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Andreev reflection in edge states of time reversal invariant Landau levels K. G. S. H. Gunawardana1,∗ and Bruno Uchoa1 1Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73069, USA† (Dated: July 14, 2015) Wedescribetheconductanceofanormal-superconductingjunctioninsystemswithLandaulevels that preserve time reversal symmetry. Those Landau levels have been observed in strained honeycomb lattices. The current is carried along the edges in both the normal and superconducting regions. WhentheLandaulevelsinthenormalregionarehalf-filled,Andreevreflectionismaximal and the conductance plateaus have a peak as a function of filling factor. The height of those peaks is quantized at 4e2/h. The interface of the junction has Andreev edge states, which form a coher- 5 ent superposition of electrons and holes that can carry a net valley current. We identify unique 1 experimental signatures for superconductivity in time reversal invariant Landau levels. 0 2 PACSnumbers: 71.21.Cd,73.21.La,73.22.Gk l u J Introduction. At zero temperature, Cooper pairs are around partial filling factors ν = 4n, n ∈ Z, when the 1 protected against phase decoherence by time reversal normal LLs are half-filled, and their height is quantized 1 symmetry (TRS). The most promising scenario to study at (n+ 1)8e2/h. 2 the coexistence of superconductivity and quantum Hall The Andreev edge states carry a finite charge current ] l states1requiresasystemthatcreateswellseparatedLan- per valley along the NS interface. The valley current l a dau levels (LLs) in the complete absence of magnetic becomes asymptotically small at large energy (ε (cid:29) µ), h flux2. In strained honeycomb lattices, uniform strain when electrons and holes move with the same group ve- - s fields can couple to the electrons as a pseudo-magnetic locity. In the opposite regime, ε (cid:28) µ, electrons and e field oriented in opposite directions in the two valleys3,4. holes have opposite group velocities along the interface, m As observed experimentally in graphene5–7 and in de- and the valley current is finite. We find transport and . formed honeycomb optical lattices8, this field can pro- spectroscopy signatures that uniquely identify proximity t a duce sharp LL quantization, and at the same time, zero induced superconductivity in TRS LLs2,12. m net magnetic flux at every lattice site. Hamiltonian. In the continuum, the electronic Hamil- - Intheusualsemi-classicalpicture,electronsmovingin tonian in the presence of a pseudo magnetic-field is13 d n cyclotronic orbits are reflected as holes at the interface (cid:18) (cid:19) H 0 (cid:88) o of a superconductor. At low energy, the holes have op- H0(A)= 0+ H = να⊗Hα, (1) c posite momentum (valley) and the same velocity of the − α=± [ incidentelectrons. Sinceparticle-holeconversionchanges 4 the sign of the Lorentz force and effective mass, the An- where να = 21(ν0 +ανz) is the projection operator into v dreevreflectedholesandtheelectronsmovecoherentlyin the valleys α=±, and 8 the same direction of the normal-superconducting (NS) H (A)=v(−i∇+αA)·(cid:126)σ −µ (2) 8 interface. The result is an Andreev edge state, a co- α α 9 herent superposition of electrons and holes, which move is the Dirac Hamiltonian in each valley. (cid:126)σ = (σ ,ασ ) 1 in alternating skipping orbits along the NS interface9,10. α x y 0 Pseudo-magnetic fields nevertheless reverse their direc- . 