Intermediate Low Spin States in a Few-electron Quantum Dot in the $\nu\le 1$ Regime
ABSTRACT We study the effects of electron-electron interactions in a circular few-electron vertical quantum dot in such a strong magnetic field that the filling factor $\nu\le 1$. We measure excitation spectra and find ground state transitions beyond the maximum density droplet ($\nu=1$) region. We compare the observed spectra with those calculated by exact diagonalization to identify the ground state quantum numbers, and find that intermediate low-spin states occur between adjacent spin-polarized magic number states. Comment: 4 pages, 4 figures
arXiv:cond-mat/0512239v1 [cond-mat.mes-hall] 12 Dec 2005
Intermediate Low Spin States in a Few-electron Quantum Dot in the ν ≤ 1 Regime
Y. Nishi,1P.A. Maksym,2D.G. Austing,3T. Hatano,4L.P. Kouwenhoven,5H. Aoki,1and S. Tarucha4,6, ∗
1Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
3Institute for Microstructural Sciences M23A, National Research Council,
Montreal Road, Ottawa, Ontario K1A 0R6, Canada
4ICORP Spin Information Project, JST, Atsugi-shi, Kanagawa 243-0198, Japan
5Department of Applied Physics, Delft University of Technology,
P.O.Box 5046, 2600 GA Delft, The Netherlands
6Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
(Dated: February 2, 2008)
We study the effects of electron-electron interactions in a circular few-electron vertical quantum
dot in such a strong magnetic field that the filling factor ν ≤ 1. We measure excitation spectra and
find ground state transitions beyond the maximum density droplet (ν = 1) region. We compare the
observed spectra with those calculated by exact diagonalization to identify the ground state quantum
numbers, and find that intermediate low-spin states occur between adjacent spin-polarized magic
PACS numbers: 73.63.Kv, 73.23.Hk
Few-electron quantum dots, or “artificial atoms”, ex-
hibit intriguing effects of quantum mechanical confine-
ment and electron-electron interactions. Both of these
effects can be significantly modified by a magnetic field.
For example, application of high magnetic fields strength-
ens electron correlation in quantum dots because the
single-particle states become nearly degenerate to form
Landau levels, and all the electrons occupy the low-
est Landau level when the magnetic field is strong
enough [1, 2, 3].As the magnetic field is increased
from zero, several ground state transitions to higher
orbital- and spin-angular momentum states are induced
because of Coulomb repulsive interactions, exchange in-
teractions [4, 5, 6, 7, 8, 9, 10, 11] and correlation ef-
fects . When the field reaches the point where the
filling factor ν = 1, the ground state becomes a spin-
polarized maximum density droplet (MDD) [5, 6, 13, 14].
This state has N electrons occupying the lowest N
angular-momentum states in the lowest Landau level and
it persists over a wide range of magnetic fields because
of the favourable balance of Coulomb repulsive and ex-
change forces. Further increase of the magnetic field
and the resultant wavefunction shrinkage increase the im-
portance of electron-electron interactions and the MDD
eventually gives way to a sequence of strongly correlated
states which occur in the ν ≤ 1 regime [14, 15]. This is
very similar to the fractional quantum Hall effect where
electron correlation plays an important role in determin-
ing the ground states.
In this strong-correlation regime beyond the MDD,
electrons in a quantum dot are predicted to form an
“electron molecule”, a state in which electrons rotate
and vibrate around a specific classical equilibrium con-
∗Electronic address: firstname.lastname@example.org
ple, the ground state angular momentum (L) of an elec-
tron molecule must belong to a specific series of values
called “magic numbers” and these values depend on both
the total spin (S) and the rotational symmetry of the
molecule [16, 17, 18, 19, 20]. For N ≤ 4 only one series
of magic numbers occurs (with interval ∆L=N) but for
N ≥ 5 multiple series of magic numbers can occur be-
cause more than one type of equilibrium configuration is
possible [16, 19, 20] and this can open a way to explore
novel physics associated with mixing of molecular con-
figurations. Thus quantum dots in the ν ≤ 1 regime are
extremely interesting for the study of strong correlation.
However, the few existing experimental studies of
quantum dots in this regime [5, 6, 14] are mainly con-
cerned with the MDD region at large N. For example,
in  the general trends of the ground state evolution
for large N are discussed in terms of charge redistribu-
tion but the small N regime is not studied. Despite its
importance as a model interacting system, a dot in the
small N, large B regime is poorly understood experimen-
tally and even its ground state quantum numbers have
not yet been identified.
