Optical manipulation of edge state transport in HgTe quantum wells in the quantum hall regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Optical manipulation of edge state transport in HgTe quantum wells in the quantumhall regime
M. J. Schmidt, E. G. Novik, M. Kindermann, and B. Trauzettel Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland Physikalisches Institut (EP3), University of W¨urzburg, 97074 W¨urzburg, Germany School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, 97074 W¨urzburg, Germany (Dated: October 16, 2018)We investigate an effective low energy theory of HgTe quantum wells near their mass inversionthickness in a perpendicular magnetic field. By comparison of the effective band structure with amore elaborated and well-established model, the parameter regime and the validity of the effectivemodel is scrutinized. Optical transitions in HgTe quantum wells are analyzed. We find selectionrules which we functionalize to optically manipulate edge state transport. Qualitatively, our findingsequally apply to optical edge current manipulation in graphene.
PACS numbers: 72.10.-d,73.61.-r
Low dimensional quantum systems with distinct topo-logical properties attract a lot of interest, not only be-cause of their possible applications in topological quan-tum computation [1], but also because they constitute aversatile playground for studying solid state realizationsof exotic phases [2]. It has recently been realized [3] thatHgTe quantum wells (QWs) exhibit very rich low en-ergy properties such as the quantum spin Hall effect andtopological phase transitions. Notably, these extraordi-nary physical properties are not only theoretical predic-tions but many of them have already been experimentallyconfirmed in HgTe nanodevices [4, 5, 6]. The remarkabletunability of parameters like, for instance, the Rashbaspin orbit coupling (SOC) strength [5] makes HgTe QWsespecially suitable for spintronics applications. Interest-ingly, the low-energy properties of electrons in HgTe QWsnear the so-called inversion point (which is related to thethickness of the HgTe layer) can be well described by theDirac equation similar to the low-energy properties ofelectrons in graphene [7]. However, the electronic spec-trum of HgTe QWs is much richer than that of grapheneand further theoretical as well as experimental researchis needed to fully characterize it.In this Rapid Communication, we first compare an ef-fective model for HgTe QWs [3] (in the presence of aperpendicular magnetic field) with a more elaborated 8-band Kane model [6, 8]. This allows us to identify the ex-perimentally relevant parameter regime for the effectivemodel. After the important energy scales are identified,we can analyse the optical transitions between Landaulevels and, even more interestingly, between (magnetic-field induced) edge states. We show that this gives riseto a new possibility of edge current reversion by pho-tons. This effect is not unique to HgTe QWs but canhappen in all systems where optical transitions betweenelectron-like and hole-like edge states are allowed whichis, for instance, the case in graphene. The bulk band structure of narrow HgTe QWs hasbeen extensively investigated before [6, 8]. It was foundthat near the mass inversion thickness d c ≃ . k -diagonal Hamiltonian [13] H = (cid:18) h ( k ) 00 h ∗ ( − k ) (cid:19) , h ( k ) = ǫ ( k ) + d a ( k ) σ a (1)with σ a the Pauli matrices, ǫ ( k ) = − Dk , d ( k ) =( Ak x , − Ak y , M − Gk ) and k the crystal momentum. A, D, M and G are parameters of the effective model(see Fig. 1). In this model, Rashba and Dresselhausspin orbit coupling (SOC) are not explicitly taken intoaccount. Rashba SOC could be treated easily by pertur-bation theory and can, in principle, be tuned to zero inthe experiments. Therefore, we neglect it here. Fur-ther, the Dresselhaus terms are known to be negligi-bly small [4]. The basis states of the model in Eq. (1)are {| E + i , | H + i , | E −i , | H −i} where E ( H ) refers tothe subband which is predominantly derived from theconduction (valence) band, and ± refers to degenerateKramers partners [6].The presence of a time reversal symmetry breakingmagnetic field perpendicular to the well is described bya corresponding vector potential - we use the Landaugauge here. After the usual transformations, it is foundthat the Hamiltonian which describes the Landau levelsis obtained by the replacement h ( k ) → h + , h ∗ ( − k ) → h − (2)in Eq. (1) with h ± = h HO + h ± JC . The harmonic os-cillator part of the Hamiltonian h HO = − DB ( a † a + ) + (cid:2) M − BG ( a † a + ) (cid:3) σ is diagonal in the electron-hole space as well as in the Kramers space, while theJaynes-Cummings terms h + JC = −√ BA ( a † σ + + aσ − )and h − JC = + √ BA ( aσ + + a † σ − ) couple different har-monic oscillator levels and lift the ± degeneracy. a and a † are bosonic operators, i.e. [ a, a † ] = 1. Thus, the2 × h ± are easily diagonalized by introduc-ing | n i + = | n i ⊗ | E + i , | n i + = | n − i ⊗ | H + i and | n i − = | n i⊗| H −i , | n i − = | n − i⊗| E −i , respectively,where | n i are harmonic oscillator states corresponding tothe Landau level n . n=2n=2n=0 n=1n=1 E F B (T) b)a)
