The scaling theory of the integer quantum hall effect

Huckestein, Bodo

doi:10.1007/bfb0106018

Cited by 47 publications

(84 citation statements)

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“…Thouless showed that localized states have zero Chern numbers 12 , which is easy to understand, because localized states are rather insensitive to changes in the boundary conditions used to calculate the Chern number 9 . This is verified by numerical calculations 10,11 , which find that non-zero Chern numbers appear near the centre of the Landau level, with a distribution in perfect agreement with the scaling theory of the integer quantum Hall effect [13][14][15] (IQHE).…”

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confidence: 59%

Spectral weight transfer in the integer quantum Hall effect and its consequences

Zhou

Berciu

2007

Nature Phys

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The energy spectrum of a two-dimensional electron gas placed in a transversal magnetic field B consists of quantized Landau levels. In the absence of disorder, the degeneracy of each Landau level is N = B A/φ 0 , where A is the area of the sample and φ 0 = h/e is the magnetic flux quantum. With disorder, localized states appear at the top and bottom of the broadened Landau level, whereas states in the centre of the Landau level (the critical region) remain delocalized. This single-electron theory adequately explains most aspects of the integer quantum Hall effect 1 . One unnoticed issue is the location of the new states that appear in the Landau level with increasing B. Here, we show that they appear predominantly inside the critical region. This situation leads to a 'spectral ordering' of the localized states, which explains the stripes observed in measurements of the local inverse compressibility 2,3 , of two-terminal conductance 4 and of Hall and longitudinal resistances 5 without the need to invoke interactions as done in previous work 6-8 .The spectrum and eigenstates of a disorder-broadened Landau level can be studied with the well-established approach of diagonalizing the single-electron hamiltonianwhere V (r) is the disorder. We choose A(r) = (0, Bx) and apply periodic boundary conditions (PBCs) to a system of area A = L × L. Owing to the PBCs, the degeneracy of each Landau level, N = BL 2 /φ 0 , is an integer. Usually, in theoretical studies the magnetic field is kept fixed and the electron density n e or, equivalently, the filling factor ν = n e A/N, is swept by adjusting the Fermi energy E F .As the experiments mentioned above investigate the behaviour of various quantities in the (n e , B) plane, we need to understand how the spectrum changes when B is also tuned. Given the constraint that an integer number of fluxes must penetrate the sample, B can only change in discrete steps of φ 0 /L 2 . Therefore, we ask the following question: how do single-electron wavefunctions evolve when one more magnetic flux is inserted?Let |i, N , 1 ≤ i ≤ N be the eigenstates of a spin-polarized Landau level corresponding to a given disorder V (r) and a magnetic field B = N φ 0 /L 2 . The states are ordered by their energies E 1 < E 2 < ··· < E N (accidental degeneracies can be lifted with minute changes in V (r)).To see how the wavefunctions evolve when B increases, we calculate their disorder-averaged overlaps:where 1 ≤ i ≤ N and 1 ≤ j ≤ N + 1 label two eigenstates of the hamiltonian (1) with the same disorder potential but different magnetic fields. The overline indicates a disorder average. In the results presented here we typically average over 1,000 disorder realizations, and show results only for the spin-polarized lowest Landau level. Similar results are expected in higher Landau levels. The disorder potential V (r) is modelled as a sum of many shortrange, randomly placed gaussian scatterers. We show four sets of data, for L = 250, 400, 500, 750 nm, N = 50, 128, 200, 550 and therefore B = 1.654 T for the first t...

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confidence: 59%

Spectral weight transfer in the integer quantum Hall effect and its consequences

Zhou

Berciu

2007

Nature Phys

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“…Thus, entering the Hall plateau, the 2DES undergoes a percolation transition with the network of merged droplets impeding conduction across the sample, and thus producing a quantum Hall plateau 1,4 . However, in the absence of longrange potential fluctuations, theories do not require the existence of droplets at fixed positions [23][24][25] . The local charging observed in Fig.…”

