Controlled coupling of spin-resolved quantum Hall edge states
Biswajit Karmakar, Davide Venturelli, Luca Chirolli, Fabio Taddei, Vittorio Giovannetti, Rosario Fazio, Stefano Roddaro, Giorgio Biasiol, Lucia Sorba, Vittorio Pellegrini, Fabio Beltram
CControlled coupling of spin-resolved quantum Hall edge states
Biswajit Karmakar , Davide Venturelli , , , Luca Chirolli , Fabio Taddei , Vittorio Giovannetti , RosarioFazio , Stefano Roddaro , Giorgio Biasiol , Lucia Sorba , Vittorio Pellegrini , and Fabio Beltram NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza San Silvestro 12, I-56127 Pisa, Italy, Institut NEEL, CNRS and Universit´e Joseph Fourier, Grenoble, France, International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy, Istituto Officina dei Materiali CNR, Laboratorio TASC, Basovizza (TS), Italy. (Dated: December 6, 2018)We introduce and experimentally demonstrate a new method that allows us to controllably coupleco-propagating spin-resolved edge states of a two dimensional electron gas (2DEG) in the integerquantum Hall regime. The scheme exploits a spatially-periodic in-plane magnetic field that is createdby an array of Cobalt nano-magnets placed at the boundary of the 2DEG. A maximum charge/spintransfer of 28 ±
1% is achieved at 250 mK.
PACS numbers: 73.43.-f, 03.67.-a, 72.25.Dc, 72.10.-d
Topologically-protected edge states are dissipationlessconducting surface states immune to impurity scatter-ing and geometrical defects that occur in electronic sys-tems characterized by a bulk insulating gap [1]. Oneexample can be found in a clean two-dimensional elec-tron gas (2DEG) under high magnetic field in the quan-tum Hall (QH) regime [2]. In the integer QH case, spin-resolved edge states (SRESs) at filling fraction ν = 2(number of filled energy levels in the bulk) are char-acterized by very large relaxation [3] and coherence [4]lengths. This system is a promising building block forthe design of coherent electronics circuitry [4–8]. It rep-resents also an ideal candidate for the implementation ofdual-rail quantum-computation architectures [9] by en-coding the qubit in the spin degree of freedom that labelstwo distinct co-propagating, energy-degenerate SRESs ofthe same Landau level (LL) at the same physical edgeof the 2DEG [10]. A key element for the realization ofsuch architecture [10–12] is a coherent beam splitter thatmakes it possible to prepare any superposition of thetwo logic states, thus realizing one-qubit gate transfor-mations. This requires the ability to induce controlledcharge transfer between the two co-propagating SRESs,a goal which up to date has not been yet achieved. Herewe solve the problem by targeting a resonant condition,in analogy with the periodic poling technique adopted inoptics [13].In the integer QH regime the SRESs are single-particleeigenstates ψ nks ( x, y ) = | s (cid:105) e ikx χ nk ( y ) / √ L of theHamiltonian H = ( p + e A ) / m ∗ + V c ( y ) − g ∗ µ B Bσ z which describes a 2DEG in the ( x, y )-plane, subject toa strong magnetic field B in the z -direction and con-fined transversely by the potential V c ( y ) [14]. Here p ≡ ( p x , p y ) and (cid:126)σ ≡ ( σ x , σ y , σ z ) are respectively, theparticle momentum and spin operators, A is the vectorpotential, L is the longitudinal length of the Hall bar,while m ∗ and g ∗ are the effective electron mass and g-factor of the material. Each ψ nks ( x, y ) represents an elec-tron state of the n th LL with spin projection s ∈ {↑ , ↓} along z -axis, which is characterized by a transverse spa-tial distribution χ nk ( y ), and which propagates along thesample with longitudinal wave-vector k . In our analysiswe