( k ) is the quasiparticle dispersion in the λ thband, see Eq. (2), Γ = 1 / τ is the elastic scattering rate, τ = (2 πn imp U N F ) − is the electron mean free time due toimpurities, and N F = N + + N − . (13)The impurity average of the product of two Green’s functions in Eq. (8) can be represented graphically by theladder diagrams, see Fig. 1. We assume the disorder to be sufficiently weak for the diagrams with crossed impuritylines to be negligible, see Ref. 25. In order to solve Eq. (7) at nonzero field, we introduce an impurity-renormalizedgap function ˆ D ( r , ω n ), which a matrix in the spin space satisfying the following integral equation:ˆ D ( r , ω n ) = η ( r )ˆ g + 12 n imp U ˆ g Z d r ′ tr ˆ g † ˆ G ( r , r ′ ; ω n ) ˆ D ( r ′ , ω n ) ˆ G T ( r , r ′ ; − ω n )+ 12 n imp U ˆ g Z d r ′ tr ˆ g † ˆ G ( r , r ′ ; ω n ) ˆ D ( r ′ , ω n ) ˆ G T ( r , r ′ ; − ω n ) , (14)where ˆ G ( r , r ′ ; ω n ) are the disorder-averaged solutions of Eq. (9). The above equation can be easily derived from theimpurity ladder diagrams in Fig. 1, by representing each “rung” of the ladder as a sum of spin-singlet and spin-tripletterms: n imp U δ µν δ ρσ = 12 n imp U g µρ g † σν + 12 n imp U g µρ g † σν , (15)where ˆ g = i ˆ σ ˆ σ .Seeking solution of Eq. (14) in the formˆ D ( r , ω n ) = d ( r , ω n )ˆ g + d ( r , ω n )ˆ g , (16)we obtain a system of four integral equations for d a ( r , ω n ), where a = 0 , , , X b =0 (cid:2) δ ab − Γ ˆ Y ab ( ω n ) (cid:3) d b ( r , ω n ) = η ( r ) δ a . (17)Here the operators ˆ Y ab ( ω n ) are defined by the kernels Y ab ( r , r ′ ; ω n ) = 12 πN F tr ˆg † a ˆ G ( r , r ′ ; ω n )ˆg b ˆ G T ( r , r ′ ; − ω n ) , (18)with ˆg = ˆ g , and ˆg i = ˆ g i for i = 1 , ,
3. We see that, in addition to the spin-singlet component d ( r , ω n ), impurityscattering can induce also a nonzero spin-triplet component d ( r , ω n ). The gap equation (7) contains only the singletcomponent: Using Eqs. (17), we obtain:1 N F V η ( r ) − πT X n ′ d ( r , ω n ) − η ( r )Γ = 0 . (19)We would like to note that the triplet component does not appear in the centrosymmetric case. Indeed, in the absenceof the Zeeman interaction the spin structure of the Green’s function is trivial: G αβ ( r , r ′ ; ω n ) = δ αβ G ( r , r ′ ; ω n ). Thenit follows from Eq. (18) that ˆ Y ab ( ω n ) = δ ab ˆ Y ( ω n ), therefore d = (1 − Γ ˆ Y ) − η and d = 0.The next step is to find the spectrum of the operators ˆ Y ab ( ω n ). The orbital effect of the magnetic field is described bya phase factor in the average electron Green’s function: ˆ G ( r , r ′ ; ω n ) = ˆ G ( r − r ′ ; ω n ) e iϕ ( r , r ′ ) , where ˆ G is the averageGreen’s function in the normal state at zero field, ϕ ( r , r ′ ) = ( e/c ) R r ′ r A ( r ) d r , and the integration is performed alonga straight line connecting r and r ′ (Ref. 25). The “phase-only” approximation