1 tion in opposite valleys, preserving TRS. As a conse- a) b) 0 quence, whentheenergyofthequasiparticles, ε, issmall S S 5 compared to the Fermi energy, µ, reflected holes can re- 1 : trace the cyclotronic path of the incident electrons, and y y v either form a bound state or counterpropagate along the i X same insulating edge, as shown in Fig. 1a. r In this Rapid Communications, we study the trans- x x a port across a NS junction in a honeycomb lattice with FIG.1. Semi-classicalpictureoftheedgestatesinTRSLLsat a discrete spectrum of TRS LLs. The current is carried theinterfacewithasuperconductingregion. Solidblacklines: throughskippingorbitsalongtheedgeofthesystemboth incident electrons with skipping orbits. Red lines: propagat- in the normal11 and in the superconducting regions, as ing electron pairs at the edge of the superconductor; dashed depicted in Fig. 1. We address the regime where the lines: Andreev reflected holes. a) ε (cid:28) µ regime: holes are coherence length is longer than the magnetic length. In retroreflected and retrace the path of the incident electrons, the limit where the energy of the quasiparticles is small preservingtheirguidingcenter. Theycanformaboundstate compared to the separation of the LLs, we show that or counterpropagate along the same edge. b) ε > µ regime: particle-hole conversion produces a peak in each longi- Andreev holes are specularly reflected and propagate along tudinal conductance plateau. Those peaks are centered the NS interface as a superposition of electrons and holes. 2 is a vector of Pauli matrices, v is the Fermi velocity and a) b) µ is the chemical potential. 1 Near the NS interface, the Bogoliubov-deGennes (BdG) Hamiltonian is gy er 0 n (cid:18)H (A) ∆ˆ(r) (cid:19) E H = 0 (3) BdG ∆ˆ∗(r) −TH (A)T−1 0 -1 where14 -5 0 5 -5 0 5 (cid:18) (cid:19) kl kl 0 σ xB xB T = z C =ν ⊗σ C (4) σ 0 x z z FIG. 2. Energy spectrum versus guiding center k (cid:96) at the x B is the time reversal symmetry operator, and C the edge (y = 0) for √µ = 1.5 (ν = 4) and ∆ = 0.5. Energy charge conjugation. Because the field A preserves TRS, scales in units of 2v/(cid:96)B. a) Normal region (x < 0). b) TH (A)T−1 = H (A). In a singlet state, the elec- superconducting region (x>0). 0 0 tronspairsymmetricallyacrossthevalleys. Thesimplest Ansatz for the off-diagonal term can be written as2 Fory ≥0,whereM =0,theenergyspectrumoftheLLs (cid:18) 0 σ (cid:19) at the edge is ∆ˆ(r)=∆(r) 0 =∆(r)ν ⊗σ . (5) σ0 0 x 0 √ (cid:112) (cid:15)(k )=s( 2v/(cid:96) ) |n(k )|−µ (10) x B x Assuming a sharp NS interface, the superconductor gap where n(k ) is a real number and s = sign[n(k )]. The varies abruptly at x=0, x x corresponding eigenvectors in the two valleys are of the (cid:40) ∆ , for x>0 form ∆(r)= (6) 0 , for x<0,  D (cid:16)√ξ (cid:17)  Ψα,n(ξ)= n(kx)−1(cid:16) 2(cid:17), (11) which separates the normal (x < 0) from the supercon- sαD √ξ n(kx) 2 ducting region (x>0). In the Landau gauge, A=(−By,0), the Hamiltonian where D (x) are parabolic cylinder functions. The n(kx) in the valleys can be written as determination of n(k ) follows from enforcing boundary x v conditions at the edge and results in a discrete number Hα = √2(cid:96) α(i∂ξσy+ξσx)−µ (7) ofedgestatesshowninFig. 2a. Fordefiniteness,w√econ- B sider the case of a zigzag edge, where φ (k(cid:96) / 2) = B,+ B whereξ =(cid:96) k−y/(cid:96) isadimensionlesscoordinatewith 0, although similar conclusions apply to any choice of B B guiding center X =(cid:96) k, and (cid:96) =(cid:112)(cid:126)/Be is the mag- boundary conditions. α B B netic length (restoring (cid:126)). By convention, we define the In the superconducting region, we can decompose the momentum along the edge for each valley as k = αk, BdG Hamiltonian (3) into two identical blocks of 4×4 x with k > 0. The electronic