Spin-valley half-metal in systems with Fermi surface nesting

A. L. Rakhmanov, A. O. Sboychakov, K. I. Kugel, A. V. Rozhkov, and Franco Nori

Phys. Rev. B 98, 155141 – Published 25 October 2018

Abstract

Half-metals have fully spin-polarized charge carriers at the Fermi surface. Such polarization usually occurs due to strong electron-electron correlations. Recently [Phys. Rev. Lett. 119, 107601 (2017)] we have demonstrated theoretically that adding (or removing) electrons to systems with Fermi surface nesting also stabilizes the half-metallic states even in the weak-coupling regime. In the absence of doping, the ground state of the system is a spin or charge density wave, formed by four nested bands. Each of these bands is characterized by charge (electron/hole) and spin (up/down) labels. Only two of these bands accumulate charge carriers introduced by doping, forming a half-metallic two-valley Fermi surface. Analysis demonstrates that two types of such half-metallicity can be stabilized. The first type corresponds to the full spin polarization of the electrons and holes at the Fermi surface. The second type, with antiparallel spins in electronlike and holelike valleys, is referred to as a “spin-valley half-metal” and corresponds to the complete polarization with respect to the spin-valley operator. We analyze spin and spin-valley currents and possible superconductivity in these systems. We show that spin or spin-valley currents can flow in half-metallic phases.

Received 23 April 2018
Revised 26 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.155141

Physics Subject Headings (PhySH)

Antiferromagnets Half-metals

Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. L. Rakhmanov^1,2,3,4, A. O. Sboychakov^1,2, K. I. Kugel^2,5, A. V. Rozhkov^1,2,3,6, and Franco Nori^1,7

¹Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
²Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow 125412, Russia
³Moscow Institute for Physics and Technology (State University), Dolgoprudnyi 141700, Russia
⁴Dukhov Research Institute of Automatics, Moscow 127055, Russia
⁵Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow 101000, Russia
⁶Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Moscow 143026, Russia
⁷Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

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Vol. 98, Iss. 15 — 15 October 2018

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Images

Figure 1
Electron bands of the model when the electron-electron coupling is neglected and doping is zero. (a) Electron band $ɛ^{a} (k)$ and hole band $ɛ^{b} (k)$ are shown by solid curves. The dashed parabola is the hole band translated by the nesting vector $Q_{0}$ . The vertical axis is energy and the horizontal axis is momentum, while the Fermi level $μ$ is shown by the horizontal dash-dot line. (b) Spherical Fermi surfaces of the electron and hole bands. The spheres coincide if we translate one of them by the nesting vector.
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Figure 2
Electron band structure for the insulating and half-metallic states. The vertical axis is the energy, while the horizontal axis is the momentum. The Fermi level $μ$ is shown by the horizontal dash-dot lines. (a) If doping is zero $(x = 0)$ , the ground state is an insulating SDW or CDW depending on the model parameters, with degenerate sectors $(Δ_{↑} \equiv Δ_{↓})$ . The energies of electron and hole bands $E_{σ}^{(1, 2)}$ are given by Eq. (9). (b) and (c) If $x > 0$ , the sectors are no longer degenerate $(Δ_{↑} < μ < Δ_{↓} \equiv Δ_{0})$ , with the charge accumulating in sector $↑$ , in which a Fermi surface appears. The spin polarizations (arrows) of the Fermi surface sheets in (b) correspond to the spin-valley half-metal, and in (c) to the CDW half-metal.
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Figure 3
Dependence of $Δ F = F_{σ} (x_{σ}) + F_{σ} (x - x_{σ}) - 2 F (x / 2)$ on the partial doping $x_{σ}$ calculated at $T = 0$ and fixed total doping: $x = 0.75 x_{0}$ [(green) dashed curve], $x = 1.5 x_{0}$ [(red) solid curve], and $x = 1.9 x_{0}$ [(blue) dash-dot curve]. The free energy curves for all three doping values have a global maximum at $x_{σ} = x / 2$ , implying that the usual metallic phase is unstable. The free energy is the lowest for either $x_{σ} = 0$ or $x_{σ} = x$ : the free energy minimum at $x_{σ} = 0 (x_{σ} = x)$ represents a half-metallic state with empty (filled) sector $σ$ and filled (empty) sector $\bar{σ}$ .
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Figure 4
Phase diagram of the system: (a) commensurate and (b) incommensurate ordering. (a) Spin-valley half-metals exist within the doping range $0 < x < 2 x_{0}$ . At $x = x_{0}$ (vertical dashed line) order parameter $Δ_{↑}$ vanishes and a second-order phase transition occurs. However, a characteristic polarization of the charge carriers at the Fermi surface (half-metallicity) is not destroyed. When $x = 2 x_{0}$ [vertical (red) solid line] a first-order transition occurs from the spin-valley half-metal phase to the PM phase. (b) Spin-valley half-metal exists within the doping range $0 < x ≲ 1.8 x_{0}$ . At $x \approx 1.8 x_{0}$ [(red) solid line] a second-order phase transition occurs from the spin-valley phase to the usual SDW incommensurate state. If $x \approx 3 x_{0}$ [vertical thin (black) solid line] a first-order phase transition occurs to the PM phase. The dashed vertical line shows the point $(x \approx 0.83 x_{0})$ when the incommensurate SDW order can exist as a metastable phase.
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Figure 5
Dependence of $Δ F_{0}^{ic} (x_{σ}, x - x_{σ}) \equiv F_{0}^{ic} (x_{σ}) + F_{0}^{ic} (x - x_{σ}) - 2 F_{0}^{ic} (x / 2)$ on the partial doping $x_{σ}$ , calculated at $T = 0$ and fixed total doping $x = 1.4 x_{0}$ [(red) solid curve], $x = 1.76 x_{0}$ [(green) dashed curve], and $x = 2.0 x_{0}$ [(blue) dash-dot curve]. At high doping $(x = 2.0 x_{0})$ , the state at $x_{σ} = x_{\bar{σ}} = x / 2$ has the lowest free energy, therefore, the usual metal, with even distribution of the doped charges among the sectors, is a stable phase. When the doping is low $(x = 1.4 x_{0})$ , the half-metal is stable. In this situation, the free energy minimum at $x_{σ} = 0 (x_{σ} = x)$ represents a half-metallic state with empty (filled) sector $σ$ and filled (empty) sector $\bar{σ}$ . At some intermediate doping $1.4 x_{0} < x^{*} < 2.0 x_{0}$ , a first-order transition from the usual metal to the half-metal occurs. Near the transition, one of the phases may become metastable. For example, a well-defined local (but not global) minimum of the free energy at $x_{σ} = x_{\bar{σ}} = x / 2$ is clearly seen for the (green) dashed curve. This implies that for $x = 1.76 x_{0}$ , the usual metal is metastable, while the half-metal is truly stable. The activation barriers for the transition into the more stable half-metallic phase are shown by the vertical arrows. The presence of the metastable phase is marked in Fig. 4.
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