Resonant transmission through an open quantum dot
ABSTRACT We have measured the low-temperature transport properties of a quantum dot formed in a one-dimensional channel. In zero magnetic field this device shows quantized ballistic conductance plateaus with resonant tunneling peaks in each transition region between plateaus. Studies of this structure as a function of applied perpendicular magnetic field and source-drain bias indicate that resonant structure deriving from tightly bound states is split by Coulomb charging at zero magnetic field. Comment: To be published in Phys. Rev. B (1997). 8 LaTex pages with 5 figures
arXiv:cond-mat/9701060v1 [cond-mat.mes-hall] 9 Jan 1997
Resonant transmission through an open quantum dot
C.–T. Liang, I. M. Castleton, J. E. F. Frost, C. H. W. Barnes, C. G. Smith, C. J. B. Ford, D. A. Ritchie, and
Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom
(June 21, 2011)
We have measured the low-temperature transport properties of a quantum dot formed in a
one-dimensional channel. In zero magnetic field this device shows quantized ballistic conductance
plateaus with resonant tunneling peaks in each transition region between plateaus. Studies of this
structure as a function of applied perpendicular magnetic field and source-drain bias indicate that
resonant structure deriving from tightly bound states is split by Coulomb charging at zero magnetic
PACS numbers: 73.40.Gk, 73.20.Dx, 73.40.-c
Advancing technology has made it possible to define
artificial semiconductor microstructures which confine
electrons in all three spatial dimensions  with discrete
zero-dimensional states.Such structures, often called
quantum dots, provide uniquely simple systems for the
study of few electron physics. In particular, the Coulomb
blockade (CB) of single electron tunneling through quan-
tum dots  has been extensively investigated .
has been demonstrated  that transport through small
quantum dots is determined by charging effects [5,6] as
well as quantum confinement effects [7–9]. Quantum dots
can also be formed by impurities which are either di-
rectly in the electron gas, as for Si devices , or are
remote ionized donors in a spacer layer  as for the
GaAs/AlxGa1−xAs heterojunction . The CB effects
in such unintentionally defined quantum dots have been
studied extensively [10,12,13].
Within a non-interacting picture Tekman and Ciraci
 have predicted that resonant tunneling (RT) may
occur through energy states bound to an attractive im-
purity potential in a split-gate device even when some
one-dimensional (1D) channels are perfectly transmit-
ted.Therefore in addition to 1D quantized conduc-
tance steps [15,16], replicated resonant features between
plateaus should be observed when a quantum dot formed
by an impurity potential is present in a split-gate device.
In this paper, we report the first observation of such res-
onant structure from a quantum dot formed by an impu-
rity potential in a split-gate device. We show how these
RT features develop in a perpendicular magnetic field B
and we investigate the energy spacings between different
resonant states using source-drain bias measurements.
The Schottky gate pattern shown in the inset to Fig. 1
was defined by electron beam lithography on the surface
of a GaAs/Al0.3Ga0.7As heterostructure, 90 nm above a
two-dimensional electron gas (2DEG). The carrier con-
centration of the 2DEG was 3.3 × 1015m−2with a mo-
bility of 90 m2/Vs. Experiments were performed in a
dilution refrigerator at 100 mK and the two-terminal dif-
ferential conductance G = dI/dV was measured using an
ac excitation voltage of 10 µV.
Figure 1 shows the differential conductance as a func-
tion of the voltage Vg1 on gate 1, for various voltages
Vg2 on gate 2.For Vg2=–1.7 V (trace 3) we observe
replicated resonant peaks in G(Vg1), reminiscent of those
predicted . As the temperature was increased, these
structures became broader but were still discernible up
to 650 mK. When the conduction channel through the
split-gate structure was moved sideways by varying the
voltage on gate 2  away from Vg2=–1.7 V, the sharp
RT features gradually diminished until at Vg2=–2.6 V
(trace 7) only quantized 1D ballistic conductance steps
were seen. In a subsequent cooldown in a3He cryostat,
we did not observe identical RT structure. Although the
surface Schottky gate pattern was intended to define a
quantum dot in the 2DEG electrostatically, both obser-
vations suggest that ionized impurities in the spacer layer
 played an important role in determining the trans-
port properties through the channel defined by the sur-
face gate. Since we observe conductance peaks (resonant
tunneling) rather than resistance peaks (resonant reflec-
tion), we believe that in our system an attractive impu-
rity potential helped create a quantum dot. Previously
McEuen et al.  claimed that two resonant transmis-
sion peaks they observed for G < 2e2/h in a disordered
split-gate device derived from the formation of a quan-
tum dot by a single hydrogenic impurity. In this experi-
ment only two peaks were observed because the electrons
which filled the impurity bound states acted to screen it
so that at higher energies only quantized conductance
with no resonant structure was seen. In our experiment
we observe at least fourteen resonant peaks (see trace 3
in Fig. 1), implying that the impurity potential does not
become screened even after accommodating 14 electrons.
