Andreev magneto-interferometry in topological hybrid junctions
AAndreev magneto-interferometry in topological hybrid junctions
Pierre Carmier
CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, Grenoble F-38054, France (Dated: November 8, 2018)We investigate the influence of the superconducting (S) proximity effect in the quantum Hall(QH) regime by computing the charge conductance flowing through a graphene-based QH/S/QHjunction. This situation offers the exciting possibility of studying the fate of topological edge stateswhen they experience tunneling processes through the superconductor. We predict the appearanceof conductance peaks at integer values of the Landau level filling factor, as a consequence of thequantum interferences taking place at the junction, and provide a semiclassical analysis allowing fora natural interpretation of these interferences in terms of electron and hole trajectories propagatingalong the QH/S interfaces. Our results suggest that non-trivial junctions between topologicallydistinct phases could offer a highly tunable means of partitioning the flow of edge states.
PACS numbers: 74.45.+c, 73.43.-f, 03.65.Sq, 72.80.Vp
Quantum Hall (QH) and superconducting (S) proxim-ity effects are two prominent mesoscopic phenomena, yettheir interplay has received little attention so far due tothe widespread assumption that their respective rangesof validity are incompatible. However, in modern highmobility two-dimensional electron gases (with mean freepaths typically exceeding the micron scale), magneticfield values B required to enter the QH regime have be-come sufficiently small to allow the fabrication of QH/Sjunctions, using high critical field superconductors suchas Nb compounds [1]. These junctions feature chiral edgestates of mixed electron-hole nature due to the Andreevreflection experienced by carriers at the interface with theS region [2]. Evidence for such edge states was success-fully demonstrated a few years ago in InAs semiconduct-ing heterostructures [1], following seminal experiments[3, 4] and earlier theoretical proposals [2, 5–7]. The ad-vent of graphene, in which both QH and S proximityeffects have been routinely observed [8–10] owing to thematerial’s low cost and tunability, has led to a revival ofexperimental activity in the field very recently [11–13].The interface between a QH insulator and a supercon-ductor actually provides a particularly interesting andnon-trivial realization of a topological junction, which isa junction between bulk insulating phases (from a single-particle perspective) characterized by different topolog-ical invariants [14, 15]. A defining property of thesejunctions is the existence of topologically protected edgestates propagating along their interface. Partition of theinformation carried by these states in the available out-going channels of a mesoscopic system is an importantproblem, both from a fundamental point of view and inthe perspective of exploiting the robustness of topolog-ical phases to build novel electronic devices. The pur-pose of this Letter is to highlight the potentially crucialrole played by quantum interferences in such topologicaljunctions, a spectacular example of which is depicted inFig. 1 where the charge conductance flowing through aQH/S/QH junction is plotted as a function of the Lan- ν G / g - π - π /2 0 π /2 π k S L [2 π ] G / g Figure 1: (Color online): Conductance G flowing through aQH/S/QH junction of length L/ξ S = 2 and width W/l B = 40as a function of the Landau level filling factor ν for k S L = π/ π ] (green circles). Transmission peaks occur at integervalues of ν . Regular QH plateaus (when the S region turnsnormal) and classical expectation are plotted for compari-son (thick black and dashed black lines, respectively). Inset:Value of the peak at ν = 6 (top blue) and of the dip at ν = 5 . k S L . dau level filling factor ν . Instead of featuring plateaus atodd values of the spin-degenerate conductance quantum g = 2 e /h (thick black line), as would be the case inthe absence of the S region (or equivalently for supercrit-ical magnetic fields), the conductance is seen to oscillatebetween extremal values, from a situation where currentis essentially blocked to one where it is fully transmit-ted through the S region. In the following, I will providea quantitative understanding of this phenomenon, usingthe classical picture of skipping orbits to describe QHedge states in the ballistic regime [7, 16–18]. By express-ing the conductance as a sum over the various semiclassi-cal trajectories contributing to the transmission probabil- a r X i v : . 