# Electrical operation of planar Ge hole spin qubits in an in-plane magnetic
field

Abhikbrata Sarkar  School of Physics, The University of New South Wales,
Sydney 2052, Australia  Zhanning Wang  School of Physics, The University of
New South Wales, Sydney 2052, Australia  Matthew Rendell  School of Physics,
The University of New South Wales, Sydney 2052, Australia  Nico W. Hendrickx
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
2628 CJ Delft, The Netherlands  Menno Veldhorst  QuTech and Kavli Institute of
Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
Giordano Scappucci  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, 2628 CJ Delft, The Netherlands  Mohammad Khalifa
Department of Electrical and Computer Engineering. University of British
Columbia, Vancouver, B.C., Canada.  Quantum Matter Institute, University of
British Columbia, Vancouver, B.C., Canada.  Joe Salfi  Department of
Electrical and Computer Engineering. University of British Columbia,
Vancouver, B.C., Canada.  Quantum Matter Institute, University of British
Columbia, Vancouver, B.C., Canada.  Andre Saraiva  School of Electrical
Engineering and Telecommunications, The University of New South Wales, Sydney
2052, Australia  A. S. Dzurak  School of Electrical Engineering and
Telecommunications, The University of New South Wales, Sydney 2052, Australia
A. R. Hamilton  School of Physics, The University of New South Wales, Sydney
2052, Australia  Dimitrie Culcer  School of Physics, The University of New
South Wales, Sydney 2052, Australia

###### Abstract

Hole spin qubits in group-IV semiconductors, especially Ge and Si, are
actively investigated as platforms for ultrafast electrical spin manipulation
thanks to their strong spin-orbit coupling. Nevertheless, the theoretical
understanding of spin dynamics in these systems is in the early stage of
development, particularly for in-plane magnetic fields as used in the vast
majority of experiments. In this work we present a comprehensive theory of
spin physics in planar Ge hole quantum dots in an in-plane magnetic field,
where the orbital terms play a dominant role in qubit physics, and provide a
brief comparison with experimental measurements of the angular dependence of
electrically driven spin resonance. We focus the theoretical analysis on
electrical spin operation, phonon-induced relaxation, and the existence of
coherence sweet spots. We find that the choice of magnetic field orientation
makes a substantial difference for the properties of hole spin qubits.
Furthermore, although the Schrieffer-Wolff approximation can describe electron
dipole spin resonance (EDSR), it does not capture the fundamental spin
dynamics underlying qubit coherence. Specifically, we find that: (i) EDSR for
in-plane magnetic fields varies non-linearly with the field strength and
weaker than for perpendicular magnetic fields; (ii) The EDSR Rabi frequency is
maximized when the a.c. electric field is aligned parallel to the magnetic
field, and vanishes when the two are perpendicular; (iii) The orbital magnetic
field terms make the in-plane $g$-factor strongly anisotropic in a squeezed
dot, in excellent agreement with experimental measurements; (iv) Focusing on
random telegraph noise, we show that coherence sweet spots do not exist in an
in-plane magnetic field, as the orbital magnetic field terms expose the qubit
to all components of the defect electric field. These findings will provide a
guideline for experiments to design ultrafast, highly coherent hole spin
qubits in Ge.

##  I Introduction

Solid state spin qubits are prime candidates for scalable, highly coherent
quantum computing platforms. Kane (1998); Loss and DiVincenzo (1998); Petta
_et al._ (2005); Hanson  _et al._ (2007); Hanson and Awschalom (2008);
Zwanenburg  _et al._ (2013); Chatterjee  _et al._ (2021); Scappucci  _et al._
(2021); Fang  _et al._ (2023) Among these group IV materials such as Ge and Si
stand out thanks to the absence of piezo-electric interaction with phonons
Cardona and Peter (2005) and the possibility of isotopic purification, which
eliminates the contact hyperfine coupling to the nuclear field,Itoh  _et al._
(1993, 2003) with the maturity of semiconductor microfabrication as an added
advantage. Recent years have witnessed a concerted push towards all-electrical
qubit operation, since electric fields are easier to apply and localise than
magnetic fields, and electrically operated qubit gates offer significant
improvements in speed and power consumption as compared to magnetic gates. A
series of theoretical predictionsBulaev and Loss (2007); Kloeffel  _et al._
(2011, 2013); Chesi  _et al._ (2014) as well as experimental leaps in growth
techniques and sample quality Dobbie  _et al._ (2012); Sammak  _et al._
(2019); Lodari  _et al._ (2019) have led to a surge in interest in spin-3/2
hole systems in group IV materials. The strong and multifaceted spin-orbit
coupling experienced by holes, Cardona and Peter (2005); Winkler (2003);
Winkler  _et al._ (2008); Durnev  _et al._ (2014); Marcellina  _et al._ (2017)
their anisotropic and tunable $g$-tensor, Danneau  _et al._ (2006); Miserev
and Sushkov (2017); Hung  _et al._ (2017); Qvist and Danon (2022); Abadillo-
Uriel  _et al._ (2022) and the absence of a valley degree of freedom makes
them ideal for electrical spin manipulation, with Ge offering the additional
advantages of a small effective mass Terrazos  _et al._ (2021) and ease of
ohmic contact formation. Compared to electrons, the weaker hyperfine coupling
for holesKeane  _et al._ (2011); Chekhovich  _et al._ (2011) due to the
absence of the contact interaction significantly reduces the nuclear field
contribution to spin decoherence, while the hole spin-3/2 is responsible for
physics with no counterpart in electron systems, Culcer  _et al._ (2006);
Winkler  _et al._ (2008); Liu  _et al._ (2018); Abadillo-Uriel  _et al._
(2018); Cullen  _et al._ (2021) which may offer flexibility in future design
strategies – for example magic angles have been predicted for acceptor qubits,
Abadillo-Uriel  _et al._ (2018) at which dipole-dipole entanglement can be
switched off without switching off the electric dipole moments of single
qubits.

Remarkable progress on hole spin qubits in several architectures has spanned
more than a decade, with an overwhelming focus on Ge and Si.Chatterjee  _et
al._ (2021); Scappucci  _et al._ (2021); Fang  _et al._ (2023) Initial work
focused on measuring hole spin states, Roddaro  _et al._ (2008); Zwanenburg
_et al._ (2009); Li  _et al._ (2015); Liles  _et al._ (2018) relaxation and
dephasing times, Hu  _et al._ (2012); Higginbotham  _et al._ (2014); Vukušić
_et al._ (2018) single spin electrical control, Pribiag  _et al._ (2013)
readout and control of the $g$-tensor Ares  _et al._ (2013a, b); Brauns  _et
al._ (2016); Watzinger  _et al._ (2016); Voisin  _et al._ (2016); Srinivasan
_et al._ (2016); Mizokuchi  _et al._ (2018); Marcellina  _et al._ (2018); Wei
_et al._ (2020); Zhang  _et al._ (2021); Liles  _et al._ (2021) and of spin
states in multiple dots, Li  _et al._ (2015); Bohuslavskyi  _et al._ (2016);
Wang  _et al._ (2016); van Der Heijden  _et al._ (2018); Ezzouch  _et al._
(2021) and achieving single-spin spin qubits. Maurand  _et al._ (2016);
Watzinger  _et al._ (2018) In recent years the development of strained
germanium in SiGe heterostructures Lodari  _et al._ (2019, 2021); Terrazos
_et al._ (2021) provided a low-disorder environment, which supported the
development of single-hole qubits, Hendrickx  _et al._ (2020a) singlet-triplet
qubits, Jirovec  _et al._ (2021) universal quantum logic, Hendrickx  _et al._
(2020b) and a four-qubit germanium quantum processor.Hendrickx  _et al._
(2021) Experiments have demonstrated ultrafast spin manipulation using the
spin-orbit interaction Hendrickx  _et al._ (2020b, a); Froning  _et al._
(2021); Wang  _et al._ (2022a) and EDSR Rabi oscillations as fast as $540$
MHz, Wang  _et al._ (2022a) electrical control of the underlying spin-orbit
coupling Gao  _et al._ (2020); Liu  _et al._ (2022) and charge sensing using a
superconducting resonator, Ungerer  _et al._ (2022) while relaxation times of
up to 32 ms have been measured in Ge dots. Lawrie  _et al._ (2020) Hole spins
in Ge have been used as quantum simulators Wang  _et al._ (2022b) and control
of an array of 16 Ge hole dots has been demonstrated. Borsoi  _et al._ (2022)
The development of hybrid structures offers another path towards entanglement,
with the demonstrations of superconductivity in planar Ge, Hendrickx  _et al._
(2018); Aggarwal  _et al._ (2021); Valentini  _et al._ (2023) hole coupling to
a superconducting resonator, Li  _et al._ (2018) dipole coupling to a
microwave resonator, Xu  _et al._ (2020) charge sensing using a
superconducting resonator Ungerer  _et al._ (2022) and devices such as
transistors and interferometers. Vigneau  _et al._ (2019) Theoretically, the
interplay of spin-orbit coupling and superconductivity in hybrid
semiconductor-superconductor structures is only now being studied in the
context of quantum computing. Lidal and Danon (2023)

Concomitantly, Si qubits have also registered remarkable recent progress, with
coherence times of up to 10ms in Si:B acceptors Kobayashi  _et al._ (2021) and
the detection of sweet spots as a function of the top gate field in quantum
dots, Piot  _et al._ (2022) sweet spots having been the subject of a number of
theoretical studies. Kloeffel  _et al._ (2011); Salfi  _et al._ (2016a, b);
Abadillo-Uriel  _et al._ (2018); Kloeffel  _et al._ (2018); Marcellina  _et
al._ (2018); Terrazos  _et al._ (2021); Wang  _et al._ (2021); Bosco  _et al._
(2021a); Bosco and Loss (2022); Wang  _et al._ (2022c); Malkoc  _et al._
(2022) Anisotropic exchange was also used to entangle two hole qubits, Geyer
_et al._ (2022) hole coupling to a superconducting resonator has been
demonstrated, Yu  _et al._ (2022) and progress has been made towards higher-
temperature operation with the observation of Coulomb diamonds up to 25K
Shimatani  _et al._ (2020) and single-qubit operation above 4K. Camenzind  _et
al._ (2022)

Despite experimental advances on many fronts, constructing an all-encompassing
theory to describe hole physics in group-IV semiconductor quantum dots is
challenging. In particular, owing to different effective masses and intrinsic
spin-orbit gap in the valence band of these materials, the wide range of QD
sizes as well as spin manipulation timescales exhibited in experiments render
it difficult to provide a theory of spin qubits common to Si and Ge. In this
work we will focus on Ge, which is somewhat more tractable analytically than
Si, and describe electrical qubit operation in an in-plane magnetic field.
This choice is motivated by the observation that experiments has
overwhelmingly favoured in-plane magnetic fields, Watzinger  _et al._ (2018);
Hendrickx  _et al._ (2020a) partly because a transverse magnetic field makes
it easier to suppress hyperfine and cyclotron quantization effects, and partly
in order to avoid the strong orbital coupling of the perpendicular field,
which can cause a large diamagnetic shift and also affect tunnel rates. For
spin-3/2 holes, in-plane magnetic fields are highly non-trivial, because the
spin and orbital degrees of freedom are intertwined. The in-plane $g$-factor
is very small, and in Ge most of it comes from the octupole interaction with
the magnetic field.Winkler (2004) Whereas in-plane magnetic fields have been
considered for realistic devices in recent theoretical studies, Ciocoiu  _et
al._ (2022); Wang  _et al._ (2022c); Martinez  _et al._ (2022) this has been
largely from an engineering point of view, and the orbital effect of an in-
plane magnetic field in planar quantum dots has largely been neglected, though
they have been shown to play an important role in nanowires. Adelsberger  _et
al._ (2022) It has thus not been possible to date to construct a full picture
of spin dynamics and electrical spin operation in planar Ge hole dots in an
in-plane magnetic field. This has left several outstanding questions
unanswered: What determines the speed of EDSR, as well as the relaxation time
$T_{1}$? Is there an optimal magnetic field orientation for driving a spin-
orbit qubit? Do coherence sweet spots exist when the magnetic field is in the
plane?

Figure 1:  A prototype double quantum dot in a strained germanium hole system.
The strained quantum well is grown epitaxially on a strain-relaxed SiGe layer.
Gate B2 and T1 control the inter-dot tunneling. The growth direction is
$\hat{\bm{z}}$.

In this work we seek to answer these questions. Our focus will be on gate-
defined Ge quantum dots, with the parent 2DHG exhibiting very high
mobilitySammak  _et al._ (2019); Lodari  _et al._ (2022), low percolation
density,Lodari  _et al._ (2021); Stehouwer  _et al._ (2023), and a low
effective mass of $m_{h}{=}\,0.05\,m_{e};$Hendrickx  _et al._ (2018); Lodari
_et al._ (2019), which all aid the formation of quantum dots. One reason for
our choice of Ge is its band alignment, which makes it the only group IV
material suitable for growing quantum wells. Another reason is pragmatic – it
is easier to describe theoretically. This is because the spin-orbit splitting
in Ge ($\Delta_{SO}{=}325\,\text{meV}$) is stronger than that in Si
($\Delta_{SO}{=}44\,\text{meV}$), resulting in a large separation of the spin-
orbit/split-off (SO) band from the heavy- and light-hole subspaces. This
ensures that the $4\\!\times\\!4$ Luttinger Hamiltonian formalism for
spin-$3/2$ is adequate for the topmost Ge valence band, as opposed to Si where
the Rashba mediated electrical control features both heavy hole-light hole
(HH-LH) coupling contributions and heavy hole-split off (HH-SO) coupling
contributions. Ge has a noticeable cubic-symmetry contribution to the
Luttinger Hamiltonian. It is strong enough to enable electrical spin
manipulation in planar dots, Terrazos  _et al._ (2021); Wang  _et al._ (2021)
making Ge ideal for electrical spin operation, but not as strong as in Si,
such that it can be treated perturbatively.

Figure. 1 provides a schematic of the device architecture for studying a
planar germanium hole quantum dot qubit. In this paper we concentrate on
developing the key formalisms for describing the spin physics of hole spin
qubits in an in-plane magnetic field. To avoid unneccessary complexity, and to
keep our results generally applicable, we avoid making the analysis overly
device specific; therefore we do not consider effects of gate electrode
induced non-uniform strain which can lead to a significant modification of the
spin-orbit interaction;Liles  _et al._ (2021); Corley-Wiciak  _et al._ (2023)
or Fowler-Norheim tunnelling of hole states through the SiGe barrier leading
to charge accumulation at the interface between the semiconductor and the gate
dielectric; or light-hole penetration through the SiGe barrier. Addressing
these would require detailed finite element numerical simulation techniques on
top of the theory developed here.

For an applied in-plane magnetic field operation of planar Ge hole QD, in the
presence of uniaxial strain but neglecting shear strain, we show that: (i)
EDSR is linear in the magnetic field with nonlinear corrections emerging at
larger fields, and is driven by Rashba spin-orbit coupling rather than by the
orbital magnetic field terms; the picture that emerges is that the orbital B
terms give rise to the finite Zeeman splitting between the qubit energy
levels, while the Rashba spin-orbit coupling gives rise to transitions between
them. The EDSR Rabi frequency is a maximum when the electric field is parallel
to the magnetic field, and vanishes when the two are perpendicular; it can be
sizable despite the smallness of the in-plane $g$-factor, and has a non-
monotonic dependence on $B$; (ii) The relaxation rate is due to bulk acoustic
phonons and is proportional to $B^{3}$ in the leading power; Li  _et al._
(2020) (iii) For a squeezed (elliptical) dot with an aspect ratio
$L_{y}/L_{x}{=}\,2$ the spin-flip frequency is an order of magnitude faster
than for a circular dot allowing Rabi time $\sim\\!10$ ns with $\sim\\!10^{5}$
operations within time $T_{1}$; (iv) For a squeezed dot, due to the effect of
the magnetic field vector potential terms, the in-plane $g$-factor exhibits a
strong anisotropy resulting in oscillatory behavior as the magnetic field is
rotated in the plane of the dot, an observation supported by new experimental
measurements shown in Sec. V of this paper; (v) Although extrema in the qubit
Zeeman splitting as a function of the top gate voltage exist in the same way
as for a perpendicular magnetic field,Wang  _et al._ (2021) coherence sweet
spots do not exist for in-plane magnetic fields for noise induced by charge
defects in the plane of the quantum well. The physics underlying this process
is associated with the orbital magnetic field terms and is not captured by the
Schrieffer-Wolff approximation (although electrical qubit operation including
EDSR can be described in a simplified Schrieffer-Wolff picture).

The outline of this paper is as follows. Section II lays down the foundation
of the semi-analytical model of a Ge hole quantum dot qubit withing the
framework of effective mass theory. Next we discuss the properties of
circularly symmetric dots in Sec. III, including the qubit Zeeman splitting,
EDSR, relaxation and dephasing. In Sec. IV we focus on elliptical dots and
study their EDSR and coherence properties, while in Sec. V we discuss the
consequences of $g$-factor anisotropy and compare our predictions with recent
experimental results. We end with a summary and outlook in Sec. VI.

