Observation of Spin-Orbit Berry’s Phase in Magnetoresistance of
a Two-Dimensional Hole Anti-dot System
Ning Kang∗ and Eisuke Abe, Yoshiaki Hashimoto, Yasuhiro Iye, and Shingo Katsumoto
Institute for Solid State Physics, University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
(Dated: February 6, 2020)
Abstract
We report observation of spin-orbit Berry’s phase in the Aharonov-Bohm (AB) type oscillation
of weak field magnetoresistance in an anti-dot lattice (ADL) of a two-dimensional hole system.
An AB-type oscillation is superposed on the commensurability peak, and the main peak in the
Fourier transform is clearly split up due to variation in Berry’s phase originating from the spin-
orbit interaction. A simulation considering Berry’s phase and the phase arising from the spin-orbit
shift in the momentum space shows qualitative agreement with the experiment.
PACS numbers: 73.61.-r, 73.23.-b, 73.50.-h
http://arxiv.org/abs/0704.0724v1
Spin-orbit interaction in two-dimensional system is predicted to introduce an additional
geometric phase (Berry’s phase) [1, 2, 3, 4, 5]. Here we consider the case of two-dimensional
hole system (2DHS), in which the spin-orbit interaction is strong in III-V compound semi-
conductors due to the p-orbital nature of the valence bands. In a 2DHS with the Rashba-type
spin-orbit coupling, the additional phase shifts arise in two ways. The spin-orbit interac-
tion causes split of the parabolic dispersion, producing two Fermi circles of opposite spins
(Fig.1(a),(b)) [6, 7, 8, 9, 10]. This means, the hole system has two different Fermi wave vec-
tors, of which the difference (∆kF) gives a phase shift and modifies the interference through,
e.g., a ring structure. Another shift in phase comes from the spin part of the wavefunction.
When a hole moves along a closed contour in the real space, the same occurs in k-space and
it feels effective magnetic field Beff due to the spin-orbit interaction. Beff is in the 2D plane
and orthogonal to the velocity. Hence when Beff is strong enough, the spin placed in the 2D
plane turns by 2π giving Berry’s phase of 0 or π.
When an external magnetic field Bext is applied perpendicular to the 2DHS plane, the
hole moving along a loop acquires an Aharonov-Bohm (AB) phase. At the same time,
the spin gets some component perpendicular to the plane, which means some diminishing
in the in-plane component (Fig.1(c)). This hence affects the above two additional phases.
The variation in Berry’s phase (∆θB) and that in the phase due to ∆kF (∆θk) are given
as[1, 2, 3, 4, 5, 12]
∆θB = π(1− cos θ), (1)
∆θk = πr∆kF sin θ, (2)
where θ ≡ arctan(Beff/Bext), and r is the radius of the AB ring. Though significant warping
of the outer Fermi contour is reported[6, 7] as shown in Fig.1(b), we can consider ∆kF
as an average over the perimeter in the present discussion. Hence the phase of an AB
oscillation in a ring made of such a 2DHG should show continuous shift giving a nontrivial
peak shape in its Fourier transform (FT) amplitude versus the frequency when the FT is
performed in a finite field range. In experiments, such phases have been observed as splitting
of AB magnetoresistance oscillation frequency in single AB rings[11, 12, 13]. Especially the
authors of Ref. 12 observed nontrivial variation of the main FT peak due to the phase shift
represented in Eqs.1 and 2.
In an anti-dot structure, the carriers are excluded from a dot region and magnetoresis-
tance oscillation similar to that in an AB ring is expected. In an anti-dot lattice (ADL),
AB oscillations of paths around single anti-dots should be averaged out because of ran-
dom phasing, whereas the effect of the AB phase remains in the density of states resulting
in magnetoresistance oscillation called AB-type oscillation[14, 15, 16, 17]. It is, then, of
strong interest whether the effect of spin-orbit Berry’s phase appears in the density of states
responsible for the AB-type oscillation.
In this letter, we report observation of the predicted Berry’s phase through clear splitting
of the main peak in FT of the AB-type oscillation in the magnetoresistance of an ADL. The
shift in k-space is independently estimated from the Shubnikov-de Haas (SdH) oscillation.
Because the ADL automatically performs the disorder average, random specific effects in
h/e oscillation are suppressed and intrinsic information can be obtained.
