Signatures of Rashba spin-orbit interaction in the superconducting proximity effect in helical Luttinger liquids
SSignatures of Rashba spin-orbit interaction in the superconducting proximity effect inhelical Luttinger liquids
Pauli Virtanen
Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, D-97074 W¨urzburg, Germany
Patrik Recher
Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, D-97074 W¨urzburg, Germany andInstitute for Mathematical Physics, TU Braunschweig, 38106 Braunschweig, Germany (Dated: November 3, 2018)We consider the superconducting proximity effect in a helical Luttinger liquid at the edge of a 2Dtopological insulator, and derive the low-energy Hamiltonian for an edge state tunnel-coupled to a s -wave superconductor. In addition to correlations between the left and right moving modes, thecoupling can induce them inside a single mode, as the spin axis of the edge modes is not necessarilyconstant. This can be induced controllably in HgTe/CdTe quantum wells via the Rashba spin-orbitcoupling, and is a consequence of the 2D nature of the edge state wave function. The distinction ofthese two features in the proximity effect is also vital for the use of such helical modes in order to splitCooper-pairs. We discuss the consequent transport signatures, and point out a long-ranged featurein a dc conductance measurement that can be used to distinguish the two types of correlationspresent and to determine the magnitude of the Rashba interaction. PACS numbers: 74.45.+c, 71.10.Pm, 73.23.-b
I. INTRODUCTION
The helical edge states of a 2D topological insulator(TI) consist of a Kramers pair of right- and left-movingelectron modes of opposite spin situated inside the bulkgap , and they have so far been observed in HgTe/CdTequantum wells (HgTe-QW) . In 3D topological insula-tors, the edge states cover the surface of the material andconsist of a single-valley Dirac cone with spin-momentumlocking, which leads to unique electromagnetic proper-ties and quantum interference effects . In both 2D-TIand 3D-TI the coupling of spin and orbital motion canlead to interesting effects when combined with supercon-ductivity. Superconducting correlations induced by theproximity of a singlet s -wave superconductor can insidethe TI obtain a p -wave character, which can be used toengineer Majorana bound states. A somewhat sim-ilar induction of non-conventional correlations has alsobeen proposed to occur in other semiconductor systemsin the combined presence of the spin-orbit interaction andsuperconductivity. When the edge state of a 2D-TI is coupled to a sin-glet superconductor, the transfer of electrons betweenthe systems can, first of all, induce singlet-type prox-imity correlations between electrons in the right and leftmoving modes (the + − channel). This already leads toseveral effects of interest. For instance, the helicity of theelectron liquid lifts the spin degeneracy and enables Ma-jorana states, causes Cooper pairs to split, and affectstransport properties. Tight-binding calculations study-ing the pair amplitude have also been made . There is,however, also a possibility of inducing correlations onlywithin the right-moving (or the left-moving) channel at a nonzero total momentum (the ++ and −− channels).Such a channel is not forbidden by symmetries in theproblem: due to the spin-orbit coupling, the spin axisof the edge state is not necessarily constant, so that theelectrons forming a Cooper pair singlet can both enterthe same mode on the TI edge, even when spin is con-served in the tunneling process and time-reversal symme-try is present. In HgTe-QW, a non-constant spin axis canbe induced externally by the Rashba spin-orbit couplingthat breaks inversion symmetry. Momentum conserva-tion is required to be broken, but this can occur e.g. dueto inhomogeneity or a finite size of a tunneling contact.Moreover, unlike in metals, in 2D-TI the momentum non-conservation can in principle be made arbitrarily small bytuning the Fermi level near the Dirac point ( k = 0).A straightforward way to probe the existence of su-perconducting correlations is to observe the Josephsoneffect or other interference effects that can be mod-ulated with superconducting phase differences. TheJosephson effect has been studied previously in vari-ous one-dimensional Luttinger liquid systems. Thefinite-momentum channel has, however, received limitedattention, and is usually negligible. As shown below,certain experiments with superconducting contacts at-tached to the helical edge states can nevertheless probesuch microscopic aspects of the tunneling, including therole of the Rashba interaction.Here, we first derive a low-energy Hamiltonian de-scribing the superconducting proximity effect in the edgestates of a 2D TI coupled to a conventional superconduc-tor by tunnel contacts. We use it to find the signaturesof both types of tunneling events in a transport exper-iment. Because of the reduced number of propagating a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n modes in the helical liquid, correlations within the samechannel occur at a finite momentum and, as in chiralliquids, are affected by the exclusion principle. It turnsout that although this component of the proximity effectgives a negligible correction to the dc Josephson effect,in the NS tunneling conductance [see Fig. 1(c)] it man-ifests as a long-ranged interference effect, oscillating asa function of the superconducting phase difference, andunlike the + − part, is not exponentially suppressed atlength scales longer than the thermal wavelength. Theratio of the contributions of the two possible channelsscales as δG ++ /δG + − ∝ ( z / (cid:126) v F ) ( k B T /M ) e πT d/ (cid:126) v F (in the noninteracting case), where z characterizes thestrength of Rashba interaction, v F is the Fermi velocityof the edge channels, M is the energy gap of the TI, and d the distance between two superconducting contacts form-ing the interferometry setup. The amplitude of the effectis proportional to the amount of spin rotation achievedby Rashba interaction, and the quadratic temperaturedependence is due to the exclusion principle. We alsodiscuss how e - e interactions modify this result.This paper is organized as follows. In Section II, we in-troduce the model for the helical Luttinger liquid (HLL),the coupling to the superconductors, and the electronicstructure of HgTe-QWs. Section III discusses the effec-tive low-energy Hamiltonian, and Section IV transportsignatures in the dc and ac Josephson effects and the NSconductance. Section V concludes the manuscript witha discussion on the results and remarks on experimentalrealizability. II. MODEL
We consider the setup depicted in Fig. 1. The edgestates of a 2D-TI are coupled to two superconductingterminals via two tunnel junctions. Below, we in generalassume that the distance d between the contacts is longerthan the superconducting coherence length ξ .The left- and right moving edge states | + , x (cid:105) and |− , x (cid:105) have a linear dispersion, and are described by thebosonized Hamiltonian H = 12 (cid:90) ∞−∞ d x u [ g − ( ∂ x ϑ ) + g ( ∂ x φ ) ] (1)where the Fermi field operator is ψ α ( x ) = (2 πa ) − / U α e iαk F x e iφ α ( x ) =(2 πa ) − / U α e iα [ k F x + √ πϑ ( x )]+ i √ πφ ( x ) , the stan-dard boson fields ϑ ( x ), φ ( x ) satisfy [ φ ( x ) , ϑ ( x (cid:48) )] =( i/
