Valley Splittings in Si/SiGe Heterostructures from First Principles

TL;DR

The paper addresses valley splittings, $E_{VS}$, between the $\pm Z$ valleys in planar Si/SiGe heterostructures, a key factor limiting spin-qubit coherence. It uses first-principles density functional theory (DFT) to benchmark valley splittings against tight-binding models and the semi-empirical $2k_0$ theory across smooth interfaces and wiggle wells, incorporating atomistic disorder and strain. The main findings are that DFT supports the qualitative trends of the $2k_0$ theory but reveals limitations of EM and TB in handling alloy disorder and nonlocal electronic effects, and that valley-orbit mixing can become significant when $E_{VS}$ is large. A conduction-band offset around $225$ meV in DFT calculations (larger than the commonly used EM value of about $170$ meV) highlights nonlocal corrections. Overall, TB and the $2k_0$ framework provide useful valley-splitting statistics for many Si/SiGe heterostructures, but fully first-principles insights are essential to capture disorder and valley-orbit physics that influence spin-qubit device performance.

Abstract

We compute valley splittings in Si/SiGe superlattices using ab initio density functional theory (DFT). This first-principle approach is expected to provide an excellent description of interfaces, strains, and atomistic disorder without empirically fitted parameters. We benchmark atomistic tight-binding (TB) and the ``$2k_0$'' theory within the effective mass (EM) approximation against DFT. We show that DFT supports the main conclusions of the 2$k_0$ theory, but reveals some limitations of semi-empirical methods such as the EM and TB, in particular about the description of atomistic disorder. The DFT calculations also highlight the effects of strong valley-orbit mixing at large valley splittings. Nevertheless, TB and the 2$k_0$ theory shall provide reasonable valley splitting statistics in many heterostructures of interest for spin qubit devices.

Figures (11)

• Figure 1: (a)Ge concentration profile (blue circles) for smooth Si/SiGe interfaces with width $w_\mathrm{int}=4$ MLs in a supercell with size $n=3$. At the interfaces, the Ge concentration varies from 0 to 27.8% by steps of $\approx5.56\%$ ($w_\mathrm{int}$ corresponds to the number of MLs with intermediate concentrations). The orange line shows the envelope of the DFT ground-state conduction band wave function at zero electric field.(b)Atomistic 3D model (side view) of a SiGe/Si/SiGe heterostructure with abrupt interfaces ($w_\mathrm{int}=0$). The supercell (with size $n=3$) is 1.649$\times$1.649$\times$18.5$\,$nm$^3$ and contains 2448 atoms (Si in yellow, Ge in red). The bottom panel shows an iso-density surface (blue) of the DFT wave function, highlighting the penetration in the disordered SiGe alloy.
• Figure 2: (a) Cross-section averaged ground-state valley wave functions $\tilde{\Psi}_{1,2}(z)$ (blue, red) confined in a quantum well with sharp interfaces characterized by the Hartree potential $V_\mathrm{H}(z)$ (black). The electron is squeezed on the bottom interface by an external electric field $F_\mathrm{z}=8.7$ mV/nm. The wave functions are real and have similar envelopes $\tilde{\Phi}_{1,2}(z)$ but are phase shifted with respect to each other. (b) DFT LDOS in the same quantum well as in (a). The orbital sub-bands with $n=0,1,\dots$ nodes can be identified within the Si layer. Valley splittings in the range of 0.1 meV are too small to be clearly resolved on this energy scale. (c) TB LDOS in the same quantum well as in (a) and (b). The external potential extracted from the DFT calculation in (a) has been transferred to the TB Hamiltonian as explained in Appendix \ref{['app:field']}. The DFT and TB LDOS are in excellent agreement.
• Figure 3: (a) DFT and TB valley splittings $E_\mathrm{VS}$ calculated in Si wells with smooth interfaces characterized by the width $w_\mathrm{int}$ (see text). The atomic positions relaxed with the PBE exchange-correlation functional are used as input for DFT and TB calculations. The plot shows the distribution of ten random realizations of each $w_\mathrm{int}$, with the DFT and TB valley splittings of the same structures connected by dashed lines. The median value of each distribution is indicated by a horizontal bar. (b) Correlation between the DFT and TB valley splittings in Si wells with smooth interfaces (all $w_\mathrm{int}$'s and samples). The dashed line is a guide to the eye with unity slope.
• Figure 4: (a)TB valley splittings calculated for the same smooth interfaces as in Fig. \ref{['fig:comp_if']}, starting from either Keating's VFF or PBE atomic positions. The valley splittings of the same structures are connected by dashed lines. The median value of each distribution is indicated by a horizontal bar.(b)Correlation between the TB valley splittings computed with Keating's VFF and DFT geometries. The dashed line is a guide to the eye with unity slope.
• Figure 5: DFT and TB valley splittings $E_\mathrm{VS}$ calculated in wiggle wells with wave number $q_w$ (see text). The atomic positions relaxed with the PBE exchange-correlation functional are used as input for DFT and TB calculations. The plot shows the distribution of five random realizations of each $q_w$, with the DFT and TB valley splittings of the same structures connected by dashed lines. The median value of each distribution is indicated by a horizontal bar.