  In this letter, carrier transport in graded
Al$\mathrm{_x}$Ga$\mathrm{_{1-x}}$N with a polarization-induced n-type doping
as low as ~ 10$\mathrm{^{17}}$ cm$\mathrm{^{-3}}$ is reported. The graded
Al$\mathrm{_x}$Ga$\mathrm{_{1-x}}$N is grown by metal organic chemical vapor
deposition on a sapphire substrate and a uniform n-type doping without any
intentional doping is realized by linearly varying the Al composition from 0%
to 20% over a thickness of 600 nm. A compensating center concentration of
~10$\mathrm{^{17}}$ cm$\mathrm{^{-3}}$ was also estimated. A peak mobility of
900 cm$\mathrm{^2}$/V$\mathrm \cdot$s at room temperature is extracted at an Al
composition of ~ 7%, which represents the highest mobility achieved in
n-Al$\mathrm{_{0.07}}$GaN with a carrier concentration ~10$\mathrm{^{17}}$
cm$\mathrm{^{-3}}$. Comparison between experimental data and theoretical models
shows that, at this low doping concentration, both dislocation scattering and
alloy scattering are significant in limiting electron mobility; and that a
dislocation density of <10$\mathrm{^7}$ cm$\mathrm{^{-2}}$ is necessary to
optimize mobility near 10$\mathrm{^{16}}$ cm$\mathrm{^{-3}}$. The findings in
this study provide insight in key elements for achieving high mobility at low
doping levels in GaN, a critical parameter in design of novel power electronics
taking advantage of polarization doping.
