Corresponding author: Vitaly A. Tkachenko ( vtkach@isp.nsc.ru ) © 2020 Vitaly A. Tkachenko, Olga A. Tkachenko, Dmitry G. Baksheev, Oleg P. Sushkov.
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Citation:
Tkachenko VA, Tkachenko OA, Baksheev DG, Sushkov OP (2020) Effect of surface charge selforganization on gateinduced 2D electron and hole systems. Modern Electronic Materials 6(3): 101106. https://doi.org/10.3897/j.moem.6.3.63361

A simple model has been suggested for describing selforganization of localized charges and quantum scattering in undoped GaAs/AlGaAs structures with 2D electron or hole gas created by applying respective gate bias. It has been assumed that these metal / dielectric / undoped semiconductor structures exhibit predominant carrier scattering at localized surface charges which can be located at any point of the plane imitating the GaAs / dielectric interface. The suggested model considers all these surface charges and respective image charges in metallic gate as a closed thermostated system. Electrostatic selforganization in this system has been studied numerically for thermodynamic equilibrium states using the Metropolis algorithm over a wide temperature range. We show that at T > 100 K a simple formula derived from the theory of singlecomponent 2D plasma yields virtually the same behavior of structural factor at small wave numbers as the one given by the Metropolis algorithm. The scattering times of gateinduced carriers are described with formulas in which the structural factor characterizes frozen disorder in the system. The main contribution in these formulas is due to behavior of the structural factor at small wave numbers. Calculation using these formulas for the case of disorder corresponding to infinite T has yielded 2–3 times lower scattering times than experimentally obtained ones. We have found that the theory agrees with experiment at disorder freezing temperatures T ≈ 1000 K for 2D electron gas specimen and T ≈ 700 K for 2D hole gas specimen. These figures are the upper estimates of freezing temperature for test structures since the model ignores all the disorder factors except temperature.
undoped structures, gateinduced 2D systems, surface charge, disorder freezing temperature.
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 V_{g} between the metallic gate and metallic contacts connected to the GaAs working layer [
${n}_{\sigma}={\epsilon}_{0}{\epsilon}_{\text{ins}}\frac{{A}_{e}}{ed}$. (1)
Here ε_{0} is the dielectric constant, ε_{ins} ≈ 8 is the dielectric permeability of Al_{2}O_{3}, A_{e} ≈ 0.3 eV, and e > 0 is the elementary charge. Upon cooling the earthed gate structure to T ~ 1 K its band diagram and the A_{e} and n_{σ} parameters remain the same as in equilibrium. The concentration n_{σ} ~ 5 · 10^{11} cm^{2} is assumed to be constant even if V_{g} ≠ 0 and 2D gas is formed at a low temperature (Fig.
Equation (1) describes the areaaverage concentration of charges trapped by point defects (traps) at V_{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 (z = 30–60 nm) [
Schematic images of the object of study: (a, b) metaldielectricundoped semiconductor structure and band diagram in thermodynamic equilibrium: M – metal (Ti), 1 – Al_{2}O_{3}, 2, 4, 6 – GaAs, 3, 5 – AlGaAs, Φ_{m} – work function of Ti, χ_{1}, χ_{s} – electron affinity of Al_{2}O_{3} and GaAs, working layer 4 is empty; (c, d) a variant of the operating mode corresponding to a 2DHG (T ~ 1 K, eV_{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 [
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 singlecomponent 2D plasma [
The point surface charge distribution determined by the radius vectors r_{i} = (x_{i},y_{i}) can be conveniently described with a Fourier transform:
${\mathrm{\rho}}_{q}=\sum _{i}{e}^{i{\mathbf{qr}}_{i}}$. (2)
We assume that the number of charges N and the system area A tend to infinity and the disorder at ρ_{q} is isotropic:
${\left{\rho}_{\mathrm{q}}\right}^{2}=N{n}_{\sigma}\left(2\pi {)}^{2}\delta \right(\mathbf{q})+{\left{\stackrel{~}{\rho}}_{\mathrm{q}}\right}^{2},{\left{\stackrel{~}{\rho}}_{\mathrm{q}}\right}^{2}={n}_{\sigma}A\times {F}_{\mathrm{q}}$. (3)
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 F_{q}. Given mutually independent and completely random r_{i} we have F_{q} = 1, and then Eq. (3) describes white noise.σ
Deviations of F_{q} from 1 due to Coulomb’s charge interaction can be taken into account within the theory of weakly nonideal singlecomponent 2D plasma [
