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Valley physics in the two bands k.p model for SiGe heterostructures and spin qubits

Tancredi Salamone, Biel Martinez Diaz, Jing Li, Lukas Cvitkovich, Yann-Michel Niquet

TL;DR

This work addresses the challenge of valley splitting in Si/SiGe spin qubits by extending a two-band $oldsymbol{k}oldsymbol{ullet}oldsymbol{p}$ framework with a nonperturbative inter-valley potential that captures valley mixing and dipole matrix elements. The authors derive and parameterize the inter-valley interaction from the $2k_0$ theory, implement it on a finite-difference mesh, and incorporate strains, spin-orbit coupling, and magnetic fields, including alloy disorder modeled through Ge distribution. Validation against atomistic TB calculations across uniform, spike, and wiggle Ge profiles shows excellent agreement for valley splittings and inter-valley dipoles, confirming the model’s accuracy in disordered SiGe heterostructures. They then apply the approach to a realistic spin qubit device, predicting valley splittings up to ~200 μeV, large Rabi frequencies near spin–valley anti-crossings, and detailed phonon-related relaxation and dephasing behavior, thereby enabling efficient, large-scale simulations of silicon-based spin and valley qubits with disorder effects. The framework offers a computationally tractable pathway to design and optimize SiGe devices where valley physics is essential, including shuttling and readout scenarios, while preserving quantitative agreement with atomistic methods. $ ext{Valley splitting} ightarrow ext{Δ}=2|J|$, with $J=raket{ ilde{c+}{V}{ ilde{c-}}}$, and valley couplings governed by the $q_z=2k_0$ component of the confinement potential. $V_{ ext{inter}}(oldsymbol r)$, parameterized by $V_{ ext{inter}}^{ ext{Ge}} ext{ and }Y(oldsymbol r)$, captures alloy effects that TB methods resolve but at far greater efficiency for device-scale simulations. $2k_0$ theory$,$ $A o-0.26$, and $V_{ ext{inter}}^{ ext{Ge}} o ext{514 meV}$ are key anchors for matching TB valley splittings across SiGe heterostructures.

Abstract

We discuss the choice and implementation of inter-valley potentials in the so-called two bands k.p model for the opposite X, Y or Z valleys of silicon. We focus on the description of valley splittings in Si/SiGe heterostructures for spin qubits, with a particular attention to alloy disorder. We demonstrate that the two bands k.p model reproduces the valley splittings of atomistic tight-binding calculations in relevant heterostructures (SiGe spikes, wiggle wells...), yet at a much lower cost. We show that the model also captures the effects of valley-orbit mixing and yields the correct inter-valley dipole matrix elements that characterize manipulation, dephasing and relaxation in spin/valley qubits. We simulate a realistic Si/SiGe spin qubit device as an illustration, and discuss electron-phonon interactions in the two bands k.p model. Beyond spin qubits, this model enables efficient simulations of SiGe heterostructure devices where spin and valley physics are relevant.

Valley physics in the two bands k.p model for SiGe heterostructures and spin qubits

TL;DR

This work addresses the challenge of valley splitting in Si/SiGe spin qubits by extending a two-band framework with a nonperturbative inter-valley potential that captures valley mixing and dipole matrix elements. The authors derive and parameterize the inter-valley interaction from the theory, implement it on a finite-difference mesh, and incorporate strains, spin-orbit coupling, and magnetic fields, including alloy disorder modeled through Ge distribution. Validation against atomistic TB calculations across uniform, spike, and wiggle Ge profiles shows excellent agreement for valley splittings and inter-valley dipoles, confirming the model’s accuracy in disordered SiGe heterostructures. They then apply the approach to a realistic spin qubit device, predicting valley splittings up to ~200 μeV, large Rabi frequencies near spin–valley anti-crossings, and detailed phonon-related relaxation and dephasing behavior, thereby enabling efficient, large-scale simulations of silicon-based spin and valley qubits with disorder effects. The framework offers a computationally tractable pathway to design and optimize SiGe devices where valley physics is essential, including shuttling and readout scenarios, while preserving quantitative agreement with atomistic methods. , with , and valley couplings governed by the component of the confinement potential. , parameterized by , captures alloy effects that TB methods resolve but at far greater efficiency for device-scale simulations. theory , and are key anchors for matching TB valley splittings across SiGe heterostructures.

