Semiconductors, minimum conductivity

[leonmate]

Homework Statement

Show that the minimum conductivity of a semiconductor occurs when n₀ = n_i√(μ_h/μ_e)

Derive an equation for the minimum conductivity

Calculate the (i) minimum and (ii) intrinsic conductivity for silicon. State the assumptions you make and the origin of any material properties used

Homework Equations

Some equations I've dug out of my notes that look revelant:

conductivity, σ = enμ_e + epμ_h = σ_e + σ_h
Number of electrons in conduction band: n₀ = n_iexp(E_i - E_f / kT)

The Attempt at a Solution

I'm a little stuck on where to start with this one,
I know that n_i is the intrinsic carrier concentration, n₀ is the carrier concentration in the conduction band.

μ_h and μ_e are hole and electron mobility. This is the constant of proportionality between the drift velocity and the electric field. So the some of the electrons drift into the conduction band

Seems obvious that conductivity is at a minimum when there is as few electrons as possible in the conduction band, so I guess I need to do something with that. But I really need a hint or two here as I'm struggling to solve this problem. It's due tomorrow! >_<

Cheers

 

[phyzguy]

Do you remember that in a semiconductor n*p = n_i²?

 

[leonmate]

yeah, i have that eq written down, trying to work out where to go from that...

 

[phyzguy]

So write n = n₀ and p = n_i² / n₀, then write σ in terms of n₀ (eliminating n and p), then find the value of n₀ that minimizes σ.

 

[leonmate]

σ = e(n₀μ_e + n_i²/n₀ μ_h)

I'm not really sure what to do with this, also where did n = n₀ come from?

 

[leonmate]

Ah, brainwave...

So write n = n₀ and p = n_i² / n₀, then write σ in terms of n₀ (eliminating n and p), then find the value of n₀ that minimizes σ.

σ = e(n₀μ_e + n_i²/n₀ μ_h)

Then differentiate wrt n and set equal to zero to find the minimum. Rearrange and got it!

 

[phyzguy]

Great!

 

[nishant619]

a. Calculate conductivity σ as function of temperature σ(T) for following semiconductors:Eg=0.4 eV, Nd=1e15 cm-3; Ec-Ed=0.02, Nc=Nv=1e19;
Use temperature from 80K to 500K, step 1K. For mobility, use µ=µoT-0.6 with µo=20,000.

can anybody help me out with this question.