Transverse Josephson effect due to spin-orbit coupling: Generation of transverse current without time-reversal symmetry breaking
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Transverse Josephson effect due to spin-orbit coupling: Generation of transversecurrent without time-reversal symmetry breaking
Takehito Yokoyama
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: October 30, 2018)We investigate transverse Josephson current in superconductor/normal metal/superconductorjunctions where the normal metal has Rashba type spin-orbit coupling. It is shown that trans-verse current arises from the spin-orbit coupling in the normal metal. This effect is specific tosuperconducting current and the transverse current vanishes in the normal state. In addition, thistransverse Josephson effect is purely stationary and applied magnetic field is unnecessary to realizethis effect, in contrast to the Hall effect in the normal state. We also discuss physical interpretationof this effect, comparing with the spin Hall effect.
PACS numbers: 73.43.Nq, 72.25.Dc, 85.75.-d
When magnetic field is applied to electrons, theLorentz force acts on the electrons and a voltage dropappears in the direction perpendicular to the appliedcurrent. This is the celebrated (classical) Hall effect.About 100 years later, the quantum Hall effect has beendiscovered : When a strong magnetic field is appliedto 2D electron gas perpendicularly, the longitudinal re-sistance vanishes while the Hall conductance is quantizedto a rational multiple of e /h . The classical and quan-tum Hall effects arise by applying magnetic field. There-fore, the Hall effect occurs together with the time-reversalsymmetry breaking. Several authors have proposed themodels which exhibit a nonzero quantization of the Hallconductance by a spatially inhomogeneous magnetic fluxwith zero average and hence without Landau levels . Inthese models, time-reversal symmetry is also broken dueto the inhomogeneous magnetic flux. On the other hand,there have been a few efforts to realize the Hall effectwithout any magnetic field (flux). It has been shown thata circularly polarized light radiation can induce Hall cur-rent in Rashba spin-orbit coupled metal and graphene .Since circularly polarized light is described by ac electricfield, the Hall effect predicted in these works also requiresthe time-reversal symmetry breaking.In this paper, we study transverse Josephson current insuperconductor/normal metal/superconductor junctionswhere the normal metal has Rashba type spin-orbit cou-pling. It is shown that transverse current arises fromthe spin-orbit coupling in the normal metal under phasegradient, and the transverse current vanishes in the nor-mal state. In addition, this transverse Josephson effect ispurely stationary and applied magnetic field is unneces-sary to realize this effect, in contrast to the Hall effect inthe normal state. We also discuss physical interpretationof this effect, comparing with the spin Hall effect wherethe transverse spin current is generated by the spin-orbitcoupling in 2D electron system .We consider a superconductor/normalmetal/superconductor junction. The Hamiltonianof the superconductor and the normal metal are givenby H S = H + H ∆ and H N = H + H so + H ϕ + H so − ϕ ,respectively. The superconductor and the normal metal are coupled via the tunneling Hamiltonian H T and the junction is described by a two barrier model.The total Hamiltonian of the system is thus given by H S + H N + H T . The H , H ∆ and H so represent thekinetic energy, the superconducting order, and theRashba type spin-orbit coupling, respectively: H = X k φ † k ξσ ⊗ τ φ k , (1) H ∆ = X k φ † k ∆ σ ⊗ τ φ k , (2) H so = − X k E so · φ † k ( k × σ ) ⊗ τ φ k (3)with ξ = ε k − ε F = ¯ h k m − ε F and φ † k =( c † k ↑ , c † k ↓ , ic − k ↓ , − ic − k ↑ ) where σ and τ are Pauli matri-ces in spin and Nambu spaces, respectively. ε F , ∆, and E so are the Fermi energy, the gap function, and the vec-tor pointing in the direction of the inversion symmetrybreaking which characterizes the Rashba type spin-orbitcoupling, respectively. Note that we adopt the basis inRef. such that singlet pairing is proportional to the unitmatrix in spin space. We consider Josephson current in-duced by phase gradient. The phase gradient along j direction, ∇ j ϕ , enters the Hamiltonian as follows H ϕ = X k φ † k ¯ h m k j ∇ j ϕσ ⊗ τ φ k , (4) H so − ϕ = − X k E so · φ † k ∇ j ϕ ( e j × σ ) ⊗ τ φ k (5)where ϕ is half the phase of superconducting correlationand ∇ j ϕ is assumed to be spatially constant. We willtreat H so , H ϕ , and H so − ϕ perturbatively. We schemat-ically show the model in Fig. 1.With the above Hamiltonians, the charge current op- (cid:1)(cid:0)(cid:2)(cid:3)(cid:4)(cid:5) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) (cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !"
