Landau Level Collapse in Gated Graphene Structures
Nan Gu, Mark Rudner, Andrea Young, Philip Kim, Leonid Levitov
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Landau Level Collapse in Gated Graphene Structures
Nan Gu, Mark Rudner, Andrea Young, Philip Kim, and Leonid Levitov Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139 Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138 Department of Physics, Columbia University, New York, NY10027
We describe a new regime of magnetotransport in two dimensional electron systems in the pres-ence of a narrow potential barrier imposed by external gates. In such systems, the Landau levelstates, confined to the barrier region in strong magnetic fields, undergo a deconfinement transi-tion as the field is lowered. We present transport measurments showing Shubnikov-de Haas (SdH)oscillations which, in the unipolar regime, abruptly disappear when the strength of the magneticfield is reduced below a certain critical value. This behavior is explained by a semiclassical analysisof the transformation of closed cyclotron orbits into open, deconfined trajectories. Comparison toSdH-type resonances in the local density of states is presented.
Electron cyclotron motion constrained by crystalboundaries displays many interesting phenomena, suchas skipping orbits and electron focusing, which haveyielded a wealth of information on scattering mecha-nisms in solids [1, 2]. Since the 1980s, many groupshave investigated electron transport in semiconductingtwo-dimensional electron systems (2DES), where gate-induced spatially varying electric fields can be used toalter cyclotron motion. A variety of interesting phenom-ena were explored in these systems, including quenchingof the quantum Hall effect [3, 4], Weiss oscillations dueto commensurability between cyclotron orbits and a pe-riodic grating [5], pinball-like dynamics in 2D arrays ofscatterers [6], and coherent electron focusing [7].The experimental realization of graphene [8], a newhigh-mobility electron system, affords new opportunitiesto explore effects that were previously inaccessible. Inparticular, attempts to induce sharp potential barriers inIII-V semiconductor quantum well structures have beenlimited by the depth at which the 2DES is buried—typically about 100nm below the surface [9]. In contrast,electronic states in graphene, a truly two-dimensionalmaterial, are fully exposed and thus allow for potentialmodulation on ∼
10 nm length scales using small localgates and thin dielectric layers [10–13]. Significantly,such length scales can be comparable to the magneticlength ℓ B = p ~ c/eB , which characterizes electronicstates in quantizing magnetic fields. In this Letter wefocus on one such phenomenon, the transformation ofthe discrete Landau level spectrum to a continuum ofextended states in the presence of a static electric field.The behavior which will be of interest for us is illus-trated by a toy model involving the Landau levels of amassive charged particle in the presence of an invertedparabolic potential U ( x ) = − ax . Competition betweenthe repulsive potential and magnetic confinement givesrise to a modified harmonic oscillator spectrum ε n ( p y ) = ~ em p B − B c ( n + 1 / − ap y e ( B − B c ) (1)for B > B c , where m is the particle mass, p y is the y FIG. 1: (a) Differentiated conductance, dG/dV tg , of a nar-row top gate graphene device, pictured in (c). Fabry-Perot(FP) oscillations appear in the presence of confining pn junc-tions. (b) dG/dV tg as a function of B and V tg . Shubnikov-deHaas (SdH) oscillations are observed at high B . The fan-like SdH pattern is altered by the barrier: in the pp ′ p regionit curves, weakens, and is washed out at fields | B | . B c ,Eq.(7), while in the pnp region a crossover to FP oscilla-tions occurs. Data shown correspond to V bg = −
70V [dashedline in (a)]. (c) Top gated graphene device micrograph andschematic; top gate width is ∼ dN/dε , which corresponds to the measured quantity dG/dV tg , is shown. Dashed parabola marks the critical field,Eq.(8). Oscillations in the DOS modulate the rate of scatter-ing by disorder, resulting in the SdH effect [18]. component of momentum, and B c = √ ma/e is the crit-ical magnetic field strength. For strong magnetic field, B > B c , the spectrum consists of discrete (but disper-sive) energy bands indexed by an integer n , whereas for B ≤ B c the spectrum is continuous even for fixed p y .This behavior can be understood quasiclassically in termsof transformation of closed cyclotron orbits into open or-bits, which occurs when the Lorentz force is overwhelmedby the repulsive barrier potential.For massless Dirac charge carriers in single-layergraphene, Landau levels subject to a linear potential U ( x ) = − eEx manifest an analogous collapse of the dis-crete spectrum below the critical field B c = E/v F [15]: ε n ( p y ) = ± v F √ n ~ eB (cid:0) − β (cid:1) / − βv F p y , (2)where n = 0 , , ... and β = E/v F B . The transitionat