Currently at ]Department of Electronic and Electrical Engineering, University
of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom.

# Temperature dependence of spin-split peaks in transverse electron focusing

Chengyu Yan<EMAIL_ADDRESS>Sanjeev Kumar  Michael Pepper  London Centre
for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, United Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Patrick See
National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW,
United Kingdom
Ian Farrer  [  David Ritchie  Jonathan Griffiths  Geraint Jones  Cavendish
Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE, United Kingdom

###### Abstract

We present experimental results of transverse electron focusing measurements
performed using n-type GaAs. In the presence of a small transverse magnetic
field (B⟂), electrons are focused from the injector to detector leading to
focusing peaks periodic in B⟂. We show that the odd-focusing peaks exhibit a
split, where each sub-peak represents population of a particular spin branch
emanating from the injector. The temperature dependence reveals the peak
splitting is well defined at low temperature whereas it smears out at high
temperature indicating the exchange-driven spin polarisation in the injector
is dominant at low temperatures.

## Background

The electron transport through a quasi one-dimensional (1D) system realised
using the two-dimensional electron gas (2DEG) formed at the interface of
GaAs/AlGaAs heterostructure has been extensively studied. A 1D system provides
an outstanding platform to envisage not only the non-interacting quantum
mechanical system where the conductance quantisationTPA86 ; DTR88 ; WVB88  is
in the units of n$\times$$\frac{2e^{2}}{h}$, where n=1,2,3… are different 1D
energy subbsands, but also a venue to explore many-body physicsTNS96 ; SFP08 ;
HTP09 ; KTS14 ; YSM17 ; YSP17 . Recently, the progress in the physics of many-
body 1D system has gained momentum due to prediction and experimental
demonstration of rich-phases in low-density 1D system leading to incipient
Wigner crystallisationHTP09 ; KTS14 ; JKM09 . Moreover the origin of the 0.7
conductance anomaly in the frame work of many-body 1D system is still
debatedWB96 ; WB98 ; RDJ02 ; RDJ05 ; BCF01 . The 0.7 anomaly has two major
features: first, in the presence of in-plane magnetic field, the 0.7 anomaly
evolves into 0.5$\times$$\frac{2e^{2}}{h}$ plateau, which indicates it is
spin-relatedTNS96 ; second, the 0.7 anomaly was found to weaken (strengthen)
with decreasing (increasing) temperatureTNS96 . These remarkable observations
have led to a volume of theoretical and experimental attempts to probe the
intrinsic spin polarisation associated with the 0.7 anomaly, however there is
no consensus as such on the origin of this anomalyWB96 ; WB98 ; RDJ02 ; RDJ05
; BCF01 . Therefore, to shed more light on the 0.7 anomaly, it is essential to
perform a direct measurement on the spin polarisation within a 1D channel.

Figure 1: The experiment setup and device characteristic. (a) A representative
plot of transverse electron focusing with both the injector and detector set
to G0 (2e2/h). Vcc is the voltage drop across the detector. Focusing peaks are
well defined with positive magnetic field and the signal is negligible with
negative magnetic field. The first peak shows pronounced splitting. The two
sub-peaks have been highlighted as peak I and peak II. The inset shows an SEM
image of the device. The separation between the injector and detector is 1.5
$\mu$m. Red squares form the Ohmic contacts whereas two pairs of grey-coloured
gates, left and top, form the injector and detector, respectively. The scale
bar is 2 $\mu$m. (b) Conductance characteristics of the injector and detector.

A scheme based on transverse electron focusing (TEF) was proposed to address
the spin polarisationGC04 ; AGC07 , and was validated in p-type GaAsLPR06 ;
SC11  and n-type InSbARD06 . Within this scheme, the spin polarisation can be
extracted from the asymmetry of the two sub-peaks of the first focusing peak.
Recently, we showed that injection of 1D electrons whose spins have been
spatially separated, can be detected in the form of a split in the first
focusing peak, where the two sub-peaks represent the population of detected
spin statesYSM16 . In the present work, we report the temperature dependence
of spin-split first focusing peak, and analyse the results based on the spin-
gap present between the two spin species.

## Method

The devices studied in the present work were fabricated from the high mobility
two dimensional electron gas (2DEG) formed at the interface of
GaAs/Al0.33Ga0.67As heterostructure. At 1.5 K, the measured electron density
(mobility) was 1.80$\times$1011cm-2 (2.17$\times$106cm2V-1s-1), therefore the
mean free path is over 10 $\mu$m which is much larger than the electron
propagation length. The experiments were performed in a cryofree dilution
refrigerator with a lattice temperature of 20 mK using the standard lockin
technique. The range of temperature dependence measurement was from 20 mK to
1.8 K.

