Spin-orbit enhancement in SiSiGe heterostructures with oscillating Ge concentration Benjamin D. Woods1M. A. Eriksson1Robert Joynt1and Mark Friesen1 1Department of Physics University of Wisconsin-Madison Madison Wisconsin 53706 USA

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Spin-orbit enhancement in Si/SiGe heterostructures with oscillating Ge concentration

Benjamin D. Woods,1M. A. Eriksson,1Robert Joynt,1and Mark Friesen1

1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

We show that Ge concentration oscillations within the quantum well region of a Si/SiGe het-

erostructure can signiﬁcantly enhance the spin-orbit coupling of the low-energy conduction-band

valleys. Speciﬁcally, we ﬁnd that for Ge oscillation wavelengths near λ= 1.57 nm with an aver-

age Ge concentration of ¯nGe = 5% in the quantum well region, a Dresselhaus spin-orbit coupling

is induced, at all physically relevant electric ﬁeld strengths, which is over an order of magnitude

larger than what is found in conventional Si/SiGe heterostructures without Ge concentration oscil-

lations. This enhancement is caused by the Ge concentration oscillations producing wave-function

satellite peaks a distance 2π/λ away in momentum space from each valley, which then couple to

the opposite valley through Dresselhaus spin-orbit coupling. Our results indicate that the enhanced

spin-orbit coupling can enable fast spin manipulation within Si quantum dots using electric dipole

spin resonance in the absence of micromagnets. Indeed, our calculations yield a Rabi frequency

ΩRabi/B > 500 MHz/T near the optimal Ge oscillation wavelength λ= 1.57 nm.

I. INTRODUCTION

Following the seminal work of Loss and DiVincenzo

[1], quantum dots in semiconductors have emerged as a

leading candidate platform for quantum computation [2–

5]. Gate-deﬁned quantum dots in silicon [6,7] are par-

ticularly attractive due to their compatibility with the

microelectronics fabrication industry. Moreover, in con-

trast with GaAs, for which coherence times are limited

by unavoidable hyperﬁne interactions with nuclear spins

[8], isotropic enrichment dramatically suppresses these

interactions in Si, enabling long coherence times [9].

While recent progress in Si quantum dots has been

quite promising, many of the leading qubit architectures

rely on synthetic spin-orbit coupling arising from micro-

magnets [10–13], leading to challenges for scaling up to

systems with many dots. An alternative approach is to

use intrinsic spin-orbit coupling for qubit manipulation,

for example, through the electric dipole spin resonance

(EDSR) mechanism [14,15]. While this possibility has

been considered for Ge and Si hole-spin qubits, where the

degeneracy of the p-orbital-dominated valence band leads

to strong spin-orbit coupling [16,17], the weak spin-orbit

coupling of the Si conduction band appears unfavorable

for electron-spin qubits.

In this work, we show how the spin-orbit coupling in

Si/SiGe quantum well heterostructures can be enhanced

by more than an order of magnitude by incorporating

Ge concentration oscillations inside the quantum well,

leading to the possibility of exploiting intrinsic spin-orbit

coupling in Si quantum dots for fast gate operations. Fig-

ure 1(a) shows a schematic of the system which consists of

a Si-dominated quantum well region sandwiched between

Si0.7Ge0.3barrier regions, where the growth direction is

taken along the [001] crystallographic axis. In contrast to

“conventional” Si/SiGe quantum wells, the quantum well

region contains a small amount of Ge with concentration

oscillations of wavelength λ, as shown in Fig. 1(b). For

comparison, Fig. 1(c) shows the Ge concentration pro-

ﬁle of a conventional Si/SiGe quantum well. Previous

FIG. 1. (a) Schematic of the Si/SiGe heterostructure consid-

ered here, consisting of a Si-dominated quantum well sand-

wiched between Si0.7Ge0.3 barrier regions. Note that the

growth direction is along the [001] crystallographic axis. (b)

