Aharonov-Bohm oscillations in the presence of strong spin-orbit
  interactions
Aharonov-Bohm oscillations in the presence of strong spin-orbit interactions
Boris Grbić∗, Renaud Leturcq∗, Thomas Ihn∗, Klaus Ensslin∗, Dirk Reuter +, and Andreas D. Wieck+
Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
Angewandte Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
We have measured highly visible Aharonov-Bohm (AB) oscillations in a ring structure defined by
local anodic oxidation on a p-type GaAs heterostructure with strong spin-orbit interactions. Clear
beating patterns observed in the raw data can be interpreted in terms of a spin geometric phase.
Besides h/e oscillations, we resolve the contributions from the second harmonic of AB oscillations
and also find a beating in these h/2e oscillations. A resistance minimum at B = 0 T, present in
all gate configurations, is the signature of destructive interference of the spins propagating along
time-reversed paths.
Interference phenomena with particles have challenged
physicists since the foundation of quantum mechanics. A
charged particle traversing a ring-like mesoscopic struc-
ture in the presence of an external magnetic flux Φ
acquires a quantum mechanical phase. The interfer-
ence phenomenon based on this phase is known as the
Aharonov-Bohm (AB) effect [1], and manifests itself in
oscillations of the resistance of the mesoscopic ring with
a period of Φ0 = h/e, where Φ0 is the flux quantum. The
Aharonov-Bohm phase was later recognized as a special
case of the geometric phase [2, 3] acquired by the orbital
wave function of a charged particle encircling a magnetic
flux line.
The particle’s spin can acquire an additional geomet-
ric phase in systems with spin-orbit interactions (SOI)
[4, 5, 6]. The investigation of this spin-orbit (SO) in-
duced phase in a solid-state environment is currently the
subject of intensive experimental work [7, 8, 9, 10, 11, 12].
The common point of these experiments is the investiga-
tion of electronic transport in ring-like structures defined
on two-dimensional (2D) semiconducting systems with
strong SOI. Electrons in InAs were investigated in a ring
sample with time dependent fluctuations [7], as well as in
a ring side coupled to a wire [9]. An experiment on holes
in GaAs [8] showed B-periodic oscillations with a relative
amplitude ∆R/R < 10−3. These observations [7, 8] were
analyzed with Fourier transforms and interpreted as a
manifestation of Berry’s phase. Further studies on elec-
trons in a HgTe ring [10] and in an InGaAs ring network
[11] were discussed in the framework of the Aharonov-
Casher effect.
In systems with strong SOI, an inhomogeneous, mo-
mentum dependent intrinsic magnetic field Bint, perpen-
dicular to the particle’s momentum, is present in the
reference frame of the moving carrier [13]. The total
magnetic field seen by the carrier is therefore Btot =
Bext + Bint, where Bext is the external magnetic field
perpendicular to the 2D system and Bint is the intrinsic
magnetic field in the plane of the 2D system present in
the moving reference frame (right inset Fig. 1(a)). The
particle’s spin precesses around Btot and accumulates an
additional geometric phase upon cyclic evolution.
Effects of the geometric phases are most prominently
expressed in the adiabatic limit, when the precession fre-
quency of the spin around the local field Btot is much
faster than the orbital frequency of the charged particle
carrying the spin [4]. In this limit the ring can be consid-
ered to consist of two uncoupled types of carriers with op-
posite spins [14]. The total accumulated phase, composed
of the AB phase and the SO induced geometric phase, is
different for the two spin species, φtot = φAB ±φSO, and
the magnetoresistance of the ring is obtained as the su-
perposition of the oscillatory contributions from the two
spin species. Such a superposition is predicted to produce
complex, beating-like magnetoresistance oscillations with
nodes developing at particular values of the external B-
field, where the oscillations from the two spin-species
have opposite phases [5]. Both h/e and h/2e peaks in
the Fourier spectrum of the magnetoresistance oscilla-
tions are predicted to be split in the presence of strong
SOI [5, 15].
The interpretation of the split Fourier signal in Ref.
[7] has been challenged [16]. The data on p-GaAs rings
[8] stirred an intense discussion [17]. Our raw data di-
rectly displays a beating of the h/e Aharonov-Bohm os-
cillations. No Fourier transform is required to verify this
effect. As additional evidence we directly measure a beat-
ing of the h/2e oscillations and a pronounced and per-
sistent zero field magnetoresistance minimum due to de-
structive interference of time-reversed paths.
