### Abstract

We review some important aspects of the doping dependence of many physical properties of the high- cuprates based on a Fermi liquid-like approach. In particular, we show that the spin-fluctuation mechanism of superconductivity on the basis of a microscopic Eliashberg approach supports the idea that the symmetry of the superconducting order parameter is of the -wave type. Furthermore, the renormalization of the quasiparticle spectrum caused by the scattering on spin fluctuations results in the so-called kink feature seen in ARPES. The peculiar momentum dependence of the spin fluctuations will result in a strong anisotropy of the renormalization at different parts of the first Brillouin zone and thus will lead to a strong anisotropy of the kink. Another important achievement of the microscopic Eliashberg approach is that the spin excitation spectrum renormalizes strongly below due to the occurrence of superconductivity with a -wave order parameter, which yields to the formation of the so-called resonance peak that can be viewed as a spin exciton. The topology of the Fermi surface and the momentum dependence of the superconducting gap explains the peculiar dispersion of the resonance peak, in good agreement with experiments.

We are thankful to M. Sigrist and A. Chubukov for helpful discussions. D. M. acknowledges financial support from the Alexander von Humboldt Foundation.

I. INTRODUCTION

A. Generic phase diagram

B. Superconducting state

II. THEORY: GENERALIZED ELIASHBERG EQUATIONS

III. RESULTS AND DISCUSSION

A. Spin excitations

B. Elementary excitations

IV. CONCLUSIONS

### Key Topics

- Superconductivity
- 79.0
- Fermi liquid theory
- 28.0
- Superconductivity models
- 27.0
- Doping
- 26.0
- Antiferromagnetism
- 25.0

## Figures

Schematic phase diagram of hole-doped cuprates. High- superconductivity occurs in the vicinity of an antiferromagnetic phase transition. The corresponding superconducting order parameter below the superconducting transition temperature is of -wave symmetry. In the overdoped region, i.e., , the cuprates behave like a conventional Fermi liquid, whereas in the underdoped regime below the pseudogap temperature one finds strong antiferromagnetic (AF) correlations. As we will discuss below, Cooper pairing can basically be described by the exchange of AF spin fluctuations (often called paramagnons). The doping region between and (shaded region) may be due to local phase-incoherent Cooper-pair formation. Only below do these pairs become phase coherent.

Schematic phase diagram of hole-doped cuprates. High- superconductivity occurs in the vicinity of an antiferromagnetic phase transition. The corresponding superconducting order parameter below the superconducting transition temperature is of -wave symmetry. In the overdoped region, i.e., , the cuprates behave like a conventional Fermi liquid, whereas in the underdoped regime below the pseudogap temperature one finds strong antiferromagnetic (AF) correlations. As we will discuss below, Cooper pairing can basically be described by the exchange of AF spin fluctuations (often called paramagnons). The doping region between and (shaded region) may be due to local phase-incoherent Cooper-pair formation. Only below do these pairs become phase coherent.

Picture of the Fermi surface for the layered cuprates in the first BZ. The arrows show the electronic states connected by the antiferromagnetic wave vector . The dotted lines and signs refer to the -wave symmetry of the superconducting gap.

Picture of the Fermi surface for the layered cuprates in the first BZ. The arrows show the electronic states connected by the antiferromagnetic wave vector . The dotted lines and signs refer to the -wave symmetry of the superconducting gap.

INS results for optimally doped taken from Ref. 13 at the antiferromagnetic wave vector for the normal and superconducting states.

INS results for optimally doped taken from Ref. 13 at the antiferromagnetic wave vector for the normal and superconducting states.

Measured dispersion of the resonant excitations away from the antiferromagnetic wave vector along the diagonal of the first BZ as taken from Ref. 17. The vertical dashed region indicates the position of the silent bands.

Measured dispersion of the resonant excitations away from the antiferromagnetic wave vector along the diagonal of the first BZ as taken from Ref. 17. The vertical dashed region indicates the position of the silent bands.

