# Coupling to an electric field in a tight binding model

1. Feb 14, 2015

### Qturtle

Hi
i'm looking for some references (prefer books) or explanations as to how one couple electrons so an EM field in a second quantized formalism tight binding model.
from what i know, one need to replace the hopping parameter with the same parameter multiplied by an exponent of the line integral of the vector potential along the hopping path. this is called the peierls substitution. after that in order to find the current - one need to calculate the derivative of the Hamiltonian with respect to the vector potential.
Can someone refer me to a book where this formalism is explained, or maybe explain here? i actually need this for a work i'm doing. I know that this is the method to couple to an EM field and finding the resulted current but i don't really understand why.

thank you very much!

2. Feb 14, 2015

### DrDu

http://www.wsi.tum.de/Portals/0/Med...883d-474a-b8a0-7569651515fb/PRB51_4940_95.pdf
and this I find also helpful:
http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1509&context=nanopub

Last edited: Feb 14, 2015
3. Feb 14, 2015

### Qturtle

Thanks DrDu
i'm actually looking for a good derivation of the peierls substitution (Eq. 9 and rf. 16 in the first paper)
I wasn't able to find the original paper of peierls though ):

4. Feb 16, 2015

### DrDu

Don't equation 9 and 10 of Vogl and Graf suffice as a derivation?

5. Feb 16, 2015

### Qturtle

It is if you understand why an exponent of a line integral of A(x) can transform a function of p to a function of p-eA(x)

6. Feb 16, 2015

### DrDu

Equation 9 is an exact relation for any function of p and x. Just use $p=\frac{\hbar}{i} \partial_\vec{r}$.

7. Feb 16, 2015

### Qturtle

yes I can see now that this is more general. Thanks