Download Shell Structures and Level Statistics of a Quantum Dot

Journal of the Korean Physical Society, Vol. 34, No. , May 1999, pp. S175∼S179
Shell Structures and Level Statistics of a Quantum Dot
Min-Chul Cha
Department of Physics, Hanyang University, Ansan 425-791
S.-R. Eric Yang
Department of Physics, Korea University, Seoul 136-701
Asia Pacific Center for Theoretical Physics, Seoul
We have investigated the level fluctuations of a two-dimensional quantum dot reflected in the
addition energy spectrum. The level degeneracy consistent with the rotational symmetry of the
confining potential determines the structure of the energy shells. In the limit that the energy
level spacings are significant compared to the electron-electron interaction energy, we find that the
addition energy spectrum clearly displays the atomic magic numbers. Even in the opposite limit a
closed shell state of a dot can give rise to large level fluctuations. When a strong magnetic field is
present these fluctuations are significantly reduced.
I. INTRODUCTION
Recent advances in semiconductor technology allows
the fabrication of zero-dimensional quantum dots called
“artificial atoms” [1,2]. Typically their shape is similar
to a pancake with the thickness and radius about several hundred angstroms and a few thousand angstroms,
respectively. Such structures contain a discrete number
of electrons and have a discrete spectrum of energy levels, which is resolvable at low temperatures. One way
to investigate the energy state of artificial atoms is to
measure the energy needed to add or subtract electrons,
analogous to the electron affinity or the ionization energy for real atoms. For this purpose one measures the
current through quantum dots as the voltage Vg between
the gate and the quantum dots is varied [3–6]. Results
display conductance resonances, which are almost periodic as a function of Vg . This phenomenon is known as
Coulomb oscillation.
The period of Coulomb oscillation roughly corresponds
to e/Cg [1], where e is the electronic charge and Cg is
the gate capacitance. When no current flows through
the dot the condition Cg Vg = N e (N integer) is satisfied. By varying Vg the number of electrons can be
increased by one so that the condition Cg Vg = (N + 1)e
is satisfied. The change in Vg must be sufficiently large
so that the tunneling electron can overcome the Coulomb
charging energy. The current starts to flow when Cg Vg =
(N + 1/2)e is fulfilled, in which case, the state with N
electrons and that with N + 1 electrons are degenerate.
The conductance peaks are occurring with the period
of e/Cg in gate voltage, reflecting the discrete nature of
the charge quantization in quantum dots. However the
conductance peak spacings also reveal the effect of the
discrete energy quantization since extra electrons added
must occupy quantum states of higher energy levels due
to the Pauli exclusion principle.
Taking into account the effect of the charge quantization and the energy quantization, we can calculate the
conductance peak spacings from the ground state energies, EN . The peaks appear at the specific values of Vg .
For example, Vgj for the j-th peak is determined by the
condition that [5]
EN +1 − EN + αeVgN +1 = µ,
(1)
where µ is the Fermi energy of the electrode, and α =
Cg /Cdot is the ratio of gate capacitance to total dot capacitance. The conductance peak spacing is thus given
by
αe(VgN +1 − VgN +1 ) = EN +1 + EN −1 − 2EN .
(2)
If we neglect the energy level spacings and take the
ground state energy to be EN = (N e)2 /(2Cdot ), Eq.
(2) yields that ∆Vg = e/Cg , as discussed above. This
approximation, known as the Coulomb blockade model,
describes well the situation of a metal quantum dot
which has many electrons so that the energy level spacing
at the Fermi energy is sufficiently small. The dots
formed in the two-dimensional electron gas by application of an external confining field or by lithography,
however, contain typically less than 100 electrons. In
this case, the finite energy level spacings at the Fermi
energy cannot be neglected. These are considered in a
simple way by the constant interaction model in which
PN
EN = (N e)2 /(2Cdot ) + k=0 k with a slowly varying
dot capacitance Cdot . Here k is the energy of the kth single-particle level. Substituting this expression into
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Journal of the Korean Physical Society, Vol. 34, No. , May 1999
Fig. 1. Level spacing plotted as a function of electron
number. The peaks in ∆ reflect closed shell states of the dot.
Eq. (2), we can find that
αe(VgN +1 − VgN +1 ) =
e2
+ N +1 − N .