D=-730 meV nm A= 375 meV nmM=-3.1 meVG=-1120 meV nm E F E ( m e V ) k (nm -1 ) -20-15-10-505101520 -20-15-10-505101520 FIG. 1: (Color online) (a) Energy spectra of the symmetri-cally doped QW at zero magnetic field for a hole density of0 . · cm − and a well thickness d = 6 . D = − , A =375meV nm , M = − . , G = − . . (b) Lan-dau levels in the effective model (dashed, red lines) and inthe Kane model (solid, black lines), using the same parame-ters as in (a). In order to find the proper parameters for the effec-tive model (1) we perform extensive calculations withina well established 8-band k · p approach for the HgTeQW [6, 8]. The parameters which enter this numericalcalculation are the well known band structure parametersfor HgTe and CdTe, the width of the QW, the strengthof the magnetic field B and the charge density in the2DEG. The parameters A, M, D and G of the effectivemodel are determined in the limits | k | → B → | k | states. Thus, the effective model description of Lan-dau levels becomes worse at higher magnetic fields. For B . T , however, the effective model is still reasonablyaccurate.In the following, we focus on h + since there is no cou-pling between h + states and h − states. The energy spec- trum of the h + eigenstates is plotted in Fig. 2. Thetreatment of the h − block works analogously. For com-pleteness, these h − energy states are drawn in light grayin Fig. 2.Since we aim at an investigation of finite 2D structures,an important issue is the modelling of the edges and thecorresponding edge states. We introduce an edge intoour Hamiltonian by a mass term M → M + V edge ( y ) thatvaries slowly on the scale of the typical extent of a Landauwave function perpendicular to the edge ( y -direction). Inthe then justified adiabatic approximation one obtains anextra contribution V edge ( k/B ) σ to the electron energies,where k is the crystal momentum parallel to the edge.The resulting energy diagram of the electron states nearan edge with quadratic confinement potential V edge ( y ) =( y/l e ) meV is shown in Fig. 2. l e is the typical lengthscale of the variation of the confinement potential. Wefurther assume that the two edges at opposite sides aresufficiently away from each other such that the finite sizeeffects of overlapping edge states, recently analyzed inRef. [9], do not matter. FIG. 2: (Color online) Landau level dispersion and opticaltransitions for h + for B = 1 T in bulk and near an edge.The electron- (hole-) like levels are labeled by n e ( n h) [12].The thick (red) lines are the interband transitions. The thin(blue) lines are intraband transitions of smaller energy in thebulk. The thin (blue) line in the edge region represents thetransition which we choose (see text). The energy levels of h − are drawn in light gray. Irradiation by a classical electromagnetic field is de-scribed by a time dependent vector potential A ( r , t ) =2 | A | ˆ ǫ cos( ω/c ˆ n · r − ωt ). We choose ω in the far infrared(FIR) regime, corresponding to the typical excitation en-ergies in our system. Consider linearly polarized light(ˆ ǫ = ˆ e y ) shining on the sample from the + z direction(ˆ n = − ˆ e z ), so that we can write in the Landau gauge k x → p x − By, k y → p y + 2 | A | cos( − ωt ) . (3)To linear order in the radiation field, two new terms ap-pear in the Hamiltonian h + → h + + h ω + h ω with h ω = − i | A | cos( ωt ) √ B ( a † − a ) (cid:2) DI × + Gσ (cid:3) (4) h ω = − | A | A cos( ωt ) σ . (5)For typical parameters as taken from Fig. 1, h ω is oneorder of magnitude smaller than h ω and thus, we neglectit [14]. By employing a rotating wave approximation wefind transitions between (cid:12)(cid:12) ψ + i ( n ) (cid:11) and (cid:12)(cid:12) ψ + j ( n ± (cid:11) where (cid:12)(cid:12) ψ + i ( n ) (cid:11) , n = 1 , ..., ∞ , i = e , h are the eigenstates of h + (see Fig. 2). In addition, there is a nonzero optical ma-trix element between the zero mode and the electron- andhole-like states of the first Landau level. This situationis very similar to the graphene case [10]. The presenceof an edge which is modelled by a spatially varying massterm does not change the selection rules ∆ n = ±
1. Also amore elaborate calculation within the 8 band Kane modelleads qualitatively to the same optical transitions.We now turn to the question how irradiation of a clas-sical light field affects the charge transport through theedge states. Consider one single pair of edge states,namely the 1h and the zero mode of the h + sector (seeFig. 