B = B c can be linked to the classical dynamics of amassless particle, characterized by closed orbits at B >B c and open trajectories at B < B c [16].A simple but intuitive picture of the spectrum (2) canbe obtained from the Bohr-Sommerfeld quantization con-dition Z x x p x ( x ) dx = π ~ ( n + 1 / − γ ) , (3) p x ( x ) = q ( ε − U ( x )) /v F − ( p y − eBx ) , (4)where x and x are the turning points, p y is the con-served momentum parallel to the barrier, and the Berryphase contribution γ is 1 / U ( x ), this gives the Landau level spectrum (2) for B > B c . As B approaches B c , one of the turning pointsmoves to infinity, indicating a transformation of closedorbits into open trajectories.To realize the collapse of Landau levels in an electronsystem, several conditions must be met. First, it must bepossible to create a potential barrier that is steep on thescale of the cyclotron orbit radius. Second, the systemmust be ballistic on this length scale, in order to sup-press the broadening of Landau levels due to disorder.Graphene, which is a truly two-dimensional material withhigh electron mobility, fulfills both conditions. Crucially,as demonstrated by the recent observation of Fabry-Perot(FP) oscillations in gated graphene structures [13], trans-port can remain ballistic even in the presence of a gate-induced barrier. Thus graphene is an ideal system forstudying the Landau level collapse.Transport data taken from a locally gated devicesimilar to that described in Ref.[13] are shown inFig.1. Graphene was prepared via mechanical exfoli-ation and contacted using electron beam lithographybefore being coated with a 7/10 nm thick hydrogensilsesquioxane/HfO dielectric layer. Narrow ( ∼
16 nm)palladium top gates were then deposited, and the electri-cal resistance measured at 1.6 Kelvin. Due to the simi-larity between the top gate width and its distance fromthe graphene 2DES, the top gate-induced density is wellapproximated by that of a thin wire above the sampleplane and an infinite conducting plane below it: eρ ( x ) = C tg V tg x /w + C bg V bg , (5) where C tg(bg) and V tg(bg) are the top (bottom) gate ca-pacitance and applied voltage and w is the distance be-tween the wire and the top gate. For such a thin gate,the modulation of the total resistance due to the gate issmall; in order to subtract the series resistances of thegraphene leads, the numerical derivative of the conduc-tance with respect to the top gate voltage, dG/dV tg , wasanalyzed.At zero magnetic field (Fig.1a), dG/dV tg shows distinctbehavior in four regions in the ( V bg , V tg ) plane, corre-sponding to pp ′ p, pnp, npn, and nn ′ n doping, where n(p) refers to negative (positive) charge density and ′ in-dicates different density. The appearance of FP interfer-ence fringes when the polarity of charge carriers in thelocally gated region and graphene leads have oppositesigns indicates that the mean free path is comparable tothe barrier width, l mf ∼ w .In high magnetic field, in both the bipolar and unipo-lar regimes, we observe a fan of SdH resonances corre-sponding to Landau levels (see Fig.1b). At lower fields,different behavior is observed depending on the polarityunder the gate. In the bipolar regime, as B is lowered, theSdH resonances smoothly evolve into FP resonances. Thehalf-period shift, clearly visible in the data at B ≈ B c ≈ ε = ε F and p y = 0 gives a good estimate for the po-sitions of the transport resonances (see also Fig.1d). Fora generic barrier potential, the quantization conditioncan be written directly in terms of experimental con-trol parameters. Using the Thomas-Fermi approxima-tion, and ignoring the effects of ‘quantum capacitance’and nonlinear screening [17], we define the position-dependent Fermi momentum k F ( x ) = p πρ ( x ) /g , where g = 4 is the spin/valley degeneracy. Substituting the re-lation ε − U ( x ) = ~ v F k F ( x ) into Eq.(4), we obtain Z x x q π ~ ρ ( x ) /g − ( p y − eBx ) dx = π ~ n. (6)We see that the resonance condition is controlled solelyby the density profile ρ ( x ), Eq.(5), and is insensitiveto the specifics of the single-particle Hamiltonian up toslight modifications due to degree of pseudospin degen-eracy and Berry phase. In particular, the quantizationcondition assumes the same form, apart from the Berryphase contribution, for massless Dirac particles (mono-layer graphene) and massive particles with g = 4 (bilayergraphene); the spectrum would be only trivially modifiedfor GaAs quantum wells ( g = 2 and γ = 0).A simple estimate for the critical field can be ob-tained by comparing the curvature of ρ ( x ) at x = 0 withthe B x term in Eq.(6). Using the device parameters C bg = 115 aF /µ m , V bg = −
70 V, and w = 50 nm, wefind B c = ( ~ /ew ) ( πC bg V bg /e ) / ≈ . C bg V bg + C tg V tg = 0, which defines the boundaryat which polarity reversal occurs (white dashed line inFig.1a).The actual density profile ρ ( x ) is nonparabolic, flatten-ing out on a length scale 2 w ≈