will focus on a ν = 2 configuration, where the longitu-dinal electron transport occurs through the SRESs of thelowest LL, i.e. Ψ ↑ ≡ ψ ,k ↑ , ↑ ( x, y ) and Ψ ↓ ≡ ψ ,k ↓ , ↓ ( x, y )(the values k ↑ , k ↓ being determined by the degeneracycondition at the Fermi energy E F = (cid:15) k ↑ = (cid:15) k ↓ of thecorresponding eigenenergies). Specifically in our schemethe two SRESs are separately contacted, grounding Ψ ↓ and injecting electrons on Ψ ↑ via a small bias gate V .The spin resolved currents I ↑ and I ↓ of the two SRESsare then separately measured at the output of the de-vice, after an artificial charge transfer from Ψ ↑ to Ψ ↓ is induced during the propagation. Since in general∆ k ≡ k ↑ − k ↓ (cid:54) = 0, Ψ ↑ and Ψ ↓ support electrons at differ-ent wave vectors. Hence any external perturbation capa-ble of inducing charge transfer between them must bothflip the spin and provide a suitable momentum transferto match the wave-vector gap ∆ k . In our scheme weachieve this by introducing a spatially-periodic in-planemagnetic fringing field (cid:126)B (cid:107) ( x, y ) [6] generated by an arrayof Cobalt nano-magnet ( magnetic fingers ) placed alongthe longitudinal direction of the 2DEG, see Fig. 1a. Thesystem Hamiltonian acquires thus a local perturbationterm ∆ H = − g ∗ µ B (cid:126)B (cid:107) ( x, y ) · (cid:126)σ/
2, which at first orderinduces a transferred current I ↓ = ( e V /h ) | t ↑↓ | , where t ↑↓ = ( L/i (cid:126) v ) (cid:104) Ψ ↓ | ∆ H | Ψ ↑ (cid:105) is the associated scatteringamplitude, and v is the group velocity of the SRESs.To capture the essence of the phenomenon, consider forinstance an array of periodicity λ and longitudinal ex-tension ∆ X described by a (cid:126)B (cid:107) ( x, y ) field of the form B y ( y ) cos(2 πx/λ )ˆ y for x ∈ [ − ∆ X/ , ∆ X/
2] and zero oth-erwise (here for simplicity x and z component of (cid:126)B (cid:107) havebeen neglected). The corresponding transmission ampli-tude computed at lowest order in the T-matrix expan-sion [16] is t ↑↓ = ig ∗ µ B (cid:104) B y (cid:105) ∆ X (cid:126) v sinc[(2 π/λ − ∆ k )∆ X/ , (1) a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec G I ↓ ν = 2 V I ↑ G2G1 V g ν = 1 ν = 1 ► ►► λ = 286 333 400 500 nm c dba coupling area FIG. 1: (Color online) a) Schematics of the device. TheCobalt fingers (blue bars) produce a fringing field (yellowlines) resulting in an in-plane, oscillatory, magnetic field (cid:126)B (cid:107) atthe level of the 2DEG (textured gray) residing below the topsurface. The field induces charge transfer between the spinup Ψ ↑ SRES (red line) and spin down Ψ ↓ SRES (blue line).b) Density plot of the modulus (cid:126)B (cid:107) in the proximity of themagnetic fingers on 2DEG plane. The dashed line indicatesthe end of the finger array at 0 . µ m from the physical edgeof the mesa (white stripe). c) Measurement set-up: The Ψ ↑ channel is excited by a bias voltage V , while Ψ ↓ is groundedat the contact denoted by G. The SRESs can be reversiblydecoupled by negatively biasing the array with a voltage V a (G1 and G2 are contacts for the top gates). d) Optical im-age of the device showing four sets of magnetic fingers withdifferent periodicity λ placed serially at the mesa boundary(the yellow elements are gold eletrical contacts). Zoomed re-gion is the scanning electron microscopic image of the arrayof periodicity λ = 400 nm: it is nearly 6 µ m long and has anoverlap on the mesa of 0 . µ m. with sinc[ · ] ≡ sin[ · ] / [ · ] being the sine cardinal functionand (cid:104) B y (cid:105) ≡ (cid:82) dyB y ( y ) χ ,k ↑ ( y ) χ ,k ↓ ( y ). The expressionclearly shows that even for small values of longitudinalfield a pronounced enhancement in inter-edge transfer oc-curs when λ matches the wave-vector difference of the twoSRESs (i.e. λ res = 2 π/ ∆ k ), the width of the resonancebeing inversely proportional to ∆ X .The quantity ∆ k that defines the resonant conditiondepends on the Zeeman energy gap and on the details ofthe confinement potential V c ( y ). An estimate based onnumerical simulations (see Supplemental Material (SM))leads to an approximate value λ res ≈
