Single- and few-electron dynamic quantum dots in a perpendicular magnetic field

Abstract

We present experimental studies of the current pumped through a dynamic
quantum dot over a wide range of magnetic fields. At low fields we observe
repeatable structure indicating increased confinement of the electrons in the
dynamic dot. At higher fields (B>3T), we observe structure which changes
markedly from device to device suggesting that in this regime the transport is
sensitive to local disorder. The results are significant for the development of
dynamic quantum dot pumps as quantum standards of electrical current.

Full text

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Single- and few-electron dynamic quantum dots in a perpendicular magnetic field
S. J. Wright1,3, A. L. Thorn1,2, M. D. Blumenthal1,2, S. P. Giblin2, M. Pepper4, T. J. B. M. Janssen2,
M. Kataoka2, J. D. Fletcher2, G. A. C. Jones1, C. A. Nicoll1, Godfrey Gumbs5, D. A. Ritchie1
1Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK.
2National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK.
3Toshiba Research Europe Ltd, Cambridge Research Laboratory,
208 Science Park, Milton Road, Cambridge CB4 0WE, UK.
4University College London, Torrington Place, London WC1E 7JE, UK.
5Department of Physics and Astronomy, Hunter College of the City University of New York,
695 Park Avenue, New York, New York 10065 USA.
(Dated: September 2, 2010)
We present experimental studies of the current pumped through a dynamic quantum dot over a
wide range of magnetic fields. At low fields we observe repeatable structure indicating increased
confinement of the electrons in the dynamic dot. At higher fields (B > 3T), we observe structure
which changes markedly from device to device suggesting that in this regime the transport is sensitive
to local disorder. The results are significant for the development of dynamic quantum dot pumps
as quantum standards of electrical current.
PACS numbers: Valid PACS appear here
I.INTRODUCTION
A quantized charge transport device can generate elec-
trical current given by I = nef, where f is the repetition
frequency of an applied potential, e is the electron charge
and n is the number of charges transported in one cycle.
This type of device is of great interest to electrical metrol-
ogists because it could form the basis of a new definition
of the SI base unit ampere, linking current to frequency
via a defined value of the electron charge [1]. Pumps
based on chains of metal-oxide tunnel barriers have been
researched extensively, and have demonstrated pumping
accuracy at the 10−8level required by metrological appli-
cations [2]. Unfortunately the time constant of the tunnel
junctions limits the current in these devices to the level of
a few pA. Recently, a new type of pump based on metal-
oxide-superconductor barriers has demonstrated parallel
scaling of 10 devices [3], but this device must be oper-
ated at finite bias voltage, thereby requiring stringent
control of leakage currents if metrological accuracy is to
be reached.
Thesemiconductor-based
(DyQD) pump, in contrast, can be operated at zero bias,
and relatively high frequency [4]. The DyQD pump, like
earlier Surface Acoustic Wave (SAW)-based pumps [5, 6],
transports electrons between a source and drain lead by
modulation of the electrostatic potential in a reduced-
dimensional semiconductor system. In the SAW pumps
the potential modulation is produced by a SAW launched
from a tuned transducer, whereas in the DyQD pump
the modulation signal is applied directly to one of the
potential-defining gates. The DyQD pump avoids heat-
ing effects present in the SAW pumps [7], and can be
driven at a wide range of frequencies. Under the applica-
tion of a perpendicular magnetic field, the performance
of the DyQD pump was shown to be enhanced [8, 12].
Recent measurements at B = 5T and f = 340MHz did
not resolve any error in the pump current within the
dynamicquantumdot
15 parts per million uncertainty in the current measure-
ment system [13]. Furthermore, parallel operation of two
pumps has been demonstrated with no noticeable loss
of accuracy [9]. The DyQD pump is therefore a strong
candidate for the realization of a quantum standard of
electrical current.
