Using Landau quantization to probe disorder in semiconductor heterostructures

Abstract

Understanding scattering mechanisms in semiconductor heterostructures is crucial to reducing sources of disorder and ensuring high yield and uniformity in large spin qubit arrays. Disorder of the parent two-dimensional electron or hole gas is commonly estimated by the critical, percolation-driven density associated with the metal-insulator transition. However, a reliable estimation of the critical density within percolation theory is hindered by the need to measure conductivity with high precision at low carrier densities, where experiments are most difficult. Here, we connect experimentally percolation density and quantum Hall plateau width, in line with an earlier heuristic intuition, and offer an alternative method for characterizing semiconductor heterostructure disorder.

Figures (3)

• Figure 1: (a) Fit to the longitudinal conductivity ($\sigma_{\mathrm{xx}}$) as a function of density ($n$) for three different strained Ge quantum well heterostructures (A, B, C) with increasing percolation density $n_\mathrm{c}$. Light red curve (heterostructure A) is from Ref. stehouwer2023germanium, $n_\mathrm{c} = \mathrm{1.2\times 10^{10}\ cm^{-2}}$; red curve (heterostructure B) from Ref. costa2025buried, $n_\mathrm{c} = \mathrm{1.4\times 10^{10}\ cm^{-2}}$; dark red curve (heterostructure C) from Ref. lodari2022lightly, $n_\mathrm{c} = \mathrm{1.8\times 10^{10}\ cm^{-2}}$. Inset: a schematic representation of the classically measured percolation density. (b) For the same heterostructures, measurements of the transverse resistance ($R_\mathrm{xy}$) as a function of the carrier density deviation $\Delta n= (eB/h-n)$ from the integer filling factor condition $\nu = 1$. Samples with lower percolation densities show narrower quantum Hall plateau. Measurements for heterostructures A and C are offset for clarity. Inset: a schematic representing how the cyclotron orbits are pinned around the same defects represented in the schematic in panel (a).
• Figure 2: (a), (b) Width of the quantum Hall plateaus ($n_{\mathrm{LL}}$) at different filling factors ($\nu$) as a function of magnetic field ($B$), shown for holes in Ge/SiGe (red, heterostructure C from Ref. lodari2022lightly) and electrons in Si/SiGe heterostructures (purple, heterostructure G from Ref. degli2022wafer), respectively. The fit to Eq.\ref{['eq:Plateau']} (black lines) gives an asymptotic width of the quantum Hall plateaus ($n_{\mathrm{LL_{0}}}$), not far from the experimental values for $\nu=1$. The fit for electron in silicon considers only the odd filling factors, which correspond to valley split levels. The dashed lines are placed at twice the percolation density $n_{\mathrm{c}}$ measured for each sample. (c) Asymptotic width of the QHE plateau $n_{\mathrm{LL_{0}}}$ as a function of percolation density $n_{\mathrm{c}}$ for Ge/SiGe heterostructures (red), GaAs/AlGaAs heterostructures (green) and Si/SiGe heterostructures (purple). Heterostructure A is from Ref. stehouwer2023germanium; heterostructure B is from Ref. costa2025buried; heterostructure C is from Ref. lodari2022lightly; heterostructure D is from Ref. tosato_highmobility_2022; heterostructure E is from Ref. sammak_low_2019; heterostructure F is from Ref. paquelet2020multiplexed; heterostructure G is from Ref. degli2022wafer and heterostructure H is from Ref. manfra2007transport. The black solid line corresponds to $n_{\mathrm{LL_0}} = 2n_{\mathrm{c}}$ as suggested by Efros efros1989metal.
• Figure 3: (a) Percolation density $n_\mathrm{c}$ as a function of inverse temperature $1/T$. The data points are obtained from percolation fits, with conductivity measurements performed on a device from heterostructure A, Ref. stehouwer2023germanium. The solid line is a fit to Eq. \ref{['eq:temp']}. (b) Width of the quantum Hall plateau for the first filling factor $n_\mathrm{LL_{1}}$ as a function of inverse temperature (same sample as in panel (a)). The solid line is an activation energy fit considering the three coldest temperatures. At higher temperature, Landau level broadening smears out the quantum Hall plateaus.