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L S l QH W R QH TT AA (n) (n) (n)(n)(n) n θ nR rr t AA Figure 2: (Color online): Cartoon of a QH/S/QH junctionwith N = 3 vertices. Full blue lines are for electron trajecto-ries and dashed red lines for hole trajectories. Semiclassically,states along the QH/S interfaces can be described by skippingorbits propagating between equidistant vertices. Given an in-coming mode ( n ) on the upper left edge, we seek how theoutgoing probabilities R ( n ) , R ( n ) A , T ( n ) , T ( n ) A scale with W . ity through the QH/S/QH junction, we will see that themagneto-oscillations featured in Fig. 1 can be naturallyinterpreted in terms of interferences between electron andhole paths propagating along the QH/S interfaces. Whilesuspended graphene should be a well-suited candidateto test these predictions, as demonstrated by recent ex-perimental evidence supporting phase-coherent ballistictransport in this system [19–21], the obtained results areessentially independent of graphene’s band structure andshould therefore be observable in other two-dimensionalelectron gases as well.Let us consider the geometry depicted in Fig. 2, wherea spin-singlet superconductor (connected to a hidden Sreservoir) is deposited on top of a two-terminal grapheneribbon of width W in the QH regime, thereby opening aproximity-induced superconducting gap ∆ S (which willbe assumed constant) in a strip of length L inside theribbon. Assuming phase-coherent ballistic transport in-side the system, finite temperature k B T < k B T c (where T c ≈ . S /k B is the critical temperature of the super-conductor) should play no role beyond renormalizing thevalues of the superconducting gap and the critical field,and k B T will thus henceforth be set to zero. In (linear)response to a subgap bias voltage eV < ∆ S applied inthe left lead, tunneling processes through the S regionallow for a charge current to flow in the right lead, char-acterized by the electrical conductance [22, 23] G = g (cid:88) n (cid:16) T ( n ) − T ( n ) A (cid:17) , (1)with T ( n ) the probability for mode n to be transmittedas an electron and T ( n ) A the probability to be transmitted as a hole (see Fig. 2). Mode n can also be reflected asan electron or as a hole with probabilities R ( n ) and R ( n ) A ,such that R ( n ) + R ( n ) A + T ( n ) + T ( n ) A = 1.In order to derive semiclassical approximations forthese probabilities, one must describe both the incom-ing QH edge states and their dynamics at the interfacewith the S region semiclassically. The first part is ratherstraightforward. Provided ν = ( k F l B ) (cid:29)
1, where k F is the Fermi wavevector and l B = (cid:112) (cid:126) / ( eB ) the mag-netic length, the classical skipping orbit picture can betranslated into a more rigorous semiclassical descriptionby applying a Bohr-Sommerfeld quantization procedureto the periodic motion of the electrons, which effectivelyresults in turning the continuous family of possible clas-sical trajectories into a set of edge modes, each charac-terized by a quantized angle 0 < θ n, ± < π . For an arm-chair edge, this quantization condition can be expressedas 2 θ n, ± − sin 2 θ n, ± = (2 π/ν )( n ± /
4) [18, 24], where n can be identified with the Landau level index, and the ± sign refers to the lifting of the two-fold valley degeneracyin graphene (which will henceforth be implicit). Howeverthe choice of boundary condition at the edges of the rib-bon is qualitatively unimportant for our purposes in thesemiclassical regime ν (cid:29)
1, where the low-energy selec-tion rules imposed by the valley-polarization constraintscharacterizing the single channel case can be relaxed [25].Let us now address the dynamics along the interface.Because the Lorentz force acting on a hole is the sameas that acting on an electron, both particles rotate inthe same direction with the magnetic field, leading tounidirectional motion along a given QH/S interface (seeFig. 2). This is due to the fact that even though a holecarries an opposite charge from that of an electron, thisis compensated by the hole’s direction of motion beingopposite to its momentum. The situation becomes morecomplicated in a QH/S/QH setup, as tunneling processesthrough the S region give rise to states localized along thesecond interface which counter-propagate with respect tothose on the first interface (see Fig. 2). The problemwe face therefore boils down to describing the dynam-ics of coupled counter-propagating QH states (of mixedelectron-hole nature). For simplicity, we shall restrict ouranalysis to the zero-bias limit, for which the cyclotronradii of electron and hole channels match. In this case,the semiclassical formalism is more tractable, since theensemble of classical trajectories reduces to scattering be-tween equidistant vertices along the interface (see Fig. 2)– assuming a sufficiently doped S region that the positionmismatch between scattering vertices on both sides of theS region can be neglected. This approximation remainsvalid when a non-vanishing bias voltage eV (cid:46) (cid:126) v F /W is taken into account, and the results can in principlebe extended to arbitrary eV [26] using the more techni-cal Green’s function approach based on the Fisher-Leeformula [18, 27, 28].To proceed further, it is convenient to introduce thevector ( e i , h i ) T composed of electron and hole probabilityamplitudes of leaving vertex i along the left side of the in-terface. Its evolution is governed by the 2 x 2 matrix W i ,according to the equation ( e