400 nm at B =4.5 T, which we assumed as a starting point in design-ing our setup. The device was fabricated on one-sidedmodulation-doped AlGaAs/GaAs heterostructure grownby molecular beam epitaxy. The 2DEG resides at the AlGaAs/GaAs heterointerface located 100 nm below thetop surface. A spacer layer of 42 nm separates the2DEG from the Si δ -doping layer above it. The 2DEGhas nominal electron density of 2 × / cm and low-temperature mobility nearly 4 × Vcm / s. The Cobaltnano-magnet array was defined at the mesa boundary ofthe 2DEG using e-beam lithography and thermal evap-oration of 10 nm Ti followed by 110 nm Co. Eightnano-magnet arrays at different periodicities (specifically λ = 500 , , , , , ,
200 and 182 nm) werefabricated, keeping the total spatial extension of the mod-ulation region nearly constant, ∆ X (cid:39) . µ m (four ofthem are on the other side of the mesa and therefore notvisible in the microscope image of Fig. 1 d). The magne-tization of the Cobalt fingers is aligned along the appliedperpendicular magnetic field B (Fig. 1a), if B is largeenough [6]. The actual value of the oscillatory (cid:126)B (cid:107) canreach 50 mT in the proximity of the fingers and it decaysaway from the array (see Fig. 1b). Importantly, couplingbetween the SRESs and a chosen set of fingers can beactivated by increasing the voltage bias V a of the arrayfrom − V a = − (cid:126)B (cid:107) . Transport measurements werecarried out in a He3 cryo-system with a base temperatureof 250 mK equipped with 12 T superconducting magnet.An ac voltage excitation of 25 . µ V at 17 Hz was appliedto the electrode V of Fig. 1 c) and the transmitted cur-rent was measured by standard lock-in techniques usingcurrent to voltage preamplifiers.We first measured the two-terminal magneto-currentat T = 250 mK in order to locate the plateau associ-ated with a number of filled LLs in the bulk ν equal to2 (see Fig. 2a). The working point was set in the cen-ter of the plateau, i.e. at B = 4 .
75 T. The two SRESscan be separately contacted as schematically shown inFig. 1c by negatively biasing the gates G1 and G2 at avoltage V ∗ G , such that the filling factor below the corre-sponding top gates becomes ν = 1 and one edge channelonly is allowed underneath the gates. The actual V ∗ G value can be determined by measuring the currents I ↑ and I ↓ as a function of V G (see Fig. 2b). When inter-edge coupling is suppressed by applying V a = − I ↑ of about 1 nA,as expected for a single channel of unit quantized resis-tance h/e ≈ . I ↓ is nearly zero for V ∗ G = − . ± .
08 V (see Fig. 2b).In agreement with [3], this implies the absence of signifi-cant spin flip processes over the distance of about 100 µ mtraveled by the co-propagating SRESs when the mag-netic fingers are deactivated. For completeness, Fig. 2cshows the dependence of the currents I ↑ and I ↓ on tem-perature: SRESs fully relax only for T ∼ . / ( k B T) ≈ . − ), while edge mixing becomes neg- - 0 . 9 - 0 . 6 - 0 . 3 0 . 51 . 01 . 5 Current (nA) V G ( V ) V *G Current (nA) T ( m e V - 1 ) T = 2 5 0 m K b a B ( T ) n = 2 c FIG. 2: (Color online) a) Plot of the two terminal magneto-current (2TMC) measured at 250 mK. The value of magneticfield B = 4 .