##  II Hamiltonian and model

The topmost valence band in Ge has orbital angular momentum $l{=}1$. When the
hole spin $s{=}\frac{1}{2}$ is taken into account the resultant states at the
$\mathbf{\Gamma}$-point are eigenstates of the total angular momentum
$\mathbf{J}{=}(\mathbf{L}+\mathbf{S})$. The four-fold degenerate
$j{=}\frac{3}{2}$ states are separated by the spin-orbit gap $\Delta_{0}$ from
the $j{=}\frac{1}{2}$ two-fold degenerate split-off states. For Ge the spin-
orbit gap $\Delta_{0}{=}325\,\text{meV}$, so the split-off band is safely
disregarded in describing hole motion in the topmost valence bands. The
$\left|\frac{3}{2}\right\rangle$ and $\left|-\frac{3}{2}\right\rangle$ states
constitute the heavy-hole (HH) manifold while the
$\left|\frac{1}{2}\right\rangle$ and $\left|-\frac{1}{2}\right\rangle$ states
represent the light-hole (LH) manifold. The Luttinger Hamiltonian describes
the hole motion in the topmost valence bands and has the following form in the
$j{=}\frac{3}{2}$ basis
$\left\\{\left|\frac{3}{2}\right\rangle,\left|-\frac{3}{2}\right\rangle,\left|\frac{1}{2}\right\rangle,\left|-\frac{1}{2}\right\rangle\right\\}$:

$H_{LK}(k)=\begin{pmatrix}P^{\prime}+Q^{\prime}&0&L^{\prime}&M^{\prime}\\\
0&P^{\prime}+Q^{\prime}&M^{\prime*}&-L^{\prime*}\\\
L^{\prime*}&M^{\prime}&P^{\prime}-Q^{\prime}&0\\\
M^{\prime*}&-L^{\prime}&0&P^{\prime}-Q^{\prime}\\\ \end{pmatrix}$ (1)

where the matrix elements of the Luttinger Hamiltonian comprise the
$k$-dependent part and strain-induced perturbations:
$P^{\prime}{=}P(\mathbf{k})+P_{\varepsilon},\,Q^{\prime}{=}Q(\mathbf{k})+Q_{\varepsilon},\,L^{\prime}{=}L(\mathbf{k})+L_{\varepsilon},\,M^{\prime}{=}M(\mathbf{k})+M_{\varepsilon}$.
The kinetic energy terms are
$P\,{=}\frac{\hbar^{2}\gamma_{1}}{2m_{0}}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})$,
$Q\,{=}\frac{\hbar^{2}\gamma_{2}}{2m_{0}}(k_{x}^{2}+k_{y}^{2}-2k_{z}^{2})$,
$L\,{=}\frac{-\sqrt{3}\hbar^{2}\gamma_{3}}{m_{0}}(\\{k_{x},k_{z}\\}-i\\{k_{y},k_{z}\\})$
and
$M\,{=}\frac{\sqrt{3}\hbar^{2}}{2m_{0}}\\{-\gamma_{2}(k_{x}^{2}-k_{y}^{2})+2i\gamma_{3}\\{k_{x},k_{y}\\}\\}$.
The strain terms are:
$P_{\varepsilon}{=}-a(\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz})$,
$Q_{\varepsilon}{=}-b(\varepsilon_{xx}+\varepsilon_{yy}-2\varepsilon_{zz})$,
$L_{\varepsilon}{=}d(\varepsilon_{xz}-i\varepsilon_{yz})$ and
$M_{\varepsilon}{=}\frac{\sqrt{3}}{2}b(\varepsilon_{xx}-\varepsilon_{yy})-d\varepsilon_{xy}$.
Here $m_{0}$ is the bare electron mass while $\gamma_{1}$,$\gamma_{2}$ and
$\gamma_{3}$ are Luttinger parameters. The constant $a{=}2\,\text{eV}$ is the
hydrostatic deformation potential, $b{=}{-}2.16\,\text{eV}$ is the uniaxial
deformation potential, and $d{=}{-}6.06\,\text{eV}$ accounts for the shear
deformation potential. In most Ge/GeSi samples there is considerable strain in
the quantum well, which significantly increases the splitting between light
and heavy holes compared to silicon - here we take the compressive strain to
be $0.6\%$.Sammak  _et al._ (2019) The strain tensor components in the plane
are: $\varepsilon_{xx}{=}\varepsilon_{yy}{=}-0.006$. The in-plane compressive
strain elongates the out-of-plane lattice constant via
$\varepsilon_{zz}{=}-2\frac{C_{12}}{C_{11}}\varepsilon_{xx}=0.004$; with
$C_{12}{=}44$ GPa, $C_{11}{=}126$ GPa.Wortman and Evans (1965) We assume the
off-diagonal shear elements of the strain tensor to be
$\varepsilon_{ij|_{i\neq j}}{=}0$. The out-of-plane confinement is described
by a one-dimensional infinite square well potential

$V(z)=\begin{cases}\infty&z\in\left\\{-\frac{L_{z}}{2},\frac{L_{z}}{2}\right\\}\\\
0&\text{otherwise}\\\ \end{cases}.$ (2)

The coupling to the top-gate electric field, denoted by $F_{z}$, gives an
additional term $eF_{z}z$ in the Hamiltonian. The in-plane confinement is
modelled by a parabolic potential
$V_{x,y}{=}\frac{1}{2}(\lambda_{x}^{2}x^{2}+\lambda_{y}^{2}y^{2})$, where
$\lambda_{x},\,\lambda_{y}$ are determined by the dot dimensions
$L_{x},\,L_{y}$ in the plane. The effective hole QD Hamiltonian is given by:

$H_{0D}=H_{LK}(\mathbf{k})+eF_{z}z+V_{\text{conf}},$ (3)

where $V_{\text{conf}}=V_{x,y}+V(z)$ is the total confinement potential. The
Zeeman interaction is given by,

$H_{Z}=-2\kappa\mu_{B}\mathbf{B}\cdot\mathbf{J}-2q\mu_{B}\mathbf{B}\cdot\mathcal{J},$
(4)

where $\mathbf{J}=\\{J_{x},\,J_{y},\,J_{z}\\}$,
$\mathcal{J}=\\{J_{x}^{3},J_{y}^{3},J_{z}^{3}\\}$; and $J_{x},\,J_{y},\,J_{z}$
are the $4{\times}4$ Pauli matrices. The $\mathcal{J}$-terms are vital in
order to obtain the correct in-plane $g$-factor $\approx 0.25$.

In the presence of a magnetic field, the canonical momentum of holes in
topmost valence band becomes
$\mathbf{k}\rightarrow\left(\mathbf{k}+\frac{e\mathbf{A}}{\hbar}\right)$. The
hole spin in a planar Ge quantum dot is then described by

$H_{QD}=H_{LK}\\!\\!\left(\mathbf{k}+\frac{e\mathbf{A}}{\hbar}\right)+eF_{z}z+V_{\text{conf}}+H_{Z}.$
(5)

The resultant modifications in
$H_{LK}\left(\mathbf{k}+\frac{e\mathbf{A}}{\hbar}\right)$ lead to non-trivial
contributions to the effective spin-orbit interaction in the HH and LH
manifolds, as well as between them. To check the gauge invariance of our
theoretical framework, we consistently diagonalise the effective quantum dot
Hamiltonian using two different gauges:

  1. 1.

$\mathbf{A}{=}-\frac{1}{2}B_{z}y\mathbf{\hat{e}_{x}}+\frac{1}{2}B_{z}x\mathbf{\hat{e}_{y}}+(B_{x}y-B_{y}x)\mathbf{\hat{e}_{z}}$.

  2. 2.

The symmetric gauge:
$\mathbf{A}\,{=}\,\frac{1}{2}\,\mathbf{B}\times\mathbf{r}\,{=}\,\frac{1}{2}(B_{y}z-B_{z}y)\,\mathbf{\hat{e}_{x}}+\frac{1}{2}(B_{z}x-B_{x}z)\,\mathbf{\hat{e}_{y}}+\frac{1}{2}(B_{x}y-B_{y}x)\,\mathbf{\hat{e}_{z}}$.

All calculations have been performed in both gauges, yielding consistent
results.

The eigenstates of the hole QD can be expressed as linear combinations of
states belonging to a basis in which the bare QD Hamiltonian is diagonal, i.e.
$\left|\Psi_{QD}\right\rangle=\sum_{i}c_{i}\psi_{i}(x,y,z)\left|\phi_{j}\right\rangle$,
where the bare Hamiltonian refers to $H_{QD}$ of Eq. 5 with its off-diagonal
elements set to zero, following the practice of ${\bm{k}}\cdot{\bm{p}}$
theory, as well as the external magnetic field set to zero. We choose the
spatial wave functions as
$\psi(x,y,z){=}\psi_{n}(x)\,\psi_{m}(y)\,\psi_{l}(z)$, where the in-plane
basis states are $\mathbf{1}$-D Harmonic oscillator states for $x$ and $y$ and
the out-of-plane basis states are given by solutions of the infinite potential
well:

$\displaystyle\psi_{n}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2^{n}\,n!}}\frac{1}{\sqrt{L_{x}\sqrt{\pi}}}e^{-\frac{x^{2}}{2L_{x}^{2}}}H_{n}\\!\\!\left(\frac{x}{L_{x}}\right),\
x\in\\{-\infty,\infty\\}$ $\displaystyle\psi_{m}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2^{m}\,m!}}\frac{1}{\sqrt{L_{y}\sqrt{\pi}}}e^{-\frac{y^{2}}{2L_{y}^{2}}}H_{m}\\!\\!\left(\frac{y}{L_{y}}\right),\
y\in\\{-\infty,\infty\\}$ $\displaystyle\psi_{l}(z)$ $\displaystyle=$
$\displaystyle\begin{cases}\cos\\!\left(\frac{(l+1)\pi z}{L_{z}}\right)&l\
\text{is even}\\\ \sin\\!\left(\frac{(l+1)\pi z}{L_{z}}\right)&l\ \text{is
odd}\\\ \end{cases}\ ,z\in\left\\{-\frac{L_{z}}{2},\frac{L_{z}}{2}\right\\}$
(6)

The indices $n,m,l$ in Eqn. II can take integer values $0,1,2,3..$ etc. The
hole spinors represent the $j{=}\frac{3}{2}$ spin states:
$\left|\phi_{j}\right\rangle\in\left\\{\left|\frac{3}{2}\right\rangle,\,\left|-\frac{3}{2}\right\rangle,\,\left|\frac{1}{2}\right\rangle,\,\left|-\frac{1}{2}\right\rangle\right\\}$.
When operated at low in-plane $\mathbf{B}$ we are in the
$\omega_{c}\ll\omega_{0}$ limit, with $\omega_{c}$ the cyclotron frequency, so
any effect of the in-plane magnetic field on the dot size are generally
irrelevant (they are taken into account in our formalism). This means the
Fock-Darwin solutions have a one-on-one analogy to the 2D Harmonic oscillator
solutions in Eqn. II. The in-plane basis states $\Psi_{n}(x)\Psi_{m}(y)$ are
ordered according to their energy $\propto(n+m+1)$. We find converging
solutions to Eqn. 5 by considering $55$ in-plane basis states, i.e.
$(n+m)\in[0,9]$; and $15$ out-of-plane $\Psi_{l}(z)$ basis states, i.e.
$l\in[0,14]$. We note that the $(n+m)=0$ level has no degeneracy; but
considering the degeneracies of $(n+m)=1,2....9$; the simulation spans $55$
in-plane levels. The numerical diagonalisation of the resultant $3300\times
3300$ Hamiltonian yields the energy levels of the hole quantum dot:
$H_{QD}\left|\Psi_{QD}\right\rangle{=}\lambda_{E}\left|\Psi_{QD}\right\rangle$.

In the Schrieffer-Wolff approximation, which assumes the longitudinal
confinement is much stronger than the in plane confinement, theoretical models
show that the in-plane magnetic field gives rise to an effective spin-orbit
coupling whose magnitude is proportional to $B$.Kloeffel  _et al._ (2018);
Wang  _et al._ (2021) Recent experimental works have shown unambiguously that
the $g$-factor of a 2D hole gas is a strong function of density,Marcellina
_et al._ (2018); Akhgar  _et al._ (2019), which can be captured by effective
2D theoretical modelsWinkler (2004); Miserev and Sushkov (2017); Miserev  _et
al._ (2017); Marcellina  _et al._ (2017). Moreover, the effective spin-orbit
interaction due to the orbital magnetic field terms has a highly non-trivial
interplay with the Rashba spin-orbit interaction stemming from the top-gate
potential.Winkler (2003, 2004); Miserev  _et al._ (2017) For a hole QD in an
in-plane $B$, as considered here, the orbital magnetic field terms also
contribute to the spin dynamics. Nevertheless, this contribution, as the
following sections make clear, cannot be captured by a naive Schrieffer-Wolff
transformation, because the orbital magnetic field terms couple the in-plane
and out-of-plane dynamics in a way that makes them inseparable: if one first
reduces the 3D Hamiltonian to an effective $2\times 2$ Hamiltonian for a 2D
hole gas, and then attempts to understand QD dynamics based on this effective
2D Hamiltonian (in analogy with electron systems) all the physics of the
orbital magnetic field terms is lost. Hence a full 3D theoretical model is
essential to understand the full spin dynamics of a hole quantum dot in an in-
plane magnetic field. The model we present in this work can treat arbitrary
quantum dot sizes, magnetic field strengths and orientations.

We comment briefly on the choice of spatial Ge basis functions. Variational
analyses of the $z$-wave function incorporating the top gate potential have
successfully described Ge hole QDs in Refs. Marcellina  _et al._ (2017); Wang
_et al._ (2021), but the variational model is hard to extend to a full 3D
numerical analysis due to the complicated form of the variational excited
states. For example, the Airy functionWang  _et al._ (2022c); Li  _et al._
(2020) provides the exact solution if the $z$ confinement is modelled as a
triangular potential well but can yield a residual Rashba spin-orbit
interaction at non-zero top gate potential ($F_{z}\\!>\\!0$), requiring a
careful choice and implementation of boundary conditions. In the present paper
we describe the $z$ confinement using an infinite square well augmented by a
linear electrostatic potential that accounts for the top gate, and consider
top gate fields up to 50MV/m (although values up to 100 MV/m can also be
studied with this method). Ref. Wang  _et al._ , 2022c used a sophisticated
model that incorporates Fowler-Nordheim tunnelling, which is beyond the scope
of the present study, as explained below. We stress that the range of $F_{z}$
is at the lower end of what we consider here (up to 2.5 MV/m), and thus our
studies can be regarded as complementary.

##  III Circular Quantum dot

#### Qubit Zeeman splitting.

We solve the full 3D Hamiltonian in an external magnetic field along $x$ for a
QD with following dimensions:
$L_{x}{=}50\,\text{nm}$,$L_{y}{=}50\,\text{nm}$,$L_{z}{=}11\,\text{nm}$.

(a)

(b)

Figure 2:  Qubit Zeeman splitting of hole spin-qubit in $B_{\parallel}$. a)
Variation in the qubit Larmor frequency $\Delta\mathbb{E}$ ($\mu\text{eV}$)
with the in-plane magnetic field $B_{x}$(T). The Zeeman splitting shows a
monotonic linear trend with $B_{x}$ at $F_{z}{=}2\,\text{MV/m}$,
$F_{z}{=}18\,\text{MV/m}$ and $F_{z}{=}35\,\text{MV/m}$ top-gate potentials
respectively; with in-plane $g_{\parallel}{\sim}\,0.22$. No significant
difference occurs in the qubit Larmor frequency for different top gate field.
b) The in-plane $g$-factor exhibits an extremum as a function of the top gate
at $F_{z}{=}18\,\text{MV/m}$ and $F_{z}{=}21\,\text{MV/m}$, respectively for
$B_{x}{=}0.1\text{T}$ and $B_{x}{=}2\text{T}$.