The sample was fabricated by atomic force micro-
scope (AFM) oxidation lithography on a p-type carbon
doped (100) GaAs heterostructure, with a shallow two-
dimensional hole gas (2DHG) located 45 nm below the
surface [18]. An AFM micrograph of the ring structure is
shown in the inset of Fig. 1(a). The average radius of the
circular orbit is 420 nm, and the lithographic width of the
arms is 190 nm, corresponding to an electronic width of
60−70 nm. The hole density in an unpatterned sample is
3.8×1011 cm−2 and the mobility is 200 000 cm2/Vs at a
temperature of 60 mK. Therefore the Fermi wavelength is
about 40 nm, and the mean free path is 2 µm. Since the
circumference of the ring is around 2.5 µm, the transport
through the ring is quasiballistic. From the temperature
dependence of the AB oscillations we extract the phase
coherence length of the holes to be Lϕ = 2 µm at a base
temperature of T = 60 mK.
The presence of strong spin-orbit interactions in the
http://arxiv.org/abs/0704.1264v2
heterostructure is demonstrated by a simultaneous ob-
servation of the beating in Shubnikov-de Haas (SdH) os-
cillations and a weak anti-localization dip in the mea-
sured magnetoresistance of the Hall bar fabricated on
the same wafer [19]. In p-type GaAs heterostuctures,
Rashba SOI is typically dominant over the Dresselhaus
SOI [20]. The densities N1= 1.35×10
11 cm−2 and N2=
2.45×1011 cm−2 of the spin-split subbands, deduced from
SdH oscillations, allow us to estimate the strength of the
Rashba spin-orbit interaction ∆SO ≈ 0.8 meV assum-
ing a cubic wave-vector dependence [13]. Due to the
large effective mass of the holes, the Fermi energy in the
system, EF = 2.5 meV, is much smaller than that in
electron systems with the same density. The large ratio
∆SO/EF ≈ 30% documents the presence of strong SOI.
We have measured the four-terminal resistance of the
ring in a 3He/4He dilution refrigerator at a base temper-
ature of about 60 mK with lock-in techniques. A low ac
current of 2 nA and 31 Hz frequency was applied in order
to prevent sample heating.
Fig. 1(a) shows the magnetoresistance of the ring (fast
oscillating curve, red online). The low-frequency back-
ground resistance is indicated by a smooth curve (blue
online). The observed Aharonov-Bohm (AB) oscillations
with a period of 7.7 mT (frequency 130 T−1) correspond
to a radius of the holes’ orbit of 415 nm, in excellent
agreement with the lithographic size of the ring. The
peak-to-peak amplitude of ∼ 200 Ω on a background of
about 6 kΩ corresponds to a visibility larger than 3%.
We restrict the measurements of the AB oscillations to
magnetic fields in the range from -0.2 T to 0.2 T in or-
der to prevent their mixing with SdH oscillations, which
start to develop above 0.2 T. Throughout all measure-
ments quantum point contact gates 3,4,5 and 6 are kept
at the same values. Plunger gates pg1 and pg2 are set to
Vpg1 = −145 mV and Vpg2 = −95 mV in the measure-
ments presented in Fig. 1(a).
After subtracting the low-frequency background from
the raw data, a clear beating pattern is revealed in the
AB oscillations with a well defined node at ∼ 115 mT
[Fig. 1(b)], where a phase jump of π occurs [arrow in
Fig. 2(c)]. The position of the beating node indicates
the presence of two oscillation frequencies differing by
1/0.115 ≈ 9 T−1.
The Fourier spectrum of the AB oscillations, taken in
the symmetric magnetic field range (-0.2 T, 0.2 T), re-
veals an h/e peak around 130 T−1 [Fig. 1(c)]. Zooming
in on the h/e peak, a splitting into 3 peaks at the fre-
quencies 127 T−1, 136 T−1 and 143 T−1 is seen. We have
carefully checked that this splitting is genuine to the ex-
perimental data and not a result of the finite data range,
by reproducing it with different window functions for the
Fourier transform. The differences of the oscillation fre-
quencies agree with that anticipated from the position of
the beating node in the raw data.