In this paper, we describe the effect of a perpendicu-
lar magnetic field on the current produced by a DyQD
pump. For fields of B ≤ 3T the pumps exhibit phenom-
ena that are reproducible from device to device. The
risers between plateaus become sharper and the plateaus
become flatter, indicating enhanced quantization. Tran-
sitions in the number of electrons transported per cycle
shift in gate voltage, demonstrating the ability of the field
to act as an extra control parameter to tune the pump
system. At fields of B > 5T an anomalous structure is
observed in the quantized current that is reminiscent of
earlier single-electron capacitance spectroscopy (SECS)
measurements with static quantum dots (QDs) [10, 11].
The observation that the details of this structure are
device-dependent suggests that they originate from local
disorder which is unique to each device. The magnetic
field appears to strengthen the effect of disorder on the
measured pumped current.
II. TUNABLE-BARRIER ELECTRON PUMP
The DyQDpump devicesare fabricated ina
GaAs/AlGaAs high electron mobility transistor (HEMT)
heterostructure where a two-dimensional electron gas
(2DEG) exists 90nm below the surface. A scanning elec-
tron microscope (SEM) image of a similar device to the
ones tested in this work is presented in Fig. 1. Ohmic
contacts were made to the source (S) and drain (D) areas
of 2DEG. Transverse confinement was provided by the
horizontal narrow channel, created through shallow wet
chemical etching. Metallic gates were deposited on the
arXiv:1009.0203v1 [cond-mat.mes-hall] 1 Sep 2010

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FIG. 1: SEM image of the device and schematic of electrical
connections. The oscillating voltage signal VRFis added to
the static DC voltage Vent and applied to the left (entrance)
gate. Vexitis applied to the middle (exit) gate. The right gate
is grounded and not used. A DyQD is periodically formed in
the channel at the position of the red dot. Electrons are trans-
ported by the DyQD from source (S) to drain (D) reservoirs
in the direction of the white arrow.
surface of the device, perpendicular to the channel. The
left-most gate will be referred to hereafter as the entrance
gate, and the middle gate as the exit gate. The right-most
gate was grounded and not used. A sinusoidal radio fre-
quency (RF) voltage signal VRFwas added to the static
DC offset voltage Ventusing a bias tee, as shown, result-
ing in a total instantaneous entrance gate voltage VTOT
When tuned correctly, a DyQD is periodically formed at
the position of the red dot in Fig. 1 at the repetition fre-
quency of VRF. A well-defined number of electrons can
be captured by the DyQD from the source. As the pump
cycle progresses and the potential is tilted, a controlled
number of the captured electrons are ejected over the exit
gate and into the drain, contributing to the measured
current. The direction of electron transport is shown by
the white arrow in the figure.
A plot of the numerical derivative of the pumped cur-
rent in Vexitand Vent,
??dIpump
ent
.
dVexit
?2
+
?dIpump
dVent
?2
,
is presented in the main left panel of Fig. 2. Here, VRF
was set to a frequency of f = 73MHz with an amplitude
at the source of −9dBm. All measurements in this work
were performed in a dilution refrigerator with a base tem-
perature of ∼ 50mK. Transitions in the number of elec-
trons transported per cycle manifest in dark lines in the
plot. We will refer to this type of plot as a pump map
hereafter. The blue dashed lines mark directions of line
scans in Ventand Vexit, seen to the right and bottom of
the main panel respectively. Ipumpis plotted in each case.
The line scans exhibit plateaus at values corresponding
to an integer number of electrons being transported per
cycle of the RF signal. This is the signature of quantized
charge transport.
The value of the current on the plateaus is proportional
to the number of electrons neejected into the drain per
cycle of the RF signal. Aspects of a model for the mecha-
nism of operation of DyQD pumps have been discussed in
previous works [4, 15, 16]. In order to correctly interpret
the results presented in this paper we present a detailed
description of a model to explain the features seen in the
pump map of Fig. 2.
