[Paper Review] Germanium under high tensile stress: nonlinear dependence of direct band gap vs. strain

K. Guilloy, N. Pauc+16 more|arXiv (Cornell University)|Jun 6, 2016

Semiconductor Quantum Structures and Devices31 references3 citations

TL;DR

This study measures the direct band gap of germanium under uniaxial tensile strain along [100] up to 3.3%, revealing a significant nonlinear dependence that invalidates standard linear deformation potential theory beyond 2% strain. Using electro-absorption spectroscopy and tight-binding simulations, the authors extract nonlinear deformation potentials, showing that first-order models lead to >50 meV errors in bandgap prediction at high strain—critical for designing CMOS-compatible lasers and optoelectronic devices.

ABSTRACT

Germanium is a strong candidate as a laser source for silicon photonics. It is widely accepted that the band structure of germanium can be altered by tensile strain so as to reduce the energy difference between its direct and indirect band gaps. However, the conventional deformation potential model most widely adopted to describe this transformation happens to have been investigated only up to 1 % uniaxially loaded strains. In this work, we use a micro-bridge geometry to uniaxially stress germanium along [100] up to $\varepsilon_{100}$=3.3 % longitudinal strain and then perform electro-absorption spectroscopy. We accurately measure the energy gap between the conduction band at the $Γ$ point and the light- and heavy-hole valence bands. While the experimental results agree with the conventional linear deformation potential theory up to 2 % strain, a significantly nonlinear behavior is observed at higher strains. We measure the deformation potential of germanium to be a = -9.1 $\pm$ 0.3 eV and b = -2.32 $\pm$ 0.06 eV and introduce a second order deformation potential. The experimental results are found to be well described by tight-binding simulations. These new high strain coefficients will be suitable for the design of future CMOS-compatible lasers and opto-electronic devices based on highly strained germanium.

Motivation & Objective

• To measure the direct band gap of germanium under high uniaxial tensile strain along [100] up to 3.3%.
• To test the validity of linear deformation potential theory at high strain levels, where it has been experimentally verified only up to 1%.
• To extract hydrostatic and tetragonal shear deformation potentials from experimental data and compare them with tight-binding simulations.
• To quantify nonlinear corrections to the band gap shift, which are critical for accurate modeling of highly strained germanium in optoelectronic devices.

Proposed method

• Fabricated Ge micro-bridges from Ge-on-insulator substrates using reactive ion etching and HF vapor release to induce uniaxial tensile strain.
• Applied electro-absorption spectroscopy using a 12 kHz modulated electric field and lock-in detection to measure optical transitions with high precision.
• Used Raman spectroscopy and X-ray micro-diffraction (at ESRF BM32) to calibrate strain levels, with XRD providing model-free strain measurements.
• Performed tight-binding simulations to model strain-induced band structure changes and extract second-order deformation potentials.
• Fitted experimental data to a second-order deformation potential model to describe the nonlinear band gap evolution.
• Compared experimental band gap shifts with theoretical predictions to validate the nonlinear model and extract precise deformation potential coefficients.

Experimental results

Research questions

• RQ1 Does the direct band gap of germanium exhibit a nonlinear dependence on uniaxial tensile strain beyond 2%?
• RQ2 How do the hydrostatic and tetragonal shear deformation potentials of germanium deviate from linear behavior at high strain?
• RQ3 Can tight-binding simulations accurately reproduce the experimental electro-absorption spectra of highly strained germanium?
• RQ4 To what extent does the conventional linear deformation potential model fail in predicting band gap shifts at strains above 2%?
• RQ5 What are the second-order deformation potential coefficients for germanium under [100] uniaxial strain?

Key findings

• The direct band gap of germanium exhibits a significant nonlinear dependence on uniaxial tensile strain beyond 2%, deviating strongly from linear deformation potential theory.
• The hydrostatic deformation potential of germanium is measured as a = -9.1 ± 0.3 eV, and the tetragonal shear deformation potential as b = -2.32 ± 0.06 eV.
• A second-order deformation potential model is required to accurately describe band gap shifts at strains above 2%, as linear models predict errors exceeding 50 meV at 3.3% strain.
• The experimental data show excellent agreement with tight-binding simulations that include second-order terms, validating the theoretical framework.
• The nonlinear behavior leads to a wavelength error of approximately 300 nm in laser emission predictions if linear models are used at high strain.
• The study demonstrates that high-precision band structure modeling in highly strained Ge requires inclusion of second-order deformation potentials for accurate device design.