2). Further assume that the Fermi energy is tunedsuch that it crosses the topmost counterclockwise movingedge state 1h [15]. One expects that the irradiation of ashort, strong FIR pulse, tuned to the transition energy, isable to scatter an electron of the counterclockwise mov-ing edge state (1h) into the clockwise moving edge state(0) with a higher energy and reversed direction of mo-tion. Note that the crystal momentum of the scatteredelectron stays constant during this process. Counterin-tuitively, in this system light may thus backscatter elec-trons by reversing their group velocity at constant mo-mentum through transitions from a hole-like band intoan electron-like band [12].A more realistic scenario is the continuous illuminationof the whole QW by light, tuned to the selected transi-tion. The relevant properties of an edge are its length L ,the radiation strength profile A ( x ) as a function of theedge coordinate x and its Fermi energy ε F , defined by theelectrochemical potential of the upstream reservoir. Inthe following, we assume that the frequency of the radi-ation is tuned to resonance with the transition indicatedin Fig. 2. More precisely, we assume that the energeticdifference ∆ ε between the state in the 1h mode with theFermi energy and the zero mode state with identical mo-mentum equals ω , the radiation frequency. We focus onthis pair of states and neglect all other edge states as theyare all highly off-resonant. We also assume a vanishinglaser line width. The transport through the edge is thendescribed by a 2 × T E ( x, x ′ ) [11] whichsatisfies the equation[ E − H ( − i∂ x )] T E ( x, x ′ ) = 0 , (6) where H ( k ) is well approximated by H ( k ) ≃ v k + Q ( x ) σ , v = (cid:18) − v v (cid:19) . (7)Here, Q ( x ) = γA | A ( x ) | characterizes the x -dependentintensity of the FIR source and E measures the energyof the scattering states relative to the Fermi energy ε F . γ . γ into A . Thesolution of (6) with the initial condition T E (0 ,
0) = 1yields M E = T E (0 , L ) = T x exp − i Z L dx v − ( E − Q ( x ) σ ) ! . (8)The symbol T x indicates a spatial ordering of operators,in analogy to the time ordering operator in the quantummechanical time evolution operator. v and v are theabsolute velocities of the clockwise mover and the coun-terclockwise mover, respectively. In the linear responseregime at zero temperature, when the energy E of therelevant edge state is E = 0, we find an exponentialsuppression of the propagation through the edge. Thetransmission amplitude at E = 0 reads t ( a ) = 1( M ) = sech (cid:18) a A √ v v (cid:19) , (9)where a = R L dx A ( x ) is the FIR intensity integratedover the length of the edge.The off-resonant edge transmission, E >
0, is shown inFig. 3 for a steplike radiation intensity profile A ( x ) = A [Θ( L − x ) − Θ( − x )] (where Θ( x ) is the Heaviside func-tion) and v = v . [16] As expected, the transport ceasesto be exponentially suppressed at | E | > A | A | , where A | A | ∼ µ eV for typical parameter values. Transitionsbetween pairs of states with energy difference ∆ ǫ thus donot block the current if (∆ ǫ − ω ) /A | A | ≫
1. This is typ-ically the case for all pairs of states at equal momentumbut the one that the radiation frequency ω is tuned to.We now turn to a discussion of two possible experi-mental realizations of optical manipulation of edge statetransport. For this, we assume that the whole sampleis uniformly illuminated by a laser of constant intensity | A | , such that a = L | A | . Experimental setup I.
Consider a quadratic structureof a HgTe quantum well with four contacts at the corners(see Fig. 4). The electrochemical potentials of the con-tacts µ i are tuned such that all µ i correspond to energiesbetween the bulk Landau levels 1 h and 2 h (see Fig. 2).Within these bounds, assume that µ > µ = µ = µ ,the difference of µ and µ being sufficiently small tobe in the linear response regime. Furthermore we as-sume equal side lengths L = L . Because of µ > µ ,a net current I ∝ µ − µ due to a counterclockwise FIG. 3: (color online) Transmission coefficient | t E ( A L, v ) | away from the linear response regime for counterpropagatingmodes with identical absolute group velocities v = v = v . A ( x ) = A [Θ( L − x ) − Θ( − x )] is assumed. The energy ofthe incident electron is varied from the linear response regime E ≃ ε F . The thin (red) arrow represents the net transportmode with energy larger than ε F due to a slightly elevatedelectrochemical potential µ > µ i , i = 2 , , moving transport mode flows from contact 1 to contact2. Irradiation of laser light, tuned to the transition en-ergy indicated in Fig. 2, results in a suppression of I by scattering the counterclockwise movers into clockwisemoving edge states which exit back into contact 1, as dis-cussed above. The photons of the FIR source supply therequired energy for this scattering process. According toEq. (9) the current I will be exponentially suppressed I ∼ sech (cid:18) | A | A L v (cid:19) (10)in the zero temperature, linear response regime. Notethat Eq. (9) has been derived for a laser profile that isfocused onto a finite segment of the edge. One, however,expects qualitatively the same behavior also for uniformillumination, which irradiates not only the edge, but alsothe reservoirs. This is because the origin of the effect,the opening of a transport gap through coupling of the modes 1h and 0, is present in both scenarios. For 5 µ mof illuminated edge, v ≃ and a FIR power of 4mWfocused onto an area of 1mm the parameter | A | AL /v in Eq. (9) is of order 1. We estimate the temperature re-quired for the zero temperature regime assumed above, kT ≪ | A | A , to be of order 100mK. Thus, the inter-esting, exponentially suppressed, regime is well withinexperimental reach. Experimental setup II.
