Thermal properties in mesoscopics: physics and ... - ResearchGate

other types of contacts, it is typically useful to find
the observable for arbitrary T and weight it with ρ(T ),
e.g., ρ(T ) = πGN /[(2e 2 )T √ 1 − T ] for a diffusive contact
(Nazarov, 1994), ρ(T ) = GN /[e 2 T 3/2√ 1 − T ] for
a dirty interface (Schep and Bauer, 1997) or ρ(T ) =
2GN/[e 2 T (1 − T )] for a chaotic cavity (Baranger and
Mello, 1994). This way, the observables can be related
to the normal-state conductance GN of the junction.
Equation (6) yields a boundary condition both for the
”spectral” functions ˆ G R/A and for the distribution functions.
In the absence of superconductivity, we simply
have ˆ G R/A = ±ˆτ3, and the boundary condition for the
distribution functions becomes independent of the type
of the constriction,
j L/T = GN (f L/T
R
L/T
− fL ). (7)
In this case, the two currents can be included in the same
function by defining j(±E) = (j L (E) ± j T (E))/2. This
yields the spectral current through the constriction
j(E) = GN (fL(E) − fR(E)), (8)
where f L/R is the energy distribution function on the
left/right side of the constriction.
Another interesting yet tractable case is the one where
a superconducting reservoir (on the ”left” of the junction)
is connected to a normal metal (on the ”right”)
and the proximity effect into the latter can be ignored.
The latter is true, for example, if we are interested in the
distribution function at energies far exceeding the Thouless
energy of the normal-metal wire, or in the presence
of strong depairing. In this case, the spectral energy and
charge currents are
j L = 2e2
h
j T = 2e2
h
TnML(Tn)θ( Ē)(f L R − f L L ) (9a)
n
Tn(M 1 T (Tn)θ(−Ē) + M 2 T (Tn)θ( Ē)f T R .
n
(9b)
Here Ē = |E| − ∆ and θ(E) is the Heaviside step function,
and the energy-dependent coefficients are
ML(T ) = 2 (2 − T )|E|Ω + T Ω 2
((2 − T )Ω + T |E|) 2
M 1 T (T ) =
M 2 T (T ) =
2T ∆2 4(T − 1)E2 + (T − 2) 2∆2 2|E|
(2 − T )Ω + T |E|
(10a)
(10b)
(10c)
Here we defined Ω ≡ √ E 2 − ∆ 2 . In the tunneling limit
T ≪ 1, we get
j L/T = NS(E)(f L/T
R
L/T
− fL ), (11)
where
NS(E) =
Re
E + iΓ
Γ→0
→ θ(Ē)|E|/Ω
(E + iΓ) 2 − ∆2 (12)
is the reduced superconducting density of states (DOS).
The first form of Eq. (12) assumes a finite pair-breaking
rate Γ, which turns out to be relevant in some cases discussed
in Sec. V.C.1 Unless specified otherwise, we assume
that the superconductors are of the conventional
weak-coupling type and the superconducting energy gap
∆ at T = 0 is related to the critical temperature Tc by
∆ ≈ 1.764kBTc (Tinkham, 1996).
C. Collision integrals for inelastic scattering
The collision integral Iin in Eq. (3) is due to electron–
electron, electron–phonon interaction and the interaction
with the photons in the electromagnetic environment.
1. Electron-electron scattering
For the electron–electron interaction, the collision integral
is of the form
I e−e
coll
= κ(d)
e−e
dωdE ′ Ĩ
α in
ω coll(ω, E, E ′ ) − Ĩout coll(ω, E, E ′
) ,
5
(13)
where α and κ (d)
e−e depend on the type of scattering and
the ”in” and ”out” collisions are
Ĩ in
coll = [1 − f(E)][1 − f(E ′ )]f(E − ω)f(E ′ + ω)
(14a)
Ĩ out
coll = f(E)f(E ′ )[1 − f(E − ω)][1 − f(E ′ + ω)].
(14b)
Electron-electron scattering can be either due to the direct
Coulomb interaction (Altshuler and Aronov, 1985),
or mediated through magnetic impurities which can flip
their spin in a scattering process (Kaminski and Glazman,
2001) or other types of impurities with internal dynamics.
In practice, both of these effects contribute to
the energy relaxation (Anthore et al., 2003; Pierre et al.,
2000). Assuming the electron–electron interaction is local
on the scale of the variations in the distribution function,
the direct interaction yields (Altshuler and Aronov,
1985) Eq. (13) with α = −2 + d/2 for a d-dimensional
wire. In a diffusive wire, the effective dimensionality of
the wire is determined by comparing the dimensions to
the energy-dependent length LE ≡ D/E. The pref-