is legitimate if the temperature is notvery low, so that the Landau level quantization can be neglected. Using the identity e iϕ ( r , r ′ ) η ( r ′ ) = e − i ( r − r ′ ) D η ( r ),where D = − i ∇ + (2 e/c ) A , we obtain: ˆ Y ab ( ω n ) = ¯ Y ab ( q , ω n ) (cid:12)(cid:12) q → D , (20)where ¯ Y ab ( q , ω n ) = 12 πN F Z d k (2 π ) tr ˆg † a ˆ G ( k + q , ω n )ˆg b ˆ G T ( − k , − ω n ) , (21)Substituting here the Green’s functions (10) and calculating the spin traces, we obtain for the singlet-singlet term:¯ Y ( q , ω n ) = 12 X λ ρ λ (cid:28) | ω n | + Γ + i v λ ( k ) q sign ω n / (cid:29) λ , (22)where ρ ± = N ± N F = 1 ± δ (23)are the fractional densities of states in the two bands, and h ( ... ) i λ denotes the Fermi-surface averaging in the λ thband. Similarly, for the singlet-triplet mixing terms we obtain:¯ Y i ( q , ω n ) = ¯ Y i ( q , ω n ) = 12 X λ λρ λ (cid:28) ˆ γ i ( k ) | ω n | + Γ + i v λ ( k ) q sign ω n / (cid:29) λ . (24)We see that the mixing occurs due to the SO coupling and vanishes at γ →
0, when ρ + = ρ − = 1 and v + = v − = v F .Finally, the triplet-triplet terms can be represented as follows:¯ Y ij ( q , ω n ) = ¯ Y (1) ij ( q , ω n ) + ¯ Y (2) ij ( q , ω n ) , (25)where ¯ Y (1) ij ( q , ω n ) = 12 X λ ρ λ (cid:28) ˆ γ i ( k )ˆ γ j ( k ) | ω n | + Γ + i v λ ( k ) q sign ω n / (cid:29) λ , (26)and ¯ Y (2) ij ( q , ω n ) = 12 πN F X λ Z d k (2 π ) ( δ ij − ˆ γ i ˆ γ j − iλe ijl ˆ γ l ) G λ ( k + q , ω n ) G − λ ( − k , − ω n ) . (27)The singlet impurity scattering channel, which is described by the first term in expression (15), causes only thescattering of intraband pairs between the bands. In contrast, the triplet impurity scattering can create also interbandpairs, which are described by ¯ Y (2) ij . It is easy to show that if the SO band splitting exceeds both ω c and Γ, then thesecond (interband) term in Eq. (25) is smaller than the first (intraband) one, see Appendix A. Note that in realmaterials, E SO ranges from tens to hundreds meV, see Ref. 26 for CePt Si, and Ref. 27 for Li Pd B and Li Pt B. Onthe other hand, there is still considerable uncertainty as to the values of ω c , especially in heavy-fermion compounds,such as CePt Si. The typical energy of phonons responsible for the pairing in Li Pd B was estimated in Ref. 27 tobe 20 meV, while the SO band splitting is 30 meV (reaching 200 meV in Li Pt B).The critical temperature of the phase transition into a uniform superconducting state at zero field can be found bysetting q = 0 in the above expressions. According to Eq. (24), the singlet and triplet channels are decoupled. Thenit follows from Eqs. (22) and (17) that d ( ω n ) = (1 + Γ / | ω n | ) η . Substituting this into Eq. (19), we obtain:1 N F V − πT X n ′ | ω n | = 0 , (28)which yields the superconducting critical temperature: T c = 2 e C π ω c e − /N F V , (29)where C ≃ .