2) sgn( x − x (cid:48) ), and U ± are the Klein factors. u = v F /g is the renormalized Fermi velocity. Here andbelow, we let (cid:126) = k B = e = 1, unless otherwise men-tioned. The parameter a is the short-distance cutoff.In the noninteracting case, the Luttinger interactionparameter g = 1, and with repulsive electron-electroninteractions one has g < d ( ) φ t ( ) φ t =0 μ ( ) I t S S ( ) I t ∼ ξ (b) (c) Φ VI I S S S (a) FIG. 1. (a) The setup considered: a 2D topological insulator,whose edge state is coupled to two superconductors via tun-nel contacts. In response to phase ϕ or voltage V differencesbetween the superconductors, Josephson currents can flow viathe edge channel, or one can probe the NS transmission by in-jecting current from the superconductors to the edge channel.If the spin axis of the edge state is not constant spatially andas a function of energy, a Cooper pair singlet can enter theedge state in two possible ways: either electrons enter modespropagating to opposite directions (left) or the same direc-tion (right). In the latter case, exclusion principle requirestemporal (or spatial) separation of the two, which can be ofthe order of the superconducting coherence length ξ = (cid:126) v F / ∆still preserving the correlation. (b) Configuration for the mea-surement of the Josephson current. (c) Configuration for themeasurement of interference in the NS conductance. a tunneling Hamiltonian H T = (cid:88) α = ± ,σ (cid:48) = ↑ , ↓ (cid:90) d x d r (cid:48) t ασ (cid:48) ( x, (cid:126)r (cid:48) ) ψ † α ( x ) ψ Sσ (cid:48) ( (cid:126)r (cid:48) ) + h . c . , (2)where the tunneling amplitude t ασ (cid:48) ( x, (cid:126)r (cid:48) ) describes thetunneling from the state | σ (cid:48) , (cid:126)r (cid:48) (cid:105) in the superconductor tostate | α, x (cid:105) in the edge mode. For what follows, it is use-ful to introduce also the corresponding one-particle oper-ator ˆ h T , in terms of which, t ασ (cid:48) ( x, (cid:126)r (cid:48) ) ≡ (cid:104) α, x | ˆ h T | σ (cid:48) , (cid:126)r (cid:48) (cid:105) .The momentum k along the edge is a good quantum num-ber for straight TI edges, and we define the state | α, x (cid:105) inthe momentum representation: | α, x (cid:105) = (cid:80) k e − ikx | α, k (cid:105) ,where | α, k (cid:105) is the edge eigenstate with momentum k andpropagation direction α = ± .We assume that the Hamiltonian is time-reversal sym-metric, which implies that the tunneling operator in gen-eral satisfies T ˆ h T T − = ˆ h T . Here, we choose the phasesof the wave functions so that the time reversal operationsread T | σ (cid:48) , (cid:126)r (cid:48) (cid:105) = σ (cid:48) |− σ (cid:48) , (cid:126)r (cid:48) (cid:105) and T | α, k (cid:105) = α |− α, − k (cid:105) . Wealso assume that the tunneling is spin-conserving, thatis, written in terms of real electron spin states in the TIand the superconductor, we have (cid:104) σ, (cid:126)r | ˆ h T |− σ, (cid:126)r (cid:48) (cid:105) = 0.To describe tunneling to HgTe-QWs, we need someknowledge of the structure of the edge states. This canbe obtained from the four-band model used in Ref. 2.In this approach, the low-energy properties of the TIare described using a 2D envelope function in the basisof four states {| E (cid:105) , | H (cid:105) , | E −(cid:105) , | H −(cid:105)} localized inthe quantum well. The edge states at the boundariesof the TI can be solved within this four-band model; forwhich we give a full analytical solution in Appendix A.We assume the terminals are conventional spin-singletsuperconductors. As usual, they are characterizedby the correlation function F ( (cid:126)r , σ , τ ; (cid:126)r , σ , τ ) ≡(cid:104) T [ ψ σ ( (cid:126)r , τ ) ψ σ ( (cid:126)r , τ )] (cid:105) that has a singlet symmetry F ( (cid:126)r , σ , τ ; (cid:126)r , σ , τ ) = σ δ σ , − σ F ( (cid:126)r , τ ; (cid:126)r , τ ). In thebulk, the correlation function obtains its equilibriumBCS form, which in imaginary time can be written as F ( (cid:126)r , (cid:126)r ; ω ) = (cid:90) d k (2 π ) e − i ( (cid:126)r − (cid:126)r ) · (cid:126)k ∆ ω + ξ k + | ∆ | , (3)with ξ k = k / (2 m ) − µ the dispersion relation and ∆ thegap of the superconductor. III. EFFECTIVE HAMILTONIAN
Integrating out the superconductors using perturba-tive renormalization group theory (RG) and consider-ing only energies | E | (cid:28) | ∆ | reduces the Hamiltonian H + H T of the total system to one concerning only theone-dimensional edge states: H = H + (cid:90) d x [Γ + − ( x ) ψ + ( x ) ψ − ( x ) (4)+ Γ ++ ( x ) ψ + ( x ) ψ + ( x + a )+ Γ −− ( x ) ψ − ( x ) ψ − ( x + a ) + h . c . ] . Here, Γ αβ describe the coupling to the superconductor,and a = (cid:126) v F / ∆ is the new short-distance cutoff in thetheory. Details of the derivation are discussed in Ap-pendix B.The coupling factors in the noninteracting case ( g = 1)are given by the expressions (see Appendix B for generaldiscussion):Γ ++ ( x ) = π (cid:90) d r (cid:48) d r (cid:48) (cid:88) K e iKx F ( r (cid:48) , r (cid:48) ; 0) (5) × ∆ v − F ∂ k P ++ ( K k, r (cid:48) ; K − k, r (cid:48) ) ∗ | k =0 , where F is given in Eq. (3), andΓ + − ( x ) = π (cid:90) d r (cid:48) d r (cid:48) (cid:88) K e iKx F ( r (cid:48) , r (cid:48) ; 0) (6) × P + − ( K − k F , r (cid:48) ; K k F , r (cid:48) ) ∗ , with Γ −− = Γ ∗ ++ in the presence of time reversal sym-metry. The main contributions should arise around E F (a) (b) E k kε ε − ε − ε k k k k = + α =− α FIG. 2. Schematic depiction of the edge mode spectrumand the states participating in 2-particle tunneling from theFermi level in the superconductor to states in the TI. (a) Inthe + − channel, the momenta corresponding to energy ε are k = k F − ε/ (cid:126) v F , k = − k F − ε/ (cid:126) v F . (b) In the ++ channel,one has k = − k F + ε/ (cid:126) v F , k = − k F − ε/ (cid:126) v F . Singlet pairtunneling into the ++ channel can occur if the electron spinaxis is different for the states at k and k . K = − k F for Γ ++ (due to the 2 k F oscillations in theFermi operators), and around K = 0 for Γ + − .The coupling is proportional to the factor P α α ( k , (cid:126)r (cid:48) ; k , (cid:126)r (cid:48) ) ≡ [ t α ↓ ( k , (cid:126)r (cid:48) ) t α ↑ ( k , (cid:126)r (cid:48) ) (7) − t α ↑ ( k , (cid:126)r (cid:48) ) t α ↓ ( k , (cid:126)r (cid:48) )] + [ (cid:126)r (cid:48) ↔ (cid:126)r (cid:48) ] , which describes two-particle tunneling of a singlet, fromtwo points (cid:126)r (cid:48) and (cid:126)r (cid:48) in the superconductor, to momen-tum states | α , − k (cid:105) , | α , − k (cid:105) in the TI edge modes(cf. Fig. 2). Here, t ασ (cid:48) ( k, (cid:126)r (cid:48) ) = (cid:82) d x e ikx t ασ (cid:48) ( x, (cid:126)r (cid:48) ) = (cid:104) α, − k | ˆ h T | σ (cid:48) , (cid:126)r (cid:48) (cid:105) is a Fourier transform of the tunnelingmatrix element.One can also verify that in the absence of interactions,the expression for the Γ + − amplitude coincides with theleading term in the zero-bias conductance in the normalstate, up to a replacement F (cid:55)→ πv F ) − Im G R . Withina quasiclassical approximation in the superconductor, one then finds a relation to the normal-state conductanceper unit length, g ( x ), of the tunnel interface:Γ + − ( x ) (cid:39) (cid:126) v F R K g ( x ) = (cid:126) v F l T R K R N . (8)Such a relation is typical for NS systems, and connectsthe amplitude Γ + − to observable quantities. The latterexpression assumes the total resistance R N is uniformlydistributed in a junction of length l T . When the inter-face resistance decreases, the effective pairing amplitudeΓ grows — and although not included in our perturba-tive calculation, one expects that this increase is cut offwhen the effective gap reaches the bulk gap of the super-conductor, Γ + − = ∆.Unlike Γ + − , the Γ ++ / −− amplitudes do not have adirect relation to the normal-state conductance, and theydepend on the factors P ++ / −− , which are proportional tothe spin rotation between the states involved in the pairtunneling (see Fig. 2). Estimating this factor is necessaryfor determining how large the same-mode tunneling is ina given system. A. Two-particle tunneling
