Exponent Of Localization Length Research Articles

Magnetic-field dependence of the quantum-Hall-liquid–insulator transition

We present measurements of the magnetic field dependence of the insulator transition in a gated ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As/GaAs heterostructure. The resulting phase diagram shows reentrant behavior as the magnetic length decreases below the zero-field electron mean free path. By fitting our data to a predicted phase diagram, we obtain a value for the localization length exponent governing the quantum-Hall-liquid--insulator transition. As has been recently suggested, this value is in agreement with that determined for the critical exponent between quantum Hall plateaus.

Conductivity peaks in the quantum Hall regime: Broadening with temperature, current, and frequency

We argue that it is hopping transport that is responsible for broadening of the σ xx peaks in low-mobility samples. Explicit expressions for the width Δν of a peak as a function of the temperature T, current J, and frequency ω are found. It is shown that Δ v grows with T as ( T T 1) K , where k is the inverse localization-length exponent. The current J is shown to act like an effective temperature T eff( J) α J 1 2 if T eff( J) ⪢ T. Broadening of the oh peaks with frequency ω is found to be determined by the effective temperature T eff(ω) ≈ ℏω k B .

Corrections to scaling in the integer quantum Hall effect.

Finite-size corrections to scaling laws in the centers of Landau levels are studied systematically by numerical calculations. The corrections can account for the apparent nonuniversality of the localization length exponent \ensuremath{\nu}. In the second lowest Landau level the irrelevant scaling index is ${\mathit{y}}_{\mathrm{irr}}$=-0.38\ifmmode\pm\else\textpm\fi{}0.04. At the center of the lowest Landau level an additional periodic potential is found to be irrelevant with the same scaling index. These results suggest that the localization length exponent \ensuremath{\nu} is universal with respect to the Landau level index and an additional periodic potential.

Conductivity peaks broadening in the quantum Hall regime

We argue that it is the hopping transport that is responsible for broadening of the σ xx peaks. Explicit expressions for the width Δν of a peak as a function of the temperature T, current J and frequency ω are found. It is shown that Δν grows with T as ( T/ T 1) κ, where κ is the inverse localization-length exponent. The current J is shown to act like the effective temperature T eff(J) ∝ J 1 2 if T eff(J) ▪T . Broadening of the ohmic ac-conductivity peaks with frequency ω is found to be determined by the effective temperature T eff(ω)∼ h ̵ ω.

Directed polymer localization in a disordered medium.

The localization of a directed polymer onto an extended defect (such as a line or a plane) in the presence of competing bulk disorder is examined. Based on scaling ideas and exact analysis on a hierarchical lattice, we develop a new renormalization scheme to study the directed polymer localization problem. We establish absence of delocalization transition for attractive columnar defect in the marginal dimension ${\mathit{d}}_{\mathit{c}}$=2, and for attractive planar defect in d=3. For columnar defect in three dimensions, our simulations yield a localization length exponent ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\perp}}}$=1.8\ifmmode\pm\else\textpm\fi{}0.6.

Microwave frequency dependence of integer quantum Hall effect: Evidence for finite-frequency scaling.

We present ac measurements of the diagonal conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$, in the integer quantum Hall regime for frequency f between 0.2 and 14 GHz and temperature T\ensuremath{\ge}50 mK. Re(${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$) was obtained from the measured attenuation of a coplanar transmission line on the sample surface. For f\ensuremath{\gtrsim}1 GHz, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$ peaks between IQHE minima broaden as f increases, roughly as (\ensuremath{\Delta}B)\ensuremath{\propto}${\mathit{f}}^{\ensuremath{\gamma}}$, where \ensuremath{\Delta}B is the peak width. \ensuremath{\gamma}=0.41\ifmmode\pm\else\textpm\fi{}0.04 for spin-split peaks, and for spin-degenerate peaks \ensuremath{\gamma}=0.20\ifmmode\pm\else\textpm\fi{}0.05. We interpret our data in terms of dynamic scaling. Our \ensuremath{\gamma}, when compared with existing estimates of the localization length exponent \ensuremath{\nu}, is consistent with assigning a dynamic exponent z=1.

Localization phase diagram for a disordered system in a magnetic field in two dimensions

A phase diagram for the localization-delocalization transition of a two-dimensional disordered semiconducting system in a perpendicular magnetic field B is investigated with a numerical method. Disorder originates from a random distribution of shallow impurities, measured in units of the impurity concentration c. Starting with a tight-binding Hamiltonian and an impurity state basis, the localization criterion is defined by means of the quantum connectivity of impurities. Finite-size scaling is employed to study the transition in the B-c parameter space. On this footing a phase diagram of the localization-delocalization transition in the B-c parameter space is calculated. At low concentrations c<c1 approximately=0.246a-2 where a is the impurity radius, all states are localized. Above c1 two nose-shaped areas of a phase of delocalized states exist, the tips of which are found at (c1,B1)=(0.246+or-0.004a-2, 0.013+or-0.001) and (c3,B3)=(0.67+or-0.03a-2, 0.76+or-0.07) with the magnetic field given in terms of a2l-2, where l is the Lamor length. Both areas join at (c2,B2)=(1.2+or-0.2a-2, 0.233+or-0.009). States are well localized at B=0. An estimate of the localization length exponent is given. The transition is discussed in terms of orbital shrinking and interference effects, which are safely distinguished. The latter mechanism can account for a re-entrant behaviour with respect to the magnetic field. The metal-insulator transition is discussed as a function of the electron density in conjunction with the phase diagram. Results are compared with previous calculations within the zero differential overlap approximation.

