The p-wave superconductivity in the presence of Rashba interaction in 2DEG

We investigate the effect of the Rashba interaction on two dimensional superconductivity. The presence of the Rashba interaction lifts the spin degeneracy and gives rise to the spectrum of two bands. There are intraband and interband pairs scattering which result in the coupled gap equations. We find that there are isotropic and anisotropic components in the gap function. The latter has the form of cos φk where . The former is suppressed because the intraband and the interband scatterings nearly cancel each other. Hence, −the system should exhibit the p-wave superconductivity. We perform a detailed study of electron-phonon interaction for 2DEG and find that, if only normal processes are considered, the effective coupling strength constant of this new superconductivity is about one-half of the s-wave case in the ordinary 2DEG because of the angular average of the additional in the anisotropic gap function. By taking into account of Umklapp processes, we find they are the major contribution in the electron-phonon coupling in superconductivity and enhance the transition temperature Tc.

Topological superconductivity with the Majorana edge channels was suggested to appear in noncentrosymmetric superconductors 30 . For electron system with Rashba interaction, in addition to charge plasmon, the chiral spin modes and their mutual coupling were investigated in ref. 31. Electron transport in p-wave superconductor-normal metal junctions affected by interface SOI was studied in ref. 32. It was suggested that certain p-wave electron pairs can be tuned via the SOI and tunnel to the normal metal at a distance longer than mean free path of singlet pairing electrons.
In the presence of the Rashba interaction, the superconducting gap function depends on momentum p through its phase, Δ p = exp(− iϕ p )Δ 0 where Δ 0 is an isotropic gap energy, as derived in ref. 24. However, an approximation on the interaction potential had been made. We find that without the approximation, the magnitude of gap function is modulated by the extra cos ϕ p factor and is anisotropic. We also make a detailed analysis of the effective interaction between electrons mediated by phonons. The results is summarized as the following. There are cancellations between different channels of scattering if the interaction is spin-independent. Approximation has to be made carefully in order to isolate the terms of cancellation. As a result, we find that the gap function is not only gauge-dependent but also has a factor ϕ = cos p p p x . Thus, the p-wave gap dominates in the presence of Rashba interaction. The interaction strength of p-wave superconductivity is only half of that of conventional BCS s-wave superconductivity. Hence, its calculated T c is as low as 0.6 K for lead film if only normal processes are considered. Only by including Umklapp processes, our results are comparable with experimental results.
This article is organized as following.The first section is the introduction. In the second section, we analyze Hamitonian in the Rashba eigen-spinor basis and obtain two coupled gap equations. In the third section, the gap equations are analyzed. The phonon mediated interaction and direct Coulomb interaction contributions are discussed separately. The corresponding dimensionless coupling strength constants λ and μ* are defined while solving the gap equations and they can be estimated by generalizing the discussion in ref. 33 to the 2DEG case. In the fourth section, we estimate the effective electron-phonon coupling constant by the model proposed by Scalapino et al. under the strong coupling approximation [34][35][36] and Umklapp processes are considered. In the fifth section, we suggest that the p-wave superconductivity can be oberved in certain experiments. A conclusion is given in the last section.

The Effect of Rashba Interaction in Superconductivity
In this section, we derive the effective Hamiltonian of superconductivity in the Rashba eigen-spinor basis. By diagonalizing the effective Hamiltonian, we can write down the ground state wave function and obtain the two coupled gap equations.
Hamiltonian in the Rashba eigen-spinor basis. The 2D model Hamiltonian for the system with screened Coulomb interaction and electron-phonon interaction in the second quantization form is given by Ω is the area of the system. The first term of H int is the Coulomb interaction between electrons. π = + V e q q 2 (4) C TF q 2 is the 2D screened electrostatic Coulomb potential energy where q TF is the Thomas-Fermi wave vector and is given by is the electron density of states at the the Fermi level. The second term of H int is the electron-phonon cou- . N c is the atomic density, Z is the valence of the ions, M c is the mass of an ion and ω q is 2D dressed phonon frequency.