We draw the reader’s attention to the four areas of
the pump map along the entrance gate line scan (direc-
tion of constant Vexit). These areas are labeled by the
number of electrons captured and ejected in each case,
(nc,ne). Schematic diagrams of the barriers defined by
the entrance and exit gates in each area are presented in
the right panel of Fig. 2. Here, EF is the Fermi energy
in the source (S) and drain (D) of the channel. In order
to generate pumped current, it is necessary for Vexitto
be negative enough at all points in the pump map for
the barrier defined by the exit gate to always be opaque.
The left and right schematics for each area represent the
minimum and maximum barrier heights defined by the
entrance gate during the pump cycle respectively.
In area (0,0), the barrier defined by Vent is too large
over the whole pump cycle to allow electrons to enter the
DyQD from the source. As the DyQD is never populated,
we measure Ipump= 0 in this region.
As Ventis made less negative the pump transitions into
the (2,2) area where the entrance barrier drops enough to
allow electrons to enter the DyQD from the source. When
the entrance barrier subsequently rises as the pump cycle
progresses we reach a point where the DyQD is isolated
from the source. We refer to this point in the pump cycle
as the capture point, shown by the middle schematic. In
this case the DyQD captures two electrons. By changing
Vexit, the size of the DyQD at the capture point can be
altered and hence more or fewer electrons are captured.
The captured electrons are subsequently ejected into the
drain as the entrance barrier rises to its highest point.
This results in a measured current in the (2,2) area of
Ipump= 2ef.
As Ventbecomes even less negative the pump switches
to the (2,1) area where Ipump= ef is measured for the
same exit gate voltage. The size of the DyQD at the cap-
ture point is expected to be the same as the previous case,
so we assume that two electrons are again captured here
but only one is ejected, with the other remaining confined
within the DyQD. We therefore measure Ipump= ef in
this area.
Finally, in area (2,0), the entrance barrier never rises
high enough to push any of the captured electrons over
the exit barrier and into the drain. The current measured
in this region is therefore Ipump= 0.
The applied Ventnecessary for the entrance barrier to
drop low enough to allow population of the dot (going

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FIG. 2: Left: the response of the pumped current to changes in Vent and Vexit. The main panel shows the numerical derivative
in Vexit and Vent of the pumped current, highlighting transitions in the number of pumped electrons. The blue dashed lines
show directions of line scans in Vent and Vexit, seen to the right and bottom of the main panel respectively. Dashed black lines
correspond to the expected plateau values. Right: schematic diagrams of the barriers defined by the gates in each of the four
(nc,ne) regions, where nc is the number of electrons captured by the DyQD and ne is the number ejected into the drain.
from (0,0) to (2,2) in Fig. 2) should be independent of
Vexit. The measured slope of the transition highlighted
by the green dashed line in Fig. 2 arises from capacitive
coupling between the gates.
We measure a different slope in the transitions cor-
responding to when ne differs from nc. The transition
from ne = nc to ne = nc− 1 (going from pumping all
to pumping all but one electrons) is highlighted by the
red dashed line in Fig. 2. We believe this slope arises as
a result of the shape of the potential at the stage in the
pump cycle where the electrons are ejected into the drain
being controlled by both Ventand Vexit.
III. PUMPING IN B⊥
We next present data from measurements of the
pumped current under the application of a perpendicular
magnetic field to the device. We propose that informa-
tion about the dynamics of the system may be extracted
by monitoring changes in the pump map. Figure 3 shows
the evolution of the pump map upon increasing B⊥. The
pumping frequency was set to 73MHz and the amplitude
of VRFat the source was −9dBm, as before.
