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

104 M. Pustilnik and L.I. Glazman
As discussed below, for EC ≫ δE the characteristic energy scale relevant
to the Kondo effect, the Kondo temperature TK, is small compared to the
mean level spacing: TK ≪ δE. This separation of the energy scales allows us
to simplify the problem even further by assuming that the conductances of the
dot-lead junctions are small. This assumption will not affect the properties of
the system in the Kondo regime. At the same time, it justifies the use of the
tunneling Hamiltonian for description of the coupling between the dot and
the leads. The microscopic Hamiltonian of the system can then be written as
a sum of three distinct terms,
H = Hleads + Hdot + Htunneling , (17)
which describe free electrons in the leads, isolated quantum dot, and tunneling
between the dot and the leads, respectively. The second term in (17), the
Hamiltonian of the dot Hdot, is given by (15). We treat the leads as reservoirs
of free electrons with continuous spectra ξk, characterized by constant
density of states ν, same for both leads. Moreover, since the typical energies
ω � EC of electrons participating in transport through a quantum dot in the
Coulomb blockade regime are small compared to the Fermi energy of the electron
gas in the leads, the spectra ξk can be linearized near the Fermi level, ξk =
vF k; here k is measured from kF . With only one electronic mode per junction
taken into account, the first and the third terms in (17) havetheform
Hleads = �
αks
Htunneling = �
ξ k c †
αks c αks , ξk = −ξ−k, (18)
αkns
t αn c †
αks d ns +H.c. (19)
Here tαn are tunneling matrix elements (tunneling amplitudes) “connecting”
the state n in the dot with the state k in the lead α (α = R, L for the right/left
lead).
Tunneling leads to a broadening of discrete levels in the dot. The width
Γαn that level n acquires due to escape of an electron to lead α is given by
�
� (20)
Γαn = πν � � t 2 αn
Randomness of single-particle states in the dot translates into the randomness
of the tunneling amplitudes. Indeed, the amplitudes depend on the values of
the electron wave functions at the points rα of the contacts, tαn ∝ φn(rα). For
kF |rL − rR| ∼kF L ≫ 1 the tunneling amplitudes [and, therefore, the widths
(20)] in the left and right junctions are statistically independent of each other.
Morover, the amplitudes to different energy levels are also uncorrelated, see
(7):
t ∗ αn t α ′ n ′ = Γα
πν δ αα ′δ nn ′ , t αn t α ′ n ′ = Γα
πν δ β,1δ αα ′δ nn ′ , (21)
The average value Γα = Γαn of the width is related to the conductance of the
corresponding dot-lead junction