Ge concentration proﬁle along the growth (z) direction of a

wiggle well. Ge concentration oscillations of wavelength λin-

side of the quantum well region lead to spin-orbit coupling

enhancement for a proper choice of λ. (c) Ge concentration

proﬁle of a “conventional” Si/SiGe quantum well, for compar-

ison.

works [18,19] have studied such a structure, which has

been named the wiggle well, and found that the periodic

Ge concentration leads to an enhancement of the valley

splitting. Here, we develop a theory of spin-orbit cou-

pling within such structures and show that the periodic

nature of the device, along with the underlying diamond

crystal structure and degeneracy of the Si zvalleys, also

gives rise to an enhancement in spin-orbit coupling. Im-

portantly, we ﬁnd that the wavelength λmust satisfy a

resonance condition to give rise to this spin-orbit cou-

pling enhancement. As discussed in detail in Sec. IV,

this involves a two-step process that can be summarized

as follows. First, the periodic potential produced by

the Ge concentration oscillations produces wave-function

satellites a distance 2π/λ away in momentum space from

each valley. Then, a satellite of a given valley couples

strongly to the opposite valley through Dresselhaus spin-

arXiv:2210.01700v2 [cond-mat.mes-hall] 16 Jan 2023

orbit coupling, provided that the satellite-valley separa-

tion distance in momentum space is 4π/a, corresponding

to the condition λ= 1.57 nm.

From the outset, it is important to remark that the

spin-orbit coupling introduced by the Ge concentration

oscillations is fundamentally distinct from the spin-orbit

coupling of conventional Si/SiGe quantum wells. For a

given subband of a conventional Si/SiGe quantum well

immersed in a vertical electric ﬁeld, the C2vpoint group

symmetry of the system allows for both Rashba and

“Dresselhaus-type” linear-kkterms of the form [20,21],

HSO =α(ky¯σx−kx¯σy) + β(kx¯σx−ky¯σy),(1)

where ¯σjare the Pauli matrices acting in (pseudo)spin

space and αand βare the Rashba and Dresselhaus coef-

ﬁcients, respectively, of the subband. The presence of

Rasbha spin-orbit coupling is unsurprising due to the

structural asymmetry provided by the electric ﬁeld [22],

while the presence of the Dresselhaus-type term is ini-

tially surprising since the diamond lattice of Si/SiGe

quantum wells possesses bulk inversion symmetry [23].

However, these systems still support a Dresselhaus-type

term of the same form β(kx¯σx−ky¯σy), due to the broken

inversion symmetry caused by the quantum well inter-

faces [20,21,24,25]. This is in stark contrast to the

true Dresselhaus spin-orbit coupling in III-V semicon-

ductors, where the asymmetry of the anion and cation

in the unit cell leads to bulk inversion asymmetry [23].

Importantly, we ﬁnd in Sec. III B that the spin-orbit cou-

pling of the wiggle well does not rely upon the presence

of an interface. Rather, it is an intrinsic property of a

bulk system with Ge concentration oscillations. In this

sense, the spin-orbit coupling investigated here is more

akin to the true Dresselhaus spin-orbit coupling of III-

V semiconductors than the Dresselhaus-type spin-orbit

coupling of conventional Si/SiGe quantum wells brought

about by interfaces. Indeed, the only requirement for

linear-kkDresselhaus spin-orbit coupling in a wiggle well

with an appropriate λis conﬁnement in the growth di-

rection (even symmetric conﬁnement), to allow for the

formation of subbands. For simplicity in the remainder

of this work, we simply refer to this form of spin-orbit

coupling as Dresselhaus.

The rest of this paper is organized as follows. In Sec. II

we describe our model used to study the quantum well

heterostructure. Section III then presents our numerical

results for the spin-orbit coeﬃcients. This also includes

the calculation of the EDSR Rabi frequency and studies

the impact of alloy disorder on the spin-orbit coeﬃcients.