We do not believe that such a potential could be gener-
ated by a single ionized impurity, only a cluster would be
capable of this.
Figure 2 shows G(Vg1) for Vg2=–1.7 V at different B.
For G < 2e2/h, the two RT peaks have a weak B de-
pendence and persist to B = 4 T. As the magnetic field
is increased, the conductance plateaus and the RT peak
positions for 2e2/h < G < 8e2/h move to more positive
Vg1as a result of the formation of hybrid magnetoelectric
subbands . At B ≈ 2 T these features are no longer
seen. A broad RT peak adjacent to the sharp RT peaks
for 4e2/h < G < 6e2/h develops at B = 0.6 T and splits
into two at higher magnetic fields, as indicated by arrows.
Similar but less pronounced results can be also seen for
2e2/h < G < 4e2/h.
The sharp resonances correspond to tightly bound
states and the broad resonances to weakly bound states
within the picture of Tekman and Ciraci . The ap-
plication of a perpendicular magnetic field strengthens
the confinement of states in a quantum dot by localiz-
ing electron wavefunctions to the sample boundaries .
This is consistent with the disappearance of the tightly
bound states (they become immeasurably small) and the
strengthening of the resonant structure from the weakly
bound states at high field seen experimentally - Fig. 2.
When the sharp RT structures for 2e2/h < G < 6e2/h
have disappeared, oscillations in G(Vg1) are still ob-
served. Their structure is more complicated and possibly
derives from a combination of resonant transmission and
resonant reflection  from bound states.
We now discuss the separation in gate voltage ∆Vg1
between each pair of tightly bound RT peaks at various
magnetic fields (marked as square, circle, triangle and
cross in Fig 2). Figure 3 shows ∆Vg1(B) for RT fea-
tures which occur with different numbers of transmitted
1D channels n1D. For n1D=0, ∆Vg1shows only a weak
magnetic field dependence. For n1D=1, 2, and 3, ∆Vg1
shows saturation at low B and a linear B dependence at
At B = 0, ∆Vg1decreases as n1Dincreases, as shown
in the inset to Fig. 4.To obtain the energy spac-
ing ∆E(n1D) between pairs of tightly bound RT peaks,
we have used a standard source-drain bias technique
[6,9,12,21]. ∆E decreases dramatically from n1D=0 to
n1D=1 (see Fig. 4). Note that we were not able to mea-
sure ∆E between pairs of RT peaks for G > 8e2/h,
perhaps because the application of a dc bias caused the
quantum dot to break down.
Within the non-interacting picture  at B = 0
the energy states through which RT occurs are spin-
degenerate. As B is increased, if there is no spin-
splitting, states with different angular momentum in the
same Landau level become closer in energy . If the
Zeeman energy is included, electrons in the same Lan-
dau level with the same angular momentum but differ-
ent spin move apart in energy causing individual reso-
nant transmission peaks to split into two peaks.
ing the minimum possible value 0.44 for the Land´ e g-
factor in our system, we estimate the Zeeman energy to
be ≈ 0.1 meV at B = 4 T, a factor of twelve larger
than thermal smearing at 100 mK, and equal the full-
width-half-maximum of the tightly bound peak closest
to pinch-off, suggesting that such splitting would be ob-
servable in our system. However, as shown in Fig. 2,
the individual peaks in each pair of tightly bound RT
peaks do not split at any magnetic field. In addition
pairs of peaks do not come closer together and each pair
of peaks remains in the same transition region. These fac-
tors imply that each pair of peaks derives from the same
single-particle state. They split at zero magnetic field
due to the energy difference between single and double
occupation of a single state . However, the case of
charging-induced splitting in mesoscopic devices where
two adjacent single-electron tunneling peaks are related
to states with different spin quantum numbers , is
only well understood in the Coulomb blockade regime
 for G < 2e2/h. Assuming that the relations ∆E =
30.1∆Vg1 meV/V (n1D = 0), ∆E = 14.8∆Vg1 meV/V
(n1D = 1), ∆E = 10.3∆Vg1 meV/V (n1D = 2), and
∆E = 7.27∆Vg1 meV/V (n1D = 3) (determined from
the data shown in Fig. 4 and the inset) which hold at
B = 0 are still valid at high field, ∆E(B) for the tightly
bound peaks with n1D=1, 2, and 3 shown in Fig. 5 also
implies charging-induced splitting at B = 0. If the split-
ting arose solely from Zeeman splitting, then one would
expect ∆E(B) → 0 as B → 0. Instead ∆E(B) shows
saturation at low B, suggesting that the splitting at low
fields is due to some effect other than Zeeman splitting.
