Strain-induced enhancement of electric quadrupole splitting in resistively detected nuclear magnetic resonance spectrum in quantum Hall systems
ABSTRACT We show electrical coherent manipulation of quadrupole-split nuclear spin states in a GaAs/AlGaAs heterostructure on the basis of the breakdown of quantum Hall effect. The electric quadrupole splitting in nuclear spin energy levels is intentionally enhanced by applying an external stress to the heterostructure. Nuclear magnetic resonance spectra with clearly separated triple peaks are obtained, and Rabi oscillations are observed between the nuclear spin energy levels. The decay of the spin-echo signal is compared between the cases before and after the enhancement of quadrupole splitting. Comment: 4 pages, 4 figures
arXiv:0912.4313v1 [cond-mat.mes-hall] 22 Dec 2009
Strain-induced enhancement of electric quadrupole splitting in resistively detected
nuclear magnetic resonance spectrum in quantum Hall systems
M. Kawamura,1,2,3, ∗T. Yamashita,1H. Takahashi,1S. Masubuchi,1
Y. Hashimoto,4S. Katsumoto,4,5and T. Machida1,5, †
1Institute of Industrial Science, University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
2Advanced Science Institute, RIKEN, 2-1 Wako, Saitama 351-0198, Japan
3PRESTO, Japan Science and Technology Agency, 4-1-8 Kawaguchi, Saitama 333-0012, Japan
4Institute for Solid State Physics, University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa 277-8581, Japan
5Institute for Nano Quantum Information Electronics,
University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
(Dated: December 22, 2009)
We show electrical coherent manipulation of quadrupole-split nuclear spin states in a
GaAs/AlGaAs heterostructure on the basis of the breakdown of quantum Hall effect. The elec-
tric quadrupole splitting in nuclear spin energy levels is intentionally enhanced by applying an
external stress to the heterostructure. Nuclear magnetic resonance spectra with clearly separated
triple peaks are obtained, and Rabi oscillations are observed between the nuclear spin energy levels.
The decay of the spin-echo signal is compared between the cases before and after the enhancement
of quadrupole splitting.
PACS numbers: 74.43.-f
Nuclear spins in semiconductors have recently at-
tracted considerable attention because their extremely
long coherence time is suitable for the implemen-
tation of quantum bits/memories1,2,3.
manipulate nuclear spin quantum states coherently,
all-electrical4,5,6/optical7,8,9nuclear magnetic resonance
(NMR) techniques have been developed on the basis of
the hyperfine interaction between nuclear spins and elec-
tron spins. However, all of these new NMR techniques
have been successfully applied in GaAs. Since all the
constituent atoms in GaAs have nuclear spins I = 3/2,
the nuclear spin states split into four-level Izeigen states
|±3/2? and |±1/2? in a magnetic field as shown on the
left-hand side of Fig. 1(a). Such a four-level system can
be regarded as coupled quantum bits if transitions be-
tween any pairs of the levels are controlled selectively10.
In the presence of a local electric field gradient, the
electric quadrupole interaction produces non-equidistant
nuclear spin energy levels11as shown on the right-hand
side of Fig. 1(a). Although the electric quadrupole split-
ting energy ∆Qis zero in GaAs because of the cubic sym-
metry of the GaAs crystal, it is possible to increase the
amplitude of ∆Q by applying an external stress to the
crystal12,13,14, because ∆Q is proportional to the local
electric field gradient. Such an external stress can be ap-
plied using pressure cells or piezoelectric devices12,13or
by coating the surface of GaAs with different material14.
In this paper, we show intentional enhancement of elec-
tric quadrupole splitting and selective control of a four-
level nuclear spin system.
nomenon of quantum Hall effect, nuclear spins are po-
larized and NMR are detected. We apply an external
stress to a Hall-bar device by coating its surface with
a polyimide film. NMR spectra with clearly separated
In order to
Using the breakdown phe-
FIG. 1: (a) Energy diagram of I = 3/2 nuclear spin system
for ∆Q = 0 (left) and ∆Q ?= 0 (right). (b) Micrograph of the
Hall-bar device. A pulsed rf-magnetic field is irradiated using
the metal strip covering the 5-µm-wide conduction channel.
triple peaks are obtained in the polyimide-coated devices.
Splitting of NMR spectra enables us to show the selec-
tive and coherent manipulation of a four-level nuclear
spin system using pulsed NMR techniques. Furthermore,
the decay of spin-echo signal is compared between the
cases before and after the enhancement of quadrupole-
GaAs/Al0.3Ga0.7As single heterostructure wafer grown
by molecular beam epitaxy on a (001) oriented GaAs
substrate. The 2DEG is located at 230 nm below the
surface. The mobility and density of the 2DEG at 4.2
K are 220 m2/Vs and 1.6 × 1015m−2, respectively.
Figure 1(b) shows an optical micrograph of the Hall-bar
device used in the present study. A 10-µm-wide Ti/Au
Schottky gate electrode that was used for tuning electron
density also functioned as a local coil for generating
radio-frequency (rf) magnetic fields Brf parallel to the
device before it was coated with a polyimide film. B = 4.92 T
(ν = 1.08). Values of f0 are 35.520 MHz (a), 49.808 MHz (b),
and 63.286 MHz (c). (d) - (f) NMR spectra obtained using
the Hall-bar device after it was coated with the polyimide
film. B = 5.82 T (ν = 1.00). Values of f0 are 41.963 MHz
(d), 58.847 MHz (e), and 74.771 MHz (f). The solid curves
are the fitting curves.
(a) - (c) NMR spectra obtained using the Hall-bar
2DEG. External static magnetic field B was applied
perpendicular to the 2DEG, hence parallel to the 
direction of the GaAs crystal.
were performed at 50 mK using a
refrigerator. The sample chip (1 mm × 1 mm × 0.5 mm)
was glued backside to a ceramic chip carrier using silver
paste. After wiring to the Hall-bar device, the ceramic
package was held against the cold finger plate of the
dilution refrigerator to make a good thermal contact.
NMR signals were obtained by dynamic nuclear polar-
ization (DNP) and resistive detection (RD) techniques
in a breakdown regime of integer quantum Hall effect
(QHE), as already demonstrated in our earlier studies15.
As an initialization process, nuclear spins are dynam-
ically polarized through the hyperfine interaction be-
tween nuclear spins and electron spins under a breakdown
regime of a quantum Hall state with the Landau level fill-
ing factor ν = 1. By applying a bias current larger than
the critical current of the QHE breakdown, electrons are
excited to the upper Landau subband, accompanied by
flips of electron spins. The flips of electron spins cause
flops of nuclear spins via the hyperfine interaction, re-
sulting in positive nuclear polarization15?Iz? > 0. Then,
the nuclear spin states are manipulated by applying Brf.
All the measurements
The manipulated nuclear spin state is read out by mea-
suring the longitudinal voltage Vxx of the Hall-bar de-
vice. The read-out procedure is based on the fact that
the positively polarized nuclear spins (?Iz? >0) reduce
the Zeeman splitting energy of electrons, which increases
First, we measured the NMR spectrum using an un-
coated Hall-bar device. In Figs. 2(a), (b), and (c), the
changes in the longitudinal voltage ∆Vxxinduced by the
Brf irradiation are plotted as a function of the Brf fre-
quency. Each curve corresponds to the NMR spectrum
observed for all the three nuclear species. These single-
peak spectra indicate that the nuclear spin levels are dis-
tributed almost equidistantly as illustrated in the left-
hand panel in Fig. 1(a).
Next, the Hall-bar device was warmed up to room tem-
perature; at this temperature, a droplet of polyimide so-
lution was dropped onto the surface of the device. Then,
the polyimide coating was baked in N2atmosphere at 180
◦C for 15 min. The polyimide-coated device was cooled
down again for the NMR measurements.16Since the ther-
mal shrinkage rate of the polyimide film is considerably
higher than that of the GaAs film, the subsequent cool-
ing of the polyimide-coated device is expected to induce
a large strain in the device.
Figure 2(d) shows the NMR spectrum of75As after
it was coated with the polyimide film. The NMR spec-
trum is split into three peaks. These peaks correspond to
transitions A, B, and C shown on the right-hand side of
Fig. 1(a). The NMR spectra of69Ga and71Ga are also
split as shown in Figs. 2(e) and (f), respectively. Ampli-
tudes of the splitting are ∆f = 36 kHz, 18 kHz, and 11
kHz for75As,69Ga, and71Ga, respectively. The ratio of
∆f is in good agreement with the ratio of the quadrupole
moment Q:∆f(69Ga)/∆f(71Ga) = 1.6 agrees with
Q(69Ga)/Q(71Ga) = 0.19×10−28m2/0.12×10−28m2=
1.6. This indicates that the splitting of the NMR spectra
is attributed to the electric quadrupole interaction. We
observed the splitting in the spectrum of75As in another
device; the single-peak spectrum before the polyimide
coating is split to three peaks (∆f = 16 kHz) after the
polyimide coating. The NMR peak splitting of 7.5 kHz
for75As was also observed in yet another device after
coating its surface with PMMA electron-beam resist17.
We consider that the polyimide film produces a strain
in the GaAs/AlGaAs heterostructure, resulting in the
generation of a large electric field gradient at nuclear spin
sites, and the induced electric field gradient enhances ∆Q.
Comparing the observed splitting of 36 kHz in the NMR
spectrum of75As [Fig. 2(d)] with the earlier measure-
ments in GaAs quantum wells12, the strain in our device
is estimated as 1.7 × 10−4. The estimated value of the
strain seems consistent with the results of electron trans-
port measurements of 2DEG under a strain-induced pe-
riodic potential modulation18. From the FWHM of the
NMR spectrum of75As [Fig. 2(a)], the strain in the de-
vice before the polyimide film coating is estimated to be
69Ga, and71Ga.A single-peak spectrum is
with various pulse durations τpulse at B = 5.82 T (ν = 1.00).
The frequencies of Brf are 41.952 MHz (A), 41.987 MHz (B),
and 42.022 MHz (C) in (a), and 41.969 MHz (D) and 42.007
MHz (E) in (b). The curves are offset for clarity. The inset
in (a) shows schematic energy diagram for single- and two-
photon absorption/emission. The inset in (b) shows the NMR
spectrum of75As with the input rf-voltage Vrf = 4.8 V.
Changes in Vxx induced by applying a pulse of Brf
not larger than 3.9 × 10−5, even if the broadening of the
spectrum is attributed to ∆Q. Therefore, contribution of
the other sources of strain, such as Ti/Au Schotky gate
or the silver paste on the backside of the sample chip, is
small compared to that of the polyimide film.
Figure 3(a) shows the changes in Vxxinduced by apply-
ing a pulse of Brfwith various pulse durations τpulseat B
= 5.82 T (ν = 1). We note that the amplitude of Brfin
the pulsed NMR measurements (Fig. 3) is 12 times larger
than that used to obtain the continuous-wave NMR spec-
tra (Fig. 2)19. The Brf frequencies for the curves A, B,
and C are 41.952 MHz, 41.987 MHz, and 42.022 MHz,
respectively, as indicated in the inset in Fig. 3(b). The
oscillatory changes in ∆Vxx denoted A, B, and C cor-
respond the Rabi oscillations of75As for transitions A
(| + 3/2? ↔ | + 1/2?), B (| + 1/2? ↔ | − 1/2?), and C
(| − 1/2? ↔ | − 3/2?), respectively. These results clearly
show that the intentional enhancement of ∆Qenables the
selective and coherent control of the four-level nuclear
spin system. Additional two peaks (D and E) are seen in
the spectrum at the middle frequencies between the peaks
A and B, and B and C as similar to the work by Yusa
et al5. These additional peaks correspond to the two-
photon absorption/emission processes (|+3/2? ↔ |−1/2?
and | + 1/2? ↔ | − 3/2?) induced by the irradiation of
Brf with a large amplitude. The oscillations D and E in
Fig. 3(b) correspond to the two-photon Rabi oscillations
taken at the Brf frequencies of 41.969 MHz and 42.007
MHz, respectively. The observed frequency of the two-
photon Rabi oscillation Ω∆m=2= 3.8 kHz nearly agrees
with the calculated value20Ω∆m=2 ∼ Ω2
(12.5 kHz)2/36 kHz = 4.3 kHz.
?????? ??? ???
FIG. 4: (a) Decay of spin-echo signal obtained in the Hall-bar
device before the polyimide coating. B = 4.92 T (ν = 1.08).
(b)-(c) Decays of spin-echo signals obtained in the Hall-bar
device after the polyimide coating. B = 5.82 T (ν = 1.00). In
the case of (c), the electrons were depleted during the rf pulse
irradiation. The inset of (a) shows a schematic of the pulse
sequence for the spin-echo measurements. The inset of (b)
shows a representative spin-echo signal obtained by changing
τ2 with a fixed τ1 = 100 µs.
We verify the effect of electric quadrupole splitting
on the nuclear spin coherence time by performing spin-
echo experiments. We applied a sequence of π/2-π-π/2
rf pulses21, as shown in the inset of Fig. 4(a). The in-
set of Fig. 4(b) shows a representative spin-echo signal
in the device after the polyimide film coating obtained
by changing the second waiting time τ2with a fixed first
waiting time τ1= 100 µs with the Brffrequency of 42.022
MHz. The coherence time T2is estimated from the decay
of the spin-echo signal by changing the total waiting time
τ1+τ2under the condition τ1= τ2. Figure 4(a) shows the
decay of the spin-echo signals for75As in the device before
the polyimide film coating. The Brffrequency was tuned
to 41.963 MHz, the peak frequency in Fig. 2(a), where
all the three NMR transitions occur simultaneously. The
value of T2 is estimated to be no longer than 0.2 ms,
and the signal decays non-monotonically. In contrast, af-
ter coating the Hall-bar device with the polyimide film,
the spin-echo signal decays exponentially as shown in
Fig. 4(b). The Brf frequency was tuned to 42.022 MHz,
the peak C in the inset of Fig. 3(b). The value of T2is
estimated as 0.42 ms, which is almost twice longer than
that obtained before the polyimide film coating. The de-
cay time of the Rabi oscillations is also increased after
the polyimide film coating (not shown). In addition, as
shown in Fig. 4(c), the value of T2is further increased to
1.1 ms by decoupling the nuclear system from the elec-
tron system during nuclear-spin manipulation22,23; elec-
trons are depleted by applying negative dc voltage to the
Schottky gate electrode during the rf-pulse irradiation.
In summary, we have demonstrated strain-induced en-
hancement of the electric quadrupole splitting and elec-
trical coherent manipulation in I = 3/2 nuclear spin en-
ergy levels in GaAs/GaAs heterostructure. The DNP
and RD techniques used in the present study can be em-
ployed at temperatures higher than 1 K and even in a
2DEG with a relatively low electron mobility6,15, because
the techniques are based on the breakdown phenomena
This work was supported by a Grant-in-Aid from
MEXT, the Sumitomo Foundation, and the Special Co-
ordination Funds for Promoting Science and Technology.
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
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