An (Al,Ga)As/GaAs heterostructure was grown by molecular beam epitaxy on a (001)
GaAs substrate. Delta-doped Be layers produced a 2DHS with the hole concentration of
2.3×1011cm−2 and the mobility of 6.8×104 cm2/Vs at 4.2K. SdH oscillation in the magne-
toresistance of the 2DHS without ADL was measured at 60mK. From the plot of the peak
index versus the inverse magnetic field, the concentrations of holes with smaller Fermi con-
tour and larger one as 0.78×1011cm−2 and 1.56×1011cm−2 respectively. In this letter, we call
the former and the latter as l-holes and h-holes, respectively. In the simple effective mass
model, these two spin-split hole bands correspond to the two Fermi contours in Fig.1(a),
though significant warping in h-hole band is reported in more realistic band calculation[6, 7].
The grown film was wet-etched into Hall bars with [110] current direction. Circular anti-dot
shape is adopted and ADLs were defined by electron beam lithography and wet-etching to
a depth of 50nm. An ADL is characterized mainly by the dot diameter d, the lattice struc-
ture, and the lattice period a. In this study, we designed a square lattice aligned along [110]
([1̄10]) with d=250nm and a=500nm (see Fig.1(d)).
The sample was directly immersed in 3He-4He mixture in a mixing chamber of a dilution
refrigerator and cooled down to 60mK. Bext up to 6T was applied by a superconducting
solenoid. The resistance was measured in four-terminal configuration by standard lock-in
technique with a frequency of 80Hz.
The solid line in Fig.2 shows the resistance of the sample as a function of Bext up to 0.5T.
In higher field region, formation of the edge states and alignment of hole spins perpendicular
to the plane bring about different physics. Therefore, we deal mainly with this low-field
region. The magnetoresistance shows a clear commensurability peak marked as A by an
arrow in the Fig.2. Generally such peaks in resistance appear when the carrier cyclotron
orbits are commensurate with an ADL and localized in the lattice. For a circular Fermi
contour, the classical cyclotron radius Rc is expressed as
eBext
, (3)
where the Fermi wavevector kF is written as
4πp with the hole concentration p (here we
assume the spin degeneracy is lifted). Bext at peak position A in Fig.2 can be well ascribed to
localized orbits of l-hole encircling single anti-dot. For h-hole, as noted above, we should take
into account the effect of warped Fermi contour in Fig.1(b). This effect will be discussed in
a separate paper. For a small hump structure around 0.05T, we do not have any appropriate
interpretation at present.
As shown in Fig.2, an oscillation with a short period superposed on the commensurability
peak is visible. In order to extract the oscillating component, we subtract the background
commensurability structure by fitting slowly-varying functions. An example of extracted
resistance variation is shown with the broken line in Fig.2, which exhibits a clear fast os-
cillation. Here for the background subtraction, a 12th order polynomial for each set of 100
consecutive data points is adopted. Although the selection of fitting function affects the
outlook of the oscillatory component, little is affected in the main and sub peak structures
in the Fourier space.
The observed oscillations are nearly periodic in Bext. The SdH oscillation is not visible
in this field range due to the restriction of the space for free cyclotron motion. The FT
amplitude of the extracted oscillation is displayed in Fig.3. The magnetic field range is
taken from −0.5T to 0.5T. At the main peak around 50T−1, clear splitting into three sub-
peaks, marked as A, A’ and B’, is observed. Sub-peak A has a shoulder structure at lower
frequency marked as B. From these four splitting frequency positions, the position of the
main peak structure is determined to be 52T−1, where a sharp dip marked as C is located.
If we measure the oscillation frequency with the magnetic flux on the circle with radius a/2,
position C just corresponds to the flux quantum h/e, confirming that the peak structure
originates from the AB effect around single anti-dots through the modulation in the density
of states.
As noted above, the selection of functional form for background subtraction does not affect
the characteristics shown in Fig.3 as long as the fitting function is slowly varying versus Bext.
We have even performed direct FT without subtraction. Though the background variation
results in enhancement of unphysical low frequency component, the splitting of the main
peak structure still exists in the result, confirming the above result.
Another characteristic structure in Fig.3 is a dip-peak structure around 110T−1, which
corresponds to about twice the frequency of the main peak, and originates from the orbits
with the winding number n=2. The signal-to-noise ratio is not high enough to resolve
detailed peak structure, nevertheless we may view it as a dip and two side peaks as indicated
by arrows. Though there is no remarkable structure at the positions at around n =3, a
surprisingly clear peak is observed at the position for n =4 as shown in the inset of Fig.3.
Figure 4 displays the temperature dependence of the FT spectrum. The three peaks (A,
A’ and B’) show almost identical temperature dependence as seen in the inset of Fig.4, which
can be fitted by the Dingle function,
f(T ) ∝
sinh κT
, κ ≡
2π2kB
, (4)
where T is the temperature and ∆E the energy difference between the peaks in the density
of states modulation caused by the AB effect. The fitting gives the value of ∆E as 86µeV.
