Plasmon dispersion in a layered crystal with one conducting layer in the unit cell, as a function of the longitudinal momentum upon a continuous variation of the transverse momentum in the interval from to . The lower curve is the upper boundary of the region of quantum Landau damping (a). Plasmon dispersion in a layered crystal with two conducting layers in the unit cell. In the region between the lower boundary of the plasmon band and the upper boundary of the damping region is an additional quasi-acoustic plasma branch due to anti-phase collective oscillations of the electron density in different sublattices formed by 2D layers shifted with respect to each other by a distance (Ref. 45) (b).
Plasmon spectral function averaged over the transverse momentum [Eq. (8)], divided by , for a layered crystal with one sublattice of conducting layers and with band spectrum in Eq. (9). The high- and low-frequency plasmon peaks correspond to maxima of the density of states at the upper and lower boundaries of the dispersion region of plasma oscillations [Eq. (3)] (a). The low-frequency region of the plasmon spectrum, showing in more detail the structure of the function near zero frequency (b). The calculations were done for .
Dependence of the function on for different values of for plasmons in a layered crystal with one sublattice and with an anisotropic band spectrum, which corresponds to the dispersion of the antibonding branch of the conduction band in YBaCuO.7 The solid curve is for , the dashed curve is for , and the dot-and-dash curve is for (a). Structure of the plasmon spectrum in the low-frequency region, which corresponds to an additional quasi-acoustic plasma branch due to the presence of ESPS36,37 (b)
Plasmon spectral function in a layered crystal with two sublattices of 2D layers and the band spectrum of Ref. 7 at . The inset shows a more detailed view of the low-frequency region of the plasma spectrum. Here, together with the features due to the presence of ESPS, one can also see a structure of additional peaks due to antiphase oscillations of the electron density in different sublattices.
Plasmon spectral function in a 2D metal with an anisotropic band spectrum7 at . The inset shows the main plasma peak in more detail.
Concentration dependence of the critical temperature of the transition to the SC state in the Cooper channel for a layered crystal with the band spectrum (9) (curve 1) and also for crystals with one (curve 2) or two (curve 3) conducting layers in the unit cell with the band spectrum of the antibonding branch from Ref. 7. Curve 4 demonstrates the dependence for a 2D metal with the band spectrum of Ref. 7. Here is the dimensionless concentration of holes (number per unit cell), and the value corresponds to a position of the Fermi level at the ESPS.
Concentration dependence of calculated in different approximations for a layered crystal with one layer in the unit cell and an anisotropic spectrum:7 the RPA (see curve 2 in Fig. 6) (1); the RPA with renormalization of the self-energy and Coulomb vertex in first order in the interaction (2); complete self-consistent calculation39,40 including renormalization of the self-energy and Coulomb vertex (but without the exchange correlations) (3).
Concentration dependence of the critical temperature for a layered crystal with one layer in the unit cell and with the band spectrum of Ref. 7, calculated in the -wave (solid curves) and -wave (dashed curves) Cooper channels in the RPA (curves 1 and ) and with corrections of first order in the interaction (curves 2 and ) (a). Concentration dependence of (solid curve) and (dashed curve) obtained on the basis of self-consistent calculations for the band spectrum of Ref. 7 without taking into account the exchange correlations (the curve corresponds to curve 3 in Fig. 7) (b).
Diagrams with crossing interaction lines, which describe exchange-correlation effects in the anomalous self-energy part: simplest diagram with two crossing lines of the retarded SCI (a); diagram of higher order in the SCI (b); sequence of “ladder” diagrams with local interaction (larger dots), the role of which is played by the momentum-averaged static SCI (c).
Concentration dependence of (solid curve) and (dashed curve), obtained for (a) and (b) from self-consistent numerical calculations with the “ladder” diagrams (see Fig. 9c), which approximate the exchange effects, taken into account.
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