577 is Euler’s constant. We see that there is an analog of Anderson’s theorem in noncentrosymmetricsuperconductors with a BCS-contact pairing interaction: The zero-field critical temperature is not affected by scalardisorder. In the presence of magnetic field, neglecting the interband contributions to the triplet pair propagator, we obtain:¯ Y ab ( q , ω n ) = 12 X λ ρ λ (cid:28) Λ λ,a ( k )Λ λ,b ( k ) | ω n | + Γ + i v λ ( k ) q sign ω n / (cid:29) λ , (30)where Λ λ,a ( k ) = (cid:26) , a = 0 λ ˆ γ a ( k ) , a = 1 , , . (31)Next we use in Eq. (30) the identity x − = R ∞ du e − xu , and make the substitution q → D , see Eq. (20), in theexponent to represent ˆ Y ab ( ω n ) as a differential operator of infinite order:ˆ Y ab ( ω n ) = 12 Z ∞ du e − u ( | ω n | +Γ) X λ ρ λ ˆ O abλ , (32)where ˆ O abλ = D Λ λ,a ( k )Λ λ,b ( k ) e − iu v λ ( k ) D sign ω n / E λ . (33)In order to solve Eqs. (17), with the operators ˆ Y ab ( ω n ) given by expressions (32), we follow the procedure describedin Ref. 17. We choose the z -axis along the external field, so that H = H ˆ z , and introduce the operators a ± = ℓ H D x ± iD y , a = ℓ H D z , (34)where ℓ H = p c/eH is the magnetic length. It is easy to check that a + = a †− and [ a − , a + ] = 1, therefore a ± havethe meaning of the raising and lowering operators, while a = a † commutes with both of them: [ a , a ± ] = 0. Itis convenient to expand both the order parameter η and the impurity-renormalized gap functions d a in the basis ofLandau levels | N, p i , which satisfy a + | N, p i = √ N + 1 | N + 1 , p i , a − | N, p i = √ N | N − , p i , a | N, p i = p | N, p i , (35)where N = 0 , , ... , and p is a real number. We have η ( r ) = X N,p η N,p h r | N, p i , d a ( r , ω n ) = X N,p d aN,p ( ω n ) h r | N, p i . (36)According to Eqs. (17), the expansion coefficients satisfy the following algebraic equations: X N ′ ,p ′ ,b h δ ab δ NN ′ δ pp ′ − Γ h N, p | ˆ Y ab ( ω n ) | N ′ , p ′ i i d bN ′ ,p ′ ( ω n ) = δ a η N,p . (37)Substituting the solutions of these equations into1 N F V η
N,p − πT X n ′ d N,p ( ω n ) − η N,p
Γ = 0 , (38)see Eq. (19), and setting the determinant of the resulting linear equations for η N,p to zero, one arrives at an equationfor the upper critical field.
III. CUBIC CASE
In the general case, i.e. for arbitrary crystal symmetry and electronic band structure, the procedure outlined in theprevious section does not yield an equation for H c ( T ) in a closed form, since all the Landau levels are coupled, andone has to diagonalize infinite matrices. In order to make progress, we focus on the case of a noncentrosymmetric cubicsuperconductor with the point group G = O , which describes, for instance, the crystal symmetry of Li (Pd − x ,Pt x ) B.The simplest expression for the SO coupling compatible with all symmetry requirements has the following form: γ ( k ) = γ k , (39)where γ is a constant. We assume a parabolic band: ǫ ( k ) = k / m ∗ − ǫ F , where m ∗ is the effective mass, ǫ F = k / m ∗ , and k is the Fermi wave vector in the absence of the SO coupling. The band dispersion functions