Introduction:
A P-N junction is formed when a P-type semiconductor (with excess holes) is brought into contact with an N-type semiconductor (with excess electrons). This junction creates a diffusion potential across it due to the movement of charge carriers from one region to another. The diffusion potential is the potential difference that exists across the junction when there is a concentration gradient of charge carriers.

Explanation:
The diffusion potential across a P-N junction depends on various factors, including doping concentration and band gap. Let's analyze each option to understand why option 'D' is the correct answer.

a) Decrease with increasing doping concentration:
When the doping concentration is increased, the number of charge carriers in the semiconductor increases. As a result, the concentration gradient at the P-N junction decreases. This reduces the diffusion potential across the junction. Therefore, option 'a' is incorrect.

b) Increase with decreasing band gap:
The band gap of a semiconductor material determines the energy required for an electron to move from the valence band to the conduction band. A smaller band gap allows more electrons to jump from the valence band to the conduction band, increasing the concentration gradient at the P-N junction. This leads to a higher diffusion potential across the junction. Hence, option 'b' is incorrect.

c) Doesn't depend on doping concentration:
The doping concentration directly affects the number of charge carriers in the semiconductor material. As mentioned earlier, a higher doping concentration reduces the concentration gradient and, consequently, the diffusion potential. Therefore, option 'c' is incorrect.

d) Increase with an increase in doping concentration:
Option 'd' is the correct answer. An increase in doping concentration leads to an increase in the number of charge carriers in the semiconductor. This results in a higher concentration gradient at the P-N junction, leading to an increased diffusion potential across the junction. Therefore, as the doping concentration increases, the diffusion potential across the P-N junction also increases.

Conclusion:
The diffusion potential across a P-N junction increases with an increase in doping concentration. This is because a higher doping concentration results in a higher concentration gradient, leading to a greater diffusion potential across the junction.