Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

3.2 Introduction to quantum dots and various derivations 59
theory. Then, we treat the charging properties by extending the theory to the
few-electron case using the constant interaction model. This simple but powerful
model provides a basic understanding of a few-electron dot and enables one to
comprehend the effects of SO interaction which are reviewed later in this chapter.
The one-electron spectra of a QD are well described by means of the Fock–
Darwin theory. This exactly solvable model assumes a parabolically confined QD
in two dimensions with perpendicular magnetic field (B = Bê z ),
(
p +
e
H 0 = c
A ) 2
+ m 2m 2
0( ω2 x 2 + y 2) , p = (p x , p y ,0). (3.10)
To understand the effect of SO coupling which we shall present in Sec. 3.2.3, we
derive the spectrum of Hamiltonian (3.10) in some detail. By choosing symmetric
gauge for the vector potential, A = (−y,x,0)B/2, we can rewrite Eq. (3.10) as
H 0 = p2 x + p 2 y
2m
+ m 2 ˜ω2( x 2 + y 2) + ω c
2 (xp y − yp x ), (3.11)
with ω c = eB/mc and ˜ω 2 = ω 2 0 + ω2 c/4. We now define oscillator operators,
a x = √ 1 (
2
x˜l + i ˜l )
p x , a † x = √ 1 (
2
x˜l − i ˜l )
p x , [a i ,a † j ] = δ i j, (3.12)
where ˜l = (/m˜ω) 1/2 (y-component is analogously defined). This leads to
(
)
H 0 = ˜ω a † xa x + a † ya y + 1 + ω ( )
c
a y a † x − a x a † y . (3.13)
2i
The introduction of
a ± = 2 − 1 2 (ax ∓ ia y ) (3.14)
decouples x and y oscillators into eigenmodes of frequency ω ± = ˜ω ± ω c /2,
(
H 0 = ω + n + + 1 ) (
+ ω − n − + 1 )
, n ± = a †
2
2
±a ± . (3.15)
Although this representation of the Hamiltonian is convenient to discuss the effect
of SO coupling in the next section, it is customary to rewrite the decoupled modes
in terms of radial and angular momentum quantum numbers,
n := min(n + ,n − ), l := n − − n + , (3.16)
leading to a spectrum known as the Fock–Darwin states [123, 124],
E nl = (2n + |l| + 1)˜ω − 1 2 l ω c. (3.17)