Cdot
Fig. 3. Level spacing plotted as a function of electron
number. The peaks in ∆ reflect closed shell states of the dot.
(3)
The conductance peak spacings of Eq. (3) have fluctuations whose source is the single-particle level fluctuations
near the Fermi energy.
It has been argued that disordered quantum systems
display a statistically universal level fluctuations that are
well described by the random matrix theory [7]. Recently
the ground state energy of disordered quantum dots is
found to display fluctuations considerably larger than
those predicted by random matrix theory [2,3,5,8]. The
electron-electron interactions are expected to be important in explaining the experimental data [5,9–12]. In the
constant interaction model the relaxation of charges due
Table. 1. The number of spin-up and -down electrons for
0 = 1 meV. In the absence of a magnetic field the dot becomes spin-unpolarized as the electron number increases. In
the presence of a magnetic field the dot is almost completely
spin-polarized. For other values of 0 the degree of spin polarization is different.
N
Fig. 2. Schematic energy diagrams for N = 12 and 13.
Energy is measured in units of 0 = 1 meV. Note that for
N = 12 spin-down (-up) shells p and d are non-degenerate
(degenerate). The explanation of this effect is given in the
text. As N changes from 12 to 13 the ordering of 2s1 and 1d2
shells in the spin-up band is reserved while in the spin-down
band split 1p states become degenerate.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B=0T
(N+ , N− )
(1,0)
(1,1)
(3,0)
(3,1)
(4,1)
(5,1)
(4,3)
(5,3)
(6,3)
(7,3)
(8,3)
(8,4)
(8,5)
(8,6)
(8,7)
B=1T
(N+ , N− )
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
(6,0)
(7,0)
(8,0)
(9,0)
(10,0)
(11,0)
(12,0)
(13,0)
(14,0)
(15,0)
N
16
17
18
19
20
21
22
23
24
25
26
27
28
29
B=0T
(N+ , N− )
(10,6)
(10,7)
(10,8)
(12,7)
(12,8)
(13,8)
(12,10)
(12,11)
(12,12)
(13,12)
(14,12)
(16,11)
(16,12)
(15,14)
B=1T
(N+ , N− )
(16,0)
(17,0)
(17,1)
(18,1)
(19,1)
(20,1)
(20,2)
(21,2)
(22,2)
(23,2)
(23,3)
(24,3)
(25,3)
(26,3)
Shell Structures and Level Statistics of a Quantum Dot – Min-Chul Cha and S.-R. Eric Yang
to the newly added extra electrons is not included.
In this work we investigate fluctuations that the
electron-electron interaction can give rise to in the conductance peak spacings given by Eq. (2), which are often
referred as the addition energy. Our calculation is based
on a Hartree-Fock (HF) approximation at zero temperature.
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χs is spin wavefunction and the constant a2 = h̄/(2m∗ ω),
where ω 2 = ω02 + 41 ωc2 . The eigenenergies are
1
0nls = h̄ω(2n + |l| + 1) − h̄ωc l − gµB Sz B,
2
(7)
Note that in the presence of a magnetic field 0nls 6= 0nl−s .
The interaction is given by
II. MODEL AND PROCEDURE
Here we adopt a model quantum dot formed in twodimensional electron gas systems where electrons are
confined by a parabolic potential. The HF single particle basis vectors are chosen to be the eigenstates of a
quantum dot in a magnetic field described by the Hamiltonian
p2
1
1
H0 =
+ m∗ ω02 r2 − ωc L · B̂ − gµB S · B, (4)
2m∗
2
2
where m∗ , B, ωc and ω0 are the effective electron mass,
magnetic field perpendicular to the 2D plane, the cyclotron frequency, and the frequency of harmonic confining potential of the dot. A vector potential in a symmetric gauge is used here. The operators L and S denote
angular momentum and spin of an electron. The eigenstate wavefunctions of this Hamiltonian are labeled by
orbital, angular momentum, and spin quantum numbers
n, l, s:
φnls (r) = √
1
r2
e−ilθ Rnl ( 2 )χs
2a
2πa
(5)
|l|
n!
e−x/2 x 2 L|l|
n (x).
(n + |l|)!