The ground state is given by
$\left|\Psi_{QD}\right\rangle=\sum_{i}c_{i}\psi_{i}(x,y,z)\left|\phi_{j}\right\rangle$.
We evaluate the QD ground state and first excited state, labelling them as
state $\left|\mathbb{0}\right\rangle$ with eigenenergy $\mathbb{E_{0}}$ and
$\left|\mathbb{1}\right\rangle$ with eigenenergy $\mathbb{E_{1}}$ repectively.
The ground state and the first excited state are of heavy hole-type with
admixtures due to heavy hole-light hole coupling, and split by an amount
$\Delta\mathbb{E}{=}\mathbb{E_{1}}-\mathbb{E_{0}}$, which we refer to as the
qubit Larmor frequency, while continuing to express it in units of energy. The
HH-LH splitting $\Delta_{LH}\sim 60\text{meV}$ is governed by the $z$-energy
bands in the quasi 2D dot limit , so that many in-plane (quantum dot) levels
are contained between any two out-of-plane (quantum well) levels. This allows
us to write
$\left|\mathbb{0}\right\rangle{=}(c_{1}\left|0,0,0,\frac{3}{2}\right\rangle+\text{admixtures})$
and
$\left|\mathbb{1}\right\rangle{=}(c_{1}^{\prime}\left|0,0,0,-\frac{3}{2}\right\rangle+\text{admixtures})$,
where the four indices denote $n,\,m,\,l,\,J_{z}$. When plotted against the
top-gate voltage, the qubit Larmor frequency exhibits an extremum as a
function of $F_{z}$.Wang  _et al._ (2021) The in-plane magnetic field gives
rise to a finite Zeeman splitting, which shows a linear trend with $B_{x}$
(figure 2a). The extrema in the qubit Larmor frequency as a function of
$F_{z}$ in an in-plane $\mathbf{B}$ are explained by the same mechanism as for
out-of-plane magnetic field operation (Figure 2b). At small values of $F_{z}$
the matrix elements connecting the HH and LH states, which give rise to Rashba
spin-orbit coupling, increase linearly with the gate field, while the change
in the HH-LH splitting is negligible. At large values of $F_{z}$ the increase
in the HH-LH splitting outweighs all other effects and the Rashba spin-orbit
coupling decreases as a function of $F_{z}$. These competing effects give rise
to an extremum in the qubit Larmor frequency at a certain value of the top
gate electric field, where the qubit is insensitive to $z$-electric field
fluctuations. The qubit Zeeman splitting is linear in $\mathbf{B}$,
$\Delta\mathbb{E}{=}g_{\parallel}\mu_{B}B_{x}$, where the effective in-plane
$g$-factor ranges between $0.215-0.219$, expected to be much smaller than out-
of-plane $g$-factor; $g_{\parallel}\ll g_{\perp}$.

#### EDSR.

An alternating electric field $\tilde{\mathbf{E}}(t)$ can induce spin-flip
transition between the qubit states $\left|\mathbb{0}\right\rangle$ and
$\left|\mathbb{1}\right\rangle$ via electron dipole spin resonance (EDSR) when
the frequency of the ac electric field matches the Zeeman splitting of the
hole spin qubit, $\Delta\mathbb{E}{=}h\nu$.

5

(a)

(b)

(c)

Figure 3: Electric dipole spin resonance of hole qubit in $B_{\parallel}.$ a)
Dependence of the EDSR Rabi frequency $f_{\text{EDSR}}$(GHz) on the top gate
field $F_{z}$(MV/m) and in-plane magnetic field $B_{x}$(T). The strength of
the in-plane EDSR driving ac electric field is $E_{0}=10\,\text{kV/m}$. The
$z$-axis representing $f_{\text{EDSR}}$ in the $3$D plot is logarithmic. b)
Variation of $f_{\text{EDSR}}$ with $B_{x}$ showing a non-monotonic dependence
on the applied $B$-field as:
$f_{\text{EDSR}}{=}a_{f}B_{x}{+}b_{f}B_{x}^{2}{+}c_{f}B_{x}^{3}$. The points
on the plot signify results from the numerical calculation while the fitting
is denoted by the dashed lines. At $F_{z}{=}2\,\text{MV/m}$ the fitting
parameters are $a_{f}=0.0002,\,b_{f}=0.00002,\,c_{f}=0.003$; at
$F_{z}{=}18\,\text{MV/m}$ they are $a_{f}=0.005,\,b_{f}=0.00006,\,c_{f}=0.01$
and at $F_{z}{=}35\,\text{MV/m}$ top-gate potential the fitting parameters are
$a_{f}=0.01,\,b_{f}=0.00004,\,c_{f}=0.01$. c) The Rabi frequency exhibits
maxima as a function of the top gate at $F_{z}{=}18\,\text{MV/m}$ only for
higher magnetic field ($B_{x}{=}2\text{T}$), over the range of top-gate
potentials used in this study.

The EDSR Rabi frequency is calculated as the matrix element of the ac field
between the qubit ground state (GS) and excited state (ES):

$f_{\text{EDSR}}=\left\langle\mathbb{0}\left|e\tilde{\mathbf{E}}(t)\cdot\mathbf{r}\right|\mathbb{1}\right\rangle$
(7)

Here we will focus on the scenario in which the alternating electric field is
in the plane. In the 3D model, the effective interaction between the qubit GS
and ES wavefunctions lies in the LH admixture to
$\left|\pm\frac{3}{2}\right\rangle$ HH states. The admixture is the results of
the two primary spin-orbit interactions in the system: structure inversion
asymmetry (SIA) due to the top-gate potential $F_{z}$ gives rise to the first
Rashba term, which stems primarily from HH-LH coupling. The second
contribution to the spin-orbit interaction comes from the orbital terms due to
the in-plane magnetic field $\mathbf{B}$. As $\mathbf{B}$ is ramped up, the
magnitude of this latter contribution increases. For an applied oscillatory
electric field of strength $E_{0}{=}10\text{kV/m}$, figure 3a presents the
spin-flip Rabi frequency variation with the top gate field and applied
$B_{x}$, with the key features of the EDSR frequency exhibiting a maximum at a
certain value of $F_{z}$, as well as a nonlinear dependence of
$f_{\text{EDSR}}$ on $B_{x}$. In figure 3b, $f_{\text{EDSR}}$ shows a non-
linear variation as a function of the in-plane magnetic field. The best fit
corresponds to
$f_{\text{EDSR}}{=}a_{f}B_{x}{+}b_{f}B_{x}^{2}{+}c_{f}B_{x}^{3}$ at a constant
top gate field. At $B_{x}{=}0.1\,\text{T}$ the Rabi frequency increases slowly
with the top gate field. On the other hand at ($B_{x}{=}2\,\text{T}$) the Rabi
frequency increases more rapidly with $F_{z}$, and the maximum shifts towards
lower values of $F_{z}$ (Fig. 3c). We find the EDSR rate is a maximum when the
electric field is parallel to the magnetic field, and vanishes when the two
are perpendicular.

(a)

(b)

Figure 4: Phonon induced hole spin-qubit relaxation for $B_{\parallel}$
operation. a) Relaxation rate $T_{1}^{-1}\,(\text{s}^{-1})$ vs. applied
magnetic field $B_{x}$ (T) at the top-gate field of $F_{z}{=}18\,\text{MV/m}$.
The B-dependence of the relaxation rate has been fitted using the fitting
parameters shown in the top inbox. b) The relaxation rate $T_{1}^{-1}$ vs. the
top gate field $F_{z}$ (MV/m) for $B_{x}{=}0.67\,\text{T}$.

The nonlinearity in $F_{z}$ in figure 3c reflects the quadrupole Rashba spin-
orbit coupling unique to spin-3/2 systems,Winkler (2004) having no counterpart
in electron spin qubits. Hole qubits in Ge quantum dots can exhibit fast EDSR
of the order of $\sim\\!100\,\text{ns}$, despite the fact that the in-plane
$g$-factor is two orders of magnitude smaller than the out-of-plane
$g$-factor. The linear term in $B_{x}$ in the Rabi frequency (figure 3b) stems
from the Zeeman terms cubic in angular momentum $\propto q$ in Eq. 4. To see
this one can write an effective $2{\times}2$ qubit Hamiltonian using the
Schrieffer-Wolff (SW) transformation up to third order in small quantities:

$H_{\text{eff}}^{2{\times}2}=H_{0}+H_{SO}+g_{\parallel}\mu_{B}B_{x}^{3}\sigma_{x}+V(x,y)+eE_{x}(t)x$
(8)

Here
$H_{0}\\!(\mathbf{k}{\rightarrow}(\mathbf{k}{+}e\mathbf{A}/\hbar)){=}[\mathcal{A}k^{2}\\!-\\!\mathcal{B}k^{4}\\!-\\!D(k_{+}^{2}\\!-\\!k_{-}^{2})^{2}]\\!I_{2{\times}2}$
is the kinetic energy including a correction $\propto k^{4}$,Winkler (2004)
and the $D$ term represents warping of the energy contours.Marcellina  _et
al._ (2017) The spin-orbit Hamiltonian $H_{SO}$ comprises two important
$k$-cubic Rashba terms induced by the top gate field, a spherical term
$\propto\alpha_{R2}$ and a cubic-symmetric correction $\propto\alpha_{R3}$.
The next three terms represent the Zeeman splitting, confinement energies and
the ac electric potential along the $x$-direction, respectively. Starting from
this effective Hamiltonian and projecting it onto the in-plane states, one may
apply the SW approximation again to obtain an effective $2\times 2$ qubit
Hamiltonian, in which the EDSR driving term appears on the off-diagonal.
Reading off the EDSR Rabi frequency we find $f_{\text{EDSR}}{\propto}B_{x}$ to
leading order, while the next order in the expansion gives
$f_{\text{EDSR}}\propto B_{x}^{3}$ (Appendix. B).

The significant quadratic $B_{x}^{2}$ dependence of the EDSR Rabi frequency at
large $F_{z}$ (Fig. 3c) is a signature of the orbital magnetic field terms.
One can determine possible paths connecting the lowest energy
$\left|\mathbb{0}\right\rangle{\leftrightarrow}\left|\mathbb{1}\right\rangle$
HH-qubit states, mainly composed of $\left|0,0,0,\frac{3}{2}\right\rangle$ and
$\left|0,0,0,-\frac{3}{2}\right\rangle$ with admixtures, respectively.
Choosing the symmetric gauge for $\mathbf{B}{=}(B_{x},0,0)$, the magnetic
field induces the following transitions in the Luttinger-Kohn picture:
$\left\\{\left|\frac{3}{2}\right\rangle\\!{\xrightarrow{B_{x}}}\\!\left|\frac{1}{2}\right\rangle,\,\left|-\frac{1}{2}\right\rangle\\!{\xrightarrow{B_{x}}}\\!\left|-\frac{3}{2}\right\rangle\right\\}$.
The gate potential gives
$\left\\{n_{z}{\xrightarrow{eF_{z}z}}n_{z}{\pm}1\right\\}$ which is spin
conserving. The Luttinger terms $L(\mathbf{k}),\,M(\mathbf{k})$ couple HH and
LH states e.g.
$\left\\{\left|n_{x},n_{y},n_{z},\frac{3}{2}\right\rangle\\!{\xrightarrow[L(\mathbf{k})]{k_{x}k_{z}}}\\!\left|n_{x}\pm
1,n_{y},n_{z}\pm 1,\frac{3}{2}\right\rangle\right\\}$. We consider the
complete SO picture in the Ge hole QD following eqn. 5 to sketch a few example
paths as follows:

  * •

$\left|0,0,0,\frac{3}{2}\right\rangle\\!{\xrightarrow[L]{B_{x}}}\\!\left|0,0,0,\frac{1}{2}\right\rangle\\!{\xrightarrow[L]{k_{y}k_{z}}}\\!\left|0,1,1,\frac{3}{2}\right\rangle\\!{\xrightarrow[M]{k_{x}k_{y}}}$
$\left|1,2,1,-\frac{1}{2}\right\rangle\\!{\xrightarrow{eF_{z}z}}\\!\left|1,2,0,-\frac{1}{2}\right\rangle\\!{\xrightarrow{eE_{x}x}}\\!\left|0,2,0,-\frac{1}{2}\right\rangle\\!{\xrightarrow[J\rightarrow
J]{k_{y}^{2}}}$
$\left|0,0,0,-\frac{1}{2}\right\rangle\\!{\xrightarrow[L]{B_{x}}}\\!\left|0,0,0,-\frac{3}{2}\right\rangle$.

  * •

$\left|0,0,0,\frac{3}{2}\right\rangle\\!{\xrightarrow[L]{k_{y}k_{z}}}\\!\left|0,1,1,\frac{1}{2}\right\rangle\\!{\xrightarrow{eE_{x}x}}\\!\left|1,1,1,\frac{1}{2}\right\rangle\\!{\xrightarrow[J\rightarrow
J]{B_{x}yk_{z}}}$
$\left|1,0,2,\frac{1}{2}\right\rangle\\!{\xrightarrow{(eF_{z}z)^{2}}}\\!\left|1,0,0,\frac{1}{2}\right\rangle\\!{\xrightarrow[M]{k_{x}k_{y}}}\\!\left|0,1,0,-\frac{3}{2}\right\rangle\\!{\xrightarrow[J\rightarrow
J]{B_{x}yk_{z}}}$
$\left|0,0,1,-\frac{3}{2}\right\rangle\\!{\xrightarrow{eF_{z}z}}\\!\left|0,0,0,-\frac{3}{2}\right\rangle$.

These paths explain the quadratic $B_{x}^{2}$ dependence at large $F_{z}$ in
the presence of the ac electric field.

#### Phonon mediated Relaxation.

The relaxation mechanism in Ge hole QD qubits is well explained by acoustic
phonon coupling to the hole spins through the valence band deformation
potential $\mathcal{D}_{i,j}$ of Ge. There are no piezo-electric phonons in
Ge, but the hole spin interacts with the thermal bath of the bulk phonons via
the hole-phonon Hamiltonian $H_{h-ph}$, where $\alpha\in\\{l,t,w\\}$ denotes
the polarization directions of the phonon and $\mathbf{q}$ is the phonon wave
vector. We write
$H_{h-ph}{=}\sum_{i,j}\mathcal{D}_{i,j}\varepsilon^{\alpha}_{i,j}(\bm{r})$
with
$\varepsilon^{\alpha}_{i,j}(\bm{r})=\sum_{q}\frac{i}{2}\sqrt{\frac{\hbar}{2\rho
NV\omega_{\mathbf{q},\alpha}}}\left(q_{i}\hat{c}_{j}+q_{j}\hat{c}_{i}\right)e^{i\bm{q}\cdot\bm{r}}\sqrt{N_{q}^{\alpha}+1}$.
The relaxation rate $1/T_{1}$ is written as:

$\frac{1}{T_{1}}=\frac{2\pi}{\hbar}\sum_{\alpha,\mathbf{q}}\left|\left\langle\mathbb{0}\right|H_{h-ph}\left|\mathbb{1}\right\rangle_{\alpha}\right|^{2}\delta\left(\Delta\mathbb{E}-\hbar\omega_{\alpha,\mathbf{q}}\right)$
(9)

(a)

(b)

Figure 5: Qubit dephasing in an in-plane magnetic field: a) Variation of qubit
dephasing time $T_{2,\text{RTN}}^{*}$ (ms) due to random telegraph noise (RTN)
from a nearby in-plane single charge defect 200 nm away from the qubit w.r.t.
top gate field $F_{z}$ (MV/m). b) Qubit Zeeman splitting $\Delta
E\,(\mu\text{eV})$ vs. applied top-gate field $F_{z}$ (MV/m). The applied
magnetic field is in the plane with the magnitude $B_{x}{=}0.67\,\text{T}$.
The dephasing time reaches a minimum at $F_{z}{=}18\,\text{MV/m}$.

We calculate the relaxation rate using our semi-analytical 3D QD model by
computing in real space the overlap integral
$\left\langle\mathbb{0}\right|H_{h-ph}\left|\mathbb{1}\right\rangle_{\alpha}$
due to position-dependent local strain, followed by the scattering integral in
the phonon wave vector $q$-space for a specific polarization direction. The
full analytical integrations and dipole approximation calculations are
detailed in Appendix. C. Figure 4a shows the nonlinear variation of the
relaxation rate $T_{1}^{-1}$ w.r.t. the external magnetic field $B_{x}$, with
$B^{3}$, $B^{4}$ and $B^{5}$ terms present in the fitting obtained from the
full 3D numerical model. While the $B^{3}$ and $B^{5}$ dependence are
explained from the first two terms in the dipole approximation, the $B^{4}$
term is understood from the orbital $B$-admixture. A minimum relaxation time
$T_{1}\sim 80\,\text{ms}$ at an in-plane magnetic field
$B_{x}{=}0.67\,\text{T}$ is obtained. This result from the theory compares
well with a single hole relaxation time measurement in Ge of over 30 ms and a
five-hole relaxation time of approximately 1 ms by Lawrie et al.Lawrie  _et
al._ (2020) The magnetic field in the experimental setupHendrickx  _et al._
(2020a); Lawrie  _et al._ (2020) is $B{=}0.67\,\text{T}$, as also used in Fig.
4a. There exists a minimum in $T_{1}$ in the range of $F_{z}$ considered here
for $B_{x}{=}0.67\,\text{T}$ (figure 4b). At this minimum the Rabi ratio
$T_{1}/T_{\pi}{\approx}2\times 10^{5}$, where $T_{\pi}$ is the time required
for an EDSR $\pi$-rotation, demonstrating that fast Rabi oscillations can be
achieved without sacrificing $T_{1}$.

(a)

(b)

Figure 6: Qubit dephasing in an out-of-plane magnetic field: a) Variation of
qubit dephasing time $T_{2,\text{RTN}}^{*}$ (ms) due to random telegraph noise
(RTN) from a nearby in-plane single charge defect 200 nm away from the qubit
w.r.t. top gate field $F_{z}$ (MV/m). b) Qubit Zeeman splitting $\Delta
E\,(\mu\text{eV})$ vs. applied top-gate field $F_{z}$ (MV/m). The applied
magnetic field is in the plane with the magnitude $B_{x}{=}0.1\,\text{T}$. In
this case, the dephasing time exhibits a maximum in top-gate field.