Cooper pairing in the cuprates due to coupling of carriers (holes or electrons) in planes to antiferromagnetic spin fluctuations characterized by the spin susceptibility ( refers to the matrix Green’s function of quasiparticles).

Cooper pairing in the cuprates due to coupling of carriers (holes or electrons) in planes to antiferromagnetic spin fluctuations characterized by the spin susceptibility ( refers to the matrix Green’s function of quasiparticles).

ARPES results for the renormalized energy dispersion along the direction of the first Brillouin zone, taken from Ref. 22. The horizontal axis shows the distance in the normalized units .

ARPES results for the renormalized energy dispersion along the direction of the first Brillouin zone, taken from Ref. 22. The horizontal axis shows the distance in the normalized units .

Calculated Fermi surface for the for the first few BZs using a tight-binding energy dispersion from Eq. (12) with and optimal doping. The arrows indicate three different scattering processes as described in the text.

Calculated Fermi surface for the for the first few BZs using a tight-binding energy dispersion from Eq. (12) with and optimal doping. The arrows indicate three different scattering processes as described in the text.

Frequency dependence of at the antiferromagnetic wave vector . It reveals a jump because of the -wave symmetry of the superconducting gap.

Frequency dependence of at the antiferromagnetic wave vector . It reveals a jump because of the -wave symmetry of the superconducting gap.

Numerical results for the resonance peak in the weak-coupling limit from Ref. 34 for optimal doping. Imaginary part of the RPA spin susceptibility (in units of states/eV) at wave vector for , 2, 3, and 4 (from bottom to top). The resonance frequency was found at .

Numerical results for the resonance peak in the weak-coupling limit from Ref. 34 for optimal doping. Imaginary part of the RPA spin susceptibility (in units of states/eV) at wave vector for , 2, 3, and 4 (from bottom to top). The resonance frequency was found at .

Imaginary part of the RPA spin susceptibility at calculated within the FLEX approximation, taken from Ref. 34 for optimal doping. For the normal state one gets and in the superconducting state . Assuming , one finds that .

Imaginary part of the RPA spin susceptibility at calculated within the FLEX approximation, taken from Ref. 34 for optimal doping. For the normal state one gets and in the superconducting state . Assuming , one finds that .

Calculated results taken from Ref. 34 for the resonance frequency versus doping. In the overdoped regime one finds a constant ratio of .

Calculated results taken from Ref. 34 for the resonance frequency versus doping. In the overdoped regime one finds a constant ratio of .

Schematic behavior of and within the RPA, , for various momenta .

Schematic behavior of and within the RPA, , for various momenta .

RPA results for magnetic excitations in a -wave superconductor. is obtained from Eq. (36) as a function of momentum (along the diagonal ) and frequency in the superconducting state, taken from Ref. 35. The arrows indicate the positions of the resonance modes.

RPA results for magnetic excitations in a -wave superconductor. is obtained from Eq. (36) as a function of momentum (along the diagonal ) and frequency in the superconducting state, taken from Ref. 35. The arrows indicate the positions of the resonance modes.

Intensity patterns of the resonance excitations for and taken from Ref. 35.

Intensity patterns of the resonance excitations for and taken from Ref. 35.

Illustration of the possible drastic changes of the Fermi surface due to various parameters of the orthorhombicity, as adopted from Ref. 37. The arrows show the change of the phase space for the bond scattering.

Illustration of the possible drastic changes of the Fermi surface due to various parameters of the orthorhombicity, as adopted from Ref. 37. The arrows show the change of the phase space for the bond scattering.

Calculated normalized two-dimensional intensity plot of for constant energy of taken from Ref. 37, without and with inclusion of orthorhombicity . Note that two of the four incommensurate peaks are suppressed.

Calculated normalized two-dimensional intensity plot of for constant energy of taken from Ref. 37, without and with inclusion of orthorhombicity . Note that two of the four incommensurate peaks are suppressed.