Making use of the time-reversal symmetry, it is possibleto rewrite the P factors in a more transparent form: P α α ( k , (cid:126)r (cid:48) ; k , (cid:126)r (cid:48) ) = (cid:104) α , − k | ˆ Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) T | α , − k (cid:105) , (9)ˆ Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) ≡ ˆ h T [1 σ ⊗ ( | (cid:126)r (cid:48) (cid:105)(cid:104) (cid:126)r (cid:48) | + | (cid:126)r (cid:48) (cid:105)(cid:104) (cid:126)r (cid:48) | )]ˆ h T , (10)where 1 σ is the identity matrix in the spin space of thesuperconductor. Unlike the starting point, this expres-sion is explicitly independent of the choice of the spinquantization axis. We also note the symmetry: P α α ( k , r (cid:48) ; k , r (cid:48) ) = − P α α ( k , r (cid:48) ; k , r (cid:48) ) , (11)following from the definition Eq. (7).We can now make some remarks on the possibilityof ++ tunneling. First, suppose that the state | α, k (cid:105) describes an electron wave function with a fixed k -independent and spatially constant spin part, and thatthe tunneling is spin conserving. In this case it is easyto see that P ++ = 0, as the inner product of a spinorand its time reversed counterpart vanishes. Such a situa-tion is realized, for instance, within the plain Kane-Melemodel. Breaking such conditions can, however, lead to P ++ (cid:54) = 0. We demonstrate in the next section that thiscan occur in HgTe-QW. B. Effect of Rashba interaction in HgTe/CdTequantum wells
We now discuss a simple model for tunneling into thehelical edge states of a HgTe-QW, taking spin axis ro-tation from the Rashba interaction into account. Wemake the following assumptions: the tunneling is spin-conserving and local [ (cid:104) (cid:126)r | ˆ h T | (cid:126)r (cid:48) (cid:105) ∝ δ ( (cid:126)r − (cid:126)r (cid:48) )] on the lengthscales of the four-band model. This results to all contri-butions to P ++ coming solely from the Rashba mixing.While we cannot estimate the actual values of P + − or P ++ within this simplified a model, we can study theirrelative magnitudes, which is now determined by the low-energy four-band physics only.Under the locality and spin-conservation assump-tions, the tunnel matrix element introduced above ob-tains the following form in terms of envelope spinorwave functions ˆΨ in the four-band basis {| j (cid:105)} = {| E (cid:105) , | H (cid:105) , | E −(cid:105) , | H −(cid:105)} : P α α ( k , (cid:126)r (cid:48) ; k , (cid:126)r (cid:48) ) = ˆΨ α , − k ( x (cid:48) , y (cid:48) ) † ˆ Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) × T ˆΨ α , − k ( x (cid:48) , y (cid:48) ) + [ (cid:126)r (cid:48) ↔ (cid:126)r (cid:48) ] , (12)ˆ Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) jj (cid:48) = (cid:104) j | h T [1 σ ⊗ | (cid:126)r (cid:48) (cid:105)(cid:104) (cid:126)r (cid:48) | ] h T | j (cid:48) (cid:105) . (13)Time reversal for the four-band spinor reads T = − iτ y K with K the complex conjugation, and the τ matrix actson the Kramers blocks (+, − ). For simplicity, we use now v F k [meV]0.60.40.20.00.20.40.6 z [ m ] z ( k ) z ( k ) ×nm FIG. 3. Projections z ( k ) of the additional spin-orbit cou-plings R and T on the edge state basis, for the parametersof Ref. 24 with M = −
10 meV. The dashed lines indicate lin-ear approximations z ≈ z , k and z ≈ z , k with z , = 0 . z , = 0 . − . a length-scale separation between the scales appearing inthe four-band model (Ψ) and the atomic ones (tunneling Z , k F,S in the superconductor, unit cell). We consideronly the long-wavelength part of P , and replace Z with aconstant describing the tunnel coupling to the quantumwell basis states, obtained by averaging it together with F [cf. Eqs. (5), (6)] over (cid:126)r (cid:48) and (cid:126)r (cid:48) :[ Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) + Z ( (cid:126)r (cid:48) , (cid:126)r (cid:48) )] F ( (cid:126)r (cid:48) , (cid:126)r (cid:48) ) ∼ A ( (cid:126)r (cid:48) ) C ( (cid:126)r (cid:48) ) 0 D ( (cid:126)r (cid:48) ) C ( (cid:126)r (cid:48) ) ∗ B ( (cid:126)r (cid:48) ) −D ( (cid:126)r (cid:48) ) 00 −D ( (cid:126)r (cid:48) ) ∗ A ( (cid:126)r (cid:48) ) C ( (cid:126)r (cid:48) ) ∗ D ( (cid:126)r (cid:48) ) ∗ C ( (cid:126)r (cid:48) ) B ( (cid:126)r (cid:48) ) F (0) δ ( (cid:126)r (cid:48) − (cid:126)r (cid:48) ) , (14)with A and B real-valued. This form follows from thetime reversal symmetry and hermiticity of the matrix el-ements of the operator in Eq. (10). We have also assumedhere that the decay length for the F function ( ∼ k − F,S )is short on the scales of the 4-band model. Finally, wefor simplicity neglect the coupling to the H B = C = 0. For a lateral contact (SC on top ofHgTe-QW), the main tunnel coupling is expected to in-volve the E1 band, which extends deeper into the CdTebarrier than H1. Including additional couplings wouldhowever cause no essential qualitative differences in theestimated ratio between the ++ and + − terms.Without additional spin axis rotation from the Rashbainteraction, the edge states are in separate Kramersblocks (see Appendix A), ˆΨ + ∝ ( ˆΦ + ,
0) and ˆΨ − ∝ (0 , ˆΦ − ), and we can see that P ++ = 0 whereas P + − ∝ A .Note that a contribution proportional D does not arise:the unperturbed edge state wave functions are both pro-portional to the same constant real-valued spinor, ˆΦ ± ∝ ˆ χ , so that the D -dependent contribution would be pro-portional to ˆ χ † iσ y ˆ χ = 0. This structure also implies thatcontributions proportional to D do not arise in the lead-ing order of the Rashba coupling.Rashba and other related spin-orbit interactions in thefour-band model can be represented as H R = (cid:18) h R h † R (cid:19) , h R = i (cid:18) − R k − δ + iS k − − δ − iS k − T k − (cid:19) , (15)where k ± = k ± ik y . For the QW parameters used inRef. 24, R ≈ − . × eE z , iS ≈ − .
10 nm × eE z ,and T ≈ − .
91 nm × eE z , where E z is the electric fieldperpendicular to the QW plane. The model could alsoinclude the bulk inversion asymmetry terms δ . To obtain the effect of the Rashba interaction on thewave functions, we find the low-energy eigenstates of H = H + H R numerically. For given k , this is a 1-D eigenvalueproblem in the y -direction, which can be discretized andsolved by standard approaches. Analytical results can beobtained by perturbation theory in H R restricted to thelow-energy subspace spanned by the unperturbed edgestates. For typical experimental parameters, the wholewave functions ˆΨ however turn out to have a significantcomponent also in the continuum of bulk modes abovethe gap, which is not adequately captured by such anapproach. Our estimates for the matrix elements P αβ below are therefore based on the numerical solutions forthe eigenstates.However, qualitative understanding can be obtained onthe basis of the model restricted to the low-energy sub-space. Projecting H R to this basis (see Appendix A), wefind the effective low-energy Hamiltonian of the system H (cid:48) R = (cid:18) − i [ R w ( k ) + T w ( k )]c . c . (cid:19) , (16) w ( k ) = χ (cid:90) ∞−∞ d y f + ,k ( y )[ k + ∂ y ] f − ,k ( y ) (17) w ( k ) = χ (cid:90) ∞−∞ d y f + ,k ( y )[ k + ∂ y ] f − ,k ( y ) , (18)where ˆΦ ± ,k ( y ) = f ± ,k ( y ) ˆ χ . This result is valid to theleading order in h R . The constant and quadratic in k terms (proportional to δ and S ) give no contribution, asˆ χ † σ y ˆ χ = 0. Using typical HgTe-QW parameters, theintegrals evaluate to w ( k ) ≈ w , k and w ( k ) ≈ w , k near the Dirac point, as illustrated in Fig. 3. The prefac-tor w , ≈ .
