- #1
Matthew_
- 5
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- TL;DR Summary
- I am failing to understand why it is necessary to correct the jellium model eigenvalues due to electronic screening. A similar process is done for screening due to oscillating ions: in this case, electron phonon-interaction can even lead to a change in the sign of the exchange term, due to "overscreening" (A. &M.). This is said to be the same principle behind the formation of cooper pairs, and I don't understand why no instability of the Fermi sea is predicted here.
In the Jellium model, it is customary to evaluate the exchange term of the Hartree-Fock equation for plane waves ##\varphi_{\mathbf{k}_i}## as a correction to the energy of the non-interacting electron gas obtaining $$\hat{U}^{ex} \varphi_{\mathbf{k}_i}=-e^2 \left( \int \dfrac{\mathrm{d}^3k}{2 \pi^2} \dfrac{1}{\| \mathbf{k}+\mathbf{k}_i \|^2} \right) \varphi_{\mathbf{k}_i}$$
This correction always lowers the energy as the integral is surely positive.
In some textbooks (e.g. Ashcroft and Mermin) it is said that the jellium model can be perfected via the introduction of screening effects, assuming the integrand as the Fourier transform of coulomb interaction and thus dividing this term by a certain dielectric function. Formal derivation of this substitution was never introduced in the course and I believe this is the main reason behind my doubts.
At first, this was done by the Thomas-Fermi dielectric function, giving rise to the substitution $$ \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k}_i \|^2} \longmapsto \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k}_i \|^2+k_{TF}^2}.$$ What I struggle to understand here is why it is possible to correct the energy eigenvalues associated with a one-electron wavefunction arguing (as a heuristic picture) that moving electrons screen each other's coulomb potential. The Hartree-Fock equation is derived from the variational principle, and I thought that it was supposed to give the best approximation of the interacting electron's wavefunction assuming it can actually be expressed as a Slater determinant. With this in mind, I would rather expect that all the corrections that do not arise from electron-phonon interaction should be performed on the total energy of the system rather than the actual eigenvalues associated with one-electron states.
Then, after introducing as valid approximation for the total dielectric function in metal the quantity $$\dfrac{1}{\epsilon}=\left( \dfrac{q^2}{q^2+k_{TF}^2}\right) \left( \dfrac{\omega^2}{\omega^2-\omega(q)^2}\right), \qquad \mathbf{q}\equiv \mathbf{k}-\mathbf{k'}, \quad \omega \equiv \dfrac{(\mathcal{E}_\mathbf{k}-\mathcal{E}_\mathbf{k'})}{\hbar}$$ it is argued that screening effects from both moving electrons and oscillating ions lead to the correction $$ \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k'} \|^2} \longmapsto \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k'} \|^2+k_{TF}^2} \left( 1+ \dfrac{\omega(\mathbf{k}-\mathbf{k'})^2}{[(\mathcal{E}_\mathbf{k}-\mathcal{E}_\mathbf{k'})/\hbar]^2-\omega(\mathbf{k}-\mathbf{k'})^2}\right).$$
This is a subtle way of considering the correction of energy eigenvalues due to electron-phonon interaction and phonon-mediated electron-electron interaction. Overscreening happens when the energy separation between electrons is lower than the phonon energy: this changes the sign of the integrand in the exchange term, which makes me think that both repulsive and attractive interactions enter the equation. In the chapter related to superconductivity (A. & M.) it is said that the very same condition is linked to the rise of an attractive interaction between electrons. I know that when an attracting pair of electrons is introduced at the fermi level (Cooper problem), one finds that this leads to instability of the fermi sea, as a bound state between paired electrons is favorable. Why is it that the previous substitution can account for phonon-electron interactions while resulting in roughly the same ground state of the free electron model (and obviously not predicting the formation of cooper pairs)?
I know that these questions may sound naive, but the lecturer rushed this topic with no satisfactory explanation and I am just confused about the actual physical meaning of these corrections.
This correction always lowers the energy as the integral is surely positive.
In some textbooks (e.g. Ashcroft and Mermin) it is said that the jellium model can be perfected via the introduction of screening effects, assuming the integrand as the Fourier transform of coulomb interaction and thus dividing this term by a certain dielectric function. Formal derivation of this substitution was never introduced in the course and I believe this is the main reason behind my doubts.
At first, this was done by the Thomas-Fermi dielectric function, giving rise to the substitution $$ \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k}_i \|^2} \longmapsto \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k}_i \|^2+k_{TF}^2}.$$ What I struggle to understand here is why it is possible to correct the energy eigenvalues associated with a one-electron wavefunction arguing (as a heuristic picture) that moving electrons screen each other's coulomb potential. The Hartree-Fock equation is derived from the variational principle, and I thought that it was supposed to give the best approximation of the interacting electron's wavefunction assuming it can actually be expressed as a Slater determinant. With this in mind, I would rather expect that all the corrections that do not arise from electron-phonon interaction should be performed on the total energy of the system rather than the actual eigenvalues associated with one-electron states.
Then, after introducing as valid approximation for the total dielectric function in metal the quantity $$\dfrac{1}{\epsilon}=\left( \dfrac{q^2}{q^2+k_{TF}^2}\right) \left( \dfrac{\omega^2}{\omega^2-\omega(q)^2}\right), \qquad \mathbf{q}\equiv \mathbf{k}-\mathbf{k'}, \quad \omega \equiv \dfrac{(\mathcal{E}_\mathbf{k}-\mathcal{E}_\mathbf{k'})}{\hbar}$$ it is argued that screening effects from both moving electrons and oscillating ions lead to the correction $$ \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k'} \|^2} \longmapsto \int \dfrac{\mathrm{d}^3k}{(2 \pi)^3} \dfrac{4 \pi e^2}{\| \mathbf{k}+\mathbf{k'} \|^2+k_{TF}^2} \left( 1+ \dfrac{\omega(\mathbf{k}-\mathbf{k'})^2}{[(\mathcal{E}_\mathbf{k}-\mathcal{E}_\mathbf{k'})/\hbar]^2-\omega(\mathbf{k}-\mathbf{k'})^2}\right).$$
This is a subtle way of considering the correction of energy eigenvalues due to electron-phonon interaction and phonon-mediated electron-electron interaction. Overscreening happens when the energy separation between electrons is lower than the phonon energy: this changes the sign of the integrand in the exchange term, which makes me think that both repulsive and attractive interactions enter the equation. In the chapter related to superconductivity (A. & M.) it is said that the very same condition is linked to the rise of an attractive interaction between electrons. I know that when an attracting pair of electrons is introduced at the fermi level (Cooper problem), one finds that this leads to instability of the fermi sea, as a bound state between paired electrons is favorable. Why is it that the previous substitution can account for phonon-electron interactions while resulting in roughly the same ground state of the free electron model (and obviously not predicting the formation of cooper pairs)?
I know that these questions may sound naive, but the lecturer rushed this topic with no satisfactory explanation and I am just confused about the actual physical meaning of these corrections.