  Stimulated by the recent implementation of a three-port Hall-effect microwave
circulator of Mahoney et al. (MEA), we present model studies of the performance
of this device. Our calculations are based on the capacitive-coupling model of
Viola and DiVincenzo (VD). Based on conductance data from a typical Hall-bar
device obtained from a two-dimensional electron gas (2DEG) in a magnetic field,
we numerically solve the coupled field-circuit equations to calculate the
expected performance of the circulator, as determined by the $S$ parameters of
the device when coupled to 50$\Omega$ ports, as a function of frequency and
magnetic field. Above magnetic fields of 1.5T, for which a typical 2DEG enters
the quantum Hall regime (corresponding to a Landau-level filling fraction $\nu$
of 20), the Hall angle $\theta_H=\tan^{-1}\sigma_{xy}/\sigma_{xx}$ always
remains close to $90^\circ$, and the $S$ parameters are close to the analytic
predictions of VD for $\theta_H=\pi/2$. As anticipated by VD, MEA find the
device to have rather high (k$\Omega$) impedance, and thus to be extremely
mismatched to $50\Omega$, requiring the use of impedance matching. We
incorporate the lumped matching circuits of MEA in our modeling and confirm
that they can produce excellent circulation, although confined to a very small
bandwidth. We predict that this bandwidth is significantly improved by working
at lower magnetic field when the Landau index is high, e.g. $\nu=20$, and the
impedance mismatch is correspondingly less extreme. Our modeling also confirms
the observation of MEA that parasitic port-to-port capacitance can produce very
interesting countercirculation effects.