03 is essentially independent of the mass pa-rameter M , and w , ≈ . − × ( | M | /
10 meV). Notehere that the matrix element 0 . R k of the Rashba in-teraction with the edge states is significantly smaller thanthe R k ± appearing in the bulk Hamiltonian. The 2 × + ,k (cid:39) (cid:32) ˆΦ + ,kiw v F ˆΦ − ,k (cid:33) , ˆΨ − ,k (cid:39) (cid:32) iw v F ˆΦ + ,k ˆΦ − ,k (cid:33) , (19)where w = R w , + T w , . The Rashba interac-tion mixes the two Kramers blocks, but in the lead-ing order does not modify the energy dispersion. Al-though the mixing angle of the ˆΦ ± ,k spinors is indepen-dent of k , the total four-band spinor is not: the decay y [nm]0.000.050.100.150.200.250.300.350.40 | + , k , E | v F / w FIG. 4. The E − component of | Ψ + , − k | at k = | M | / (6 (cid:126) v F ). It is linear in the Rashba parameter w /v F ,provided | w /v F | (cid:46)
1. Shown are numerical results (solidline), the component in the unperturbed edge state subspace(dashed line), and the result of Eq. (19) (dotted). The be-havior for y (cid:38)
20 nm depends mainly on the linear in k Rashba term R . Here, M = −
10 meV. Inset: results for M = −
50 meV. lengths 1 /λ / ( k, α ) of ˆΦ ± ,k in the y -direction depend on k and are different for the α = + and α = − states:time-reversal symmetry only guarantees λ / ( k, +) = λ / ( − k, − ). This makes the electron spin axis to ro-tate both spatially and with energy ε , which ultimatelyis required for a finite P ++ .For comparison, we show in Fig. 4 the E − compo-nent of the numerically computed total edge state wavefunction Ψ + ,k ( y ), and its projection to the low-energysubspace, which can be seen to match Eqs. (19) to avery good accuracy. The E − component is propor-tional to the Rashba coupling and contributes to P ++ .As is clearly visible in the figure, neglecting the bulkstates underestimates the total amount of spin rotation,for experimentally relevant parameters. For a larger (butunphysical) value for the gap | M | , the low-energy theoryworks slightly better, as visible in the inset of Fig. 4.We can now estimate the relative order of magnitudebetween P ++ and P + − within this model. From the re-sults above, one can see that the representative quantitiesto be compared are c ++ ( K, (cid:126)r (cid:48) , (cid:126)r (cid:48) ) = ∆ i (cid:126) v F ∂ k P ++ ( K k, (cid:126)r (cid:48) ; K − k, (cid:126)r (cid:48) ) | k =0 , (20)and c + − ( K, (cid:126)r (cid:48) , (cid:126)r (cid:48) ) = P + − ( K − k F , (cid:126)r (cid:48) ; K k F , (cid:126)r (cid:48) ) . (21)In Fig. 5 we show the ratio of these amplitudes for (cid:126)r (cid:48) = (cid:126)r (cid:48) = (0 , y ) (i.e., the value at a distance y from the edge).The c ++ amplitude increases when the energies of theedge states involved approach the TI energy gap edge.The general order of magnitude of the c ++ / + − factorscan be estimated to be of the order c ++ ∼ ∆ | M | z (cid:126) v F c + − . (22) y [nm]012345 ( c ++ / c + ) × ( v F | M | / z ) M [meV]101 FIG. 5. Relative magnitude of the two types of tunneling asa function of location and parameters, at K = − k F for P ++ and K = 0 for P + − , as obtained from numerically computedˆΨ ± ,k . The scaling with the electric field is given by z ≡ × E z / (500 mV / nm ). Solid and dashed lines indicate (cid:126) v F k F = ±| M | / (6 (cid:126) v F ), i.e., energies E = E Dirac ∓ | M | / ≈ (0 . ∓ . | M | for the states involved in the c ++ / + − factors.For z /v F (cid:46) c ++ is linear in z . A similar relation is then expected also between the Γ ++ and Γ + − factors for surface contacts to area near y =0. Here and below, we characterize the strength of theRasbha interaction with the quantity z ≡ × E z / (500 mV / nm ).Using the above results for the order of magnitude of c ++ and c + − we find (see Appendix B) the estimates forthe general case with e - e interactions:Γ + − (cid:39) ( a ∆) g + g − − Γ g =1+ − (23a)Γ ++ (cid:39)
12 ( a ∆) g + g − − z ∆ v F | M | Γ g =1+ − , (23b)corresponding to cutoff a = (cid:126) v F / ∆. The relation (8)for the noninteracting value Γ g =1+ − (cid:39) ( (cid:126) v F /l T ) R K / (4 R N )fixes the magnitudes relative to experimental parameters.With finite electron-electron interactions ( g (cid:54) = 1) in thehelical liquid, all the effective tunnel rates obtain identi-cal scaling in the original short-distance cutoff a . Thisreflects the renormalization of the single-particle tunnel-ing elements t ασ by the electron-electron interactions.We can also estimate the Rashba coupling factor ap-pearing in Γ ++ . With a typical TI gap | M | ∼
10 meV ∼
100 K, we see that the factor of ∆ / | M | can be made ofthe order of 0 . . . . . z / (cid:126) v F ∼ E z / (100 mV / nm ), and as vis-ible in Eq. (19), measures the rotation of the spin axiscaused by the Rashba mixing. An upper limit for thefield that can be applied in practice is likely of the or-der E z ∼ mV / nm , as for fields larger than that, thepotential difference across the QW becomes compara-ble to the energy gap of the barrier material (CdTe).Based on this we find an estimate for the achievable ra-tio, Γ ++ ∼ . . . . + − . Finally, let us remark that tunneling that is localin real space, (cid:104) (cid:126)r | ˆ h T | (cid:126)r (cid:48) (cid:105) ∝ δ ( (cid:126)r − (cid:126)r (cid:48) ), does not lead totunneling that is local in the edge state Hamiltonian, t ασ ( x, (cid:126)r (cid:48) ) ∝ δ ( x − x (cid:48) ). This follows in a straightforwardway from the extended 2-D nature of the edge statesand the k x , k y mixing due to the spin-orbit interactions: (cid:104) α, x | ˆ h T | σ, (cid:126)r (cid:105) ∝ (cid:80) k e ik ( x − x (cid:48) ) Y α,k ( σ, y (cid:48) , z (cid:48) ) ∗ . If the spa-tial profile Y of the wave function has k -dependence onthe scale k , the sum resembles a rounded δ function ofwidth k − . For HgTe QW edge states, k − ∼ (cid:126) v F / | M | is a low-energy length scale. Because of this, a pointlikecontact to a superconductor can produce a finite P ++ ,even though assuming t ( x, (cid:126)r (cid:48) ) ∝ δ ( x − x (cid:48) ) in Eq. (7) leadsto the opposite conclusion. IV. TRANSPORT SIGNATURES
To study the experimental signatures implied by theabove model, we consider the transport problem in thesetups depicted in Fig. 1. There, two superconductingcontacts are coupled to a helical liquid, whose potentialis tuned by additional terminals at the ends. There arethree related transport effects one can study here: theequilibrium dc Josephson effect, the ac Josephson effect,and the NS conductance.We consider a general nonequilibrium case of a time-dependent pair potential ∆( t ) = | ∆ | e iϕ ( t ) in the leftcontact and ∆( t ) = | ∆ | e iϕ ( t ) in the right one, with ϕ ( t ) = ϕ / V t and ϕ ( t ) = − ϕ / V t . InEq. (4), the factors Γ inherit this time dependence. Wealso assume that only sub-gap energies are involved inthe transport, so that the quasiparticle current to thesuperconductors remains exponentially suppressed by thesuperconducting gap.The current is obtained as an expectation value of acurrent operator ˆ I = i [ H, ˆ N ] where ˆ N is the particlenumber in the HLL. From the effective Hamiltonian, weidentifyˆ I = ˆ I S + ˆ I S (24)ˆ I S = (cid:88) αβ (cid:90) S d x i Γ αβ ( x ) ψ α ( x ) ψ β ( x ) + h . c . , (25)ˆ I S = (cid:88) αβ (cid:90) S d x i Γ αβ ( x ) ψ α ( x ) ψ β ( x ) + h . c . , (26)where ˆ I S and ˆ I S must be interpreted as the parts cor-responding to currents injected through the interfaces at S S