#### Random Telegraph Noise (RTN) Dephasing.

The large spin-orbit coupling exposes the hole spin qubit to charge noise. The
dephasing time $T^{*}_{2,\text{RTN}}$ is evaluated from the fluctuation in the
qubit energy gap, denoted by $\delta\omega$, caused by the screened potential
$U_{s}(\bm{r})$ of a nearby single charge defect in the 2DHG.Maman  _et al._
(2020); Culcer  _et al._ (2009); Ramon and Cywiński (2022); Roszak  _et al._
(2019); Malkoc  _et al._ (2022) The mathematical formulation of
$U_{s}(\bm{r})$ is given in Appendix. D. The matrix elements $\left\langle
n,m,l|U_{s}(\bm{r})|n^{\prime},m^{\prime},l^{\prime}\right\rangle$ are added
to the full Hamiltonian, and the $3300\times 3300$ matrix is diagonalised to
evaluate the qubit energy splitting in the presence of the charge defect as
$\Delta\mathbb{E}+\delta\omega$. The dephasing rate is:

$\left(T^{*}_{2,\text{RTN}}\right)^{-1}{=}\frac{(\delta\omega)^{2}\tau}{2\hbar^{2}}$
(10)

where the defect switching time is taken as $\tau{=}10^{3}\,t_{Rabi}$. This
picture assumes the most significant contribution to RTN comes from charge
defects away from the top gate, close to the qubit plane; hence we consider a
single charge defect in the qubit plane situated $200$ nm away from the center
of the qubit. Fluctuating single charge defects right above the qubit will be
screened by the presence of the top-gate, where the image charge changes the
interaction to a much weaker dipole interaction. In contrast fluctuating
charges in the plane of the quantum well are less effectively screened by
surface gates, and may be the dominant source of charge noise. Culcer and
Zimmerman (2013)

It is evident that for hole qubit in an in-plane magnetic field, the dephasing
time $T_{2,RTN}^{*}$ actually decreases as a function of the top gate electric
field $F_{z}$, and reaches a minimum at a certain value of this field (Fig.
5a), in other words, a coherence hot spot. The location of this dephasing time
hotspot is closely related to the extremum in the qubit Zeeman splitting (Fig.
5b). This behaviour is in sharp contrast to hole spin qubits in a
perpendicular magnetic field, where the qubit exhibits a sweet spot at a
certain value of the top gate field (Fig. 6a), at which its sensitivity to
noise vanishes to leading order in the noise strength, and dephasing time
$T_{2,RTN}^{*}$ reaches a maximum. The location of the sweet spot in $F_{z}$
for out-of-plane qubit operation is closely related to an extremum in the
qubit Zeeman splitting (Fig. 6b).

####  $B_{\perp}$ vs. $B_{\parallel}$ coherent qubit operation.

In the context of qubit coherence one must distinguish between extrema in the
qubit Zeeman splitting and actual sweet spots in the coherence time. It is
important to recall that in a spin qubit the dephasing time $T_{2}^{*}$
depends on the magnitude of the magnetic field. This follows from time-
reversal symmetry considerations, since the combination of charge noise and
spin-orbit coupling cannot give rise to an energy difference between qubit
states that form a Kramers doublet. The magnetic field dependence involves
both the Zeeman terms and the orbital vector potential terms, a fact that is
responsible for the main difference between in-plane and out-of-plane magnetic
fields with regard to qubit dynamics: the make up of the ground and first
excited states is very different when the magnetic field is in the plane and
when it is out of the plane.

For an out of plane magnetic field the hole $g$-factor is large, having a
textbook value of 20.4 for Ge.Winkler (2003); Lu  _et al._ (2017) With the
magnetic field out of the plane one can understand the physics qualitatively
by considering an approximate decomposition of in-plane and out-of-plane
dynamics by means of a Schrieffer-Wolff transformation Wang  _et al._ (2021)
The picture that emerges is that the top gate electric field primarily affects
spin dynamics in the plane by enabling a Rashba term. In a quantum dot this
Rashba term is responsible for a renormalization of the $g$-factor. In other
words, one can think of the magnetic field terms as providing the qubit Zeeman
splitting, and the Rashba spin-orbit terms as renormalizing this Zeeman
splitting. Background charge fluctuations generating an electric field
perpendicular to the plane are the biggest danger for this qubit, because they
directly affect the Rashba interaction and through it the $g$-factor,
generating pure dephasing. A more detailed analysis of hole spin qubit in
$B_{\perp}$ Wang  _et al._ (2021) reveals that in-plane charge fluctuations do
not produce pure dephasing to leading order. Hence for $B_{\perp}$ operation,
when the qubit Zeeman splitting is at an extremum with respect to the top gate
electric field (Fig. 6b), the qubit is protected against noise and one also
observes a sweet spot in $T_{2}^{*}$ in the vicinity of this point (Fig. 6a).

On the other hand, for hole spin qubit in $B_{\parallel}$, we recall that to a
first approximation the in-plane $g$-factor is zero, hence the entire qubit
Zeeman splitting is given by coupling to the excited states. This coupling
involves Luttinger spin-orbit terms, the orbital magnetic field terms, the top
gate electric field, and any other electric fields present in the system. The
orbital terms due to the magnetic field mix the in-plane and out-of-plane
coordinates regardless of the gauge choice. There is no clear separation
between in-plane and out-of-plane dynamics, and no suitable Schrieffer-Wolff
transformation from the 3D picture to the asymptotic 2D limit. One may at best
envisage a combined Rashba-Zeeman interaction with contributions from all the
components of the electric field, not just the top gate. The qubit states
contain a strong admixture of all the higher orbital excited states in all
three directions, which exposes the qubit to all components of the electric
field of the defect. Thus, even though one can still identify an extremum in
the qubit Zeeman splitting as a function of the top gate field (Fig. 5b), this
does not offer protection against noise and does not constitute a sweet spot
(Fig. 5a). It only protects against noise fields perpendicular to the plane,
without offering any protection against the in-plane electric field of a
defect. We check explicitly that for a defect that produces only an out-of-
plane electric field at the qubit location the sensitivity to this out-of-
plane noise is minimised at the extremum in the Zeeman splitting. We have also
checked that the qubit is not shielded from the in-plane electric field of the
defect at this extremum: there is nothing special about the extremum from this
perspective. We note that in an experimental sample exposed to an ensemble of
defects it is possible for the net in-plane electric field to cancel out, or
nearly cancel out. Hence, to achieve a more complete understanding of
coherence, it is vital to consider a realistic configuration leading to $1/f$
noise. In light of this, and of additional complexities identified recently in
modelling hole spin coherence,Shalak  _et al._ (2023) we defer the full theory
of hole spin coherence in the presence of $1/f$ noise to a future publication.

We note that our findings appear to agree with recent experimental work
reporting sweet spot operation of a Ge hole spin qubit Hendrickx  _et al._
(2023) as well as strong anisotropy in the noise sensitivity. Sensitivity to
charge noise is found to increase significantly when the qubit is operated in
an in-plane magnetic field. This is in agreement with the finding of the
present paper that in-plane magnetic fields expose the qubit to noise much
more strongly than out-of-plane magnetic fields, leading to the coherence hot
spot seen in Fig. 5a. Remarkably, the dominant source of noise in Ref.
Hendrickx  _et al._ , 2023 is believed to lie directly above the qubit,
implying charge fluctuations predominantly in the perpendicular electric field
component, and suggesting the qubit was not operated in the sweet spot for
out-of-plane charge fluctuations. Nevertheless, a full comparison between
theory and experiment is premature at this stage, given that tilting of the
$g$-tensor and local strain have not been considered in the present work.

(a)

(b)

(c)

Figure 7: a)Variation of the EDSR Rabi frequency $f_{\text{EDSR}}$ with the
aspect ratio $L_{y}/L_{x}$ of the hole QD. The confinement in x is fixed at
$L_{x}{=}50\text{nm}$ while $L_{y}$ is varied. The calculation is done for
$B_{x}{=}0.7\text{T}$. The EDSR driving field is applied along the
$\hat{x}$-direction. $f_{\text{EDSR}}$ is a minimum when the dot is circular,
i.e. $L_{x}{=}L_{y}{=}50\text{nm}$, while elliptical dots show an improvement
of $>\\!1$ order of magnitude in the EDSR spin-flip rate at high
ellipticities. b) Linear monotonic dependence of $f_{\text{EDSR}}$ on the
applied in-plane magnetic field $B_{x}$ for an elliptical dot with
$L_{y}/L_{x}{=}2,\,L_{x}{=}50\text{nm}$ for three different values of the top
gate potential: $F_{z}{=}2\,\text{MV/m}$, $F_{z}{=}18\,\text{MV/m}$ and
$F_{z}{=}35\,\text{MV/m}$. c) Variation in $f_{\text{EDSR}}$ with the top gate
electric field for an elliptical dot at low ($0.1\,\text{T}$) and high
($2\,\text{T}$) in-plane magnetic fields, displaying a maximum EDSR Rabi
frequency at $F_{z}{=}18\,\text{MV/m}$ and $F_{z}{=}21\,\text{MV/m}$,
respectively.

##  IV Elliptical Quantum dot

Introducing asymmetry into the planar confinement, i.e. having one lateral
confinement potential stronger than the other will bring in additional sources
of structure inversion asymmetry (SIA). For such elliptical hole QDs, the
resultant Rashba spin-orbit interaction is stronger, bridging the gap between
planar QD and nanowires Bosco  _et al._ (2021b) in terms of fast gate
operations. A theoretical understanding of QD ellipticity for holes is thus
important. Insight into our numerical results can be obtained from the
effective $2{\times}2$ spin-orbit Hamiltonian in Eqn. 8, which we rewrite as:

$\displaystyle H_{SO}$ $\displaystyle{=}$
$\displaystyle\alpha_{R2}\\!\left[k_{+}^{3}\sigma_{-}{-}k_{-}^{3}\sigma_{+}\right]\\!+\\!\alpha_{R3}\\!\left[\\{k_{+}^{2},k_{-}\\}\sigma_{+}{-}\\{k_{+},k_{-}^{2}\\}\sigma_{-}\right]$
(11) $\displaystyle\text{where,}\ \,\,\
\alpha_{R2}{=}\frac{e\hbar^{4}}{m_{0}^{2}}\gamma_{3}\overline{\gamma}S_{hh};\
\,\alpha_{R3}{=}\frac{e\hbar^{4}}{m_{0}^{2}}\gamma_{3}\delta S_{hh};$

with the spherical Luttinger parameter
$\overline{\gamma}{=}(\gamma_{2}+\gamma_{3})/2$, the cubic-symmetry parameter
$\overline{\delta}{=}(\gamma_{3}-\gamma_{2})/2$, and $S_{hh}$ the sub-band
interaction in $3^{\text{rd}}$ order perturbation theory (Appendix. B). In a
circular dot the cubic-symmetry correction $\alpha_{R3}$ is responsible for
EDSR, while the spherical term $\alpha_{R2}$ does not contribute. In contrast,
in an elliptical dot the $\alpha_{R2}$ term is nonzero. From Eqn. 11,
$\alpha_{R3}=(\delta/\overline{\gamma})\,\alpha_{R2}$; using the lattice
parameters of Ge we evaluate $\alpha_{2}{\approx}10\,\alpha_{3}$. We present
the results for a dot size of $L_{z}{=}11\,\text{nm}$,
$L_{x}{=}50\,\text{nm}$, and varying $L_{y}$. Fig. 7a shows the variation of
the EDSR Rabi frequency with the aspect ratio $L_{x}/L_{y}$, showing that an
increase in the aspect ratio results in a larger Rabi frequency. The qubit
Zeeman splitting is linear in the applied in-plane magnetic field, similar to
the circular case. The relaxation rate varies as $B^{3}$. The EDSR Rabi
frequency is linear in $B$ (Fig. 7b), which is reminiscent of out-of-plane
$B$-field operation in presence of strong SIA; and the Rabi frequency exhibits
a maximum as a function of $F_{z}$ (Fig. 7c).

##  V $\bm{g}$-factor anisotropy of elliptical QD and comparison with
experiment

The in-plane $g$-factor of an elliptical dot is strongly anisotropic and
exhibits an oscillatory behaviour as the magnetic field is rotated in the
plane. We compare the predicted variation of the $g$-factor with
experimentally measured values from EDSR in a planar germanium hole qubit. The
qubit sample is a gate-defined double quantum dot in a Ge/Si0.2Ge0.8
heterostructure. (See Ref. 64 for further sample info). The dots are assumed
to be elliptical, with a slight misalignment of their semi-major axes.

Figure 8: a) Sample scanning electron microscope (SEM) image of a gate-defined
Ge double dot. b), c) Bias triangles (dashed) in forward and reverse bias
respectively. Pauli spin blockade is visible in the transition marked by a
yellow circle. d) g-factors measured via EDSR for the magnetic field direction
indicated in part a) (black solid arrow). e) Comparison of the g-factor
measured via EDSR with the theoretical prediction. The experimental results
for $g$-factor anisotropy in Ref.115 are presented as scatter plots of
effective $g$-factor ($g^{*}$) vs. the in-plane magnetic field angular
orientation w.r.t. the double dot transport direction ($\theta$). We use
triangular markers to denote the experimental data in scatter plots, while the
solid fit lines are generated using the theoretical model in Sec. II. The
$g$-factor for the left dot is denoted by $g_{L}$ (empty blue triangular
experimental data points, solid red theoretical fit line), while the right dot
$g$-factor is denoted by $g_{R}$ (filled yellow triangular experimental data
points, solid purple theoretical fit line).

Fig. 8a shows a false colour SEM of the gate design. Plunger gates (purple)
are used to define the two dots, while the barrier gates (green) are used to
control coupling to the leads and between the dots. Metal ohmic contacts
(yellow) act as a reservoir for holes. By negatively biasing the barrier and
plunger gates, the sample can be tuned to the few hole regime. The relative
angle between the applied magnetic field direction in the plane and the double
dot transport direction is denoted by $\theta$. Bias triangles of the double
dot measured via transport for positive and negative bias are shown in Fig. 8b
and Fig. 8c. A region of Pauli spin blockade is visible at the base of one
charge transition as indicated by a yellow circle. By applying an external
magnetic field and a microwave tone to the P2 gate, we are able to drive spin
rotations via EDSR when the microwave frequency matches the Larmor frequency
($\hbar f{=}g\mu_{B}B$). These spin rotations lift the Pauli spin blockade,
causing a change in the current through the double dot. Using a lock-in
amplifier, we measure the difference in current through the double dot when
the microwave is on vs off. Fig. 8d shows the change in the leakage current
for the double quantum dot with the external magnetic field applied in the
direction indicated in Fig. 8a. Clear EDSR lines are visible for both dots,
and both single and multi photon lines can be seen. From the slope of these
resonance lines the g-factor can be calculated for each dot. Using this
technique, we measure the g-factor as a function of field angle by rotating a
magnetic field of $B{=}0.7\,\text{T}$ in the 2D plane. Fig. 8e shows the
results of this measurement for both quantum dots, revealing an oscillatory
variation in the g-factor as a function of in-plane magnetic field angle. The
direction of $\theta$ is shown in Fig. 8a.

Using the model developed in Sec. II, we fit to the experimental data. For
both dots we use the same size, shape and strain. We are able to account for
the difference in g-factor between the dots by considering only a rotation of
the dot axes in-plane and a change in the magnitude of the vertical electric
field. A full list of fitting parameters is given in Table 1. The maximum
value of the g-factor is not aligned with the external magnetic field or
sample axes, and is also different for the left and right dots. To account for
this, we introduce a phase shift angle ($\theta_{ps}$) which effectively
rotates the axes of the quantum dots. Here $\theta_{ps,l}{=}\frac{3\pi}{4}$
and $\theta_{ps,r}{=}\frac{5\pi}{8}$ The magnitude of the g-factor is also
different for each dot. This is accounted for by changing the vertical
electric field applied to each dot. Here we use 10MV/m for the left dot and
45MV/m for the right dot. The results of the fits for both dots are shown by
the solid lines in Fig 8 e).

Fitting Parameters | values
---|---
Left QD in-plane dimensions | $40\,\text{nm}{\times}30\,\text{nm}$
Right QD in-plane dimensions | $40\,\text{nm}{\times}30\,\text{nm}$
Left dot perpendicular confinement ($L_{z,l}$) | $10\,\text{nm}$
Right dot perpendicular confinement ($L_{z,r}$) | $10\,\text{nm}$
Left dot Top-gate voltage ($F_{z,l}$) | $10\,\text{MV/m}$
Right dot Top-gate voltage ($F_{z,r}$) | $45\,\text{MV/m}$
Left dot phase shift $\theta_{ps,l}$ | $\frac{3\pi}{4}$
Right dot phase shift $\theta_{ps,r}$ | $\frac{5\pi}{8}$
Left dot uni-axial compressive strain $\varepsilon_{xx,l}$ | $-0.006$
Right dot uni-axial compressive strain $\varepsilon_{xx,r}$ | $-0.006$
Applied magnetic field magnitude $B$ | $0.7\,\text{T}$
Table 1: List of the parameters used to fit the analytical model to the
experimental $g$-factor anisotropy data in Fig. 8e. While the choices of phase
shift angles ($\theta_{ps}$) explain the experimental artifact of relative
misalignment of the two dots; the top-gate voltages ($F_{z}$) are tuned to
match the $g$-factor oscillation amplitudes of the dots. We stress however
that the choice of fitting parameters given in this table for Fig. 8e is not
unique. A possible set of parameters that yields a similar theoretical fitting
of the experimental data in Fig. 8e are listed in appendix E.

