Corresponding author: Vitaly A. Tkachenko (vtkach@isp.nsc.ru)

A simple model has been suggested for describing self-organization of localized charges and quantum scattering in

Charging of surface and interface defects is one of the key physical phenomena in semiconductor electronics [

In contrast to the standard modulation doping method, the 2D gas is in this case is created at low temperatures by bias _{g} between the metallic gate and metallic contacts connected to the GaAs working layer [_{g} = 0 and a high temperature, along with the image charge in the metal, and a common Fermi level is established in the metal / dielectric / undoped semiconductor structure (Fig. _{F} level is pinned by the defect states near the band gap center at the boundaries of the epitaxial heterostructure with the insulator and the GaAs semi-insulating substrate. This provides for flat bands in the semiconductor. In accordance with the Gauss theorem the Al_{2}O_{3} gate dielectric thickness _{e} for the adjacent GaAs layer and the Ti gate (Fig. _{σ} at the GaAs protective layer and the positive charge concentration at the metal/dielectric interface:

Here ε_{0} is the dielectric constant, ε_{ins} ≈ 8 is the dielectric permeability of Al_{2}O_{3}, _{e} ≈ 0.3 eV, and _{e} and _{σ} parameters remain the same as in equilibrium. The concentration _{σ} ~ 5 · 10^{11} cm^{-2} is assumed to be constant even if _{g} ≠ 0 and 2D gas is formed at a low temperature (Fig.

Equation (1) describes the area-average concentration of charges trapped by point defects (traps) at _{g} = 0 and a sufficiently high temperature on the surface of the GaAs protective layer. The distribution of the surface charges does not change below some “freezing temperature”. This temperature is determined by the energy of electron transition to the leakage level from the deep traps, but this freezing temperature is not known for undoped structures. Frozen disorder in the locations of surface charges and hence image charges in the metal, together produce static fluctuations of electrostatic potential at which mobile carriers are scattered in 2D gas at ~ 1 K if the gas is located close to the surface (

Schematic images of the object of study: (_{2}O_{3}, _{m} – work function of Ti, χ_{1}, χ_{s} – electron affinity of Al_{2}O_{3} and GaAs, working layer 4 is empty; (_{TG} < 0 is the difference of the Fermi levels in the working layer and the upper gate).

We simulate this disorder and calculate carrier scattering times in 2D gas within a simple model [^{13} cm^{-2} [_{σ}. It is assumed that electron exchange between adjacent traps in thermodynamic equilibrium on the surface is much more intense than between the surface, the gate and the semiconductor bulk. It is accepted for simplicity that point charges can be trapped at any point of the ideal plane imitating the real GaAs/Al_{2}O_{3} interface.

The state distribution of this thermostated system of charges with a fixed number of particles obeys canonical Gibbs distribution and the system in thermodynamic equilibrium is similar to classic single-component 2D plasma [

The point surface charge distribution determined by the radius vectors _{i}_{i}_{i}

We assume that the number of charges _{q}

In the solution of Poisson’s equation the delta function term yields a constant potential that can be neglected. Of interest are only the fluctuations of potential caused by the isotropic structural factor _{q}. Given mutually independent and completely random _{i}_{q} = 1, and then Eq. (3) describes white noise.σ

Deviations of _{q} from 1 due to Coulomb’s charge interaction can be taken into account within the theory of weakly non-ideal single-component 2D plasma [_{σ}. The following simple formula has been derived within the case-adapted theory:

Here _{1} = ε_{GaAs}, ε_{2} = ε_{ins}. When deriving Eq. (4) we ignored the difference in the dielectric permeabilities of the GaAs and AlGaAs layers and by analogy with an earlier work [

Note that for standard structures formed by remote doping, a theory was developed long time ago describing the effect of ultrathin charged impurity layers on the low-temperature parameters of high-mobility 2D carriers [_{q} for the charged impurity distribution in the delta layer. For the undoped structures in question we developed a similar theory. In this theory the quantum (τ_{q}) and transport (τ_{t}) carrier scattering times in 2D gas can be expressed with the following formulas:

Here _{F} is the Fermi momentum in the 2D gas, and _{q} is given by Eq. (3) for _{i} distributions frozen at some unknown equilibrium temperature. Note that Eqs. (5) follow from Poisson’s equation with account of gate screening and self-screening of carriers in the 2D gas in the Thomas–Fermi approximation and from Fermi’s golden rule where the interaction matrix element is found from unperturbed carrier wave functions in the 2D gas.