75 T, indicated by an arrow, is used to placethe 2DEG approximately at the center of the ν = 2 plateau.b) Plot of the currents I ↑ (red) and I ↓ (blue) measured atthe current terminals red and blue respectively (Fig. 1c) withthe voltage V G applied to the gates G1 and G2, while thenano-magnets are deactivated by applying a voltage bias of V a = − V G is set to V ∗ G , indi-cated by an arrow, for separately contacting the spin-resolvededge states (see Fig. 1c). c) Temperature dependence of I ↑ (red) and I ↓ (blue) currents shows enhancement of relaxationbetween SRESs with increasing temperature. Thermally me-diated mixing of currents becomes negligible at T = 250 mK. ligible at our working point T = 250 mK. Moreover, an-alyzing our data as in Refs. [3] we can conclude that therelaxation length is of the order of 1 cm at T = 250 mK.The upper panel of Fig. 3 shows the measured I ↑ and I ↓ when coupling occurs at several different individualarrays (one at a time) as identified by their 2 π/λ value.Since inter-edge coupling leads to charge transfer betweenthe two spin-resolved edge channels it results in a de-crease of I ↑ , with the consequent increase of I ↓ while thetotal current remains constant at about 1 nA. Note thatcurrent transfer is significant only for a specific intervalof λ values: indeed a resonance peak appears to occurat λ res between 400 and 500 nm. Such behavior is con-sistent with Eq. (1) and with a more refined theoreticalanalysis based on the Landauer-B¨uttiker transport for-malism [17] which we have solved numerically in orderto go beyond the result of first-order perturbation the-ory [18] (see inset of the upper panel of Fig. 3 and SM).Static disorder and/or inelastic mechanisms induced, e.g.by the finite temperature and Coulomb interactions, mayaffect the resonance, resulting in a broadening of the cur-rent peak versus 2 π/λ . Importantly, if the fingers were anincoherent series of scatterers one should expect a mono-tonic λ -dependence of the charge transfer [19], while the / l ( m m - 1 ) Current (nA)
Transferred current (nA) B ( T ) 1 2 1 4 1 60 . 00 . 10 . 2 Transferred currnet (nA) p / l ( m m - 1 ) FIG. 3: (Color online) Upper panel: Plot of the transmittedcurrents I ↑ (red) and transferred current I ↓ (blue) as a func-tion of the inverse periodicity of the activated (by applying V a = 0) set of nano-fingers at the working point B = 4 .
75T and T = 250 mK. The measured current I ↑ and I ↓ areguided by the dashed line which demonstrates selectivity ofnano-magnet at periodicity between λ = 400 nm and 500 nm.The inset shows a numerical simulation of transferred currentwhich in the absence of the static disorder and/or inelasticmechanisms predicts a width of the peak that scales inverselyon ∆ X as in Eq. (1). Lower Panel: measured transferred cur-rent I ↓ as a function of the perpendicular magnetic field B for the nano-magnet array of periodicity λ = 400 nm. observed non-monotonic selective behavior of the currentsuggests an underlying constructive interference effect.For the case of λ = 400 nm, the lower panel of Fig. 3shows the dependence of transferred current I ↓ on theperpendicular magnetic field B when the latter spans the ν = 2 plateau (see Fig. 2a). The monotonic decrease of I ↓ is a consequence of at least three combined effects: (i)the ratio | (cid:126)B (cid:107) | /B decreases as B is increased, so that thenet effect of the in-plane magnetic modulation is weak-ened; (ii) the magnetic length decreases with increasing B , causing the reduction of the spatial overlap of thetransverse wavefunctions; (iii) the change of SRES spa-tial configuration with increasing magnetic field due tointeraction effects [20, 21].Apart from activating/deactivating the various nano-finger sets, the voltage V a can also be used as an externalcontrol to adjust the resonant mixing condition. Figure 4shows the measured transferred current I ↓ as a functionof V a and B for the array of periodicity λ = 400 nm(similar data were obtained for different λ , see SM). Thepronounced features present for intermediate values of V a show that the coupling between SRESs can be controlled - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 54 . 64 . 74 . 84 . 9 Transferred current (nA) V g ( V ) B (T) Current (nA)
FIG. 4: (Color online) Upper panel: Dependence of the trans-mitted current I ↑ (red) and transferred current I ↓ (blue) uponthe voltage V a applied to activated nano-finger of periodicity λ = 400 nm at B = 4 .