The Rashba interaction mixes the spin-up and spin-down states of the free electrons.
, is the creation(annihilation) operator of the electron in the σ band with momentum k. σ = + and σ = − represent the χ + and χ − spinor states respectively. These second quantized operators satisfy the commutation , , , , and = = .
The kinetic energy of the system relative to the Fermi level μ is Combining the Coulomb interaction and electron-phonon interaction, the effective electron-electron interacting Hamiltonian is given as where V C q is the screened Coulomb potential in Eq. (4) and the phonon-mediated potential energy V ph q is In the effective interaction Hamiltonian, there are interband and intraband interaction. However, while considering the scattering of the pairing electrons near the Fermi surface, the most important pairing configuration is that paired electrons being in the same band. The spinor bands and pairing of electrons are shown in Fig. 1. The zero-momentum pairing states uniformly distributed in either one of the two Rashba bands at Fermi level as shown in Fig. 1(b,c). The electrons in the same band can be scattered into any other unoccupied pairing states of zero momentum. This kind of scattering is the dominant scattering channel in superconductivity. Two electrons from different bands cannot form a pair with zero-momentum as shown in Fig. 1   The second and third terms represent the intraband and interband pair scattering respectively. Here, the intraband scattering means interacting Cooper pairs stay in the same band and interband scattering means interacting Cooper pairs are in the different bands as shown in Fig. 1(b,c). We note that there are factors 1 ± cos(ϕ k − ϕ l ) in the second and third terms. These factors favor p-wave superconductivity as shown below.
The two-band coupled gap equations. The Hamiltonian in Eq. (14) is to be diagonalized. The expectation value of pair creation operators on the ground state is defined as The Hamiltonian in Eq. (14) is expanded with respect to the fluctuations (a −lσ a lσ − A lσ ) up to first order. Then we obtain where ∑ ϕ ϕ ϕ ϕ with constraints |u k,σ | 2 + |ν k,σ | 2 = 1, u −k,σ = u k,σ and v −k,σ = − v k,σ 37 . The Fermi statistics of the γ operators satisfy . The mean field Hamiltonian in Eq. (16) can be diagonalized as 2 is the excitation energy of quasi-particles. Δ k,σ is the gap energy.
. Hence, the ground state wave function is We note in passing that ϕ k i should be in the range π ϕ > ≥ 0 k i in ground state wave function, or the ground state can not be properly normalized. If the range of ϕ k is from 0 to 2π, then the wave function Because Δ k,σ is an odd function of k, it can also be written as factor in the first summation term means the cancellation between intraband and interband pair scattering contribution and its contribution to the gap energy almost vanishes. It will be shown later in Fig. 2 Comparison with previous theoretical investigation. In As shown in ref. 24.
where U(|p − p′|) is the interaction strength and the scalar product of spinors is equal to The same definition of Green functions in ref. 24 read where τ λ  a p ( , ) is a λ (p) in the Heisenberg representation. With τ − τ′ = 0 + , we find and The same processes were given in Eqs (7-13) of ref. 24. f λ (p) can be related to the mean field A pλ we used in Eq. (15), From the Gor'kov equations in Eqs (37,38), the gap function without approximation can be written as  With the approximation  We have to note that both assumptions, ζ = 1 and U(|p − p′|) ≈ U(|p + p′|) ≈ U(0) are necessary for the cos(ϕ k − ϕ l ) term to vanish. But, the isotropic assumption U(|p − p′|) ≈ U(|p + p′|) ≈ U(0) is usually harmful to traditional triplet p-wave superconductivity and p-wave superfluidity [37][38][39][40] . The consequence of the approximation is the cancellation in Eq. (44). The definition of f λ (p, τ − τ′ ) in Eq. (32) results in a factor λ in front of the bracket. Hence, f + (p) and f − (p) have opposite signs. The summation over λ in Eq. (44) produces cancellation. The resulting gap function is greatly suppressed. Therefore, we concluded that the approximation U(|p − p′|) ≈ U (|p + p′|) ≈ U(0) of Eq. (14) in ref. 24 is not a good approximation for spin-independent interaction which we are dealing with in this work. On the other hand, the second term in the brace on the right hand side in Eq. (40) has two terms with opposite signs. We take advantage of that and reach Eq. (25). Our choice of phase of Rashba basis enable us to identify the possible cancellation as shown in Eq. (24). As a result, the gap equation Eq. (25) contains clearly the dominant interaction without any cancellation.