2. The sums over αβ run over ++, + − , and −− .Considering only the Cooperon terms [cf. Fig. 7(a)],using perturbation theory up to second order in Γ we find I J,S ( t ) = − (cid:88) αβ (cid:90) S d x (cid:90) S d x X αβ ( x , x ) (27) × e iϕ ( t ) (cid:90) ∞ d t (cid:48) e − iϕ ( t − t (cid:48) ) Im[ χ αβ ( x − x , t (cid:48) )] ,I J,S ( t ) = I J,S ( t ) | ϕ ↔ ϕ , (28)where X αβ ( x , x ) ≡ e ik F ( α + β )( x − x ) Γ αβ ( x )Γ αβ ( x ) ∗ (29) χ αβ ( x ; t ) = (cid:104) e iφ α ( x,t ) e iφ β ( x,t ) e − iφ α (0 , e − iφ β (0 , (cid:105) (2 πa ) . (30)The αβ = + − component of the current coincides withthe result obtained in Ref. 18. Note that the terms in-cluded here contain the leading order of the dependencein the phase difference ϕ − ϕ .The above correlation functions can be evaluated viastandard bosonization techniques: χ αα ( x, t ) = (2 πa ) − B α ( x, t ) g + g − +2 B − α ( x, t ) g + g − − , (31a) χ + − ( x, t ) = (2 πa ) − B + ( x, t ) /g B − ( x, t ) /g , (31b) B ± ( x, t ) = − iaz sinh[ z ( ut − ia ∓ x )] , (31c)where z = πT /u .In the noninteracting case ( g = 1), we can evaluate thetime integrals analytically, to order O ( a ): I J,S ( t ) = (cid:90) S d x (cid:90) S d x [ j ++ J,S + j −− J,S + j + − J,S ] (32) j ++ J,S = | X ++ | πv F V [( V / ∆) + 4 π ( T / ∆) ] (33) × cos (cid:16) ϕ + 2 V t − V ( t − | x − x | /v F ) + φ (cid:17) ,j −− J,S = 0 , (34) j + − J,S = − | X + − | πv F z sinh(2 | x − x | z ) (35) × sin (cid:16) ϕ + 2 V t − V ( t − | x − x | /v F ) (cid:17) , where φ ( x , x ) ≡ k F ( x − x )+arg[Γ ++ ( x )Γ ++ ( x ) ∗ ]is a dynamical phase shift.Below, we discuss the implications of these results firstat equilibrium and then at finite biases. A. Equilibrium
At equilibrium, the leading contribution to the super-current comes from the + − channel. As shown in Fig. 6,the supercurrent is finite at zero temperature, and de-cays exponentially as the temperature is increased above g I / I (a) 0 1 2 3 4 Td / v F FIG. 6. Equilibrium Josephson current between the su-perconducting contacts, relative to its noninteracting zero-temperature value I . (a) Dependence ∝ ∆ − /g ( T / ∆ = 0 . T (cid:28) (cid:126) v F /d . (b) Temperaturedependence for different values of g . (cid:126) v F /d , in a way that depends on the strength of electron-electron interactions. The qualitative features are thesame as those found in Ref. 18.The contribution from the ++ and −− channels to theequilibrium current is not more relevant than + − even inthe interacting case, unlike in Ref. 19. Based on scalingdimensions in the effective Hamiltonian (dim ψ + ψ − = g − , dim ψ + ψ + = g + g − ), one finds the scaling I + − ∝ ( E/ ∆) /g − and I ++ / −− ∝ ( E/ ∆) g + g − ) − for the low-energy scale E = max( T, v F /d ), which implies that I + − will be more relevant than I ++ / −− whatever the inter-action parameter. This difference arises from the ex-clusion principle, which makes the + + / − − channelless favorable for the supercurrent, although note thatwith decreasing g (larger repulsive e-e interaction), the(+ + / − − ) contribution grows relative to the + − one.However, as noted in Section B, the scaling with the bareshort-distance cutoff a as opposed to ∆ − is identical forΓ ++ / −− and Γ + − . B. Nonequilibrium
When the superconductors are biased with a finite volt-age, currents generically start to flow between all the ter-minals, and they may also be time dependent due to theac Josephson effect. To fully understand these effects, itis illuminating to compute the spatial distribution of thecurrents in the system.The spatial dependence of the currents in the helicalliquid can be obtained by making use of the followingexpression for the current operator ˆ I = v F √ π ∂ x φ ( x, t ) inthe Heisenberg picture (see App. C): ˆ I ( x, t ) = ˆ I ( x, t ) + v F (cid:90) ∞−∞ d x (cid:48) d t (cid:48) (cid:88) α = ± α (cid:16) (36)1 + g g D α ( x, t ; x (cid:48) , t (cid:48) ) − − g g D − α ( x, t ; x (cid:48) , t (cid:48) ) (cid:17) ˆ j α ( x (cid:48) , t (cid:48) ) , ˆ j α ( x (cid:48) , t (cid:48) ) = δV ( t (cid:48) ) δφ α ( x (cid:48) ) . (37)This applies to any Hamiltonian of the form H = H + V ( t ), where H is the bosonized Hamiltonian in Eq. (1);ˆ I is the current operator evolving in time with the unper-turbed Hamiltonian H . The operator j α ( x (cid:48) , t (cid:48) ) can beinterpreted as the current density injected to the mode α = ± at position x (cid:48) at time t (cid:48) . The functions D +( − ) are initially right(left)-propagating δ pulses originatingat point x (cid:48) at time t (cid:48) .Let us for simplicity assume that the two supercon-ducting contacts are pointlike in the low-energy model,that g = 1, and that the helical liquid is homogeneous.Then, D ± ( x, t ; x (cid:48) , t (cid:48) ) = θ ( t − t (cid:48) ) δ ( x − x (cid:48) ∓ v F ( t − t (cid:48) )) andwe find I ( x, t ) = (cid:88) j =1 , (cid:2) θ ( x − x j ) l T (cid:104) ˆ I + ,j ( t − | x − x j | v F ) (cid:105) (38) − θ ( x j − x ) l T (cid:104) ˆ I − ,j ( t − | x − x j | v F ) (cid:105) (cid:3) . ˆ I + ,j = ∂∂φ + (cid:104) Γ ++ ψ + ψ + + Γ −− ψ − ψ − + Γ + − ψ + ψ − + h . c . (cid:105) | x = x j (39)= 2 i Γ ++ ψ + ψ + + i Γ + − ψ + ψ − + h . c . , ˆ I − ,j = 2 i Γ −− ψ − ψ − + i Γ + − ψ + ψ − + h . c . , (40)where l T is the small contact size, and the expectationvalues (cid:104)·(cid:105) closely correspond to the different parts of theinjection currents I J,S /S evaluated in the previous sec-tion. Indeed, (cid:104) ˆ I + , ( t ) (cid:105) = j ++ S ( t ) + 12 j + − S ( t ) , (41a) (cid:104) ˆ I − , ( t ) (cid:105) = j −− S ( t ) + 12 j + − S ( t ) , (41b) (cid:104) ˆ I + , ( t ) (cid:105) = j ++ S ( t ) + 12 j + − S ( t ) , (41c) (cid:104) ˆ I − , ( t ) (cid:105) = j −− S ( t ) + 12 j + − S ( t ) . (41d)The physical interpretation is particularly simple: thecontacts at x and x inject current to the helical liq-uid. The component due to + − tunneling splits evenlyto the left and right-moving modes, whereas the ++ and −− components end up solely in the + and − modes, re-spectively. Within each edge mode, the injected currentpropagates with the Fermi velocity, as indicated by theretarded time arguments. The calculations done in the previous section indicatedthat in this case j −− S = 0 and j ++ S = 0 to leading order in a . Therefore, essentially all of the current injected by the++ and −− tunneling in fact flows only to the reservoirsthat maintain the chemical potential of the helical liquidat µ = 0, rather than between the two superconductingcontacts, which can be verified by computing the currentat x < x , x and at x > x , x . The effect essentiallyamounts to a modulation of the NS conductance betweenthe superconductors and the normal leads by the (time-dependent) phase difference ϕ ( t ) − ϕ ( t ) between thesuperconducting contacts.Based on the above results, we can write down an ex-pression for the part of the NS current [see Fig. 1(c)]that depends on the phase difference, in the configura-tion V = V = V : δI NS = 2 l T | X ++ | π