Magnetic surface levels

Khaikin, M. S.

doi:10.1080/00018736900101257

Cited by 47 publications

(16 citation statements)

References 24 publications

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“…This quantization rule h a s (he simple geomelrical interprefalion [20] (hat the flux enclosed by one arc of the skipping orbit and the boundary equals (n-1/4) times the flux quantum h\e (see insets in Fig. 6).…”

Section: Edge Slalcs and Skipping Orbitsmentioning

confidence: 99%

“…Magnetic edge states [19,20] are transverse inodes of a wave guide of width ~ / cyc] formed by thc magnelic fielcl at tlie 2DEG boundary. The edge states at tlie Fermi level are labclled by a quantum number n = 1,2 ... N, with W = /c r .· / cyd /2 tlie total number of propagating modes or edge channels (for simplicity we ignore here tlie discreleness of N).…”

Section: Edge Slalcs and Skipping Orbitsmentioning

confidence: 99%

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Coherent electron focusing

Beenakker

Houten

Wees

1989

Advances in Solid State Physics

198

326

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Summnry: Thcory and cxpcrimcnl arc rcvicwcd οΓ Ihc clussical and quantum mcchanical focusing by a niagncüc fickl of ballislic clcctrons injcctcd tlirough a point contact in a Uvo-dimcnsional electron gas. Two altcrnalivc poinLs of vicw arc cmphasizcd. On thc onc band, thc cxpcrimcnL i s n rcalizntion ofclcclron oplics in Ihc solid stalc. Thc thrcc basic building blocks arc a cohcicnt and monochromalic point sourcc/dctcctor, an elcctrostalic mirror wilh littlc diffuse scatlcring, and a magnclic Icns. On thc othcr band, cohcrent clcclron focusing is a rcsislancc mcasurcmcnl in thc quantum ballistic transport rcgime, which cxhibits Ihc charactcristic featurcs of this rcgime in a most extreme way. For cxamplc, largc magnclorcsi.stancc oscillations occur (up to 9.5% amplitudc modulalion is obscrvcd), with a pcriodicily which is non-locally dclcrmincd by thc Separation bctwccn currcnl and vollage point contacLs. Λ WKB calculation of Ihc liansmission probabililics shows lliat Ihis cffect is thc rcsult of thc intcrfcrcncc of cohcrcnlly cxcitcd magnctic edgc slalcs al the electron gas boundary, Anothcr cxamplc is thc abscncc of local cquilibrium: Thc mcasurcmcnts shovv that thc point conlacts can sclectivcly populatc (and clelcct) specific Land au Icvcls, and that Ihis highly non-cquilibrium population is mainlaincd ovcr dislances of microns.

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Section: Edge Slalcs and Skipping Orbitsmentioning

confidence: 99%

Section: Edge Slalcs and Skipping Orbitsmentioning

confidence: 99%

Coherent electron focusing

Beenakker

Houten

Wees

1989

Advances in Solid State Physics

198

326

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“…This behavior is explained by a semiclassical analysis of the transformation of closed cyclotron orbits into open, deconfined trajectories. Comparison to SdH-type resonances in the local density of states is presented.Electron cyclotron motion constrained by crystal boundaries displays many interesting phenomena, such as skipping orbits and electron focusing, which have yielded a wealth of information on scattering mechanisms in solids [1,2]. Since the 1980s, many groups have investigated electron transport in semiconducting two-dimensional electron systems (2DES), where gateinduced spatially varying electric fields can be used to alter cyclotron motion.…”

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confidence: 99%

Collapse of Landau Levels in Gated Graphene Structures

Rudner

Young

et al. 2011

Phys. Rev. Lett.

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We describe a new regime of magnetotransport in two dimensional electron systems in the presence of a narrow potential barrier imposed by external gates. In such systems, the Landau level states, confined to the barrier region in strong magnetic fields, undergo a deconfinement transition 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 magnetic field is reduced below a certain critical value. This behavior is explained by a semiclassical analysis of the transformation of closed cyclotron orbits into open, deconfined trajectories. Comparison to SdH-type resonances in the local density of states is presented.Electron cyclotron motion constrained by crystal boundaries displays many interesting phenomena, such as skipping orbits and electron focusing, which have yielded a wealth of information on scattering mechanisms in solids [1,2]. Since the 1980s, many groups have investigated electron transport in semiconducting two-dimensional electron systems (2DES), where gateinduced spatially varying electric fields can be used to alter cyclotron motion. A variety of interesting phenomena were explored in these systems, including quenching of the quantum Hall effect [3,4], Weiss oscillations due to commensurability between cyclotron orbits and a periodic grating [5], pinball-like dynamics in 2D arrays of scatterers [6], and coherent electron focusing [7].The experimental realization of graphene [8], a new high-mobility electron system, affords new opportunities to explore effects that were previously inaccessible. In particular, attempts to induce sharp potential barriers in III-V semiconductor quantum well structures have been limited by the depth at which the 2DES is buriedtypically about 100nm below the surface [9]. In contrast, electronic states in graphene, a truly two-dimensional material, are fully exposed and thus allow for potential modulation on ∼ 10 nm length scales using small local gates and thin dielectric layers [10][11][12][13]. Significantly, such length scales can be comparable to the magnetic length ℓ B = c/eB, which characterizes electronic states in quantizing magnetic fields. In this Letter we focus on one such phenomenon, the transformation of the discrete Landau level spectrum to a continuum of extended states in the presence of a static electric field.The behavior which will be of interest for us is illustrated by a toy model involving the Landau levels of a massive charged particle in the presence of an inverted parabolic potential U (x) = −ax 2 . Competition between the repulsive potential and magnetic confinement gives rise to a modified harmonic oscillator spectrum ε n (p y ) = e m B 2 − B 2 c (n + 1/2) − 2ap 2 y e 2 (B 2 − B 2 c ) (1) for B > B c , where m is the particle mass, p y is the y component of momentum, and B c = √ 2ma/e is the critical magnetic field strength. For strong magnetic field, B > B c , the spectrum consists of discrete (but dispersive) energy ban...

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“…The periodic motion perpendicular to the boundary leads to the formation of discrete quantized states (with quantum number n). These states are known äs magnetic edge states (and in metals äs magnetic surface states [35,36]). The wave number along the boundary is a plane wave with wave number ky Ξ ρ /U, which can be expressed in the guiding center coordinate X according to X = -(l m } 2 ky.…”

Section: Skipping Orbits and Magnetic Edge Statesmentioning

confidence: 99%