i +1 , h i +1 ) T = W i ( e i , h i ) T .All of the information regarding the semiclassical dynam-ics at vertex i is encoded in W i which will be referred toas the local propagator. Noting N = [ W/l n ] the (integer)number of vertices, where l n = 2 l B √ ν sin θ n is the dis-tance separating consecutive scattering events along theinterface, it is clear that (cid:18) e N h N (cid:19) = N (cid:89) i =1 W i (cid:18) (cid:19) . (2)Transmission probabilities can then easily be obtainedby summing over all possible coordinates for the initialscattering vertex: for example, R ( n ) = 1 l n (cid:90) l n dl | e N ( l ) | , (3)with N ( l ) = 1+[( W − l ) /l n ], such that N ≤ N ( l ) ≤ N +1.The main task we are left with is to compute the localpropagator W i . In order to do so, let us now look inmore detail at the scattering processes taking place atthe QH/S interfaces between consecutive vertices. Foran incoming particle with angle θ n at a given vertex,these processes are described by the 2 x 2 matrices R = (cid:18) re iφ r (cid:48) A e iφ r A e iφ (cid:48) r (cid:48) e iφ (cid:48) (cid:19) , T = (cid:18) te iφ t (cid:48) A e iφ t A e iφ (cid:48) t (cid:48) e iφ (cid:48) (cid:19) , (4)where R corresponds to reflection along a given interfaceand T to transmission from one interface to the other.Phases φ and φ (cid:48) account for the action, Maslov index andBerry phase [17, 18] respectively acquired by electron andhole channels during their propagation between scatter-ing events on the interface, such that δφ = φ − φ (cid:48) = 2 πν .The scattering coefficients in Eq. (4) are the local prob-ability amplitudes of normal reflection ( r ), Andreev re-flection ( r A ), elastic cotunneling ( t ) and crossed Andreevreflection ( t A ). Assuming that the QH/S interfaces areabrupt on the scale of l B , the existence of the magneticfield can be locally ignored and the scattering coefficientscan be determined by matching the quantum mechanicalwavefunctions at the interfaces and making use of mo-mentum conservation arising from translational invari-ance in the transverse direction. In the limit L (cid:38) ξ S ,where ξ S = (cid:126) v F / ∆ S is the superconducting coherencelength, one obtains r = − cos θ n ,r A = − i sin θ n ,t = 2 sin θ n (cos k S L sin θ n + i sin k S L ) e − L/ξ S ,t A = − i sin 2 θ n cos k S L e − L/ξ S , (5)assuming once more k S (cid:29) k F , with k S the Fermiwavevector in the S region (primed coefficients in Eq. (4) can be obtained from those of Eq. (5) by reversing thesign of k S ). These coefficients carry the signature ofgraphene’s unusual band structure through their angu-lar dependence [29]. In particular, r and t A vanish un-der normal incidence as a consequence of the absenceof backscattering (so-called Klein tunneling). Also, notethat t and t A are, as expected, exponentially suppressedon the scale of ξ S . Therefore, in order for tunneling pro-cesses to play a role, typical lengths of the S region will belimited to sizes such that diamagnetic screening currentscan be neglected [30]. As a consequence, the validity ofthe plane wave approximation to tunneling coefficients inEq. (5) will require magnetic lengths l B (cid:38) L .We now have all the necessary ingredients to computethe local propagator. Before presenting the general so-lution, let us familiarize ourselves with it by comput-ing the first couple of terms. The first one, W = R ,is rather obvious: incoming carriers at the first vertexmust necessarily be reflected, else they will be transmit-ted through the S region and irrevocably leave the in-terface (see Fig. 2). The second term can be expressedas an infinite sum, W = R (cid:80) + ∞ m =0 T m , where the m th contribution takes into account trajectories where chargecarriers have tunneled 2 m times through the S region. Itcan be conveniently rewritten in a self-consistent form, W = R + W T , the meaning of which is the follow-ing: unless carriers incoming at the vertex are directlyreflected ( R ), they must tunnel twice through the S re-gion ( T ), at which point one is back to the starting point( W ). Elaborating on this idea, a general recurrence re-lation can be derived for i ≥ W i = R + i − (cid:88) j =0 ( j (cid:89) k =0 W i − k ) TR j T . (6)Likewise, an equation similar to Eq. (2) can be writ-ten down for the electron and hole probability ampli-tudes to leave the QH/S/QH junction on the right side,( e (cid:48) N , h (cid:48) N ) T = W (cid:48) N (1 , T , with W (cid:48) = T and, for N ≥ W (cid:48) N = T + N − (cid:88) j =0 ( j (cid:89) k =0 W N − k ) TR j +1 . (7)In the limit L/ξ S (cid:29) W i = R de-scribing the periodic skipping orbit motion along theleft interface, and one thus retrieves the solution inde-pendently obtained by Chtchelkatchev in non-relativistictwo-dimensional electron gases [7, 16] and by the authorin graphene QH bipolar junctions [17, 18].Eqs. (6, 7) are the central results of this Letter and canbe solved numerically. There are special cases, however,where they can be exactly solved, an important one forour purposes being when the condition t + t (cid:48) e − iδφ = 0 is G / g B R A ( ) Figure 3: (Color online): Upper panel: G as a function of W for L/ξ S = 2, k S L = π/ π ], and ν = 6 (top blue), ν = 5 . n = 2) to be reflected as a hole for ν = 6 (bottomblue), ν = 5 . fulfilled, which is equivalent totan πν = sin θ n tan k S L . (8)In this case, the local propagator can be shown to takethe simple form W i = α i R , with 0 < α i ≤