75 T. For V a < − . V a (cid:39) V a a series of pronounced peaks in I ↓ areevident. Lower panel: contour plot of I ↓ upon V a and B forthe nano-finger set of periodicity λ = 400 nm. The horizontalline indicates the center of the ν = 2 plateau ( B = 4 .
75 T). and amplified. Remarkably, a charge transfer of 28 ± B = 4 .
5T with V a ≈ − . V a ’s the SRESs are pushed away from the re-gion where the magnetic fringe field is present and, asexpected, the coupling vanishes. The same Fig. 4 revealsadditional resonances occurring at specific values of V a .A non-monotonic dependence of the local value of ∆ k on V a , can be invoked to explain these features. A systemsimulation shows that a local change of the confinementpotential in the proximity of the associated nano-fingersmodifies the relative distance of the SRESs and hencethe local value of ∆ k in a non-monotonic way (see SM).More precisely, for low V a the fingers act as top gatesfor the underlying edge states: the transverse distancebetween SRESs can locally increase and reach a maxi-mum as V a gets negative, since Ψ ↓ and Ψ ↑ are pushedaway from the finger region, one after the other. Aswe further increase V a the transverse distance betweenthe SRESs increases again. It is worth stressing, how-ever, that the process just described is not necessarilysmooth: electron-electron interaction may in fact induceabrupt transitions in SRESs distances when the slopeof the effective local potential decreases below a certaincritical value which depends on the details of the sampleproperties [21] (also the gate voltage can influence theFermi velocity, as shown in edge magnetoplasmons time- of-flight experiments [22]). The trajectories of SRESs areunknown and (differently from what shown in the graph-ical rendering of Fig. 1a) are likely to be outside the re-gions corresponding to the projections of the fingers whena significant voltage is applied. Nevertheless non-linearrepulsive effect is expected to be effectively active in theexperiment where the electrostatic potential profiles ex-tends much beyond the length of the fingers. Moreover,the functional dependence of the potential induced by V a upon the longitudinal coordinate x presents also an os-cillatory behavior with periodicity λ . As a consequenceof the adiabatic evolution of the edges, their transversedistance will also show such oscillations. A detailed mod-eling of the observed resonance features would require totake fully into account these effects and is beyond thescope of the present paper. However it clearly deservesfurther investigation as it represents a positive feature ofthe system, since any value of the modulation periodicity λ has typically more than one value of V a that can fulfillthe resonant condition.Our proposal provides a way to realize beam splittersfor flying qubit using topologically protected SRESs. Itemploys a nanofabricated periodic magnetic field oper-ated at a resonant condition which enhances quite signif-icantly the weak magnetic field produced by the Cobaltnanomagnets. Already at T = 250 mK the effect is sig-nificant and should be enhanced at lower temperatures.This work was supported by MIUR through FIRB-IDEAS Project No. RBID08B3FM and by EU throughProjects SOLID and NANOCTM. We acknowledge use-ful discussions with N. Paradiso and S. Heun. [1] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005); B. A. Bernevig, T. L. Hughes, and S. C. Zhang,Science , 1757 (2006); D. Hsieh, et al. , Nature ,970 (2008); S. Das Sarma, M. Freedman, and C. Nayak,Phys. Rev. Lett. , 166802 (2005).[2] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. , 494 (1980); D. C. Tsui, H. L. Stormer, and A. C.Gossard, ibid. , 1559 (1982).[3] G. M¨uller, et al. , Phys. Rev. B , 3932 (1992); S.Komiyama, H. Hirai, M. Ohsawa, and Y. Matsuda, ibid. , 11085 (1992).[4] Y. Ji, et al. , Nature , 415 (2003); I. Neder, et al. , Na-ture , 333 (2007); Phys. Rev. Lett. , 016804, (2006); ibid. , 036803 (2007).[5] P. Roulleau, et al. , Phys. Rev. B , 161309(R) (2007);Phys. Rev. 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Here we describe the theoretical approach to numerically simulate the transport properties of the proposed device.On this basis we discuss a mechanisms that, for a given periodicity of the fingers, produces multiple resonances inthe transferred current as we vary the gate voltage V a . We also report a comparison between the transferred currentcomputed at first order and the exact numerical solution, showing that indeed the former is able to detect the resonantcondition and we provide an estimate of the resonant periodicity of the array. Finally we present some extra data ofthe measured transferred current as a function of V a and B for different periodicities λ of the fingers. Numerical Simulations
We performed numerical simulations by modeling the device through a tight-binding Hamiltonian describing a HallBar about 100 nm wide (hard wall confinement potential), with lattice spacing a = 2 . B = 4 .
