Chapter 7: Conductance of quantum point contacts and half-plane scattering

The quantum point contact (QPC)[20,65] has played a central role in the understanding of mesoscopic conductance. It is the simplest example of a 2DEG system where the quantum coherent nature of the electron controls the bulk transport properties. The Landauer-Büttiker (LB) formalism [129,130,41,55,65] reduces the calculation of quantum conductance in the linear response regime to the evaluation of single-particle wavefunction transmission amplitudes. Traditionally, these amplitudes are measured between travelling wave basis states in the `leads'. Far from the scattering system the leads have constant profiles of finite-width, and support a finite number of transverse modes (channels). Eventually the leads are impedance-matched (that is, without reflection) into `reservoirs' which act as thermalized sources of electrons at their respective potentials; these potentials are taken to reflect the measured bias voltage (hence the resulting conductance is between points deep within the reservoirs). Such theoretical constructs have been remarkably successful at describing transport phenomena, for instance conductance quantization [193,20,65], because the scattering systems involved have generally had good lead-to-reservoir matching.

We consider `open' 2-terminal mesoscopic systems,
namely those
where a QPC is *non-adiabatic*
(possessing rapid longitudinal variation in
transverse profile^{7.1} [20]) and has short or
nonexistent leads
(for instance if it suddenly abutts onto the `reservoir' regions),
or those where there can be scattering off nearby
objects in the `reservoir' region.
We call such systems `open' because the fully two-dimensional (2D)
nature of the `reservoirs'
(*i.e. *the surrounding semi-infinite regions of free space) is important,
and therefore they cannot be reduced using the quasi-1D approach described
above.
This includes a variety of recent
mesoscopic experiments, for example the combination of QPCs with nearby
resonator structures [112] or with a nearby
depletion region underneath
an AFM tip [192].
It also includes any QPC system where elastic backscattering from
disorder in the reservoirs is significant, or generally where the
lead-reservoir matching is bad.
In such systems, the conventional quasi-1D picture
does not apply: the scattering system is not coupled to leads
in the usual sense, indeed the
distinction between leads and reservoirs
is no longer clear.
(The quasi-1D approach can be retained by attaching very wide
leads to such systems, following [189],
although this has both numerical and conceptual limitations).
The aim of the present work is to
introduce a 2D scattering approach
which can handle such systems, and to consider some of the consequences
for mesoscopic quantum conductance.
We will stay within the non-interacting quasiparticle picture,
consider zero applied magnetic field, and assume spin degeneracy of 2
throughout.

We imagine a geometry where a 2DEG exists in two semi-infinite half-plane regions, separated by an impenetrable potential barrier which we align with the -axis (see Fig. 7.1a). Our general `QPC scattering system' is any gap in this barrier which allows coupling of the wavefunction on the left and right sides. This gap can be defined by an arbitrary form of the elastic potential, and may include other nearby scattering objects, disorder, etc. (which would all be placed within the box shown in Fig. 7.1a). The only important limitation is that this coupling region (the `system') be of finite extent, so that electrons which leave the system do so via a well-defined terminal: either the left () or the right (). We also assume that the system size is much smaller than both the dephasing length and the momentum relaxation (elastic scattering) length . The former requirement allows treatment using a coherent wavefunction across the system; the latter allows free-space elastic scattering concepts to be applied.

The conventional distinction between `reservoir' and `lead'
is no longer applicable, however
at short distances outside the system
( but and ) the two semi-infinite
free space regions behave like leads, since they support scattering-free
`channels' (see Section 7.2).
At large distances the *same* regions behave as reservoirs:
for ergodicity ensures that the momentum distribution
is uniform in angle, and
for the energy is redistributed to ensure equilibrium at
the relevant (experimentally-measured) chemical potential of each terminal.
In the intermediate region, there is a broad cross-over from lead to reservoir.

In this work we first derive a general relation between
transmission cross section
(a concept we define using scattering in the half-plane)
and conductance for this open geometry, in Section 7.1.
In Section 7.2 we show that partial-wave type states,
defined in the half-plane regions,
can take the place of transverse lead modes in the Landauer formula.
In Section 7.3 we examine
an idealized, highly non-adiabatic QPC
(a hole in a thin hard wall), for which we discuss
the small hole (tunneling), large hole (semiclassical)
and intermediate regimes.
In Section 7.4 we
discuss the maximum conductance through this QPC,
which is reached when a resonator is placed on one side of the QPC.
We find a universal result *regardless* how small the hole is.
This illuminates the findings
of a recent experiment[112] in such an open geometry.
In Section 7.5 we discuss attempts to exceed the universal
quantum of conductance in a single channel.
A reciprocity relation for cross section is derived in Section 7.6,
and the possibility of breaking this
reciprocity, due to a non-thermal reservoir occupation, is described.
We discuss an application to matter-wave `conductance' through
a 3D QPC.
Finally we conclude in Section 7.7.

- Conductance in terms of cross section
- Partial-wave channel modes for a 2-terminal system
- Idealized `slit' aperture point contact
- Point contact coupled to a resonator
- What is the maximum conductance of a single quantum channel?
- Reciprocity and `conductance' of atom waves
- Conclusions