The linear fits ∆E = 0.636B (solid line), ∆E = 0.507B
(dotted line), and ∆E = 0.424B (dashed line) shown
in Fig. 5 yield Land´ e g-factors of 10.9, 8.7, and 6.9 for
n1D= 1,2, and 3, respectively. Such large g-factors have
been measured in the quantum Hall regime where ex-
change energy is important . For the case n1D=0,
∆E(B) has a weak B dependence since near pinch-off
the Coulomb charging effect is much stronger than the
Having established the role of Coulomb charging ef-
fects in our system, we can now explain the splitting of
the broad resonant tunneling peak, indicated by arrows
in Fig. 2, which occurs as B is increased. For B = 0.6 T,
the broad resonant tunneling peak is spin-degenerate, as
the state through which RT occurs is weakly bound and
the Coulomb charging arising from confinement is not
pronounced. At higher B this state becomes more tightly
bound, increasing the contribution of Coulomb charging
effects. Therefore when the applied magnetic field is in-
creased from B = 0.6 T, both the Zeeman term and the
Coulomb charging lift the electron spin-degeneracy, caus-
ing the broad resonant tunneling peak to split into two.
The decrease of ∆E(n1D) as n1D is increased, at
B = 0, shown in Fig. 4, arises from two mechanisms: the
Coulomb force between electrons bound in the quantum
dot is increasingly screened as n1Dis increased; and the
conduction channel defined by the surface Schottky gates
becomes wider, increasing the spatial extent of the bound
state wavefunctions, and hence reducing the Coulomb
charging energy as n1Dis increased.
Although we can explain our results in terms of
Coulomb charging effects qualitatively, it is important
to note that ascribing the pairs of sharp RT features to
zero-field splitting, for G > 2e2/h, requires an extension
of the Coulomb charging picture to the metallic regime
where some 1D channels are transmitted, and that the
Coulomb interaction between pairs of electrons are par-
tially screened by these 1D channels. In principle the
results we present here are able to give information on
the ability of 1D states to screen 0D states.
In conclusion, we have reported an observation of
transmission resonances through an open quantum dot.
The magnetic field dependence of pairs of tunneling peaks
provides experimental evidence for Coulomb charging ef-
fects at zero-field magnetic field even when some one-
dimensional channels are perfectly transmitted through
the open quantum dot.
This work was funded by the United Kingdom (UK)
Engineering and Physical Sciences Research Council. We
thank D.H. Cobden, J.H. Davies, D.E. Khmelnitskii, P.C.
Main, S.E. Ulloa, and in particular J.T. Nicholls for
helpful discussions. C.T.L. acknowledges financial sup-
port from Hughes Hall College, the Committee of Vice–
Chancellors and Principals, UK, and the C.R. Barber
Trust Fund. C.H.W.B. acknowledges support from the
Isaac Newton Trust.
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Figure 1. G(Vg1) when the conduction path is elec-
trostatically shifted by applying various gate voltages to
gate 2.Trace 1 to 7: Vg2= −1.3, −1.5, −1,7, −1.9,
−2.1, −2.3, and −2.6 V, respectively. The inset shows
the Schottky gate geometry.
Figure 2. G(Vg1) for Vg2=−1.7 V at various magnetic
fields. The corresponding magnetic fields are, from bot-
tom to top: 2.5, 2.4, 2.3, 2.2, 2.1, 2, 1.9, 1.8, 1.7, 1.6,
1.5, 1.4, 1.2, 1, 0.8, 0.6, 0.4, 0.2, and 0 T. Traces are
vertically offset for clarity.
evolution of the RT features for G < 2e2/h (square),
2e2/h < G < 4e2/h (circle), 4e2/h < G < 6e2/h (tri-
angle), and 6e2/h < G < 8e2/h (cross) as the applied
magnetic field is increased from 0 T. Arrows serve as a
guide to the eye indicating a single resonant peak splits
into two for 2e2/h < G < 4e2/h and 4e2/h < G < 6e2/h,
Figure 3. ∆Vg1(B) for various n1D.
Figure 4. The energy spacing ∆E between pairs of
RT peaks as a function of n1Ddeduced from the dc bias
measurements. The inset shows ∆Vg1(n1D).
Figure 5. The energy spacing ∆E between pairs of RT
peaks as a function of B determined from data shown in
Fig. 3 and 4. The straight line fits are discussed in the
The symbols indicate the
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