The upper panel of Fig. 3 shows that the transitions
between plateaus become sharper (i.e. darker) as B⊥is
increased. This suggests an enhancement of the quanti-
zation [8, 12]. The lower panel of Fig. 3 supplements this
observation, where linescans in Vexitfor Vent= −0.17V
at each field increment are shown. It follows that the
error mechanisms that give rise to deviations from per-
fectly quantized current at zero field must be suppressed
at higher fields. A recent theoretical framework predicts
the contribution of back-tunneling errors arising during
FIG. 3: Upper panel: evolution of the pump map in the main
left panel of Fig. 2 under the application of a perpendicular
magnetic field B⊥. Lower panel: line scans for Vent = −0.17V
at each field increment. The dashed black lines mark the
expected values for each plateau.
the capture process [17]. In a perpendicular magnetic
field we expect that the increased confinement of the cap-
tured electrons would lead to a reduction in the radial
extent of the wave function [8]. We therefore expect a
smaller overlap of the wave function with the leads and
thus a lower probability of back-tunneling, resulting in
an enhanced pumping accuracy.

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The lower panel of Fig. 3 shows line scans in Vexitat
each magnetic field for Vent= −0.17V. The plateaus are
flatter in higher fields, as discussed. They are also longer,
indicating enhanced robustness of the pumping mecha-
nism [8]. In a field of B⊥ = 5T these DyQD pumps,
operating at f = 340MHz, were shown to be accurate
to better than 1.5 parts in 105[13]. This result is im-
portant for quantum metrology and the development of
a quantum standard for current.
At higher fields we observe quantized current plateaus
corresponding to a larger number of electrons robustly
transported per cycle. The blue and red crosses in Fig. 3
are placed at the same coordinates in each plot, and they
highlight the use of the field as an effective tuning param-
eter. In the case of the blue crosses the field is able to turn
the pumping on in an area of the pump map where our
model suggests that the dot is too small to capture elec-
trons at 0 T. As illustrated in Fig. 2, during the first part
of the pump cycle the DyQD is coupled to the source, and
so electrons are able to easily leave the DyQD and return
to the source as the RF cycle progresses and the DyQD
becomes smaller. The perpendicular magnetic field has
the effect of increasing the effective confinement poten-
tial experienced by electrons in the DyQD, and so there
is an enhanced probability of an electron remaining in
the DyQD at the capture point. This explains the grad-
ual increase in the pumped current from zero to ef at
the point indicated by the blue cross in Fig. 3 as B⊥is
increased from 0 T to 3 T.
A similar explanation can be applied to the pumped
current at the point indicated by the red cross in Fig. 3.
At 0 T, the red cross resides in the (1,1) region of the
map, indicating that no electrons remain in the DyQD
at the end of the pump cycle. Conversely, at a field of
3 T the red cross is in the (2,1) region. Here, we see that
the increased confinement has led to a transition from
capturing ncelectrons to capturing nc+ 1 electrons, as
above, whilst also enabling the DyQD to confine a single
electron at the end of the pump cycle (nc− ne = 0 at
0 T, but nc− ne= 1 at 3 T).
The green dots in the B = 0T and 2 T pump maps
of Fig. 3 serve to further illustrate this behaviour. In
zero field the DyQD captures and ejects three electrons
per cycle. Upon increasing the field to B = 2T the
DyQD was able to capture and eject five electrons for
the same electrically defined DyQD. A full explanation
of the evolution of the pump map in a magnetic field will
require a more detailed computational study of electron
dynamics in this device [14].
IV. PUMPING IN HIGH B⊥
At fields of B > 3T the pump maps begin to exhibit
phenomena that are no longer reproducible from device
to device. Figure 4 shows pumping maps at fields of B =
5T and 9T for two different samples. The data presented
earlier in this work was collected with sample B. Sample
FIG. 4: Pump maps for large B⊥. Lower panel: continuation
of the data presented in Fig. 3. The upper panel shows data
collected with a different device, processed using a different
HEMT wafer. VRFfor sample A was set to f = 306.7MHz
at an amplitude of −9.6dBm. Numbers in the plateaus cor-
respond to the number of electrons transported per cycle in
those regions.
A was fabricated using a different HEMT wafer and had
a slightly different etched channel geometry. Sample A’s
pumping frequency was f = 306.7MHz.