(6)
with
Rnl (x) =
s
hnls|VH |n0 lsi =
X
Hint =
1X
e2
p
,
2
κ |ri − rj |2 + d2
(8)
i6=j
where κ is the dielectric constant and d is the thickness
of the planar quantum dot. The HF Hamiltonian has a
block diagonalized form. Each block may be labeled by
[l, s] and it has the following form
[l,s]
Dnn0 = hnls|H0 |n0 lsi
+hnls|VH |n0 lsi − hnls|VX |n0 lsi.
(9)
In order to find the Hartree and exchange matrix elements we need to write HF eigenstates as a linear combination of the HF basis vectors
|αlsi =
X
c(αls)
|mlsi.
m
(10)
m
Invoking rotational invariance about the z-axis we may
take the HF single-particle eigenstates, |α, l, si, to be
eigenstates of the z-component of angular momentum
operator: Lz |α, l, si = h̄l|α, l, si, where α denotes HF
orbital quantum numbers. We find
hnls; αps0 |Hint |n0 ls; αps0 ifαps0
m,p,s0
0
X
=
(α,p,s0 )
∗(α,p,s )
cm
cm0
hnls; mps0 |Hint |n0 ls; m0 ps0 ifαps0
(11)
α,p,s0 ,m,m0
and
hnls|VX |n0 lsi =
X
hnls; αps0 |Hint |αps0 ; n0 lsifαps0
α,p
=
X
(α,p,s)
∗(α,p,s)
cm
cm0
hnls; mps|Hint |m0 ps; n0 lsifαps .
(12)
α,p,m,m0
where fαps are Fermi functions. As in atomic physics
it is possible to introduce energy shells in a quantum
dot. A shell consists of two angular momentum states l
and −l, and is denoted by α|l|s with α = 1, 2, .. and
|l| = s, p, d, f, ... The two angular momentum states
in a shell may not always be doubly degenerate. It is
degenerate only when there are equal number of occupied states with positive and negative signs of angular momenta below the Fermi level. The degeneracy is lifted when this number is unequal. The lifting
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Journal of the Korean Physical Society, Vol. 34, No. , May 1999
is due to the exchange field which couples states with
different sign of angular momentum differently: since
hn, l; α, p|Hint |α, p; n, li =
6 hn, −l; α, p|Hint |α, p; n, −li we
find
hn, l, s|VX |n, l, si =
6 hn, −l, s|VX |n, −l, si.
(13)
Such an effect is absent in the Hartree field since
hα, p; n, l|Hint |α, p; n, li = hα, p; n, −l|Hint |α, p; n, −li
leads to
hn, l, s|VH |n, l, si = hn, −l, s|VH |n, −l, si.
(14)
Note also that αl+ and αl− need not be degenerate since
the dot may be partially spin-polarized.
III. RESULTS
The addition energy is calculated using Eq. (2).
Among different (N+ , N− ) with N = N+ + N− the
ground state is chosen to be the one which gives the
lowest value of E(N+ , N− ). In such a state the Fermi
levels for spin-up and -down electrons match. Our basis
set consists of the 120 lowest energy states of a noninteracting single-particle.
An atom with a three-dimensional spherical potential
has atomic magic numbers 2, 10, 18, ... due to the shell
structure. Similarly, a two-dimensional quantum dot in a
harmonic potential has magic numbers 2, 6, 12, .... This
shell structure of the artificial atom is prominent when
the confining is strong so that the energy level spacings
are significant compared to the electron-electron interaction energy. Fig. 1 shows this behavior. In this figure the
single-particle energy spacing is set to be 0 ≡ h̄ω0 = 4
meV. The parameters used in this figure are κ = 12.4,
d = 200 Å. We have peaks of the addition energy at
the atomic magic numbers. These results are consistent
with experimental observations [4] and theoretical calculations [12].
The results obtained in this paper apply to quantum dots in which the disorder broadening of energy
levels is smaller than the level spacing. When a significant amount of disorder is present the level degeneracy will be lifted. Like in a real atom the energy ordering of shell energies does not follow a simple rule.
The physical origin of this effect is the relaxation of
the HF potential. If the single particle wavefunctions
in the N - and (N − 1)-particle states are the same,
energy ordering will be unchanged. However as N
changes a new level may appear below the Fermi level.
When the electron-electron interactions are comparable to the energy level spacing the energy ordering will
be changed as the number of electrons increases. For
example when 0 = 1 meV, for N = 8 the spin-up
(-down) ground configuration is 1s1 1p2 1d2 (1s1 1p2 ).
