Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

Low-Temperature Conduction of a Quantum Dot 99
2 Model of a Lateral Quantum Dot System
The Hamiltonian of interacting electrons confined to a quantum dot has the
following general form,
Hdot = � �
� �
s
ij
hijd †
isd 1
js +
2
ss ′
ijkl
h ijkl d †
is d†
js ′d ks ′d ls . (1)
Here an operator d †
is
φi(r) (the wave functions are normalized according to � drφ∗ i (r)φj hij = h∗ ji
creates an electron with spin s in the orbital state
(r) =δij);
is an Hermitian matrix describing the single-particle part of the
Hamiltonian. The matrix elements hijkl depend on the potential U(r − r ′ )of
electron-electron interaction,
�
hijkl = dr dr ′ φ ∗ i (r)φ ∗ j (r ′ )U(r − r ′ )φk(r ′ )φl (r) . (2)
The Hamiltonian (1) can be simplified further provided that the quasiparticle
spectrum is not degenerate near the Fermi level, that the Fermi-liquid
theory is applicable to the description of the dot, and that the dot is in the
metallic conduction regime. The first of these conditions is satisfied if the dot
has no spatial symmetries, which implies also that motion of quasiparticles
within the dot is chaotic.
The second condition is met if the electron-electron interaction within the
dot is not too strong, i.e. the gas parameter rs is small,
rs =(kF a0) −1 � 1 , a0 = κ� 2 /e 2 m ∗
Here kF is the Fermi wave vector, a0 is the effective Bohr radius, κ is the
dielectric constant of the material, and m ∗ is the quasiparticle effective mass.
The third condition requires the ratio of the Thouless energy ET to the
mean single-particle level spacing δE to be large [13],
(3)
g = ET /δE ≫ 1 . (4)
For a ballistic two-dimensional dot of linear size L the Thouless energy ET is
of the order of �vF /L, whereas the level spacing can be estimated as
δE ∼ �vF kF /N ∼ � 2 /m ∗ L 2 . (5)
Here vF is the Fermi velocity and N ∼ (kF L) 2 is the number of electrons in
the dot. Therefore,
g ∼ kF L ∼ √ N, (6)
so that having a large number of electrons N ≫ 1 in the dot guarantees that
the condition (4) is satisfied.
Under the conditions (3), (4) theRandom Matrix Theory (for a review
see, e.g., [14, 15, 16, 17]) is a good starting point for description of noninteracting
quasiparticles within the energy strip of the width ET about the