In this Letter we report on ground state transitions in
the electronic energy spectra in the ν ≤ 1 regime and,
to identify the ground state configurations, we compare
the observed spectra with exact calculations. To do this
we have developed a new theoretical model of a pillar
dot, whose parameters are derived from experiment, so
that data for individual dots can be analysed . We
focus on N = 5 and N = 3 quantum dots, since the
N = 5 dot is the smallest system with two types (pen-
tagonal and square+center) of molecular configuration
(hence two series of magic numbers), while the N = 3
dot has the simplest non-trivial molecular configuration
Our sample is a circular vertical quantum dot mesa of
To satisfy the Pauli exclusion princi-
FIG. 1: (a) Schematic diagram of quantum dot mesa. (b)
B dependence of the Coulomb oscillation peaks. (c) Exper-
imental and theoretical ∆µN against B for N = 3, 4 and
diameter 520nm (Fig. 1(a)). Details of the layer struc-
ture and device mesa can be found in Ref. . Unlike for
a lateral quantum dot geometry, tunneling rates between
the dot and contacts are not drastically suppressed even
in high magnetic fields (up to 15 T) because the mag-
netic field is applied parallel to the current in our device
geometry. Our measurements are performed in a dilution
refrigerator and the electron temperature is estimated to
be below 100mK.
The current I flowing vertically through the dot is mea-
sured as a function of the gate voltage Vg at a fixed dc
source-drain voltage Vsd. For small Vsd(< 100µV), cur-
rent flows only through the ground states. Then the posi-
tion of the Nth current (Coulomb oscillation) peak Vg(N)
is a measure of the Nth ground state electro-chemical po-
tential µ(N). For finite Vsd, on the other hand, the ex-
cited states in the transport window as well as the ground
state contribute to the current. We can then obtain ex-
citation spectra in which the excited states appear as
step-like features within a broad current stripe .
The magnetic field (B) evolution of the Coulomb oscil-
lation peaks for N = 1 ∼ 20 is shown in Fig. 1(b). This
result is similar to that reported earlier . As discussed
in refs. [2, 14], atomic features such as a shell structure
and deviations due to Hund’s first rule indicate that our
dot has high rotational symmetry. Regions of ν > 2,
2 > ν > 1, ν = 1 and ν < 1 can be distinguished by
the overall behavior of the peaks . The peak spacing,
∆Vg= Vg(N) − Vg(N − 1), is not strongly N-dependent
in the MDD region. This is no longer the case when
the ν < 1 region is entered, which implies that a new
energetic structure is induced by correlation. Now we
focus on the evolution of ∆Vgto study the ground state
transitions, which has not been done before for ν < 1.
The peak spacings are related to the N-electron
ground state energy EN via the “addition energy”
∆µN = µ(N) − µ(N − 1) = EN− 2EN−1+ EN−2 =
e(αN(Vg,B)Vg(N) − αN−1(Vg,B)Vg(N − 1)) ∼ e¯ α∆Vg
where α is an electrostatic leverage factor  and ¯ α =
(αN + αN−1)/2. We use experimental values of ¯ α to
convert ∆Vgto ∆µNand compare the results with ∆µN
calculated from ENvalues obtained by exact diagonaliza-
tion. The effects of electron-electron interactions, screen-
ing, finite dot thickness, image charges, and the Vg de-
pendence of the confinement potential are incorporated
realistically. The parameters in the theoretical model,
the confinement energy, ?ω = 4.80 + 0.021(N − 1) meV
and screening length λ = 15.3 nm, are found by fitting
data up to 10 T  and the g-factor, |g∗| = 0.3 is esti-
mated from the Zeeman splitting observed at high fields.
Figure 1(c) shows experimental (black) and theoretical
(red) ∆µN values for N = 3,4 and 5. There is generally
excellent agreement especially for B < 10 T where the
RMS deviation is less than ∼ 0.15meV. For B < 8 T (i.e.