In contrast to the h/e-periodic AB oscillations, which
are very sensitive to phase changes in the ring arms,
Altshuler-Aronov-Spivak (AAS) h/2e oscillations, orig-
Gate1
-0.2 -0.1 0 0.1 0.2
0 200 400
frequency (1/T)
100 120 140 160
250 270 290
frequency (1/T)
(c) (d)
FIG. 1: (color online) (a) Measured magnetoresistance of the
ring (strongly oscillating curve, red online) together with the
low-frequency background resistance (smooth curve, blue on-
line); Left inset: AFM micrograph of the ring with designa-
tions of the in-plane gates. Bright oxide lines fabricated by
AFM oxidation lithography lead to insulating barriers in the
2DHG. Right inset: Scheme of a carrier travelling around the
ring in the presence of the external field Bext and SO induced
intrinsic fieldBint. (b) AB oscillations obtained after subtrac-
tion of the low-frequency background from the raw data. A
clear beating pattern is revealed in the AB oscillations. (c)
Fourier transform spectra of the AB oscillations, revealing h/e
and h/2e peaks. (d) Splitting of the h/e Fourier peak. (e)
Splitting of the h/2e Fourier peak.
inating from the interference of time reversed paths, are
expected to be more robust if the microscopic configu-
ration of the arms is changed. Besides, h/2e oscillations
are less susceptible to the details how the spin rotates
when it enters the ring than h/e oscillations. This is due
to the fact that the geometric phase accumulated along
the paths contributing to the h/2e oscillations is larger
than that in the case of the h/e oscillations and cannot
be completely cancelled by the spin rotations in the con-
tacts, as in the latter case [9]. In Fig. 1(c) we can identify
B(mT)
0-10 10 20-20
B(mT)
0-10 10 20-20
Rd - Rb
0 20-20 40 60 80
B(mT)
100 120 140 160
(a) (b)
Rd - Rb - Rh/e
Rh/2e
Rd - Rb - Rh/e
Rh/2e
FIG. 2: (color online) (a) Measured magnetoresistance of the
ring after subtracting the low-frequency background, Rd−Rb
(full line, red online), together with the filtered h/e oscilla-
tions Rh/e (dashed line). (b) Difference Rd −Rb −Rh/e (full
line red online) together with the inverse Fourier transform
of the h/2e peak Rh/2e (dashed line). (c) Beating in filtered
h/e oscillations. The width of the gray and white rectangles
corresponds to the period of 7.7 mT. The arrow points to the
beating node where a phase jump of π occurs. (d) Beating
in filtered h/2e oscillations with arrows indicating possible
nodes.
the peak at about 270 T−1 in the Fourier spectrum, cor-
responding to h/2e oscillations. If we zoom in on it [Fig.
1(e)], we see a splitting with the two main peaks having a
separation of about 8 T−1, similar to the h/e peak split-
ting. The splitting of the h/2e Fourier peak arises due to
the frequency shift of the main peak by ±1/Bint [5], and
the obtained splitting of 8 T−1 allows to estimate the SO
induced intrinsic field to be Bint ≈ 0.25T.
We now focus directly on the magnetic field-dependent
resistance. In Fig. 2(a) we present the raw data after
subtracting the low-frequency background (full line, red
online) together with the filtered h/e oscillations (dashed
line) [21]. The h/e contribution to the signal is the in-
verse Fourier transform of the h/e peak in the Fourier
spectrum. We will use the following notation below:
Rd denotes the raw data, Rb is the low-frequency back-
ground, Rh/e is the inverse Fourier transform of the h/e
peak and Rh/2e is the inverse Fourier transform of the
h/2e peak in the Fourier spectrum. One can see in Fig.
2(a) that Rd − Rb contains additional resistance modu-
lations, beyond the h/e oscillations. In order to demon-
strate that those additional features are due to h/2e oscil-
lations we plot in Fig. 2(b) the difference Rd−Rb−Rh/e
(full line, red online) and the curve Rh/2e obtained by
inverse Fourier transform of the h/2e peak (dashed line)
and find excellent agreement.
We further plot in Fig. 2(d) the difference Rd − Rb −
Rh/e (full line, red online), together with the filtered h/2e
oscillations Rh/2e (dashed line) in a larger range of mag-
netic fields. A beating-like behavior in the h/2e oscilla-
tions is observed. Possible nodes develop around 40 mT,
115 mT, and 175 mT [arrows in Fig. 2(d)]. The appear-
ance of these unequally spaced nodes is in agreement with
the complex split-peak pattern in Fig. 1(e). In the plot
of the filtered h/e oscillations [Fig. 2(c)] we notice that
only the node around 115 mT is common for both, h/e
and h/2e oscillations, while the other two nodes in the
h/2e oscillations correspond to maxima in the beating of
h/e oscillations. This kind of aperiodic modulation of the
envelope function of the h/2e oscillations, rather than a
regular beating, is predicted for the case of diffusive rings
in the presence of Berry’s phase [5], since the latter also
changes with increasing external magnetic field.
