Hole-phonon interactions in quantum dots: Effects of phonon confinement andencapsulation materials on spin-orbit qubits

Jing Li,1, ∗ Benjamin Venitucci,1, † and Yann-Michel Niquet1, ‡

1Univ. Grenoble Alpes, CEA, IRIG-MEM-L Sim, F-38000, Grenoble, France(Dated: August 18, 2020)

Spin-phonon interactions are one of the mechanisms limiting the lifetime of spin qubits made insemiconductor quantum dots. At variance with other mechanisms such as charge noise, phononsare intrinsic to the device and can hardly be mitigated. They set, therefore fundamental limitsto the relaxation time of the qubits. Here we introduce a general framework for the calculationof the spin (and charge) transition rates induced by bulk (3D) and strongly confined 1D or 2Dphonons. We discuss the particular case of hole spin-orbit qubits described by the 6 bands k · pmodel. We next apply this theory to a hole qubit in a silicon-on-insulator device. We show that spinrelaxation in this device is dominated by a band mixing term that couples the holes to transverseacoustic phonons through the valence band deformation potential d, and optimize the bias pointand magnetic field orientation to maximize the number of Rabi oscillations Q that can be achievedwithin on relaxation time T1. Despite the strong spin-orbit coupling in the valence band, the phonon-limited Q can reach a few tens of thousands. We next explore the effects of phonon confinement in1D and 2D structures, and the impact of the encapsulation materials on the relaxation rates. Weshow that the spin lifetimes can depend on the structure of the device over micrometer-long lengthscales and that they improve when the materials around the qubit get harder. Phonon engineeringin semiconductor qubits may therefore become relevant once the extrinsic sources of relaxation havebeen reduced.

I. INTRODUCTION

Spins in semiconductor quantum dots are consideredas an attractive platform for quantum computing.1,2

Silicon3,4 has, in particular, garnered much interest asa host material for spin qubits because it can be isotopi-cally purified in order to get rid of the nuclear spins thathave undesirable hyperfine interactions with the electronspins. As a matter of fact, very long electron spin life-times T1 have been measured on donors5 and electrostati-cally defined quantum dots in silicon.6 This promoted thedemonstration of high fidelity single and two qubit gatesin this material.7–13

Among other specific features of silicon, intrinsic spin-orbit interactions are known to be weak in the conductionband.4 This decouples the electron spins from electricaland charge noise as well as phonons and further enhancesspin lifetimes. However, this hinders the electrical manip-ulation of electron spins through electric dipole spin res-onance (EDSR).14,15 Artificial spin-orbit coupling needs,therefore, to be engineered for that purpose using, e.g.,micro-magnets next to the qubits.7,9,16

This fostered the interest in hole qubits that can be ef-ficiently tuned and manipulated electrically owing to themuch stronger spin-orbit coupling in the valence band.17

Electrically driven Rabi oscillations with frequencies upto hundreds of MHz have been reported in hole qubitsmade of silicon18–20 and germanium.21,22 Two qubit gateshave also been demonstrated recently.23 Modeling showsthat hole spins couple to electric fields mostly through“direct” spin-orbit interactions in the heavy-hole/light-hole/split-off valence band manifold,17,24–26 with littlecontributions from “remote” coupling to other bands.27

One of the downside of the strong spin-orbit interac-

tions in the valence band is the increased device-to-devicevariability and the shorter spin lifetimes resulting fromthe enhanced coupling to the electrical fluctuations andphonons. It is, therefore, essential to understand the fun-damental limits of hole spin-orbit qubits. In this work, weaddress specifically the coupling between hole spins andphonons. Phonons are indeed intrinsic to the qubits andare much more difficult to mitigate than other scatteringmechanisms such as charge noise, which can, in princi-ple, be reduced thanks to a better design of the devices.Recent works have also shown that phonons can domi-nate the relaxation, especially at high temperature28,29

and magnetic field. Most studies so far have focusedon spin-phonon coupling in electron quantum dots,29–33

while those on hole quantum dots have made use of sim-plified strain Hamiltonians or have considered specificgeometries.28,34–38 We focus here on one-phonon pro-cesses, but give equations that account for the full setof deformation potentials and are applicable to any ge-ometry.

We also discuss two important issues that are also rel-evant for electron spin qubits. First, we investigate howthe spin lifetime depends on the dimensionality of thephonon band structure.39 We highlight the differencesbetween bulk-like (3D) and strongly confined 2D and 1Dphonons. We show that the transition from the 1D or 2Dregime to the 3D regime takes place in “large” microm-eter size structures because qubits usually couple to lowenergy phonons that can probe the structure over verylong length scales. Second, we explore as a consequencethe impact of the encapsulation materials on the spin life-times. We show, as a general trend, that the relaxationrate decreases when the qubits are embedded in hardermaterials.

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The paper is organized as follows: in section II, wepresent the framework for the calculation of phonon-induced relaxation rates, then discuss the case of holespin-orbit qubits described by the six bands k · pmodel,40–42 and the extension to strongly confined 1Dand 2D phonons. In section III, we apply this theoryto a hole spin-orbit qubit on silicon-on-insulator (SOI)such as the one measured in Ref. 19 and modeled inRefs. 25 and 26. We identify the dominant contributionsto the relaxation, and the optimal operation point forthis qubit. We then analyze in section IV the impact ofphonon confinement and of the encapsulation materialson the qubit lifetime.

II. THEORY

In this section, we introduce the general frameworkfor the calculation of the phonon-limited relaxation time,then discuss the application to a prototypical 3D phononband structure, and to the 6 bands k · p model forhole spin-orbit qubits. We next extend the formalismto strongly confined 2D and 1D phonon band structures,and highlight, in particular, the different dependences ofthe relaxation time on the Larmor frequency. We finallydiscuss the numerical calculation of the acoustic phononband structure in complex, realistic qubit structures.

A. General framework for 3D phonons

We consider a qubit based on two eigenstates |0〉 and|1〉 of a Hamiltonian H0 (with energies E0 and E1 re-spectively). In a spin-orbit qubit, these two states forma Kramers-degenerate doublet at zero magnetic field (ei-

ther the ground or an excited one), but the followingtheory applies to any pair of states. This qubit interactswith a thermal bath of bulk acoustic phonons with ener-gies ~ωαq, where α is a branch index and q is a 3D wavevector. We assume that the qubit and phonons are cou-pled by a Hamiltonian ∆H[εij(r)] that depends linearlyon the local strains εij(r) (i, j ∈ x, y, z). These strainscan be calculated from the displacement field u(r) of thephonons:43,44

εij(r) =1

2

(∂ui(r)

∂rj+∂uj(r)

∂ri

). (1)

The displacement field operator in branch α furtherreads:

uαq(r, t) = Aαq(r, t)cαq , (2)

where cαq is the unit phonon polarization vector and:

Aαq(r, t) =

√~

2ρΩωαqeiq·r(aαqe

−iωαqt + a†α,−qeiωαqt) ,

(3)with ρ the density of the host material, Ω the volume ofthe system, and a†αq the phonon creation operators. Thestrain tensor that derives from Eq. (2) is therefore:

εαq(r, t) = iqAαq(r, t)εαq , (4)

where:

εαq =1

2

2cxqx cxqy + cy qx cxqz + cz qxcy qx + cxqy 2cy qy cy qz + cz qycz qx + cxqz cz qy + cy qz 2cz qz

, (5)

with c ≡ cαq and q the unit vector along q.

Assuming E1 > E0, the rate of transitions Γ3D01 from state |0〉 to state |1〉 due to phonon absorption is given by

Fermi’s golden rule:33–35

Γ3D01 =

2π

~∑α

∫d3q ρq

~q2Nαq2ρΩωαq

∣∣∣〈0|eiq·r∆H(εαq)|1〉∣∣∣2δ(E1 − E0 − ~ωαq) , (6)

where ρq = Ω/(2π)3 is the density of states in reciprocal space, and Nαq = 1/(eβ~ωαq − 1) is the thermal populationof phonons with energy ~ωαq (with β = 1/(kBT ) and T the temperature). This equation accounts exclusively forone-phonon processes and is, therefore, valid only at low temperature.28,29 The rate of transitions Γ3D

10 from state |1〉to state |0〉 due to spontaneous and stimulated phonon emission is given by the same expression with Nαq replaced

by Nαq + 1. Therefore, the total relaxation rate Γ3Dph = T−11 = Γ3D

01 + Γ3D10 reads:

Γ3Dph =

1

8π2ρ~ωcoth

(~ω

2kBT

)∑α

∫d3q q2

∣∣∣〈0|eiq·r∆H(εαq)|1〉∣∣∣2δ(ω − ωαq) , (7)

Only phonons matching the Larmor frequency ω = (E1 − E0)/~ couple to the qubit as highlighted by the deltafunction. This equation can be transformed as:

Γ3Dph =

1

8π2ρ~ωcoth

(~ω

2kBT

)∑α

∫Sα(ω)

d2qq2

|vαq|

∣∣∣〈0|eiq·r∆H(εαq)|1〉∣∣∣2 , (8)

3

where Sα(ω) is the surface ωαq = ω and vαq = ∇qωαq is the group velocity of the phonons. The integration can befinalized once a model has been chosen for the phonon and electronic band structures of the qubit.

B. Model 3D phonon band structure

In the following, we consider a prototypical, isotropic3D phonon band structure with one longitudinal acoustic(LA) branch ωlq = vlq, and two degenerate transverseacoustic (TA1 and TA2) branches ωt1q = ωt2q = vtq,where vl and vt1 = vt2 = vt are the longitudinal andtransverse sound velocities. We specify the orientation

of the phonon wave vector by the azimuthal and polarangles θ and ϕ so that q = (sin θ cosϕ, sin θ sinϕ, cos θ).The polarization vector of the LA branch cl = q can,therefore, be characterized by the angles (θl = θ, ϕl = ϕ),while the polarization vectors ct1 and ct2 of the TAbranches can be characterized by the angles (θt1 =θ + π/2, ϕt1 = ϕ) and (θt2 = π/2, ϕt2 = ϕ + π/2),respectively. Then,

Γ3Dph =

ω3

8π2~ρcoth

(~ω

2kBT

) ∑α∈l,t1,t2

1

v5α

∫ π

0

dθ sin θ

∫ 2π

0

dϕ∣∣∣〈0|eiqαq(θ,ϕ)·r∆H[εα(θ, ϕ)]|1〉

∣∣∣2, (9)

where vαqα = ω, and:

εl(θ, ϕ) =1

2

2 sin2 θ cos2 ϕ sin2 θ sin 2ϕ sin 2θ cosϕsin2 θ sin 2ϕ 2 sin2 θ sin2 ϕ sin 2θ sinϕsin 2θ cosϕ sin 2θ sinϕ 2 cos2 θ

(10a)

εt1(θ, ϕ) =1

2

sin 2θ cos2 ϕ 12 sin 2θ sin 2ϕ cos 2θ cosϕ

12 sin 2θ sin 2ϕ sin 2θ sin2 ϕ cos 2θ sinϕcos 2θ cosϕ cos 2θ sinϕ − sin 2θ

(10b)

εt2(θ, ϕ) =1

2

− sin θ sin 2ϕ sin θ cos 2ϕ − cos θ sinϕsin θ cos 2ϕ sin θ sin 2ϕ cos θ cosϕ− cos θ sinϕ cos θ cosϕ 0

. (10c)

In the single band effective mass approximation for sili-con, we recover the expressions of Ref. 33 for the phonon-induced relaxation rate. We further discuss the case ofhole qubits within the six bands k · p model in the nextsubsection.