In Sec. IV we provide an extensive explanation of the

mechanism behind the spin-orbit coupling enhancement.

Finally, we conclude in Sec. V.

II. MODEL

In this section, we outline the model used to study

our Si/SiGe heterostructure along with the methods used

to calculate the spin-orbit coeﬃcients. In Sec. II A, we

describe the tight binding model used to model generic

SiGe alloy systems. Next, in Sec. II B we employ a vir-

tual crystal approximation to impart translation invari-

ance in the plane of the quantum well, allowing us to

reduce the problem to an eﬀective one-dimensional (1D)

Hamiltonian parametrized by in-plane momentum kk. In

Sec. II C, we expand the model around kk= 0 to separate

out the Hamiltonian components that give rise to Rashba

and Dresselhaus spin-orbit coupling, respectively, and we

explain the important diﬀerences between the two com-

ponents. Finally, in Sec. II D, we transform the Hamil-

tonian into the subband basis, which allows us to obtain

expressions for the Rashba and Dresselhaus spin-orbit

coeﬃcients in each subband.

A. Model of SiGe alloys

To study the spin-orbit physics of our system we use

the empirical tight-binding method [26], where the elec-

tronic wave function is written as a linear combination

of atomic orbitals:

|ψi=X

n,j,ν,σ |njνσiψnjνσ .(2)

Here, hr|njνσi=φν(r−Rn,j )|σiis an atomic orbital

centered at position Rn,j , corresponding to atom jof

atomic layer nalong the growth direction [001],νis a

spatial orbital index, and |σiis a two component spinor

with σ=↑,↓indicating the spin of the orbital. We

use an sp3d5s*basis set with 20 orbitals per atom, on-

site spin-orbit coupling, nearest-neighbor hopping, and

strain. Note that nearest-neighbor sp3d5s*tight-binding

models are well established for accurately describing the

electronic structure of semiconductor materials over a

wide energy range [27]. Explicitly, νis a spatial orbital

index from the set including s,s∗,pi(i=x, y, z), and

di(i=xy, yz, zx, z2, x2−y2) orbitals, which are meant

to model the outer-shell orbitals of individual Si and

Ge atoms that participate in chemical bonding. Addi-

tionally, these orbitals possess certain spatial symmetries

that, combined with the diamond crystal structure of the

SiGe alloy, dictate the forms of the nearest neighbor cou-

plings, as ﬁrst explained in the work of Slater and Koster

[26]. The free parameters of the tight-binding model (in-

cluding onsite orbital energies, nearest-neighbor hopping

energies, strain parameters, etc.) are then chosen such

that the band structure of the system agrees as well as

possible with experimental and/or ab initio data. In this

work, we use the tight binding model and parameters

of Ref. [28], which allows for the modeling of strained,

random SiGe alloys with any Ge concentration proﬁle.

The Hamiltonian of an arbitrary SiGe alloy takes the

form,

Hmi,nj

µσ,νσ =δnj

mi hδνσ0

µσ ε(nj)

ν+Vn+δσ0

σC(nj)

µν +S(nj)

µσ,νσ0i

+δσ0

σδn+1

mT(n)

iµ,jν +δn−1

mT(m)†

iµ,jν ,(3)

where Hmi,nj

µσ,νσ0=hmiµσ|H|njνσ0iand δequals 1if its

subscripts match its superscripts and 0otherwise. The

ﬁrst line in Eq. (3) contains intra-atomic terms, where

ε(nj)

νis the onsite energy of orbital ν, for atom jin

atomic layer n,Vnis the potential energy due to the ver-

tical electric ﬁeld, C(nj)

µν accounts for onsite energy shifts

and couplings caused by strain, and S(nj)

µσ,νσ0accounts for

spin-orbit coupling. The matrix C(nj)

µν is determined by

the deformation of the lattice due to strain as detailed

in Ref. [28] and arises from changes in the onsite poten-

tial of the atom due to the displacement of its neighbors.