V cos(2 V d + φ ) cos( ϕ ) (42) × [( V / ∆) + 4 π ( T / ∆) ]+ 4 l T | X + − | π sin(2 V d ) cos( ϕ ) z sinh(2 zd ) . Note that the modulation of the NS conductance fromthe + − channel decays exponentially as the tempera-ture increases, whereas the ++ contribution does not.The same situation should persist in all orders of pertur-bation in the effective Hamiltonian for the + − tunnel-ing: the terms coupling to ϕ contain inequal numbers of ψ + ( d ) ψ − ( d ) and ψ † + ( d ) ψ †− ( d ), which implies that the cor-relation function is of the form [ B + ( d, t ) B − ( d, t )] × O (1)and thus has an overall exponential prefactor e − πT d/v F .Therefore, there in principle is a temperature regime at T (cid:29) (cid:126) v F /k B d in which the leading contribution to the ϕ dependence of the NS current comes mainly from the++ tunneling, despite the power-law suppression of thischannel in helical liquids. The physical reason for thedifference can be seen in Fig. 2: for the + − channelan electron pair injected to energies ± ε and traversingthrough the junction obtains an energy dependent phasefactor e i ( k + k ) d = e − iεd/v F , typical of Andreev reflec-tion, which averages towards zero when a finite energywindow is considered. For the ++ channel, because of thelinear spectrum, the corresponding phase factor e ik F d isenergy-independent and no such averaging occurs.The above result requires validity of the perturbationtheory, i.e., l T Γ + − ∼ R K / R N (cid:46) l T is the con-tact length. If this condition is not satisfied, an addi-tional contribution decaying only as 1 /T in temperaturearises in the + − channel. This additional proximityeffect contribution is similar to what occurs in metallicsystems of a more macroscopic size, although there itcan be much amplified as the electrons can stay a longtime near the NS interface due to impurity scattering.With finite repulsive interactions ( g < − contribution to the NS conductance obtains a power-lawprefactor according to the scaling dimensions, I + − ∝ ( E/ ∆) /g − with E = max( T, V ), and the prefactor ofthe ++ part is modified, I ++ / −− ∝ ( E/ ∆) g + g − ) − .According to the correlation functions (31), exponentialdecay will also appear in the ++ part due to charge frac-tionalization, but it will be weaker than in the + − partfor all values of g .A second distinguishing feature of the ++ contributionto the NS current is that it is expected to oscillate notonly as a function of the bias, but also as a function ofthe Fermi wave vector appearing in the dynamical phase φ . In metals or other systems where k F is large, thewavelength of such oscillations would be on the atomiclength scales, and the contribution would average to zero[as ∼ sinc( k F w ) ] over any practical contact size w . However, this needs not be the case in HgTe-QW (or innanotubes, see Ref. 19) when the Fermi level lies closeto the Dirac point: for example assuming | µ − E Dirac | ∼ M/ ∼ . /k F ∼
150 nm. Such lengthscales are likely experimentally accessible.One should also note that the finite wave velocity com-bined with the ac Josephson effect causes some additionaleffects. The current propagates at the (renormalized)Fermi velocity v F /g , rather than at the substantiallyhigher speed of light c at which electromagnetic exci-tations propagate. Assuming only the + − channel con-tributes, one can find the spatial dependence of currentbetween the two contacts: I ( t ) ∝ sin[2 V t − V xv F ] + sin[2 V t − V ( d − x ) v F ] . (43)Based on this, it is clear that for biases V (cid:38) (cid:126) v F /ed be-tween the two superconducting electrodes, the ac Joseph-son effect must be associated with appreciable standingwave oscillations in the charge density. This behavior isnot specific to helical liquids: a similar spatially resolvedcalculation as above for the spinful liquid ac Josephson ef-fect of Ref. 18 should also produce this feature. Whethersuch effects are observable in reality, however, dependson how realistic the model assumptions about screeningare in the systems studied (see also Ref. 31). V. DISCUSSION AND CONCLUSIONS
In this work we considered the proximity effect inducedin a helical edge state, taking into account a spatially andenergetically non-constant spin quantization axis. Suchrotation of the spin axis naturally arises from the spin-orbit interaction in real materials such as the HgTe QWs,for example in a controlled way by structure inversionsymmetry breaking Rashba terms. This has the conse-quence that the singlet correlations in an s-wave super-conductor can also induce a proximity effect in the samechannel of left and right-movers (Γ ++ / −− ) in additionto the usual term where the correlation is between op-posite chiral states (with amplitude Γ + − ). We deriveda description of the proximity effect in both channels inthe presence of Rashba interaction using a simple modelfor the tunneling between the superconductor and the helical edge state, respecting spin conservation and timereversal symmetries.The extra transport channels (+ + / − − ) describe pro-cesses that are in principle parasitic for the splitting ofa Cooper pair into two electrons propagating into differ-ent directions (the + − channel) . For a single super-conducting contact to the helical liquid, the scaling withtemperature (or bias voltage) at low energies however al-ways favors the + − channel. In Ref. 32, the two-particletunneling into the bulk of a spinful Luttinger liquid wasfound to be suppressed in a power law in 1 / ∆ similarly ashere, but there the tunneling into the + + / − − channelwas found to be dominant. The difference arises becausein a spinful liquid the two opposite spins can tunnel intodifferent spin channels, and therefore no Pauli-blockingfactors appear.Observing effects related to the same-mode tunneling(Γ ++ ) is likely rather challenging, as they can be sup-pressed relative to Γ + − by several factors: the power lawsuppression ( T / ∆) + ( V / ∆) from exclusion principle,suppression of the tunneling factor Γ ++ itself, and aver-aging effects related to contacts if they are larger than1 /k F (i.e. ∼
150 nm for parameters in Fig. 5). How-ever, by observing the dependence of the NS conductanceon the superconducting phase, the relative difference canbe reduced due to the exponential dephasing of the + − contribution at high temperatures. (In the case that theonly mode of transport is via the + − channel, the mod-ulation would still contain features distinct to ballistictransport, such as oscillations as the bias voltage is in-creased.) The question is therefore more on how smallsignals can be detected in the conductance, oscillatingwith the phase difference ϕ , and how large the thermalfactor 2 πk B T d/ (cid:126) v F can be made before inelastic interac-tion effects (e.g. electron-phonon scattering), which wehave neglected, start to play a role.We find that controlling the spin axis via external elec-tric fields in HgTe-QW in general requires very strongfields, because of the weak coupling of the additionalspin-orbit interactions to the edge states. For our case,this makes achieving a large Γ ++ more difficult, andmay in general pose problems to proposals relying onthe control of the spin axis. Making an optimistic es-timate, we find from Eqs. (23) and (42) that the ra-tio of the two contributions to the amplitude of phase-dependent oscillations in the conductance is ( V (cid:28) T , g = 1, E z ∼ mV / nm ) δG ++ ( ϕ ) δG + − ( ϕ ) = 2 π (cid:12)(cid:12)(cid:12)(cid:12) Γ ++ Γ + − (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) k B T ∆ (cid:19) sinhc (cid:18) πk B T d (cid:126) v F (cid:19) (44) ∼ (cid:18) k B TM (cid:19) sinhc(2 πk B T d/ (cid:126) v F ) , where sinhc( x ) = sinh( x ) /x . With finite e - e inter-actions [cf. Eq. (B25)], the ratio is multiplied by(∆ /T ) − g , making the result depend only on ∆ /M inthe limit g →
