# Homework Help: Electronic Specific Heat constant

1. Nov 2, 2012

### Sturnn17

1. The problem statement, all variables and given/known data

Hi all. This isn't homework per say, more lack of understanding of something when reading around the notes.My problem is in trying to derive the electronic specific heat constant $γ$ in 2 dimensions.

2. Relevant equations

I know the general formula for specific heat is $c^{el} = {\frac{π^2 g(ε_F) k^2 T}{3}}$

I also derived the density of states in 2D (g(ε)) to be ${\frac{m^*}{ ħ^2 π}}$

And that $c^{el} = γ~T$

3. The attempt at a solution

I know that I could trivially combine my equations together to give $c^{el} ={\frac{π m^* k^2 T}{3ħ^2}}$

However I am confused to what this actually represents, is this the contribution to the specific heat by each electron?

I am looking for $γ$ in units of $J~mol^{-1}~K^{-2}$ or something similar so I would think I need to introduce avogadros number into my equation somehow but I have failed so far.

I tried to include it by working out n (the number of electrons per unit area)

$n = ~^{ε_F}_{0}\int g(ε) dε =~ ^{ε_F}_{0}\int{\frac{m^*}{ ħ^2 π}} dε = {\frac{m^*}{ ħ^2 π}}~ε_F$

$πk^{2}_{F}= {\frac{N}{ 2}}({\frac{2π}{ L}})^2$ => $k_F= (2nπ)^{1/2}$

$ε_F= {\frac{ħ^2 (2nπ)}{ 2m^*}} = {\frac{ħ^2 nπ}{m^*}}$

$n = {\frac{m^*}{ ħ^2 π}}~{\frac{ħ^2 nπ}{m^*}}= n...$ This evidently wasn't very helpful.

Thanks for taking time to read my post and for any input you may have.

Last edited: Nov 2, 2012