2D superconductivity with strong spin-orbit interaction

We consider superconductivity confined at a two-dimensional interface with a strong surface spin-orbit (Rashba) interaction. Some peculiar properties of this system are investigated. In particular, we show that an in-plane Zeeman field can induce a supercurrent flow. PACS numbers: 74.20.-z, 74.25.-q, 74.20.Rp, 73.20.-r Most superconductors have their underlying crystal structures and the normal states obeying inversion symmetry. This symmetry allows the classification of superconductors [1–3] into singlet and triplet pairing, and correspondingly even and odd symmetry of the order parameter under sign change of momentum ~ p → −~ p, i.e. the opposite sides of the Fermi surface. This classification has played an important role in our current understanding of superconductors and their properties. Most ”conventional” superconductors such as Nb and Pb are singlet s-wave [4], oxide superconductors are likely to be singlet d-wave [5], whereas superfluid He is triplet p-wave [6]. When inversion symmetry is absent in the normal state, such classification is no longer possible. The superconducting pairing can thus be neither singlet nor triplet [7], and the order parameter neither even nor odd under ~p → −~ p. The superconductor can therefore have rather peculiar physical properties when compared with those where the above mentioned classification can be made. This absence of inversion symmetry may be relevant to some known superconductors. (see also references cited in [8]) An examination of the list of superconductors in Table 6.1 of [9] shows that, e.g., Mo3Al2C (symmetry P4132), La5B2C6 (symmetry P4) and Mo3P (symmetry I 4̄) are all without inversion centers. Furthermore, 1 two-dimensional (2d) surface superconductivities have been induced by gate electric potentials in C60 and some molecular crystals in the field-effect-transistor geometry [10,11]. There is no inversion symmetry in these cases since ”up” and ”down” are different due to the electric gates, substrates etc. Some properties of superconductors without inversion centers have already been studied theoretically before (see [7,8] and references therein). For definiteness and motivated by the last mentioned examples above, we here consider, as in [7,8], a 2d superconductor at an interface with no ”up-down” symmetry. As pointed out there, one potentially important effect due to the lack of inversion symmetry in such a geometry is the existence of a surface spin-orbit coupling or Rashba [12] term in the Hamiltonian of the form −αn̂ × ~p · ~σ. Here n̂ is the surface normal and ~σ are the Pauli spin matrices. This term acts like an effective magnetic field along n̂ × ~p and thus splits the spin degeneracy of the electrons at a given momentum ~p. The energy difference near the Fermi level can be large: in some systems it is known to be of order 0.1eV [13], and is therefore expected to be much larger than the superconducting gap ∆ even for a transition temperature ∼ 100K. Rashba splitting of this magnitude hence is expected to have dramatic effects on the superconducting properties in these systems. Some physical consequences due to this spin-orbit coupling term have been considered in [7,8] using Green’s function approach. Gor’kov and Rashba [7] calculated the spin susceptibility in this system. Edelstein [8] pointed out an interesting magnetoelectric effect, that a spin-polarization can be induced by a supercurrent flow. Here we shall reconsider these physical properties under the most probable case where pF 2m >> αpF >> |∆| (1) using simple physical arguments. [ Here pF is the Fermi momentum and m is the effective mass. The definition of pF will be made more precise below]. In addition, we give a more complete description of the magneto-electric effect in this system. More precisely, we shall show the existence of an inverse effect, i.e., a supercurrent can be induced by an applied Zeeman field. The relation of this effect to that proposed by Edelstein and the possibility 2 of its experimental observation is discussed. We shall then consider a two-dimensional electronic system lying in the x-y plane. The one-body part of the Hamiltonian is given by H = p 2m − αn̂× ~p · ~σ (2) with n̂ = ẑ. We shall first summarize some consequences of eq (2) which we shall need below. As mentioned, the effect of the Rashba term is like a Zeeman field along n̂× ~ p. The eigenstates of this spin-dependent part of the Hamiltonian thus correspond to states with spins along and opposite to this direction. We shall label these spin states by |~ p,+ > and |~ p,− > respectively. The spinors for these states can be chosen to be ( by rotating those for an up and down spin by − 2 along p̂),