C. Application to the six bands k · p model

In the six bands k · p model, the hole wave functionsare expanded as:40–42

ψ(r) = FX↑(r)uX↑(r) + FX↓(r)uX↓(r)

+ FY ↑(r)uY ↑(r) + FY ↓(r)uY ↓(r)

+ FZ↑(r)uZ↑(r) + FZ↓(r)uZ↓(r) , (11)

where the Fiσ’s are envelope functions and the uiσ’s arethe Bloch functions at Γ. They are bonding combinations

of atomic pX , pY or pZ orbitals with spin σ = ↑ or ↓ alongZ (the X ‖ [100], Y ‖ [010], Z ‖ [001] axes being thecubic axes).

Alternatively the u’s (hence the F ’s) can be mappedby a unitary transform onto the eigenstates |J,mJ〉 of J2

and Ju, where J = L+S is the total angular momentumof the Bloch functions and u is an arbitrary quantizationaxis. Actually, any six bands (J = 3/2 & J = 1/2 multi-plets) or four bands (J = 3/2 multiplet) flavor of the k ·pmodel for the valence band can be put in the form of Eq.(11).40–42 In the |X ↑〉, |Y ↑〉, |Z ↑〉, |X ↓〉, |Y ↓〉, |Z ↓〉basis set, the Hamiltonian ∆H(ε) is:42,45,46

∆H(ε) =

[∆Hs(ε) 03×3

03×3 ∆Hs(ε)

], (12)

where the 3×3 up or down spin sub-block ∆Hs(ε) reads:

∆Hs(ε) =

lεXX +mεY Y +mεZZ nεXY nεXZnεXY mεXX + lεY Y +mεZZ nεY ZnεXZ nεY Z mεXX +mεY Y + lεZZ

(13)

4

with:

l = a+ 2b ; m = a− b ; n = d√

3 . (14)

a ≡ av is the hydrostatic deformation potential, b theuniaxial deformation potential, and d the shear deforma-tion potential of the valence band.47 The above modelaccounts for spin relaxation due to “direct” spin-orbit in-teractions within the valence band manifold;17,24,48 Evenif ∆H is diagonal in spin, it can drive relaxation betweentextured spin states such as the mixed heavy- and light-hole states encountered in spin-orbit qubits.

We next write:

〈r|0〉 =∑

i∈X,Y,Z

∑σ∈↑,↓

aiσ(r)uiσ(r) (15a)

〈r|1〉 =∑

i∈X,Y,Z

∑σ∈↑,↓

biσ(r)uiσ(r) , (15b)

and assume that the extension of the envelope functionsaiσ(r) and biσ(r) is much smaller than the typical wavelength of the phonons at the Larmor frequency. We canthen complete the integration of Eq. (9) in the dipoleapproximation for the phase factor eiq·r (see AppendixA), and get:

Γ3Dph =

ω3

8π~ρcoth

(~ω

2kBT

)×∑α=l,t

(ω2

v7α

9∑n=1

AnΛAnα +

1

v5α

2∑n=1

BnΛBnα

), (16)

where the Λ’s depend on material parameters and theAn’s and Bn’s can be expressed as a function of the fol-lowing moments of the hole envelopes:

Sij =

∫d3r

[a∗i↑(r)bj↑(r) + a∗i↓(r)bj↓(r)

](17a)

Rkij =

∫d3r

[a∗i↑(r)bj↑(r) + a∗i↓(r)bj↓(r)

]rk (17b)

T kk′

ij =

∫d3r

[a∗i↑(r)bj↑(r) + a∗i↓(r)bj↓(r)

]rkrk′ (17c)

and:

Omnijkl = RmijRn∗kl −

1

2

(Tmnij S∗kl + Tmn∗kl Sij

). (18)

Namely,

A1 =∑i

Oiiiiii (19a)

A2 =∑i 6=j

Oiiiijj +Oiijjii (19b)

A3 =∑i 6=j

Oiijjjj (19c)

A4 =∑i 6=j 6=k

Oiijjkk (19d)

A5 =∑i 6=j

Oiiijij +Oiiijji +Oiijiij +Oiijiji (19e)

A6 =∑i 6=j 6=k

Oiijkjk +Oiijkkj (19f)

A7 =∑i 6=j

Oijiiij +Oijiiji +Oijijii +Oijjiii + c.c. (19g)

A8 =∑i6=j 6=k

Oijijkk +Oijjikk + c.c. (19h)

A9 =∑i6=j 6=k

Oijikjk +Oijkijk +Oijikkj +Oijkikj

+Oijjkik +Oijkjik +Oijjkki +Oijkjki , (19i)

where c.c. stands for complex conjugate, and:

B1 =∑i

SiiS∗ii (20a)

B2 =∑i6=j

Sij(S∗ij + S∗ji) . (20b)

The sums over i, j, k run over X,Y, Z. The ΛAnl param-

eters for the longitudinal phonons are:

ΛA1l =

140a2 + 224ab+ 176b2

105(21a)

ΛA2l =

140a2 + 56ab− 88b2

105(21b)

ΛA3l =

140a2 − 112ab+ 80b2

105(21c)

ΛA4l =

140a2 − 112ab+ 8b2

105(21d)

ΛA5l =

12d2

35(21e)

ΛA6l =

4d2

35(21f)

ΛA7l =

28ad+ 8bd

35√

3(21g)

ΛA8l =

28ad− 16bd

35√

3(21h)

ΛA9l =

4d2

35(21i)

5

while the ΛAnt parameters for transverse phonons are:

ΛA1t =

72b2

35(22a)

ΛA2t = −36b2

35(22b)

ΛA3t =

48b2

35(22c)

ΛA4t = −12b2

35(22d)

ΛA5t =

16d2

35(22e)

ΛA6t =

2d2

7(22f)

ΛA7t =

2√

3bd

35(22g)

ΛA8t = −4

√3bd

35(22h)

ΛA9t =

3d2

35. (22i)

Finally, the ΛBnl and ΛB

nt parameters are:

ΛB1l =

24b2

5(23a)

ΛB2l =

4d2

5, (23b)

ΛB1t =

36b2

5(24a)

ΛB2t =

6d2

5. (24b)

The TA1 and TA2 branches have been summed up in thetransverse phonon parameters ΛA

nt and ΛBnt. The two Bn

terms are “band mixing” terms whose prefactor scaleswith the phonon strains (∝ q2/ω ≡ ω) and density ofstates (∝ ω2) at the Larmor frequency. They result fromthe zero-th order expansion eiq·r ∼ 1 in Eq. (8) (homo-geneous component of the strain), and are ruled by the

overlap between the different envelope functions. The 9An terms are dipole-like terms that follow from the first-order expansion of eiq·r in Eq. (8). They show, therefore,an additional ∝ q2 ≡ ω2 dependence. Although the Anterms also mix bands, we point out that the Bn termsdo not exist in one-band models where the envelopes ofdifferent states are, by design, orthogonal. Note that theAn and Bn terms are also dependent on ω (see next sec-tion). We will sort the different contributions in sectionIII A.

D. Model 2D & 1D phonon band structures

The 3D model discussed up to now is suitable for“large” enough structures where the phonons (yet notnecessarily the electrons or holes) are weakly confined(bulk-like dispersion). In confined structures such asnanowires or thin films, the vibrational modes may, how-ever, be more adequately described by a 1D or 2D phononband structure.39,49–51 The integration over ϕ and/or θin Eq. (9) is then replaced by a sum over phonon sub-bands. In general, many sub-bands can contribute to therelaxation rate. Yet the splitting between phonon sub-bands increases with lateral confinement so that only the1D or 2D acoustic branches will ultimately couple to thequbit once the phonons get confined enough. The condi-tions in which this strongly confined regime is achievedwill be explored in section IV A.

We can derive expressions similar to Eq. (16) in thestrongly confined regime by making simple assumptionsfor the acoustic phonon wave functions. Namely, weassume that the displacements are homogeneous in thethickness of the film (2D) or in the cross section of thenanowire (1D). This corresponds to a choice of peri-odic Born-von-Karman instead of free-standing boundaryconditions at the surface of the film or wire, or equiva-lently to a sampling of the 3D phonon band structure atq⊥ = 0, where q⊥ is the component of the wave vectorperpendicular to the film or wire. The relevance and va-lidity of this approximation will be discussed in sectionIV A. We then reach the following expressions for the re-laxation rate in thin films (see Supplementary Material):

Γ2Dph =

ω2

4~ρLcoth

(~ω

2kBT

) ∑α=l,t

(ω2

v6α

∑n

AnΛAnα +

1

v4α

∑n

BnΛBnα

), (25)

where L is the thickness of the film, and in nanowires:

Γ1Dph =

ω

2~ρScoth

(~ω

2kBT

) ∑α=l,t

(ω2

v5α

∑n

AnΛAnα +

1

v3α

∑n

BnΛBnα

), (26)

where S is the cross sectional area of the wire.

The An’s, Bn’s and Λ’s depend on the dimensionality and on the crystallographic orientation of the nanostructure.

6

They are given for (100), (110), and (111) films, and for[100], [110], and [111] oriented wires in the Supplemen-tary Material.

With respect to bulk phonons [Eq. (16)], the relax-ation rate is inversely proportional to the characteristicsize of the system, and the dependence on ω is reducedby one power each time the phonons get confined in anadditional direction. This results from the scaling of thephonon density of states with dimensionality. We empha-size, however, that the net dependence on the relaxationrates on the Larmor frequency ω is typically two ordersof magnitudes stronger than suggested by Eqs. (16), (25)and (26), because the An’s and Bn’s also depend on ω.Indeed, they vanish at zero magnetic field if |0〉 and |1〉form a Kramers degenerate pair linked by time-reversalsymmetry (as is the case in a spin-orbit qubit), and in-crease as ω2 ∝ B2 once a magnetic field B breaks time-reversal symmetry.

Also, Eqs. (16), (25) and (26) can actually be used tocalculate the transition rates between any pair of states(provided the dipole approximation applies at the transi-tion energy), in order to set up master equations for spinand charge relaxation in the system for example.36,52 Thecoth term shall be replaced with N(ω) = 1/(eβ~ω−1) forabsorption rates and by N(ω) + 1 for emission rates.

E. Numerical phonon band structures

Real qubit devices are often made of nanostructuredstacks of materials with complex phonon band struc-tures. In order to validate the above models and ad-dress, e.g., the impact of encapsulation materials on therelaxation rates, we have developed a numerical approachto the acoustic phonon band structure based on contin-uum elasticity theory.43,44,50 The latter is expected tohold at small strains in semiconductor heterostructureswith characteristic sizes around 10 nm;53,54 Non-localand surface effects have been shown to change the bend-ing rigidity of 10 nm × 10 nm Si and SiO2 beams byless than 10%.55 We consider, in particular, 1D struc-tural models with arbitrary cross-section and materials.This cross-section is meshed and the dynamical matrixis computed from a finite-element discretization of thecontinuum elasticity theory.