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confidence: 98%

Subsurface charge accumulation imaging of a quantum Hall liquid

Tessmer

Glicofridis

Ashoori

et al. 1998

Nature

165

131

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The unusual properties of two-dimensional electron systems that give rise to the quantum Hall effect have prompted the development of new microscopic models for electrical conduction. The bulk properties of the quantum Hall effect have also been studied experimentally using a variety of probes including transport, photoluminescence, magnetization, and capacitance measurements. However, the fact that two-dimensional electron systems typically exist some distance (about 100 nm) beneath the surface of the host semiconductor has presented an important obstacle to more direct measurements of microscopic electronic structure in the quantum Hall regime. Here we introduce a cryogenic scanning-probe technique-- subsurface charge accumulation imaging-- that permits very high resolution examination of systems of mobile electrons inside materials. We use it to image directly the nanometer-scale electronic structures that exist in the quantum Hall regime.Comment: 6 pages, 4 figure

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“…The basic physical picture is that at the coercive field, the chiral edge states at the magnetic domain walls may tunnel into each other when the distance between them is shorter than the spatial decay length of the edge state. Therefore, the plateau transition at the coercivity in a QAH insulator can be mapped to the network model of the integer QH plateau transition in the lowest Landau level (LL) 8,[15][16][17] . It was predicted that an intermediate plateau with zero Hall conductivity could occur at H C , and the longitudinal conductivity will show two peaks 7 .…”

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Observation of the Zero Hall Plateau in a Quantum Anomalous Hall Insulator

Feng

et al. 2015

Phys. Rev. Lett.

119

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Quantum anomalous Hall (QAH) effect in magnetic topological insulator (TI) is a novel transport phenomenon in which theThe realization of QAH effect requires that a two-dimensional (2D) material must be FM, topological, and insulating simultaneously 9 . Magnetically doped TIs have been proposed 1, 2, 10-12 and experimentally proved 3-6 to be an ideal material system for fulfilling these stringent requirements. For a 3D TI, the inverted bulk band structure ensures topologically protected metallic surface states (SSs), which become 2D when the film is sufficiently thin 13 . The spontaneous FM order induced by magnetic doping not only leads to the anomalous Hall effect, but also opens an energy gap at the Dirac point. When the Fermi level (E F ) lies within this gap, the only remaining conduction channel is the quasione-dimensional chiral edge state, which gives rise to quantized Hall resistance and vanishing longitudinal resistance at zero magnetic field 3, 14 . Up to date, the QAH effect has been observed in Cr or V doped (Bi,Sb) 2 Te 3 TI thin films with accurately controlled chemical composition and thickness grown by molecular beam epitaxy (MBE) 3-6 .The MBE-grown QAH insulator film studied here has a chemical formula Fig. 1a is a schematic drawing of the transport device, which is similar to that reported previously 3 .The film is manually scratched into a Hall bar geometry, and the SrTiO 3 substrate is used as the bottom gate oxide due to its large dielectric constant at low temperature. The Cr concentration, hence the density of local moment, is higher than that in the sample where the QAH effect was originally discovered 3 . As a result, the FM order forms at a higher Curie temperature T C = 24 K as determined by the temperature dependent anomalous Hall effect (supplementary Fig. S1). Another important consequence of higher Cr doping is that the sample becomes more disordered, which is crucial to the physics that will be discussed in this work.We first demonstrate the existence of QAH effect in this sample. Fig. 1b displays the gate voltage (V g ) dependence of the Hall resistance  yx (blue curve) and longitudinal resistance  xx (red curve) measured at T = 10 mK in a strong magnetic field B = 12 T applied perpendicular to the film. The  yx exhibits a plateau for -10 V < V g < 10 V with its maximum value close to 99.1% of the quantum resistance h/e 2 ~ 25.8 k. In the same V g range  xx shows a pronounced dip with its minimum value close to 0.1 h/e 2 . To show that the apparent Hall quantization in Fig. 1b is due to the QAH effect rather than conventional QH effect in high magnetic field, in Fig. 1c we display the field dependence of  yx measured at V g = -5 V, when  xx reaches a minimum in Fig. 1b. The Hall trace shows an abrupt jump at zero magnetic field, characteristic of the anomalous Hall effect.With increasing magnetic field, the  yx value increases gradually and approaches h/e 2 at 12 T. The  xx shown in Fig. 1d exhibits two sharp peaks at the coercive field H C , and decreases rapidly on b...

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The scaling theory of the integer quantum hall effect

Cited by 47 publications

References 117 publications

Spectral weight transfer in the integer quantum Hall effect and its consequences

Spectral weight transfer in the integer quantum Hall effect and its consequences

Subsurface charge accumulation imaging of a quantum Hall liquid

Observation of the Zero Hall Plateau in a Quantum Anomalous Hall Insulator

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