0, and the exponential dependence be-0comes ∼ exp(2 πg [2 − g ] k B T d/ (cid:126) v F ). Taking junctionlength d = 3 µ m, the temperature scale of the exponen-tial suppression factor is E T ≡ (cid:126) v F / (2 πd ) ≈ . T ∗ ≈ E T { log[2( M/E T ) ] − log log[2( M/E T ) ] } ≈ . M (here M = −
10 meV).Given a suitable superconducting material, this shouldbe achievable.Another option for amplifying the same-mode tunnel-ing could be to break the time-reversal symmetry andintroduce additional spin flips or spin rotation, for exam-ple via magnetic impurities or ferromagnets. The effectcould still be detected in the NS conductance, as thatconclusion is only based on the generic form of the low-energy effective Hamiltonian.Observe that in our analysis the true 2D nature ofthe edge states in HgTe-based QWs was important. The(+ + / − − ) proximity channel cannot be found in a com-pletely 1D description, as in such a picture the spin quan-tization axis is simply rotated globally by the Rashbaterms (cf. Refs. 13 and 33). Such rotations can haveno consequences for Cooper pair injection into a singleedge, due to the s -wave symmetry of the pairing [cf.Eq. (9)]. Spatially inhomogeneous Rashba interaction, could, however, induce a finite + + / − − amplitude.As in other systems with small critical currents, also here thermal fluctuations in the superconductingphase difference are a problem for measurements of thetemperature-dependence of the Josephson effect: thetemperature scale relevant for the phase fluctuations, E J = (cid:126) I c / e , is smaller than the intrinsic one, E T = (cid:126) v F /d . More complicated measurement schemes than the simple current-biased setup in Fig. 1(b) maynevertheless help in overcoming this problem. One should, however, note that only the Josephson currentis a problematic observable in this respect. The mea-surement of phase oscillations of the NS conductance inthe setup of Fig. 1(c) is expected to suffer much less fromphase fluctuations, as there the phase difference is lockedby the magnetic flux and the large critical current of thesuperconducting loop itself.In summary, starting from a tunneling Hamiltonian,we derived an effective low-energy theory describing thesuperconducting proximity effect in the helical edge stateof a 2D topological insulator. We showed that in thesesystems, despite the s-wave symmetry of the supercon-ductor, correlations can occur both in (++ / −− ) and be-tween (+ − ) the left and right moving modes, and withina simple model, we estimated the expected magnitudesfor the effective proximity gap parameters in HgTe/CdTequantum wells. Based on the effective Hamiltonian, westudied the dc and ac Josephson effects in the helical liq-uid, and considered phase-dependent oscillations of theNS conductance. In nonequilibrium, we found that cor-relations within the same mode can give rise to a long-ranged interference effect, which could act as a signatureof their presence. Our results also shed light on the mean-ing of ”spin” in the helicity of these edge states which isof importance if one intends to use these edge states forspin-injection or spin-detection. ACKNOWLEDGMENTS
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Quantum Transport in Semi-conductor Submicron Structures , edited by B. Kramer(Kluwer, 1995) arXiv:cond-mat/9605014. F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys.Rev. B , 165309 (2005). Appendix A: HgTe/CdTe QW edge states
The edge states of a HgTe-QW can be described withinthe four-band model introduced in Ref. 2. Here, we deriveexplicit analytical expressions for the edge states in asingle edge following the approach of Ref. 21, for usein Section. III B, and to demonstrate that the directionwhere the 4-band spinors point is independent of k and M , in the absence of inversion symmetry breaking terms.The four-band Hamiltonian reads H = (cid:18) h ( k ) 00 h ( − k ) ∗ (cid:19) , (A1) h ( k ) = (cid:15) ( k ) σ + (cid:126)d ( k ) · (cid:126)σ , (cid:15) ( k ) = C − D | (cid:126)k | , (A2) (cid:126)d ( k ) = ( Ak, − Ak y , M − Bk ) . (A3)For the parameters A , B , C , D we use values fromRef. 24: A = 365 meV nm, B = −
706 meV nm , D = (a) (b) tF Γ F + t Γ + FIG. 7. (a) Cooperon giving a contribution to the Josephsoncurrent. (b) Integrating out the relative coordinates gives aneffective description of Andreev reflection. −
532 meV nm and take C = 0 (it only shifts the Diracpoint). For an edge with QW lying at y >
0, with thewave function vanishing at y = 0, the edge eigenstatesare:ˆΨ + ,k = (cid:18) ˆΦ + ,k (cid:19) , ˆΨ − ,k = (cid:18) − ,k (cid:19) , (A4)ˆΦ α,k = N e − ikx (cid:2) e − λ y − e − λ y (cid:3) (cid:18) −√ D − B √− D − B (cid:19) , (A5)where E = − DMB − αk A √ B − D B , (A6) λ / = (cid:113) k + F ∓ (cid:112) F − Q , (A7) F = A − BM + DE )2( B − D ) , Q = M − E B − D , (A8)and N is a normalization constant.Note that the spinor points to a single direction inde-pendent of k or energy E , so that the above results areof the formˆΨ + ,k = (cid:18) ˆ χ (cid:19) f + ,k , ˆΨ − ,k = (cid:18) χ (cid:19) f − ,k , (A9)where the constant spinor ˆ χ is normalized ( ˆ χ † ˆ χ = 1) anddepends only on the parameters B and D . The envelope f α,k ( x, y ) = N (cid:48) e ikx [ e − λ y − e − λ y ] is a scalar function.Using the above parameters we find ˆ χ = (0 . , . T ,so that the spinor has the main contribution in the H Appendix B: Low-energy Hamiltonian
In this Appendix, we derive the effective Hamilto-nian in Eq. (4) via perturbative renormalization group(RG), aiming to approximate the Cooperon term [seeFig. 