Conductance in terms of cross section

We consider scattering of a single-quasiparticle wavefunction
from the general 2-terminal system described in the
Introduction (see Fig. 7.1a).
The Hamiltonian is
,
for a quasiparticle mass .
The elastic scattering potential
completely defines the system.
We imagine a monochromatic unit plane wave
incident
from the free-space left-hand region^{7.2}.
The wavevector is
in polar coordinates,
being the angle of incidence.
The free-space wavevector magnitude is taken as
(corresponding to a total energy
equal to the Fermi energy),
unless stated otherwise.

We are at liberty to choose our definition of the `unscattered' wave
.
We take it to be the wavefunction
which would result from reflection of the incident wave
off a wall uniform in the direction.
We can imagine creating such a wall by replacing the `system box' shown
in Fig. 7.1a by the surrounding -invariant
wall profile.
Note that
exists only on the left side.
In the left free-space region it is

where the change in reflected wave exists only on the left side, and the new transmitted wave exists only on the right. These scattered waves have the asymptotic ( and ) forms of 2D scattering theory (see [128], and Chapter 7 of [174])

See Fig. 7.1a for definitions of and .

The transmission cross section
is the ratio of
, the transmitted particle flux (number per unit time), to
, the incident particle flux per unit length
normal to the incident beam:

where the line integral encloses the entire transmitted wave, and the (rightwards-pointing) surface normal is . Applying this and (7.4) to the asymptotic form gives

familiar from scattering theory apart from the restriction to the right half-plane. There is a corresponding form

for the reflective cross section (removal from the unscattered wave without being transmitted).

We will calculate the conductance by
assuming the chemical potential is slightly higher on the left side
than the right,
and as is usual[20,55]
consider only the left-to-right transport of the states in this
narrow energy range.
We take the left region to be a large () closed region
of area
containing single-particle states,
and find their decay rate through the QPC into the
right side.
Semiclassically each single-particle state occupies a
phase-space volume , where we have .
Therefore the phase-space density in the 2DEG Fermi sea
is where the factor of 2 comes from the spin degeneracy.
We can project this density onto momentum space
in order to find the effective
number of plane-wave states impinging on the wall^{7.5}:
this corresponds to a uniform density of states in -space given by

(7.8) |

(7.9) |

where the last step incorporated the linear-response assumption that is constant over the range .

When a potential difference is applied across the QPC, the energy
range carrying current is
,
which we can equate
with
using the dispersion relation.
This can be used with (7.10) to write the conductance

where the particle wavelength is . The latter form is written in terms of the angle-averaged cross section at the Fermi energy. The weighting of this average is uniform because of the ergodic assumption that incoming states are uniformly distributed in angle.

Eq.(7.1) is a key result of this chapter. Like the Landauer formula, it directly connects conductance and scattering. In a scattering measurement from the left side, appears to be the QPC's inelastic cross section (since the transmitted waves never return to this side). In a current measurement the corresponding conductance is given by (7.1). An independent verification is provided by the result (L.4) of Appendix L, when combined with the Landauer formula Eq.(7.14). Our derivation was for temperature , but it applies at a finite as long as does not change significantly over the energy range . This can be seen by generalizing the above to include integration over the Fermi distribution.

In the limit where a QPC is adiabatic, its conductance is known to be quantized [193,20,65]: where is the integer number of open channels at the Fermi energy. Looking at (7.11), this corresponds to quantization of the angular integral of the cross section in units of .