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Cooper Pairs

Electron pairs bound together by interactions with the crystal lattice, crucial for mediating the superconducting state.
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The statement of the theorem

Define the pairing potential Δk\Delta_{\mathbf{k}} in the context of the BCS mean-field approximation. The formation of Cooper pairs is characterized by the condensation of electron pairs with total momentum 0\mathbf{0} and spin 0\mathbf{0}. The gap parameter Δk\Delta_{\mathbf{k}} is defined by the expectation value of the pairing interaction: \nΔk=−∑k′Vk,k′⟨ck′,↑c−k′,↓⟩\Delta_{\mathbf{k}} = -\sum_{\mathbf{k}^{\prime}} V_{\mathbf{k}, \mathbf{k}^{\prime}} \langle c_{\mathbf{k}^{\prime}, \uparrow} c_{-\mathbf{k}^{\prime}, \downarrow} \rangle \nIn the weak coupling limit, this leads to the pairing gap equation, relating Δk\Delta_{\mathbf{k}} to the attractive interaction VV and the density of states N(0)N(0) at the Fermi level.