The evolution of the AB oscillations upon changing
plunger gate voltages Vpg1 and Vpg2 is explored in Fig.
3(a). Plunger gate voltages are changed antisymmetri-
cally: Vpg1 = −120mV −V ; Vpg2 = −120 mV +V . Two
distinct features are visible: there is always a local min-
imum in the AB oscillations at B = 0 T, and the os-
cillations experience a phase jump by π around V = 27
mV. In order to understand the origin of these two fea-
tures we analyze the filtered h/e (not shown) and h/2e
oscillations [Fig. 3(b)] as a function of V . The h/e os-
cillations experience a phase jump of π, while the h/2e
oscillations do not [Fig. 3(b)]. We have explored this
behavior in several other gate configurations and always
found the same result. The reason for such a behavior is
that the h/e oscillations are sensitive to the phase differ-
ence ∆ϕ = k1l1 − k2l2 between the two arms, which can
be changed by the plunger gates, while the AAS h/2e
oscillations are not. We observe a resistance minimum at
B = 0 T for all gate configurations [Fig. 3(a)], which is
due to a minimum at B = 0 T in the h/2e oscillations
[Fig. 3(b)]. It indicates that time reversed paths of the
holes’ spinors interfere destructively due to strong SOI, in
contrast to n-type GaAs systems where h/2e oscillations
produce a resistance maximum at B = 0T [22]. This
effect has the same origin as the weak anti-localization
(WAL) effect. However, the observed minimum is not
caused by WAL in the ring leads, since the WAL dip
in bulk 2D samples has a much smaller magnitude (less
than 1Ω, [19]) than the minimum at B = 0T in the ring.
The resistance minimum at B = 0T is a result of the
destructive interference of the holes’ spins in the ring.
The adiabatic regime is reached when ωB/ωorbit >> 1,
where ωB = gµBBtot/2~ is the spin precession frequency
around Btot, while ωorbit = vF /r is the orbital frequency
of the holes around the ring in the ballistic regime. p-
type GaAs systems have strong SOI, and therefore large
Bint, which, together with a small vF (due to the large
effective mass of the holes) makes p-types systems very
favorable for reaching the adiabatic regime compared to
other systems. Using the estimated value for Bint of
0.25T and assuming a holes’ g factor of 2, we obtain
-20 -10 20100
B (mT)
-20 -10 20100
B (mT)
(a) (b)
FIG. 3: (color online) (a) Evolution of the AB oscillations
upon changing the plunger gate voltages Vpg1 = −120mV
−V ; Vpg2 = −120 mV +V . (b) Filtered h/2e oscillations
as a function of the plunger gate voltages, showing the local
minimum at B = 0 T at all gate voltages.
ωB/ωorbit ≈ 0.2 − 0.3 for the measured range of Bext
up to 0.2T. Therefore the adiabatic regime is not fully
reached in our measurements.
There remains a pronounced discrepancy between the
internal magnetic field obtained from the beating of the
SdH oscillations of 7T (converting the corresponding en-
ergy scale ∆SO ≈ 0.8 meV to a magnetic field via the
Zeeman splitting) and the field scale of 0.25 T obtained
from the beating of the AB oscillations. The latter evalu-
ation is strictly valid in the diffusive regime [5] while our
sample is at the crossover to the ballistic regime. It is
also not clear how the limited adiabaticity in our samples
will influence these numbers.
In a straightforward picture one would expect that the
node of the beating in the h/2e oscillations occurs at
half the magnetic field as the node in the h/e oscillations
since the accumulated phase difference between the two
spin species should be proportional to the path length
travelled in the ring. Within experimental accuracy the
data in Fig. 1 (d) and (e) suggests that the splitting in
the corresponding Fourier transforms is the same.
In conclusion, we have measured Aharonov-Bohm os-
cillations in a ring defined on a 2D hole gas with strong
spin-orbit interactions. We observe a beating in the mea-
sured resistance which arises from an interplay between
the orbital Aharonov-Bohm and a spin-orbit induced ge-
ometric phase. In addition we resolve h/2e oscillations
in the ring resistance, and find that they also show a
beating-like behavior, which produces a splitting of the
h/2e peak in the Fourier spectrum. A resistance mini-
mum at B = 0, present in all in-plane gate configura-
tions, demonstrates the destructive interference of the
hole spins propagating along time reversed paths.
We thank Daniel Loss and Yigal Meir for stimulating
discussions. Financial support from the Swiss National
Science Foundation is gratefully acknowledged.
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