FIG. 1: (Color online) Schematic picture of the model of su-perconductor/normal metal/superconductor junctions wherethe normal metal has the Rashba type spin-orbit couplingcharacterized by the vector E so . The charge current j c flowsperpendicularly to the phase gradient ∇ ϕ . erator ( j c ) in i -direction reads j c,i = − e ¯ hm k i σ ⊗ τ − δ ij e ¯ hm ∇ j ϕσ ⊗ τ + e ¯ h ( σ × E so ) i ⊗ τ ≡ j ,i + j ϕ,i + j so,i (6)where − e is the electron charge.Before proceeding to the explicit calculation, let us dis-cuss transverse current qualitatively based on the time-reversal symmetry. Now, consider the Ohm’s law j c = σ · E (7)where j c , and E are, respectively, the charge current andthe applied electric field. The charge current is time-reversal odd while the electric field is even under time-reversal. Since the Ohm’s law relates quantities of differ-ent symmetries under time-reversal, the charge conduc-tivity σ breaks the time-reversal symmetry and describesthe inevitable joule heating and dissipation. A transversecurrent can flow under the applied electric field, which ishence dissipationless, but the time-reversal symmetry iscompensated by the external magnetic field. Therefore,the Hall effect in the normal state occurs, inevitably ac-companied by the time-reversal symmetry breaking.On the other hand, let us consider the London equa-tion, i.e., the response equation of supercurrent, j c = − e m ρ · A . (8)where ρ and A are, respectively, the superfluid den-sity and the vector potential. Since charge current andvector potential are time-reversal odd, ρ describes thereversible and dissipationless flow of the supercurrent.Thus, transverse superconducting current can flow with-out breaking time-reversal symmetry. Similary, we seethat since spin current is time-reversal even, supercon-ducting spin current is absent without breaking time-reversal symmetry. Now, we calculate transverse Josephson current andgive the analytical expression. We consider the unper-turbed advanced Green’s functions in the normal metal ./ 01234 567 89 :;<= >? @ABCDEFGHIJKL MNO
FIG. 2: Diagrammatic representations of the current densi-ties with second-order contributions of the spin-orbit coupling E so and the first-order contributions of the phase gradient ∇ ϕ . The wavy lines denote the spin-orbit interaction anddotted lines represent the phase gradient. The wavy and dot-ted lines intersecting at one point correspond to H so − ϕ . of the form g a k ,ω = g a , k ,ω σ ⊗ τ + g a , k ,ω σ ⊗ τ + f a k ,ω σ ⊗ τ ≡ σ ⊗ g aτ, k ,ω (9)where g a , k ,ω and g a , k ,ω are normal Green’s functionswhile f a k ,ω is anomalous Green’s function. The anoma-lous Green’s function is in general nonzero in the normalmetal due to the proximity effect. We perform pertur-bative calculation with respect to E so and ∇ j ϕ up to asecond and a first order, respectively. Diagrammatic rep-resentation of the transverse current is shown in Fig. 2.The transverse Josephson current can be expressed as j c,i = i ¯ h emV X k Tr k i σ ⊗ τ G < k , k ( t, t )+ δ ij i ¯ h emV ∇ j ϕ X k Tr σ ⊗ τ G < k , k ( t, t ) − ieV X k Tr( σ × E so ) i ⊗ τ G < k , k ( t, t ) (10)where V is the total volume and Tr is taken over spin andNambu spaces. G < k , k ( t, t ) is the lesser Green’s functionof the total Hamiltonian. Performing perturbation withrespect to H so , H ϕ and H so − ϕ , we expand the lesser com-ponent using the advanced Green’s functions by the Lan-greth theorem. Noting that g < k ,ω = f ω h g a k ,ω − ( g a k ,ω ) † i with the lesser Green’s function g < k ,ω and the Fermi dis-tribution function f ω , and δ ij = ∂k i ∂k j , we can compute thetransverse Josephson current. Then, the leading term ofthe transverse current ( i = j ) is given by the second or-der expansion with respect to