Partial-wave channel modes for a 2-terminal system

In traditional scattering theory, partial waves form a basis in which to decompose the asymptotic ( $r \rightarrow \infty$) form of the full wavefunction $\psi$ into incoming and outgoing states of definite angular momentum $l$. The basis functions are the cylindrical (in 2D) solutions to the free-space wave equation; the $S$-matrix which takes incoming to outgoing waves can then be written in this basis [128]. Because there is only a single set of incoming channels and a single set of outgoing channels, this is equivalent to a scattering system (a stub) connected to a single `lead', with an infinite number of open channel modes. This contrasts the open two-terminal geometry we study, where we need to account for two new facts: 1) in the $r \rightarrow \infty$ limit the potential $V$ no longer becomes angular-momentum-preserving, and 2) there are now distinct ways the particle can enter and exit the system, via different leads.

We define a `half-plane partial-wave basis' as the subset of the cylindrical free-space solutions which go to zero on the entire $y$-axis. This gives independent basis functions existing on either the left or right side of the $y$-axis. The basis is expressed in terms of Hankel functions[7] on either side:

$$\phi^{-L}_l (kr) \; \equiv \; H^{(2)}_l (kr) \sin [ l(\mbox{\small$\frac{\pi}{2}$}- \theta )]$$

$$\phi^{+L}_l (kr) \; \equiv \; H^{(1)}_l (kr) \sin [ l(\mbox{\small$\frac{\pi}{2}$}- \theta )]$$

$$\phi^{-R}_l (kr) \; \equiv \; H^{(2)}_l (kr) \sin [ l(\mbox{\small$\frac{\pi}{2}$}- \theta' )]$$

$$\phi^{+R}_l (kr) \; \equiv \; H^{(1)}_l (kr) \sin [ l(\mbox{\small$\frac{\pi}{2}$}- \theta' )]$$ (7.13)

where on the left ($L$) side $\theta$ is the angle from the negative $x$-axis and on the right ($R$) side $\theta'$ is the angle from the positive $x$-axis (see Fig. 7.1a). The channel index is $l = 1,2, \cdots \infty$, and $+$($-$) refers to outgoing (incoming) travelling waves. We note that the s-wave $l=0$ is excluded because of the $y$-axis barrier, leaving the first channel as the p-wave $H_1(kr)\cos(\theta)$. Assuming the width of the barrier is finite and constant as $\vert y\vert \rightarrow \infty$ (see Fig. 7.1a), then any wavefunction in the $r \rightarrow \infty$ limit can be written as a sum of the above basis functions (this is discussed further in Appendix L). The separability of this basis in $(r,\theta)$ is directly analogous to the separability of conventional (constant-width) lead basis states [129,130,41] into a product of transverse modes and longitudinal travelling waves.

The corresponding outgoing (incoming) amplitude coefficients are $p^+_l$ ($p^-_l$) on the left and $q^+_l$ ($q^-_l$) on the right. Applying (7.5) to the aymptotic forms of the Hankel functions $H_l^{(1,2)}(x) \approx (2/\pi x)^{1/2} e^{i (x + {\mbox{\tiny phase}})}$ shows that a unit amplitude carries a flux of $\hbar/m$, independent of channel index $l$. This results from the fact that all channels have the same asymptotic particle velocity, equal to that in free space. This contrasts the usual constant-width lead case where the non-zero transverse energy results in an $l$-dependent asymptotic velocity, and thus care is needed to include a velocity factor in the normalisation [74,187,55].

Now we give a brief reminder of the recipe for the conductance of a general completely-open single channel [20,65,55]. The channel is a 1D Fermi gas corresponding to the longitudinal degree of freedom. This can correspond to a conventional (constant-width) transverse mode, to a Hankel (cylindrical) mode defined above, or indeed to any situation where the Hamiltonian is separable into (orthogonal) transverse modes and longitudinal motion in the asymptotic regime ^7.6. For a 1D Fermi gas the rightward-moving density of states per unit length (including spin) is $\rho (E) = (\pi \hbar v)^{-1}$, where $v$ is the (group) velocity in the channel. The current carried is $\delta I = e v \, \rho(E)\, \delta E$ in the relevant energy range $\delta E = e \, \delta V$, giving a single `quantum' of conductance $G \equiv \delta I/\delta V = 2e^2/h$. The important point is that the velocity factors cancel, giving the equipartition rule that an equal current is carried by all open channels regardless of their channel velocities.

If the basis is chosen such that unit amplitude coefficients carry equal fluxes in all incoming and outgoing channels, then flux conservation implies the unitarity of the $S$-matrix when written in this basis. Then the results of the previous paragraph generalise to the familiar Landauer formula

$$G \; = \; \frac{2e^2}{h} \mbox{Tr}( t^\dag t) ,$$ (7.14)

in the case of general transmission matrix $t$ [187,55]. Our partial-wave basis (7.13) has this required flux normalisation. The transmission matrix (from incoming left to outgoing right channels) is defined by $q^+_l = \sum_m t_{lm} p^-_m$. Thus we have shown that the partial-wave basis, appropriate for open 2-terminal systems, is equivalent to the conventional transverse lead mode basis as far as the Landauer formula is concerned. Note that is possible to `mix and match' different basis set types (for instance define a transmission matrix between transverse lead modes on the left side and partial-wave modes on the right), as long as equal-flux normalisation and orthogonality are preserved.

Figure 7.2: a) Transmission cross section $\sigma_{{\mbox{\tiny T}}}(k,\phi)$ as a fraction of the classical normal-incidence cross section $2a$, vs dimensionless slit size $2ka$, for our idealised slit QPC. Four different incident angles $\phi$ are shown (each curve is labelled with its corresponding $\phi$). b) The incident-angle-averaged transmission cross section as a fraction of $2a$ (solid line), compared to its small $ka$ (dotted line) and large $ka$ (dashed line) asymptotic forms. The conductance is proportional to $\langle \sigma_{{\mbox{\tiny T}}}(k,\phi)\rangle$, and would show very weak quantization steps.