Single-electron charging in quantum dots with large dielectric mismatch
A. Orlandi, M. Rontani, G. Goldoni, F. Manghi, and E. Molinari
Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica, Universita ` di Modena e Reggio Emilia,
Via Campi 213/A, I-41100 Modena, Italy
Received 13 September 2000; published 8 January 2001
Semiconductor quantum dots characterized by a strong dielectric mismatch with their environment are
studied theoretically through direct diagonalization of the many-body Hamiltonian. The enhancement of the
electron-electron Coulomb interaction, arising from polarization effects, is found to induce a strong increase in
addition energies with increasing dielectric mismatch. For large dielectric mismatch, the excited many-body
states can undergo reconstructions as the dot is filled with carriers even in the absence of external magnetic
fields.
DOI: 10.1103/PhysRevB.63.045310 PACS numbers: 73.61.-r, 85.35.Be, 85.35.Gv
I. INTRODUCTION
Semiconductor quantum dots QD’s are structures where
carrier confinement at the nanometer scale is achieved in all
spatial directions; the energy spectrum is therefore discrete,
as in natural atoms, and the Coulomb interaction among
electrons is enhanced owing to the large spatial overlap of
their wave functions. In recent years, a large experimental
and theoretical effort has lead to the demonstration of QD
devices where the strong Coulomb interaction is exploited to
allow control of charge injection at the single-electron
level:
1–3
Single-electron transistors SET’s are a fascinating
manifestation of many-body physics at work. A reason for
the importance of SET’s is in the extremely low power re-
quired for their operation of the order of nanowatts. This
opens the way to their possible integration in bioenviron-
ments, which is generally incompatible with the large power
dissipation of current microelectronic transistors at least six
orders of magnitude larger.
A specific characteristic of organic environments is their
huge dielectric mismatch with typical inorganic-
semiconductor QD structures. Quantum dots with a large di-
electric mismatch to the surrounding medium have been pre-
pared and studied by different techniques in recent years.
They comprise QD’s embedded in glasses
4
and organic ma-
terials, including those fabricated by colloidal techniques
5–8
or inserted in biological environments.
9
In most of these sys-
tems, photons have been the primary probe for investigating
or modifying the electronic properties of the dots. More re-
cently, it was shown that spherical nanocrystals in low-
dielectric-constant matrices can be built into a single-
electron device, and that capacitance or tunneling
spectroscopies can be used to obtain their addition
spectra.
6–8,10
The effects of the large dielectric mismatch
have been, however, overlooked in the interpretation of
single-electron charging phenomena in these dots.
In this article we present a theoretical calculation of these
effects, based on the direct solution of the exact few-particle
Hamiltonian for the dots. For large electron numbers, where
the dimension of the Hilbert space becomes excidingly big,
we also make use of a Hubbard-like approximation to the
full Hamiltonian.
11
We show that the enhancement of the
electron-electron Coulomb interaction, arising from the
buildup of large polarization charges at the interfaces, pro-
duces new features in the addition spectra, namely: i a
strong increase in the absolute values of addition energies,
and ii the possible occurrence of ‘‘reconstructions’’ of the
electronic configurations as the dot is filled with electrons.
The operation of single-electron devices in organic environ-
ments is therefore expected to involve phenomena that
modify the relevant energy scale and charge distribution with
respect to conventional QD devices embedded in inorganic
semiconductor matrices. These phenomena may be relevant
for applications of SET’s as sensors of dielectric properties
at the nanoscale.
II. THEORETICAL APPROACH
The key quantities that are used to characterize these sys-
tems experimentally are the addition energies E
add
( N ) the
variation of the energy required to add an electron to a QD
containing N electrons, analogous to the differences between
electron affinities in natural atoms. These are defined as
differences between the QD chemical potentials, , as one
electron is added to the dot: E
add
( N ) = ( N +1) - ( N ). In
turn, the chemical potential of the QD with N electrons is
( N ) =E
0
( N ) -E
0
( N -1), where E
0
( N ) are the ground-
state energies in the dot.
To calculate E
0
( N ) we need to know the ground-state
configuration of the many-electron system. The full many-
body Hamiltonian is
H
ˆ
=
a
a
c ˆ
a
†
c ˆ
a
+
1
2
abcd '
V
abcd
c ˆ
a
†
, c ˆ
b
'
†
c ˆ
c
'
c ˆ
d
,
1
where c ˆ
a
†
( c ˆ
a
) is the fermionic creation destruction op-
erator in the eigenstate | a of the single-particle Hamil-
tonian a stands for the set of orbital quantum numbers, for
spin;
a
are the single-particle energies; V
abcd
are the inter-
action integrals
V
abcd
=
a
* r
b
* r' V r, r'
c
r'
d
r d r d r'
2
where
a
( r) are the single-particle envelope functions.
12
PHYSICAL REVIEW B, VOLUME 63, 045310
0163-1829/2001/634/0453106/$15.00 ©2001 The American Physical Society 63 045310-1