# Ohmic contacts in a diode (metal-semiconductor junction) - contact difference of potential $V_0$ - how to compute

I'm reading "Microelectronics" by Millman, Grabel.

Premise:

The book developed, for a pn step-graded junction, the contact potential as $$V_0 = V_T \text{ln} \frac{p_{p0}}{p_{n0}} = - V_T \text{ln} \frac{n_{p0}}{n_{n0}} \qquad (1)$$ where $p_{p0}$ is the hole concentration in the p-side, $p_{n0}$ is the hole concentration in the n-side, $n_{p0}$ the electron concentration in the p-side and $n_{n0}$ the electron in the n-side.

To develop the previous equation, it made use of the following $$J_p = q \mu_p p \mathscr{E} - q D_p \frac{\text{d} p}{\text{d} x} \\ D_p = \mu_p V_T \\ \mathscr{E}(x) = - \frac{\text{d}V}{\text{d} x}$$ where $\mu$ is the mobility, $\mathscr{E}$ the electric field, $D$ the diffusion constant and $p$ ($n$) the concentration of holes (electrons). For electrons, the formulas are very similar, substituting $p$ with $n$ and changing the sign of the diffusion term in $J_n$.

I was wondering if one could compute the contact potential of a metal-semiconductor junction (terminals of the diode) with the same formula $(1)$. The derivation of the formula $(1)$ does not make any particular assumption regarding the conduction medium (at least in my opinion), provided the Einstein relation is valid for a metal. Is the Einstein relation $\frac{D_n}{\mu_n} = V_T$ valid for a metal conductor? What is a typical value for mobility of electrons in a metal conductor?

In the Example 1.1 the book worked out the electron concentration for a conducting line of an IC chip, that is $4.38 \times 10^{21} ~ \text{cm}^{-3}$. Let's say that the electron concentration of the p-type silicon is a typical $4.2 \times 10^{5}$. Using the formula $(1)$

$$V_0 = − 0,0259× \text{ln} \frac{4,2×10^5}{4,38×10^{21}} = 0,955 \text{V}$$

It seems to be an extremely big number considering that the cut-in voltage for a diode is $0.6 \text{V}$ and that the contact potential for a metal-semiconductor junction should be negligible.

In the same example it states that the mobility of electrons in the conducting line is $500 ~ \text{cm}^2 / (V \cdot s)$. How can be that small considering that the electron mobility in silicon is $1500 ~ \text{cm}^2 / (V \cdot s)$?

Thanks, Luca

• You may benefit by breaking this into two separate posts. The first question "Is the Einstein relation valid for a metal conductor?" should be one post. The question about mobilities should be another. – DanielSank Dec 16 '14 at 18:36
• I put in the same post because they are strictly related. Knowing the mobility of an electron is strictly related with (part of) knowing if the Einstein relation is valid. – the_eraser Dec 16 '14 at 19:11
• I understand and that makes sense. I'm just trying to help you get the most out of this site. You will get more and higher quality answers if you make your questions as simple and specific as possible. This is just advice meant to help you. Take it or leave it. – DanielSank Dec 16 '14 at 20:50