75 T, corresponding to a cyclotron gap (cid:126) ω c ≈ .
85 meV, is introduced by Peierls phase factors on the hoppingamplitudes. The Zeeman gap has been taken to be E Z = (cid:126) ω c /
17, corresponding to an effective value of g ∗ = 1 . λ res consistent with the experimental value reported in Fig.3 of the maintext). This value, which is larger than the usual value for GaAs (i.e. | g ∗ | = 0 . λ res one gets by using the following simple heuristic analysis. Indeed assumingadiabatic following of the confinement potential by the Landau levels, the transverse spatial separation ∆ Y betweenthe SRESs can be evaluated by an energy-balance argument: the external confinement field must work against theenergy gap ∆ (cid:15) in order to make the channels degenerate in energy. For a linear confinement potential V c ( y ) = eEy ,this yields eE ∆ Y = ∆ (cid:15) . (2)While it is experimentally very tricky to measure the edge separation for SRESs, several solid hints can be extrapolatedfor the separation ∆ Y c of cyclotron-resolved edge states [2]. More specifically, we could use the experimental valueof ∆ Y c for ∆ (cid:15) = (cid:126) ω c , to infer by proportionality the ∆ Y for spin-resolved channels: ∆ Y (cid:39) (cid:15) z (cid:126) ω c ∆ Y c . By employingthe measurements of Ref. [3] performed at ν = 4 in similar experimental conditions, we are lead to λ res (cid:39)
360 nmwhich, considering the different approximations adopted in the two approaches (e.g. linear confinement vs. sharppotential), is in good agreement with the number we obtained from the simulations. The above analysis give usalso the opportunity of stressing that the mixing of spin-degenerate, energy-degenerate edge channels belonging todifferent LLs would require a spatial modulation of the perturbative field with periodicity on order of few ˚A thatis practically impossible to engineer. In the absence of perturbation potentials, this yields a wave-vector differenceof the two spin resolved edges of the order of ∆ k (cid:39) . − corresponding to a value of the resonant condition λ res = 2 π/ ∆ k (cid:39)
450 nm and to an overlap integral of the wavefunctions γ = (cid:82) dyχ ,k ↑ ( y ) χ ,k ↓ ( y ) of (cid:39) .
