Determination of spin-orbit interaction in semiconductor nanostructures via non-linear transport Renato M. A. Dantas1Henry F. Legg1Stefano Bosco1Daniel Loss1and Jelena Klinovaja1y

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Determination of spin-orbit interaction in semiconductor

nanostructures via non-linear transport

Renato M. A. Dantas,1, ∗Henry F. Legg,1Stefano Bosco,1Daniel Loss,1and Jelena Klinovaja1, †

1Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

(Dated: October 12, 2022)

We investigate non-linear transport signatures stemming from linear and cubic spin-orbit inter-

actions in one- and two-dimensional systems. The analytical zero-temperature response to external

ﬁelds is complemented by ﬁnite temperature numerical analysis, establishing a way to distinguish

between linear and cubic spin-orbit interactions. We also propose a protocol to determine the rele-

vant material parameters from transport measurements attainable in realistic conditions, illustrated

by values for Ge heterostructures. Our results establish a method for the fast benchmarking of

spin-orbit properties in semiconductor nanostructures.

Introduction - Engineering spin-orbit interactions

(SOIs) in semiconductor nanostructures is a crucial chal-

lenge in several branches of physics, ranging from spin-

tronics [1] and topological materials [2–4] to quantum

information processing [5–7]. Notably, large values of

SOIs emerge in nanowires [8–14] and two-dimensional

heterostructures [15–19], where the charge carriers are

holes in the valence band rather than electrons in the

conduction band [20]. In these systems, the SOIs are

also completely tunable by external electric ﬁelds [21–28],

yielding sweet-spots against critical sources of noise [29–

33] and on-demand control of the interaction between

qubits and resonators [34–39].

In particular, hole gases in planar germanium (Ge)

heterostructures are emerging as highly promising can-

didates for processing quantum information [40]. Their

large SOI enables ultrafast qubit operations at low power

in a highly CMOS compatible platform [15–17] and re-

moves the need for additional bulky micromagnets [41–

47], oﬀering a clear practical advantage for scaling up the

next generation of quantum processors [48–50]. Remark-

ably, in these structures, the SOI can be designed to be

linear or cubic in momentum [51–55], greatly impacting

the response of the material to external ﬁelds [56]. De-

spite its potential, an eﬃcient and simple way to measure

the SOI in these materials remains elusive.

From the early classiﬁcation of solid-state systems in

insulators, conductors, and semiconductors [57] to the

more recent discovery of the role of geometry and topol-

ogy in non-trivial band structures [3, 4, 58, 59], transport

experiments have been at the core of condensed matter

physics, providing arguably the most practical yet in-

sightful way to probe the physics of solid-state systems.

While most of the seminal eﬀects, such as the integer

quantum Hall [60–63] or the anomalous Hall [64–68] ef-

fect, depend linearly on the external ﬁelds, lately, an in-

creasing number of novel transport properties in mate-

rials with non-trivial band structure have been reported

in the non-linear regime [69–82]. A particularly fruitful

direction has been the application of dc non-linear re-

sponses, such as magnetochiral anisotropy (also known as

bilinear magnetoresistance) and non-linear Hall eﬀects,

to gain insight into the electronic structure of the sys-

tem [83–91].

In this work, we employ Boltzmann transport theory

to study the response of one- (1D) and two-dimensional

(2D) nanostructures with linear and cubic SOI and show

that these eﬀects leave distinct signatures in the non-

linear response of the system. Moreover, numerical anal-

yses for realistic material parameters and small ﬁnite

temperatures support the zero-temperature analytics and

conﬁrm that these signatures can be measured in state-

of-the-art experiments. This work paves the way for a

functional and time-eﬃcient experimental characteriza-

tion of SOI in these semiconductor nanostructures, en-

abling fast benchmarking already at the material level.

1D - First, we consider 1D systems, e.g. nanowires,

described by the eﬀective Hamiltonian [92]

H=~2k2

2m∗−αkx+βk3

xσy+ ∆jσj,(1)

where we assume Einstein summation convention, αand

β,resp., correspond to SOI linear and cubic in momen-

tum kx,m∗is the eﬀective mass, σiare the elements

of the Pauli vector σ= (σx, σy, σz) acting in (pseudo-)

spin space, and ∆i=1

2µBgiBi(no summation implied)

is the Zeeman ﬁeld, written in terms of the components

of the diagonal g-tensor and the external magnetic ﬁeld

B[93]. The cubic SOI βis often neglected but it can

yield signiﬁcant anisotropies in the spectrum of quantum

dots [92, 94] and, as we shall see, can be determined via

transport measurements. The energy dispersion for such

a two-band Hamiltonian is given by

ε(±)

kx=~2k2

2m∗±q∆2

x+ (αkx+βk3

x−∆y)2+ ∆2

z,(2)

The response of the system to a static external electric

ﬁeld, aligned with the nanowire axis (E=Eˆ

x), is ob-

tained by solving perturbatively the Boltzmann equation

within the relaxation time approximation

f(s)(k) = f0ε(s)

k−τ˙

k(s)· ∇kf(s)(k),(3)

arXiv:2210.05429v1 [cond-mat.mes-hall] 11 Oct 2022

-0.08 0.00 0.08

-0.5

0.0

0.5

1.0

kx(nm-1)

ε(meV)

ℓ(T)

1(0K)

2(0K)

1(0.5 K)

2(0.5 K)

0.0 0.5 1.0 -0.5

0.0

0.5

1.0

σ(ℓ)/|σ(ℓ)max

(meV)

-0.08 0.00 0.08

-0.5

0.0

0.5

1.0

kx(nm-1)

ε(meV)

ℓ(T)

1(0K)

2(0K)

1(0.5 K)

2(0.5 K)

-1.0 -0.5 0.0 0.5 1.0 -0.5

0.0

0.5

1.0

σ(ℓ)/|σ(ℓ)max

(meV)

FIG. 1. Band dispersions (left) and normalized 1st and 2nd order conductivities (right), σ(1) and σ(2),resp., for a 1D system with

linear and cubic SOIs, see Eq. (1), with (a) B= 5 ˆ

yT and (b) slightly rotated towards the SOI direction [B= (5 ˆ

y+ 1 ˆ

z) T].