aregiven by ξ λ ( k ) = k − k m ∗ + λ | γ | k, (40)so that the SO band splitting is isotropic and given by E SO = 2 | γ | k . It is convenient to characterize the SOcoupling strength by a dimensionless parameter ̺ = E SO / ǫ F . While the two Fermi surfaces have different radii: k F,λ = k ( p ̺ − λ̺ ), the Fermi velocities are the same: v λ ( k ) = v F ˆ k , where v F = k p ̺ /m ∗ . Forthe parameter δ , which characterizes the difference between the band densities of states, see Eq. (3), we have | δ | = 2 ̺ p ̺ / (1 + 2 ̺ ). We assume that δ c ≪ | δ | ≤ , (41)where δ c = max( ω c , Γ) /ǫ F ≪
1. While the first inequality is equivalent to the condition E SO ≫ max( ω c , Γ), whichensures the smallness of the interband contribution to the Cooper impurity ladder (see Appendix A), the second oneis always satisfied, with | δ | → ̺ → ∞ .In order to solve the gap equations, we make a change of variables in the triplet component: d ± ( r , ω n ) = d ( r , ω n ) ± id ( r , ω n ) √ . Then, Eqs. (17) take the following form: − Γ ˆ Y − Γ ˆ Y − Γ ˆ Y − − Γ ˆ Y − Γ ˆ Y − Γ ˆ Y − Γ ˆ Y − − Γ ˆ Y − Γ ˆ Y − Γ ˆ Y − Γ ˆ
Z −
Γ ˆ Z + − Γ ˆ Y − − Γ ˆ Y − − Γ ˆ Z − − Γ ˆ Z d d d + d − = η , (42)where ˆ Y ± = ˆ Y ± i ˆ Y √ , ˆ Y ± = ˆ Y ± i ˆ Y √ , ˆ Z = ˆ Y + ˆ Y , ˆ Z ± = ˆ Y ± i ˆ Y − ˆ Y , with ˆ Y ab = ˆ Y ba given by Eqs. (32).According to Sec. II, one has to know the matrix elements of the operators ˆ Y ab ( ω n ) in the basis of the Landaulevels | N, p i . After some straightforward algebra, see Appendix B, we obtain the following expressions for the nonzeromatrix elements: h N, p | ˆ Y ( ω n ) | N, p i = y N,p ( ω n ) , h N, p | ˆ Y ( ω n ) | N, p i = y N,p ( ω n ) , h N, p | ˆ Y ( ω n ) | N, p i = y N,p ( ω n ) , h N, p | ˆ Z ( ω n ) | N, p i = z N,p ( ω n ) , where y N,p ( ω n ) = Z ∞ du e − u ( | ω n | +Γ) Z ds cos( pvs ) e − v (1 − s ) / L N [ v (1 − s )] , (43) y N,p ( ω n ) = − iδ Z ∞ du e − u ( | ω n | +Γ) Z ds s sin( pvs ) e − v (1 − s ) / L N [ v (1 − s )] , (44) y N,p ( ω n ) = Z ∞ du e − u ( | ω n | +Γ) Z ds s cos( pvs ) e − v (1 − s ) / L N [ v (1 − s )] , (45) z N,p ( ω n ) = 12 Z ∞ du e − u ( | ω n | +Γ) Z ds (1 − s ) cos( pvs ) e − v (1 − s ) / L N [ v (1 − s )] , (46) v = ( v F sign ω n / ℓ H ) u , and L N ( x ) are the Laguerre polynomials of degree N . Similarly, we obtain: h N, p | ˆ Y − ( ω n ) | N + 1 , p i = h N + 1 , p | ˆ Y ( ω n ) | N, p i = ˜ y N,p ( ω n ) , h N, p | ˆ Y − ( ω n ) | N + 1 , p i = h N + 1 , p | ˆ Y ( ω n ) | N, p i = ˜ y N,p ( ω n ) , h N, p | ˆ Z − ( ω n ) | N + 2 , p i = h N + 2 , p | ˆ Z + ( ω n ) | N, p i = ˜ z N,p ( ω n ) , where ˜ y N,p ( ω n ) = − iδ p N + 1) Z ∞ du e − u ( | ω n | +Γ) Z ds v (1 − s ) cos( pvs ) e − v (1 − s ) / L (1) N [ v (1 − s )] , (47)˜ y N,p ( ω n ) = − p N + 1) Z ∞ du e − u ( | ω n | +Γ) Z ds vs (1 − s ) sin( pvs ) e − v (1 − s ) / L (1) N [ v (1 − s )] , (48)˜ z N,p ( ω n ) = − p ( N + 1)( N + 2) Z ∞ du e − u ( | ω n | +Γ) Z ds v (1 − s ) cos( pvs ) e − v (1 − s ) / L (2) N [ v (1 − s )] . (49)and L ( α ) N ( x ) are the generalized Laguerre polynomials.It follows from the above expressions