Spin polarisation and spin dependent scattering of holes in transverse magnetic focussing M. J. Rendell1S. D. Liles2A. Srinivasan2O. Klochan3 1I. Farrer4 5D. A. Ritchie5and A. R. Hamilton1y

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Spin polarisation and spin dependent scattering of holes in transverse magnetic

focussing

M. J. Rendell,1, ∗S. D. Liles,2A. Srinivasan,2O. Klochan,3, 1 I. Farrer,4, 5 D. A. Ritchie,5and A. R. Hamilton1, †

1School of Physics and Australian Research Council Centre of Excellence in Future Low-Energy Electronics Technologies,

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

2School of Physics, University of New South Wales, Sydney, NSW 2052, Australia

3University of New South Wales Canberra, Canberra, ACT 2600, Australia

4Department of Electronic and Electrical Engineering, University of Sheﬃeld, Sheﬃeld, S1 3JD, UK

5Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK

(Dated: October 10, 2022)

In 2D systems with a spin-orbit interaction, magnetic focussing can be used to create a spatial

separation of particles with diﬀerent spin. Here we measure hole magnetic focussing for two diﬀerent

magnitudes of the Rashba spin-orbit interaction. We ﬁnd that when the Rashba spin-orbit magni-

tude is large there is signiﬁcant attenuation of one of the focussing peaks, which is conventionally

associated with a change in the spin polarisation. We instead show that in hole systems with a k3

spin-orbit interaction, this peak suppression is due to a change in the scattering of one spin state,

not a change in spin polarisation. We also show that the change in scattering length extracted from

magnetic focussing is consistent with results obtained from measurements of Shubnikov-de Haas

oscillations. This result suggests that scattering must be considered when relating focussing peak

amplitude to spin polarisation in hole systems.

I. INTRODUCTION

In a magnetic focussing experiment, a collimated beam

of charge is focussed into a circular orbit by a transverse

magnetic ﬁeld, analogous to a mass spectrometer. Mag-

netic focussing was originally proposed as a method of

studying the Fermi surface of metals [1, 2], and has also

been used to measure band structures in graphene [3],

and electron-electron scattering lengths in GaAs/AlGaAs

[4].

In systems with a spin-orbit interaction (SOI), the

magnetic focussing trajectories become spin dependent

as the spin states are now coupled to momentum. If the

SOI is suﬃciently large, magnetic focussing can spatially

separate the spin states and create a spin-dependent mass

spectrometer [5–12]. The high mobility and large SOI

of 2D hole systems in GaAs has made them an ideal

candidate for spin-dependent magnetic focussing experi-

ments. Experimental work has used magnetic focussing

to measure spatial separation of spin [5], spin ﬁltering

by quantum point contacts (QPCs) [7] and interactions

between 1D subbands in a QPC [13]. Magnetic focussing

of holes has also been proposed as a way to measure

g-factor anisotropies [14], and complex spin dynamics

[9, 15], which are not visible in other measurements of

2D systems such as Shubnikov-de Haas oscillations.

Here, we concentrate on the use of magnetic focussing

peak amplitude as a measure of the spin polarisation

[5, 7, 16–18]. It has been proposed that the relative am-

plitudes of the spin-split magnetic focussing peaks is de-

termined by the spin polarisation of the injected charge.

∗M. J. Rendell and S. D. Liles contributed equally to this work

†alex.hamilton@unsw.edu.au

This technique has been used in hole systems to observe

spontaneous polarisation in QPC transmission [7] and

spin-dependent transmission of QPCs [5, 13]. Despite

magnetic focussing being used for these techniques, there

has been limited study of the eﬀect of changing the mag-

nitude of the Rashba SOI on hole magnetic focussing.

A recent study investigated magnetic focussing using a

device where the Rashba SOI magnitude could be tuned

in situ using a top gate voltage (VTG) [19]. This tech-

nique revealed an increase in the spatial separation of

the spin-split focussing trajectories as the Rashba SOI

was increased. However, there is a limit to the amount

the Rashba SOI can be changed using this method. In

addition, any change to VT G will also change the 2D hole

density and conﬁning potential in addition to the Rashba

SOI magnitude. As such, further study requires a diﬀer-

ent method of changing the Rashba SOI.

In this work we study magnetic focussing in two litho-

graphically identical samples which diﬀer only in the

magnitude of the Rashba SOI. We change the Rashba

SOI by changing the heterostructure used to conﬁne the

2D system, allowing us to create a large change in the

magnitude of the Rashba SOI for a similar VTG and 2D

density. By comparing the two samples, we observe a

change in the amplitude of the magnetic focussing peaks,

which is typically associated with a change in the spin po-

larisation. However, we instead ﬁnd that the change in

peak amplitude is consistent with an increase in scatter-

ing of one spin state rather than a change in spin polar-

isation. We measure the scattering length of each spin

state from the focussing peak amplitude, and ﬁnd good

agreement with scattering lenghts found from Shubnikov-

de Haas measurements. We conclude that the change in

focussing peak amplitude is due to the k3Rashba term

causing a diﬀerent eﬀective mass and hence scattering

length of each spin state, rather than a change in spin po-

arXiv:2210.03383v1 [cond-mat.mes-hall] 7 Oct 2022

k+k-

Injector

Detector

FIG. 1. a) Magnetic focussing in the presence of a spin-

orbit interaction. The red and blue lines correspond to the

spin split focussing trajectories, which result in a splitting

of the ﬁrst focussing peak. The dashed line corresponds to

the classical focussing trajectory. c) The ﬁrst 2D subband

for a hole system. Here the Rashba SOI term depends on k3

which causes a change in the slope (and hence m∗) of the spin

resolved subbands.