The analysis of the interaction
In this section, we obtain the p-wave like gap energies (i.e., an additional cos(ϕ p ) modulation to the gap function comparing with Eq. (43)) through the analysis of the gap equations. There are two dimensionless coupling constants. One is for electron-phonon interaction and the other is for electron-electron interaction. Following a procedure similar to the model proposed by Morel and Anderson 33 , one can evaluate the gap energies. The needed parameters can be found in Table 1. The finite temperature case and transition temperature are also discussed. The evolution of transition temperature relative to Rashba strength for Pb film on Si(111) is also shown.   The superconducting state parameters. In this subsection, we give the outlines of how the electron-phonon interaction and Coulomb interaction are considered. The details can be found in Appendix C of Supplementary Information. For the free electron case, both spin-up and spin-down electrons have the equivalent contribution to the gap energy. However, in the presence of the Rashba interaction, the spin degeneracy is lifted and the gap energy is spin-band dependent.
From Eq. (25), the first term and second term in the brace of the summation are the intraband and the interband pair scattering contribution. We replace is the electron density of states for spin χ ± state at the Fermi energy and is the deviation of density of states due to Rashba splitting. Thus Small variation of Δ k,σ in E k,σ will be neglected such that , , we discuss the phonon mediated scattering process and the Coulomb interaction contribution to the gap energy separately. We use the model proposed by Morel and Anderson 33 to analyze the parameters in the gap equations.
The effective potential of the phonon-mediated scattering, in view of Eq. (13) and Eqs (C3, C6, C7 and C10) in Appendix C of Supplementary Information. is plays the role of dimensionless electron-phonon coupling constant "N(0)V" in BCS theory 41,42 . Here q D is the Debye wave vector. Interested readers can find derivation in Appendix C of Supplementary Information. Here and c is the velocity of longitudinal phonon. Since k σ and l σ′ are close to Fermi wave vector of σ and σ′ band respectively, |a kσ, lσ′ | < |b kσ, lσ′ | and the principle value for|a| < |b| will be applied. The effective Coulomb interaction potential is taken as  The finite temperature gap energy and transition temperature T c . The finite temperature gap energy equation can be obtained through Eqs (15) and (18). At transition temperature T = T c , Δ σ → 0, it can be further simplified as The first term usually dominates the right side of Eq. (57). Here again, the effective strength in the Rashba case is roughly half of the 2DEG case because of the angular dependence of Δ . As a result the critical temperature in the Rashba case becomes much smaller than that of conventional s-wave superconductor due to the exponential relation to the inverse of the coupling strength.
Lead film. The Rashba effect [14][15][16] and superconductivity [20][21][22] had been reported seperately for the Pb film grown on Si(111) with T c ranged from 1.5 K ~ 7K. Therefore, our model may be realized in lead thin film. The large effective mass of electrons in the quantum well state is taken to be 10 m e 45 . The lattice constatnt for Pb(111) film is = . a 3 50 Å  Table 1. We solve Eq. (54) numerically for the case of Pb-film. The relation between gap energy and the Rashba strength α is shown in Fig. 2(a). In the Pb-film case, Eq. (54) surely dominates as we expect in section B. The relation between the transition temperature and Rashba strength for the Pb-film is also evaluated numerically and shown in Fig. 2(b). The estimated transition temperatures for bulk lead and lead film in which the normal processes are considered are shown in Table 1(b). We find that transition temperature T c of p-wave superconductivity in the presence of the Rashba interaction is roughly 0.63 K as shown in Table 1(b). Hence, our calculation up to now can not account for the experimental finding in refs 20-22.

2-dimensional umklapp processes
In order to make perform a more accurate calculation and be able to explain exprimental results, we consider Umklapp processes in this section.