The relaxation rate is still calculated from Fermi’sgolden rule:

Γph =1

2~ωcoth

(~ω

2kBT

)×∑α

∑q∈qα

1

|vαq|

∣∣∣〈0|∆H[εαq(r)]|1〉∣∣∣2 , (27)

where α run over all 1D sub-bands, qα are the solutionsof ωαqα = ω (if any), and:

[εαq(r)]ij =1

2

(∂ui(r)

∂rj+∂uj(r)

∂ri

)(28)

~Vfg

Vac

Vbg

Gate

Si

SiO2

30 nm 10 nm

HfO2

x y

z

θ

φ

B

FIG. 1. Schematics of the hole qubit device on silicon-on-insulator: a silicon nanowire channel (red) with widthW = 30nm and height H = 10 nm lies on top of the buried oxide(blue). Top gates (orange) with length Lg = 30 nm andspacing Sg = 30 nm partly cover the silicon nanowire (over 20nm). The gate stack is made of 2 nm of SiO2 and 2 nm of HfO2

(green). The central front gate defines the hole quantum dot.An iso-density surface of the confined hole ground-state wavefunction is depicted in yellow. The device axes are defined onthe figure.

with:

u(r) =1√ρ(y, z)

c(y, z)eiqx (29)

and c ≡ cαq the eigenvector of the dynamical matrix(normalized so that

∫dydz |c(y, z)|2 = 1). The 1D wave

vector is aligned along the x axis. Note that we do notmake the dipole approximation on the phase factor here.The convergence of Eq. (27) is discussed in Appendix B.

III. APPLICATION TO HOLE-SPIN QUBITSON SILICON-ON-INSULATOR

In this section, we apply the models introduced in sec-tion II to a hole spin-orbit qubit on silicon-on-insulator.The device, represented in Fig. 1, is the same as the onemodeled in Ref. 25, and is similar to the one measuredin Ref. 19. It is made of a [110]-oriented silicon nanowirechannel with width W = 30 nm [(110) facets] and heightH = 10 nm [(001) facets] lying on top of a 25 nm thickburied oxide and silicon substrate. A 30 nm long centralgate partly overlapping the nanowire controls an electro-statically defined quantum dot. The two other gates onthe left and right mimic neighboring qubits. The gatestack is made of 2 nm of SiO2 and 2 nm of HfO2. Thewhole device is embedded in Si3N4. The qubit can be fur-ther controlled using the substrate as a back gate, whichallows to tune separately the chemical potential and theelectric field in the dot. We bias the central front gate atVfg = −0.1 V and ground the other front gates in order toconfine holes. Rabi oscillations can be electrically drivenby a radio-frequency signal on the central front gate.

7

The potential landscape in the device is computed witha finite volumes Poisson solver and the qubit states (cho-sen as the topmost valence band states) with a finite dif-ferences discretization of the six bands k ·p model.25 Forthe model phonon band structures, the group velocitiesare vl = 9000 m/s and vt = 5400 m/s. The numericalphonon band structures are computed with the elasticconstants listed in Table I. The deformation potentialsof the valence band of silicon are a = 2.38 eV, b = −2.1eV, and d = −4.85 eV.56,57 The amplitude of the mag-netic field is adjusted so that the Larmor frequency ofthe qubit sticks to ω/(2π) = 10 GHz. At that frequency,the wave length of bulk acoustic phonons is greater thanλt = 2πvt/ω = 540 nm, hence is much longer than thedot size (∼ 30 nm) and within the range of validity of thedipole approximation. The temperature is set to T = 100mK. The results are, however, weakly dependent on tem-perature T ≤ 100 mK, as 1 < coth(~ω/2kBT ) < 1.017 inthis range (base cryostat temperature was T ' 15 mK inRefs. 18 and 19). Also, the splitting between the groundqubit states and the first excited orbital states is alwayslarger than 2 meV in this device (Fig. 8 of Ref. 25).From the scaling laws given in Ref. 28, we do not expectsignificant two-phonon corrections at such low tempera-tures, although we did not compute them explicitly atthis stage.

As discussed in Ref. 25, the Larmor and Rabi fre-quencies of this device are strongly dependent on theorientation of the magnetic field and on the back gatevoltage Vbg. This is a fingerprint of the action of thespin-orbit coupling on the holes, which (in the absence ofstrains) have a dominant heavy-hole character along thestrong confinement axis z = [001].17,24,48 In particular,the Rabi frequency exhibits a dip at back gate voltageVbg ' −0.15 V where the hole wave functions show anapproximate inversion center that hampers the action ofspin-orbit coupling (the hole spins decouple from the gateelectric field). This will be further investigated in sectionIII B.

We first discuss which terms do play a role in Eq. (16),then the optimal bias point and magnetic field orientationfor this qubit.

A. Which terms do play a role ?

As the nanowire is embedded in other materials, weassume that the phonons are weakly confined in siliconand thus make use of the 3D model [Eq. (16)] for thecalculation of the relaxation time. The actual impactof dimensionality and encapsulation materials will be in-vestigated in sections IV A and IV B. Eq. (16) containsmany different terms that can be categorized as a func-tion of i) their nature [An (dipole) or Bn (homogeneousband mixing) terms]; ii) the polarization of the phonons(LA or TA) and iii) the deformation potentials involved.In order to sort Eq. (16) into these categories, the re-laxation rate Γ3D

ph is plotted as a function of the Larmor

Material ρ c11 c12 c44Si (aniso)58 2.329 166.0 64.0 79.6Si (iso) 2.329 188.6 52.8 67.9SiO2

59 2.200 77.5 15.7 30.9Si3N4

60,61 2.500 193.0 65.0 64.0Diamond62 3.500 1076.0 125.0 577.0

TABLE I. Density ρ (g/cm3) and elastic constants c11, c12and c44 (GPa) of the different materials considered in thiswork.58–62 The anisotropic (aniso) elastic constants of sili-con are those measured in this material.58 They give rise toanisotropic phonon bands with two non-degenerate transverseacoustic branches in bulk. The isotropic (iso) elastic constantshave been modified in order to get isotropic phonon bandswith degenerate transverse acoustic branches consistent withthe model 3D phonon band structure (vl = 9000 m/s andvt = 5400 m/s). The elastic models for amorphous SiO2 andSi3N4 are isotropic (c11 − c12 = 2c44).

frequency ω of the qubit in Figs. 2a-c (Vbg = −0.15 V)and decomposed (a) into An and Bn contributions, (b)LA and TA contributions, and (c) a, b and d contribu-tions (the other deformation potentials being set to 0). Itis clear from Fig. 2 that the band mixing Bn terms dom-inate the relaxation through the coupling to TA phononsby the shear deformation potential d. The B2ΛB

2t ∝ d2

term of Eq. (16) actually makes the largest contribu-tion to the relaxation rate, followed by the B1ΛB

1t ∝ b2

term. This implies that we can make the approxima-tion eiq·r ∼ 1 in Eq. (8) and drop all dipole An termsarising from the first-order expansion of this phase fac-tor. This is opposite to electrons in the single band effec-tive mass approximation, whose relaxation is exclusivelyruled by such dipole terms. The action of spin-orbit cou-pling within the valence band is, indeed, very dependenton the heavy- and light-hole balance and is, therefore,sensitive to band mixing by phonons. This will be fur-ther discussed in section III B. We also emphasize thatthe dipole terms can be dominant for charge relaxationbetween states whose main envelopes are orthogonal.

The relaxation rate is ∝ ω4 at low Larmor frequencyand ∝ ω5 at large Larmor frequency. The ∝ ω5 behaviorresults from the ∝ ω3 prefactor of the Bn terms (phononstrains and density of states) and from the ∝ ω2 depen-dence of the Bn terms themselves. Indeed, as discussedin section II D, the Bn terms vanish at zero magneticfield when |0〉 and |1〉 are time-reversal symmetrics one ofeach other, and increase as ω2 once a finite magnetic fieldbreaks time-reversal symmetry. Accordingly, the (how-ever negligible) An term show a ∝ ω7 behavior (owing tothe additional q2 ∝ ω2 dependence of the dipole terms).At small magnetic field, the relaxation rate departs fromthe ∝ ω5 behavior due to the coth[~ω/(2kBT )] prefac-tor that accounts for the larger population of acousticphonons at the Larmor frequency (enhancement of ab-sorption and stimulated emission processes).

The contributions of LA and TA phonons to the relax-ation rate are plotted as a function of back gate voltage

8

10 1 100 101 102

/2 (GHz)10 8

10 5

10 2

101

104

107

ph (H

z)

(a)

TotalB terms

4

5

10 1 100 101 102

/2 (GHz)10 8

10 5

10 2

101

104

107

(b)

TotalLATA

10 1 100 101 102

/2 (GHz)10 8

10 5

10 2

101

104

107

(c)

Totalabd

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

0

50

100

150

ph (H

z)

(d)TAB termsabd

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

0

2

4

6

8

ph (H

z)

(e)LAB termsabd

FIG. 2. (a-c) Contribution of the different terms of Eq. (16) to the relaxation rate as a function of the Larmor frequency ω(Vbg = −0.15 V). The relaxation is dominated by (a) the homogeneous band mixing Bn terms, (b) the transverse acousticphonons, and (c) the shear deformation potential d. The above conclusions are valid in a wide range of back gate voltagesVbg, as shown in panels (d) and (e). The orientation of the magnetic field is θ = 45, ϕ = 0 in all panels and its amplitude isadjusted so that the Larmor frequency is ω/(2π) = 10 GHz in panels (d, e).

in Figs. 2d,e. The conclusions drawn above remain validover the whole range of investigated back gate voltages.We discuss in more detail the dependence of Γ3D

ph on Vbgin the next section.

B. Optimal operation point

The total relaxation rate is plotted as a function of Vbgin Fig. 3a. We have added for comparison the relaxationrate Γjn due to Johnson-Nyquist noise on the central frontgate (zero-point and thermal fluctuations in the circuitconnected to that gate):32,63

Γjn = 4πR

R0

∣∣∣〈0|Dfg|1〉∣∣∣2ωcoth

(~ω

2kBT

), (30)

where R is the resistance connected to the gate, R0 =h/e2, and Dfg(r) = ∂Vt(r)/∂Vfg is the derivative of thetotal potential Vt(r) in the device with respect to thefront gate voltage Vfg. We assume R = 200 Ω.

The relaxation rate Γ3Dph is plotted along with the Rabi

frequency fR in Fig. 3b. The Rabi oscillations are driven

by a radio-frequency modulation δVfg = 1 mV on thefront gate. The quality factor Q1 = fR/Γ

3Dph is also plot-

ted on that figure. It gives the number of Rabi oscil-lations that can be achieved within one relaxation timeT1 = Γ−1ph . We have only accounted for phonons in thisfigure, which therefore provides an upper limit to thequality factor of the qubit. Other mechanisms for re-laxation (Johnson-Nyquist and charge noise...) are in-deed extrinsic to the qubit and are, in principle, moreamenable to optimization (e.g., by the reduction of cir-cuit impedances for Johnson-Nyquist noise). Multi-phonon and photon processes (relevant at high enoughtemperature),28,29 and additional mechanisms for spin-orbit coupling neglected in this study (through remotecoupling to the conduction bands in particular27) mayalso degrade this figure of merit. The impact of Johnson-Nyquist noise on the quality factor and coherence timeis discussed in Appendix C.

The relaxation rate shows modulations as a functionof Vbg that mimic those of the Rabi frequency. In par-ticular, Γ3D

ph displays a dip near Vbg = −0.15 V wherethe hole wave functions feature an approximate inver-

9

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

100

101

102

(Hz)

(a)

phjnph+jn

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

0

20

40

60

80

100

f R (M

Hz)

fRph

Q1

0

50

100

150

200

ph (H

z)

0

10

20

30

40

50

Q1 (

×104 )

(b)

0 30 60 90 120 150 180 (°)

0

30

60

90

120

150

180

(°)

(c)

500

1000

1500

2000ph

(Hz)

0 30 60 90 120 150 180 (°)

0

30

60

90

120

150

180

(°)

(d)

20

40

60

80

100

120

f R (M

Hz)

0 30 60 90 120 150 180 (°)

0

30

60

90

120

150

180

(°)

(e)

10

20

30

40

Q1 (

×104 )

FIG. 3. (a) Johnson-Nyquist (Γjn), phonon-induced (Γ3Dph ) and total relaxation rate as a function of back gate voltage Vbg.