larisation. This result suggests that care must be taken

when relating the amplitude of spin-split focussing peaks

to the spin polarisation in 2D hole systems.

II. MAGNETIC FOCUSSING WITH A CUBIC

RASHBA SPIN-ORBIT INTERACTION

Fig 1a) shows a schematic of a hole magnetic focussing

device. A constant current is applied through an injec-

tor, where an out-of-plane perpendicular magnetic ﬁeld

causes the holes to follow cyclotron orbits. The voltage

across the detector is measured, and a peak is observed

when the focussing diameter is equal to the spacing be-

tween injector and detector (black dashed line in Fig 1a).

Peaks in the focussing signal occur when the magnetic

ﬁeld is an integer multiple of [20]

B=2~kF

Where kFis the Fermi momentum and dis the distance

between injector and collector QPC (focussing diameter).

In the presence of a spin-orbit interaction (SOI) the hole

trajectories become spin dependent, resulting in a spa-

tial separation of spin (blue and red lines in Fig 1a). The

spatial spin separation causes the ﬁrst magnetic focussing

peak to split into two, with each peak corresponding to

a diﬀerent spin chirality. The relative amplitude of these

spin peaks has been used as a measure of the spin polar-

isation in 2D hole systems.[5]

The form of the Rashba spin-orbit term for 2D hole

systems is fundamentally diﬀerent to equivalent electron

systems. This diﬀerence can have a dramatic impact on

spin resolved focussing peaks. In GaAs, the subband

dispersion for 2D holes with a Rashba SOI is given by

[21]

Eh=~2k2

2m∗±βEz

∆HH-LH

k3(1)

where Ezis the electric ﬁeld in the out-of-plane direc-

tion and ∆HH-LH is the splitting between the heavy hole

(HH) and light hole (LH) subbands. Fig 1b) shows the

resulting HH subband dispersion for a 2D hole system

with Rashba SOI. The SOI causes the momentum of the

holes to become spin dependent, with two values of k(k+

and k−) at the Fermi energy (horizontal dashed line). In

a magnetic focussing measurement, this results in sepa-

rate cyclotron orbits for each spin and creates a spatial

spin separation, splitting the ﬁrst focussing peak. Previ-

ous work has demonstrated the ability to detect a change

in peak splitting as the magnitude of the Rashba SOI is

changed [19].

The k3structure of the Rashba SOI term for holes also

causes the curvature of the 2D subbands to become spin

dependent. This results in a diﬀerence in eﬀective mass

for each spin chirality in addition to the diﬀerence in k

[22]. The spin dependent eﬀective mass has been used

to demonstrate electrical control of the Zeeman split-

ting [23], and proposed as a way to detect and gener-

ate topological properties in a 2D hole system [24, 25].

The change in eﬀective mass is also possible to detect via

focussing peaks. If the Rashba SOI term is suﬃciently

large, the diﬀerence in eﬀective mass can be observed

as a diﬀerence in scattering. Since focussing peak am-

plitude is exponentially sensitive to scattering [26, 27],

the change in eﬀective mass will therefore impact the fo-

cussing peak amplitude. This analysis does not include

contributions from k-linear Rashba SOI terms for 2D

holes [28–30]. These terms do not cause a spin-dependent

change in the curvature of the 2D subbands and should

not aﬀect the diﬀerence in eﬀective mass between the

spin chiralities.

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SpinpolarisationandspindependentscatteringofholesintransversemagneticfocussingM.J.Rendell,1,S.D.Liles,2A.Srinivasan,2O.Klochan,3,1I.Farrer,4,5D.A.Ritchie,5andA.R.Hamilton1,y1SchoolofPhysicsandAustralianResearchCouncilCentreofExcellenceinFutureLow-EnergyElectronicsTechnologies,UniversityofNewSouthWa...

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Spin polarisation and spin dependent scattering of holes in transverse magnetic focussing M. J. Rendell1S. D. Liles2A. Srinivasan2O. Klochan3 1I. Farrer4 5D. A. Ritchie5and A. R. Hamilton1y.pdf

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Spin polarisation and spin dependent scattering of holes in transverse magnetic focussing M. J. Rendell1S. D. Liles2A. Srinivasan2O. Klochan3 1I. Farrer4 5D. A. Ritchie5and A. R. Hamilton1y

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