In the estimations of transition temperature and gap energy, the dimensionless electron-phonon coupling strength λ is an important factor. In 3-dimension case, Morel and Anderson proposed that λ is a state parameter and can be expressed as λ = In that approximation, λ is usually smaller than 1 2 and is suitable for the weak coupling case. However, the Umklapp scattering was not included in that approximation and λ is usually underestimated 47 . For the strong-coupled superconductor, the self energy calculations is usually treated with the Eliashberg equation 48,49 and the effective coupling strength can be properly renormalized. The transition temperature equation obtained by Macmillan 50 can be applied to calculate of a number of metals and alloys. The electron-phonon coupling constant λ is defined as q q q q 0 2 in Macmillan's analysis 50 . Scalapino, Wada and Swihart 34 estimated α 2 (ω q ) by including both normal processes and umklapp contribution with appropriate F(ω q ), and then evaluated the coupling strength λ. They were able to get results close to the experimental data [34][35][36] . In the previous section, we discuss the superconducting state parameter mainly following the model proposed by Morel and Anderson 33 in the absence of the Umklapp processes. In this section, we estimate the 2D superconducting state parameters λ by revising the the model of Scalapino, Wada and Swihart 34 for 2DEG.
The phonon coupling kernel where V 2D (q + K) is the effective Coulomb pseudo-potential and the phonon polarization is denoted by ν. The phonon density of states F(ω) is and the effective phonon coupling α 2 (ω) can be defined from Eqs (59) and (61). There is one longitudinal (l) mode and one transverse (t) mode for the phonon polarization and Scientific RepoRts | 6:29919 | DOI: 10.1038/srep29919 The first Brillouin zone is approximated by a circle of radius q D and Thus the effective phonon coupling is where K is a reciprocal lattice vector. Because electron-phonon coupling has contribution only when |q + K| < 2k F , there are 18 reciprocal lattice vectors K involved in the Umklapp processes listed in Table 2.
We follow the procedures in refs 34-36 to evaluate α ω ν ( ) 2 and F ν (ω) for lead. The potential in Eq. (60) of Harrison's form is where β = 60 Ry-atomc unit of area (a ) B 2 34 . F ν (ω) is assumed to vary as ω at low frequency regime in order to obtain the linear dispersion. There are two peaks in phonon density of states, one transverse peak at 4.4 meV and one longitudinal peak at 8.5 meV for the bulk Pb. These peaks are also adopted in Pb-film case. We use the cutoff Lorentzians to approximate the peak of F ν (ω). Normalization of F ν (ω) and continuity of F ν (ω) are used to determine A ν and ω ν0 . ω ν3 = 3ω ν2 is used in the Lorentzians. The parameters used in the calculation, the calculated coupling strength and the transition temperature for 2DEG are listed in Table 3. α 2 (ω) is a smooth function of ω and the value of α 2 (ω ν0 ) can be adopted for α 2 (ω) to evaluate λ. The effective phonon coupling strength λ = 1.05 and effective Coulomb interaction strength μ* = 0.1 for Pb film in 2DEG case. In this strong coupling case, the effective coupling strength is renormalized by Z(0) = 1 + λ and the renormalized coupling strength constants are λ = λ λ + re 1 In the precence of Rashba interaction in 2DEG, the coupling strength constants are about one-half of those for the free electron case and λ = 0.525 and μ* = 0.05 are taken for the presence of Rashba interaction case. Considering the renormalization effect, the renormalized coupling strength constants λ re = 0.344 and μ re = 0.033 are adopted in Eqs (54) and (57), the relation between gap energy Δ and to Rashba strength α and transition temperature T c relative to Rashba strength α are shown in Fig. 3(a,b) respectively. T 4K c in the presence of Rashba interaction case. The experimental T c ranged from 1.5~7 K for lead film on Si(111) are reported in refs 20-22. We have to note that as shown in the Table 3, the transverse phonon coupling α ω = .