Figure _{q} and τ_{t} with _{q} determined by the adapted single-component plasma theory (Eq. (4)). Evidently, it is sufficient to calculate the integrals in Eqs. (5) for the 0 < ^{-1} range and substitute the dielectric permeabilities of Al_{2}O_{3} and GaAs with their average.

The behavior of the integrands in formulas (_{σ} = 5 · 10^{11} cm^{-2}.

For thermostated systems with a constant number of interacting particles, a universal, efficient and powerful modification of the Monte-Carlo method has existed for a long time, i.e., the Metropolis algorithm [_{i}. Noteworthy this algorithm uses data on _{i} and the internal energy of the system, i.e., it does not require calculating the system’s entropy and free energy. The calculated ratio of the probabilities of the system being in the two different states obeys the canonical Gibbs distribution [_{0} located in the ideal plane and having respective image charges in the metal, equilibrium states may exist at arbitrarily low

We found these states numerically using the Metropolis algorithm and simultaneously observed the formation of a 2D Wigner crystal and its melting at _{q}^{2} and contained up to _{σ}_{σ} ~ 5 × 10^{11} cm^{-2}).

Taking into account the image charges in the gate the total system energy is the sum of pairwise interactions between vertical dipoles having the length 2

The multiplier 1/2 in _{ij}_{ij} on the torus for _{ij}_{i} we set a displacement in an arbitrary direction through a random distance which was not greater than the average distance between the charges and then recalculated the total system energy. If the energy decreased (Δ_{i} was accepted, whereas for Δ^{1/2} ~ 0.01. For each _{i} distributions and calculated ρ_{q}_{q}^{2} using Eqs. (2) and (3). Then we found the mean |ρ_{q}^{2} distribution for multiple iterations. This allowed us to imitate |ρ_{q}^{2} for a far greater system than the test area. Example of this mean |ρ_{q}^{2} distribution is shown in Fig. _{q}^{2} distribution we found the dependence of the structural factor on the absolute value of the wave number _{q}^{-1} and 100 K ≤ _{q} = 1 demonstrates the electrostatic self-organization of surface charge within the suggested simple model.

(_{q}^{2} in the case of _{σ} = 5 ·10^{11} cm^{-2} in an equilibrium state with a temperature _{q} for the same _{σ} and indicated

Our colleagues from the University of New South Wales, Australia, experimentally studied structures with gate-induced 2D electron gas (_{q} and τ_{t} calculated using Eqs. (4) and (5) and the experimental scattering times for these specimens. The experimental scattering times (lifetimes) were estimated using a standard method [_{t} in the experiment was determined from the carrier mobility µ, and the quantum time τ_{q} was found from the measured magnetic field dependences of the Shubnikov–de Haas oscillations amplitude. The parameter in these cases was the experimental density of the gate-induced 2D gas, i.e., the Fermi momentum in Eqs. (5). The data and details of their processing for obtaining experimental τ_{q} will be published separately. This work focuses on the key simulation results. For the typical densities of 2D electron gas (_{e} ~ 1.4 × 10^{11} cm^{-2}) and 2D hole gas (_{h} ~ 0.6 × 10^{11} cm^{-2}) the experimental τ_{t} were 2–3 times greater than those calculated in the assumption of a fully disordered surface charge distribution (_{q} = 1) which corresponds to the infinite temperature _{q} also was greater than that calculated in the _{q} = 1 assumption. The underestimation of the theoretical scattering times τ_{theor} in comparison with the experimental ones τ_{exp} cannot be eliminated by making the natural assumption that experimental results are affected by theory-unaccounted scattering mechanisms with the characteristic time τ_{0}. Indeed taking these mechanisms into account in the theory would further underestimate the calculated times (τ_{new} < τ_{theor}< τ_{exp}) in accordance with the common rule:

On the contrary, it is clear from Eqs. (5) that the self-organization of surface charges reduces _{q} = 1 case. At _{q} ≠ 1 Eqs. (4) and (5) contain only one free parameter, i.e., the _{i} distribution freezing temperature _{i} disorder corresponding to the resultant temperatures

Summing up we considered self-organization of localized charges at the interface between gate dielectric and undoped semiconductor heterostructure containing gate-induced 2D electron or hole gas. In the suggested analytical formulas and Monte-Carlo calculations we used

This work was supported by Grant No. 19-72-30023 of the Russian Research Foundation. The calculations were carried out using computing resources of the Joint Supercomputer Center of the Russian Academy of Sciences under State Assignment No. 0306-2019-0011. We are grateful to colleagues A.R. Hamilton, O. Klochan and D.Q. Wang from the University of New South Wales, Australia, for the opportunity to compare calculations and theory with experimental data.

_{x}Ga

_{l-x}As.

_{1-x}Al

_{x}As heterostructures.