Recent advances in hole-spin qubits Yinan Fang1 Pericles Philippopoulos2 Dimitrie Culcer34 W. A. Coish5 and Stefano Chesi67

2025-04-29 0 0 1.79MB 46 页 10玖币

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Recent advances in hole-spin qubits

Yinan Fang1, Pericles Philippopoulos2, Dimitrie Culcer3,4, W.

A. Coish5, and Stefano Chesi6,7

1School of Physics and Astronomy, Yunnan University, Kunming 650091, China

2Nanoacademic Technologies Inc., Montréal QC, Canada

3School of Physics, University of New South Wales, Sydney NSW 2052, Australia

4Australian Research Council Centre of Excellence in Future Low-Energy Electronics

Technologies, The University of New South Wales, Sydney NSW 2052, Australia

5Department of Physics, McGill University, Montréal QC, Canada

6Beijing Computational Science Research Center, Beijing 100193, China

7Department of Physics, Beijing Normal University, Beijing 100875, China

E-mail: stefano.chesi@csrc.ac.cn

Abstract. In recent years, hole-spin qubits based on semiconductor quantum dots

have advanced at a rapid pace. We ﬁrst review the main potential advantages of these

hole-spin qubits with respect to their electron-spin counterparts, and give a general

theoretical framework describing them. The basic features of spin-orbit coupling and

hyperﬁne interaction in the valence band are discussed, together with consequences on

coherence and spin manipulation. In the second part of the article we provide a survey

of experimental realizations, which spans a relatively broad spectrum of devices based

on GaAs, Si, or Si/Ge heterostructures. We conclude with a brief outlook.

Nearly 25 years have passed since the idea of implementing quantum information

processing with semiconductor quantum dots was put forward [1]. The spin of conﬁned

electrons (or holes) is a natural two-level system, relatively well shielded from charge

ﬂuctuations and which, as it has been now established, can be coherently manipulated

with high accuracy and speed. Research in the ﬁeld has been constantly progressing over

the years, exploring at the same time the rich physics of these semiconductor devices,

but is still restricted to the few-qubits regime. At the moment it is not yet obvious

which platform oﬀers better prospects to realize intermediate- or large-scale devices,

and recently hole-spin devices have gained prominence.

Recent rapid progress in controlling hole spins has brought breath of fresh air to

the ﬁeld. However, the potential advantages (and disadvantages) of these qubits over

their electron-spin counterparts should be carefully weighted. To get some perspective,

it is interesting to discuss how this family of spin qubits ﬁts the main line of progress,

dominated by electron-spin devices. Within about 10 years from the 1998 theoretical

proposal, all the main ingredients required to realize quantum information processing,

such as spin readout, single-spin manipulation, and controllable spin-spin exchange

interactions, were ﬁrmly established. An account of early progress can be found in

the 2007 review Ref. [2]. In this initial phase, the most successful qubits were planar

arXiv:2210.13725v2 [cond-mat.mes-hall] 6 Feb 2023

Recent advances in hole-spin qubits 2

(lateral) quantum dots in AlGaAs/GaAs heterostructures. However, these systems are

susceptible to strong dephasing caused by nuclear spins. This dephasing is unavoidable

in III-V semiconductors since every isotope of Ga and As has a ﬁnite nuclear spins.

To remedy this situation, Si electron-spin devices have gradually emerged as a better

alternative. Coherent singlet-triplet oscillations in a Si/SiGe double quantum dot were

demonstrated in 2012 [3] and a number of landmark results have been obtained so

far, such as high-ﬁdelity (>99.9%) single-qubit manipulation [4, 5], strong coupling

of spin qubits to a superconducting resonator [6–9] and, very recently, two-qubit

quantum processors with gate ﬁdelities surpassing the fault-tolerance threshold [10–12]

and universal operation of a six-qubit array [13].