( ) 1 09 . However, such transverse phonon mode is not included in the model estimation of last section in the absence of Umklapp process. The transverse mode which comes from the Umklapp processes has important contribution to the electron phonon coupling. Hence, the gap energy and transition temperature is enhanced while including the umklapp processes and the final result agrees reasonably well with experiments.
Finally, we discuss the effect of the band structure in lead. The strong-coupling superconductivity of lead was described very well by Eliashberg formalism 48,51 with practical physical quantities such as phonon spectra and electron density of states. In that and subsequent treatments, the Fermi surface was assumed to be spherical like what we have done in this work. Later, lead was reported to be a two-band superconductor 52,53 , albeit with two K(Å −1 ) Number of K

The observation of the p-wave superconductivity
We discuss how our calculation can be verified by experiments in this section. For the conventional superconductor, the excitation probability of the quasiparticles of isotropic gap energy Δ is proportional to exp[− Δ /k B T]. Thus, the power law T dependence of specific heat 54 at  T T c is inconsistent with isotropic gap prediction. This power law T dependence is due to the allowed states around the nodes in the superconducting gaps and is a feature of the unconventional superconductor. This can be applied to verify the nodes of this p-wave like gap energy.
Transverse ultrasound attenuation can be used in gap-anisotropic systems and probe the electronic gap nodes. By analyzing the quasiparticle contribution in the transverse ultrasound attenuation, the relationship between the quasiparticle gap structure and the electron viscosity tensor can be examined 55 . For temperature low enough, the quasiparticles are entirely concentrated within the gap nodes of the excitation spectrum. The attenuation due to certain node is related to the propagation direction and polarization direction of the sound wave. If neither direction is perpendicular to the node position vector in k-space, the attenuation is activated 55 . It had been used to locate the gap lines and nodes of anisotropic superconductor UPt 3 56 . The ultrasonic attenuation measurements on p-wave superconductor Sr 2 RuO 4 57 and d-wave cuprate superconductors YBa 2 CuO 3+x 58 were also performed. The tunneling spectroscopy were used to analysis gap profile in superconductivity. When an electron tunnels from a normal metal to an anisotropic superconductor, the tunneling depends on the angle between the superconducting crystal orientation and the interface because of the anisotropic gap 59 . For example, the zero-bias conductance peaks (ZBCPs) of a superconductor/insulator/normal metal tunneling conductance curve was reported in d-wave superconductors 60,61 which is the consequence of the Andreev bound states 62,63 and only exist in the     interface of junction. There should also be clear ZBCPs in the crystal node orientations in our p-wave like superconductor junctions . It is different from the flat U-shape conductance curve with no ZBCP for the s-wave case.
In addition, for both the diffusive normal metal/p-wave superconductor junction and diffusive normal metal/d-wave superconductor juction, while the gap node direction is along the interface, the injected and reflected quasiparticles feel different sign of the pair potentials and mid-gap Andreev resonant state (MARS) form. The p-wave or d-wave superconductivity at such normal metal/unconventional superconductor junction still can be distiguished by the charge transport property 64 . In d-wave superconductor, the destructive angular average of the proximity effect in MARS would result in zero proximity. However, in p-wave superconductor, the destructive average of the proximity effect in MARS is avoid and the proximity is finite. The proximity effect can be investigated from the local density of states (LDOS) in the normal metal side of the normal metal/superconductor junction. In p-wave superconductor case, there should be zero energy peak of LDOS in the normal metal side because of the penetration of the MARS from the superconductor side into the normal metal region. In d-wave superconductor case, there are zero proximity in MARS and LDOS in the normal metal side is a constant and still flat at zero energy. This difference is suggested to originate from the symmetry of the induced odd frequency pairing 65,66 . Thus, the p-wave and d-wave superconductor can be distinguished.

Conclusion
We investigate the effect in the superconductivity in the presence of the Rashba interaction. The presence of the Rashba field requires a new basis. Consider only the pairing in the same band, we obtain the coupled gap equations of two bands. Due to the partial cancellation between the intraband and interband pairs scattering, the dominant gap function is p-wave like ϕ ∆ = ∆ ϕ e cos