7(a)] appearing in the perturbation expansion in thetunneling with the effective term in Fig. 7(b). When com-puting the Josephson current, for the Γ + − channel thisapproach is compatible with that used in Ref. 18 andelsewhere in the long-junction case. For the Γ ++ andΓ −− channels we however need to pay more attention tothe tunneling elements.2We take our effective Hamiltonian to have the form: H eff = H + H T + H T = H + V , (B1) H T = (cid:88) α = ± ,σ (cid:48) = ↑ , ↓ (cid:90) d x d r (cid:48) t ασ (cid:48) ( x, (cid:126)r (cid:48) ) ψ † α ( x ) ψ Sσ (cid:48) ( (cid:126)r (cid:48) ) + h . c . (B2) H T = (cid:88) αβ (cid:90) d x Γ αβ ( x ) ψ α ( x ) ψ β ( x + a ) + h . c . , (B3)and rescale only the cutoff in the helical liquid in theprogress of RG. Because correlations in the superconduc-tor decay exponentially at distances (cid:38) ∆ − , and becausethe superconducting gap prohibits dissipation at low en-ergies inside the superconductor, H T can be neglected incalculations after the scaling to low-energy length scales (cid:29) ∆ − is done — which reduces the effective Hamilto-nian to that in Eq. (4).The scaling equations readd t ασ (cid:48) d l = [2 − η ] t ασ (cid:48) (B4)dΓ αβ d l = [2 − η ,αβ ]Γ αβ + S αβ ( l ) , (B5)where η = ( g + g − ) / e iφ ± appearing in tunneling H T , and η ,α, − α = 1 /g and η ,α,α = g + g − are scaling dimensions of the operators in H T . Although essentially a standard calculation, belowwe explain the derivation of S αβ in detail.Below, we need the factorization ψ α ( x , τ ) ψ α ( x , τ ) = ( a q ) η U α ( τ ) U α ( τ )2 πa (B6) × e ik F ( α x + α x ) × : e iφ α ( x ,τ ) e iφ α ( x ,τ ) : C α ,α ( x − x , τ − τ ) , where q = 2 π/L is the infrared cutoff, and the correla-tion functions read C ++ ( z ) = ( q z ) g − ( g ) ( q z ∗ ) g − ( − g ) (B7) C α, − α ( z ) = | q z | (1 − g ) / (2 g ) , (B8)where C αβ ( x, τ ) = C αβ ( z ), z = v F τ − ix , and C −− ( z ) = C ++ ( z ) ∗ . Observe that C α α ( z ) ∝ q η ,α α − η .To perform the RG steps, we also need the correspond-ing operator product expansions. Taking sign changesdue to Klein factors and time ordering into account, wefind [cf. Eq. (B6)]: T [ ψ α ( z ) ψ α ( z )] (B9) (cid:39) u α α ( z − z ) ψ α ( z + z ψ α ( z + z a ) , where u α α ( z ) = e i ( α − α ) k F x/ sgn( τ ) δ α ,α (B10) × C α ,α ( x sgn( τ ) , | τ | ) C α ,α (0 , a ) , and δ α ,α in the sign factor arises from the fact that U α U α = ( − δ α ,α U α U α .The source term S αβ ( l ) for the Andreev reflection pro-cesses Γ αβ appears from the second-order term in thepertubation expansion of the partition function, Z/Z = (cid:104) T e − (cid:82) β d τ λV ( τ ) (cid:105) = 1 + c λ + c λ + . . . . Combining two H T and using the operator product expansions gives acontribution to Γ αβ . We also trace out the supercon-ductors at this step, factorizing the expectation value to (cid:104) . . . (cid:105) = (cid:104) . . . (cid:105) HLL, (cid:104) . . . (cid:105) S, . This yields the resultd c d l = (cid:90) d z (cid:88) αβ (cid:104) T [ ψ α ( z ) ψ β ( z + a )] (cid:105) S αβ ( l, z ) , (B11) S αβ ( l, z ) = a [ ∂ r f αβ ( l, z, r )] r = a , (B12) f αβ ( l, z, r ) = (cid:90) | z (cid:48) |
1. The effective tun-nel rates obtain an identical scaling in the bare short-distance cutoff a related to interactions, which appearsbecause of a renormalization of the tunneling elements t ασ . The low-energy scaling with a follows the scalingdimensions in the effective Hamiltonian. Finally, Γ ++ has an additional factor z ∆ /v F | M | that is a signatureof the Rashba coupling.The expression (5) for Γ ++ deserves some comments:First, we know that F ( ω ) (cid:39) cω for ω →
0, sothat ∂ ω F | ω =0 vanishes, and from Eq. (11) we know that P ++ ( − k ) = − P ++ ( k ) which means that ∂ k P ++ is evenin k and can be finite at k = 0. Note that the gradi-ents ∂ ω , ∂ k appear because the boson correlation function (cid:104) e iφ + ( x,τ ) e iφ + (0 , (cid:105) vanishes at x, τ → ψ + ψ + than ψ + kψ + , which is in agree-ment with the results of Ref. 16.Finally, we observe that in the noninteracting case,Γ + − is related to the leading-order off-diagonal Nambucomponent of the self-energy. The factors Γ ++ (andΓ + − in the interacting case) however in general containadditional information, as the out-integration of shortlength scales captures the renormalization from interac-tions, and the effect of the exclusion principle when av-eraging ψ ( x + x (cid:48) ) ψ ( x ) over short distances x (cid:48) . Appendix C: Current operator
For completeness, we include here a derivation ofEq. (36) that shows the result obtained in Ref. 28 ap-plies also to time-dependent perturbations. Related re-sults can be found e.g. in Ref. 37, and a special case ofthe present result is given in terms of path integrals inRef. 38.4Consider the Heisenberg equation of motion under aHamiltonian H = H + V ( t ), where H is given in Eq. (1),and the perturbation V ( t ) is switched on at t >
0. Iter-ating the equation of motion for ∂ x φ twice, one obtains ∂ t ( ∂ x φ ) − ∂ x ( u ∂ x ( ∂ x φ )) = s ( x, t ) (C1) s ( x, t ) = [ H , [ H , ∂ x φ ]] − [ H, [ H, ∂ x φ ]] + i [ ˙ H, ∂ x φ ] , (C2)where ˙ H = ˙ V contains the explicit time dependence ofthe Hamiltonian, and u ( x ) = v F /g ( x ) is the renormalizedwave velocity. The solution to this linear equation canbe written in terms of the retarded Green function of thewave equation on the LHS: ∂ x φ ( x, t ) = ∂ x φ ( x, t ) (C3)+ (cid:90) ∞−∞ d x (cid:48) C R ( x, t ; x (cid:48) , i [ V, ∂ x φ ( x (cid:48) , (cid:90) ∞ d t (cid:48) (cid:90) ∞−∞ d x (cid:48) C R ( x, t ; x (cid:48) , t (cid:48) ) s ( x (cid:48) , t (cid:48) ) , where ∂ x φ evolves under H , and the second term en-sures that the initial condition ∂ t ( ∂ x φ ) = i [ H, ∂ x φ ] issatisfied — this follows from ∂ t C R ( x, t ; x (cid:48) , t (cid:48) ) | t → t (cid:48) +0 + = δ ( x − x (cid:48) ).We can also rewrite s using properties of the fields and H : s ( x, t ) = − ∂ x (cid:0) v F g ( x ) − δV ( t ) δφ ( x ) (cid:1) + ∂ t δV ( t ) δϑ ( x ) , (C4)where we noted the correspondence[ A, ∂ x φ ( x )] = − i δAδϑ ( x ) , [ A, ∂ x ϑ ( x )] = − i δAδφ ( x ) , (C5)valid for functionals A = A [ ϑ, φ ]. The expression for s can be substituted in Eq. (C3), and an integration byparts transfers the gradients to operate on C R . One of the resulting boundary terms cancels the second term inEq. (C3), and the others vanish, provided the perturba-tion s vanishes at x → ±∞ .We then find the exact result ∂ x φ ( x, t ) = ∂ x φ eq ( x, t ) + (cid:90) ∞ d t (cid:48) (cid:90) ∞−∞ d x (cid:48) (cid:104) (C6) v F g ( x (cid:48) ) − [ ∂ x (cid:48) C R ( x, t ; x (cid:48) , t (cid:48) )] δV ( t (cid:48) ) δφ ( x (cid:48) ) − [ ∂ t (cid:48) C R ( x, t ; x (cid:48) , t (cid:48) )] δV ( t (cid:48) ) δϑ ( x (cid:48) ) (cid:105) . We can simplify this further by making use of prop-erties of the 1-D wave equation. The Green function C R ( x, t ; x (cid:48) , t (cid:48) ) satisfies the initial value problem[ ∂ t − ∂ x u ∂ x ] C R = 0 , at t > t (cid:48) , (C7) C R = 0 , ∂ t C R = δ ( x − x (cid:48) ) , at t = t (cid:48) . (C8)If u is a constant, the solution is a sum of two wavefronts C R = C R + + C R − , C R ± = (4 u ) − θ ( t − t (cid:48) ) sgn[ ± ( x (cid:48) − x )+ u ( t − t (cid:48) )]. This is valid in the limit t → t (cid:48) also if u is smoothlyspatially varying — the wave equation only sees u around x (cid:48) . Since the wave equation is linear and its solution isunique, the Green function can always be decomposedto these two parts. Let us now define D ± = 2 ∂ t C R ± and F ± = 2 ∂ x (cid:48) C R ± . They satisfy the wave equation at t > t (cid:48) ,and the initial conditions are inherited from the t → t (cid:48) behavior of C R ± : D ± = δ ( x − x (cid:48) ) , ∂ t D ± = ∓ u ( x (cid:48) ) δ (cid:48) ( x − x (cid:48) ) , (C9) F ± = ± u ( x (cid:48) ) − δ ( x − x (cid:48) ) , ∂ t F ± = − δ (cid:48) ( x − x (cid:48) ) . (C10)Due to linearity, clearly F ± = ± u ( x (cid:48) ) D ± . Be-cause C R ( x, t ; x (cid:48) , t (cid:48) ) = C R ( x, x (cid:48) , t − t (cid:48) ), we then find ∂ t (cid:48) C R = − ∂ t C R = − D + + D − and ∂ x (cid:48) C R = F + + F − = u ( x (cid:48) ) D + − D − . Substituting these to Eq. (C6) and defin-ing j ± = √ π [ δVδφ ± δVδϑ ] ≡ δVδφ ±±