Γjn is computed for a resistance R = 200 Ω on the central front gate. (b) Rabi frequency fR, phonon-induced relaxation rateΓ3Dph , and quality factor Q1 = fR/Γ

3Dph as a function of Vbg. (c) Relaxation rate Γ3D

ph , (d) Rabi frequency fR, and (e) quality

factor Q1 = fR/Γ3Dph as function of the orientation of the magnetic field characterized by the azimuthal and polar angles θ and

ϕ defined on Fig. 1 (and consistent with Refs. 19, 25 and 26), at Vbg = −0.2 V. The Larmor frequency is ω/(2π) = 10 GHzin all plots. The orientation of the magnetic field is θ = 45, ϕ = 0 in panels (a) and (b). Note that panel (d) looks differentfrom Ref. 25 because it is plotted at constant Larmor frequency instead of constant magnetic field amplitude.

sion center.25 This hampers the action of spin-orbit cou-pling and decouples the holes from the radio-frequencyelectric field from the front gate. The Rabi oscillationsare therefore slow but the holes get also decoupled fromJohnson-Nyquist and charge noise at that point.

The dip in Γ3Dph is not, however, as marked as the dip in

the Rabi frequency and Γjn. This follows, in particular,from the fact that the band mixing terms can not be castas the action of an effective electric field due to phonons;only the hydrostatic ∝ a terms can be so. The decrease ofthe relaxation rate at large positive or negative back gatevoltage is due to the strong lateral confinement in thestatic electric field of the gate.25,26 Although the lifetimeis longer at large |Vbg|, the Rabi frequency is smaller,which slightly lowers the quality factor.

The phonon-limited lifetimes are typically shorter (yetstill > 5 ms) than expected in electron qubits15,32,33 ow-ing to the strong spin-orbit coupling in the valence band.This is however balanced by much larger Rabi frequen-cies, allowing for significant quality factors. Hole qubitsare also much more sensitive to Johnson-Nyquist noisethan electron qubits.15,32 As a matter of fact, electri-cal and charge noise is presumably dominating decoher-ence and relaxation in present hole qubit devices.18 Thephonon-limited quality factor is maximal near Vbg =−0.2 V. As suggested by Fig. 3a, the position of the

optimal bias point for manipulation will move furtheraway from Vbg = −0.15 V when increasing electrical andcharge noise (see Appendix C). Anyhow, the qubit maybe brought back to the “sweet spot” Vbg = −0.15 V inbetween manipulations, where the lifetime is longest.17,38

The Rabi frequency, relaxation rate Γ3Dph and quality

factor are also strongly dependent on the magnetic fieldorientation (Figs. 3c,d,e). The orientational dependenceof fR has been discussed in detail in Refs. 25 and 26and is the fingerprint of the mostly heavy-hole characterof the qubit states and of the symmetries of the device.Following the lines of Ref. 26, we reach the following ex-pressions for the dominant B1 and B2 terms [Eqs. (20b)]near the “sweet spot” Vbg = −0.15 V, at the leadingorder in the channel height H:

B1 =m2

0H4

2~2γ22(κµBB)2 sin2 θ (31a)

B2 = 2B1 , (31b)

wherem0 is the free electron mass, γ2 = 0.339 is a valenceband Luttinger parameter, κ = −0.42 is the Zeeman coef-ficient of the holes and µB is Bohr’s magneton.27 B1 andB2 behave, as expected, as ω2 ∝ B2. At this order in H,they do not depend on the width W of the channel, henceon lateral confinement. In this respect, phonon-induced

10

relaxation behaves differently than Johnson-Nyquist re-laxation and Rabi oscillations, which do require lateralconfinement.26 This results from the fact that strains cancouple directly heavy- and light-hole Bloch functions,at variance with a radio-frequency electric field. Actu-ally, the ∝ B sin θ in-plane magnetic field mixes the ma-jority, heavy |3/2,+3/2〉 component of |1〉 with a light|3/2,+1/2〉 envelope, which can then be coupled by thephonons to the majority |3/2,−3/2〉 component of |0〉through the strain Hamiltonian ∆H(ε) [Eq. (12)]. Themagnetic mixing between the |3/2,+3/2〉 and |3/2,+1/2〉envelopes of |1〉 is inversely proportional to the splittingbetween the confined heavy- and light-hole subbands,which gives rise to theH4/γ22 dependence in Eqs. (31) (asin a quantum well). The relaxation rate is hence maxi-mal near θ = π/2; the dependence on θ on Fig. 3c is evenstrenghtened because the effective g factor of heavy-holesis minimal at θ = π/2, so that larger magnetic fields areneeded to reach the target Larmor frequency ω/(2π) = 10GHz. The quality factor is weakly dependent on ϕ andis significant in a wide band of θ’s. It peaks near θ = 30

and θ = 150, close to the reference orientation chosenin Figs. 3a,b. Note that fR ∝ sin θ when θ → 0 or π sothat Eqs. (31) suggest that Q1 → ∞ (although this isirrelevant since fR → 0). Higher-order contributions toEq. (31) give rise to a finite Γph and to the dependenceon ϕ clearly visible on Fig. 3c, ensuring that Q1 → 0when θ → 0 or π as shown in Fig. 3e.

IV. EFFECTS OF PHONON CONFINEMENTAND ENCAPSULATION MATERIALS

In this section, we discuss the effects of phononconfinement39 on the relaxation rate, as well as the im-pact of encapsulation materials. We highlight how thelifetime of the qubit depends on its vibrational environ-ment over long length scales.

A. Phonon confinement

In section II, we have derived the relaxation rate forbulk phonons [Eq. (16)], strongly confined 2D [Eq. (25)]and strongly confined 1D [Eq. (26)] phonons. In thissection, we validate the 3D, 2D and 1D expressions onnumerical calculations of the phonon band structure andaddress their range of validity. We discuss the impact ofphonon confinement on the relaxation rate of the qubit.

For that purpose, we consider the same qubit as inFig. 1, but coupled to the phonon band structure of asquare [110]-oriented wire with varying side ly = lz, or ofa rectangular [110]-oriented wire with side lz = H = 10nm and varying ly. In the first case (square wire), weexpect a transition from a 1D regime at small ly = lzto a 3D regime at large ly = lz, and in the second case(rectangular wire) a transition from a 1D regime at smallly to a 2D regime at large ly. The relaxation rates are

0.00 0.02 0.04 0.06q (nm 1)

0

10

20

30

40

/2 (G

Hz)

(a)LA

TA

10 GHz

0.00 0.02 0.04 0.06q (nm 1)

0

10

20

30

40

/2 (G

Hz)

(b)LA

To Fx2

Fx1

0.00 0.01 0.02q (nm 1)

0

2

4

6

8

10

12

/2 (G

Hz)

(c)0.00 0.01 0.02

q (nm 1)

0

2

4

6

8

10

12

/2 (G

Hz)

(d)

FIG. 4. Phonon band structures of a ly = 30 nm × lz =10 nm silicon nanowire with (a) periodic boundary condi-tions and (b) free-standing boundary conditions; and of aly = 600 nm × lz = 600 nm silicon wire with (c) periodicboundary conditions and (d) free-standing boundary condi-tions. The isotropic elastic constants of silicon were used inthis calculation (table I). The gray dash-dotted line is theLarmor frequency of the qubit ω/(2π) = 10 GHz. The lineartorsional (To) and parabolic flexural (Fx1 and Fx2) modes ofthe free-standing nanowire are clearly visible in panel (b).

computed with Eq. (27).

We consider either i) free-standing boundary condi-tions (no stress perpendicular to the surfaces) or ii) peri-odic Born-von-Karman boundary conditions at the sur-face of the wires. For periodic boundary conditions,the resulting phonon band structure is nothing else thanthe bulk band structure sampled at wave vectors q =(q, 2πny/ly, 2πnz/lz) (in the device axes frame), where qis the 1D wave vector of the wire and ny, nz are inte-gers. Each pair (ny, nz) defines a set of three sub-bands(sampled in the bulk LA, TA1 and TA2 branches). Theacoustic branches (ωαq → 0 when q → 0) of the wireare the ny = nz = 0 sub-bands. In this approximation,the displacements are homogeneous in the cross-sectionof the wire. The other 1D sub-bands have finite ωα0.In the strongly confined regime (only the three acous-tic branches of the wires couple to the qubit), the re-laxation rate shall therefore exactly match the expres-sions for Γ1D

ph or Γ2Dph that were established under these

assumptions. On the opposite, in the weakly confinedregime (many sub-bands couple to the qubit), the relax-ation rate shall tend to Γ3D

ph . This provides a numericaltest of these expressions, and allows for a clear investiga-

11

tion of the transition from one regime to an other. Free-standing boundary conditions are more relevant for trulyconfined phonons with inhomogeneous displacements inthe wire cross section. Also, the phonon band structureof a free-standing wire shows specific features:50,51 thereare, in particular, four branches whose ωαq → 0 whenq → 0 (one linear longitudinal and one linear torsionalmode, and two parabolic flexural modes). This reflectsthe translational and rotational invariances of the elasticenergy of a free-standing structure. We will discuss thecontributions of each mode to the relaxation rate and therelevance of the 1D formula in this context.

The phonon band structure of a ly = 30 nm × lz =10 nm wire is plotted as an illustration in Figs. 4a,b,for both periodic and free-standing boundary conditions.We use the isotropic model for the elastic constants ofSilicon (Table I). In the free-standing case, the phononband structure is computed with the numerical finite-differences approach outlined in section II E. The twoparabolic flexural modes, as well as the torsional and lon-gitudinal branches are clearly visible on this figure. Thephonon band structures of a ly = 600 nm× lz = 600 nmwire with free-standing and periodic boundary conditionsare likewise plotted in Figs. 4c,d. The energy and wavevector range where the flexural branches are parabolic de-creases with increasing wire size and the acoustic groupvelocities outside this range get closer to those of the bulkLA and TA phonons. The number of sub-bands belowthe Larmor frequency is larger with free-standing thanwith periodic boundary conditions (with the appearanceof, e.g., breathing modes), but the sub-bands are muchmore degenerate in the latter case.

For periodic boundary conditions, the relaxation ratein the square wires is plotted as a function of ly = lzin Fig. 5a, and the relaxation rate in the rectangularwires is plotted as a function of ly in Fig. 5c. Both LAand TA contributions are displayed. The numerical dataare compared with the models for the 1D and 2D or 3Dphonon band structures.

First of all, the transitions from the 1D to 3D regime,and from the 1D to the 2D regime are clearly visible inthese figures. As expected, the analytical expressions ofsection II reproduce very well the numerical data in thesedifferent regimes. In the 1D regime, the relaxation rate isinversely proportional to the cross-sectional area S = lylzof the wire [Eq. (26)], as the overlap between the squaredacoustic phonon and qubit wave functions scales as 1/S.The relaxation rate in the 1D regime can hence be muchlarger than in the 3D regime depending on S (and onthe Larmor frequency ω). The transition from the 1Dto the 3D regime occurs in the range ly = lz ' 1000nm – which is comparable to the typical wave length ofthe 3D phonons involved in the relaxation but very largewith respect to the size of the qubit itself. This resultsfrom the fact that the splitting between the 1D phononsub-bands remains greater that the Larmor frequency (10GHz) over a wide range of dimensions, so that only the1D LA and TA branches can contribute to scattering

until ly = lz ' 500 nm. The introduction of a new sub-band gives rise to a peak in the relaxation rate (due tothe Van-Hove singularity in the density of states of aparabolic sub-band), until the number Nph of sub-bandsat the Larmor frequency is large enough to reach the3D limit. Indeed, at wide enough ly = lz, Nph 1 isapproximately given by:

Nph 'S

(2π)2

∑α∈l,t1,t2

π

(ω

vα

)2

, (32)

where vl, vt1 = vt2 = vt are the longitudinal and trans-verse sound velocities in bulk silicon. The ∝ S depen-dence of Nph balances the ∝ 1/S dependence of thesquared phonon amplitudes, so that the relaxation ratebecomes independent on S (see Appendix B for a dis-cussion on the nature of the convergence). The sameconclusions hold for the 1D to 2D transition, which alsooccurs around ly = 1000 nm.

