# 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

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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

number states.

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 [12]. 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: tarucha@ap.t.u-tokyo.ac.jp

figuration [16].

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 [14] 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 [21]. 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

(triangular).

Our sample is a circular vertical quantum dot mesa of

To satisfy the Pauli exclusion princi-

Page 2

2

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

5.

diameter 520nm (Fig. 1(a)). Details of the layer struc-

ture and device mesa can be found in Ref. [2]. 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 [22].

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 [14]. 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 [14]. 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 [24] 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 [21] 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 [23]. 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 [26]. 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

The

Page 3

3

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 [21]) 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 [25]. 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) [26].

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.

Nis the

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

Page 4

4

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 [23]. 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 [16]. 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-

molecule picture.

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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[26] A dI/dVg plot can show abrupt changes in I clearly over

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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.