E so (the first order term vanishes), which results in the form j c,i ∼ = 2 i ¯ h e V m ∇ j ϕ X k ,ω Tr τ k i k j ( E so × k ) (cid:2) ( g τ, k ,ω ) τ g τ, k ,ω τ g τ, k ,ω + τ g τ, k ,ω τ ( g τ, k ,ω ) τ g τ, k ,ω + g τ, k ,ω τ g τ, k ,ω τ ( g τ, k ,ω ) (cid:3) < + δ ij i ¯ h eV m ∇ j ϕ X k ,ω Tr τ τ ( E so × k ) (cid:2) g τ, k ,ω τ g τ, k ,ω τ g τ, k ,ω (cid:3) < − i ¯ h eV m ∇ j ϕ E iso E jso X k ,ω Tr τ k j (cid:2) ( g τ, k ,ω ) τ g τ, k ,ω + g τ, k ,ω τ ( g τ, k ,ω ) (cid:3) < = 256 e V ∇ j ϕ E iso E jso X k ,ω f ω ε k Im (cid:20) ε k ( f a k ,ω ) (cid:8) ( g a , k ,ω ) − ( f a k ,ω ) + 5( g a , k ,ω ) (cid:9) + 32 g a , k ,ω ( f a k ,ω ) (cid:21) (11)where Tr τ means the trace in Nambu space (here we havealready taken the trace in spin space). This is a generalexpression which is applicable to any Green’s function ofthe form of Eq. (9). We see that in the absence of the su-perconductivity f a k →
0, the transverse current vanishes j c,i →
0. From Eq.(11), we find that the direction of thetransverse Josephson current is determined by the vector E so , which characterizes the spin-orbit coupling, as j c,i ∝ [ E so × ( E so × ∇ ϕ )] i (12)for j c ⊥ ∇ ϕ . Longitudinal Josephson current can flow un-der phase gradient without spin-orbit coupling. Thus, inthe leading order, it is proportional to the phase gradi-ent and does not depend on spin-orbit coupling. Then,Eq.(12) can be rewritten as j c,i ∝ (cid:2) E so × ( E so × j lc ) (cid:3) i (13)where j lc is the longitudinal Josephson current parallel to ∇ ϕ . We also see that when the phase gradient is along x -axis, to obtain finite j c,y the vector E so should haveboth x and y components.When the normal metal is sufficiently thin and theinterfaces between the normal metal and the supercon-ductors are transparent, proximity effect becomes verystrong such that the Green’s functions in the normalmetal have the same form as those in the bulk super-conductor: g a k ,ω = ( ω − iγ ) σ ⊗ τ + ξσ ⊗ τ + ∆ σ ⊗ τ ( ω − iγ ) − ξ − ∆ . (14)The transverse current is then given by j c,i ∼ = 64 πeν V ∇ j ϕ E iso E jso X ω f ω Re " ∆ [( ω − iγ ) − ∆ ] / = 32 eν V ¯ h ∇ j ϕ E iso E jso γ p γ + ∆ − ! (15) PQRSTUV WXYZ [\]^_‘ abcdefghij klmnopqrs tuvwxyz{|
FIG. 3: (Color online) Physical picture of the spin Hall ef-fect (left) and the transverse Josephson effect by Cooper pair(right). at zero temperature where γ is the inelastic scatteringrate by impurities and ν is the density of states at theFermi energy.Let us discuss the mechanism of the transverse Joseph-son effect predicted in this paper, comparing with thespin Hall effect . Consider the spin-orbit couplingEq.(3) as a spin dependent potential. The spin Hall ef-fect occurs since electron with spin-up and that with spin-down feel potential with the opposite sign and hence arescattered in the opposite direction as shown in Fig. 3.On the other hand, the Josephson current is carried byCooper pair which consists of electron pair with oppositemomentum and spin, ( k , ↑ ) and ( − k , ↓ ). As seen fromEq.(3), the Rashba type spin-orbit coupling is invariantunder the sign change of both momentum and spin (sincethis term is time-reversal even). Thus, the two electronswhich constitute the Cooper pair feel exactly the samepotential and hence move in the same direction, resultingin the net transverse current.As seen from the London equation Eq.(8), since bothcharge current and vector potential are odd under spa-tial inversion, the inversion symmetry breaking is unnec-essary to obtain the charge current. In fact, the coef-ficient of the phase gradient is given by even (second)order with respect to the Rashba type spin-orbit cou-pling (see Eq.