ν ≥ 1) almost all the ground state transitions predicted
in the theory (indicated by diamonds) are observed as
upward kinks (triangles) in the experimental results. The
MDD region extends over a wide magnetic field range
between two kinks in each curve. Some of the predicted
transitions are missing probably due to the resolution of
our experiments. For B > 8 T there are discrepancies in
the magnitude of the experimental and theoretical values
of ∆µN but the positions of features agree well in both
Fig. 1(c) and excitation spectra (discussed below). To
investigate the cause of the magnitude discrepancie we
used the field range up to 14 T (instead of 10 T) to fit
the model parameters and found that the positions of
the features did not agree in either the low or high field
regimes. This suggests that the magnitude discrepancies
are caused by imprecise knowledge of ¯ α in the high field
regime . Howeverprecise knowledge of ¯ α is not needed
to interpret the positions of features in the excitation
spectra so we can apply our model to identify ground
state quantum numbers.
We begin with N = 5. The observed excitation spec-
trum for B = 4 - 12 T is shown in the upper panel
of Fig. 2, where regions of positive (zero and negative)
dI/dVg appear black (white) in the B-Vg plane.
lower edge of the stripe delineates the ground state. Since
we are interested in the ground state transitions, we focus
on the relative positions of the ground state and a few
of the lower excited states. We find the MDD ground
state (pink broken line) between two kinks of the lower
edge of the stripe at B = 5.9 T (labelled A) and 9.3 T
(B), as already seen in Fig. 1(c). The area around B
is magnified in the inset, where the magnitude of I is
color coded . As we go above the lower edge of the
stripe the color abruptly changes from green to red or
black for B ≤ 9.3 T and B ≥ 10.0 T, which indicates
an onset where an excited state can contribute to the
current. Indeed, we can see that an excited state for
B ≤ 9.3 T becomes the ground state between 9.3 T (B)
and 10.0 T (C), which then becomes an excited state
again for B > 10.0 T. Namely, an intermediate ground
FIG. 2: Top: Measured excitation spectrum for N = 5 for
B = 4 − 12 T with Vsd=1.8mV: dI/dVg is plotted to show
sharp changes. (Inset: magnitude of I boundaries between
different colors indicate excited states). Bottom: Calculated
energy spectrum. Colored lines are guides to the eye.
state emerges at the collapse of the MDD state. The tran-
sitions B and C agree well with those in the calculated
spectrum (defined as E∗
N− EN−1− µcwhere E∗
N-electron excited state energy and µcis an approxima-
tion to the contact electro-chemical potential ) which
is shown in the lower panel of Fig. 2. This shows that a
partly spin-polarized ground state with (L,S) = (14,3/2)
appears between the fully spin-polarized magic-number
states (10,5/2) (between A′and B′) and (15,5/2) (be-
yond C′). The calculation also reproduces other impor-
tant features in the experimental spectrum. For exam-
ple, parts of the excited states along the orange, blue
and green broken lines in the dI/dVg plot of Fig. 2 are
reproduced by the calculation.
Within the stripe there are several grey “patches”.
Those near the lower edge of the stripe can be attributed
to “emitter states”, which result from density of state
fluctuations in the heavily doped contacts . Emitter
state features are distinguished from excited state fea-
tures because (i) they do not smoothly connect to the
ground state and/or (ii) similar patterns can be found in
neighboring stripes (i.e. they are N-independent) and/or
(iii) they appear as fluctuations or peak-like features in
the I − Vgcharacteristics, whereas excited states appear
as step-like features (See the insets to Figs. 2 and 3) .
For example, a transition-like structure observed around
the point labelled X in Fig. 2 can be attributed to the
emitter states because (i) it “touches” the ground state
so steeply that it cannot smoothly connect to the ground
state, and (ii) similar features are found for N = 4 and 6
at the same magnetic field. The “patches” not very close
to the lower edge can arise both from emitter states or
higher lying excited states which are so dense that it is al-
most impossible to identify them experimentally. These
general statements also apply to Figs. 3 and 4.
FIG. 3: Measured excitation spectrum for N = 5 for B =11
- 15 T (top), and the calculated spectrum (bottom). Col-
ored lines in both spectra are guides to the eye. Upper inset:
magnitude of I. Lower inset: pair-correlation functions.
No ground state transitions occur for B = 10.0 - 12.0 T
in Fig. 2, so we now explore the higher magnetic-field re-
gion, B>12 T for N = 5. Fig. 3 shows the excitation
spectrum for B = 11 - 15 T, where the upper panel and
inset, respectively, show dI/dVgand I. We can immedi-
ately see that the ground state for B ≤ 12.4 T (red bro-
ken line) gives way to a new ground state (violet broken
line) at B = 12.4 T (D), which is in turn taken over by an-
other excited state (light blue broken line) that becomes
the next ground state at B = 14.0 T (E). These transi-
tions are also identified as color changes in the I-intensity
plot inset. If we turn to the calculation in the lower panel,
we can identify the ground state between D and E as the
partly spin-polarized (L,S) = (18,3/2) state between D′
and E′, and the ground state for B ≥ 14.0 T as the spin-
polarized magic-number state (20,5/2) to the right of E′.