wavefunction in the normal matrices, where region moving towards (away from) the interface in val- (cid:18) (cid:19) H ∆σ ley α = + (−) takes the form ψˆ (x,y) = eαikxΨ (y), α 0 Φ =EΦ , (12) α α ∆∗σ −H α α whereΨ (y)=(φ ,φ )isatwocomponentspinorin 0 α α A,α Bα sublattice A and B of the honeycomb lattice. is the reduced BdG equation and Φ = (Ψ Ψ ) is a α e,α h,α Edge states. In order describe the edge states, we 4 component spinor including electron and hole states. introduce a mass term potential M(y)ν0 ⊗ σz, where At y > 0, where the wavefunctions are finite (M = 0), M(y) = W for y < 0 and M = 0 for y ≥ 0. In the limit the solution follows from squaring (12) with the charge W →∞, this potential describes the edge of the system conjugatedBdGHamiltonian,whichresultsinthediffer- aty =0. Theelectronswillmoveinskippingorbitsalong ential equation the y = 0 line under the influence of a pseudo-magnetic (cid:20) (cid:21) field. In the normal side of the NS interface, 1 1 ∂2− ξ2+(cid:15)2+µ2+∆2+2µM Φ =0, (13) 2 ξ 2 α [H +M(y)σ ]Ψ =εΨ , (8) α z α α where where Ψ (y) is a two component spinor in the sublattice basis. Mαultiplying Eq. (8) on the left by the charge M=(cid:18)(cid:15)− 21σz −∆σ0 (cid:19) conjugated form of H +M(y)σ , where µ → −µ, this −∆σ −(cid:15)− 1σ α z 0 2 z equation assumes a diagonal form, is a 4×4 matrix in the particle-hole basis, and ∆ is as- (cid:20) (cid:21) 1 1 1 sumedreal. ThefourcomponentspinorthatsatisfiesEq. ∂2− ξ2−M2(y)+(ε+µ)2− σ Ψ =0. (9) 2 ξ 2 2 z α (13) and hence (12) is 3 1 (cid:18) (cid:19) β Ψ (ξ) Φ = α α,ns (14) α β−αΨα,ns(ξ) 2|r|A 0.5 (cid:113) where β = 1(1+α(cid:112)1−∆2/E2), with α=± index- α 2 ing the valleys. The energy spectrum in the supercon- 0 ducting edge is given by 10 (cid:114) (cid:16) √ (cid:112) (cid:17)2 h) E(kx)= s¯( 2v/(cid:96)B) |ns(kx)|−µ +∆2, (15) 2e/ 2 5 G ( where s¯ = sign[n (k )] and n (k ) is a real number to s x s x be found from the boundary condition at the edge (y = 0), in superconducting√side. Imposing similar boundary 00 2 44 6 88 conditions φB,+(k(cid:96)B/ 2) = 0, the energy spectrum at ν the superconducting edge is shown in Fig. 2b. Transport across the NS junction. In the normal re- FIG. 3. Top panel: Andreev reflected hole probability |r |2 A gion, the electron and hole like excitations are decou- per channel as a function of the no√rmal filling factor ν. The pled. Near the NS interface, the normal edge state can superconductorgap∆(inunitsof 2v/(cid:96)B)rangesfrom0.01 (red line) to 0.2 in steps of 0.025. Bottom: corresponding bewrittenasasuperpositionofthewavefunctionsofthe longitudinal conductance G at the edge as a function of ν. incident electron, ψ+(x,y), the reflected electron in the e Thesolidlineplateaus: normalconductance(∆=0). Dotted oppositevalley,ψ−(x,y),andtheAndreevreflectedhole, e lines: upper bound for the conductance, which is quantized ψh−(x,y). For simplicity, we assume from now on all at 4e2/h. In all curves, the temperature T =0.02(√2v/(cid:96)B). length scales to be in units of the magnetic length (cid:96) √ B and all energy scales to be in units of 2v/(cid:96) . B √ The largest contribution to scattering comes from where n =nint[(cid:0) E2−∆2+αµ(cid:1)2]∈Z. The reflected α states at integer values of n(k) and n (k), where the s electronandholeamplitudescanbecalculatedbymatch- density of states is the largest. In the four component ing the amplitudes at the interface (x=0), particle-hole basis, the electron wavefunctions are ψ++ψ−+ψ− =ψ++ψ−. (20) (cid:18) (cid:19) e e h S S Ψ (k −y) ψ+(x,y)= +,ne i eikix, (16) e,i 0 In the limit of large coherence length, ξ =v/∆(cid:29)(cid:96) , B