Hole-spin qubits ﬁt naturally into the historical path, as they can overcome the

inﬂuence of nuclear spins in the same way as electron-spin devices based on Si: For

hole-spin qubits in Si and Ge, the low natural abundance of spinful nuclei can be further

reduced by isotopic puriﬁcation. Furthermore, even in III-V semiconductors, holes are

much less aﬀected by nuclear spins than conduction-band electrons. The Fermi contact

interaction vanishes, leading to an anisotropic hyperﬁne interaction caused by dipolar

and orbital terms, weaker by about an order of magnitude. In addition to having

a weaker inﬂuence of nuclear spins, two attractive features of holes should facilitate

scalability. The ﬁrst is the absence of a valley degree of freedom. The valley splitting

of electrons in Si quantum dots depends on atomistic features of the device, which are

diﬃcult to control uniformly over large quantum-dot arrays. The second and most

important feature is the strong spin-orbit interaction of holes, allowing in-situ electric

control of spin couplings and eﬀective spin manipulation through a local electric drive.

In contrast, the spin-orbit coupling of conduction electrons in Si is weak, and single-spin

manipulation (as well as coupling of the spin to superconducting resonators) requires

large auxiliary elements, such as micromagnets [14,15].

Strong spin-orbit coupling is, of course, a double-edged sword, as it enhances the

inﬂuence of charge noise on the spin degrees of freedom. Despite this general diﬃculty,

and the late start of hole devices in terms of fabrication, in recent years hole-spin qubits

have achieved remarkable results. In many aspects concerning single-spin manipulation

[16–19], scalability [20,21], as well as coupling to superconducting circuits [22], hole-spin

qubits can approach and sometimes already surpass the performance of electron-spin

qubits.

To describe recent progress in hole-spin qubits, we have structured the article in two

main parts. Section 1 gives a theoretical perspective, introducing the general framework

for hole-spin qubits and the main interactions aﬀecting them. Section 2 is a survey of

various experimental platforms, highlighting some of the most interesting achievements.

While the present review does not aim to be complete, it should oﬀer an informative

and suﬃciently broad view of hole-spin qubits. Recently, two more specialized reviews

have appeared: Ref. [23], focusing on hole spins in GaAs lateral quantum dots, and

Ref. [24], reviewing hole-based Ge devices. For a more complete perspective on spin

qubits (including electron-spin systems), one could consult Ref. [25].

Recent advances in hole-spin qubits 3

1. General description of hole-spin qubits

Quantum dots for use in information processing are typically based on either III-V

compound semiconductors such as GaAs, which have zincblende lattices, or on group-

IV semiconductors such as Si and Ge, with diamond lattices. While the conduction band

is formed by s-like atomic wave functions, resulting in spin-1/2quasiparticles, valence-

band states in diamond and zincblende semiconductors are predominantly formed by

atomic p-orbitals. In the atomic limit, combining the L= 1 orbital angular momentum

with the electron spin-1/2we obtain two possibilities for the total angular momentum

J, henceforth also referred to as (eﬀective) spin: J= 3/2or J= 1/2. At zero crystal

momentum kthese states are separated by a gap and the band originating from the

J= 1/2manifold, which is always lower in energy, is known as the split-oﬀ band.

More precisely, since rotational symmetry is broken in the crystal, the valence-band

states at k= 0 are classiﬁed by the Γ8and Γ7representations of the tetrahedral double

group Td[26, 27]. Furthermore, the crystalline potential leads to the hybridization of p

orbitals with dstates [28–32], which can signiﬁcantly aﬀect the form of the hyperﬁne

interaction [33, 34] (see Section 1.3). Despite these caveats, the fourfold degeneracy

of atomic J= 3/2states (corresponding to Γ8) is preserved. The existence of the

split-oﬀ band makes a diﬀerence for numerical values, especially in Si, but properties of

holes are usually well understood considering the states at the top of the valence band.

Therefore, the following Luttinger Hamiltonian [35, 36] is often taken as a fundamental

starting point for the description of holes:

HL(k) = −γ1+5

2γ2~2k2

2m0

+γ2

(k2

xJ2

x+k2

yJ2

y+k2

zJ2

+γ3

2m0

({kx, ky}{Jx, Jy}+ c.p.),(1)

where {Ji}are spin-3/2matrices, γ1,γ2and γ3are the Luttinger parameters, m0is the

bare electron mass, {A, B}= (AB +BA)/2, and c.p. are terms obtained from cyclic

permutations of x, y, z. In contrast with conduction electrons, where perturbative spin-

orbit interactions arise from the valence band and are suppressed by the large band gap of

the semiconductor, the Hamiltonian describing the valence-band holes [Eq. (1)] features

a strong spin-orbit coupling between kand J. This coupling results in a splitting of the

fourfold degeneracy at k= 0 into two doubly degenerate bands at ﬁnite momentum.