Quantum percolation and plateau transitions in the quantum Hall effect.

We introduce a simple model of quantum percolation and analyze it numerically using transfer matrix methods. A central point of this paper is that 3 both integer and fractional plateau transitions in the quantum Hall effect are due to quantum percolation. Within this model, we obtained the localization length exponent \ensuremath{\nu}=2.4\ifmmode\pm\else\textpm\fi{}0.2, the dynamical exponent z=1, and the scaling functions for the conductivity tensor for both the integer and the fractional transitions. We show that our results agree extremely well with the experimental results for the integer plateau transition obtained by McEuen et al.

Edge and Bulk Landau States in Quantum Hall Regime

Backscattering probability of edge states is calculated in two-dimensional systems confined within a finite width in strong magnetic fields. The critical exponent of localization length of bulk Landau states is derived using the system-size dependence of the energy interval where the transmission probability changes from zero to unity.

Universal singularities in the integral quantum Hall effect.

The transition between localized and extended eigenstates in the integral quantum Hall regime is observable as a scaling phenomenon in the magnetoresistance tensor as a function of temperature and applied magnetic field strength. Field-theoretic studies on the quantum Hall effect are used to derive the scaling functions. The scaling behavior is shown to involve only a single critical exponent ($\ensuremath{\mu}$) which is the ratio between the inelastic-scattering exponent ($p$) and twice the localization length exponent ($\ensuremath{\nu}$). The results have recently been confirmed experimentally.

Magnetic tuning of the metal-insulator transition for uncompensated arsenic-doped silicon.

Barely insulating and barely metallic Si:As samples have been studied in the temperature range 4.2 K to 50 mK. Metallic samples show a new high-field magnetoresistance behavior explainable in terms of magnetic tuning of the critical density ${N}_{c}$. Insulating samples at fixed fields show Mott variable-range-hopping behavior and the increase of characteristic temperature ${T}_{0}(N,H)$ with $H$ results primarily from magnetic tuning of the localization-length exponent $\ensuremath{\nu}$. Magnetocapacitance results depend on field-induced changes in both ${N}_{c}$ and $\ensuremath{\nu}$.

Critical behavior of Mott variable-range hopping in Si:As near the metal-insulator transition.

dc conductivity measurements of barely insulating uncompensated Si:As samples yield results showing Mott variable-range-hopping behavior [ln\ensuremath{\sigma}\ensuremath{\propto}(${T}_{0}$/T)(1/4)] in the temperature range 0.15&lt;T&lt;10 K. The exponent (1/4) has been confirmed using the logarithmic-derivative technique. The characteristic temperature ${T}_{0}$ shows scaling behavior of the form ${T}_{0}$\ensuremath{\propto}(1-N/${N}_{c}$${)}^{p}$ with 2.3&lt;p&lt;2.9 for 1-N/${N}_{c}$0.07 and p\ensuremath{\simeq}8.4 for 0.11-N/${N}_{c}$0.16. Both the exponent (1/4) in the Mott law and the localization length exponent \ensuremath{\nu}=p/3 derived from the scaling of ${T}_{0}$ suggest a filling-in of the Coulomb gap and a reduced role of electron-electron interactions in the temperature range of these measurements.

Magnetic Tuning of the Metal-Insulator Transition for Uncompensated Arsenic-Doped Silicon

Barely insulating and barely metallic Si: As samples have been studied in the temperature range 4.2 K to 50 mK. Metallic samples show a new high-field magnetoresistance behavior explainable in terms of magnetic tuning of the critical density Nc. Insulating samples at fixed fields show Mott variable-range-hopping behavior and a strong increase in the characteristic temperature T0(N, H) with H. The ratio T0(N, H)/T0(N, 0) at fixed field shows quasicritical behavior increasing by a large field-dependent factor as N → Nc. Magnetocapacitance measurements on an 0.87Nc sample show an initial quadratic decrease with H followed by a flattening toward a linear decrease above 4 T. Both T0(N, H) and ε'(N, H) - εh can be explained in terms of the localization length decreasing with H, however the interpretation of the field-dependent decrease depends somewhat on the theoretical model employed. The decrease in ξ(N, H) cannot be explained in terms of Nc-tuning, but also seems to require some field-induced tuning in the localization length exponent which is qualitatively consistent with Hikani's theoretical prediction.