${F}_{\mathrm{q}}=\left[1\frac{{\stackrel{~}{k}}_{T}}{q+{\stackrel{~}{k}}_{T}}\right]$. (4)
Here ${\stackrel{~}{k}}_{T}={k}_{T}{C}_{d}$, ${k}_{T}=\frac{2\pi {e}^{2}{n}_{\sigma}}{\epsilon T}$, ${C}_{d}=\frac{1\mathrm{exp}(2qd)}{1\lambda \mathrm{exp}(2qd)}$, $\epsilon =\frac{{\epsilon}_{1}+{\epsilon}_{2}}{2}$, $\lambda =\frac{{\epsilon}_{1}{\epsilon}_{2}}{{\epsilon}_{1}+{\epsilon}_{2}}$, ε_{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 lowtemperature parameters of highmobility 2D carriers [
${\tau}_{\mathrm{q}}^{1}=\frac{\pi {n}_{\sigma}}{2{m}^{*}}{I}_{\mathrm{q}},{I}_{\mathrm{q}}=\frac{2}{{p}_{\mathrm{F}}}{\int}_{0}^{\infty}\frac{{F}_{\mathrm{q}}}{{\left(\frac{q}{k}+{D}_{\mathrm{z}}\right)}^{2}}{e}^{2qz}{C}_{d}^{2}\mathrm{d}q$;
${\tau}_{\mathrm{t}}^{1}=\frac{\pi {n}_{\sigma}}{2{m}^{*}}{I}_{\mathrm{t}},{I}_{\mathrm{t}}=\frac{2}{{p}_{\mathrm{F}}^{3}}{\int}_{0}^{\infty}\frac{{F}_{\mathrm{q}}}{{\left(\frac{q}{k}+{D}_{\mathrm{z}}\right)}^{2}}{e}^{2q\mathrm{z}}{C}_{d}^{2}{q}^{2}\mathrm{d}q$; (5)
$k=\frac{2{m}^{*}{e}^{2}}{\epsilon},{D}_{z}=1+{e}^{2qz}\frac{\lambda \mathrm{exp}(2qd)}{1\lambda \mathrm{exp}(2qd)}$ .
Here p_{F} is the Fermi momentum in the 2D gas, and F_{q} is given by Eq. (3) for r_{i} distributions frozen at some unknown equilibrium temperature. Note that Eqs. (5) follow from Poisson’s equation with account of gate screening and selfscreening 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
For thermostated systems with a constant number of interacting particles, a universal, efficient and powerful modification of the MonteCarlo method has existed for a long time, i.e., the Metropolis algorithm [
We found these states numerically using the Metropolis algorithm and simultaneously observed the formation of a 2D Wigner crystal and its melting at T ~ 1 K [
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 2d:
${U}_{ij}=\frac{1}{2}\frac{{q}_{0}^{2}}{2\pi \epsilon {\epsilon}_{0}}\left[{r}_{ij}^{1}{\left({r}_{ij}^{2}+4{d}^{2}\right)}^{\frac{1}{2}}\right]$,
$E=\frac{1}{2}\sum _{i\ne j}{U}_{ij}$. (6)
The multiplier 1/2 in U_{ij} takes into account the difference between the image charge and the real charge. We convoluted the simulated area into a torus and took the shortest distances r_{ij} on the torus for U_{ij} calculation. The kinetic energy of the charges was neglected. For each charge q_{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 (ΔE < 0) the new location of the charge q_{i} was accepted, whereas for ΔE ≥ 0 the new charge location was accepted only if ${e}^{\frac{\Delta E}{T}}\ge r$ where r is a random value between 0 and 1. The iterations were continued after reaching a “constant” E. Due to the finiteness of N the relative width δE/E of the system’s internal energy distribution is not infinitely small: δE/E ~ 1/N^{1/2} ~ 0.01. For each E fluctuating near the most probable value, we found the r_{i} distributions and calculated ρ_{q}, ρ_{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.
(a) An example of the averaged distribution ρ_{q}^{2} in the case of n_{σ} = 5 ·10^{11} cm^{2} in an equilibrium state with a temperature T = 100 K: Monte Carlo calculation; (b) dependence of the isotropic structural factor F_{q} for the same n_{σ} and indicated T. Thick solid lines were obtained by the Monte Carlo method. Thin solid and dotted lines are calculated according to the theory of a onecomponent plasma.
Our colleagues from the University of New South Wales, Australia, experimentally studied structures with gateinduced 2D electron gas (d = 25 nm, z = 45 nm) and 2D hole gas (d = 20 nm, z = 68 nm). We compared the scattering times τ_{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 [
$\frac{1}{{\tau}_{\text{new}}}=\frac{1}{{\tau}_{\text{theor}}}+\frac{1}{{\tau}_{0}}$. (7)
On the contrary, it is clear from Eqs. (5) that the selforganization of surface charges reduces F (q) (Fig.
Summing up we considered selforganization of localized charges at the interface between gate dielectric and undoped semiconductor heterostructure containing gateinduced 2D electron or hole gas. In the suggested analytical formulas and MonteCarlo calculations we used only one free parameter, i.e., the disorder freezing temperature. This temperature was found by comparing the calculated and experimentally measured transport and quantum scattering times for 2D carriers.
This work was supported by Grant No. 197230023 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. 030620190011. 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.