100 nm. However, since 2 w far exceeds the magnetic length for the fields of interest( B & | x | & w , thecorresponding orbits are very long. For such states, theparticle traverses the TGR, makes a partial cyclotron or-bit outside of the TGR, and finally crosses the TGR againto close the orbit (Fig.2a). The net orbit length is a few w , which is much greater than the orbit size at strongfields (i.e. a few magnetic lengths). The contributionof long orbits to SdH oscillations will be suppressed dueto spatial inhomogeneity and disorder scattering; hencethe distinction between confined and deconfined orbitsremains sharp despite the flattening of the potential.To estimate the critical field as a function of exper-imental control parameters V tg and V bg for the densityprofile (5), we consider an equation for the turning points.Setting p y = 0, we have ~ k F ( x ) = ± eBx . Solving thisequation and equating the result to barrier half-width, x = ± w , we obtain B c = ( ~ /ew ) q (2 π/eg )(2 C bg V bg + C tg V tg ) . (7)The observed behavior of B c is well described by thisresult (red line in Fig.1b).To further explore the effects of curvature, below weanalyze the inverted parabola model, U ( x ) = − ax . Asimple estimate of the collapse threshold can be obtainedby considering balance between the Lorentz force and theforce due to the electric field, v F B = − dU/dx . Thiscondition is satisfied for a particle moving parallel to thebarrier with x = ℓ = ev F B/ (2 a ). Thus we find an energy-dependent critical field, B c ( ε ) = (2 /ev F ) √− aε, (8)which increases with detuning from neutrality, as in ex-periment.A microscopic model of collapse is provided by theHamiltonian H = (cid:18) U ( x ) v F p − v F p + U ( x ) (cid:19) , p ± = − i ~ ddx ± i ( p y − eBx ) , (9) where p y is the conserved canonical momentum compo-nent parallel to the barrier. We nondimensionalize theproblem using “natural units” ε ∗ = ( ~ v F a ) / , x ∗ = (cid:18) v F ~ a (cid:19) / , B ∗ = ~ e (cid:18) av F ~ (cid:19) / . For each value of p y and magnetic field B , we representthe Hamiltonian as an M × M matrix defined on a grid inposition space, with periodic boundary conditions. Weuse the eigenvalues and eigenstates obtained from diag-onalization to evaluate the local density of states (DOS)in the middle of the barrier, N ( ε ) = Z dp y π M X n =1 γπ h| ψ n,p y ( x = 0) | i ( ε − ε n ) + γ , (10)with Landau level broadening incorporated through theLorentzian width γ = 0 . ε ∗ . In our simulation, a systemof size L = 15 x ∗ discretized with M = 1500 points wasused. Averaging with a gaussian weight was used to sup-press the effect of spurious states arising due to a vectorpotential jump at the boundary, h| ψ n,p y ( x = 0) | i = Z dx ′ e − x ′ / σ | ψ n,p y ( x ′ ) | , (11)with σ ≈ x ∗ . Oscillations in the density of states (10)modulate the rate of electron scattering by disorder, andthus show up in transport quantities measured as a func-tion of experimental control parameters, as in the canon-ical SdH effect [18].The resulting local DOS, shown in Fig.1d, exhibits os-cillations which track Landau levels at high B . At lower B , the behavior is different in the pnp and pp ′ p regimes,realized at positive and negative energies, respectively. Inthe pnp case, a crossover to FP oscillations is observed.In the pp ′ p case, discrete Landau levels give way to a con-tinuous spectrum in the region inside a parabola (dashedline) which marks the quasiclassical collapse threshold,Eq.(8).We note that the behavior of dG/dV tg observed inthe pnp region, in particular the half-period shift of FPfringes at relatively low B . T , is not captured by localDOS, Eq.(10). As discussed in Ref.[14], this half-periodshift results from FP interference due to Klein scatter-ing at pn interfaces. A proper model of this effect mustaccount for ballistic conductance in the FP regime.The collapse observed in the density of states is relatedto deconfinement of classical orbits. The orbits can beanalyzed as constant energy trajectories of the problem ε = v F q p x + ˜ p y + U ( x ) , ˜ p y = p y − eBx. (12)For parabolic U ( x ) = − ax the orbits with p y = 0 canbe easily found in polar coordinates p x + ip y = | p | e iθ : | p | p = 1sin θ (cid:18) ± r − εε c sin θ (cid:19) , ε c = ( v F eB ) a (13) FIG. 2: (a) Closed orbits for the Thomas-Fermi potential ob-tained from the density profile, Eq.(5), with B = 9 , , , , p y = 0. Long trajectories, extending far outside the gatedregion, do not contribute to SdH oscillations (see text). (b,c)Trajectories for the potential U ( x ) = − ax and p y = 0. Threetypes of trajectories are shown in momentum space (b) andposition space (c): subcritical (red), critical (black), and su-percritical (blue). The saddle points in momentum space cor-respond to motion along straight lines x = ± ℓ , where theLorentz force is balanced by the electric field. with p = v F e B / a (see Fig.2b). Only real, positivesolutions should be retained; when ε/ε c >