The higher (lower) valence band states are denoted as heavy (light) holes, with angular

momentum Jz=±3/2(±1/2) in the limit k→0. Since γ2,3are generally of the

same order as γ1, the spin-orbit splitting between heavy-hole and light-hole bands is

comparable to the total kinetic energy.

Envelope function approximation. In the presence of smooth external potentials

and magnetic ﬁelds, the microscopic Bloch wave at k= 0 is modulated by a slowly-

varying envelope that satisﬁes a multi-component Shrödinger equation [36, 37]. The

usual prescription for electronic states is to replace ~k→π=−i~∇+eA(r)(−e < 0

is the electron charge) in HL, where A(r)is the vector potential associated with the

Recent advances in hole-spin qubits 4

external magnetic ﬁeld B. As usual, the components of the canonical momentum π

do not commute, in general, thus in the second line of Eq. (1) we have speciﬁed the

appropriate symmetrized products of k(which become unnecessary when B= 0). Some

care should be taken in writing the Hamiltonian for the state of a missing electron. First,

the electronic energy is subtracted from a ﬁlled valence band, which can be simply

accounted for by a global change of sign in the Hamiltonian. This leads to the following

Hamiltonian describing holes within the envelope-function approximation:

Henv =−HL(π/~) + eV (r)+2κµBJ·B+ 2qµBJ·B,(2)

where eV (r)is the 3D conﬁning potential and the last two terms describe the Zeeman

interaction [35, 36], with J=J3

x, J3

y, J3

zand µBthe Bohr magneton. While Eq. (2)

is a legitimate choice, giving the correct energy of the hole states, it still refers to the

electronic wavefunction and this should be taken into account when computing certain

observables. Recall, in particular, that a hole with quantum numbers (k, Jz)corresponds

to a missing (−k,−Jz)electronic state. To obtain directly the hole wavefunction, whose

angular momentum corresponds to the physical state, one can consider an alternative

version of Henv, by applying time-inversion to it. Since −i~∇→+i~∇upon time

inversion, the transformed Hamiltonian recovers the Peierls substitution appropriate for

a positively charged particle, i.e., π→ −i~∇−eA(r)in Eq. (2). Furthermore, the

second and third terms of Eq. (2) change sign, since J→ −J.

Inversion-asymmetry and strain. In III-V zincblende materials, for which there is

no center of inversion symmetry, it is sometimes important to supplement Eq. (1) with

additional terms. These corrections are known as Dresselhaus spin-orbit couplings and

include the k-linear term:

H1=2

√3Ckkx{Jx, J2

y−J2

z}+ c.p.,(3)

and a more convoluted k-cubic contribution [36]:

H3=b41 {kx, k2

y−k2

z}Jx+ c.p.+b42 {kx, k2

y−k2

z}J3

x+ c.p.

+b51 {kx, k2

y+k2

z}{Jx, J2

y−J2

z}+ c.p.+b52 k3

x{Jx, J2

y−J2

z}+ c.p..(4)

In particular, the k-cubic terms were shown early on to give a signiﬁcant spin-orbit

splitting in symmetric GaAs quantum wells [38]. A recent theoretical analysis [39] has

assessed the relative importance of H1over H3in GaAs 2DHGs. Furthermore, a diﬀerent

type of coupling induced by inversion-asymmetry, but also depending on the external

potential V(r), has been discussed [39]:

HE=eaBχ

√3[Ex{Jy, Jz}+ c.p.],(5)

where E=−∇Vis the local electric ﬁeld, aBis the Bohr radius, and χis a material

parameter. Equation (5), which involves the gradient of the potential and includes

interband matrix elements between heavy- and light-hole states, goes beyond the

Recent advances in hole-spin qubits 5

usual envelope-function approximation. Such pseudospin-electric coupling physically

originates from the interaction between the electric ﬁeld and the dipole operator, which

takes the general form of Eq. (5) when expressed in the subspace of valence-band states.