In addition, spin-orbit coupling S(nj)is an intra-atomic

coupling between porbitals [29] and is the only term in

Eq. (3) that does not conserve spin σ. (See Appendix B

for the explicit form of S(nj).) Note that the superscripts

(nj), which index the atoms, are needed here because the

intra-atomic terms depend on whether an atom is Si or

Ge, as well as the local strain environment. The second

line in Eq. (3) contains inter-atomic terms describing the

hopping between atoms on adjacent atomic layers, where

T(n)is the hopping matrix from atomic layer nto atomic

layer n+1. Nearly all elements of T(n)are zero, with non-

zero hoppings occurring only between nearest-neighbor

atoms. A non-zero hopping matrix element T(n)

iµ,jν then

depends on three things: (1) the orbital indices µand

ν, (2) the types of atoms involved, and (3) the direction

and magnitude of the vector Rn+1,i −Rn,j connecting the

atoms. We then use the Slater-Koster table in Ref. [26]

along with the parameters of Ref. [28] to calculate T(n)

iµ,jν .

We note that strain aﬀects the hopping elements by al-

tering the direction and length of the nearest-neighbor

vectors (i.e., the crystalline bonds) [28,30].

In this work, we let the Ge concentration vary be-

tween layers, as shown in Figs. 1(b) and 1(c), but as-

sume it to be uniform within a given layer. Note that

the large diﬀerence in Ge concentration between the bar-

rier and well regions results in a large conduction band

oﬀset that traps electrons inside the quantum well. This

occurs naturally in the tight binding model of Eq. (3)

because ε(nj), C(nj), and T(n)

ij are diﬀerent for Si and Ge

atoms. Finally, we point the reader to Appendix Afor a

description of the lattice constant dependence on strain.

B. Virtual crystal approximation and pseudospin

transformation

While the Hamiltonian in Eq. (3) provides an accurate

description of SiGe alloys, it lacks translation invariance

when alloy disorder is present. This makes the model

computationally expensive to solve, and it obscures the

physics of the spin-orbit enhancement coming from the

averaged eﬀects of the inhomogeneous Ge concentration

proﬁle. We therefore employ a virtual crystal approxima-

tion where the Hamiltonian matrix elements are replaced

by their value averaged over all alloy realizations. Specif-

ically, we deﬁne a virtual crystal Hamiltonian HVC with

elements (HVC)mi,nj

µσ,νσ0=DHmi,nj

µσ,νσ0E, where h. . . iindicates

an average over all possible alloy realizations. The Hamil-

tonian is then translation invariant within the plane of

the quantum well. In addition, it is useful to move be-

yond the original orbital basis, where the spin is well-

deﬁned, to a pseudospin basis deﬁned by

|nj¯ν¯σi=X

νσ |njνσiU(n)

νσ,¯ν¯σ,(4)

where ¯νare the hybridized orbital states, ¯σ=⇑,⇓are the

pseudospins, and U(n)is the transformation matrix for

layer n. Full details of this basis transformation can be

found in Appendix B. Here, we mention three important

features of Eq. (4). First, the basis transformation diag-

onalizes the onsite spin-orbit coupling, making the spin-

orbit physics more transparent. Second, the pseudospin

states represent linear combinations of orbitals including

both spin ↑and spin ↓. Third, the transformation matrix

U(n)can be shown to satisfy

U(n)=(U(0), n ∈Zeven,

U(1), n ∈Zodd.(5)

This alternating structure for the transformation matrix

is crucial to the results that follow, and results from the

presence of two sublattices in the diamond crystal struc-

ture of Si, as shown in Fig. 2(a).