The theoretical fit in Fig. 8e shows good agreement between the phase shift
angle $\theta_{ps}$ parameter choices and B-field angular orientations where
the experimental $g$-factors are maximum. In other words, the largest value of
the $g$-factor occurs when the magnetic field is parallel to the semi-major
axis of the elliptical hole QD. This behaviour is consistent with the
effective in-plane $g$-factor being primarily the result of coupling to higher
excited states brought about by the orbital magnetic field terms. We note that
inhomogeneous strain in the sample, or the Ge/SiGe heterostructure interface
induced roughness and disorder, or a misalignment of the sample with respect
to the in-plane B-field (since $g_{\perp}\gg g_{\parallel}$) could potentially
lead to significant modulation of the $g$-tensor. We can rule out the latter,
since there is a different phase shift for the left and right dots in Fig. 8e.
The effects of strain and inhomogeneities on $g$-factor anisotropy will be
considered in a future publication.

##  VI Conclusions and Outlook

We have presented a generalised, semi-analytical model that fully describes
the electrical operation of a planar germanium hole qubit in presence of an
in-plane magnetic field. A comprehensive theory for spin manipulation via
electron dipole spin resonance (EDSR) is given: surface inversion asymmetry
(SIA) mediated fast EDSR is a result of the $\mathbf{k}\cdot\mathbf{p}$
coupling of the heavy hole ground state to higher energy light hole bands. The
EDSR rate is linear in $B$ with important nonlinear corrections due to orbital
mixing. Qubit relaxation is induced by acoustic phonons and the relaxation
rate $1/T_{1}$ has terms with $B^{3},\,B^{4},\,B^{5}$ dependence, again
reflecting the importance of the orbital mixing. In-plane operation
demonstrates an excellent trade-off between relaxation and EDSR. The in-plane
$g$-factor is strongly anisotropic and oscillates as the magnetic field is
rotated in the plane. Random telegraph noise from charges in the plane of the
quantum well results in decoherence, with an optimal top gate potential where
it is insensitive to $\Delta F_{z}$; although the in-plane magnetic field
exposes the qubit to $x-y$ electric field of the fluctuator. Hence, in
contrast to the case of out-of-plane magnetic fields, coherence sweet spots
cannot be identified in an in-plane $B$ for a qubit exposed to electric field
fluctuations in all spatial directions. For an elliptical QD of aspect ratio
$L_{y}/L_{x}{=}2$, EDSR is shown to be faster by an order of magnitude
compared to a circular dot of $L_{x}{=}50\,\text{nm}$ radius; and the non-
linear correction to EDSR is suppressed as rotational asymmetry induces more
SIA Rashba.

Acknowledgments. This project is supported by the Australian Research Council
Centre of Excellence in Future Low-Energy Electronics Technologies (project
number CE170100039) and Discovery Project DP200100147. We acknowledge
stimulating discussions with S. Das Sarma, M. Russ, X. Hu, and S. Liles.

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##  Appendix A Gauge Invariance

The hole motion in the topmost valence band is described by the $4{\times}4$
Luttinger-Kohn (LK) Hamiltonian, which in the general operator form is given
by:

$\displaystyle H_{LK}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}}\left[\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right)k^{2}-2\gamma_{2}(k_{x}^{2}J_{x}^{2}+k_{y}^{2}J_{y}^{2}+k_{z}^{2}J_{z}^{2})-4\gamma_{3}\left(\\{k_{x},k_{y}\\}\\{J_{x},J_{y}\\}+c.p.\right)\right]$
(A12)

Expanding the anti-commutators, LK Hamiltonian has the form:

$\displaystyle H_{LK}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}}\left[\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right)(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})-2\gamma_{2}(k_{x}^{2}J_{x}^{2}+k_{y}^{2}J_{y}^{2}+k_{z}^{2}J_{z}^{2})-4\gamma_{3}\left(\left(\frac{k_{x}k_{y}+k_{y}k_{x}}{2}\right)\left(\frac{J_{x}J_{y}+J_{y}J_{x}}{2}\right)\right.\right.$
(A13)
$\displaystyle+\left(\frac{k_{y}k_{z}+k_{z}k_{y}}{2}\right)\left(\frac{J_{y}J_{z}+J_{z}J_{y}}{2}\right)\left.\left.+\left(\frac{k_{x}k_{z}+k_{z}k_{x}}{2}\right)\left(\frac{J_{x}J_{z}+J_{z}J_{x}}{2}\right)\right)\right]$

with $m_{0}$ notifying bare electron mass, $\gamma_{1}{=}13.38$,
$\gamma_{2}{=}4.24$ and $\gamma_{3}{=}5.69$ are Luttinger parameters. We have
tried two different gauges: $\frac{1}{2}\mathbf{B}\times\mathbf{r}$ and gauge
choice from Loss et al.Kloeffel  _et al._ (2018)

###  A.1 Gauge 1:

In presence of magnetic field, the momentum correction would be:
$\mathbf{k}\rightarrow\left(\mathbf{k}+\frac{e\mathbf{A}}{\hbar}\right)$. We
use the gauge (symmetric in z, Landau in x-y)
$\mathbf{A}=-\frac{1}{2}B_{z}y\mathbf{\hat{e}_{x}}+\frac{1}{2}B_{z}x\mathbf{\hat{e}_{y}}+(B_{x}y-B_{y}x)\mathbf{\hat{e}_{z}}$.

Momentum Correction:

  1. 1.

The components of corrected momentum:

$\displaystyle k_{x}$ $\displaystyle\rightarrow$
$\displaystyle\left(k_{x}-\frac{eB_{z}}{2\hbar}y\right)\,,\,k_{y}\rightarrow\left(k_{y}+\frac{eB_{z}}{2\hbar}x\right)\,,\,k_{z}\rightarrow\left(k_{z}+\frac{eB_{x}}{\hbar}y-\frac{eB_{y}}{\hbar}x\right)$

where, $\mathbf{k}=-\iota\mathbf{\partial}$. For $B_{y}=0,\,B_{z}=0$;
$\text{we write: }\,k_{x}\rightarrow k_{x}\,,\,k_{y}\rightarrow
k_{y}\,,\,k_{z}\rightarrow\left(k_{z}+\frac{eB_{x}}{\hbar}y\right)$

  2. 2.

We evaluate $k_{i}^{2}$ terms below:

$\displaystyle k_{x}^{2}$ $\displaystyle\Rightarrow$ $\displaystyle
k_{x}^{2}\,,\,k_{y}^{2}\Rightarrow
k_{y}^{2}\,,\,k_{z}^{2}\Rightarrow\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)$
(A14)

and the cross-terms $k_{i}k_{j}$ are:

$\displaystyle k_{x}k_{y}\Rightarrow
k_{x}k_{y};\,k_{y}k_{z}\Rightarrow\left(k_{y}k_{z}+\frac{eB_{x}}{\hbar}\\{y,k_{y}\\}\right);\,k_{x}k_{z}\Rightarrow\left(k_{x}k_{z}+\frac{eB_{x}yk_{x}}{\hbar}\right)$
(A15)

  3. 3.

Using these corrections, the LK Hamiltonian is:

$\displaystyle
H_{LK}=\bigg{[}\,\overline{\gamma_{1}+\frac{5\gamma_{2}}{2}}\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)-2\gamma_{2}\left(k_{x}^{2}J_{x}^{2}+k_{y}^{2}J_{y}^{2}+\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)J_{z}^{2}\right)$
$\displaystyle-4\gamma_{3}\left(k_{x}k_{y}\\{J_{x},J_{y}\\}+\left(k_{y}k_{z}+\frac{eB_{x}}{\hbar}\\{y,k_{y}\\}\right)\\{J_{y},J_{z}\\}+\left(k_{x}k_{z}+\frac{eB_{x}yk_{x}}{\hbar}\right)\\{J_{z},J_{x}\\}\right)\bigg{]}\frac{\hbar^{2}}{2m_{0}}$
(A16)

where,

$\displaystyle H_{LK}^{11/22}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}}\bigg{[}(\gamma_{1}-2\gamma_{2})\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)+(\gamma_{1}+\gamma_{2})(k_{x}^{2}+k_{y}^{2})\bigg{]}$
$\displaystyle H_{LK}^{33/44}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}}\bigg{[}(\gamma_{1}+2\gamma_{2})\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)+(\gamma_{1}-\gamma_{2})(k_{x}^{2}+k_{y}^{2})\bigg{]}$
$\displaystyle H_{LK}^{13}=L$ $\displaystyle=$
$\displaystyle-\sqrt{3}\frac{\hbar^{2}\gamma_{3}}{m_{0}}\bigg{[}\left(k_{x}k_{z}+\frac{eB_{x}yk_{x}}{\hbar}\right)-i\left(k_{y}k_{z}+\frac{eB_{x}}{\hbar}\\{y,k_{y}\\}\right)\bigg{]}$
$\displaystyle H_{LK}^{14}=M$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}\hbar^{2}}{2m_{0}}\left[-\gamma_{2}\left(k_{x}^{2}-k_{y}^{2}\right)+2i\gamma_{3}k_{x}k_{y}\right]$

Figure A1: EDSR time $T_{\pi}$ (ns) vs. top gate field $F_{z}$ and in-plane
magnetic field $B_{x}$.

###  A.2 Gauge 2:

In this section, we use the symmetric gauge
($\frac{1}{2}\mathbf{B}\times\mathbf{r}$):
$\mathbf{A}=\frac{1}{2}(B_{y}z-B_{z}y)\mathbf{\hat{e}_{x}}+\frac{1}{2}(B_{z}x-B_{x}z)\mathbf{\hat{e}_{y}}+\frac{1}{2}(B_{x}y-B_{y}x)\mathbf{\hat{e}_{z}}$.

Momentum Correction:

  1. 1.

The components of corrected momentum with $B_{y}=0\,,B_{z}=0$:

$\displaystyle k_{x}\rightarrow
k_{x},\,k_{y}\rightarrow\left(k_{y}-\frac{eB_{x}}{2\hbar}z\right)\,,\,k_{z}\rightarrow\left(k_{z}+\frac{eB_{x}}{2\hbar}y\right)\,\text{where,}\,\mathbf{k}=-\iota\partial.$
(A18)

  2. 2.

We evaluate $k_{i}^{2}$ terms below:

$\displaystyle k_{x}^{2}$ $\displaystyle\Rightarrow$ $\displaystyle
k_{x}^{2}\,,\,k_{y}^{2}\Rightarrow\left(k_{y}^{2}-\frac{eB_{x}zk_{y}}{\hbar}+\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}\right)\,,\,k_{z}^{2}\Rightarrow\left(k_{z}^{2}+\frac{eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{4\hbar^{2}}\right)$
(A19)

and the cross-terms $k_{i}k_{j}$ are:

$\displaystyle k_{x}k_{y}$ $\displaystyle\Rightarrow$
$\displaystyle\\!\left(\\!k_{x}k_{y}-\frac{eB_{x}zk_{x}}{2\hbar}\\!\right);\\!k_{y}k_{z}\Rightarrow\\!\left(\\!k_{y}k_{z}+\frac{eB_{x}}{2\hbar}(\\{y,k_{y}\\}-\\{z,k_{z}\\})-\frac{e^{2}B_{x}^{2}}{4\hbar^{2}}yz\right)\\!;\\!k_{x}k_{z}\Rightarrow\left(\\!k_{x}k_{z}+\frac{eB_{x}yk_{x}}{2\hbar}\right)$
(A20)

  3. 3.

Using these corrections, the LK Hamiltonian terms become:

$\displaystyle
H_{LK}=\frac{\hbar^{2}}{2m_{0}}\bigg{[}\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right)\left(k_{x}^{2}+k_{y}^{2}-\frac{eB_{x}zk_{y}}{\hbar}+\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}+k_{z}^{2}+\frac{eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{4\hbar^{2}}\right)$
$\displaystyle-2\gamma_{2}\left(k_{x}^{2}J_{x}^{2}+\left(k_{y}^{2}-\frac{eB_{x}zk_{y}}{\hbar}+\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}\right)J_{y}^{2}+\left(k_{z}^{2}+\frac{eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{4\hbar^{2}}\right)J_{z}^{2}\right)-4\gamma_{3}\left(\left(k_{x}k_{y}-\frac{eB_{x}zk_{x}}{2\hbar}\right)\\{J_{x},J_{y}\\}\right.$
$\displaystyle\left.+\left(k_{y}k_{z}+\frac{eB_{x}}{2\hbar}\\{y,k_{y}\\}-\frac{eB_{x}}{2\hbar}\\{z,k_{z}\\}-\frac{e^{2}B_{x}^{2}}{4\hbar^{2}}yz\right)\\{J_{y},J_{z}\\}+\left(k_{x}k_{z}+\frac{eB_{x}yk_{x}}{2\hbar}\right)\\{J_{z},J_{x}\\}\right)\bigg{]}$
(A21)

where,

$\displaystyle
H_{LK}^{11/22}=\frac{\hbar^{2}}{2m_{0}}\left[(\gamma_{1}-2\gamma_{2})\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)+(\gamma_{1}+\gamma_{2})\left(k_{x}^{2}+k_{y}^{2}-\frac{eB_{x}zk_{y}}{\hbar}+\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}\right)\right]$
(A22) $\displaystyle
H_{LK}^{33/44}=\frac{\hbar^{2}}{2m_{0}}\left[(\gamma_{1}+2\gamma_{2})\left(k_{z}^{2}+\frac{2eB_{x}yk_{z}}{\hbar}+\frac{e^{2}B_{x}^{2}y^{2}}{\hbar^{2}}\right)+(\gamma_{1}-\gamma_{2})\left(k_{x}^{2}+k_{y}^{2}-\frac{eB_{x}zk_{y}}{\hbar}+\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}\right)\right]$
(A23) $\displaystyle
H_{LK}^{13}=L=-\sqrt{3}\frac{\hbar^{2}\gamma_{3}}{m_{0}}\bigg{[}\left(k_{x}k_{z}+\frac{eB_{x}yk_{x}}{2\hbar}\right)-i\left(k_{y}k_{z}+\frac{eB_{x}}{2\hbar}\\{y,k_{y}\\}-\frac{eB_{x}}{2\hbar}\\{z,k_{z}\\}-\frac{e^{2}B_{x}^{2}}{4\hbar^{2}}yz\right)\bigg{]}$
$\displaystyle
H_{LK}^{14}=M=\frac{\sqrt{3}\hbar^{2}}{2m_{0}}\bigg{[}-\gamma_{2}\left(k_{x}^{2}-k_{y}^{2}+\frac{eB_{x}zk_{y}}{\hbar}-\frac{e^{2}B_{x}^{2}z^{2}}{4\hbar^{2}}\right)+2i\gamma_{3}\left(k_{x}k_{y}-\frac{eB_{x}zk_{x}}{2\hbar}\right)\bigg{]}$
(A24)

##  Appendix B Schreiffer-Wolff Transformation

In the effective $2\times 2$ system Hamiltonian given by Eqn. 8,
$H_{0}+V(x,y)$ produces the quantized in-plane energies. We consider a basis
with following in-plane states:

$\left\\{\frac{e^{-\left(\\!\frac{x^{2}}{2L_{x}^{2}}{+}\frac{y^{2}}{2L_{y}^{2}}\right)}}{\sqrt{\pi
L_{x}L_{y}}},\,\frac{\sqrt{2}y\,e^{-\left(\\!\frac{x^{2}}{2L_{x}^{2}}{+}\frac{y^{2}}{2L_{y}^{2}}\right)}}{\sqrt{\pi
L_{y}^{3}L_{x}}},\,\frac{\sqrt{2}x\,e^{-\left(\\!\frac{x^{2}}{2L_{x}^{2}}{+}\frac{y^{2}}{2L_{y}^{2}}\right)}}{\sqrt{\pi
L_{x}^{3}L_{y}}}\right\\}\otimes\left\\{\uparrow,\,\downarrow\right\\}$ (B25)

where the spatial states are the first three harmonic oscillator product
states: $(n,m){=}\\{(0,0),(0,1),(1,0)\\}$, we denote the orbital energies of
these states as $E_{0},\,E_{1},\,E_{1}$. The spinors are the effective spin-up
and spin-down states of the 2D hole qubit, and gives $\pm
g_{\parallel}\mu_{B}B_{x}$ Zeeman energies for the up and down spin states.
The spin-orbit interactions can be listed as:

$H_{SO}{=}i\alpha_{R1}(k_{-}\sigma_{-}{-}k_{+}\sigma_{+})\\!+i\alpha_{R2}\left(k_{+}^{3}\sigma_{-}-k_{-}^{3}\sigma_{+}\right)\\!+i\alpha_{R3}\left(\\{k_{+},k_{-}^{2}\\}\sigma_{-}\\!-\\!\\{k_{+}^{2},k_{-}\\}\sigma_{+}\right)\\!+i\alpha_{R4}\left(\\{k_{z}^{2},k_{-}\\}\sigma_{-}{-}\\{k_{z}^{2},k_{+}\\}\sigma_{+}\right)$
(B26)

