- #1
Karthiksrao
- 68
- 0
Dear all,
In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modeled as a mass connected to neighbouring atoms through springs). He gives the green function as
## G = \frac{2}{\omega^2 + \sqrt{\omega^4 + 4 k \omega^2}} ##
From this the intention is to find the spectral density of states. He directly gives it as
## \rho = \frac{1}{2 \pi} \frac{\sqrt{\omega^4 + 4 k \omega^2}}{2 k \omega^2} ##.
I have not been able to show this.
Earlier in the chapter he uses the standard representation for the spectral density of states as
## G - G^* = 2 \pi \rho ##.
Using this I attempted the following way. Let ## \omega^2 = z ## (he had used such a representation earlier, hence I tried this), which gives:
## G(z) = \frac{2}{z + \sqrt{z^2 + 4 k z}} ##
which can also be written as:
## G(z) = \frac{ z - \sqrt{z^2 + 4 k z}}{- 2 k z} ##
Because of the ## \sqrt{z}## factor there is a branch cut along negative real axis. That's probably the only reason why ## G - G^*## would have a non-negative value - since there would be a non-zero difference in the values across the branch cut.
So what we need is :
## \lim_{\delta \rightarrow 0} G(z+ i \delta) - G(z - i \delta) ##
This gives:
## \rho = \lim_{\delta \rightarrow 0} \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k}{z+ i \delta}} - 1}{ 2 k } - \frac{ \sqrt{1 + \frac{4 k }{z - i \delta}} - 1}{ 2 k } \right) = \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k }{z+ i \delta}} - \sqrt{1 + \frac{4 k }{z - i \delta}}}{ 2 k } \right) ##
If you take the limit, this just goes to zero.
What am I doing wrong?
Thanks!
In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modeled as a mass connected to neighbouring atoms through springs). He gives the green function as
## G = \frac{2}{\omega^2 + \sqrt{\omega^4 + 4 k \omega^2}} ##
From this the intention is to find the spectral density of states. He directly gives it as
## \rho = \frac{1}{2 \pi} \frac{\sqrt{\omega^4 + 4 k \omega^2}}{2 k \omega^2} ##.
I have not been able to show this.
Earlier in the chapter he uses the standard representation for the spectral density of states as
## G - G^* = 2 \pi \rho ##.
Using this I attempted the following way. Let ## \omega^2 = z ## (he had used such a representation earlier, hence I tried this), which gives:
## G(z) = \frac{2}{z + \sqrt{z^2 + 4 k z}} ##
which can also be written as:
## G(z) = \frac{ z - \sqrt{z^2 + 4 k z}}{- 2 k z} ##
Because of the ## \sqrt{z}## factor there is a branch cut along negative real axis. That's probably the only reason why ## G - G^*## would have a non-negative value - since there would be a non-zero difference in the values across the branch cut.
So what we need is :
## \lim_{\delta \rightarrow 0} G(z+ i \delta) - G(z - i \delta) ##
This gives:
## \rho = \lim_{\delta \rightarrow 0} \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k}{z+ i \delta}} - 1}{ 2 k } - \frac{ \sqrt{1 + \frac{4 k }{z - i \delta}} - 1}{ 2 k } \right) = \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k }{z+ i \delta}} - \sqrt{1 + \frac{4 k }{z - i \delta}}}{ 2 k } \right) ##
If you take the limit, this just goes to zero.
What am I doing wrong?
Thanks!