Let us now compare the above with the N = 3 ex-
citation spectrum (Fig. 4, upper panel). A down-going
excited state (light blue broken line) changes places with
the up-going ground state (pink) at B = 5.0 T (F).
This indicates a transition to the MDD state, (L,S) =
(3,3/2), for B = 5.0 - 10.0 T, consistent with Fig. 1(c).
There are several dark patches (not reproduced by the-
ory) intersecting with the MDD region, but we attribute
them to emitter states because they shift too rapidly with
magnetic field to connect to the ground state smoothly.
However, it can be seen that a down-going excited state
(red broken line) crosses the MDD ground state at B =
10.0 T (G), for which the resulting kink is also observed
at B = 10.0 T in Fig. 1(c). This marks the collapse of the
MDD state, and a transition to a new ground state. We
FIG. 4: Top: Measured excitation spectrum for N = 3 for
B = 4 - 15 T with Vsd =2.7mV. Bottom: Calculated energy
spectrum. Colored lines in both spectra are guides to the eye.
also find another down-going excited state (green broken
line). This excited state crosses the ground state (to the
right of G) at B = 11.6 T (H). Hence an intermediate
ground state (red broken line) occurs between G and H.
Another transition occurs at B = 14.5 T (I) and one is
predicted to occur just above 15 T (not shown). The
discrepancy may be due to disorder . If we compare
the observed transitions to those in the theoretical spec-
trum in the lower panel, we can identify the intermediate
state as the low spin state (5,1/2) that appears between
two spin-polarized magic-number states (L,S) = (3,3/2)
(MDD state) and (6,3/2) (to the right of H).
To summarize, for N = 5, we have observed and iden-
tified the ground state transition from the MDD state
((L,S) = (10,5/2)) to the next spin-polarized magic-
number state (15,5/2) via the low-spin state (14,3/2),
and a higher transition to another spin-polarized magic
number state (20,5/2) via the low-spin state (18,3/2).
For N = 3, the MDD state (3,3/2) makes a transition to
the next spin-polarized magic-number state (6,3/2) via
the low-spin state (5,1/2).
The nature of the low spin states can be interpreted
by using the electron-molecule picture . Specifically,
for N = 5 and S = 5/2 the magic numbers L = 14 and
18 correspond to a ‘square + center’ configuration whilst
the pentagonal configuration is forbidden. In contrast,
when S = 3/2, electron-molecule theory predicts that
both pentagonal and square+center configurations are
possible for L = 14 and 18. This has interesting conse-
quences which we investigate by calculating ground state
pair-correlation functions. The pair correlation functions
(Fig. 3) for the spin-polarized molecular states (15,5/2)
and (20,5/2) show pentagonal symmetry but the one for
the (18,3/2) intermediate state has a plateau at the cen-
ter instead of a valley. This suggests that the (18,3/2)
state is well approximated by a mixture of molecular
states with pentagonal and ‘square + center’ symmetry.
In conclusion, we have measured energy spectra of
N = 3 and 5 quantum dots and observed various ground
state transitions. We have compared them with theoreti-
cal calculations to assign ground state quantum numbers.
We have identified intermediate low-spin states in the
ν ≤ 1 region located between spin-polarized magic num-
ber states and explained their origin using the electron-
DGA, LPK, and ST are partly supported by DARPA-
QUIST program (DAAD 19-01-1-0659). ST is grateful
for financial support from the Grant-in-Aid for Scientific
Research A (No. 40302799), SORST-JST and IT pro-
gram MEXT. PAM thanks the University of Leicester
for the provision of study leave. We are pleased to thank
H. Imamura for fruitful comments and discussions.
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 A dI/dVg plot can show abrupt changes in I clearly over
a wide B-range, but it also emphasizes I fluctuations
caused by emitter states. In a color-coded I plot it is dif-
ficult to show the spectrum over a wide B-range because
the absolute value of I can change drastically with B.
However, step-like features due to the excited states can
be highlighted in a color-coded plot over a small B range.