Cooper pairs are one of the models how superconductivity is explained.

What still baffles me is how a vibration of the crystal lattice (the so-called phonon) can interact with the electron (an actual particle), in such a way that it then creates a coupled pair with an other electron…

What is the explanation for this behaviour? What is the maths behind it?

## Best Answer

Phonons aren't particles like electrons or protons are, phonons are quasi particles, these type of particles are just used to describe excitations of a field: in phonons case, phonons are used to describe elementary lattice vibrations which have certain frequency.What are phonons?

Basically Cooper pairs are just pairs of electrons which attract each other due to electron phonon interaction, this interaction is caused by coulomb force, which exists between electrons and lattice (positively charged nucleus(protons)), because lattice vibrates(this vibration is called phonons), phonons affect electron (because electron is moving in changing potential field caused by lattice vibrations), and sometimes electron will absorb that phonon, and will gain some momentum $\vec{q}$. So, in the beginning, electron's state is described using this wavefunction: $|\vec{k_1}\rangle$ where $\vec{k_1}$ represents momentum of the electron, and it's energy is described using this Shrodinger Equation: $$ h|\vec{k_1}\rangle=\epsilon_{k_1}|\vec{k_1}\rangle $$ Where $h$ is hamiltonian for single electron and $\epsilon_k$ is it's energy. Because lattice is vibrating, electron is moving in changing potential field, and sometimes electron will absorb that vibration (electron will absorb phonon), and it will gain momentum and it's state will be described using this wavefunction: $|\vec{k_1}+\vec{q}\rangle$:Electron-Phonon Interaction:After electron gained momentum, it's energy has also changed, and is described using this equation: $$ h|\vec{k_1}+\vec{q}\rangle = \epsilon_{k_1+q}|\vec{k_1}+\vec{q}\rangle $$ It is also possible for electron to emit phonon (i.e. cause lattice to vibrate): electron starts with momentum $\vec{k_1}$ (wavefunction |$\vec{k_1}\rangle$ and then emits phonon with momentum $\vec{q}$ and due to conservation of momentum it loses momentum $\vec{q}$, and it's new wavefunction is $|\vec{k_1}-\vec{q}\rangle$ (and it's energy is $\epsilon_{k_1-q}$):

As you can see electrons can either absorb or emit phonons. Before now we have been considering single electron model (i.e. system with just one electron), now in order to explain how do electron pair forms, we have to add one more electron, as I have already mentioned electrons can exchange momentum with phonons, due to this effect, one electron can emit phonon, which will be absorbed by another electron, i.e. electrons will exchange phonon, this process can be described using this feynmann diagram:

Effective potential (for electrons) in this interaction can be written in this form $$ V = \frac{|M_{\vec{q}}|^2}{(\epsilon_\vec{k}-\epsilon_{\vec{k}+\vec{q}})^2-(\hbar\omega_\vec{q})^2} $$

Where $\omega_q$ is lattice vibration frequency and $M_q$ is probability amplitude of electron phonon absorbtion which momentum $\vec{q}$. As you can see if $|\epsilon_\vec{k}-\epsilon_{\vec{k}+\vec{q}}|<\hbar\omega_\vec{q}$ potential is negative i.e. there exists force which attracts two electrons, but if $|\epsilon_\vec{k}-\epsilon_{\vec{k}+\vec{q}}|\geq \hbar\omega_\vec{q}$ then potential is positive, and because of that, electrons aren't attracted to each other and no cooper pair is formed.

As you can see electron-phonon interaction creates potential which attracts two electrons, i.e. pairs them and forms cooper pairs. The potential equation is $V=\frac{|M_q^2|}{(\epsilon_{\vec{k}}-\epsilon_{\vec{k}+\vec{q}})^2-\hbar^2\omega_{\vec{q}}^2}$, and if the $|\epsilon_\vec{k}-\epsilon_{\vec{k}+\vec{q}}|<\hbar\omega_\vec{q}$ then potential is negative and electrons attract each other and form Cooper pairs.Conclusion: