  We present a general theory for laser-free entangling gates with trapped-ion
hyperfine qubits, using either static or oscillating magnetic-field gradients
combined with a pair of uniform microwave fields symmetrically detuned about
the qubit frequency. By transforming into a `bichromatic' interaction picture,
we show that either ${\hat{\sigma}_{\phi}\otimes\hat{\sigma}_{\phi}}$ or
${\hat{\sigma}_{z}\otimes\hat{\sigma}_{z}}$ geometric phase gates can be
performed. The gate basis is determined by selecting the microwave detuning.
The driving parameters can be tuned to provide intrinsic dynamical decoupling
from qubit frequency fluctuations. The
${\hat{\sigma}_{z}\otimes\hat{\sigma}_{z}}$ gates can be implemented in a novel
manner which eases experimental constraints. We present numerical simulations
of gate fidelities assuming realistic parameters.