The relaxation rates are plotted for free-standingboundary conditions in Fig. 5b,d. The equations forΓ1Dph and Γ2D

ph still hold surprisingly well in the stronglyconfined regime despite the differences between periodicand free-standing phonon band structures (Fig. 4). Thetransition from the 1D to 2D or 3D regime takes place atsignificantly smaller lateral dimensions (' 500 nm) dueto the larger density of low energy sub-bands. In the 1Dlimit, the qubit mostly couples to the longitudinal andflexural modes of the wire. We emphasize, though, thatthe qubit was placed near the highly symmetric centralposition of the square or rectangular wire where the ef-fects of boundary conditions are expected to be minimal.

Irrespective of the choice of boundary conditions, Fig.5 clearly highlights the effect of phonon confinement onthe relaxation rate in the qubit. Γph can be stronglydependent on the geometry far away from the qubit asthe spin couples to very long wave length energy acous-tic phonons that can probe the device over hundreds ofnanometers. We further support this conclusion in thenext section by studying the impact of the encapsulationmaterials on the relaxation in the qubit.

B. Impact of encapsulation materials

As discussed previously, the qubits are usually embed-ded in complex stacks of materials, which can have a sig-nificant impact on the long-wave length acoustic phononsthat couple to the spins. In particular, the velocity anddegree of confinement of the phonons is highly depen-dent on the hardness of the materials around the channel.In order to explore this issue, we compute the phononband structure of the device of Fig. 1 modeled as arectangular silicon nanowire with sides W = 30 nm andH = 10 nm embedded in a homogeneous material withvarying elastic constants. We apply periodic boundaryconditions over the cross-section in a supercell with sidesly = lz = Lcell = 1 µm.

12

101 102 103 104

ly, lz (nm)10 1

100

101

102

103

104

ph (H

z)

1D 3D (a)

LATA

101 102 103

ly, lz (nm)

102

103

104

ph (H

z)

(b)

Free-standing NW1Dph [Eq. (26)]3Dph [Eq. (16)]

101 102 103 104 105

ly (nm)

102

103

104

ph (H

z)

1D 2D (c)

LATA

101 102 103 104

ly (nm)103

104

105

ph (H

z)

(d)

Free-standing NW1Dph [Eq. (26)]2Dph [Eq. (25)]

FIG. 5. Relaxation rate Γph computed in a qubit embedded (from a vibrational point of view) in (a) a square nanowire (NW)with sides ly = lz and periodic boundary conditions, (b) the same square nanowire with free-standing boundary conditions,(c) a rectangular nanowire with side lz = 10 nm, varying ly, and periodic boundary conditions, and (d) the same rectangularnanowire with free-standing boundary conditions. For periodic boundary conditions, the phonon band structure is sampledfrom the bulk LA and TA branches and the corresponding contributions are displayed separately. The semi-analytical resultsfor Γ1D

ph , Γ3Dph (panels a, b) and Γ2D

ph (panels c, d) are also plotted as solid lines. The transitions from the 1D to 3D regimes(panels a, b) and from the 1D to 2D regimes (panels c, d) are clearly visible. All data were computed with the isotropic modelfor the elastic constants of silicon (see Table I). The back gate voltage is Vbg = −0.2 V, and the orientation of the magneticfield is ϕ = θ = 45.

We use both the standard (anisotropic) elastic con-stants of silicon and the isotropic modification giving rise(in bulk) to a LA branch with velocity vt = 9000 m/s andto two degenerate TA branches with velocity vt = 5400m/s, consistent with the analytical 3D model of sectionII (see Table I). The elastic constants of the encapsula-tion material are rescaled by a factor κ with respect tothose of silicon but the density is the same.

The relaxation rate Γph is plotted as a function of κ inFig. 6a. The horizontal dotted line is the relaxation ratecomputed in a free-standing W = 30 nm × H = 10 nmnanowire. As expected, Γph tends to this limit whenκ→ 0 (encapsulation material much softer than silicon).The relaxation rate then decreases continuously as theencapsulation material becomes harder and harder. Theresults obtained with the isotropic and anisotropic mod-els for the elastic constants are very close, which showsthat the moderate anisotropy of silicon does not havemuch impact on the relaxation.

In order to get further insights into these trends, weplot in Figs. 7a,b,c the 1D phonon sub-band structure

computed at κ = 0.1, κ = 1 and κ = 10 (isotropic model).The horizontal dash-dotted line on these plots is the Lar-mor frequency of the qubit ω/(2π) = 10 GHz. The longi-tudinal and transverse sound velocities (drawn from thethree branches whose ωαq → 0 when q → 0) are plottedas a function of κ in Figs. 7d, and the number of 1Dsub-bands that cross the Larmor frequency of the qubitis plotted in Fig. 7e.

The sound velocities v ' venc =√κvSi are defined

by the encapsulation material that fills 99.97% of thesupercell (vSi and venc being respectively the sound ve-locities in silicon and in the encapsulation material). Ac-cordingly, the density of 1D phonon sub-bands is essen-tially proportional to 1/v2enc, so that the number Nph ofphonon sub-bands at the Larmor frequency is given byEq. (32) with vα replaced by venc,α (and S by L2

cell).Nph therefore behaves as 1/κ. This trend is clearly vis-ible in Figs. 7a,b,c: the number of phonon sub-bandsthat cross the Larmor frequency increases with decreas-ing κ. For κ = 1, the phonon wave functions are simplythe bulk wave functions delocalized over the whole su-

13

10 1 100 101 102

Modulus relative to Si ( )10 2

10 1

100

101

102

103

104

ph (H

z)

(a)

AnisotropicIsotropic

1Dph [Eq. (26)]3Dph [Eq. (16)]

Free-standing NW

Si

1 µm

30 nm10 n

m

Encapsulation material

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

100

101

102

103

104

ph (H

z)

(b)

Free Standing NWNW in SiO2NW in Si

NW in Si3N4NW in Diamond

1 µ

m

FIG. 6. (a) Phonon-induced relaxation rate as a function of the hardness of the encapsulation material. From a vibrationalpoint of view, the system is modeled as a W = 30 nm × H = 10 nm silicon nanowire (NW) embedded in a material whoseelastic constants are rescaled by a factor κ with respect to those of silicon. Periodic boundary conditions are applied overa supercell with side Lcell = 1 µm. Data are plotted for both isotropic and anisotropic elastic models (see Table I). Thefree-standing limit, as well as the semi-analytical results for Γ3D

ph and Γ1Dph (with rescaled phonon velocities, see main text) are

displayed for comparison. The back gate voltage is Vbg = −0.2 V, and the orientation of the magnetic field is ϕ = θ = 45.(b) Phonon-induced relaxation rate as a function of back gate voltage for different encapsulation materials, that are either soft(free-standing limit, SiO2), comparable (Si, Si3N4), or harder (diamond) than silicon (θ = 45, ϕ = 0). The Larmor frequencyis ω = 10 GHz in all panels.

percell. As a consequence, the relaxation rate scales asΓ3Dph ∝ 1/v5enc ∝ κ−5/2 near κ = 1 [Eq. (16) using venc

as input]. When decreasing κ 1, Nph increases contin-uously; yet all the sub-bands do not contribute equallyto the relaxation rate. Indeed, many low-energy sub-bands mostly propagate in the encapsulation material atsmall κ and are, therefore, weakly coupled to the qubit(phonon depletion effect65–67). When κ→ 0, the LA andtwo TA branches of the silicon nanowire actually emergebehind the quasi-continuum of sub-bands propagating inthe encapsulation material, and are the only ones thatefficiently scatter the hole (see Fig. 7f). The relaxationrate hence saturates at the free-standing limit. On thecontrary, when the encapsulation material is hardened(κ 1), the splitting between phonon sub-bands in-creases until the qubit can only couple to the LA andtwo TA sub-bands whose ωαq → 0 when q → 0 (Fig. 7c).These three sub-bands are highly dispersive (with groupvelocities that scale as

√κ), and tend to relocalize in the

silicon core at finite q.68 In this regime, Γph behaves as

1/(v3encL2cell) ∝ κ−3/2, as expected from Eq. (26) for the

1D phonons model. The relaxation rate remains, how-ever, significantly larger than Eq. (26) due to the finitephonon reconfinement in the silicon core at small wavevector q ∝

√κ.68 The residual relaxation rate also be-

comes dependent on Lcell, being sensitive to details ofthe structure over the scale of the phonon wave length.69

In Fig. 6b, we compare the relaxation rates computedin Si nanowires embedded in different materials, as afunction of the back gate voltage. The elastic constantof these materials are given in Table I. They are not allmeant to be realistic encapsulation materials for a siliconnanowire qubit, but have been chosen as representativesof “soft” and “hard” materials. We have also neglectedbuilt-in strains as well as disorder and (if relevant) piezo-electric scattering in these materials. We recover thetrends discussed above: the softer the encapsulation ma-terial, the shorter the phonon-limited lifetime. In par-ticular, the bulk relaxation rate Γ3D

ph [Eq. (16)] typicallyoverestimates the lifetime as the encapsulation materials(such as SiO2) are usually softer than silicon. Phononengineering in semiconductor qubits may, therefore, ul-timately improve their performances, once all other ex-trinsic sources of scattering have been mitigated.70

V. CONCLUSIONS

We have derived the phonon-limited lifetime in holespin-orbit qubits within the 6 bands k ·p framework, ac-

14

0.00 0.01 0.02 0.03q (nm 1)

0

2

4

6

8

10

/2 (G

Hz)

= 0.1(a)

0.000 0.005 0.010q (nm 1)

= 1.0(b)

0.000 0.001 0.002 0.003 0.004q (nm 1)

= 10.0(c)

10 1 100 101 102

Modulus relative to Si ( )103

104

105

v ph (

m/s

)

(d)

LATAvl, Si

vt, Si

10 1 100 101 102

Modulus relative to Si ( )100

101

102

Nph

(e)

AnisotropicIsotropicAnalytic [Eq. (32)]

0.00 0.01 0.02 0.03 0.04q (nm 1)

0

2

4

6

8

10

12

14

/2 (G

Hz)

(f)

FIG. 7. (a, b, c) Phonon band structures computed for κ = 0.1, κ = 1 and κ = 10 (isotropic elastic constants). (d) Thetransverse and longitudinal acoustic velocities (drawn from the branches ωαq → 0 when q → 0) as a function of κ. The groupvelocities are defined by the encapsulation material that fills 99.97% of the supercell and behave, therefore, as venc =

√κvSi.