Now we consider the oppositelimit, namely a rectangular device with L ≫ L and µ i = µ j , ∀ i, j . Without laser irradiation no net currentflows between the terminals. Nevertheless, a large coun-terclockwise background current exists, as illustrated bythe gray arrows in Fig. 4. When the laser is switched onparts of the large counterclockwise current are blocked.According to Eq. (9), the blocking will be more effectivefor the long edges than for the short edges and thus anet current from terminal 3 to terminal 1 and one fromterminal 2 to terminal 4 will be measured. We had sup-pressed this effect in the scenario of setup I by choosingall edges of equal length. The action of the laser on thebackground current then cancels between edges. Alter-natively, one may observe the suppression predicted byEq. (10) also for unequal lengths L = L by measuringthe response of the current I to a small change of µ .In conclusion, we have shown that it is possible to op-tically manipulate the electronic transport in quantumhall edge states by illumination with properly tuned laserlight. Remarkably, the backscattering into counterprop-agating modes by photons is only possible if the relevantedge modes are hole-like. The scattering of hole-like edgestates to electron-like edge states reverses the group ve-locity which results in a measurable reversal of the chargecurrent direction through an edge. The HgTe QW isespecially convenient for an experimental realization ofthis proposal, since the relevant parameters are highlytunable and well under control.We acknowledge enlightening discussions with B.Braunecker, H. Buhmann, D. Loss and L.W. Molenkamp.M.J.S. was financially supported by Swiss NSF andNCCR Nanoscience. E.G.N. acknowledges financial sup-port by the German DFG via grant no. AS327/2-1. B.T.was financially supported by the German DFG via grantno. Tr950/1-1. [1] A. Kitaev, Ann. Phys. (N.Y.) , 2 (2003).[2] X.-G. Wen, Quantum Field Theory of Many-Body Sys-tems , Oxford University Press (2004).[3] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science , 1757 (2006).[4] M. K¨onig, Hartmut Buhmann, Laurens W. Molenkamp,Taylor L. Hughes, Chao-Xing Liu, Xiao-Liang Qi, andShou-Cheng Zhang, J. Phys. Soc. Jpn. , 031007 (2008).[5] M. K¨onig, A. Tschetschetkin, E. M. Hankiewicz, Jairo Sinova, V. Hock, V. Daumer, M. Sch¨afer, C. R. Becker,H. Buhmann, and L. W. Molenkamp, Phys. Rev. Lett. , 076804 (2006).[6] Y. S. Gui, C. R. Becker, N. Dai, J. Liu, Z. J. Qiu, E. G.Novik, M. Sch¨afer, X. Z. Shu, J. H. Chu, H. Buhmann,and L. W. Molenkamp, Phys. Rev. B , 115328 (2004).[7] A.K. Geim and K.S. Novoselov, Nature Mater. , 183(2007).[8] E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. La-tussek, C. R. Becker, G. Landwehr, H. Buhmann, and L.W. Molenkamp, Phys. Rev. B , 035321 (2005).[9] Bin Zhou, Hai-Zhou Lu, Rui-Lin Chu, Shun-Qing Shen,and Qian Niu, Phys. Rev. Lett. , 246807 (2008).[10] Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E.Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, Phys.Rev. Lett. , 197403 (2007). [11] P. A. Mello and N. Kumar, Quantum Transport in Meso-scopic Systems , Oxford University Press (2004).[12] The characterization of Landau levels as electron (hole)-like refers to their behavior near an edge: if the energy ofa band increases (decreases) as the edge is approached,we label it as electron (hole)-like.[13] We set ~ , e, c = 1 in the following.[14] h ω1+