- #1

Conductor

- 2

- 0

## Homework Statement

The figure shows how a resistor in an integrated circuit made of silicon has been created by doping with gallium with the concentration ##5\cdot10^{22} m^{-3}## in an area that since before contained arsenic with the concentration ##1\cdot10^{22} m^{-3}##.

The following questions concern the grey area in the figure.

a) Is the material n-type or p-type?

b) Is it electrons or holes that are minority charge carriers?

c) Calculate the minority- and majority charge carrier concentrations at ##300 K##.

d) Calculate the resistance if the dimensions of the doped are are: length ##100 µm##, width ##1 µm## and thickness ##100 nm##.

e) Calculate the distance between the valence band edge ##E_V## and the Fermi level ##E_F##. Sketch in a figure where ##E_F## is in comparison with ##E_V##, ##E_C## and ##Ei##.

(##E_i## is the intrinsic fermi energy level)

(concentration = density)

## Homework Equations

**Fermi level:**

##E_F=\frac{E_V+E_C}{2} + \frac{kT}{2} \cdot ln \frac{n}{p}##

where ##E_F## is the Fermi energy level, ##E_V## is the energy level of the valence band edge, ##E_C## is the energy level of the conduction band edge, ##n## is the electron concentration and ##p## is the hole concentration.

**Electron concentration ##n## in the conduction band:**

##n=N_C\cdot e^{\frac{E_F-E_C}{k\cdot T}}##

where ##N_C## is the "effective state density" of the conduction band

**Hole concentration ##p## in the valence band:**

##n=N_V\cdot e^{\frac{E_V-E_F}{k\cdot T}}##

where ##N_V## is the "effective state density" of the valence band

Resistivity

Resistivity

##\rho = \frac{1}{e\cdot(\mu_n \cdot n + \mu_p \cdot p)}##

where ##\mu_n## = electron mobility, ##\mu_p## = hole mobility

## The Attempt at a Solution

**a) Is the material n-type or p-type?**

P-type, since the concentration of gallium is higher than the concentration of arsenic.

**b) Is it electrons or holes that are minority charge carriers?**

Electrons, since it's p-type.

**c) Calculate the minority- and majority charge carrier concentrations at ##300 K##.**

Minority charge carriers (electrons):

##n=N_C\cdot e^{\frac{E_F-E_C}{k\cdot T}}##

##k=0.0259 eV## (Boltzmann's constant)

##T=300 K##

I need help with this question. How are the minority and majority charge carrier concentrations related to the concentrations of gallium and arsenic? (They are given as ##5\cdot10^{22} m^{-3}## and ##1\cdot10^{22} m^{-3}##)

I'm guessing one of the equations I posted can be used somehow, but it seems we lack information, since we don't know either ##E_F##, ##E_C## or ##E_V##?

**d) Calculate the resistance if the dimensions of the doped are are: length ##100 µm##, width ##1 µm## and thickness ##100 nm##.**

I'm not sure how to calculate this. I wrote an equation for resistivity above, but that requires knowledge of mobilites of electrons and holes. That equation also doesn't include any geometric variabels which are given in the question.

What is the relation (equation) between geometry (length, width and thickness) and resistivity?

**e) Calculate the distance between the valence band edge ##E_V## and the Fermi level ##E_F##. Sketch in a figure where ##E_F## is in comparison with ##E_V##, ##E_C## and ##Ei##.**

I wrote some equations above that relates these energy levels to each other, but again like in question c) it seems we need more data. I guess I've missed some relation (equation) that can be used in order to answer it. I appreciate if someone can help me find out these relations.

The questions I've answered so far, are they answered correctly? I also appreciate assistance with the questions I couldn't answer. Please see each question for specifics.