Dynamical spin susceptibility of a spin-valley half-metal

D. A. Khokhlov, A. L. Rakhmanov, A. V. Rozhkov, and A. O. Sboychakov

Phys. Rev. B 101, 235141 – Published 18 June 2020

Abstract

A few years ago we predicted theoretically that in systems with nesting of the Fermi surface the spin-valley half-metal has lower energy than the spin density wave state. In this paper we suggest a possible way to distinguish these phases experimentally. We calculate the dynamical spin susceptibility tensor for both states in the framework of the Kubo formalism. Discussed phases have different numbers of bands: four bands in the spin-valley half-metal and only two bands in the spin density wave. Therefore, their susceptibilities, as functions of frequency, have different numbers of peaks. Besides, the spin-valley half-metal does not have rotational symmetry, thus, in general the off-diagonal components of the susceptibility tensor are nonzero. The spin density wave obeys robust rotational symmetry and off-diagonal components of the susceptibility tensor are zero. These characteristic features can be observed in experiments with inelastic neutron scattering.

Received 17 February 2020
Revised 24 April 2020
Accepted 3 June 2020

DOI:https://doi.org/10.1103/PhysRevB.101.235141

Physics Subject Headings (PhySH)

Antiferromagnetism Magnetic susceptibility Spin density waves Spin fluctuations

Half-metals

Neutron scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

D. A. Khokhlov ^1,2,3, A. L. Rakhmanov ^1,2,4, A. V. Rozhkov ^2,4,5, and A. O. Sboychakov⁴

¹Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia
²Moscow Institute of Physics and Technology, Institutsky Lane 9, Dolgoprudny, Moscow region, 141700, Russia
³National Research University Higher School of Economics, Moscow 101000, Russia
⁴Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow 125412, Russia
⁵Skolkovo Institute of Science and Technology, Skolkovo Innovation Center 3, Moscow 143026, Russia

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Vol. 101, Iss. 23 — 15 June 2020

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Figure 1
The energy of single-electron and quasiparticle bands (vertical axis) versus momentum (horizontal axis) for different order parameters and doping values. (a) Bare single-particle bands. The (blue) solid line represents the electron dispersion (band $a$ ); the (red) solid line is the hole dispersion (band $b$ ). The translated hole band is shown by dashed line. Arrow $Q_{0}$ corresponds to the nesting vector. The dashed-dotted horizontal line is the Fermi energy of the undoped state, when the nesting is perfect. (b)–(d) Band structure when the effects of weak electron-electron repulsion are taken into account. (b) The band structure of the undoped SDW state. The chemical potential level (dash-dotted line) lies in the spectral gap separating filled band $E^{(1)}$ and empty band $E^{(2)}$ . Both bands are doubly degenerate. (c) The band structure of the doped SDW state. The bands are shown by solid lines. The chemical potential crosses the upper band $E^{(2)}$ which becomes partially filled. The vertical arrow in the insets indicates quasiparticle interband transition at the threshold frequency $ω_{2}^{sdw}$ , see Eq. (29). (d) The quasiparticle band structure of the SVHM phase. Double degeneracy present in the SDW state is lifted, and all four bands are distinct. The Fermi surface is polarized with respect to the spin-valley index. Up and down bold arrows illustrate this polarization. Up arrows (blue) show spin-up electrons from the valley $a$ , down arrows (red) show spin-down electrons from the valley $b$ . The vertical arrows in the inset illustrate minimum energies which are required to open a specific interband scattering channel. The frequencies $ω_{1}$ , $...$ , $ω_{5}$ are defined by Eq. (32).
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Figure 2
The diagonal components of the susceptibility tensor ${\tilde{χ}}_{α β}^{s, f} (Q, ω)$ at fixed momentum $Q$ versus frequency $ω$ in the SDW phase. Dot curves show the slow part of the susceptibility calculated for $Q = q$ ; solid curves show the fast part of the susceptibility calculated for $Q = q \pm Q_{0}$ . Panels (a) and (c) present ${\tilde{χ}}_{⊥} (Q, ω)$ , panels (b) and (d) present ${\tilde{χ}}_{x x} (Q, ω)$ . The data in panels (a) and (b) is plotted for $q = 0.1 Δ_{0} / v_{F}$ . The data in panels (c) and (d) is plotted for $q = 0.75 Δ_{0} / v_{F}$ . Peaks at low frequency exist due to intraband electron transitions in the conduction band. They start from $ω = 0$ and disappear beyond $ω_{1}^{sdw}$ [the latter frequency is defined in Eq. (28)]. Peaks close to $ω_{2}^{sdw}$ arise due to electron transitions from the valence band to the conduction band shown in the inset in Fig. 1. The frequency $ω_{2}^{sdw}$ is the threshold frequency for this process. This energy is defined in Eq. (29).
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Figure 3
The diagonal components of the susceptibility tensor ${\tilde{χ}}^{s, f} (Q, ω)$ at fixed momentum $Q$ versus frequency $ω$ in the SVHM phase. Dot curves show the slow part of the susceptibility calculated for $Q = q$ ; solid curves show the fast part of the susceptibility calculated for $Q = q \pm Q_{0}$ . Panels (a) and (b) present ${\tilde{χ}}_{x x} (Q, ω)$ , panels (c) and (d) present ${\tilde{χ}}_{y y} (Q, ω)$ , and panels (e) and (f) present ${\tilde{χ}}_{z z} (Q, ω)$ correspondingly. The data in panels (a), (c), and (e) is plotted for $q = 0.1 Δ_{0} / v_{F}$ . The data in panels (b), (d), and (f) is plotted for $q = 0.75 Δ_{0} / v_{F}$ . The (weak) peaks at the lowest frequency are due to intraband transitions within the conduction band. All other peaks are caused by transitions between bands. These transitions are marked by arrows in the inset of Fig. 1. Each threshold frequency $ω_{1}$ , $...$ , $ω_{5}$ represents the opening of a new interband scattering channel. They are determined by Eq. (32).
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Figure 4
The off-diagonal component of the susceptibility tensor ${\tilde{χ}}_{x y}^{s, f} (Q, ω)$ at fixed momentum $Q$ versus frequency $ω$ in the SVHM phase. Dot curves show the slow part of the susceptibility calculated for $Q = q$ ; solid curves show the fast part of the susceptibility calculated for $Q = q \pm Q_{0}$ . The data in panel (a) is plotted for $q = 0.1 Δ_{0} / v_{F}$ . The data in panel (b) is plotted for $q = 0.75 Δ_{0} / v_{F}$ . Each threshold frequency $ω_{1}, ..., ω_{5}$ represents the opening of a new interband scattering channel. They are determined by Eq. (32).
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