Why do the electron and hole currents vanish in equilibrium in a semiconductor?

In equilibrium, there can be no net current in a semiconductor. Accounting for both drift and diffusion current, the following relationships can be derived relating the electron density, $N_{e}$, the hole carrier density, $N_{h}$, and the electrostatic potential, $varphi$:

$$
N_{e}propto e^{frac{qvarphi}{kappa T}}
$$
$$
N_{h}propto e^{-frac{qvarphi}{kappa T}}
$$

where $q$ is the electron charge, $kappa$ is the Boltzmann constant and $T$ is the temperature. However, this relies on assuming that both the electron and hole currents separately vanish. Why should this be the case? Can’t there be a net movement of electrons and holes in the same direction in equilibrium so as to cancel out the overall current?

Thanks in advance for any help.