Biomedical Engineering Reference
In-Depth Information
Fig. 1.8
The model of a
ballistic conductor
contact
contact
T
Lead 1
Lead 2
conductor
E
FL
E
FR
where M is the number of subbands in the leads. The current that flows between
contacts at low temperatures is then I
D
.2e
2
=h/
MTV
.
In this case, the total resistance between contacts, R
D
h=.2e
2
MT
/, can be
written as a sum between h=.2e
2
M/, which has the meaning of contact resistance,
and R
s
D
h.1
T/=.2e
2
MT
/, which is the resistance of a scatterer with transmission
T . In the same way, the resistance of a succession of scatterers with transmissions
T
i
is R
s
D
P
i
R
s;i
,whereR
s;i
D
h.1
T
i
/=.2e
2
MT
i
/. This formula suggests
that a series of scatterers is equivalent to a single scatterer with total transmission
probability given by .1
T/=T
D
P
i
.1
T
i
/=T
i
, expression that results from
adding all partially transmitted waves.
We have assumed up to now that the Fermi-Dirac distribution f.E/ can be
approximated with a step function, but this assumption does not hold at higher
temperatures (see Fig.
1.2
), case in which the electrons contributing at electrical
conduction have energies in the range E
FR
E<E<E
FL
C
E,whereE is
afewk
B
T . Then, the current flowing between the left and right contacts, which have
respective Fermi-Dirac quasi-distribution functions f
L
.E/ and f
R
.E/,isgivenby
Datta
(
1997
)
Z
M.E/T.E/Œf
L
.E/
f
R
.E/dE:
2e
h
I
D
(1.14)
From (
1.14
), it follows that the calculation of the transmission probability is
essential for current estimation using the Landuaer formula. The transmission
probability can be computed using the transfer Hamiltonian formalism, the Green's
function approach, or the Kubo formalism (
Datta 1997
;
Ferry and Goodnick 2009
),
but the easiest method to determine it is the matrix formalism, described below.
The transmission probability can be calculated once the electron wavefunctions
are known. In the simplest case, when the 1D ballistic conductor is composed
of a succession of several regions with constant but different electron effective
masses and potential energies, which extend along the x direction, the solution
of the Schrodinger equation (
1.2
)intheith region, ‰
i
.x/
D
A
i
exp.ik
i
x/
C
B
i
exp.
ik
i
x/, can be regarded as a superposi
tion of waves
that propagate forward
and backward with wavenumbers k
i
D„
1
p
2m
i
.E
V
i
/. Continuity conditions
require that at each interface between layers i and i
C
1, situated at x
D
x
i
,as
shown in Fig.
1.9
, the wavefunction and (@‰=@x/=m
˛C1
are constant.
If ˛
D
0 in (
1.2
), these requirements connect the wavefunction components on
each side of the interface via a transfer matrix
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