In each sample we observe an anomalous structure in
the pumped current at high fields. For sample A the
plateau corresponding to capturing two electrons and
ejecting one electron (the last electron remaining con-
fined within the DyQD at the end of the pump cycle) is
no longer present at 9T, as can be seen in the upper-
right pump map of Fig. 4. The last two electrons ap-
pear to exit into the drain for the same entrance bar-
rier height. Similar findings have been reported in SECS
measurements where electrons were seen to tunnel into
and out of static QDs in pairs and bunches over a range
of B⊥ [10, 11]. Several theories which rely on disorder
have been developed to explain this behavior [18–20] but
the origin remains unclear.
We did not observe identical behavior in sample B,
but we did see other plateaus disappear at similar fields
as transitions in the number of ejected electrons begin
to merge. The white dashed ellipses in Fig. 3 highlight
regions in the pump map where this merging can be ob-
served. Different lines are seen to merge in each device.
This behavior is also reminiscent of earlier SECS mea-
surements where the addition spectra of different QDs
displayed pairing and bunching of certain energy lev-
els. In one experiment, artificial disorder was created
by tuning the coupling of two nearby QDs. The pair-

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ing/bunching behavior was shown to be strongly depen-
dent on the inter-dot coupling, and hence upon disor-
der [21]. Bunching behavior in our devices generally oc-
curs for magnetic fields of at least ∼ 5T. In disordered
systems it is expected that the field enhances disorder:
the wave function shrinks, leading to an enhancement
of the effects of a localization potential (for a review,
see [22]).
A full plateau structure persisted up to the maximum
readily achievable fields in our measurement system of
15T. Our results are very different from those published
by Kaestner et al. [12], where at 10.2T only one ne= 1
plateau was observed with all ne> 1 plateaus being com-
pletely suppressed. For certain frequencies, RF signal
amplitudes and field strengths we did see similar patterns
to those of Kaestner et al. which we attribute to anoma-
lous rectified biases that appear to be not only frequency
dependent but also magnetic field dependent. The origin
of rectification in our devices is not fully understood at
present but is likely to be due to a complicated response
of the sample holder, bond wires and ohmic contacts to
the applied RF signal.
V. CONCLUSIONS
In summary, we have presented experimental observa-
tions of the effect of a perpendicular magnetic field on the
quantized current produced by DyQD electron pumps.
The pumping accuracy was shown to be enhanced by
the field, suggesting a suppression of the error mecha-
nisms associated with a loss of quantization. The field
was shown to be an effective extra control parameter in
the tuning of the pump. As we increased the field to
B = 3T the pump could be turned on in a region of the
pump map where no pumped current was generated at
zero field. Our observations suggest the magnetic field is
adding an extra confinement potential to the gate-defined
DyQD. For B > 5T we detected anomalous structure in
the quantized current. We observed the onset of a pair-
ing behavior in the ejected electrons reminiscent of SECS
measurements, where several theories predict pair tunnel-
ing in QDs can arise from disorder unique to each indi-
vidual QD. Our data suggests that local disorder, unique
to each DyQD, affects the pumping more strongly for
higher magnetic fields. We hope our findings will pro-
mote DyQDs as useful tools for probing few-electron dy-
namics in many fundamental investigations.
We gratefully acknowledge Bernd Kaestner, Christoph
Leicht and Philipp Mirovsky for useful discussions. SJW
acknowledges support from the EPSRC and Toshiba
Research Europe Ltd. The work of MDB was sup-
ported by the UK National Measurement Systems Quan-
tum Metrology Programme.
supported by contract FA9453-07-C-0207 of AFRL.
CAN acknowledges support from the EPSRC QIP IRC
(GR/S82176/01).
The work of GG was
[1] M. J. T. Milton, J. M. Williams, and S. J. Bennett, Metrologia
44, 356 (2007).