When an extra electron is added a new level 2s appears
in the spin-up configuration: 1s1 1p2 2s1 1d2 . Fig. 2 shows
the change of the energy ordering as the number of electrons N = 12 is increased by 1.
The shell structures and the energy orderings are reflected in the addition energy spectrum. This is plotted
as a function of N in Fig. 3. We remark the following
features. In the absence of a magnetic field the addition
energy alternate between a trough and a peak as a function of N . However, at N = 13, 14 and N = 16, 17 this
trend is broken and the values of the addition energy are
almost unchanged. The variance of the addition energy
is 0.31h̄ω0 .
In the absence of magnetic field, the alternation of a
trough and a peak reflects closed- and open-shell structures. For N = 12 the lastly occupied spin-down shell
is 1d1 , which is open (see Fig. 2(a)). For N = 13 the
lastly occupied spin-up and -down shells are is 1f 2 and
1d2 , which are both closed (see Fig. 2(b)). When the
system is in a closed shell state ∆N is larger than the
average value while when it is in a partially filled shell
the addition energy is smaller. These fluctuations are
consequence of the energy level degeneracy and the exchange splitting, not just the twofold degeneracy of the
spin degrees of freedom [9].
When a magnetic field is applied the addition spectrum displays much smaller fluctuations. At B = 1 T we
observe an interruption of this smooth dependence of the
addition energy on N . It turns out that spin flipping is
responsible for this effect. The numbers of spin-up and
-down electrons are given in Table 1 for different values
of N . For small N Coulomb interaction dominates, and,
consequently, to minimize the interaction energy, electrons tend to spin polarize. As N increases the confinement energy dominates over the Coulomb energy and it
is energetically favorable to pack electrons closely. This
state is achieved by flipping a spin and placing it at the
center of the quantum dot [13].
IV. SUMMARY
We study the conductance peak spacings of a quantum dot. We find that even in the absence of random
disorder there can exist large fluctuations due to closed
energy shell structures. The energy shell structure is
determined by the degeneracy reflecting the rotational
symmetry of the confining potential. Like in a real atom
the ordering of energy shells in a dot does not follow a
simple rule. The physical origin of this effect is the relaxation of the HF potential, which is neglected in the
constant interaction model. When an extra electron is
added to the system the other electrons must readjust
their positions self-consistently to minimize the total energy. The new self-consistent potential may be significantly different from the old one, and, therefore, energy
ordering may change. We have also investigated level
fluctuations when a strong magnetic field is present. We
find significantly reduced fluctuations.
Shell Structures and Level Statistics of a Quantum Dot – Min-Chul Cha and S.-R. Eric Yang
The main result of our work is that even in a relatively
clean dot level fluctuations can be significant. Large level
fluctuations does not seem to be a unique signature of a
disordered dot. It will be interesting to vary the amount
of disorder systematically and measure the fluctuations
of level spacing. Such an experiment will deepen our
theoretical understanding of the subtle interplay between
disorder and electron-electron interactions.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
ACKNOWLEDGMENTS
[10]
This work has been supported by the KOSEF under grant 981-0207-085-2, and by Korea Research Foundation under grants 1998-015-D00128 and 1998-015D00114.
[11]
REFERENCES
[12]
[13]
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Marc A. Kastner, Phys. Today 46, 24 (1993).
R. C. Ashoori, Nature(London) 379, 413 (1996).
J. A. Folk et al., Phys. Rev. Lett. 76, 1699 (1996).
S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996).
U. Sivan et al., Phys. Rev. Lett. 77, 1123 (1996).
S. R. Patel et al., Phys. Rev. Lett. 80, 4522 (1998).
M. L. Mehta, Random Matrices (Academic Press, 1991).
A. M. Chang et al., Phys. Rev. Lett. 76, 1695 (1996).
Ya. M. Blanter, A. D. Mirlin and B. A. Muzykantskii,
Phys. Rev. Lett. 78, 2449 (1997).
R. Berkovits and B. L. Altshuler, Phys. Rev. B55, 5297
(1997).
Hiroyuki Tamura and Masahito Ueda, Phys. Rev. Lett.
79, 1345 (1997).
S. Nagaraja et al., Phys. Rev. B56, 15752 (1997).
S.-R. Eric Yang, A. H. MacDonald and M. D. Johnson,
Phys. Rev. Lett. 71, 3194 (1993).