and (cid:15) (cid:28) v/(cid:96) , we can neglect scattering processes be- B and tween different modes. Also, the number of edge channels is the same for electrons and holes in the two sides C(cid:88)ne (cid:18)Ψ (k −y)(cid:19) of the junction, ne = nh = nα. In this regime, the ψ−(x,y)= re −,ne j e−ikjx, (17) solution of Eq. (20) is re = 0 and rA = β /β = e,i i,j 0 i i + − (cid:112) j E/∆− E2/∆2−1. When E < ∆, rA is complex and i thetotalamplitudeoftheAndreevreflectionis|rA|2 =1. where the nearest integer function ne =nint(cid:2)((cid:15)+µ)2(cid:3)∈ In fig. 3, we show the amplitude of the Andireev re- Zsetstheindexofthehighestoccupiedband,withCne = flection versus the filling factor of the normal LLs. In 2ne+1 the number of channels crossing the Fermi level the normal region, when the LLs are well separated, the at the edge. The wavefunction of the Andreev reflected Fermidistributionofthequasiparticlesisequaltothefill- hole is ing fraction of the highest occupied LL, (e(cid:15)/T +1)−1 = f(ν)=(ν/4+ 1)mod(1)∈[0,1], where T is the temper- C(cid:88)nh (cid:18) 0 (cid:19) ature. The And2reev amplitude is rA ≡Θ(ν), where ψ− (x,y)= rA e−ikjx, (18) i h,i i,j Ψ (k −y) j −,nh j (cid:114) |(cid:15)(ν)| (cid:15)2(ν) Θ(ν)=− + +1, (21) where n = nint(cid:2)((cid:15)−µ)2(cid:3) ∈ Z, with C equivalently ∆ ∆2 h nh the number of edge channels for hole states. with In the superconducting region, the electron-like and (cid:18) (cid:19) hole-like solutions can be written as, f(ν) (cid:15)(ν)=T ln . (22) 1−f(ν) ψα =C(cid:88)nαAα (cid:18) βαΨα,n+(kj −y) (cid:19)eαikjx, (19) From the Blonder-Tinkham-Klapwijk formula15, the S,i i,j β Ψ (k −y) j −α α,n+ j differential conductance at the NS junction can be writ- 4 1 a) b) c) a) ∆ 0 / ε y -1 -4 0 4 x 1 b) FIG. 5. Andreev edge states at the NS interface. Solid lines: ∆ 0 ε/ electron cyclotronic orbits; dashed: Andreev reflected holes. a) ε/µ → 0 regime: electrons and holes form a bound state. -1 -4 0 4 b)Intermediateregime,ε<min(µ,∆): electronsareretrore- flected into holes with group velocity v = ∂ε/∂k having k l g y yB opposite sign. c) ε > µ regime: electrons are specularly reflected into holes, which propagate along the same direction. FIG. 4. Energy spectrum ε along the NS interface (x = 0) versusguidingcenterk (cid:96) . Negativeguidingcenters(k (cid:96) < y B y B 0) describe states in the normal region. k (cid:96) >0 correspond y B numbers. In the superconducting side, to states in the superconducting one. Left curves: normal edge states. Center right curves: low energy Andreev edge (cid:18) (cid:19) (cid:88) β Ψ (x−k ) states. a) µ=0 (ν =0), ε>µ regime for finite ∆. Electron ψ (x,y)= Aα γ α,nα(ε) y eαikyy, S(cid:107) (cid:107),γ β Ψ (x−k ) andholestateshavethesamegroupvelocityvg =dε/dky and α,γ −γ α,nα(ε) y propagateinthesamedirectionoftheinterface;b)µ/∆=2: (25) electrons and Andreev reflected holes have the same guiding with γ = +(−) indexing electron(hole)-like states, and center and opposite group velocities along the interface for √ (cid:0) (cid:1)2 ε(cid:28)µ, forming an Andreev bound state. nα(ε) = ε2−∆2+αµ . Matching the amplitudes ψ (0,y) = ψ (0,y) and the derivatives ∂ ψ (x,y) = (cid:107) S(cid:107) x (cid:107) ∂ ψ (x,y) at x = 0 yields eight linear equations for x S(cid:107) eight unknown coefficients V , with i=1,...,8. This set ten as i ofequationscanbeexpressedinmatrixformasQ·V =0. G= 2e2 (cid:88)Cn(1−|re|2+|rA|2)≈ 4e2(cid:18)n+ 1(cid:19)(cid:2)1+|Θ(ν)|2(cid:3). Ncoonnd-ittriiovnia,lwseolunutimonesrirceaqlluyireextthraacttDtehte(Qsp)e=ct0ru.mFroomf etxhciis- h i i h 2 i=1 tations ε(ky) near the NS interface, as