The $k$-linear Rashba terms come from the coupling of the bonding VB
$p$-orbitals to the antibonding CB $p$-orbitals, which is very small. In the
Luttinger formalism thus the $\alpha_{R1},\,\alpha_{R4}$ terms vanish. The
nonzero contributions are:

$H_{SO}{=}i\alpha_{R2}\left(k_{+}^{3}\sigma_{-}-k_{-}^{3}\sigma_{+}\right)\\!+i\alpha_{R3}\left(\\{k_{+},k_{-}^{2}\\}\sigma_{-}\\!-\\!\\{k_{+}^{2},k_{-}\\}\sigma_{+}\right)$
(B27)

the spin-orbit matrix elements are calculated below:

$\displaystyle\left\langle(0,0)\uparrow\right|H_{SO}\left|(0,1)\downarrow\right\rangle$
$\displaystyle{=}$
$\displaystyle-i\alpha_{R2}\left\langle(0,0)\uparrow\right|k_{-}^{3}\sigma_{+}\left|(0,1)\downarrow\right\rangle+i\alpha_{R3}\left\langle(0,0)\uparrow\right|k_{+}k_{-}k_{+}\sigma_{+}\left|(0,1)\downarrow\right\rangle$
(B28) $\displaystyle{=}$
$\displaystyle-i\alpha_{R2}\left\langle(0,0)\right|k_{-}^{3}\left|(0,1)\right\rangle+i\alpha_{R3}\left\langle(0,0)\right|k_{+}k_{-}k_{+}\left|(0,1)\right\rangle$
$\displaystyle{=}$
$\displaystyle\frac{-3i\alpha_{R2}}{2\sqrt{2}L_{y}^{3}}\left(1-\frac{L_{y}^{2}}{L_{x}^{2}}\right)+\frac{i\alpha_{R3}}{2\sqrt{2}L_{y}^{3}}\left(1+3\frac{L_{y}^{2}}{L_{x}^{2}}\right)=R_{c}$
$\displaystyle\left\langle(0,0)\uparrow\right|H_{SO}\left|(1,0)\downarrow\right\rangle$
$\displaystyle{=}$
$\displaystyle-i\alpha_{R2}\left\langle(0,0)\uparrow\right|k_{-}^{3}\sigma_{+}\left|(1,0)\downarrow\right\rangle+i\alpha_{R3}\left\langle(0,0)\uparrow\right|k_{+}k_{-}k_{+}\sigma_{+}\left|(1,0)\downarrow\right\rangle$
(B29) $\displaystyle{=}$
$\displaystyle-i\alpha_{R2}\left\langle(0,0)\right|k_{-}^{3}\left|(0,1)\right\rangle+i\alpha_{R3}\left\langle(0,0)\right|k_{+}k_{-}k_{+}\left|(0,1)\right\rangle$
$\displaystyle{=}$
$\displaystyle\frac{-3\alpha_{R2}}{2\sqrt{2}L_{x}L_{y}^{2}}\left(1-\frac{L_{y}^{2}}{L_{x}^{2}}\right)+\frac{\alpha_{R3}}{2\sqrt{2}L_{x}L_{y}^{2}}\left(1+3\frac{L_{y}^{2}}{L_{x}^{2}}\right)=R_{r}$

The ac electric field $eE_{x}(t)x$ is spin-conserving and connects the states
$(0,0)$ and $(1,0)$. Putting all of it together, the $6\times 6$ Hamiltonian
would be:

$H^{\prime}=\begin{pmatrix}E_{0}-\frac{1}{2}g_{\parallel}\mu_{B}B_{x}&0&0&R_{c}&eE_{x}a&R_{r}\\\
0&E_{0}+\frac{1}{2}g_{\parallel}\mu_{B}B_{x}&R_{c}^{*}&0&-R_{r}&eE_{x}a\\\
0&R_{c}&E_{1}-\frac{1}{2}g_{\parallel}\mu_{B}B_{x}&0&0&0\\\
R_{c}^{*}&0&0&E_{1}+\frac{1}{2}g_{\parallel}\mu_{B}B_{x}&0&0\\\
eE_{x}a&-R_{r}&0&0&E_{1}-\frac{1}{2}g_{\parallel}\mu_{B}B_{x}&0\\\
R_{r}&eE_{x}a&0&0&0&E_{1}+\frac{1}{2}g_{\parallel}\mu_{B}B_{x}\end{pmatrix}$
(B30)

$2^{\text{nd}}$ order perturbation theory gives the EDSR matrix element as:

$\displaystyle\tilde{H}_{12}$ $\displaystyle=$
$\displaystyle\sum_{j=3}^{6}\frac{1}{2}H^{\prime}_{1j}H^{\prime}_{j2}\left(\frac{1}{\epsilon_{1}-\epsilon_{j}}+\frac{1}{\epsilon_{2}-\epsilon_{j}}\right)=\frac{1}{2}H^{\prime}_{15}H^{\prime}_{52}\left(\frac{1}{\epsilon_{1}-\epsilon_{5}}+\frac{1}{\epsilon_{2}-\epsilon_{5}}\right)+\frac{1}{2}H^{\prime}_{16}H^{\prime}_{62}\left(\frac{1}{\epsilon_{1}-\epsilon_{6}}+\frac{1}{\epsilon_{2}-\epsilon_{6}}\right)$
(B31) $\displaystyle=$
$\displaystyle-\frac{1}{2}eE_{x}a\,R_{r}\left(\frac{1}{-\Delta_{01}}+\frac{1}{-\Delta_{01}+Z}\right)+\frac{1}{2}eE_{x}a\,R_{r}\left(\frac{1}{-\Delta_{01}-Z}+\frac{1}{-\Delta_{01}}\right)$
$\displaystyle=$ $\displaystyle
eE_{x}a\,R_{r}\left(\frac{1}{\Delta_{01}\left(1-\frac{Z}{\Delta_{01}}\right)}-\frac{1}{\Delta_{01}\left(1+\frac{Z}{\Delta_{01}}\right)}\right)=(eE_{x}a\,R_{r}Z)/\Delta_{01}^{2}$

Where we have used $Z=g_{\parallel}\mu_{B}B_{x}$, $\Delta_{01}=E_{1}-E_{0}$.
This gives $\tilde{H}_{12}{\propto}B_{x}$, explaining the linear magnetic
field dependence of EDSR.

From Eqn. B28, B29, the $\alpha_{2}$ term vanishes when $L_{x}=L_{y}$ i.e. for
a symmetric dot, while it is strongly nonzero for an elliptical dot; making
the linear $B_{x}$-dependence stronger.

##  Appendix C Bulk Phonon mediated Relaxation Rate Analytics

The total strain in the $4\times 4$ Luttinger-Kohn-Bir-Pikus formalism is
given as:

$H^{LKBP}_{strain}(\bm{r})=\begin{pmatrix}P_{\varepsilon}(\bm{r})+Q_{\varepsilon}(\bm{r})&0&L_{\varepsilon}(\bm{r})&M_{\varepsilon}(\bm{r})\\\
0&P_{\varepsilon}(\bm{r})+Q_{\varepsilon}(\bm{r})&M_{\varepsilon}(\bm{r})^{*}&-L_{\varepsilon}(\bm{r})^{*}\\\
L_{\varepsilon}(\bm{r})^{*}&M_{\varepsilon}(\bm{r})&P_{\varepsilon}(\bm{r})-Q_{\varepsilon}(\bm{r})&0\\\
M_{\varepsilon}(\bm{r})^{*}&-L_{\varepsilon}(\bm{r})&0&P_{\varepsilon}(\bm{r})-Q_{\varepsilon}(\bm{r})\\\
\end{pmatrix}$ (C32)

where,
$P_{\varepsilon}(\bm{r}){=}-a(\varepsilon_{xx}(\bm{r})+\varepsilon_{yy}(\bm{r})+\varepsilon_{zz}(\bm{r}))$,
$Q_{\varepsilon}(\bm{r}){=}-\frac{b}{2}(\varepsilon_{xx}(\bm{r})+\varepsilon_{yy}(\bm{r})-2\varepsilon_{zz}(\bm{r}))$,
$L_{\varepsilon}(\bm{r}){=}d(\varepsilon_{xz}(\bm{r})-i\varepsilon_{yz}(\bm{r}))$,
$M_{\varepsilon}(\bm{r}){=}\left(\frac{\sqrt{3}}{2}b(\varepsilon_{xx}(\bm{r})-\varepsilon_{yy}(\bm{r}))-id\varepsilon_{xy}(\bm{r})\right)$.
The non-zero static component of the strain tensor is given by
$\varepsilon_{xx}=\varepsilon_{yy}=-0.006$, and
$\varepsilon_{zz}=-\frac{C_{12}}{C_{11}}\varepsilon_{xx}$ with $C_{12}{=}44$
GPa, $C_{11}{=}126$ GPa. Considering the lattice deformation potential
$\mathcal{D}(\bm{r})$, the ’local’ strain is given by:

$\varepsilon_{i,j}^{\alpha}(\bm{r})=\frac{1}{2}\left(\frac{\partial
u_{i}(\bm{r})}{\partial r_{j}}+\frac{\partial u_{j}(\bm{r})}{\partial
r_{i}}\right),\ \ i,j\in\\{x,y,z\\}$ (C33)

Here $\bm{u}$ vector designates the deformation field at the position
$\bm{r}$. For a phonon traveling with wave vector $\mathbf{q}$ in the
polarized state $\alpha$, the strain tensor is given by,

$\displaystyle\varepsilon_{i,j}^{\alpha}(\bm{r})=\frac{i}{2}\sqrt{\frac{\hbar}{2V_{c}\rho\omega_{\mathbf{q},\alpha}}}q(\hat{c}_{i}^{\alpha}\hat{q}_{j}+\hat{c}_{j}^{\alpha}\hat{q}_{i})(e^{-i\mathbf{q}\cdot\mathbf{r}}a_{\mathbf{q},\alpha}+e^{i\mathbf{q}\cdot\mathbf{r}}a^{\dagger}_{\mathbf{q},\alpha})$
(C34)

where $\hat{c}$ is the polarisation unit vector. We consider three
polarisations with con-ordinate systems understood as
$l\\!:\\!(r,\theta,\phi);\,t\\!:\\!\left(r,\theta+\frac{\pi}{2},\phi\right);\,w\\!:\\!\left(r,\frac{\pi}{2},\phi+\frac{\pi}{2}\right)$.
Using $\omega_{\mathbf{q},\alpha}{=}\,v_{\alpha}q$,

$\varepsilon_{i,j}^{\alpha}(\bm{r})=i\sqrt{\frac{\hbar}{2V_{c}\rho
v_{\alpha}}}\sqrt{q}\mathcal{A}_{\varepsilon,ij}^{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}}(a_{-\mathbf{q},\alpha}+a^{\dagger}_{\mathbf{q},\alpha})$
(C35)

where $v_{\alpha}$ are the acoustic phonon velocities, and we assumed
$\frac{1}{2}(\hat{c}_{i}^{\alpha}\hat{q}_{j}+\hat{c}_{j}^{\alpha}\hat{q}_{i})=\mathcal{A}_{\varepsilon,ij}^{\alpha}$.
The matrix elements of $\mathcal{A}_{\varepsilon,ij}^{\alpha}$ are sketched
out below:

$\mathcal{A}_{\varepsilon}^{\alpha}=\frac{1}{2}\begin{pmatrix}2\hat{c}_{x}^{\alpha}\hat{q}_{x}&\hat{c}_{x}^{\alpha}\hat{q}_{y}+\hat{c}_{y}^{\alpha}\hat{q}_{x}&\hat{c}_{x}^{\alpha}\hat{q}_{z}+\hat{c}_{z}^{\alpha}\hat{q}_{x}\\\
\hat{c}_{y}^{\alpha}\hat{q}_{x}+\hat{c}_{x}^{\alpha}\hat{q}_{y}&2\hat{c}_{y}^{\alpha}\hat{q}_{y}&\hat{c}_{y}^{\alpha}\hat{q}_{z}+\hat{c}_{z}^{\alpha}\hat{q}_{y}\\\
\hat{c}_{z}^{\alpha}\hat{q}_{x}+\hat{c}_{x}^{\alpha}\hat{q}_{z}&\hat{c}_{z}^{\alpha}\hat{q}_{y}+\hat{c}_{y}^{\alpha}\hat{q}_{z}&2\hat{c}_{z}^{\alpha}\hat{q}_{z}\end{pmatrix}$
(C36)

The phonon wave vector has the components
$\overrightarrow{q}\rightarrow\\{q\sin\theta\cos\phi,q\sin\theta\,\sin\phi,q\cos\theta\\}$;
so the unit vectors for $\bm{q}$ are given by:
$\bm{\hat{}}{q}\rightarrow\\{\sin\theta\cos\phi,\,\sin\theta\,\sin\phi,\,\cos\theta\\}$.
The polarization wave vectors are as follows:

  * •

$\hat{c}^{l}\rightarrow
q^{-1}\\{q_{x},q_{y},q_{z}\\}=\\{\sin\theta\cos\phi,\,\sin\theta\cos\phi,\,\cos\theta\\}$

  * •

$\hat{c}^{t}\rightarrow
q^{-1}(q_{x}^{2}+q_{y}^{2})^{-\frac{1}{2}}\\{q_{x}q_{z},q_{y}q_{z},-(q_{x}^{2}+q_{y}^{2})\\}=\\{\cos\theta\cos\phi,\,\cos\theta\sin\phi,-\sin\theta\\}$

  * •

$\hat{c}^{w}\rightarrow(q_{x}^{2}+q_{y}^{2})^{-\frac{1}{2}}\\{q_{y},-q_{x},0\\}=\\{-\sin\phi,\,\cos\phi,\,0\\}$

We can write the matrix elements of $\mathcal{A}_{\varepsilon,ij}^{\alpha}$
for the three polarisations using the decompositions above:

$\displaystyle\mathcal{A}_{\varepsilon}^{l}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}2\frac{q_{x}^{2}}{q^{2}}&2\frac{q_{x}q_{y}}{q^{2}}&2\frac{q_{x}q_{z}}{q^{2}}\\\
2\frac{q_{y}q_{x}}{q^{2}}&2\frac{q_{y}^{2}}{q^{2}}&2\frac{2q_{y}q_{z}}{q^{2}}\\\
2\frac{q_{z}q_{x}}{q^{2}}&2\frac{q_{z}q_{y}}{q^{2}}&2\frac{q_{z}^{2}}{q^{2}}\end{pmatrix}=\frac{1}{q^{2}}\begin{pmatrix}q_{x}^{2}&q_{x}q_{y}&q_{x}q_{z}\\\
q_{y}q_{x}&q_{y}^{2}&q_{y}q_{z}\\\ q_{z}q_{x}&q_{z}q_{y}&q_{z}^{2}\\\
\end{pmatrix}$ $\displaystyle\mathcal{A}_{\varepsilon}^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}2\frac{q_{z}}{q}\frac{q_{x}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{x}}{q}&2\frac{q_{z}}{q}\frac{q_{x}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{y}}{q}&\frac{q_{x}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{z}}{q}-\frac{\sqrt{q_{x}^{2}+q_{y}^{2}}}{q}\frac{q_{x}}{q}\\\
2\frac{q_{z}}{q}\frac{q_{x}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{y}}{q}&2\frac{q_{z}}{q}\frac{q_{y}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{y}}{q}&\frac{q_{y}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{z}}{q}-\frac{\sqrt{q_{x}^{2}+q_{y}^{2}}}{q}\frac{q_{y}}{q}\\\
\frac{q_{x}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{z}}{q}-\frac{\sqrt{q_{x}^{2}+q_{y}^{2}}}{q}\frac{q_{x}}{q}&\frac{q_{z}^{2}q_{y}-q_{x}^{2}q_{y}-q_{y}^{3}}{q^{2}\sqrt{q_{x}^{2}+q_{y}^{2}}}&-2\frac{q_{z}}{q}\frac{\sqrt{q_{x}^{2}+q_{y}^{2}}}{q}\end{pmatrix}$
$\displaystyle\mathcal{A}_{\varepsilon}^{w}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}-2\frac{q_{y}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{x}}{q}&-\frac{q_{y}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{y}}{q}+\frac{q_{x}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{x}}{q}&-\frac{q_{y}}{\sqrt{q_{x}^{2}+q_{y}^{2}}}\frac{q_{z}}{q}\\\
\frac{q_{x}^{2}-q_{y}^{2}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}&2\frac{q_{x}q_{y}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}&\frac{q_{x}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}\\\
-\frac{q_{y}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}&\frac{q_{x}q_{z}}{q\sqrt{q_{x}^{2}+q_{y}^{2}}}&0\end{pmatrix}$
(C37)

We wish to evaluate the angular integrals, so we write the matrices in terms
of $\theta$ and $\phi$:

$\mathcal{A}_{\varepsilon}^{l}=\begin{pmatrix}\sin^{2}\theta\cos^{2}\phi&\sin^{2}\theta\sin\phi\cos\phi&\sin\theta\cos\theta\cos\phi\\\
\sin^{2}\theta\sin\phi\cos\phi&\sin^{2}\theta\sin^{2}\phi&\sin\theta\cos\theta\sin\phi\\\
\sin\theta\cos\theta\cos\phi&\sin\theta\cos\theta\sin\phi&\cos^{2}\theta\end{pmatrix}$
$\displaystyle\mathcal{A}_{\varepsilon}^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}2\cos\theta\cos\phi\sin\theta\cos\phi&2\cos\theta\cos\phi\sin\theta\sin\phi&\cos\theta\cos\phi\cos\theta-\sin^{2}\theta\cos\phi\\\
2\cos\theta\cos\phi\sin\theta\sin\phi&2\cos\theta\sin\phi\sin\theta\sin\phi&\cos\theta\sin\phi\cos\theta-\sin^{2}\theta\sin\phi\\\
\cos\theta\cos\phi\cos\theta-\sin^{2}\theta\cos\phi&\cos\theta\sin\phi\cos\theta-\sin^{2}\theta\sin\phi&-2\sin\theta\cos\theta\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\frac{1}{2}\sin
2\theta\cos^{2}\phi&\frac{1}{4}\sin 2\theta\sin 2\phi&\frac{1}{2}\cos
2\theta\cos\phi\\\ \frac{1}{4}\sin 2\theta\sin 2\phi&\frac{1}{2}\sin
2\theta\sin^{2}\phi&\frac{1}{2}\cos 2\theta\sin\phi\\\ \frac{1}{2}\cos
2\theta\cos\phi&\frac{1}{2}\cos 2\theta\sin\phi&-\frac{1}{2}\sin
2\theta\end{pmatrix}$ $\displaystyle\mathcal{A}_{\varepsilon}^{w}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}-2\sin\phi\sin\theta\cos\phi&-\sin\phi\sin\theta\sin\phi+\cos\phi\sin\theta\cos\phi&-\sin\phi\cos\theta\\\
-\sin\phi\sin\theta\sin\phi+\cos\phi\sin\theta\cos\phi&2\cos\phi\sin\theta\sin\phi&\cos\phi\cos\theta\\\
-\sin\phi\cos\theta&\cos\phi\cos\theta&0\end{pmatrix}$ (C38) $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{pmatrix}-\sin 2\phi\sin\theta&\cos
2\phi\sin\theta&-\sin\phi\cos\theta\\\ \cos 2\phi\sin\theta&\sin
2\phi\sin\theta&\cos\phi\cos\theta\\\
-\sin\phi\cos\theta&\cos\phi\cos\theta&0\end{pmatrix}$

The total hole-phonon Hamiltonian can be added as per the following equation:

$H_{h-ph}^{\alpha}=\mathcal{D}_{11}^{\alpha}\mathcal{A}_{\varepsilon,11}^{\alpha}+\mathcal{D}_{12}^{\alpha}\mathcal{A}_{\varepsilon,12}^{\alpha}+\mathcal{D}_{13}^{\alpha}\mathcal{A}_{\varepsilon,13}^{\alpha}+\mathcal{D}_{22}^{\alpha}\mathcal{A}_{\varepsilon,22}^{\alpha}+\mathcal{D}_{23}^{\alpha}\mathcal{A}_{\varepsilon,23}^{\alpha}+\mathcal{D}_{33}^{\alpha}\mathcal{A}_{\varepsilon,33}^{\alpha}$
(C39)

where $\alpha$ is the polarisation index, and $\mathcal{D}$ are the $4\times
4$ LK deformation potential matrices.

#### l-polarisation:

Putting in the local strain terms, we can write:

$H_{h-ph}^{l}=i\sqrt{q}\sqrt{\frac{\hbar}{2V_{c}\rho
v_{l}}}\begin{pmatrix}P_{\varepsilon}^{l}+Q_{\varepsilon}^{l}&0&L_{\varepsilon}^{l}&M_{\varepsilon}^{l}\\\
0&P_{\varepsilon}^{l}+Q_{\varepsilon}^{l}&M_{\varepsilon}^{l^{*}}&-L_{\varepsilon}^{l^{*}}\\\
L_{\varepsilon}^{l^{*}}&M_{\varepsilon}^{l}&P_{\varepsilon}^{l}-Q_{\varepsilon}^{l}&0\\\
M_{\varepsilon}^{l^{*}}&-L_{\varepsilon}^{l}&0&P_{\varepsilon}^{l}-Q_{\varepsilon}^{l}\end{pmatrix}e^{i\mathbf{q}\cdot\mathbf{r}}(a_{-\mathbf{q},l}+a^{\dagger}_{\mathbf{q},l})$

where,

$\displaystyle P_{\varepsilon}^{l}$ $\displaystyle=$
$\displaystyle-a(\sin^{2}\theta\cos^{2}\phi+\sin^{2}\theta\sin^{2}\phi+\cos^{2}\theta)=-a$
$\displaystyle Q_{\varepsilon}^{l}$ $\displaystyle=$
$\displaystyle-\frac{b}{2}(\sin^{2}\theta\cos^{2}\phi+\sin^{2}\theta\sin^{2}\phi-2\cos^{2}\theta)=-\frac{b}{2}(1-3\cos^{2}\theta)$
$\displaystyle L_{\varepsilon}^{l}$ $\displaystyle=$ $\displaystyle
d(\sin\theta\cos\theta\cos\phi-i\sin\theta\cos\theta\sin\phi)=\frac{d}{2}\sin
2\theta\,e^{-i\phi}$ $\displaystyle M_{\varepsilon}^{l}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}b}{2}(\sin^{2}\theta\cos^{2}\phi-\sin^{2}\theta\sin^{2}\phi)-id\sin\phi\cos\phi\sin^{2}\theta=\frac{\sqrt{3}b}{2}\sin^{2}\theta\cos
2\phi-\frac{id}{2}\sin 2\phi\sin^{2}\theta$ (C40)

#### t-polarisation:

The hole-phonon Hamiltonian in this case takes the following form in angular
co-ordinates:

$H_{h-ph}^{t}=i\sqrt{q}\sqrt{\frac{\hbar}{2V_{c}\rho
v_{t}}}\begin{pmatrix}P_{\varepsilon}^{t}+Q_{\varepsilon}^{t}&0&L_{\varepsilon}^{t}&M_{\varepsilon}^{t}\\\
0&P_{\varepsilon}^{t}+Q_{\varepsilon}^{t}&M_{\varepsilon}^{t^{*}}&-L_{\varepsilon}^{t^{*}}\\\
L_{\varepsilon}^{t^{*}}&M_{\varepsilon}^{t}&P_{\varepsilon}^{t}-Q_{\varepsilon}^{t}&0\\\
M_{\varepsilon}^{t^{*}}&-L_{\varepsilon}^{t}&0&P_{\varepsilon}^{t}-Q_{\varepsilon}^{t}\end{pmatrix}e^{i\mathbf{q}\cdot\mathbf{r}}(a_{-\mathbf{q},t}+a^{\dagger}_{\mathbf{q},t})$

where,

$\displaystyle P_{\varepsilon}^{t}$ $\displaystyle=$
$\displaystyle-a\left(\frac{1}{2}\sin 2\theta\cos^{2}\phi+\frac{1}{2}\sin
2\theta\sin^{2}\phi-\frac{1}{2}\sin 2\theta\right)=0$ $\displaystyle
Q_{\varepsilon}^{t}$ $\displaystyle=$
$\displaystyle-\frac{b}{2}\left(\frac{1}{2}\sin
2\theta\cos^{2}\phi+\frac{1}{2}\sin 2\theta\sin^{2}\phi+\sin
2\theta\right)=-\frac{3b}{4}\sin 2\theta$ $\displaystyle L_{\varepsilon}^{t}$
$\displaystyle=$ $\displaystyle d\left(\frac{1}{2}\cos
2\theta\cos\phi-\frac{i}{2}\cos 2\theta\sin\phi\right)=\frac{d}{2}\cos 2\theta
e^{-i\phi}$ $\displaystyle M_{\varepsilon}^{t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}b}{2}\left(\frac{1}{2}\sin
2\theta\cos^{2}\phi-\frac{1}{2}\sin
2\theta\sin^{2}\phi\right)-\frac{id}{4}\sin 2\theta\sin
2\phi=\frac{\sqrt{3}b}{4}\sin 2\theta\cos 2\phi-\frac{id}{4}\sin 2\theta\sin
2\phi$ (C41)

#### w-polarisation:

For the $3^{\text{rd}}$ polarisation direction:

$H_{h-ph}^{w}=i\sqrt{q}\sqrt{\frac{\hbar}{2V_{c}\rho
v_{w}}}\begin{pmatrix}P_{\varepsilon}^{w}+Q_{\varepsilon}^{w}&0&L_{\varepsilon}^{w}&M_{\varepsilon}^{w}\\\
0&P_{\varepsilon}^{w}+Q_{\varepsilon}^{w}&M_{\varepsilon}^{w^{*}}&-L_{\varepsilon}^{w^{*}}\\\
L_{\varepsilon}^{w^{*}}&M_{\varepsilon}^{w}&P_{\varepsilon}^{w}-Q_{\varepsilon}^{w}&0\\\
M_{\varepsilon}^{w^{*}}&-L_{\varepsilon}^{w}&0&P_{\varepsilon}^{w}-Q_{\varepsilon}^{w}\end{pmatrix}e^{i\mathbf{q}\cdot\mathbf{r}}(a_{-\mathbf{q},w}+a^{\dagger}_{\mathbf{q},w})$

where,

$\displaystyle P_{\varepsilon}^{w}$ $\displaystyle=$
$\displaystyle-a\left(-\frac{1}{2}\sin 2\phi\sin\theta+\frac{1}{2}\sin
2\phi\sin\theta\right)=0$ $\displaystyle Q_{\varepsilon}^{w}$ $\displaystyle=$
$\displaystyle-\frac{b}{2}\left(-\frac{1}{2}\sin
2\phi\sin\theta+\frac{1}{2}\sin 2\phi\sin\theta\right)=0$ $\displaystyle
L_{\varepsilon}^{w}$ $\displaystyle=$ $\displaystyle
d\left(-\frac{1}{2}\sin\phi\cos\theta-\frac{i}{2}\cos\phi\cos\theta\right)=-\frac{id}{2}\cos\theta
e^{-i\phi}$ $\displaystyle M_{\varepsilon}^{w}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}b}{2}\left(-\frac{1}{2}\sin
2\phi\sin\theta-\frac{1}{2}\sin 2\phi\sin\theta\right)-\frac{id}{2}\cos
2\phi\sin\theta=-\frac{\sqrt{3}b}{2}\sin 2\phi\sin\theta-\frac{id}{2}\cos
2\phi\sin\theta$ (C42)

The relaxation rate is characterized by the spontaneous and stimulated phonon
scattering, hence:

$\Gamma_{1}=\frac{1}{T_{1}}=\sum_{\alpha}\left(\frac{2\pi}{\hbar}\sum_{\mathbf{q}}\left|\left\langle\mathbb{0}\right|H_{h-ph}\left|\mathbb{1}\right\rangle_{\alpha}\right|^{2}\delta\left(\Delta\mathbb{E}-\hbar\omega_{\alpha,\mathbf{q}}\right)\right)$
(C43)

The summation over the wave vectors can be changed to continuous integral, and
the creation-annihilation operators can be approximated to produce a factor of
$N_{q}$, which denotes the number of acoustic phonons with $q$ momentum:

$\displaystyle\Gamma_{1}=\frac{1}{T_{1}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{2\pi}{\cancel{\hbar}}\frac{\cancel{V}}{(2\pi)3}\int_{V_{q}}d^{3}\mathbf{q}\,q\frac{\cancel{\hbar}}{2\cancel{V_{c}}\rho
v_{\alpha}}\left|\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\cancel{N_{q}}\left|\mathbb{1}\right\rangle_{\alpha}\right|^{2}\delta\left(\Delta\mathbb{E}-\hbar
v_{\alpha}q\right)\right)$ (C44) $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{1}{8\pi^{2}}\int_{V_{q}}q^{3}\,dq\,\sin\theta\,d\theta\,d\phi\frac{1}{\rho
v_{\alpha}}\left|\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle_{\alpha}\right|^{2}\frac{1}{\hbar
v_{\alpha}}\delta\left(\frac{\Delta\mathbb{E}}{\hbar
v_{\alpha}}-q\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{1}{8\pi^{2}\hbar\rho
v_{\alpha}^{2}}\int_{0}^{2\pi}\\!d\phi\,\int_{0}^{\pi}\\!d\theta\sin\theta\,\int_{0}^{\infty}\\!q^{3}dq\,\left|\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle_{\alpha}\right|^{2}\delta\left(\frac{\Delta\mathbb{E}}{\hbar
v_{\alpha}}-q\right)\right)$

The qubit ground state $|\mathbb{0}\rangle$ and excited state
$|\mathbb{1}\rangle$ are $1\times 4$ spinors with each component multiplied to
spatial functions.

$\displaystyle\Rightarrow$
$\displaystyle\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle=\begin{pmatrix}g_{1}^{*}&g_{2}^{*}&g_{3}^{*}&g_{4}^{*}\end{pmatrix}e^{i\mathbf{q}\cdot\mathbf{r}}\begin{pmatrix}P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha}&0&L_{\varepsilon}^{\alpha}&M_{\varepsilon}^{\alpha}\\\
0&P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha}&M_{\varepsilon}^{\alpha^{*}}&-L_{\varepsilon}^{\alpha^{*}}\\\
L_{\varepsilon}^{\alpha^{*}}&M_{\varepsilon}^{\alpha}&P_{\varepsilon}^{\alpha}-Q_{\varepsilon}^{\alpha}&0\\\
M_{\varepsilon}^{\alpha^{*}}&-L_{\varepsilon}^{\alpha}&0&P_{\varepsilon}^{\alpha}-Q_{\varepsilon}^{\alpha}\\\
\end{pmatrix}\begin{pmatrix}e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\end{pmatrix}$
(C45) $\displaystyle=$
$\displaystyle\begin{pmatrix}g_{1}^{*}&g_{2}^{*}&g_{3}^{*}&g_{4}^{*}\end{pmatrix}e^{i\mathbf{q}\cdot\mathbf{r}}\begin{pmatrix}(P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha})e_{1}+L_{\varepsilon}^{\alpha}e_{3}+M_{\varepsilon}^{\alpha}e_{4}\\\
(P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha})e_{2}+M_{\varepsilon}^{\alpha^{*}}e_{3}-L_{\varepsilon}^{\alpha^{*}}e_{4}\\\
L_{\varepsilon}^{\alpha^{*}}e_{1}+M_{\varepsilon}^{\alpha}e_{2}+(P_{\varepsilon}^{\alpha}-Q_{\varepsilon}^{\alpha})e_{3}\\\
M_{\varepsilon}^{\alpha^{*}}e_{1}-L_{\varepsilon}^{\alpha}e_{2}+(P_{\varepsilon}^{\alpha}-Q_{\varepsilon}^{\alpha})e_{4}\\\
\end{pmatrix}$ $\displaystyle=$
$\displaystyle\left[\left\\{(P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha})e^{i\mathbf{q}\cdot\mathbf{r}}g_{1}^{*}e_{1}+L_{\varepsilon}^{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}}g_{1}^{*}e_{3}+M_{\varepsilon}^{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}}g_{1}^{*}e_{4}\right\\}+\left\\{(P_{\varepsilon}^{\alpha}+Q_{\varepsilon}^{\alpha})e^{i\mathbf{q}\cdot\mathbf{r}}g_{2}^{*}e_{2}+M_{\varepsilon}^{\alpha^{*}}e^{i\mathbf{q}\cdot\mathbf{r}}g_{2}^{*}e_{3}-L_{\varepsilon}^{\alpha^{*}}e^{i\mathbf{q}\cdot\mathbf{r}}g_{2}^{*}e_{4}\right\\}\right.$
$\displaystyle\\!+\\!$
$\displaystyle\\!\left.\left\\{L_{\varepsilon}^{\alpha^{*}}e^{i\mathbf{q}\cdot\mathbf{r}}g_{3}^{*}e_{1}\\!+\\!M_{\varepsilon}^{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}}g_{3}^{*}e_{2}\\!+\\!(P_{\varepsilon}^{\alpha}-Q_{\varepsilon}^{\alpha})e^{i\mathbf{q}\cdot\mathbf{r}}g_{3}^{*}e_{3}\right\\}\\!+\\!\left\\{M_{\varepsilon}^{\alpha^{*}}e^{i\mathbf{q}\cdot\mathbf{r}}g_{4}^{*}e_{1}\\!-\\!L_{\varepsilon}^{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}}g_{4}^{*}e_{2}+\\!(P_{\varepsilon}^{\alpha}\\!-\\!Q_{\varepsilon}^{\alpha})e^{i\mathbf{q}\cdot\mathbf{r}}g_{4}^{*}e_{4}\right\\}\right]$

According to our model
$g_{i}^{*}{=}\sum_{\\{m,n,l,i^{\prime}\\}}c_{ii^{\prime}}^{g^{*}}\psi_{n}(x)\psi_{m}(y)\psi_{l}(z)$,
and
$e_{j}^{*}{=}\sum_{\\{m^{\prime},n^{\prime},l^{\prime},j^{\prime}\\}}c^{e}_{jj^{\prime}}\psi_{n^{\prime}}(x)\psi_{m^{\prime}}(y)\psi_{l^{\prime}}(z)$;
implies that the terms in Eqn. C45 have the form

$\displaystyle
e^{i\mathbf{q}\cdot\mathbf{r}}g_{i}^{*}e_{j}\\!\Rightarrow\\!\left(\sum_{i^{\prime},j^{\prime}}c^{g^{*}}_{ii^{\prime}}c^{e}_{jj^{\prime}}\right)\\!\left(\sum_{n,n^{\prime}}\sum_{m,m^{\prime}}\sum_{l,l^{\prime}}\right)\int_{-\infty}^{\infty}\\!dx\,e^{iq_{x}x}\psi_{n}(x)\psi_{n^{\prime}}(x)\int_{-\infty}^{\infty}\\!dy\,e^{iq_{y}y}\psi_{m}(y)\psi_{m^{\prime}}(y)\int_{-\infty}^{\infty}\\!dz\,e^{iq_{z}z}\psi_{l}(z)\psi_{l^{\prime}}(z)$

To our advantage, the inversion-symmetric basis wave-functions we use to
describe our hole QD, i.e. an infinite barrier in $z$ and harmonic potential
in $x{-}y$, have closed form solutions of the $e^{i\mathbf{q}\cdot\mathbf{r}}$
integrals. This allows us to evaluate the Relaxation rate $\Gamma_{1}$
analytically. We also show the dipole approximation to agree with the
analytical results for $T_{1}$; thirdly, a numerical pathway is sketched as an
alternative.