(e) The number of phonon branches Nph that match the Larmor frequency ω/(2π) = 10 GHz of the qubit. The dash-dottedline is Eq. (32) using the sound velocities venc as input. (f) The phonon band structure computed at κ = 0.001 and Lcell = 100nm (Lcell being reduced here due to computational limitations). The diameter of the dots is proportional to the density ofelastic energy Uel in the silicon core.64 The solid color lines are the longitudinal, torsional and flexural modes of a free-standingW = 30 nm ×H = 10 nm silicon nanowire. At such small κ, most of the low-energy phonons propagate in the encapsulationmaterial (phonon depletion effect) and do not couple efficiently to the qubit; yet the phonon modes of the free-standing nanowireemerge behind the quasi-continuum of states of the encapsulation material.

counting for the complete set of deformation potentials ofthe valence band. The resulting expressions for the one-phonon transition rates can actually be applied to bothspin and charge relaxation in a hole quantum dot. Wehave extended these expressions to strongly confined 1Dand 2D phonon band structures and highlighted the dif-ferent dependences on the Larmor frequency of the qubit.We have then applied this theory to a hole spin-orbitqubit on silicon-on-insulator similar to Refs. 19 and 25.We have shown that phonon-induced spin relaxation inthis qubit is dominated by a band mixing term that cou-ples the hole to transverse acoustic phonons through thevalence band deformation potential d. We have next opti-mized the bias point and magnetic field orientation look-ing for the best quality factor Q1 = fRT1 (the number ofRabi oscillations that can be performed within one relax-ation time T1). When only phonons are accounted for inthe relaxation, Q1 can reach a few tens of thousands de-spite the strong spin-orbit coupling in the valence band.Hole spin-orbit qubits are, however, very sensitive to elec-trical and charge noise, which calls for a careful designof the devices and of the electronics around. We havealso discussed the impact of confinement and encapsula-tion materials on the phonon-limited lifetimes. Indeed,the qubit couples to low-energy phonons that probe thedevice over very long length scales. The lifetime does, inparticular, increase when the materials around the qubit

get harder. This may be evidenced experimentally atmagnetic fields large enough so that phonons dominateover electrical and charge noise relaxation. Phonon engi-neering might, therefore, ultimately improve the perfor-mances of semiconductor qubits.

ACKNOWLEDGMENTS

This work was supported by the European Union’sHorizon 2020 research and innovation program undergrant agreements No 688539 MOSQUITO and 810504-QUCUBE-ERC-2018-SyG, and by the French nationalresearch agency (ANR project MAQSi).

Appendix A: The dipole approximation

If the extension of the qubit wave functions is sig-nificantly smaller than the wave length of the phononsinvolved in the relaxation, we can make the following

15

(dipole) approximation for the phase factor:

eiq·(r−r′) ≈ 1 + iq · (r − r′)− 1

2[q · (r − r′)]2

≈ 1 + iq · r − iq · r′ + (q · r)(q · r′)

− 1

2[(q · r)2 + (q · r′)2] . (A1)

The matrix element in Eq. (9) can then be expanded as:∣∣∣〈0|eiq·r∆H(εαq)|1〉∣∣∣2 ≈ SS∗ + i

∑k

qk(RkS∗ − R∗kS)

+∑k,k′

qkqk′

[RkR

∗k′ −

1

2(Tkk′ S

∗ + T ∗kk′ S)

], (A2)

where qk is the component of q on axis k ∈ x, y, z, and

S, Rk, Tkk′ are defined as:

S = 〈0|∆H|1〉 =∑i,j

∆HijSij (A3a)

Rk = 〈0|∆Hrk|1〉 =∑i,j

∆HijRkij (A3b)

Tkk′ = 〈0|∆Hrkrk′ |1〉 =∑i,j

∆HijTkk′

ij , (A3c)

with ∆H ≡ ∆Hs(εαq), and Sij , Rkij and T kk

′

ij given byEq. (17).

Appendix B: Convergence of the numericalrelaxation rate calculations

Eq. (27) is expected to diverge when ω approaches theedge of a parabolic phonon sub-band with vα0 = 0. Thisis a known pitfall of Fermi Golden rule, whose transitionrates are proportional to the phonon density of states,which then behaves as 1/

√ω − ωα0.

In order to understand the implications of this re-sult, we consider (as in section IV A) a qubit embeddedin a homogeneous nanowire with cross-sectional area S.When S increases, the density of phonon sub-bands alsoincreases ∝ S [Eq. (32)], but the effective width of each∝ 1/(S

√ω − ωα0) peak in the relaxation rate decreases

as 1/S2. Therefore, Γph does only converge “in measure”to Γ3D

ph when S →∞. In order to smooth out the conver-

gence, we have introduced a cut-off |q| > 2π/λmax in Eq.(27), where λmax = 5.431 cm is very large with respect toall dimensions of the system. This cut-off was enforcedin section IV.

From a practical point of view, the van-Hove singu-larities in the vibrational DoS may also be smoothedby phonon scattering (disorder and phonon-phonon in-teractions). The mean free path of low-energy acous-tic phonons can, however, remain very long at lowtemperature.39

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

0

20

40

60

80

100

f R (M

Hz)

fR

Q1

0

100

200

300

400

500

(Hz)

0

10

20

30

40

50

Q1 (

×104 )

FIG. 8. Rabi frequency fR, total relaxation rate Γ = Γ3Dph +Γjn

and quality factor Q1 = fR/Γ as a function of the back gatevoltage Vbg (θ = 45, ϕ = 0, ω/(2π) = 10 GHz). TheJohnson-Nyquist relaxation rate was computed for a resis-tance R = 200 Ω and scales ∝ R.

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

100

101

102

2 (Hz

)phjnph+jn

FIG. 9. Phonon Γ2,ph, Johnson-Nyquist Γ2,jn and total de-coherence rate Γ2 = Γ2,ph + Γ2,jn as a function of back gatevoltage (θ = 45, ϕ = 0, ω/(2π) = 10 GHz).

Appendix C: Figures of merit of the qubit in thepresence of Johnson-Nyquist noise

We plot in Fig. 8 the Rabi frequency fR, the totalrelaxation rate Γ = Γ3D

ph + Γjn and the quality factor

Q1 = fR/Γ as a function of the back gate voltage Vbg.This figure is the counterpart of Fig. 3b including theJohnson-Nyquist contribution computed for a resistanceR = 200 Ω. As discussed in the main text, Johnson-Nyquist noise increases the relaxation rate away fromVbg = −0.15 V and moves the optimal Q1 farther fromthat point (because the Rabi frequency is also zero atVbg = −0.15 V where the qubit decouples from the elec-trical noise).

Phonons and Johnson-Nyquist noise also induce de-coherence. The Johnson-Nyquist decoherence rate isΓ2,jn = T−12,jn = Γjn/2 + T ∗−12,jn , where:

T ∗−12,jn =2π

~R

R0|D11 −D00|2kBT , (C1)

with D00 = 〈0|Dfg|0〉 and D11 = 〈1|Dfg|1〉. As for

16

0.4 0.3 0.2 0.1 0.0 0.1Vbg (V)

0

20

40

60

80

100f R

(MHz

)fR

2Q2

0

50

100

150

200

250

300

2 (Hz

)

0

10

20

30

40

50

Q2 (

×104 )

FIG. 10. Quality factor Q2 = fR/Γ2 as a function of backgate voltage (θ = 45, ϕ = 0, ω/(2π) = 10 GHz).

phonons, the decoherence rate is simply Γ2,ph = T−12,ph =

Γph/2 because the longitudinal spectral function S(ω) of

phonons is “super-ohmic” [S(ω = 0) = 0] and T ∗−12 ∝

S(0) in the Bloch-Redfield theory32,36,38 (Strictly speak-ing, S(0) = 0 for one-phonon processes in 3D, but notnecessarily for two-phonon processes,71 which may giverise to a finite dephasing time).

The phonon Γ2,ph, Johnson-Nyquist Γ2,jn and total de-coherence rate Γ2 = Γ2,ph + Γ2,jn are plotted as a func-tion of back gate voltage in Fig. 9. The quality factorQ2 = fR/Γ2 is plotted as a function of Vbg in Fig. 10.

It is clear from Figs. 8 and 10 that the qubit is limitedby relaxation rather than decoherence with the aboveassumptions. This results from the fact that the spec-tral densities of both phonons and Johnson-Nyquist noiseare small or even zero at low frequency and temper-ature. The situation will be opposite in the presenceof a charge noise with a 1/ωα tail, as decoherence willbe much faster.72 The modeling of charge noise in suchqubits goes, however, far beyond the present work. Holespin qubits being sensitive to electrical and charge noiseowing to the strong spin-orbit coupling in the valenceband, this calls for a careful design of devices and elec-tronics around.

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∗ZZ + εY Y ε

∗ZZ + c.c.)

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),

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18

Supplementary material for “Hole-phonon interactions in quantum dots: Effects ofphonon confinement and encapsulation materials on spin-orbit qubits”

In this supplementary material, we discuss the relaxation rates for strongly confined 2D and 1D phonon bandstructures. We choose the cubic axes x ‖ [100], y ‖ [010] and z ‖ [001] as the reference frame. The azimuthal angle θis measured with respect to z, and the polar angle ϕ with respect to x.

In the sums over coordinates, the x, y and z axes are labeled by an integer (1 stands for x, 2 for y and 3 for z).

Supplementary S1: Relaxation rate in thin films

We consider a thin film with thickness L, normal to n(θ0, ϕ0). We define the two in-plane unit vectors q1(θ0 +π/2, ϕ0), and q2(π/2, ϕ0 + π/2). We assume Born-von-Karman periodic boundary conditions at the surface of thefilm. The resulting phonon band structure is then the bulk band structure sampled at wave vectors:

q =2πn

Ln + q‖(cos γq1 + sin γq2) , (S1)

where n is an integer and γ is the angle between the in-plane wave vector q‖ and q1. Each value of n defines threephonon sub-bands, sampled in the bulk LA, TA1 and TA2 branches. The acoustic branches of the film (ωαq‖ → 0

when q‖ → 0) are the n = 0 sub-bands. In this approximation, the wave function of the acoustic branches are,therefore, homogeneous in the thickness of the film. The other sub-bands tend to a finite ωα0. For strong enoughconfinement, only the n = 0 sub-bands can hence couple to the qubit. The polarization vectors of the LA and TAbranches of the film are:

cl,x = cos γ cos θ0 cosϕ0 − sin γ sinϕ0

cl,y = cos γ cos θ0 sinϕ0 + sin γ cosϕ0

cl,z = − cos γ sin θ0

ct1,x = sin θ0 cosϕ0

ct1,y = sin θ0 sinϕ0

ct1,z = cos θ0

ct2,x = cos γ sinϕ0 + sin γ cosϕ0 cos θ0

ct2,y = − cos γ cosϕ0 + sin γ sinϕ0 cos θ0

ct2,z = − sin γ sin θ0 . (S2)

The integration can then be completed in Eq. (8) of the main text, the lines ωαq‖ = ω being circles. We end up with

Eq. (25) for Γ2Dph . The value of the An’s, Bn’s and Λ parameters are given below for (001), (110) and (111) films.

1. (001) films

For a (001) film, θ0 = 0 and ϕ0 = 0. With indices i, j, k, l,m ∈ 1, 2, the 11 An terms are:

A1 = O111111 +O22

2222 (S3)

A2 = O112222 +O22

1111 (S4)

A3 = O113333 +O22

3333 (S5)

A4 =∑i 6=j

O11iijj +O22

iijj (S6)

A5 =∑i

Oiiii33 +Oii33ii (S7)

19

A6 =∑i 6=j

Oiijj33 +Oii33jj (S8)

A7 =∑

i,j 6=k,l 6=m

Oiijklm (S9)

A8 =∑

i 6=j,k 6=l,m

Oijmmkl +Ojiklmm (S10)

A9 =∑

i 6=j,k 6=l

Oij33kl +Ojikl33 (S11)

A10 =∑i

Oiii3i3 +Oiii33i +Oii3ii3 +Oii3i3i (S12)

A11 =∑

i 6=j,k 6=l

Oiij3j3 +Oiij33j +Oii3jj3 +Oii3j3j +Oijk3l3 +Oijk33l +Oij3kl3 +Oij3k3l . (S13)

The parameters for longitudinal acoustic phonons are: ΛA1l = 8a2+20ab+17b2

8 ; ΛA2l = 8a2−4ab+5b2

8 ; ΛA3l = a2 − 2ab + b2;

ΛA4l = 8a2+8ab−7b2

8 ; ΛA5l = 4a2+ab−5b2

4 ; ΛA6l = 4a2−5ab+b2

4 ; ΛA7l = 3d2

8 ; ΛA8l =

√3d 2a+b

8 ; ΛA9l =

√3da−b4 ; ΛA

10l = 0; ΛA11l = 0.