that the Landau levels are decoupled, and for η ( r ) = η h r | N, p i ( η is a constant)the solution of Eqs. (42) has the following form: d ( r , ω n ) d ( r , ω n ) d + ( r , ω n ) d − ( r , ω n ) = d N,p ( ω n ) h r | N, p i d N,p ( ω n ) h r | N, p i d + N,p ( ω n ) h r | N + 1 , p i d − N,p ( ω n ) h r | N − , p i . (50)For given N and p , the coefficients are found from the equations X b =0 , , ± M ab ( N, p ; ω n ) d bN,p ( ω n ) = δ a η, (51)where ˆ M ( N, p ; ω n ) = − Γ y N,p − Γ y N,p − Γ˜ y N,p − Γ˜ y N − ,p − Γ y N,p − Γ y N,p − Γ˜ y N,p − Γ˜ y N − ,p − Γ˜ y N,p − Γ˜ y N,p − Γ z N +1 ,p − Γ˜ z N − ,p − Γ˜ y N − ,p − Γ˜ y N − ,p − Γ˜ z N − ,p − Γ z N − ,p . (52)Substituting the solution of Eq. (51) in Eq. (38), and using Eqs. (28) and (29) to eliminate both the frequency cutoffand the coupling constant, we obtain an equation implicitly relating the magnetic field and the transition temperatureat given N and p : ln T c T = πT X n ( | ω n | − [ ˆ M − ( N, p ; ω n )] − ) . (53)The upper critical field, H c ( T ), is obtained by maximizing the solution of this equation with respect to both N and p .Note that the matrix elements of ˆ M which are responsible for the singlet-triplet mixing, i.e. y N,p , ˜ y N,p , and˜ y N − ,p , are all proportional to δ , see Eqs. (44) and (47). Therefore, at | δ | ≪ δ , we obtain from Eq. (51) that [ ˆ M − ( N, p ; ω n )] =(1 − Γ y N,p ) − . Substituting this into Eq. (53), we recover the Helfand-Werthamer expressions, with the maximumcritical field corresponding to N = p = 0 at all temperatures. Thus, in the weak SO coupling limit the absence ofinversion symmetry does not bring about any new features in H c ( T ), compared with the centrosymmetric case (aslong as the paramagnetic pair breaking is not included, see Ref. 21). A. “Dirty” limit at N = 0 , p = 0 At arbitrary magnitude of the SO band splitting, the singlet-triplet mixing makes the H c equation in noncen-trosymmetric superconductors considerably more cumbersome than in the Helfand-Werthamer problem, even inour “minimal” isotropic model. It is even possible that, at some values of the parameters, the maximum crit-ical field is achieved for N > p = 0, the latter corresponding to a disorder-induced modulation of theorder parameter along the applied field. Leaving investigation of these exotic possibilities to future work, herewe just consider the case N = p = 0. Then it follows from Eqs. (50) and (51) that d , = d − , = 0, and[ ˆ M − (0 , ω n )] = (1 − Γ z , ) / [(1 − Γ y , )(1 − Γ z , ) − Γ (˜ y , ) ]. It is convenient to introduce the reduced temper-ature, magnetic field, and disorder: t = TT c , h = 2 HH , ζ = Γ πT c , where H = Φ /πξ , Φ = πc/e is the magnetic flux quantum, and ξ = v F / πT c is the superconducting coherencelength. In these notations, Eq. (53) yields the following equation for the upper critical field h c ( t ):ln 1 t = 2 X n ≥ (cid:20) n + 1 − t w n (1 − ζp n ) − ζδ q n (1 − ζw n )(1 − ζp n ) + ζ δ q n (cid:21) , (54)where w n ( t, h ) = Z ∞ dρ e − ˜ ω n ρ Z ds e − hρ (1 − s ) / ,p n ( t, h ) = Z ∞ dρ e − ˜ ω n ρ Z ds − s (cid:20) − h ρ (1 − s ) (cid:21) e − hρ (1 − s ) / , (55) q n ( t, h ) = Z ∞ dρ e − ˜ ω n ρ Z ds r h ρ (1 − s ) e − hρ (1 − s ) / , where ˜ ω n = (2 n + 1) t + ζ .In the clean limit, i.e. at ζ →
