Idealized `slit' aperture point contact

We consider an idealized QPC model: the potential $V$ is zero everywhere except along a hard, thin wall where it is taken as infinite. (This corresponds to the reflection phase shift $\gamma_{\bf k}= 0$). The `scattering system' is an aperture (hole) in this wall, of width $2a$ centered at the origin (see Fig. 7.1b). The desired solutions to the Schrodinger equation in this system are solutions of the Helmholtz equation corresponding to Dirichlet boundary conditions on the wall. This is a paradigm system for the study of classical particle conductance[179], and because of the sharp edges it forms a very non-adiabatic point contact. However, in mesoscopic 2DEG systems where depletion regions are defined electrostatically by gates the potential cannot possess structures sharper than about $1/k_F$ due to screening[20,55]. Therefore as a model of real 2DEG systems, it is approximate.

In the limit of a small aperture ($ka \ll 1$) the transmission cross-section scales as $\sigma_{{\mbox{\tiny T}}}\sim a^4$ for constant $k$, and is dominated by p-wave (dipole) scattering. This is derived in Appendix K, including the correct prefactor. The result is

$$\sigma_{{\mbox{\tiny T}}}(k,\phi) \; = \; \frac{\pi^2}{16} k^3 a^4 \cos^2\phi . \hspace{0.6in} ka \ll 1.$$ (7.15)

Substituting this into the conductance formula (7.12) yields

$$G \; = \; \frac{2 e^2}{h}\cdot \frac{\pi^2}{64} (k a)^4 , %% \ll \; \frac{2 e^2}{h}, \hspace{0.6in} ka \ll 1 ,$$ (7.16)

which is much less than $2e^2/h$ so corresponds to a tunneling junction, and vanishes as the aperture closes. There is another way to compute the conductance, using the transmission matrix $t$ of Section 7.2. Only the dipole-dipole element $t_{11}$ will be significant (argued in Appendix K). Because transmission is very small, a unit incoming channel amplitude $\phi_1^{-L}$ will reflect with nearly unit amplitude into $\phi_1^{+L}$. The $x$-gradient of this combined wavefunction a small distance to the left of the QPC will therefore [using (7.13)] be $c_1 = k$. Substituting this into Eq.(K.4) gives the transmitted amplitude $q_1^+$, corresponding to a matrix element

$$t_{11} \; = \; \frac{i\pi}{8} \, (ka)^2 \hspace{0.6in} ka \ll 1.$$ (7.17)

Reassuringly, insertion of a transmission matrix with this single non-zero element into the Landauer formula Eq.(7.14) gives the same conductance as the cross section based derivation, Eq.(7.16).

The limit of a large aperture ($ka \gg 1$) is the semiclassical limit, and the conductance is that of a classical gas (first studied in 3D by Sharvin [179].) The cross section is a purely classical quantity (independent of $k$), being simply the projection of the aperture length $a_{{\mbox{\tiny eff}}}= 2a$ onto the incident beam direction:

$$\sigma_{{\mbox{\tiny T}}}(k,\phi) \; = \; a_{{\mbox{\tiny eff}}}\cos(\phi) \hspace{0.6in} \mbox{classical}.$$ (7.18)

Integrating over incident direction gives

$$\int_{-\pi/2}^{\pi/2} \! \! d\phi \, \sigma_{{\mbox{\tiny T}}}(k,\phi) \; = \; 2 a_{{\mbox{\tiny eff}}},$$ (7.19)

which with (7.11) gives the conductance $G = (2e^2/h)N$ where $N = 2a_{{\mbox{\tiny eff}}}/\lambda_{{\mbox{\tiny F}}}$ is the number of half-wavelengths spanning the aperture. Note that this result is identical to that of an arbitrarily-long uniform scattering-free channel of the same width. When we are no longer in the semiclassical limit, or when we consider a general open 2-terminal system, (7.19) serves as a definition of the effective classical area $a_{{\mbox{\tiny eff}}}$. Namely, $a_{{\mbox{\tiny eff}}}$ is the slit aperture width whose classical conductance is equal to that of a given quantum system.

The exact result at arbitrary $ka$ is also known. We refer the reader to Chapter 11.2 of [148] and [144] for the detailed form of the wavefunctions. They can be expressed as Mathieu functions [1] in elliptical coordinates $(\mu,\alpha)$ defined by $x = a \sinh \mu \cos \alpha$ and $y = a \cosh \mu \sin \alpha$. The Mathieu function expansion of the incoming wave is also needed. Calculation of $\sigma_{{\mbox{\tiny T}}}(k,\phi)$ requires summing the transmitted flux of the lowest few transverse Mathieu functions [144].

We apply this scheme to generate Fig. 7.2a, which shows the cross section $\sigma_{{\mbox{\tiny T}}}(k,\phi)$ for arbitrary width of the aperture and for various incident angles $\phi$. Fig. 7.2b shows the angular average $\langle \sigma_{{\mbox{\tiny T}}}(k,\phi) \rangle_\phi$ (our plot differs from that of [144] only in that we show cross section as a fraction of the normal-incidence classical cross section $2a$). This plot also shows asymptotic convergence to the small- and large-$ka$ results presented above, and gives an idea of when breakdown occurs.

The oscillations in $\langle \sigma_{{\mbox{\tiny T}}}\rangle_\phi$ have the same period as the quantization steps in an adiabatic QPC of the same minimum width, but are much weaker, even though we are at zero temperature: this is because our QPC is very nonadiabatic [189,206].