Pair susceptibility and mode propagation in superconductors: A microscopic approach

Abstract

We derive the full microscopic set of equations governing small oscillations :(1) in the magnitude of the superconducting order parameter (the Schmid mode), (2) the phase of the order parameter in a neutral superfluid (the Anderson-BogoIiubov mode), and (3) the coupled oscillations in the phase of the order parameter and in the electric field (the transverse, or CarlsonGoldman mode). The derivation is not limited by the restrictions of previous papers. No limitations are required for the magnitude of the frequency, the concentration of impurities, or the magnitude of the temperature. Special attention is given to the Carlson-Goldman (CG) mode, whose dispersion law frequency (oJ) vs. wave vector (k) and damping is calculated. The velocity of the CG mode in the propagation region oJ/Ik [ is found to equal c= [22xDx(21rTr)], where D is the diffusion constant and X is the function appearing in the theory of superconducting alloys. In the dirty (l << ~:o) and clean (l >> ~o) limits, this expression reduces to those previously derived by Schmid and Sch6n, and by Artemenko and Volkov, respectively. At large values of k, the frequency of the CG mode approaches a limiting value of 2 h. The damping is small in this limit and tends to zero as Ik[ increases.Our results are obtained by calculating the linear response of a superconductor to a perturbation in the magnitude and phase of the order parameter, and the electromagnetic potentials. The response of the superconductor to these perturbations is calculated by properly continuing the thermodynamic perturbation function of linear response from imaginary frequencies to the real ones, then inserting into the self-consistency BCS equation and Poisson's *

Multiple amplitude modes in strongly coupled phonon-mediated superconductors

“…The factor 2 appears because the present excitation does not induce any average phonon displacement. 3 This naive expectation (ω XX = 2ω r ) is satisfied in the normal state with el-ph couplings that are not too strong, while in the SC phase ω H2 (= ω XX ) drastically deviates from 2ω r (see Fig. 3).…”

“…1(a). Here,ˆ is a renormalized vertex, which can be expressed aŝ In the case of DMFT+self-consistent Migdal approximation, the vertex part is given aŝ (t,t ; t ) Here,ˆ (t,t ; t ) ≡ 1 3 . The diagrams for the vertex part are displayed in Fig.…”

“…Before this experiment [15], the amplitude Higgs mode had been observed only in a special case, 2H -NbSe 2 , where SC coexists with a charge density wave [4][5][6]18,19]. Theoretical studies of the SC order parameter dynamics have so far primarily focused on the static mean-field dynamics [1][2][3][5][6][7][8][9][10][11][12][13]17,19,[22][23][24]. One important conclusion of these works is that the frequency of the Higgs amplitude mode (ω H ) should coincide with the SC gap (2 SC ) in the BCS regime [27], which is a threshold for quasiparticle excitations.…”

Why holes are not like electrons: A microscopic analysis of the differences between holes and electrons in condensed matter

“…This 'correlated hopping' Hamiltonian has been extensively studied in recent years by approximate and exact techniques, and the reader is referred to the references for detailed information [9,[23][24][25][26][27][28][29][30][31][32]. The condition for superconductivity in the limit of low hole concentration is [32] K > (1 + u)(1 + w) − 1 (62) with K = 2z∆t/D, u = U/D, w = zV /D, with z the number of nearest neighbors to a site and D the (renormalized) bandwidth.…”

Going beyond the BCS level in the superfluid path integral: A consistent treatment of electrodynamics and thermodynamics

“…In this equation we have introduced the notation ∆ To express Eq. (4) in a more suggestive form, we define the set of two-point response functions [8,[28][29][30]:…”