Here, σ(`)for `= 1,2 were numerically obtained for T= 0 (solid lines) and T= 0.5 K (dashed lines), for ~2/2m∗= 310 meV nm2,

α= 10 meV nm, β= 116 meV nm3, and gy= 2.07 [92].

where f(s)is the out-of-equilibrium distribution for the

s=±band, f0is the Fermi-Dirac distribution, and τ

is the intra-band relaxation time. In 1D, the dynam-

ics of holes are governed solely by the electric ﬁeld, i.e.

~˙

k(s)=eE, where eis the elementary charge. The cur-

rent density can be written as a power series in the elec-

tric ﬁeld E, which at zero temperature, T= 0, is given

jx=X

j(l)

x=e

2π

∞

l=1 τeE

~l

s,i

sgnv(s)

(s)

F,i V(s)

l,k

(s)

F,i

,(4)

where ~V(s)

l,kx=∂l

kxε(s)

kx,~v(s)

kx=∂kxε(s)

kxis the group veloc-

ity, and k(s)

F,i is the ith Fermi wave vector associated with

the band s[obtained from ε(s)

(s)

(∆) = µ] [72, 83, 87].

To gain insight into the impact of the SOI, we ﬁrst con-

sider Baligned with the spin polarization axis of the sys-

tem. Due to the SOI, the otherwise degenerate quadratic

bands develop a ﬁnite spin expectation value along the y-

direction and, in the presence of a collinear magnetic ﬁeld

By, become spin-split [ε(s)

kx=~2k2

2m∗+s(∆y−αkx−βk3

x)].

In this case, the only non-zero contribution in Eq. (4),

aside from the linear term, which is dominated by the

kinetic term and hence less useful to extract information

about SOI, is given by the term quadratic in E,

j(2)

x=−3e3τ2E2β

π~3X

s=±

shk(s)

F,R −k(s)

F,Li,(5)

where k(s)

F,R(L)is the positive (negative) Fermi wave vector

associated with the band s. Note that j(2)

xis directly

proportional to the cubic SOI coupling β, providing a way

to directly determine the presence of this eﬀect from the

second-order conductivity σ(2), deﬁned by jx=σ(1)E+

σ(2)E2+O(E3). Alternatively, non-linear currents can

be induced in systems lacking cubic SOI by choosing B

such that ∆×ˆ

y6= 0 (Fig. 1) [81].

2D - We now consider the eﬀect of linear and cubic SOI

in a 2D system described by the eﬀective Hamiltonian

H=~2k+k−

2m∗+α(σ×k)·ˆ

z+iβ1k3

−σ+−k3

+σ−

+iβ2k−k2

+σ+−k+k2

−σ−+σi∆i,(6)

where k±=kx±ikyand σ±= (σx±iσy)/2. Here, α,

β1, and β2are the linear, isotropic cubic, and anisotropic

cubic SOI constants, resp. [51, 54, 95–98]. The energy

dispersion reads

ε(±)

~2k2

x+k2

y

2m∗±qn2

x+n2

y+∆2

z,(7)

where

nx=αky+i

2β1(k3

−−k3

)−β2(k2

−k+−k2

+k−)+∆x,

ny=−αkx−1

2β1(k3

−+k3

)+β2(k2

−k++k2

+k−

)+∆y.

To set the parameter values, hereafter we consider Ge

[100] and [110] for realistic experimental conditions.

While both systems have similar eﬀective masses and

cubic SOIs, assumed as ~2/2m∗= 620 meV nm2,β1=

190 meV nm3and β2= 23.75 meV nm3, they diﬀer in the

linear SOI, which is absent for Ge [100] and takes the

value α= 1.5 meV nm for Ge [110] [53]. Further, gx/y =

0.207 for Ge [100], while for Ge [110], gx/y = 1.244.

Compared to 1D, the dynamics of the hole quasiparti-

cles is not only enriched by orbital eﬀects for out-of-plane

magnetic ﬁelds but also by the band geometry, captured

by the equations of motion

r(s)=˜v(s)

k−˙

k(s)×Ω(s)

k,~˙

k(s)=eE−e˙

r(s)×B,(8)

where Ω(s)

k=ih∇ku

(s)

k|×|∇ku

(s)

kiis the Berry curvature,

with H|u

(s)

ki=ε(s)

k|u

(s)

ki, and ˜v(s)

k=v(s)

k−~−1∇

k(m(s)

k·B)

is the group velocity for the dispersion modiﬁed by the

orbital magnetic moment of the wave packet, m(s)

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Determinationofspin-orbitinteractioninsemiconductornanostructuresvianon-lineartransportRenatoM.A.Dantas,1,HenryF.Legg,1StefanoBosco,1DanielLoss,1andJelenaKlinovaja1,y1DepartmentofPhysics,UniversityofBasel,Klingelbergstrasse82,4056Basel,Switzerland(Dated:October12,2022)Weinvestigatenon-lineartranspo...

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Determination of spin-orbit interaction in semiconductor nanostructures via non-linear transport Renato M. A. Dantas1Henry F. Legg1Stefano Bosco1Daniel Loss1and Jelena Klinovaja1y

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