[Paper Review] Oblique propagation of electrons in crystals of germanium and silicon at sub-Kelvin temperature in low electric fields

Blas Cabrera, M. Pyle+3 more|arXiv (Cornell University)|Apr 8, 2010

Silicon Nanostructures and Photoluminescence3 citations

TL;DR

This paper presents a simplified anisotropic model explaining the oblique electron drift in germanium and silicon crystals at sub-Kelvin temperatures and low electric fields, where electrons remain confined to single valleys due to suppressed inter-valley scattering. By transforming momentum space to isotropize energy surfaces and simulating electron trajectories, the model successfully reproduces experimental drift velocities and explains the tenfold larger lateral straggling of electrons compared to holes in cryogenic detectors like CDMS.

ABSTRACT

We show that oblique propagation of electrons in crystals of Ge and Si, where the electron velocity does not follow the electric field even on average, can be explained using standard anisotropic theory for indirect gap semiconductors. These effects are pronounced at temperatures below ~1K and for electric fields below ~5V/cm because inter-valley transitions are energetically suppressed forcing electrons to remain in the same band valley throughout their motion and the valleys to separate in position space. To model, we start with an isotropic approximation which incorporates the average properties of the crystals with one phonon mode, and include the ellipsoidal electron valleys by transforming into a momentum space where constant energy surfaces are spheres. We include comparisons of simulated versus measured drift velocities for holes and electrons, and explain the large discrepancy between electrons and holes for shared events in adjacent electrodes.

Motivation & Objective

• To explain the experimentally observed oblique propagation of electrons in Ge and Si crystals at sub-Kelvin temperatures and low electric fields, where electron velocity deviates significantly from the applied electric field direction.
• To address the long-standing discrepancy in lateral straggling between electrons and holes in cryogenic semiconductor detectors, particularly in CDMS experiments where electrons show ten times greater spread than holes.
• To develop a simplified yet accurate model that captures the essential physics of anisotropic electron transport without requiring full-band Monte Carlo simulations, enabling faster detector simulations.
• To validate the model against measured drift velocities of electrons and holes in Ge and Si, using a single adjustable parameter (deformation potential) for each material.

Proposed method

• Transform the anisotropic momentum space of electron valleys (L-type in Ge, Δ-type in Si) into a spherical, isotropic momentum space via Herring's transformation, where constant energy surfaces become spheres.
• Apply energy and momentum conservation with isotropic longitudinal phonon emission, assuming spontaneous emission only at T < 1 K, and model electron scattering using a single deformation potential parameter Ξ.
• Simulate electron trajectories in the transformed (starred) space under a uniform electric field, then map results back to real space to recover anisotropic drift patterns and spatial spreading.
• Use the same isotropic hole model for comparison, assuming a spherical Γ-valley and isotropic phonon coupling, to enable direct comparison of electron and hole drift behavior.
• Derive transformed quantities such as electric field, velocity, and position in starred space using scaling relations based on effective mass ratios: $ E_i^* = ilde{m}_i^{-1/2} E_i $, $ v_i^* = ilde{m}_i^{1/2} v_i $, where $ ilde{m}_i = m_i / m_c $.
• Compare simulated drift velocities with experimental data from Sundqvist (2009), fitting the deformation potential $ ilde{oldsymbol{ ho}} $ to achieve agreement, with $ ilde{oldsymbol{ ho}} = 11.0 $ eV for Ge and 3.4 eV for Si.

Experimental results

Research questions

• RQ1 Why do electrons in Ge and Si crystals exhibit oblique drift at sub-Kelvin temperatures and low electric fields, while holes do not?
• RQ2 What is the origin of the tenfold larger lateral straggling of electrons compared to holes in CDMS-type cryogenic detectors?
• RQ3 How can the anisotropic electron transport in indirect gap semiconductors be accurately modeled with a simplified framework that retains physical insight?
• RQ4 To what extent does the suppression of inter-valley scattering at low temperatures and low fields lead to valley-specific electron trajectories and anisotropic spreading?
• RQ5 Can a transformation to isotropic momentum space accurately reproduce the anisotropic drift behavior of electrons in L- and Δ-valleys of Ge and Si?

Key findings

• The model successfully reproduces the measured drift velocities of electrons in Ge and Si across low electric fields (up to ~5 V/cm), with a best-fit deformation potential of 11.0 eV for Ge and 3.4 eV for Si.
• Electron trajectories in Ge crystals show four distinct, elliptically spreading beams aligned with the [111] crystal axes, with a transverse-to-parallel spread ratio of approximately $ ilde{m}_{ ext{parallel}} / ilde{m}_{ ext{perp}} = 4.41 $, matching the theoretical anisotropy ratio.
• The lateral straggling of electrons is found to be ten times larger than that of holes in CDMS detectors, consistent with experimental observations and explained by the anisotropic effective mass tensor and valley separation.
• In [100] Si crystals, oblique propagation is absent because all six Δ-valleys have one principal axis aligned with the electric field, but it would emerge in [111] Si due to misalignment.
• In [111] Ge crystals, oblique propagation persists because only one of the four L-valleys is aligned with the field, preserving anisotropic effects.
• The characteristic scattering range $ l_0 $ is 257 μm for Ge and 108 μm for Si, and the simulated electron beam width $ ilde{oldsymbol{ ho}} $ matches the experimental spread at 1 V/cm, confirming the model's predictive power.