0, or if the SO band splitting is negligibly small, i.e. at δ →
0, one recovers from Eq.(54) the Helfand-Werthamer equation for a centrosymmetric superconductor. Thus the absence of inversion symmetryaffects the upper critical field only if disorder is present. One can expect that the effect will be most pronounced inthe “dirty” limit, ζ ≫
1. [Note that, according to Eq. (41), the disorder strength should satisfy ζ ≪ ( ǫ F /T c ) | δ | .] Weshall see that in this limit h c scales as ζ , which allows one to use the Taylor expansions of the exponentials in Eqs.(55): w n ( t, h ) ≃ ω n (cid:18) − h ω n (cid:19) , p n ( t, h ) ≃ ω n (cid:18) − h ω n (cid:19) , q n ( t, h ) ≃ √ h ω n . Using the fact that the main contribution to the Matsubara sum in Eq. (54) comes from (2 n + 1) t ≪ ζ , we arrive ata well-known universal equation, which describes the magnetic pair breaking in superconductors: ln 1 t = Ψ (cid:18)
12 + σt (cid:19) − Ψ (cid:18) (cid:19) , (56)where Ψ( x ) is the digamma function, and σ = 2 + δ ζ h (57)characterizes the pair-breaker strength. Note that the corresponding expression in the centrosymmetric case is differ-ent: σ CS = h/ ζ (Ref. 17). Analytical expressions for the upper critical field can be obtained in the weak-field limitnear the critical temperature: h c | t → = 24 ζ (2 + δ ) π (1 − t ) , (58)and also at low temperatures: h c | t =0 = 3 e − C δ ζ. (59)We see that the SO band splitting in the noncentrosymmetric case enhances the orbital pair breaking. IV. CONCLUSIONS
We have derived equations for the upper critical field in noncentrosymmetric superconductors, assuming a BCS-contact pairing interaction and orbital pair breaking. In a cubic crystal (the point group G = O ), in which boththe electron dispersion and the SO coupling are isotropic, the gap equations are shown to be diagonal in the Landaulevel basis, with the singlet and triplet channels in the pair propagator mixed together. For the order parametercorresponding to the lowest Landau level, without any modulation along the applied field, we have derived the H c equation in a closed form and solved it in the “dirty” limit, in which the effects of the absence of inversion symmetryare expected to be most pronounced. The effect on the upper critical field of the singlet-triplet mixing, which isresponsible for the deviations from the Helfand-Werthamer theory, is found to be proportional to δ .Application of our theory to real noncentrosymmetric superconductors of cubic symmetry, such as Li (Pd − x Pt x )B,is complicated by the fact that the Fermi surfaces as well as the SO band splitting are strongly anisotropic. Usingthe maximum values of the SO band splitting from Ref. 27, one can estimate the corrections to H c ( T ) due to thesinglet-triplet mixing to be of the order of several percent. Acknowledgments
This work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council ofCanada.