The parameters for transverse acoustic phonons are: ΛA1t = 9b2

8 ; ΛA2t = 9b2

8 ; ΛA3t = 0; ΛA

4t = −9b28 ; ΛA

5t = 0; ΛA6t = 0;

ΛA7t = 3d2

8 ; ΛA8t = 0; ΛA

9t = 0; ΛA10t = 9d2

16 ; ΛA11t = 3d2

16 .There are 6 Bn terms:

B1 = S11S∗11 + S22S

∗22 (S14)

B2 = S33S∗33 (S15)

B3 = S11S∗22 + S22S

∗11 (S16)

B4 = S11S∗33 + S33S

∗11 + S22S

∗33 + S33S

∗22 (S17)

B5 = S12S∗12 + S12S

∗21 + S21S

∗12 + S21S

∗21 (S18)

B6 =∑i

Si3S∗i3 + Si3S

∗3i + S3iS

∗i3 + Si3S

∗3i (S19)

The parameters for longitudinal acoustic phonons are: ΛB1l = 8a2+8ab+11b2

4 ; ΛB2l = 2(a − b)2; ΛB

3l = 8a2+8ab−7b24 ;

ΛB4l = 2a2 − ab− b2; ΛB

5l = 3d2

4 ; ΛB6l = 0.

The parameters for transverse acoustic phonons are: ΛB1t = 9b2

4 ; ΛB2t = 0; ΛB

3t = −9b24 ; ΛB

4t = 0; ΛB5t = 3d2

4 ; ΛB6t = 3d2

4 .

2. (110) films

For a (110) film, θ0 = π/2 and ϕ0 = π/4. With indices i, j, k, l,m ∈ 1, 2, the 21 An terms are:

A1 =∑i,j,k

(−1)(i+j)Oijkkkk (S20)

20

A2 =∑i,j

(−1)(i+j)Oij3333 (S21)

A3 =∑i,j,k 6=l

(−1)(i+j)Oijkkll (S22)

A4 =∑i,j,k

(−1)(i+j)(Oijkk33 +Oij33kk) (S23)

A5 =∑

i,j,k,l 6=m

(−1)(i+j)(Oijkklm +Oijlmkk) (S24)

A6 =∑i,j,k 6=l

(−1)(i+j)(Oij33kl +Oijkl33) (S25)

A7 =∑i,j,k

(−1)(i+j)(Oi3jjk3 +Oi3jj3k +Oi3k3jj +Oi33kjj +O3ijjk3 +O3i

jj3k +O3ik3jj +O3i

3kjj) (S26)

A8 =∑k,l 6=m

O33kklm +O33

lmkk (S27)

A9 =∑i,j

(−1)(i+j)(Oi333j3 +Oi3333j +Oi3j333 +Oi33j33 +O3i33j3 +O3i

333j +O3ij333 +O3i

3j33)−∑i 6=j

O3333ij +O33

ij33 (S28)

A10 =∑i,j,k 6=l

(−1)(i+j)(Oijklkl +Oijkllk) (S29)

A11 =∑i,j,k 6=l

(−1)(i+j+1)(Oijk3l3 +Oijk33l +Oij3kl3 +Oij3k3l) (S30)

A12 =∑i,j,k

(−1)(i+j)(Oijk3k3 +Oijk33k +Oij3kk3 +Oij3k3k) (S31)

A13 =∑i

O33iiii (S32)

A14 = O333333 (S33)

A15 =∑i6=j

O33iijj (S34)

A16 =∑i

O33ii33 +O33

33ii (S35)

A17 =∑i

O33i3i3 +O33

i33i +O333ii3 +O33

3i3i (S36)

21

A18 =∑i 6=j

O33ijij +O33

ijji (S37)

A19 =∑i6=j

O33i3j3 +O33

i33j +O333ij3 +O33

3i3j (S38)

A20 =∑i,j 6=k,l

(−1)(i+l+1)(Oi3jkl3 +Oi3jk3l +Oi3l3jk +Oi33ljk +O3ijkl3 +O3i

jk3l +O3il3jk +O3i

3ljk) (S39)

A21 =∑i 6=j

O33ij33 +O33

33ij . (S40)

The parameters for longitudinal acoustic phonons are: ΛA1l = ΛA

3l = 32a2+8ab+5b2

64 ; ΛA2l = 8a2−4ab+5b2

16 ; ΛA4l =

16a2−2ab−5b232 ; ΛA

5l = −√

3d 12a+3b64 ; ΛA

6l =√

3d 3b−6a32 ; ΛA

7l =√

3d 4a−b32 ; ΛA

8l =√

3d b−4a32 ; ΛA9l =

√3d 2a+b

16 ; ΛA10l = 15d2

64 ;

ΛA11l = ΛA

12l = ΛA20l = 3d2

32 ; ΛA13l = ΛA

15l = 32a2−40ab+17b2

32 ; ΛA14l = 8a2+20ab+17b2

8 ; ΛA16l = 16a2+10ab−17b2

16 ;

ΛA17l = ΛA

18l = 3d2

16 ; ΛA19l = − 3d2

16 ; ΛA21l = 0.

The parameters for transverse acoustic phonons are: ΛA1t = 63b2

64 ; ΛA2t = 9b2

16 ; ΛA3t = −45b2

64 ; ΛA4t = −9b2

32 ; ΛA5t = −

√3d 3b

64 ;

ΛA6t = ΛA

7t =√

3d 3b32 ; ΛA

8t = ΛA9t = 0; ΛA

10t = ΛA11t = 3d2

64 ; ΛA12t = 9d2

64 ; ΛA13t = 27b2

32 ; ΛA14t = 9b2

8 ; ΛA15t = −9b2

32 ; ΛA16t = −9b2

16 ;

ΛA17t = 15d2

32 ; ΛA18t = ΛA

19t = 3d2

32 ; ΛA20t = 0; ΛA

21t =√

3d 3b16 .

There are 9 Bn terms:

B1 = S11S∗11 + S22S

∗22 (S41)

B2 = S33S∗33 (S42)

B3 = S11S∗22 + S22S

∗11 (S43)

B4 = S11S∗33 + S33S

∗11 + S22S

∗33 + S33S

∗22 (S44)

B5 =∑i

SiiS∗12 + SiiS

∗21 + S12S

∗ii + S21S

∗ii (S45)

B6 = S33S∗12 + S33S

∗21 + S12S

∗33 + S21S

∗33 (S46)

B7 = S12S∗12 + S12S

∗21 + S21S

∗12 + S21S

∗21 (S47)

B8 =∑i

Si3S∗i3 + Si3S

∗3i + S3iS

∗i3 + S3iS

∗3i (S48)

B9 =∑i6=j

Si3S∗j3 + Si3S

∗3j + S3iS

∗j3 + S3iS

∗3j (S49)

The parameters for longitudinal acoustic phonons are: ΛB1l = ΛB

3l = 32a2−16ab+11b2

16 ; ΛB2l = 8a2+8ab+11b2

4 ; ΛB4l =

16a2+4ab−11b28 ; ΛB

5l = −√

3d 8a+b16 ; ΛB

6l =√

3d b−4a8 ; ΛB7l = 9d2

16 ; ΛB8l = 3d2

8 ; ΛB9l = − 3d2

8 .

The parameters for transverse acoustic phonons are: ΛB1t = 45b2

16 ; ΛB2t = 9b2

4 ; ΛB3t = −27b2

16 ; ΛB4t = −9b2

8 ; ΛB5t = −

√3d 3b

16 ;

ΛB6t =

√3d 3b

8 ; ΛB7t = 3d2

16 ; ΛB8t = 3d2

4 ; ΛB9t = 0.

22

3. (111) films

For a (111) film, θ0 = arccos(√

3/3) and ϕ0 = π/4. With indices i, j, k, l,m ∈ 1, 2, 3, the 25 An terms are:

A1 =∑i

Oiiiiii (S50)

A2 =∑i 6=j

Oiijjjj (S51)

A3 =∑i 6=j

Oiiiijj +Oiijjii (S52)

A4 =∑i 6=j 6=k

Oiijjkk (S53)

A5 =∑i 6=j

Oiiiiij +Oiiiiji +Oiiijii +Oiijiii (S54)

A6 =∑i 6=j 6=k

Oiiiijk +Oiijkii (S55)

A7 =∑i 6=j

Oiijjij +Oiijjji +Oiiijjj +Oiijijj (S56)

A8 =∑i 6=j 6=k

Oiijjik +Oiijjki +Oiiikjj +Oiikijj (S57)

A9 =∑i 6=j 6=k

Oiijjjk +Oiijjkj +Oiijkjj +Oiikjjj (S58)

A10 =∑i6=j

Oiiijij +Oiiijji +Oiijiij +Oiijiji (S59)

A11 =∑i6=j 6=k

Oiijkjk +Oiijkkj (S60)

A12 =∑i 6=j 6=k

Oiiijik +Oiiijki +Oiijiik +Oiijiki (S61)

A13 =∑i 6=j 6=k

Oiiijjk +Oiiijkj +Oiijijk +Oiijikj +Oiijkij +Oiijkji +Oiikjij +Oiikjji (S62)

A14 =∑i 6=j

Oijiiii +Ojiiiii (S63)

23

A15 =∑i 6=j 6=k

Oijkkkk (S64)

A16 =∑i 6=j

Oijiijj +Ojiiijj (S65)

A17 =∑i 6=j 6=k

Oijiikk +Ojikkii +Oijkkii +Ojiiikk (S66)

A18 =∑i 6=j

Oijiiij +Oijiiji +Oijijii +Oijjiii +Ojiiiij +Ojiiiji +Ojiijii +Ojijiii (S67)

A19 =∑i 6=j 6=k

Oijiiik +Oijiiki +Oijikii +Oijkiii +Ojiiiik +Ojiiiki +Ojiikii +Ojikiii (S68)

A20 =∑

i 6=j 6=k,l 6=m,m 6=k

Oijllmk +Oijllkm +Oijmkll +Oijkmll (S69)

A21 =∑i6=j

Oijijij +Oijijji +Oijjiij +Oijjiji (S70)

A22 =∑i6=j 6=k

Oijikik +Oijikki +Oijkiik +Oijkiki +Ojiikik +Ojiikki +Ojikiik +Ojikiki (S71)

A23 =∑i6=j 6=k

Oijijik +Oijijki +Oijjiik +Oijjiki +Oijikij +Oijkiij +Oijikji +Oijkiji

Ojiijik +Ojiijki +Ojijiik +Ojijiki +Ojiikij +Ojikiij +Ojiikji +Ojikiji (S72)

A24 =∑i6=j 6=k

Oijikjk +Oijikkj +Oijkijk +Oijkikj +Ojiikjk +Ojiikkj +Ojikijk +Ojikikj (S73)

A25 =∑i 6=j 6=k

Oijkkij +Oijkkji +Oijijkk +Oijjikk (S74)

The parameters for longitudinal acoustic phonons are: ΛA1l = 2a2+2ab+b2

3 ; ΛA2l = 2a2−ab+b2

3 ; ΛA3l = 4a2+ab−b2

6 ; ΛA4l =

4a2−2ab−b26 ; ΛA

5l = −√

3d 3a+2b18 ; ΛA

6l =√

3d b18 ; ΛA

7l = −√

3d 6a+b36 ; ΛA

8l = −√

3d 6a−5b36 ; ΛA

9l =√

3d−b36 ; ΛA10l = 2d2

9 ;

ΛA11l = ΛA

12l = d2

18 ; ΛA13l = −d

2

36 ; ΛA14l = − 2a2+2ab+b2

6 ; ΛA15l = − 2a2−4ab+b2

6 ; ΛA16l = − 4a2+4ab−b2

12 ; ΛA17l = − 4a2−2ab−b2

12 ;

ΛA18l =

√3d 3a+b

18 ; ΛA19l =

√3d b

18 ; ΛA20l = −

√3d b

36 ; ΛA21l = − 7d2

36 ; ΛA22l = ΛA

22l = −d2

36 ; ΛA24l = −d

2

18 ; ΛA25l =

√3d 3a−2b

18 .