Figure 2: Mechanism of peak splitting. (a)-(b) Peak splitting in the
coordinate-space for first and second focusing peaks, respectively. The red
and white arrows represent spin-up and spin-down electrons, the coloured
blocks stand for the electrostatic potential and the red-dashed trace is with
smaller cyclotron radius while the yellow-dotted one is with larger cyclotron
radius. (c)-(d), Peak splitting in the k-space for first and second focusing
peaks, respectively. The electrons travel from (0, ky) to (-kx, 0)
anticlockwise in plot (c). In plot (d), the thick blue arrow highlights the
transition after reflection at the boundary of electrostatic potential formed
between the injector and detector.

## Results and discussion

Figure 1(a) shows the experimental setup along with a typical focusing
spectrum obtained using the device shown in the inset. The focusing device is
specially designed so that the injector and detector can be controlled
separately to avoid a possible cross-talking between themYSM16 ; RMP02 ; HVH89
. The quantum wire used for the injector and detector has a width (confinement
direction) of 500 nm and length (current flow direction) of 800 nm. Both the
injector and detector show well defined conductance plateaus as shown in Fig.
1(b). Further details on the device are given in the caption of Fig. 1.

With negative magnetic field, the measured signal is almost zero because
electrons bend into the opposite direction and thus miss the detector. It is
also evident that the Shubnikov-de Haas oscillation and quantum Hall effect do
not contribute to the observation. In the presence of a small positive
transverse magnetic field B⟂ electrons are focused from the injector to
detector leading to focusing peaks periodic in B⟂ while the detected signal is
negligible at the negative magnetic field end. The calculated periodicity of
60 mT using the relationHVH89 ,

$B_{focus}=\frac{\sqrt{2}\hbar k_{F}}{eL}$ $None$

is in good agreement with the experimental result. Here e is the elementary
charge and $\hbar$ is the reduced Planck constant, L is the separation between
the injector and detector (in the 90∘ focusing device geometry, this is the
separation along the diagonal direction). In addition to the periodic
focussing peak which is a manifestation of the semi-classical electron
cyclotron orbit, it is interesting to notice the splitting of odd-numbered
focusing peaks. It is suggested that this anomalous splitting of odd-numbered
focusing peaks arises from the spin-orbit interaction (SOI)GC04 ; AGC07  and
has been successfully observed in GaAs hole gasLPR06 ; SC11  and InSb electron
gasARD06 . We recently demonstrated splitting of odd-numbered focusing peaks
in n-GaAsYSM16  where a longer quantum wire possessing partially polarised and
spatially separated 1D electrons was used to inject the polarised 1D electrons
into the 2D regime and subsequently measured across the detector in the form
of a split in the first focusing peak. Here we are interested in investigating
the thermal effect on the spin states within the 1D channel via the transverse
electron focusing. We note that the splitting smears out when the thermal
energy kBT exceeds 2$\Delta$E ($\Delta$E is the energy difference between the
two spin branches) agreeing with the theoretical predictionAGC07 .

Figure 3: Temperature dependence of TEF. (a)-(c) The injector was set to
0.5$G_{0}$, G0 and 1.8G0, respectively. The lattice temperature was
incremented from 20 mK (top trace) to 1.8 K (bottom trace). Data have been
offset vertically for clarity. (d)-(f), zoom-in of the data in (a)-(c).

Before we discuss the temperature dependence effect, it is important to
understand the mechanism responsible for the observed peak splitting. Figure
2(a) and (b) show the potential profile of the split gates forming the
injector (bottom pair) and the detector (left pair). In the presence of SOI,
the two spin species follow different cyclotron radii as shown in Fig. 2(a)
thus resulting in two sub-peaks in the first focusing peak. However, the
situation is different for the second focusing peak where a scattering at the
boundary of electrostatic potential created by the split gates is involved as
shown in Fig. 2(b). In this case, a spin-up electron (red arrow in the
colourplots) initially follows a smaller cyclotron radius while it possesses a
larger radius after the scatteringGC04 ; AGC07  and vice-versa for the spin-
down electron (white arrow), thus the two spin species re-join at the
detector. The underneath reasoning for the peak splitting can be found in the
k-space in Fig. 2(c) and (d). Here we assume the spin-orbit interaction is of
Rashba-type, however, the analysis holds valid for Dresselhaus effect in bulk
as well. For the first focusing peak (Fig. 2(c)), the two spin-species travel
from (0, ky) to (-kx, 0) along different Fermi surfaces. For the second
focusing peak (Fig. 2(d)), the same argument holds true before the scattering,
however, the momentum changes its sign while the spin orientation remains
preserved after the scatteringGC04 . Therefore, a spin-up electron (red
arrows) initially occupying the inner Fermi surface hops to the outer Fermi
surface after the scattering to guarantee both the sign of the momentum and
the spin orientation are in the correct order (the hopping is highlighted by
the thick blue arrow in Fig. 2(d)) and vice-versa for the spin-down electron.
The cyclotron radius is proportional to the momentum, so that the alternation
in cyclotron radius occurs in the coordinate-space as a consequence of hopping
between two Fermi surfaces which leads to a single second focusing peak.

