  We have developed a simulation system for nanoscale high-electron mobility
transistors, in which the self-consistent solution of Poisson and Schr\"odinger
equations is obtained with the finite element method. We solve the exact set of
nonlinear differential equations to obtain electron wave function, electric
potential distribution, electron density, Fermi surface energy and current
density distribution in the whole body of the device. For more precision, local
dependence of carrier mobility on the electric field distribution is
considered. We furthermore compare the simulation to a recent experimental
measurement and observe perfect agreement. We also propose a graded channel
design to improve the transconductance and thereby the threshold frequency of
the device.