Abstract

We discuss the choice and implementation of inter-valley potentials in the so-called two bands k.p model for the opposite X, Y or Z valleys of silicon. We focus on the description of valley splittings in Si/SiGe heterostructures for spin qubits, with a particular attention to alloy disorder. We demonstrate that the two bands k.p model reproduces the valley splittings of atomistic tight-binding calculations in relevant heterostructures (SiGe spikes, wiggle wells...), yet at a much lower cost. We show that the model also captures the effects of valley-orbit mixing and yields the correct inter-valley dipole matrix elements that characterize manipulation, dephasing and relaxation in spin/valley qubits. We simulate a realistic Si/SiGe spin qubit device as an illustration, and discuss electron-phonon interactions in the two bands k.p model. Beyond spin qubits, this model enables efficient simulations of SiGe heterostructure devices where spin and valley physics are relevant.

Paper Structure

This paper contains 21 sections, 59 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. The valley splitting in the $2k_0$ theory
  4. The two bands k.p model
  5. Implementation of an inter-valley potential in the two bands k.p model
  6. Strains, spin-orbit coupling, and magnetic fields
  7. About the assumptions of the $2k_0$ theory and the modeling of SiGe alloys
  8. Alloy disorder
  9. Validation of the model
  10. Wells with uniform Ge concentration
  11. Ge spike in the well
  12. Wiggle wells
  13. Valley-orbit mixing and dipole matrix elements
  14. Application to spin qubits
  15. Device
  16. ...and 6 more sections

Figures (17)

  • Figure 1: The conduction band structure of silicon in an extended zone scheme around $Z$. The blue parabola is the $+Z$ valley centered around $k_z=0.85\times 2\pi/a$. The orange parabola is the replica of the $-Z$ valley centered around $k_z=-0.85\times 2\pi/a$ in the second Brillouin zone. Both bands are degenerate at $Z$. The two bands $\boldsymbol{k}\cdot\boldsymbol{p}$ model expands the wave functions on the degenerate Bloch functions $\hat{u}_\pm(\boldsymbol{r})\equiv\tilde{u}_\pm(\boldsymbol{r})$ at $Z$. $E_c$ is the conduction band edge energy.
  • Figure 2: Cross-section (in the $xz$ plane) of an illustrative finite-differences mesh used to solve the two bands $\boldsymbol{k}\cdot\boldsymbol{p}$ equations. The spacing between successive mesh lines along $z$ is $\delta z=a/4$ (the bare distance between monolayers). The inter-valley potential $V_\mathrm{inter}$ is multiplied by $+1$ (red) or $-1$ (blue) on each monolayer.
  • Figure 3: TB and 2 bands $\boldsymbol{k}\cdot\boldsymbol{p}$ valley splittings (2$\boldsymbol{k}\cdot\boldsymbol{p}$ ) computed for the model potential of \ref{['eq:modelpot']} using $A=-0.26$. The vertical dash-dotted line is the inter-valley wave number $q=2k_0$.
  • Figure 4: The different Ge concentration profiles considered in this work: (a) Wells with uniform Ge concentration (b) Wells with a Ge spike (c) Wiggle wells.
  • Figure 5: (a) Two bands $\boldsymbol{k}\cdot\boldsymbol{p}$ and (b) TB valley splittings in a quantum well with uniform Ge concentration $Y_w$ and constant $\Delta Y=Y_b-Y_w=30$%, for different vertical electric fields $E_z$. The median splittings, computed over 64 (TB) or 128 ($\boldsymbol{k}\cdot\boldsymbol{p}$) disorder configurations, are plotted with error bars giving the inter-quartile range. The interface width is $w_\mathrm{int}=10$ ML.
  • ...and 12 more figures