Figure 4: Analysis of the temperature dependence data. (a) Reconstructing the
first focusing peak with two Lorentzian peaks at 20 mK. The solid blue line is
the experimental data, the green-round marker is the fit for peak I and red-
square marker is the fit for peak II and the magenta-diamond marker highlights
the reconstructed focusing peak. (b) FHWM, $\gamma$ as a function of
temperature; the sub-peaks broaden with increasing temperature in both cases.
The markers represent the same meaning as in plot (a). (c) The polarisation
measured with Gi = G0 fluctuates around 0.6$\%$. On the other hand, the
polarisation measured with Gi = 1.8G0 follows an exponential decay.

Figure 3(a)-(c) show the temperature dependence of focusing results with
injector set to 0.5G0, G0 and 1.8G0, respectively, where the lattice
temperature is incremented from 20 mK (the electron temperature is calibrated
to be around 70 mK) to 1.8 K, and Fig. 3(d)-(f) shows the zoom-in of the data
in Fig. 3(a)-(c), respectively. For Gi = 0.5G0 (Fig. 3(a)) a single peak is
observed (as only one spin-subband is occupied), which broadens gradually at
higher temperature. In addition, the focusing peak shifts towards the center
of the spectrum and becomes more symmetric at higher temperature (see the
bottom trace, T = 1.8 K, Fig. 3(a) and (d)). This may be due to a possible
electron transition between the two spin-subbands at relatively high
temperature. In comparison, for Gi = G0 (Fig. 3(b)) the sub-peaks, each
representing a spin-state, are present from 20 mK up to 1.2 K. However, the
dip in the first focusing peak leading to two sub-peaks smears out at 1.8 K
(Fig. 3(b) and (e)). With Gi set to 1.8G0 (Fig. 3(c)), the splitting is not
well resolved and the left sub-peak (I) dominates the spectrum. We note that
on increasing the temperature the peak I gradually reduced in amplitude to
result in an asymmetric first focusing peak at 1.8 K. In n-type InSb, the
splitting was pronounced even at 10 K, which is consistent with the fact the
peak splitting was around 60 mT, an indication of strong SOI in InSbARD06 ,
which is one order larger than the peak splitting of 5.5 mT measured in the
present case.

To extract the peak width and amplitude accurately considering the two sub-
peaks may partially overlap with each other, we use two Lorentzian peaks to
reconstruct the experimental data as shown in Fig. 4(a) using the relation,

$A(B)=\sum_{i=1,2}A_{i}\times\frac{\gamma_{i}^{2}}{\gamma_{i}^{2}+(B-B_{i})^{2}}$
$None$

where Ai is the amplitude of the peak $i$ ($i$ =1, 2 for peak I and peak II,
respectively), $\gamma_{i}$ denotes the full width at half maximum (FWHM),
$B_{i}$ is the center of the peak. Two noticeable results can be extracted
from the fitting: First, it is seen from Fig. 4(b) that $\gamma$ (see caption
of Fig. 4 for details on traces and symbols representing peak I and peak II)
for both peak I and peak II increases with rising temperature regardless of
the injector conductance which indicates the thermal broadening of the sub-
peaks prevents the observation of peak splitting at high temperature. It may
be noted that peak I for Gi = 1.8G0 is relatively robust against temperature
compared to other peak (both peaks of G0 and peak II of 1.8G0). Second, the
measured spin polarization $p$ ($p=|\frac{A_{1}-A_{2}}{A1+A_{2}}|$ ) with Gi =
G0 fluctuates around 0.6$\%$ and shows no explicit temperature dependence
which agrees with the fact that spin polarisation at conductance plateau
should remain at 0 regardless of temperature (Fig. 4(c), upper plot). On the
other hand, when Gi is set to 1.8G0, the extracted spin polarisation decays
from 5$\%$ to 0.8$\%$ (Fig. 4(d), lower plot) following the relationBCF01 ,

$p=\alpha exp(-\frac{k_{B}T}{\Delta E})+c$ $None$

where $\alpha$ is a prefactor accounting for the amplitude, $k_{B}$ is the
Boltzmann constant, $\Delta E$ is the energy difference between the two spin-
branches and c accounts for the small residual value arises from the
uncertainty in the experiment. We extracted the value of $\Delta E$ to be
around 0.041 meV (corresponding to 0.5 K). The theoryAGC07  predicts the
splitting should persist until $k_{B}T$ exceeds 2$\Delta E$ (i.e. 1 K in our
case) which agrees resonablely well with our result that the peak splitting is
observable up to 1.2 K.

## Conclusion

In conclusion, we showed the temperature dependence of the transverse electron
focusing where the contribution of the two spin states manifested as two sub-
peaks in the first focusing peak. It was observed that the peak splitting is
well defined from 20 mK up to 1.2 K and beyond this temperature the peak
splitting smeared out. Moreover, the focusing peak has a tendency to become
more symmetric at higher temperature indicating a possible equilibrium between
the two spin branches due to thermal excitation.

The work is funded by the Engineering and Physical Sciences Research Council
(EPSRC), UK.

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