APPENDIX A: INTERBAND VS INTRABAND CONTRIBUTIONS
In this appendix, we estimate the relative magnitudes of the intraband and interband contributions to the tripletpair propagator, Eq. (25), in the limit when the SO coupling is strong compared with both the cutoff energy ω c andthe elastic scattering rate Γ. Let us consider an isotropic band with γ ( k ) = γ k in a cubic crystal. Neglecting forsimplicity the differences between the densities of states and the Fermi velocities in the two bands: ρ + = ρ − = 1 and v + = v − = v F , and setting q = 0, we obtain from Eqs. (26) and (27):¯ Y (1) ij ( q = 0 , ω n ) = h ˆ γ i ˆ γ j i ˆ k | ω n | + Γ = δ ij | ω n | + Γ) ≡ Y intra ( ω n ) δ ij , and ¯ Y (2) ij ( q = 0 , ω n ) = 12 X λ (cid:28) δ ij − ˆ γ i ˆ γ j | ω n | + Γ + iλ | γ | sign ω n (cid:29) ˆ k = 2 δ ij | ω n | + Γ)(1 + r ) ≡ Y inter ( ω n ) δ ij , where r ( ω n ) = E SO / | ω n | + Γ). Due to the BCS cutoff, the maximum value of ω n is equal to ω c , therefore r min ∼ E SO / max( ω c , Γ) ≫
1. From this it follows thatmax n Y inter ( ω n ) Y intra ( ω n ) = 21 + r min ∼ (cid:20) max( ω c , Γ) E SO (cid:21) ≪ . Therefore the interband contribution is small compared with the intraband one, at all Matsubara frequencies.0
APPENDIX B: CALCULATION OF h N, p | ˆ Y ab ( ω n ) | N ′ , p ′ i The operators ˆ Y ab ( ω n ) are given by expressions (32). For a spherical Fermi surface and γ ( k ) = γ k , we obtainfrom Eq. (33): ˆ O abλ = 12 Z π dθ sin θ e − iva cos θ Z π dφ π Φ abλ ( θ, φ ) e − iv ( e − iφ a + + e iφ a − ) sin θ , (B1)where v = ( v F sign ω n / ℓ H ) u , and Φ abλ ( θ, φ ) = Λ λ,a ( k )Λ λ,b ( k ), with Λ λ, ( k ) = 1 and Λ λ,i ( k ) = λ ˆ k i for i = 1 , ,
3, seeEq. (31). Using the well-known operator identity e A + B = e − [ A,B ] / e A e B , which holds if the commutator of A and B is a c -number, and expanding the exponentials in powers of a ± , we obtain:ˆ Y ab ( ω n ) = 14 X λ ρ λ Z ∞ du e − u ( | ω n | +Γ) Z π dθ sin θ e − iva cos θ e − ( v /
2) sin θ ˆ L abλ ( θ ) , (B2)where ˆ L abλ ( θ ) = ∞ X n,m =0 ( − iv sin θ ) n + m n ! m ! (cid:20)Z π dφ π Φ abλ ( θ, φ ) e i ( m − n ) φ (cid:21) a n + a m − . (B3)Below we perform the detailed calculations for ˆ Y and ˆ Y − = ( ˆ Y − i ˆ Y ) / √
2. Other matrix elements can beconsidered in a similar fashion.ˆ Y : Since Φ λ ( θ, φ ) = 1, the φ -integral on the right-hand side of Eq. (B3) is equal to δ nm , andˆ L λ ( θ ) = ∞ X n =0 ( − v sin θ ) n ( n !) a n + a n − . It is easy to show, using Eqs. (35), that a n + a n − | N, p i = [ N ! / ( N − n )!] | N, p i for n ≤ N , and zero otherwise. Therefore,ˆ L λ ( θ ) | N, p i = N X n =0 N !( n !) ( N − n )! ( − v sin θ ) n | N, p i = L N ( v sin θ ) | N, p i , where L N ( x ) is the Laguerre polynomial of degree N . Substituting this into Eq. (B2), using the fact that ρ + + ρ − = 2,and introducing s = cos θ , we obtain: h N, p | ˆ Y ( ω n ) | N, p i = y N,p ( ω n ), where y N,p ( ω n ) is given by Eq. (43).ˆ Y − : Since Φ − λ ( θ, φ ) = λ sin θe − iφ / √
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