Long-lived spin plasmons in a spin-polarized two-dimensional electron gas

Amit Agarwal, Marco Polini, Giovanni Vignale, and Michael E. Flatté

Phys. Rev. B 90, 155409 – Published 6 October 2014

Abstract

Collective charge-density modes (plasmons) of the clean two-dimensional unpolarized electron gas are stable, for momentum conservation prevents them from decaying into single-particle excitations. Collective spin-density modes (spin plasmons) possess no similar protection and rapidly decay by production of electron-hole pairs. Nevertheless, if the electron gas has a sufficiently high degree of spin polarization ( $P > 1 / 7$ , where $P$ is the ratio of the equilibrium spin density and the total electron density, for a parabolic single-particle spectrum) we find that a long-lived spin plasmon—a collective mode in which the densities of up and down spins oscillate with opposite phases—can exist within a “pseudogap” of the single-particle excitation spectrum. The ensuing collectivization of the spin excitation spectrum is quite remarkable and should be directly visible in Raman-scattering experiments. The predicted mode could dramatically improve the efficiency of coupling between spin-wave-generating devices, such as spin-torque oscillators.

Received 4 June 2014

DOI:https://doi.org/10.1103/PhysRevB.90.155409

Authors & Affiliations

Amit Agarwal^1,*, Marco Polini², Giovanni Vignale³, and Michael E. Flatté^4,†

¹Department of Physics, Indian Institute of Technology, Kanpur 208016, India
²NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
³Department of Physics, University of Missouri, Columbia, Missouri 65211, USA
⁴Department of Physics and Astronomy and Optical Science and Technology Center, University of Iowa, Iowa City, Iowa 52242, USA

^*amitag@iitk.ac.in
^†michael_flatte@mailaps.org

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Vol. 90, Iss. 15 — 15 October 2014

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Figure 1
Cartoon depicting that (a) the in-phase oscillations of the two spin fluids give rise to a charge plasmon, and (b) the out-of-phase oscillations of the two spin fluids gives rise to the spin plasmon. The ratio of the oscillation amplitudes of the two spin fluids, for the case of the spin-plasmon mode in panel (b), is given by Eq. (27). The dashed and the colored circles (blue for $↑$ -spin and red for $↓$ -spin) represent the equilibrium and displaced densities of the two spin fluids, respectively, in both the panels.
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Figure 2
Panel (a) shows the real and imaginary parts of the RPA dielectric function $ε (q, ω)$ versus $ω$ for a clean system. Panel (b) displays $Im [χ_{S_{z} S_{z}} (q, ω)]$ versus $ω$ for a clean system, with and without Coulomb interactions. Panels (c) and (d) show the real and imaginary parts of the dielectric function versus $ω$ and $Im [χ_{S_{z} S_{z}} (q, ω)]$ versus $ω$ , respectively, for a disordered system with ${(τ ɛ_{F})}^{- 1} = 0.01$ . Other parameters are chosen to be $P = 0.5$ , $q / k_{F} = 0.1$ , and $r_{s} = 2$ . The left and right vertical lines in the plots mark the upper limit of the electron-hole continuum, of the minority ( $ω_{+ ↓}$ for $P > 0$ ) and majority spin species ( $ω_{+ ↑}$ for $P > 0$ ), respectively. Note that the spin-plasmon mode arises from the second zero of the dielectric function (from the origin), in panels (a) and (c), and it always lies between $ω_{+ ↓}$ and $ω_{+ ↑}$ . In panels (b) and (d) “NI” stands for “noninteracting.' Panels (e) and (f) display the color plot of the loss function [ $- 1 / ε (q, ω)$ ] in the $q − ω$ plane for $P = 0$ and $P = 0.5$ , respectively. The inset in panel (e) shows the spin split energy bands.
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Figure 3
Panels (a) and (b) display the frequency $ω_{spl}$ and the damping rate $γ_{spl}$ of the spin-plasmon mode as a function of spin polarization and wave vector, respectively, for $r_{s} = 2$ and in the absence of disorder. The points, filled blue squares for $ω_{spl}$ and filled red circles for $γ_{spl}$ , are obtained from the numerical calculations, and the respective solid lines (blue and red) are from the analytical expressions [Eq. (15) for $ω_{coll}$ for all $q$ and Eq. (7) for $γ_{coll}$ in the $q \to 0$ limit]. In panel (a) $q / k_{F} = 0.01$ , and the thin vertical line marks the $P = 1 / 7$ line, for visual aid. Note that in panel (b), the spin-plasmon mode ceases to exist beyond a certain $q_{spl}^{\max}$ (depicted as a thin vertical line), which is given by the condition $ω_{spl} > ω_{+ ↓}$ . Here we have chosen $P = 0.5$ . Panels (c) and (d) show the dependence of $q_{spl}^{\max}$ on $P$ and $r_{s}$ , respectively.
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Figure 4
Quality factor of the spin-plasmon mode in the presence of disorder. Panel (a) shows the quality factor of the spin-plasmon mode, in the $τ^{- 1} − q$ plane for $r_{s} = 2$ and $P = 0.5$ . Panel (b) illustrates the quality factor of the spin-plasmon mode in the $τ^{- 1} − P$ plane for $r_{s} = 2$ and $q / k_{F} = 0.15$ . Panel (c) displays the quality factor of the spin-plasmon mode in the $r_{s} − P$ plane for $q / k_{F} = 0.15$ and ${(τ ɛ_{F})}^{- 1} = 0.005$ . Panel (d) shows the quality factor of the spin-plasmon mode in the $q − P$ plane for $r_{s} = 2$ and ${(τ ɛ_{F})}^{- 1} = 0.005$ . In panel (e) we present results for the quality factor of the spin-plasmon mode in the $τ^{- 1} − r_{s}$ plane for $q / k_{F} = 0.15$ and $P = 0.5$ . Finally, panel (f) shows the quality factor in the $q − r_{s}$ plane for ${(τ ɛ_{F})}^{- 1} = 0.005$ and $P = 0.5$ . In all the panels, the solid black line marks the $Q_{spl} = 0$ contour line, i.e., the boundary of existence of the spin-plasmon mode. Note that we have chosen the upper limit of the color scale to be 12 to ensure uniformity of all the panels, and to improve the contrast of the figures. The quality factor is not limited by it and is much higher in some regions.
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