Calculated spectral density from Ref. 39 of the quasiparticles in hole-doped cuprates at along and (inset) directions of the first BZ. The dashed line denotes the unrenormalized chemical potential. In both directions at energies of approximately a kink occurs, since the velocity of quasiparticles changes.

Calculated spectral density from Ref. 39 of the quasiparticles in hole-doped cuprates at along and (inset) directions of the first BZ. The dashed line denotes the unrenormalized chemical potential. In both directions at energies of approximately a kink occurs, since the velocity of quasiparticles changes.

Positions of the peaks in the spectral density versus (energy dispersion) along the direction of the BZ, calculated in Ref. 39 for optimally hole-doped cuprates. Inset: change in the peak positions of in the superconducting state. Note that in underdoped cuprates the kink feature shifts to lower frequencies due to decreasing of . In the overdoped regime the kink feature should become less pronounced and isotropic in different directions of the BZ.

Positions of the peaks in the spectral density versus (energy dispersion) along the direction of the BZ, calculated in Ref. 39 for optimally hole-doped cuprates. Inset: change in the peak positions of in the superconducting state. Note that in underdoped cuprates the kink feature shifts to lower frequencies due to decreasing of . In the overdoped regime the kink feature should become less pronounced and isotropic in different directions of the BZ.

Comparison of the quasiparticle damping in the normal state for optimal doping and the overdoped case taken from Ref. 40. The solid curves are calculated at and the dashed curves refer to . The upper curves correspond to the antinodal wave vector , whereas the lower curves correspond to the nodal one both on the Fermi line. For comparison, we also show in results for the superconducting state (, dashed-dotted curve). Note that for both doping concentrations a linear behavior for larger frequencies and a vanishing anisotropy for in the overdoped case.

Comparison of the quasiparticle damping in the normal state for optimal doping and the overdoped case taken from Ref. 40. The solid curves are calculated at and the dashed curves refer to . The upper curves correspond to the antinodal wave vector , whereas the lower curves correspond to the nodal one both on the Fermi line. For comparison, we also show in results for the superconducting state (, dashed-dotted curve). Note that for both doping concentrations a linear behavior for larger frequencies and a vanishing anisotropy for in the overdoped case.

Calculated frequency dependence of the quasiparticle self-energy at the nodal point in the first BZ taken from Ref. 39 for optimally hole-doped cuprates. The solid curves correspond to the normal state at , whereas the dashed curves to the superconducting state at . In the normal state one clearly sees approximately at a crossover from Fermi liquid behavior to a non-Fermi-liquid behavior for low-energy frequencies as a function of temperature. We show in the inset the behavior of calculated at very low temperatures without superconductivity (, dashed curve). Note that the behavior of the self-energy will become more Fermi-liquid-like in the overdoped regime, and thus the kink feature will be less pronounced.

Calculated frequency dependence of the quasiparticle self-energy at the nodal point in the first BZ taken from Ref. 39 for optimally hole-doped cuprates. The solid curves correspond to the normal state at , whereas the dashed curves to the superconducting state at . In the normal state one clearly sees approximately at a crossover from Fermi liquid behavior to a non-Fermi-liquid behavior for low-energy frequencies as a function of temperature. We show in the inset the behavior of calculated at very low temperatures without superconductivity (, dashed curve). Note that the behavior of the self-energy will become more Fermi-liquid-like in the overdoped regime, and thus the kink feature will be less pronounced.

Calculated frequency dependence of the quasiparticle self-energy at the anti-nodal point in the first BZ taken from Ref. 41 for optimally hole-doped cuprates. The solid curve correspond to the normal state at , whereas the dashed curves correspond to the superconducting state at . Inset: the corresponding superconducting gap function versus frequency.

Calculated frequency dependence of the quasiparticle self-energy at the anti-nodal point in the first BZ taken from Ref. 41 for optimally hole-doped cuprates. The solid curve correspond to the normal state at , whereas the dashed curves correspond to the superconducting state at . Inset: the corresponding superconducting gap function versus frequency.

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