Depletion-limited Effective Hall mobility in Micrometer-Scale High-Purity Germanium Crystals

Electrostatic effects can strongly constrain charge transport in thinned high-purity germanium (HPGe), with direct implications for radiation detectors and Ge-based electronic and quantum devices. We report a systematic experimental characterization of the thickness-dependent effective Hall mobility in bulk-grown, detector-grade HPGe at room temperature using Hall-effect measurements on n- and p-type samples sequentially thinned from 2.7mm to 7\textmu m. The intrinsic bulk carrier mobility remains thickness independent in this regime; the observed reduction in Hall-extracted mobility arises from electrostatic surface depletion that reduces the electrically active conducting thickness. The thickness-dependent data are accurately parameterized by an empirical extended-exponential relation, $μ(t)=μ_{0}[1-\exp(-(t/τ)^β)]$, where $τ$ is a characteristic electrostatic length scale. Comparison with boundary-scattering and depletion-based models shows that Fuchs–Sondheimer scattering is negligible, while electrostatic depletion dominates the transport behavior. The hierarchy $λ_{D}<τ\lesssim W_{0}$ directly links the apparent mobility reduction to long-range screening and near-surface electric fields. These results yield a simple design guideline: maintaining thicknesses $t\gtrsim 3τ$ preserves near-bulk transport, whereas thinner structures operate in a depletion-controlled regime with strongly reduced effective conductivity.

💡 Research Summary

**
The paper presents a systematic experimental study of how electrostatic surface depletion controls the apparent Hall mobility in high‑purity germanium (HPGe) crystals when they are thinned to micrometer and sub‑micrometer dimensions. Bulk‑grown, detector‑grade HPGe wafers were prepared with initial impurity concentrations of 2–3 × 10¹¹ cm⁻³. Samples were sequentially thinned from 2.7 mm down to 7 µm using a combination of mechanical lapping, fine polishing, and controlled HF:HNO₃ chemical etching. Thickness was measured after each etch step by a mass‑density‑area method, achieving sub‑micrometer accuracy and cross‑validated with profilometry.

Four‑point Van der Pauw Hall measurements were performed at room temperature (295 K) on both n‑type and p‑type specimens. A constant current (1–200 µA) was applied while sweeping a perpendicular magnetic field up to 0.58 T. Longitudinal and Hall voltages were recorded, allowing extraction of sheet resistance, carrier concentration, and Hall mobility. The analysis assumes uniform current flow across the full geometric thickness t; however, the authors emphasize that in the presence of surface depletion the electrically active thickness t_eff can be substantially smaller than t, leading to an “effective Hall mobility” µ_Hall that is reduced relative to the intrinsic bulk mobility µ₀.

A simple electrostatic picture is introduced: band bending at the free surface and at the Ge–PTFE interface creates depletion regions of widths W₁ and W₂, respectively. The neutral bulk region that actually conducts has thickness t_eff = t − (W₁ + W₂). Within this region the intrinsic mobility µ₀ is unchanged, so the measured Hall mobility follows µ_Hall = µ₀·(t_eff/t). As the sample becomes thinner, the depletion zones occupy a larger fraction of the total thickness, causing a pronounced drop in µ_Hall even though the microscopic scattering mechanisms (phonon, impurity) remain the same.

To quantify this behavior, the authors fit the measured µ_Hall(t) to an empirical extended‑exponential function:

µ(t) = µ₀