#### Analytical: in-plane integrals.

The matrix elements of $e^{iq_{x}x}$ between two $x$(or $y$) wavefunctions are
given by,

$\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle=\int_{-\infty}^{\infty}\\!dx\,e^{iq_{x}x}\psi_{n}(x)\psi_{n^{\prime}}(x)$
(C47)

$e^{iq_{x}x}$ can be written as an infinite expansion in Hermite polynomial
basis:

$e^{iq_{x}x}=e^{iq_{x}L_{x}\frac{x}{L_{x}}}=e^{\frac{(-iq_{x}L_{x})^{2}}{4}}\sum_{r=0}^{\infty}\frac{(iq_{x}L_{x})^{r}}{2^{r}r!}H_{r}\\!\left(\frac{x}{L_{x}}\right)$
(C48) $\Rightarrow\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle=\frac{1}{\sqrt{2^{n}n!L_{x}\sqrt{\pi}}}\frac{1}{\sqrt{2^{n^{\prime}}n^{\prime}!L_{x}\sqrt{\pi}}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}\sum_{r=0}^{\infty}\frac{(iq_{x}L_{x})^{r}}{2^{r}r!}\mathbb{I}(r,n,n^{\prime})$
(C49)

with
$\mathbb{I}(r,n,n^{\prime})\\!=\\!\int_{-\infty}^{\infty}dx\,e^{-\frac{x^{2}}{L_{x}^{2}}}H_{r}\\!\left(\frac{x}{L_{x}}\right)H_{n}\\!\left(\frac{x}{L_{x}}\right)H_{n^{\prime}}\\!\left(\frac{x}{L_{x}}\right)\\!=\\!L_{x}\int_{-\infty}^{\infty}d\\!\left(\frac{x}{L_{x}}\right)\,e^{-\frac{x^{2}}{L_{x}^{2}}}H_{r}\\!\left(\frac{x}{L_{x}}\right)H_{n}\\!\left(\frac{x}{L_{x}}\right)H_{n^{\prime}}\\!\left(\frac{x}{L_{x}}\right)$.
Substituting $\frac{x}{L_{x}}\rightarrow x$, we write:

$\displaystyle\Rightarrow\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle$ $\displaystyle=$
$\displaystyle\left(2^{n+n^{\prime}}n!n^{\prime}!\pi\right)^{-\frac{1}{2}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}\sum_{r=0}^{\infty}\frac{(iq_{x}L_{x})^{r}}{2^{r}r!}\int_{-\infty}^{\infty}dx\,e^{-x^{2}}H_{r}\\!(x)H_{n}\\!(x)H_{n^{\prime}}\\!(x)$
(C50)

The product of two Hermite polynomials can be expanded in the Hermite
polynomial basis:

$H_{n}(x)H_{n^{\prime}}(x)=2^{n}n!n^{\prime}!\sum_{k=0}^{n}\frac{H_{2k+n^{\prime}-n}(x)}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}$
(C51) $\displaystyle\Rightarrow\\!\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle\\!$ $\displaystyle=$
$\displaystyle\\!\frac{e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}}{\sqrt{2^{n+n^{\prime}}n!n^{\prime}!\pi}}\sum_{r=0}^{\infty}\frac{(iq_{x}L_{x})^{r}}{2^{r}r!}2^{n}n!n^{\prime}!\sum_{k=0}^{n}\frac{1}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}\int_{-\infty}^{\infty}dx\,e^{-x^{2}}H_{r}\\!(x)H_{2k+n^{\prime}-n}(x)$
(C52) $\displaystyle=$
$\displaystyle\\!\frac{e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}}{\sqrt{2^{n+n^{\prime}}n!n^{\prime}!\pi}}\sum_{r=0}^{\infty}\frac{(iq_{x}L_{x})^{r}}{2^{r}r!}2^{n}n!n^{\prime}!\sum_{k=0}^{n}\frac{2^{2k+n^{\prime}-n}(2k+n^{\prime}-n)!\sqrt{\pi}\delta_{r,2k+n^{\prime}-n}}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}$

where we have used the orthonormality
relation:$\int_{-\infty}^{\infty}e^{-x^{2}}H_{n}(x)H_{n^{\prime}}(x)\,dx=2^{n}n!\sqrt{\pi}\delta_{n,n^{\prime}}$.
The $\delta$-function boils the $r$-sum down to only one term, such that:

$\displaystyle\Rightarrow\\!\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle\\!=\\!\sqrt{\frac{n^{\prime}!n!}{2^{n^{\prime}-n}}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}\sum_{k=0}^{n}\frac{(iq_{x}L_{x})^{2k+n^{\prime}-n}}{\cancel{2^{2k+n^{\prime}-n}}\cancel{(2k+n^{\prime}-n)!}}\frac{\cancel{2^{2k+n^{\prime}-n}}\cancel{(2k+n^{\prime}-n)!}}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}$
(C53) $\displaystyle=$
$\displaystyle\sqrt{\frac{n!}{2^{n^{\prime}-n}n^{\prime}!}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}\sum_{k=0}^{n}\frac{n^{\prime}!(iq_{x}L_{x})^{n^{\prime}-n}(iq_{x}L_{x})^{2k}}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}=\sqrt{\frac{n!}{2^{n^{\prime}-n}n^{\prime}!}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}(iq_{x}L_{x})^{n^{\prime}-n}\sum_{k=0}^{n}\frac{n^{\prime}!((iq_{x}L_{x})^{2})^{k}}{2^{k}k!(k+n^{\prime}-n)!(n-k)!}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{n!}{2^{n^{\prime}-n}n^{\prime}!}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}(iq_{x}L_{x})^{n^{\prime}-n}\sum_{k=0}^{n}\frac{(-1)^{k}(n+n^{\prime}-n)!}{k!(k+n^{\prime}-n)!(n-k)!}\left(\frac{q_{x}^{2}L_{x}^{2}}{2}\right)^{k}$

Using the formula for associated Laguerre polynomial
$\mathcal{L}_{n}^{a}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}(n+a)!}{k!(k+a)!(n-k)!}x^{k}$,
the matrix element of $e^{iq_{x}x}$ can be analytically evaluated as:

$\Rightarrow\\!\left\langle
n\right|e^{iq_{x}x}\left|n^{\prime}\right\rangle\\!=\\!\sqrt{\frac{1}{2^{n^{\prime}-n}}\frac{n!}{n^{\prime}!}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}(iq_{x}L_{x})^{n^{\prime}-n}\mathcal{L}_{n}^{n^{\prime}-n}\\!\left(\frac{q_{x}^{2}L_{x}^{2}}{2}\right)$
$\Rightarrow\\!\left\langle
m\right|e^{iq_{y}y}\left|m^{\prime}\right\rangle\\!=\\!\sqrt{\frac{1}{2^{m^{\prime}-m}}\frac{m!}{m^{\prime}!}}e^{-\frac{q_{y}^{2}L_{y}^{2}}{4}}(iq_{y}L_{y})^{m^{\prime}-m}\mathcal{L}_{m}^{m^{\prime}-m}\\!\left(\frac{q_{y}^{2}L_{y}^{2}}{2}\right)$
(C54)

The $z$-basis wavefunctions are:

$\psi_{l}(z)=\begin{cases}\cos\\!\left(\frac{(l+1)\pi
z}{L_{z}}\right)&l=0,2,4,..\\\ \sin\\!\left(\frac{(l+1)\pi
z}{L_{z}}\right)&l=1,3,5..\\\ \end{cases}\
,z\in\left\\{-\frac{L_{z}}{2},\frac{L_{z}}{2}\right\\}$ (C55)

The matrix element $\left\langle
l\right|e^{iq_{z}z}\left|l^{\prime}\right\rangle$ is evauated as:

$\displaystyle\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz\,e^{iq_{z}z}\psi_{l}(z)\psi_{l^{\prime}}(z)=\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz\,(\cos
q_{z}z+i\sin q_{z}z)\psi_{r}(z)\psi_{r^{\prime}}(z)$ (C56)

where $r=l+1$ is the new index for z-integral (for simpler mathematical
expressions).

#### $r$ odd, $r^{\prime}$ even.

For the case of evaluating the matrix element between even and odd z-basis
functions is calculated as

$\displaystyle=$
$\displaystyle\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz(\cos q_{z}z+i\sin
q_{z}z)\sqrt{\frac{2}{L_{z}}}\cos\\!\left(\frac{r\pi
z}{L_{z}}\right)\sqrt{\frac{2}{L_{z}}}\sin\\!\left(\frac{r^{\prime}\pi
z}{L_{z}}\right)$ (C57) $\displaystyle=$
$\displaystyle\frac{2}{L_{z}}\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz(\cos
q_{z}z+i\sin
q_{z}z)\frac{1}{2}\left(\sin\\!\left(\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\right)-\sin\\!\left(\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\right)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{L_{z}}\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz\left(\cos
q_{z}z\sin\\!\left(\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\\!\right)\\!-\cos
q_{z}z\sin\\!\left(\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\\!\right)\\!+i\sin
q_{z}z\sin\\!\left(\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\\!\right)\\!-i\sin
q_{z}z\sin\\!\left(\frac{\overline{r-r^{\prime}}\pi z}{L_{z}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2L_{z}}\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz\left(\sin\left(\\!q_{z}z+\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\right)\\!-\sin\left(\\!q_{z}z-\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\right)\\!-\sin\left(\\!q_{z}z+\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\right)\\!+\sin\left(\\!q_{z}z-\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\right)\\!\right)$ $\displaystyle+$
$\displaystyle\frac{i}{2L_{z}}\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz\left(\\!\cos\\!\left(\\!q_{z}z+\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\right)\\!+\cos\\!\left(\\!q_{z}z-\frac{\overline{r+r^{\prime}}\pi
z}{L_{z}}\right)\\!-\cos\\!\left(\\!q_{z}z+\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\right)\\!-\sin\\!\left(\\!q_{z}z-\frac{\overline{r-r^{\prime}}\pi
z}{L_{z}}\right)\\!\right)$

The $\sin$ and $\cos$ terms can be now evaluated straight forwardly to give us
the final simplified expression:

$=\frac{i}{2}\left[\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}}-\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}}-\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}}\right]$
(C58)

#### $r,r^{\prime}$ both odd/even.

For the case of evaluating the matrix element between both even(both odd)
z-basis functions is calculated as

$\displaystyle=$
$\displaystyle\int_{-\frac{L_{z}}{2}}^{\frac{L_{z}}{2}}dz(\cos q_{z}z+i\sin
q_{z}z)\sqrt{\frac{2}{L_{z}}}\sin\\!\left(\frac{r\pi
z}{L_{z}}\right)\sqrt{\frac{2}{L_{z}}}\sin\\!\left(\frac{r^{\prime}\pi
z}{L_{z}}\right)$ (C59) $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}}\right]$

Finally we put the results from Eqns. C54, C59 into Eqn. C to evaluate:

$\displaystyle e^{i\mathbf{q}\cdot\mathbf{r}}g_{i}^{*}e_{j}\\!$
$\displaystyle\Rightarrow$
$\displaystyle\\!\left(\sum_{i^{\prime},j^{\prime}}c^{g^{*}}_{ii^{\prime}}c^{e}_{jj^{\prime}}\right)\\!\left(\sum_{n,n^{\prime}}\sum_{m,m^{\prime}}\sum_{l,l^{\prime}}\right)\sqrt{\frac{1}{2^{n^{\prime}-n}}\frac{n!}{n^{\prime}!}}e^{-\frac{q_{x}^{2}L_{x}^{2}}{4}}(iq_{x}L_{x})^{n^{\prime}-n}\mathcal{L}_{n}^{n^{\prime}-n}\\!\left(\frac{q_{x}^{2}L_{x}^{2}}{2}\right)$
$\displaystyle\times$
$\displaystyle\sqrt{\frac{1}{2^{m^{\prime}-m}}\frac{m!}{m^{\prime}!}}e^{-\frac{q_{y}^{2}L_{y}^{2}}{4}}(iq_{y}L_{y})^{m^{\prime}-m}\mathcal{L}_{m}^{m^{\prime}-m}\\!\left(\frac{q_{y}^{2}L_{y}^{2}}{2}\right)\times
f(l,l^{\prime},q_{z}L_{z})$

where,

$f(l,l^{\prime},q_{z}L_{z})=\begin{cases}\frac{i}{2}\left[\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}}-\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}}-\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}}\right]&l\pm
l^{\prime}=\text{odd}\\\
-\frac{1}{2}\left[\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r+r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r+r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}+\frac{(r-r^{\prime})\pi}{2}}+\frac{\sin\\!\left(\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}\right)}{\frac{q_{z}L_{z}}{2}-\frac{(r-r^{\prime})\pi}{2}}\right]&l\pm
l^{\prime}=\text{even}\\\ \end{cases}$ (C61)

Next we make the substitution $q_{x}\rightarrow q\sin\theta\cos\phi$,
$q_{y}\rightarrow q\sin\theta\sin\phi$, $q_{z}\rightarrow q\cos\theta$ to
write

$\displaystyle e^{i\mathbf{q}\cdot\mathbf{r}}g_{i}^{*}e_{j}\\!$
$\displaystyle\Rightarrow$
$\displaystyle\\!\left(\sum_{i^{\prime},j^{\prime}}c^{g^{*}}_{ii^{\prime}}c^{e}_{jj^{\prime}}\right)\\!\left(\sum_{n,n^{\prime}}\sum_{m,m^{\prime}}\sum_{l,l^{\prime}}\right)\sqrt{\frac{1}{2^{n^{\prime}-n}}\frac{n!}{n^{\prime}!}}e^{-\frac{q^{2}L_{x}^{2}\sin^{2}\theta\cos^{2}\phi}{4}}(iqL_{x}\sin\theta\cos\phi)^{n^{\prime}-n}\mathcal{L}_{n}^{n^{\prime}-n}\\!\left(\frac{q^{2}L_{x}^{2}\sin^{2}\theta\cos^{2}\phi}{2}\right)$
$\displaystyle\times$
$\displaystyle\sqrt{\frac{1}{2^{m^{\prime}-m}}\frac{m!}{m^{\prime}!}}e^{-\frac{q^{2}L_{y}^{2}\sin^{2}\theta\sin^{2}\phi}{4}}(iqL_{y}\sin\theta\sin\phi)^{m^{\prime}-m}\mathcal{L}_{m}^{m^{\prime}-m}\\!\left(\frac{q^{2}L_{y}^{2}\sin^{2}\theta\sin^{2}\phi}{2}\right)\times
f(l,l^{\prime},qL_{z}\cos\theta)$

These set of equations allow us to calculate the terms in Eqn. C45 to express
$\left|\left\langle\mathbb{0}\right|e^{i\mathbf{q}\cdot\mathbf{r}}H^{LKBP}_{strain,\alpha}\left|\mathbb{1}\right\rangle\right|^{2}$
as $I_{\alpha}(q,\theta,\phi)$. From Eqn. C44,

$\displaystyle\Gamma_{1}=\frac{1}{T_{1}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{1}{8\pi^{2}\hbar\rho
v_{\alpha}^{2}}\int_{0}^{2\pi}\\!d\phi\,\int_{0}^{\pi}\\!d\theta\sin\theta\,\int_{0}^{\infty}\\!q^{3}dq\,I_{\alpha}(q,\theta,\phi)\,\delta\\!\left(\frac{\Delta\mathbb{E}}{\hbar
v_{\alpha}}-q\right)\right)$ (C63) $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{\Delta\mathbb{E}^{3}}{8\pi^{2}\hbar^{4}\rho
v_{\alpha}^{5}}\int_{0}^{2\pi}\\!d\phi\,\int_{0}^{\pi}\\!d\theta\sin\theta\,I_{\alpha}\\!\left(q\rightarrow\frac{\Delta\mathbb{E}}{\hbar