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 region7.2. 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
exists only on the left side.
In the left free-space region it is
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:
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
containing single-particle states,
and find their decay rate through the QPC into the
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 wall7.5:
this corresponds to a uniform density of states in -space given by
When a potential difference is applied across the QPC, the energy
range carrying current is
which we can equate
using the dispersion relation.
This can be used with (7.10) to write the conductance
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 .