Here we will consider the problem of Cooper pair with spin-dependent effective masses. This can occur in certain materia

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Here We Will Consider The Problem Of Cooper Pair With Spin Dependent Effective Masses This Can Occur In Certain Materia 1 (96.82 KiB) Viewed 25 times

Here we will consider the problem of Cooper pair with spin-dependent effective masses. This can occur in certain materials under magnetic field, and has interesting consequences. In this case the centre-of-mass (COM) motion of the Cooper pair becomes non-zero and we wish to analyse this. Recall, that the electrons which scatter (interact) by the potential Vk are of opposite momenta k,-k scattering to k', -k'. Assume throughout that r,',k,k' are vectors and that the energy dispersion is eq for the wave numbers q = Q. k, k' and their respective 2m masses. 1. The COM motion was assumed to be zero in the derivation that we did in the lecture, and we only had the relative momentum k = (kı – k2)/2. Assume that the masses are equal mı = m2. The the pair wave function, for the case where COM momentum hQ, where Q = k1 + ka, is non-zero looks like U(R,r) = iQ-R (1) gheter Show how this form can be derived. (hint: review the central potential problem in Quantum Mechanics) [10 marks] 2. The gap equation which we derived in the lecture was 1 V 1 2€ - E A>ke where the k is as defined above, and the upper limit of the sum is when € > Ep + We. Show that if you add the COM kintic energy term 2/2M to the Hamiltonian, the Shrödinger equation in momentum space will have a term €Q9k added to it. Show then how this will modify the resulting gap equation above? (hint: follow the derivation of the gap equation that we did for the Cooper pairs). [10 marks a 3. Consider now the case where that the masses of the two electrons in the Cooper pair are different mı + m2. Which values in k-space the individual momenta of the electrons in the pair can have? Explain for both the case of Q = 0 and Q +0. (hints: assume that €k = € = Er, m, 7 m2, Q = ki + ky. it is also recommended to draw a 2D sketch of the Fermi sphere.) [10 marks] =