[2] M. W. Keller, J. M. Martinis, N. M. Zimmerman, and
A. H. Steinbach, Appl. Phys. Lett. 69, 1804 (1996).
[3] V. F. Maisi, Yu. A. Pashkin, S. Kafanov, J. S. Tsai, and
J. P. Pekola, New J. Phys. 11, 113057 (2009).
[4] M. D.Blumenthal,B. Kaestner,
T. J. B. M. Janssen, M. Pepper, D. Anderson, G. Jones, and
D. A. Ritchie, Nature Physics 3, 343 (2007).
[5] J. M. Shilton, V. I. Talyanskii, M. Pepper, D. A. Ritchie,
J. E. F. Frost, C. J. B. Ford, C. G. Smith, and G. A. C. Jones,
J. Phys. Cond Mat 8, L531 (1996).
[6] M. Kataoka, M. R. Astley, A. L. Thorn, D. K. L. Oi,
C. H. W. Barnes, C. J. B. Ford, D. Anderson, G. A. C. Jones,
I. Farrer, D. A. Ritchie, and M. Pepper, Phys. Rev. Lett., 102,
156801 (2009).
[7] R. J. Schneble, M. Kataoka, C. J. B. Ford, C. H. W. Barnes,
D. Anderson, G. A. C. Jones, I. Farrer, D. A. Ritchie, and
M. Pepper, Appl. Phys. Lett. 89, 122104 (2006).
[8] S. J. Wright, M. D. Blumenthal, Godfrey Gumbs, A. L. Thorn,
M. Pepper, T. J. B. M. Janssen, S. N. Holmes, D. Anderson,
G. A. C. Jones, C. A. Nicoll, and D. A. Ritchie, Phys. Rev. B
78, 233311 (2008).
[9] S. J. Wright, M. D. Blumenthal, M. Pepper, D. Anderson,
G. A. C. Jones, C. A. Nicoll, and D. A. Ritchie, Phys. Rev. B
80, 113303 (2009).
[10] R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer,
L. Li, S. Giblin,
S. J. Pearton, K. W. Baldwin, and K. W. West, Phys. Rev.
Lett. 68, 3088 (1992).
[11] N. B. Zhitenev, R. C. Ashoori,L. N. Pfeiffer, and K. W. West,
Phys. Rev. Lett. 79, 2308 (1997).
[12] B. Kaestner, C. Leicht, V. Kashcheyevs, K. Pierz, U. Siegner,
and H. W. Schumacher, Appl. Phys. Lett. 94, 012106 (2009).
[13] S. P. Giblin, S. J. Wright, J. Fletcher, M. Kataoka, M. Pepper,
T. J. B. M. Janssen, D. A. Ritchie, C. A. Nicoll, D. Anderson,
and G. A. C. Jones, New J. Phys. 12, 073013 (2010).
[14] S. J. Wright, A. L. Thorn et al., manuscript in preparation.
[15] B. Kaestner, V. Kashcheyevs, S. Amakawa, M. D. Blumenthal,
L. Li, T. J. B. M. Janssen, G. Hein, K. Pierz, T. Weimann,
U. Siegner, and H. W. Schumacher, Phys. Rev. B 77, 153301
(2008).
[16] B. Kaestner, V. Kashcheyevs, G. Hein, K. Pierz, U. Siegner,
and H. W. Schumacher, Appl. Phys. Lett. 92, 192106 (2008).
[17] V. Kashcheyevs and B. Kaestner, Phys. Rev. Lett. 104,
186805 (2010).
[18] Yi Wan, Gerardo Ortiz, and Philip Phillips, Phys. Rev. Lett.
75 2879 (1995).
[19] M. E. Raikh, L. I. Glazman, and L. E. Zhukov, Phys. Rev.
Lett. 77, 1354 (1996).
[20] C. M. Canali, Phys. Rev. Lett. 84, 3934 (2000).
[21] M. Brodsky, N. B. Zhitenev, R. C. Ashoori, L. N. Pfeiffer, and
K. W. West, Phys. Rev. Lett. 85, 2356 (2000).
[22] M. Pepper, Journal of Non-Crystalline Solids 32, 161 (1979).