97 (numberobtained by using the wave-functions χ ,k s ( y ) for hard wall confinement [4]).Transport properties are computed through the recursive-Green function algorithm of the KNIT Numerical Pack-age [5]. FIG. 5: a) vectorial field plot and density plot of the in-plane component (cid:126)B (cid:107) of the magnetic field computed for a singlemagnetic finger (indicated by the red rectangle): the direction of the field in the plane is represented by the green arrows whileits intensity is represented by the different colors of the background (lighter hue corresponding to higher intensities). Thewhite line in the plot indicates where the physical edge of the mesa. b) Schematic of a simulated Hall bar device where thespatial distribution of the x -component of the inhomogeneous magnetic field (cid:126)B (cid:107) generated by a finger array consisting of justfour fingers (short dashed rectangles in the figure) is highlighted; the top gates used for the selective injection and detectionelectrodes are represented by long dashed rectangles. c) Charge density plots of the Ψ ↓ (lower panel) and Ψ ↑ (upper panel)SRES in the proximity of the simulated top gates that guarantee their selective population. A nano-magnet finger array is placed at the right boundary of the 2DEG (short dashed rectangles in Fig. 5b) andproduces a periodic magnetic field (cid:126)B (cid:107) in the x -direction, which extends on the Hall bar in the y -direction for 40 nm.The field (cid:126)B (cid:107) induced on the 2DEG by a single rectangular magnetic finger, which can be calculated exactly, is shownthrough a vectorial plot in Fig. 5a (its maximum amplitude reaching values of the order of 50mT [6]). Selectivityof edge channels is realized by placing top gates in the injection and detection electrodes (long dashed rectangles inFig. 5b) which induce an electrostatic potential so to set a filling factor equal to 1 underneath the gates. As a result,the outer edge state only is allowed to enter the electrode. This is clearly demonstrated by the charge density plotsof the inner (Ψ ↓ ) and the outer (Ψ ↑ ) SRES shown in the upper and lower panels, respectively, of Fig. 5c. Multiple resonances
As discussed in the main text, when the negative value of the voltage V a applied to the fingers increases, theactual path followed by the edge channels is deformed so that their local separation, and hence their wave vectordifference ∆ k , can vary in a non-linear fashion. Using the method detailed in Ref. [7], we numerically determine ∆ k as a function of V a for a single long finger extending 19.7 nm in the y -direction. The resulting resonant periodicity,defined as λ res = 2 π/ ∆ k and plotted in Fig. 6 as a function of V a (blue line), first decreases, reaching a minimum,and thereafter slowly increases. Such behavior reflects the fact that the two edge channels are progressively expelledfrom underneath the finger one after the other. The process is pictorially described by the three cartoons a), b), c) onthe right side of Fig. 6. Here the arrows describe the position of the two edges in the mesa while the dashed rectanglerepresents the region of the fingers: the configuration a) corresponds to the case in which V a nullifies (both edgeslie below the fingers); configuration b) corresponds to the situation in which the inner state only is expelled (whenthis happens λ res reaches its minimum value); finally configuration c) corresponds to the case of very large negativevalue of V a when both edges are completely expelled from the region beneath the fingers: this effect is illustratedin the inset to Fig. 6a where a numerical simulation of the charge density of the outer channel Ψ ↑ shows that thecorresponding edge state has been pushed away from the finger region (in the plot the electrostatic repulsion has beentaken homogeneous).In the left panel of Fig. 6 we report also the value of the computed transferred current I ↓ for arrays of differentperiodicities λ as a function of V a (black curves of the figure — see caption for details). We notice that in correspon-dence of the matching between λ ( V a ) (blue curve in picture) with the periodicity λ of the finger, I ↓ shows a peak(otherwise it is zero). This indicates that the resonant condition discussed in the main text, can be met for more thanone value of V a depending on the periodicity. Such an effect is in qualitative agreement with Fig. 4 of the main text V a Thursday, September 15, 2011
FIG. 6: Left panel: The blue curve indicates the resonant period as function of the finger voltage λ res ( V a ), while the differentblack curves represent the transferred currents I ↓ ( V a ) for simulated devices consisting of (from top to bottom) 12 magneticfingers spaced 500 nm, 15 fingers spaced 400 nm, 18 fingers spaced 333 nm and 21 fingers spaced 286 nm. For the sake of claritythe scales of the current for each curve have been set in arbitrary unit and their zero levels have been shifted to match withthe periodicity of the associated λ . Inset: example of the simulated Hall-bar, where the charge density solution for the outeredge channel is plotted for large negative V a (here the electrostatic repulsion is considered homogeneous in the finger region).Right panel: pictorial view of the repulsion effect of the edges (represented in the picture by the arrows) at the origin of thenon-monotonic behavior of the resonance condition with the increase of V a (shaded rectangles represents the area underneaththe fingers); a) V a = 0, b) maximum separation corresponding to the minimum of the resonance curve in the right panel, c)total repulsion of the edge channels at large values of V a . (see also the last section of the Supplementary Material) which, for fixed λ , shows the presence of resonant peaks forintermediate values of V a . Comparison between first order and exact numerical solution
In the main text we used first order perturbative analysis to show the existence of a resonant condition for thefinger periodicity (i.e. λ res = 2 π/ ∆ k ). Here such result is compared with the exact solution obtained by computingthe transferred current via the tight-binding recursive Green’s function method [1, 5] detailed in the first section ofthe supplementary material.The resulting plots are shown in Fig. 7: we chose the cyclotron gap (cid:126) ω c ≈ .