Making use of the virtual crystal approximation, we

now convert our three-dimensional (3D) Hamiltonian

into an eﬀective 1D Hamiltonian. To begin, we note that

the virtual crystal Hamiltonian takes the simpliﬁed form,

(HVC)mi,nj

¯µ¯σ,¯ν¯σ0=δnj ¯σ0

mi¯σhδ¯ν

¯µ¯ε(n)

¯ν+Vn+¯

C(n)

¯µ¯νi

+δn+1

m¯

T(n)

i¯µ¯σ,j ¯ν¯σ0+δn−1

m¯

T(m)†

i¯µ¯σ,j ¯ν¯σ0,(6)

where (HVC)mi,nj

¯µ¯σ,¯ν¯σ0=hmi¯µ¯σ|HVC|nj¯ν¯σ0i,¯ε(n)

¯νis the on-

site energy of pseudospin orbital ¯ν, and ¯

C(n)and ¯

T(n)

are the onsite strain and hopping matrices, respectively,

transformed into the pseudospin basis and averaged over

alloy realizations. Note that ¯ε(n)includes contributions

from diagonalizing the onsite spin-orbit coupling. Impor-

tantly, ¯ε(n)and ¯

C(n)maintain a dependence on the layer

index ndue to the non-uniform Ge concentration pro-

ﬁle. In contrast, the dependence of intra-atomic terms

on the intra-layer atom index jhas vanished due to the

virtual crystal approximation. Moreover, the translation

invariance of the virtual crystal approximation implies

that hopping matrix elements between any two layers

only depend upon the relative position of atoms, i.e.,

T(n)

i¯µ¯σ,j ¯ν¯σ0=¯

T(n)

¯µ¯σ,¯ν¯σ0(Rn+1,i −Rn,j ). We therefore intro-

duce in-plane momentum kkas a good quantum number

and Fourier transform our Hamiltonian. To do so, we

deﬁne the basis state

kkn¯ν¯σ=1

pNkX

eikk·Rnj |nj¯ν¯σi,(7)

where kk= (kx, ky)and Nkis the number of atoms

within each layer. The Hamiltonian has matrix elements

Hmn

¯µ¯σ,¯ν¯σ0(kk) = δn¯σ0

m¯σhδ¯ν

¯µ¯ε(n)

¯ν+Vn+¯

C(n)

¯µ¯νi

+δn+1

T(n)

¯µ¯σ,¯ν¯σ0kk+δn−1

T(m)†

¯µ¯σ,¯ν¯σ0kk,(8)

where e

Hmn

¯µ¯σ,¯ν¯σ0(kk) = kkm¯µ¯σHVCkkn¯ν¯σ0, and

T(n)

¯µ¯σ,¯ν¯σ0kkis the Fourier-transformed hopping matrix

given by

T(n)

¯µ¯σ,¯ν¯σ0kk=

l=1

e−ikk·r(n)

l¯

T(n)

¯µ¯σ,¯ν¯σ0(r(n)

l),(9)

where r(n)

lis a nearest-neighbor vector from a reference

atom in layer nto one of its nearest neighbors in layer

n+ 1. For a diamond lattice, each atom has only has

two such bonds, as indicated in Fig. 2(a). Note that

the Hamiltonian matrix elements vanish between states

with diﬀerent momenta due to translational invariance.

Hence, we obtain an eﬀective 1D Hamiltonian as a func-

tion of kk.

An important feature of the Fourier-transformed hop-

ping matrix e

T(n)(kk)is that it depends on the layer index

nfor two reasons. First, the inhomogeneous Ge concen-

tration along the growth axis causes the hopping param-

eters to change slightly from layer to layer. Second, and

more importantly, the diamond crystal structure is com-

posed of two interleaving face-centered-cubic sublattices

which each contribute an inequivalent atom to the prim-

itive unit cell. This is illustrated in Fig. 2(a) where the

atoms belonging to the two sublattices are colored red

and blue, respectively. Indeed, the atoms for n∈Zeven

and n∈Zodd belong to sublattice 1 and 2, respectively,

and have diﬀerent nearest neighbor vectors. It is there-

fore useful to deﬁne

T(n)kk=(e

T(n)

+kk, n ∈Zeven

T(n)

−kk, n ∈Zodd

(10)

as the hopping matrices for the two sublattices. We stress

that the dependence of e

T(n)

+(kk)and e

T(n)

−(kk)on the

layer index nis due to the inhomogeneous Ge concentra-

tion proﬁle, and that e

T(n)

+(kk)and e

T(n)

−(kk)diﬀer due

to the diamond crystal structure having two inequiva-

lent atoms in its primitive unit cell. We can therefore

visualize the system, for any given kk, as a 1D, multi-

orbital tight binding chain, as shown in Fig. 2(b), where

the hopping terms alternate in successive layers.

