LATHIOTAKIS ET AL.FIGURE 1. Energies vs <strong>for</strong> an s-wave pair-potential with 1 (left panel) and energies vs <strong>for</strong> a pair-potentialwith 0.01 (right panel). Both panels were calculated <strong>for</strong> r s 1 and T 0K.the <strong>functional</strong> <strong>for</strong> an s-wave pairing potential havingthe shape of a Gaussian s k s exp k/k F 1 2 2 (20)where k F is the Fermi momentum, and and aredimensionless parameters. To gain some insightinto the relative importance of the anomalous Hartreeenergy [the energy term associated with theanomalous Hartree potential (8)]f AH 4 31 d3 k d3 k 4 k k2 2 3 k k 2 E k E kthis calculation was set to zero. The dependence ofthe energies on the parameters and turns out tobe rather smooth. The largest positive contributioncomes from the anomalous Hartree term, and isalmost canceled by the RPA correlation energy difference( f RPA f NRPA ). The exchange part is positive,Sbut much smaller (almost an order of magnitude)than the other two terms. The sum of the three ispositive everywhere. The same statement holdstrue <strong>for</strong> 0.01 100 1, 0.01 1, and r s 0.1, 1, 2, 3, 4, 5. In the conventional s-wave <strong>superconductors</strong>,the pairing mechanism is phononic,and the above Coulombic positive-energy contributionsreduce superconductivity. tanh 2 E ktanh 2 E k, (21)versus the exchange and the RPA contributions, wecalculated these quantities numerically <strong>for</strong> a widerange of parameters.The results are summarized in Figure 1, wherewe plot the difference of exchange energies, f S x f N x ,in the superconducting (S) and normal (N) states;Sthe negative difference ( f RPA f NRPA ) of the correspondingRPA correlation energies; the anomalousHartree energy, f AH and the sum, f cond , of thesethree terms representing the potential part of thecondensation energy. In the left panel of Figure 1these energies are plotted versus the parameter <strong>for</strong> 1, and in the right panel the same quantitiesare plotted as functions of <strong>for</strong> 0.01. These twoplots were obtained <strong>for</strong> r s 1. The temperature inPhonon-Induced SuperconductivityFor a proper description of real materials, boththe electron–phonon and electron–electron interactionshave to be included. A suitable xc <strong>functional</strong>has been constructed via Kohn–Sham perturbation<strong>theory</strong> taken to the second order in powers ofe, the electron charge, and g, the electron–phononcoupling constant. The resulting gap equationwas then simplified by taking the values of K ijand Z i at 0. Two terms were found to contributeto the kernel K ij , the first of phononicorigin, K ph ij , and the second of electronic origin,el . They can be written asK ijK el ij w ij ,794 VOL. 99, NO. 5
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