Diffusion current causes drift current?

I was studying about currents in an injected semiconductor (by illumination) from Integrated Electronics by Millman and Halkias. They consider the diffusion hole current in an injected n-type open-circuit uniformly doped semiconductor, assuming that hole concentration is negligible compared to electron concentration, and hence, the hole drift current can be neglected as it is proportional to the hole concentration.

Since the semiconductor does not have any external potential difference, drift current due either charge carrier should be zero. But it is said that the diffusion current density of the holes causes an internal electric field leading to the drift current of electrons, and the relation $$J=\sigma E$$ is used to calculate the field (which is non-uniform) where $$\E\$$ is the electric field caused by the diffusion current density, $$\\sigma\$$ is the conductivity and $$\J\$$ is the diffusion current density which is not constant as it depends on distance from illumination. The net current (which is the sum of hole diffusion current, electron diffusion current and electron drift current) is zero as it is an open circuit. How, physically, does the current density cause an internal electric field?

• $\sigma$ is the conductivity and $\mathscr{E}$ is an externally imposed field expressed in volts per meter. $J$ is just the current density, usually assuming uniform current in the device due to the imposed field intensity and a uniform area from one end to the other (of course.) Open circuit, there cannot be any current flow (except for a very short time after being opened, as there's no closed loop for the current.) There can be gradients in the doping, for example. But I'm not sure you are discussing that situation. Can you add more to your question? – jonk Oct 14 at 2:46
• @jonk I have edited to the question. – Anubhab Das Oct 14 at 5:32