A ﬁrst-principles evaluation gives a relatively large value χ'0.2in GaAs [39].

For Si and Ge, the above Dresselhaus terms are absent in the bulk. However, it is

important to account for ﬁnite strain when modelling, e.g., high-mobility SiGe/Ge/SiGe

quantum wells [40, 41] as well as core-shell nanowires [42, 43]. In general, the eﬀect of

strain is described by the well-known Bir-Pikus Hamiltonian Hε=−(av+5

4bv)(εxx +

εyy +εzz ) + bv(J2

xεxx + c.p.) + 2

√3dv({Jx, Jy}εxy + c.p.), where {εij }are the strain tensor

elements and av, bv, dvare material-speciﬁc deformation potentials [44]. Assuming that

only hydrostatic and uniaxial strain are present, i.e., εxy =εyz =εzx = 0 and εxx =εyy,

the eﬀect of Hεis to induce diﬀerent oﬀsets for the heavy-hole (Jz=±3/2) and light-hole

(Jz=±1/2) states [40–42].

1.1. Spin-orbit interaction of conﬁned holes

The description of conﬁned hole systems can be based directly on the general four-band

formalism just described, eventually supplemented by the split-oﬀ band. However,

many of the hole-spin qubits of interest are fabricated from low-dimensional systems

such as quantum wells or nanowires, where additional gates realize the zero-dimensional

conﬁnement. It is therefore useful to describe the eﬀective Hamiltonians governing such

2D or 1D systems. Because such eﬀective models are relatively simple, they are often

taken as starting points to characterize various important eﬀects, such as spin relaxation

or electrical manipulation of spin qubits.

Quantum wells. For quasi-2D holes the potential in Eq. (2) may be taken as V(z)

if the growth direction is along a main crystal axis. Then, for B= 0 and zero in-

plane wavevector, kk= (kx, ky) = (0,0), Eq. (2) simpliﬁes considerably, giving two

decoupled Schrödinger equations in 1D. The two sectors correspond to the Jz=±3/2

and Jz=±1/2subspaces, characterized by diﬀerent eﬀective masses m0/(γ1±2γ2).

The lowest-energy subband is formed by heavy-holes (Jz=±3/2) and the energy gap

to the lowest light-hole subband can be estimated as ∆LH '2γ2(π~/L)2/m0, assuming

an inﬁnite potential well of width L. For more realistic scenarios, the value of ∆LH

depends of the detailed shape of V(z)and, as discussed already, is modiﬁed by uniaxial

strain. In general, the four-dimensional degeneracy of the Luttinger Hamiltonian has

been broken by the conﬁning potential, generating naturally a two-level pseudospin. For

the low-energy states, we can identify the eﬀective spin up/down eigenstates with the

Jz=±3/2eigenvalues, i.e., |⇑i =|+3/2iand |⇓i =|−3/2i.

Despite the simple decoupling described above (which, we stress, is exact only at

zero in-plane momentum), the light-hole/heavy-hole mixing terms still play a crucial role

in the eﬀective 2D Hamiltonian. As seen from Eq. (1), ﬁnite values of kx,y will induce

matrix elements between the two subspaces, through the action of the Jx,y operators.

It is this coupling between light- and heavy-hole subbands what generates the desired

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Recentadvancesinhole-spinqubitsYinanFang1,PericlesPhilippopoulos2,DimitrieCulcer3;4,W.A.Coish5,andStefanoChesi6;71SchoolofPhysicsandAstronomy,YunnanUniversity,Kunming650091,China2NanoacademicTechnologiesInc.,MontréalQC,Canada3SchoolofPhysics,UniversityofNewSouthWales,SydneyNSW2052,Australia4Australi...

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Recent advances in hole-spin qubits Yinan Fang1 Pericles Philippopoulos2 Dimitrie Culcer34 W. A. Coish5 and Stefano Chesi67

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