This is in similar order with other measurement in 2D electron systems[17]. The agreement
in the temperature dependence manifests the three peaks have the same physical origin.
There, we stressed the nature of the ensemble average in AB-type oscillations in ADLs, as
opposed to the ordinary AB effect in single ring structure. The AB-type oscillation manifests
the oscillatory fine structure of the density of state, and the temperature dependence of the
oscillation amplitude can be explained by thermal broadening of the energy levels of the
orbits.
We attribute the origin of the splitting to shifts in additional phases expressed in Eqs.1
and 2. In the simplest approximation the oscillatory resistance ∆R is written as the sum of
four terms:
∆R = cos(n(θAB+∆θB))+cos(n(θAB−∆θB))+cos(n(θAB+∆θk))+cos(n(θAB−∆θk)), (5)
where θAB is the AB phase given by 2πBext · πr
2/(h/e). Because the phases in Eqs.1 and
2 are not linear in Bext, each term in Eq.5 is not sinusoidal oscillation and hence the FT
spectrum strongly depends on the region of transformation. By the same token, the FT
lineshape for higher harmonics (n > 1) is not simply n-times enlargement in frequency axis
of the main peak structure.
In order for a qualitative comparison with experiment, we calculate the FT spectra of
Eq.5 in the magnetic field ranging from −0.5 to 0.5T with ∆kF=4.1×10
7m−1, which is the
value measured from the SdH oscillation, and r is taken to be a/2. We adopt 0.55T for
Beff , which is reported for 2DHSs with conditions close to the present ones[9]. The results
are shown in Fig.5. Despite the crude approximation, the solid lines in Fig.5 bears striking
resemblance to the experiment. For n=1, clear splitting into 4 sub-peaks is observed. With
increasing n, the distribution of the peaks increases but the variation of the center peak
structure is not monotonic. For n=2, there is a sharp dip at the center while only a broad
peak exists for n = 3. When n is increased to 4, the peak distribution still broadens though
the center peak resharpens. These changes in n = 3 and 4 explain the disappearance and
reappearance of peak structure in the experiment.
As shown by the dotted lines in Fig.5, when there is no contribution of Berry’s phase,
the characteristics of the center peak are completely different. No splitting appears even for
n =2; Rounding and sharpening for n =3 and 4 do not appear either. These results hence
support the interpretation that the splitting of FT peaks is due to the phases in Eqs.1 and
2 arising from the spin-orbit coupling, added to the AB phase. Equation 5 is too crude
an approximation for further quantitative discussion. Especially, in ADLs, orbits encircling
anti-dots may vary their sizes with Bext, which is smaller effect than Berry’s phase and
results in simple shift of peak positions. Shift of peak positions, e.g., for n = 4 may be
explained with this effect.
It should be remarked that the mean free path of the present 2DHG is about 600nm,
which is much shorter than even the perimeter of a closed trace encircling a single anti-dot.
This fact clearly emphasizes that the mean free path is essentially different from the phase
coherence length, and also from the circumference of cyclotron orbit. The former difference
has been repeatedly mentioned in literatures[18] while the latter manifests that the major
origin of the scattering is impurity potential, which allows free space for ballistic cyclotron
motions.
Another experimental fact that supports our interpretation is that the peak splitting in
the FT spectra disappears in the AB-type oscillation in high fields, where the spin is fixed
perpendicular to the plane and neither variation in Berry’s phase nor in ∆kF occurs. This
phenomenon will be reported in a separate paper.
In conclusion, we have observed spin-orbit Berry’s phase through splitting of Fourier
transform peaks in the Aharonov-Bohm type oscillation of weak field magnetoresistance in
an anti-dot lattice of a two-dimensional hole system. A simple simulation with adopting
parameters obtained from independent measurements shows fair qualitative agreement with
the experimental observations.
The authors would like to thank K. Suzuki for help during this experiment. This work is
supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture,
Sports, Science, and Technology of Japan and also supported by Special Coordination Funds
for Promoting Science and Technology.
∗ Electronic address: ning.kang@issp.u-tokyo.ac.jp
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h-hole
l-hole
[100]
[010]
FIG. 1: (a) Surface plot of spin subbands explains two Fermi contours with different kF appear
as a result of spin-orbit interaction. (b) According to more realistic calculation[6, 7], the outer
subband (h-hole) is significantly warped. (c) Schematic illustration of Beff seen by holes moving
around a ring structure in the presence of spin-orbit interaction. (d) Scanning electron micrograph
of the ADL sample.
