  We employ matrix-product state techniques to numerically study the
zero-temperature spin transport in a finite spin-1/2 XXZ chain coupled to
fermionic leads with a spin bias voltage. Current-voltage characteristics are
calculated for parameters corresponding to the gapless XY phase and the gapped
N\'eel phase. In both cases, the low-bias spin current is strongly suppressed
unless the parameters of the model are fine-tuned. For the XY phase, this
corresponds to a conducting fixed point where the conductance agrees with the
Luttinger-liquid prediction. In the N\'eel phase, fine-tuning the parameters
similarly leads to an unsuppressed spin current with a linear current-voltage
characteristic at low bias voltages. However, with increasing the bias voltage,
there occurs a sharp crossover to a region where a current-voltage
characteristic is no longer linear and the smaller differential conductance is
observed. We furthermore show that the parameters maximizing the spin current
minimize the Friedel oscillations at the interface, in agreement with the
previous analyses of the charge current for inhomogeneous Hubbard and spinless
fermion chains.
