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

6.2 Details 99
with the Hamiltonian of the decoupled components
H 0 = E L n L + E R n R + H p + H leads , (6.9)
the tunnelling Hamiltonian between the dots
H T = T c (p + p † ), (6.10)
the phonon system
the Hamiltonian of the leads
H p = ∑ω q a † qa q , (6.11)
q
the electron-phonon coupling
H leads = ∑(ε L k c† k c k + ε R k d† k d k), (6.12)
k
H ep = ∑(α q n L + β q n R )(a q + a † −q), (6.13)
q
and the coupling between the leads and the dots
H ld = ∑(V k c † k s L +W k d † k s R + h.c.). (6.14)
k
The coupling between the two dots is parametrised with the tunnel matrix element
T c . The electronic reservoirs of the leads (H leads ) and the phonon environment
(H p ) are assumed to be in thermal equilibrium. H ep describes the interaction † of
the electrons in the dots with acoustic phonons, α q and β q are the matrix elements
of the interaction potential in the representation of |L〉 and |R〉 (see appendix D).
Introduction of pseudo-spin operators σ x and σ z leads to the effective spin-boson
Hamiltonian (6.1) with ε = E L − E R .
Applying a dc bias between source and drain, the current through the DQD
is determined by the expectation value of the current operator Î = ieT c (p − p † ).
The current as a function of the energy difference of the dot levels consists of
two parts [147, 148], I = I el + I inel . The elastic contribution I el which arises from
resonant tunnelling [181] is dominating the transport around ε = 0. For ε > 0, the
current is determined by inelastic contributions due to spontaneous emission of
acoustic phonons [147, 148].
† The influence of the non-diagonal part of the electron-phonon interaction has been shown to
be negligible [149, 180].