Calculating Phonon Self Energy for Electron-Phonon Interaction in Graphene

[Physicslad78]

Hello guys... Can anyone tell me where I can find a detailed calculation of the phonon self energy for an electron phonon interaction in graphene... The expression I need to get is in T. Ando paper:

Journal of the Physical Society of Japan
Vol. 75, No. 12, December, 2006, 124701

Equation (3.4)...

or even just hiow to get the self energy (phonon) fr an electron phonon interaction using perturbation theory...


Thanks very much

 

[DrDu]

Maybe you can give us the formula, as I do not have access to that journal.
Generally in lowest order in the coupling constant, the self energy is due to the phonon getting annihilated in a process where an electron is excited. At a later time, the electron hops back to the ground state and the phonon re-emerges.
Hence the self energy of an (eventually virtual) phonon of energy omega and momentum p is something like (up to constants) $$g^2 \int dk d \epsilon G(\omega+\epsilon, k+p)G(-\epsilon,-k)$$
where g is the electron phonon coupling constant and G is the electronic Greens function (which should be of Dirac form near the conical points.)

 

[Physicslad78]

Thanks very much for the reply. I am trying to get the self energy for graphene (monolayer). I will give you the formula...I just can't get my greens functions right... The result they get in paper is

$$\Pi$$(q,$$\omega$$)=C$$~\sum_{s,s_{1}}$$ $$\int$$$$\frac{dk}{(2\pi)^2}$$$$\frac{f[\epsilon(s,k)]-f[\epsilon(s_{1},k)]}{\hbar\omega-\epsilon(s,k)+\epsilon(s_{1},k)+i0}$$

where $$f[\epsilon(s,k)]$$ and $$f[\epsilon(s_{1},k)]$$are the fermi dirac statistics for electrons in a state k in bands s and s´(conduction and valence bands) and C are some constants which i managed to calculate. The e-phonon interaction Hamiltonian is:

H=$$\sum c^{+}_{k+q}c_{k}(b_{q}+b^{+}_{-q})$$
where c are electronic operators and b are phonon operators...The self energy is coming from retarded Green functions but i am trying to get matsubara Green function and I obtain strange results and I do not get energies within different bands...Can anyone please help..Thanksss

 

[DrDu]

I managed to have a look at that paper. However, I have never done any Matsubara calculation myself, so that I can offer only very limited help.
First we need the electronic Greens functions. They should be of the form $$(\epsilon-\epsilon_s)^{-1}$$ and $$(\epsilon-\epsilon_s'+\hbar \omega)^{-1}$$ (very schematically and dropping all infinitesimal imaginary constants). We need the product of the two GF's and have to integrate over epsilon.
To do this, the product has to be written as a sum of two partial fractions. When doing this partial fraction decomposition, I get a term $$(\hbar\omega-\epsilon(s',k)+\epsilon(s,k))^{-1}$$ as appears in the denominator of 3.2. The integrals over the partial fractions yield probably the Fermi occupation probabilities (although I would have expected an $$\hbar \omega$$ to appear in one of these probabilities), so that 3.2 does not appear implausible. I hope that helps somehow.

 

[Physicslad78]

Thanks very much for your help. I managed to get the Green functions and drew the Feynman diagrams and discarded the disconnected ones and I got a product of two Green functions of the form


G($$k lambda$$, $$\tau$$-$$\tau1$$)*G($$k+q \lambda1$$, $$\tau1$$).

where (k $$\lambda$$) is a state of electron with momentum k and in valence band $$\lambda$$, and the other one is an a electron in state k+q and conduction band $$\lambda1$$,. of course $$\tau$$ and $$\tau1$$ are the variables of the Fourier Transform.

Now I need to Fourier transform this to the Matsubara frequency. I cannot manage to do this. I know we need the convolution theorem but how do we appoly it here because we have got two times $$\tau$$ and $$\tau1$$. Then I guess we can apply the partial fraction method and use sums over frequencies or Residue theorem. I am happy to get the Fourier transform now as I can continue the rest of the calculations. If you have any suggestions or a method to get this Fourier transform, please let me know..

Thanks in advance again

Regards

 

[DrDu]

Why don't you work direktly with the Greensfunktions and rules in frequency space?
But anyhow, what we want to describe is this: a phonon anihilates into an electron and a hole at time $$\tau_1$$ and resurrects at time $$\tau_2$$ hence both Greens functions depend on the same time difference $$\tau_2-\tau_1$$. Using this time-difference as a new variable, the Fourier transform gives you a convolution of the two Greensfunctions in frequency space, that is, the equivalent of what is written in my post #2.

 

[DrDu]

Last evening I had a look at Fetter, Walecka, "Quantum Theory of Many Body Systems", Dover. It contains the calculation of the self energy Pi_0. The Matsubara Greens functions result principally from the ones of my post #4 upon replacing $$\epsilon$$ with $$i \nu$$ where $$\nu$$ are the Matsubara frequencies (the Fourier transform of tau). The sum over the Matsubara frequencies can be evaluated after partial fraction decomposition, but one has to introduce some exponential convergence factor for the sums to be well defined. They become the occupation probabilities as said above.