The parameters for transverse acoustic phonons are: ΛA1t = 4b2

3 ; ΛA2t = 5b2

6 ; ΛA3t = − 2b2

3 ; ΛA4t = −b2

6 ; ΛA5t = ΛA

8t =

ΛA9t =

√3d b

36 ; ΛA6t = ΛA

7t = −√

3d b18 ; ΛA

10t = 11d2

72 ; ΛA11t = 7d2

36 ; ΛA12t = −d

2

18 ; ΛA13t = − 7d2

72 ; ΛA14t = − 2b2

3 ; ΛA15t = − b

2

6 ;

ΛA16t = 7b2

12 ; ΛA17t = b2

12 ; ΛA18t = ΛA

20t =√

3d b36 ; ΛA

19t = −√

3d b18 ; ΛA

21t = −d2

18 ; ΛA22t = − 7d2

72 ; ΛA23t = d2

36 ; ΛA24t = 5d2

72 ;

ΛA25t = −

√3d b

18 .There are 6 Bn terms:

B1 =∑i

SiiS∗ii (S75)

24

B2 =∑i 6=j

SiiS∗jj (S76)

B3 =∑i6=j

SiiS∗ij + SiiS

∗ji + SijS

∗ii + SjiS

∗ii (S77)

B4 =∑i6=j 6=k

SiiS∗jk + SjkS

∗ii (S78)

B5 =∑i 6=j

SijS∗ij + SijS

∗ji (S79)

B6 =∑i 6=j 6=k

SijS∗ik + SijS

∗ki + SjiS

∗ik + SjiS

∗ki (S80)

The parameters for longitudinal acoustic phonons are: ΛB1l = 2a2+b2; ΛB

2l = 4a2−b22 ; ΛB

3l = −√

3d 2a+b6 ; ΛB

4l =√

3d b−a3 ;

ΛB5l = d2

2 ; ΛB6l = 0.

The parameters for transverse acoustic phonons are: ΛB1t = 3b2; ΛB

2t = −3b22 ; ΛB

3t = ΛB4t = 0; ΛB

5t = d2

2 ; ΛB6t = −d2

4 .

Supplementary S2: Relaxation rate in nanowires

We consider a wire oriented along u(θ0, ϕ0), with width l1 along q1 and height l2 along q2 (q1 ⊥ u, q2 ⊥ u). Weagain assume Born-von-Karman periodic boundary conditions at the surface of the wire. The resulting phonon bandstructure is then the bulk band structure sampled at wave vectors:

q = q‖u +2πn1l1

q1 +2πn2l2

q2 , (S1)

where n1 and n2 are integers. Each pair (n1, n2) defines three phonon sub-bands, sampled in the bulk LA, TA1 andTA2 branches. The acoustic branches of the film (ωαq‖ → 0 when q‖ → 0) are the n1 = n2 = 0 sub-bands. For strongenough confinement, only the n1 = n2 = 0 sub-bands can couple to the qubit. The integration can then be completedin Eq. (8) of the main text. We end up with Eq. (26) for Γ1D

ph . The value of the An’s, Bn’s and Λ parameters

are given below for [001], [110] and [111]-oriented wires. Note that the shape of the nanowire is not relevant as thewave function of the acoustic branches are homogeneous in the cross-section of the wire in this approximation. Therelaxation rate only depends on the area of this cross-section.

1. [001] wires

For a [001] wire, θ0 = 0 and ϕ0 = 0. With indices i, j ∈ 1, 2, the 4 An terms are:

A1 =∑ij

O33iijj (S2)

A2 = O333333 (S3)

A3 =∑i

O33ii33 +O33

33ii (S4)

A4 =∑i

O33i3i3 +O33

i33i +O333ii3 +O33

3i3i . (S5)

25

The parameters for longitudinal acoustic phonons are: ΛA1l = 2a2−4ab+2b2; ΛA

2l = 2a2+8ab+8b2; ΛA3l = 2a2+2ab−4b2;

ΛA4l = 0.

The parameters for transverse acoustic phonons are: ΛA1t = ΛA

2t = ΛA3t = 0; ΛA

4t = 3d2

2 .There are 4 Bn terms:

B1 = S11S∗11 + S22S

∗22 + S11S

∗22 + S22S

∗11 (S6)

B2 = S33S∗33 (S7)

B3 = S11S∗33 + S33S

∗11 + S22S

∗33 + S33S

∗22 (S8)

B4 =∑i

Si3S∗i3 + Si3S

∗3i + S3iS

∗i3 + S3iS

∗3i . (S9)

The parameters for longitudinal acoustic phonons are: ΛB1l = 2a2−4ab+2b2; ΛB

2l = 2a2+8ab+8b2; ΛB3l = 2a2+2ab−4b2;

ΛB4l = 0.

The parameters for transverse acoustic phonons are: ΛB1t = ΛB

2t = ΛB3t = 0; ΛB

4t = 3d2

2 .

2. [110] wires

For a [110] wire, θ0 = π/2 and ϕ0 = π/4. With indices i, j, k, l,m, n ∈ 1, 2, the 8 An terms are:

A1 =∑i,j,k

Oijkkkk (S10)

A2 =∑i,j

Oij3333 (S11)

A3 =∑i,j,k 6=l

Oijkkll (S12)

A4 =∑i,j,k

Oijkk33 +Oij33kk (S13)

A5 =∑

i,j,k,l 6=m

Oijkklm +Oijlmkk (S14)

A6 =∑i,j,k 6=l

Oij33kl +Oijkl33 (S15)

A7 =∑

i,j,k 6=l,m6=n

Oijklmn (S16)

A8 =∑i,j,k,l

Oijk3l3 +Oijk33l +Oij3kl3 +Oij3k3l . (S17)

The parameters for longitudinal acoustic phonons are: ΛA1l = 4a2+4ab+b2

4 ; ΛA2l = a2 − 2ab + b2; ΛA

3l = 4a2+4ab+b2

4 ;

ΛA4l = 2a2−ab−b2

2 ; ΛA5l =

√3d 2a+b

4 ; ΛA6l =

√3da−b2 ; ΛA

7l = 3d2

4 ; ΛA8l = 0.

The parameters for transverse acoustic phonons are: ΛA1t = 9b2

4 ; ΛA3t = − 9b2

4 ; ΛA8t = 3d2

8 ; ΛA2t = ΛA

4t = ΛA5t = ΛA

6t =

26

ΛA7t = 0.

There are 8 Bn terms:

B1 = S11S∗11 + S22S

∗22 (S18)

B2 = S33S∗33 (S19)

B3 = S11S∗22 + S22S

∗11 (S20)

B4 = S11S∗33 + S33S

∗11 + S22S

∗33 + S33S

∗22 (S21)

B5 =∑i6=j

SiiS∗ij + SiiS

∗ji + SijS

∗ii + SjiS

∗ii (S22)

B6 = S33S∗12 + S33S

∗21 + S12S

∗33 + S21S

∗33 (S23)

B7 = S12S∗12 + S12S

∗21 + S21S

∗12 + S21S

∗21 (S24)

B8 =∑ij

Si3S∗j3 + Si3S

∗3j + S3iS

∗j3 + S3iS

∗3j . (S25)

The parameters for longitudinal acoustic phonons are: ΛB1l = 4a2+4ab+b2

2 ; ΛB2l = 2a2 − 4ab + 2b2; ΛB

3l = 4a2+4ab+b2

2 ;

ΛB4l = 2a2 − ab− b2, ΛB

5l =√

3d 2a+b2 ; ΛB

6l =√

3d(a− b); ΛB7l = 3d2

2 ; ΛB8l = 0.

The parameters for transverse acoustic phonons are: ΛB1t = 9b2

2 ; ΛB3t = − 9b2

2 ; ΛB8t = 3d2

4 ; ΛB2t = ΛA

4t = ΛA5t = ΛA

6t =

ΛA7t = 0.

3. [111] wires

For a [111] wire, θ0 = arccos(√

3/3) and ϕ0 = π/4. With indices i, j, k, l,m, n ∈ 1, 2, 3, the 6 An terms are:

A1 =∑i,j,k

Oijkkkk (S26)

A2 =∑i,j,k 6=l

Oijkkll (S27)

A3 =∑i,j,k 6=l

Oijkkkl +Oijkklk +Oijklkk +Oijlkkk (S28)

A4 =∑

i,j,k 6=l 6=m

Oijkklm +Oijlmkk (S29)

A5 =∑i,j,k 6=l

Oijklkl +Oijkllk (S30)

A6 =∑

i,j,k 6=l 6=m

Oijklkm +Oijklmk +Oijlkkm +Oijlkmk . (S31)

27

The parameters for longitudinal acoustic phonons are: ΛA1l = 2a2

3 ; ΛA2l = 2a2

3 ; ΛA3l =

√3d 2a

9 ; ΛA4l =

√3d 2a

9 ; ΛA5l = 2d2

9 ;

ΛA6l = 2d2

9 .

The parameters for transverse acoustic phonons are: ΛA1t = 4b2

3 ; ΛA2t = − 2a2

3 ; ΛA3t =

√3d b9 ; ΛA

4t = −√

3d 2b9 ; ΛA

5t = d2

9 ;

ΛA6t = −d

2

18 .There are 6 Bn terms:

B1 =∑i

SiiS∗ii (S32)

B2 =∑i 6=j

SiiS∗jj (S33)

B3 =∑i 6=j

SiiS∗ij + SiiS

∗ji + SijS

∗ii + SjiS

∗ii (S34)

B4 =∑i6=j 6=k

SiiS∗jk + SjkS

∗ii (S35)

B5 =∑i6=j

SijS∗ij + SijS

∗ji (S36)

B6 =∑i 6=j 6=k

SijS∗ik + SijS

∗ki + SjiS

∗ik + SjiS

∗ki . (S37)

The parameters for longitudinal acoustic phonons are: ΛB1l = 2a2; ΛB

2l = 2a2; ΛB3l =

√3d 2a

3 ; ΛB4l =

√3d 2a

3 ; ΛB5l = 2d2

3 ;

ΛB6l = 2d2

3 .

The parameters for transverse acoustic phonons are: ΛB1t = 4b2; ΛB

2t = −2b2; ΛB3t =

√3d b3 ; ΛB

4t = −√

3d 2b3 ; ΛB

5t = d2

3 ;

ΛB6t = −d

2

6 .