Bext (T)
0 0.1 0.2 0.3 0.4 0.5
-0.02
-0.01
FIG. 2: Upper solid line: Resistance of the sample as a function of the external magnetic field
Bext for T=60 mK. Lower broken line: Extracted oscillating part of the magnetoresistance. For
the background subtraction, a 12th order polynomial for each set of 100 consecutive data points is
adopted.
Frequency (T −1)
T =60mK
20 40 60 80 100 120 140
160 180 200 220
FIG. 3: Fourier spectrum of the AB-type oscillation in the magnetoresistance at 60mK. The main
peak splits into four (B, A, A’, B’) and also the 2nd harmonic splits into two (A∗, A’∗). (The other
two are within the noise level.) The inset shows peak structures around four times frequency of the
main peak structure as indicated by arrows. The scales of FT amplitude are shown by the arrows
marked as u in the respective figures.
Frequency (T −1)
T=60mK
120mK
150mK
210mK
250mK
40 60 80 100 120 140
T (mK)
100 200 300
FIG. 4: FT results at temperatures from 300mK down to 60mK. The data are offset for clarity.
The inset shows the peak heights normalized at 60mK for peaks A, A’ and B’. The solid line in
the inset is the fit to the Dingle function (4).
40 50 60
100 110 120
140 150 160 170
Frequency (T −1)
190 200 210 220 230 240
FIG. 5: Solid lines are the FT spectra of the function (5) for the winding number n = 1,2,3,4.
The magnetic field region is taken as −0.5T< B <0.5T and the parameter ∆kF is taken from SdH
measurement as 4.1×107m−1. The scale of abscissa is modified with n to show the entire peak
structures. Dotted lines are the results when Berry’s phase ∆θB is set to zero. Significant difference
between solid and dotted lines around the center while they are almost identical in surrounding
regions.
	Acknowledgments
	References

  We report observation of spin-orbit Berry's phase in the Aharonov-Bohm (AB)
type oscillation of weak field magnetoresistance in an anti-dot lattice (ADL)
of a two-dimensional hole system. An AB-type oscillation is superposed on the
commensurability peak, and the main peak in the Fourier transform is clearly
split up due to variation in Berry's phase originating from the spin-orbit
interaction. A simulation considering Berry's phase and the phase arising from
the spin-orbit shift in the momentum space shows qualitative agreement with the
experiment.

Introduction to Mesoscopic Physics 2nd ed. (Oxford, 2001).
h-hole
l-hole
[100]
[010]
FIG. 1: (a) Surface plot of spin subbands explains two Fermi contours with different kF appear
as a result of spin-orbit interaction. (b) According to more realistic calculation[6, 7], the outer
subband (h-hole) is significantly warped. (c) Schematic illustration of Beff seen by holes moving
around a ring structure in the presence of spin-orbit interaction. (d) Scanning electron micrograph
of the ADL sample.
Bext (T)
0 0.1 0.2 0.3 0.4 0.5
-0.02
-0.01
FIG. 2: Upper solid line: Resistance of the sample as a function of the external magnetic field
Bext for T=60 mK. Lower broken line: Extracted oscillating part of the magnetoresistance. For
the background subtraction, a 12th order polynomial for each set of 100 consecutive data points is
adopted.
Frequency (T −1)
T =60mK
20 40 60 80 100 120 140
160 180 200 220
FIG. 3: Fourier spectrum of the AB-type oscillation in the magnetoresistance at 60mK. The main
peak splits into four (B, A, A’, B’) and also the 2nd harmonic splits into two (A∗, A’∗). (The other
two are within the noise level.) The inset shows peak structures around four times frequency of the
main peak structure as indicated by arrows. The scales of FT amplitude are shown by the arrows
marked as u in the respective figures.
Frequency (T −1)
T=60mK
120mK
150mK
210mK
250mK
40 60 80 100 120 140
T (mK)
100 200 300
FIG. 4: FT results at temperatures from 300mK down to 60mK. The data are offset for clarity.
The inset shows the peak heights normalized at 60mK for peaks A, A’ and B’. The solid line in
the inset is the fit to the Dingle function (4).
40 50 60
100 110 120
140 150 160 170
Frequency (T −1)
190 200 210 220 230 240
FIG. 5: Solid lines are the FT spectra of the function (5) for the winding number n = 1,2,3,4.
The magnetic field region is taken as −0.5T< B <0.5T and the parameter ∆kF is taken from SdH
measurement as 4.1×107m−1. The scale of abscissa is modified with n to show the entire peak
structures. Dotted lines are the results when Berry’s phase ∆θB is set to zero. Significant difference
between solid and dotted lines around the center while they are almost identical in surrounding
regions.
	Acknowledgments
	References