(a)z(b)

T(0)

+k∥

T(1)

−k∥

T(2)

+k∥

T(3)

−k∥

FIG. 2. (a) Diamond crystal structure of silicon. The dashed

lines outline the conventional unit cell of the face centered

cubic lattice. Both red and blue atoms are silicon but be-

long to diﬀerent sublattices. Notice that the vectors connect-

ing an atom to its four nearest neighbors are fundamentally

diﬀerent for the red and blue atoms, giving rise to the al-

ternating hopping structure shown in (b). (b) Eﬀective 1D

tight-binding chain, with hopping matrix terms alternating

between

T(n)

+(kk)and

T(n)

−(kk). Note that each site has 20

orbitals, and only the forward hopping terms are shown. On-

site and backward hopping terms are not shown. For a SiGe

alloy in the virtual crystal approximation, the two-sublattice

structure is retained, but the atoms are replaced by virtual

atoms, with averaged properties consistent with the Ge con-

centration of a given layer.

C. Expansion around kk= 0

Our goal is to understand the spin-orbit physics of low-

energy conduction band states near the Fermi level. In

strained Si/SiGe quantum wells, these derive from the

two degenerate valleys near the Z-point of the strained

Brillouin zone [31]. Therefore, the low-energy states have

small |kk|, and we can understand the spin-orbit physics

by expanding the Fourier-transformed hopping matrices

T(n)

±kkto linear order. We ﬁnd that

T(n)

±(kk) = e

T(n)

0+e

T(n)

R(kk)±e

T(n)

D(kk) + O(k2

k),(11)

where e

T(n)

0is the hopping matrix for kk= 0, and e

T(n)

and e

T(n)

Dcontain the linear kkcorrections. These hop-

ping matrix components are found to be

T(n)

0= Ω(n)¯σ0,(12)

T(n)

R(kk)=Φ(n)(ky¯σx−kx¯σy),(13)

T(n)

D(kk)=Φ(n)(kx¯σx−ky¯σy),(14)

where Ω(n)and Φ(n)are real-valued 10 ×10 matrices,

and ¯σjare the Pauli matrices acting on pseudospin space,

with j= 0, x, y, z.

There are several features to remark on in Eqs. (12)-

(14). First, the momentum-spin structure of e

T(n)

Rand

T(n)

Dhave the familiar forms of Rasbha and Dresselhaus

spin-orbit coupling; hence, we apply the subscript labels

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Spin-orbitenhancementinSi/SiGeheterostructureswithoscillatingGeconcentrationBenjaminD.Woods,1M.A.Eriksson,1RobertJoynt,1andMarkFriesen11DepartmentofPhysics,UniversityofWisconsin-Madison,Madison,Wisconsin53706,USAWeshowthatGeconcentrationoscillationswithinthequantumwellregionofaSi/SiGehet-erostructur...

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Spin-orbit enhancement in SiSiGe heterostructures with oscillating Ge concentration Benjamin D. Woods1M. A. Eriksson1Robert Joynt1and Mark Friesen1 1Department of Physics University of Wisconsin-Madison Madison Wisconsin 53706 USA.pdf

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Spin-orbit enhancement in SiSiGe heterostructures with oscillating Ge concentration Benjamin D. Woods1M. A. Eriksson1Robert Joynt1and Mark Friesen1 1Department of Physics University of Wisconsin-Madison Madison Wisconsin 53706 USA

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