85 meV, corresponding to a magneticfield B = 4 .
75 T, and Zeeman gap E Z = (cid:126) ω c /
17, such to produce the resonant peak between 2 π/ . π/ . µ m − .The longitudinal extent of the nano-magnet array is ∆ X = 6 µ m and the in-plane component of the associatedmagnetic field are calculated exactly. The analytic expression of the transferred current, following the perturbativeapproximation detailed in the main text, has been evaluated for ∆ k ≈ . µ m − and properly rescaled to matchthe result of the simulation. The good agreement between the two curves shows that the perturbative approach issufficient to capture the resonant behavior of the device. Transferred current as a function of magnetic field and gate voltage
The reproducibility of controlled coupling of spin-resolved edge states at filling factor ν = 2 is shown in Fig. 8.The measurements are performed under the same procedure and experimental conditions described in the main paperand several weeks after the experiments reported in the main paper. The data for the array with λ = 400nm referto a different cool down of our sample and a slightly different electron density. However, the generic features of themeasured transferred current I ↓ are similar to the color plot reported in the main paper. We emphasize that at zerobias voltage ( V a = 0), the transferred current I ↓ for the namomagnet array of periodicity λ = 400 nm is maximumcompared to the other nanomagnet arrays and decays with increasing applied perpendicular magnetic field B, spannedover the ν = 2 QH plateau. Therefore, the result is consistent with our theoretical understanding and the reportedexperimental results in Figs. 3 and 4 of the main paper. Moreover, at intermediate values of V a from 0 to -2.25 V,several resonant peaks in the transferred current I ↓ for both the nanomagnet arrays of periodicities of λ = 400 nm m ) -1 T r a n s f e r c u rr e n t ( n A ) FIG. 7: Comparison between the transfer current I ↓ obtained from the perturbative approach Eq. (1) of the main text (reddashed curve) and from a numerical calculation (black solid curve). The red dashed curve is rescaled in order to match themaximum value obtained with the numerical calculation. l = 4 0 0 n m B (T) l = 4 0 0 n m - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 54 . 64 . 85 . 0 l = 3 3 3 n m V g ( V ) B (T) Transfer current (nA)
FIG. 8: Color plot of measured transferred current I ↓ in the magnetic field (B) - gate voltage ( V a ) plane for nanomagnet arraysof periodicities λ = 400nm and λ = 333nm. The measurement is performed on the same sample after a different cool down. and 333 nm appear and shift quasi linearly towards higher values of V a with increasing perpendicular magnetic fieldB. The transferred current I ↓ becomes significantly low as the artificial coupling induced by the nanomagnet arraysvanishes for V a less than -2.25 V. It is also notable that the signal strength of the transferred current I ↓ is lowerthan that reported in the experiments described in the main paper (Figs. 3 and 4). This signal reduction indicatesoxidation of Cobalt nanomagnet fingers leading to a reduction of the parallel fringing field (cid:126)B (cid:107) due to formation ofantiferromagnetic CoO [6]. [1] D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge, 2009).[2] G. M¨uller, et al. Phys. Rev. B , 3932 (1992).[3] N. Paradiso, et al. Physica E , 1038 (2010).[4] A. H. MacDonald and P. Streda, Phys. Rev. B , 1616 (1984).[5] K. Kazymyrenko and X. Waintal, Phys. Rev. B , 115119 (2008).[6] A. Ursache, J. T. Goldbach, T. P. Russell, and M. T. Tuominen, J. of Appl. Phys. , 10J322 (2005).[7] S. Sanvito, et al. Phys. Rev. B59