(11)), and hence does not break the inver-sion symmetry. This implies that when the system re-spects the inversion symmetry, the transverse Josephsoneffect predicted in this paper would also emerge due tospin-orbit coupling. Further investigation of the trans-verse Josephson effect based on such model as the Lut-tinger Hamiltonian , the spin-orbit coupling of whichpreserves the inversion symmetry, will be an interestingfuture work.The present formalism assumes the translational sym-metry of the system and hence the length of the nor-mal metal does not appear in the results. To investigatethe effect of the length of the normal metal, investiga-tion based on, for example, the Usadel equation includingspin-orbit coupling is necessary. To test the transverse Josephson effect predicted in thispaper experimentally, one may use BiTeI, a recently dis-covered 3D Rashba system , as a spin-orbit couplednormal metal. For this material, we have ν ∼ × − states/eV/unit cell, the lattice constant ∼ k F E so ∼
100 meV, and the Fermi wavevector k F ∼ − . Taking ∇ j ϕ ∼ (100nm) − , γ ∼
10 meV and ∆ ∼ (cid:12)(cid:12) j c,i (cid:12)(cid:12) ∼ × − A/cm . Our prediction could be confirmed by Joseph-son junctions with this material in four-terminal geome-try by injecting longitudinal current. Since transverseJosephson current gives a phase shift along the trans-verse direction, one can detect the predicted effect byusing superconducting quantum interference device at-tached to the normal metal transversely. The transverseJosephson effect is reflected as a shift in the interferencepattern.In summary, we have investigated transverseJosephson current in superconductor/normal metal/superconductor junctions where the normalmetal has Rashba type spin-orbit coupling. It hasbeen shown that transverse current arises from thespin-orbit coupling in the normal metal. This effect isspecific to superconducting current and the transversecurrent vanishes in the normal state. In addition, thistransverse Josephson effect is purely stationary andapplied magnetic field is unnecessary to realize thiseffect, in contrast to the Hall effect in the normal state.We have also presented physical interpretation of thiseffect, comparing with the spin Hall effect.The author thanks S. Murakami for helpful discussion.This work was supported by Grant-in-Aid for Young Sci-entists (B) (No. 23740236) and the ”Topological Quan-tum Phenomena” (No. 23103505) Grant-in Aid for Sci-entific Research on Innovative Areas from the Ministryof Education, Culture, Sports, Science and Technology(MEXT) of Japan.
Appendix.
Here, we will derive Eq.(6). The velocityoperator in i -direction, v i , is given by the derivative ofthe Hamiltonian with respect to the momentum: v i = ∂H N ¯ h∂k i = ¯ hm k i σ ⊗ τ + δ ij ¯ hm ∇ j ϕσ ⊗ τ − h ( σ × E so ) i ⊗ τ . (16)One obtains the current operator by multiplying the ve-locity operator by the charge operator − eσ ⊗ τ : j c,i = − eσ ⊗ τ v i = − e ¯ hm k i σ ⊗ τ − δ ij e ¯ hm ∇ j ϕσ ⊗ τ + e ¯ h ( σ × E so ) i ⊗ τ . (17)Note that the charge is opposite for electron and hole. K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. , 494 (1980). D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.Lett. , 1559 (1982). The Quantum Hall Effect , edited by R. E. Prange and S.M. Girvin (Springer-Verlag, New York, 1987). F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988). E. Tang, J-W. Mei, and X-G. Wen, Phys. Rev. Lett. ,236802 (2011). T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys.Rev. Lett. , 236804 (2011). K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Phys. Rev.Lett. , 236803 (2011). V. M. Edelstein, Phys. Rev. Lett. , 156602 (2005). T. Oka and H. Aoki, Phys. Rev. B , 081406(R) (2009). S. Murakami, N. Nagaosa and Shou-cheng Zhang, Phys.Rev. B , 235206 (2004). H. Haug and A.-P. Jauho,
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