shown in Fig. 4. (23) The interface states are an admixture of two type of The peaks in the differential conductance are shown in modes: i) normal edge states formed by conventional the bottom panel of Fig. 3 as a function of the filling skipping orbits moving in one direction, and ii) Andreev factor of the normal region ν. The solid line plateaus states,whicharecoherentsuperpositionsofparticlesand represent the conductance in the absence of Andreev re- holes. ThefirstmodeisconnectedtobulkLLenergiesin flections. The different curves show the conductance for the normal region as k → −∞. The second one has no y different values of the normalized gap ∆ ranging from correspondence with the bulk LLs in the normal region, 0.01 (red line) to 0.2 in 0.025 steps. The peaks appear and appears around k (cid:96) ∼0 and also inside the super- at partial fillings ν = 4n, n ∈ Z, and their height is conducting region, forypBositive guiding centers (k >0). y quantized at (2n+1)4e2/h. At those fillings, the normal In panel 4a, we plot the interface modes at µ = 0 LLs are particle-hole symmetric and Andreev reflection (ν =0),forfinite∆. Inthisregime,ε>µ=0,thegroup is maximal. At integer fillings ν = 4(n+ 1), when the 2 velocity of the Andreev edge states vg =∂ε/∂ky >0 for Fermi level is in the middle of the LL gap, Andreev re- both electrons and holes (center right curves), which are flection is suppressed and the conductance is quantized specularly reflected at the interface16 and move in alter- by half, at (2n+1)2e2/h. natingskippingorbitsinthesame direction(seefig. 5c). Andreev edge states. More insight on Andreev reflec- In the limit ε (cid:29) µ , electrons and holes have the same tion and the electronic states near the interface can be velocityandguidingcenter,andcarryzeronetchargeper obtained by considering the current flowing parallel to valley. Inallothercases,theirvelocitiesaredifferent,re- the interface. In the configuration where the insulating sulting in a net valley current along the NS interface. In edge is zigzag, the NS interface has armchair character. panel 4b, we plot the energy of the modes for µ/∆ = 2. Itsstatesareformedbyasuperpositionofeigenstateson Intheregimeε<min(µ,∆)wefindnumericallythatthe both valleys. Using the Landau gauge A = (0,Bx), the Andreev reflected holes and electrons have group veloci- wavefunction in the normal side of the interface can be tiesv withoppositesigns(fig. 5b). Inthelimitε/µ→0, g written as theholesretracethepathoftheincidentelectrons,forming an Andreev bound state schematically shown in fig. (cid:18) (cid:19) (cid:88) a Ψ (x−k ) ψ (x,y)= e,α α,ne(ε) y eαikyy, (24) 5a. (cid:107) a Ψ (x−k ) α h,α α,nh(ε) y In summary, we have derived transport and spec- troscopysignaturesofproximityinducedsuperconductiv- where n (ε) = (ε+µ)2 and n (ε) = (ε−µ)2 are real ity in TRS LLs. We found the longitudinal conductance e h 5 as a function of the filling factor across an NS junction, Acknowledgements. We thank K. Mullen and P. and showed that it is quantized at (2n+1)4e2/h in the Carmier for discussions. BU acknowledges University of n-th LL at half-filling, when Andreev reflection is maxi- OklahomaandNSFCareergrantDMR-1352604forsup- mal. We also showed that the NS interface has Andreev port. edge states, with unique spectroscopic features. ∗ Currently at Ames Laboratory, Iowa State University, 1649 (2011). USA. 8 M.C.Rechtsman,J.M.Zeuner,A.Tu¨nnermann,S.Nolte, † [email protected], [email protected] M. Segev and A. Szameit, Nat. Photonics 7, 153 (2013). 1 Z. Tesanovic, M. Rasolt, L. 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