Spin excitations in layered antiferromagnetic metals and superconductors

W. Rowe, J. Knolle, I. Eremin, and P. J. Hirschfeld

Phys. Rev. B 86, 134513 – Published 11 October 2012

Abstract

The proximity of antiferromagnetic order in high-temperature superconducting materials is considered a possible clue to the electronic excitations which form superconducting pairs. Here we study the transverse and longitudinal spin excitation spectrum in a one-band model in the pure spin density wave (SDW) state and in the coexistence state of SDW and the superconductivity. We start from a Stoner insulator and study the evolution of the spectrum with doping, including distinct situations with only hole pockets, with only electron pockets, and with pockets of both types. In addition to the usual spin-wave modes, in the partially gapped cases we find significant weight of low-energy particle-hole excitations. We discuss the implications of our findings for neutron scattering experiments and for theories of Cooper pairing in the metallic SDW state.

Received 16 July 2012

DOI:https://doi.org/10.1103/PhysRevB.86.134513

Authors & Affiliations

W. Rowe^1,*, J. Knolle², I. Eremin^3,†, and P. J. Hirschfeld¹

¹Department of Physics, University of Florida, Gainesville, Florida 32611, USA
²Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
³Institut für Theoretische Physik III, Ruhr-Universität Bochum, D-44801 Bochum, Germany

^*wwang@ufl.edu
^†ieremin@tp3.rub.de

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Vol. 86, Iss. 13 — 1 October 2012

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Images

Figure 1
Three possible types of Fermi surface topology in the commensurate SDW state in layered cuprates either for hole (a) or electron doping (b) and (c) analyzed in this paper. Due to the mean-field SDW order the original large Fermi surface (green curve) folds, yielding small hole pockets (blue curves) centered around $(\pm π / 2, \pm π / 2)$ and electron pockets (red curves) centered around $(\pm π, 0)$ , $(0, \pm π)$ points of the BZ. For larger doping and smaller sizes of the AF gap both types of the pockets can be present. As argued in the text, the commensurate SDW order becomes unstable once the hole pockets appear around $(\pm π / 2, \pm π / 2)$ points of the BZ. This occurs due to overall negative spin stiffness of the commensurate spin waves in this case [see Fig. 3a].Reuse & Permissions
Figure 2
Calculated spin-wave dispersion $Ω$ vs $q$ in units of $π / a$ along the symmetry route $(0, 0) \to (π, π) \to (0, π) \to (0, 0)$ of the BZ for a Stoner insulator at zero doping ( $x$ $=$ 0), $W = 0.75$ $t$ and $t^{'} = 0$ (a), $t^{'} = 0.2 t$ (b), and $t^{'} = 0.35 t$ (c). Here, we fixed the magnitude of $W$ by using $U = 1.40 t$ and the chemical potential $μ = 0.00 t$ (a), $μ = - 0.30 t$ (b), and $μ = - 0.68 t$ (c), accordingly. The intensity for Im $χ_{RPA}^{+ -} (q, q, Ω)$ is shown on the log scale.Reuse & Permissions
Figure 3
Calculated transverse Im $χ_{RPA}^{+ -} (q, q, Ω)$ (upper panel), and longitudinal Im $χ_{RPA}^{z z} (q, q, Ω)$ (lower panel) spin excitation spectra $Ω$ vs $q$ in units of $π / a$ for the metallic commensurate SDW state with $t^{'} / t = 0.35$ and $U = 1.3875 t$ . (a) and (d) Refer to the hole doping $x$ $=$ $-$ 0.05, $W = 0.61 t$ , $μ = - 0.8819 t$ ; (b)and (e) refer to the electron doping $x = 0.10$ , $W = 0.4404 t$ , $μ = - 0.4284 t$ ; (c)and (f) refer to the electron doping $x = 0.14$ , $W = 0.12 t$ , $μ = - 0.302 t$ . The corresponding Fermi surface topology is shown in Fig. 1. The intensity in $states / t$ is shown on a log scale. The white arrows in (a) denote the incommensurate momentum associated with $2 k_{F}$ scattering due to hole FS pockets shown in Fig. 1a. Observe also the difference in the intensity maps between upper and lower panels.Reuse & Permissions
Figure 4
Calculated mean-field phase diagram of coexisting $d$ -wave superconductivity and commensurate AF order in the electron-doped cuprates for the case $Δ_{1} = 0$ . The solid and dashed blue curve refer to the Lifshitz transition indicating the change of the Fermi surface topology from two types of pockets to the one type of pocket in the pure AF and AF+dSC state, respectively. The open circle for $x = 0$ refers to the AF semimetal with two equal-sized electron and hole pockets (high $T$ ) to AF insulator (low $T$ ) transition.Reuse & Permissions
Figure 5
Calculated transverse, Im $χ_{RPA}^{+ -} (q, q, Ω)$ spin excitation spectra $Ω$ vs $q$ in units of $π / a$ for three different electron dopings, $n = 1.06$ , $n = 1.09$ , and $n = 1.12$ (from upper to lower panel) for the coexisting SDW+dSC state and $Δ_{1} = 0$ (right panel). For comparison the left panel shows the results for the pure SDW state. The blue lines denote $1 = U Re χ_{0}^{+ -} (q, Ω)$ condition. The intensity is shown on the log scale. The following parameters are used in the units of $t$ for (a) $μ = - 0.5362$ , $W = 0.5617$ ; for (b) $μ = - 0.4573$ , $W = 0.4617$ ; for (c) $μ = - 0.3694$ , $W = 0.3456$ ; for (d) $μ = - 0.5349$ , $W = 0.5530$ , $Δ_{0} = 0.0727$ ; for (e) $μ = - 0.4509$ , $W = 0.4555$ , $Δ_{0} = 0.0705$ ; and for (f) $μ = - 0.3624$ , $W = 0.3458$ , $Δ_{0} = 0.0618$ .Reuse & Permissions
Figure 6
Calculated longitudinal Im $χ_{RPA}^{z z} (q, q, Ω)$ spin excitation spectra $Ω$ vs $q$ in units of $π / a$ for three different electron dopings, $n = 1.06$ , $n = 1.09$ , and $n = 1.12$ (from upper to lower panel) for the coexisting SDW+dSC state and $Δ_{1} = 0$ (right panel). For comparison the left panel shows the behavior of the longitudinal susceptibility for the pure SDW state. The intensity is shown on the absolute scale.Reuse & Permissions

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