1, and itis then a simple task to show that this implies R ( n ) + R ( n ) A ≤ e − N ( | t | + | t A | ) : in other words, full transmissionof current through the S region is achieved exponentiallyfast with W (blue curves in Fig. 3). One can also easilyprove that W (cid:48) N = α (cid:48) N T if Eq. (8) holds, thereby yieldingfor the asymptotic value of transmission in the S region T ( n ) − T ( n ) A = | t | − | t A | | t | + | t A | . (9)In particular, for cos k S L = 0, Eqs. (5, 8, 9) imply thatperfect electronic transmission is achieved at integer val-ues of ν , which translates into the conductance peaksshown in Fig. 1. The blue curve in the inset of Fig. 1interestingly suggests that these peaks should survive ifcos k S L is not strictly zero.A closer look at Fig. 1 shows that the conductancepeaks are all the more visible that they are accompaniedby dips at half-integer values of ν . The simplificationbrought by the vanishing of the amplitude of crossed An-dreev reflection when cos k S L = 0 (see Eq. (5)) allows tosupport this observation by an analytical statement, asone can then show that | h (cid:48) N | = 0, while | e (cid:48) N | = | t | if N is odd, and zero otherwise. This yields T ( n ) − T ( n ) A ≤ | t | :in other words, G this time remarkably does not increasewith W (red curves in Fig. 3) and only charge carriers having tunneled through the S region at the first ver-tex may actually contribute to G . This even-odd effectis a signature of the destructive interference of electronand hole paths ( δφ = π ) when ν is half-integer, whichis already manifest in the limit of a single QH/S in-terface [2, 5–7]. Indeed, taking advantage of the uni-tarity of the local propagator in this limit, its effect onvector ( e i , h i ) T can then be interpreted as rotating thelatter on the Bloch sphere with a frequency ω given bycos ( ω/
2) = cos θ n cos πν , such that, for half-integer val-ues of ν , the rotation frequency is ω = π and the sameeven-odd effect is displayed. The red curve in the inset ofFig. 1 however clearly demonstrates that this interpreta-tion breaks down when cos k S L is no longer zero, as ex-pected from the explicit coupling between ν and k S L dis-played in Eq. (8). In fact, for cos k S L = 1, the positionsof the conductance peaks and dips are exchanged with re-spect to the previous situation (see inset of Fig. 1), withdips occuring at integer values of ν and peaks at half-integer values (in agreement with Eq. (8)). The valueof the peaks will however be of lesser magnitude in thiscase, since t A (cid:54) = 0 (see Eq. (9)).The above predictions are in clear contrast with whatwould be expected from a purely classical point of view inthis situation (dashed lines in Figs. 1 and 3): the classicaldynamics along the left interface indeed follows a one-dimensional random walk with backward hopping prob-ability p = | t | + | t A | , for which Eq. (6) can be solved byrecurrence, yielding R ( n ) + R ( n ) A ≤ (1 − p ) / (1 − p + N p ).In other words, transmission through the S region is clas-sically expected to increase – albeit only slowly (alge-braically) – with W , which can be understood as arisingfrom the fact that charge carriers will have to experi-ence an ever larger number of scattering events to crossthe system (see Fig. 2). That this is not always the casesemiclassically (as discussed above) illustrates the crucialrole played by quantum interferences in this setup.As a closing remark, let us briefly comment on thesensitivity to disorder of these interference effects. Whiledisorder away from the S region should bring no meaning-ful change to the results, the presence of strong enoughdisorder in the vicinity of the QH/S interfaces will essen-tially randomize the phases of the charge carriers, includ-ing the phase difference δφ acquired by electron and holecarriers between consecutive vertices, thus likely spoilingthe conductance oscillations depicted in Fig. 1. How-ever, experimental measurements of the conductance ingraphene would still be valuable in the disordered regime,even in the limit of a single QH/S interface, drawingon an analogy between QH/S and bipolar QH junctionswhich shall be discussed elsewhere [26]: they could indeedallow estimating the relevance of charge density fluctu-ations [31] regarding the equipartition of charge carriersobserved in bipolar QH junctions [32–34].To summarize, we have seen that spectacular quantuminterference effects can arise at the interface between aQH insulator and a superconductor, and that these ef-fects can be quantitatively understood using an intuitivetrajectory-based semiclassical approach. 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