Plasmons in spin-polarized graphene: A way to measure spin polarization

Amit Agarwal and Giovanni Vignale

Phys. Rev. B 91, 245407 – Published 8 June 2015

Abstract

We study the collective charge excitations (plasmons) in spin-polarized graphene, and derive explicit expressions for their dispersion in the undamped regime. From this, we are able to calculate the critical wave vector beyond which the plasmon enters the electron-hole continuum, its quality factor decreasing sharply. We find that the value of the critical wave vector is strongly spin-polarization dependent, in a way that has no analog in ordinary two-dimensional electron gases. The origin of this effect is in the coupling between the plasmon and the interband electron-hole pairs of the minority spin carriers. We show that the effect is robust with respect to the inclusion of disorder and we suggest that it can be exploited to experimentally determine the spin polarization of graphene.

Received 5 May 2015

DOI:https://doi.org/10.1103/PhysRevB.91.245407

Authors & Affiliations

Amit Agarwal^1,* and Giovanni Vignale^2,†

¹Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
²Department of Physics, University of Missouri, Columbia, Missouri 65211, USA

^*amitag@iitk.ac.in
^†vignaleg@missouri.edu

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Vol. 91, Iss. 24 — 15 June 2015

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Figure 1
(a) Real and imaginary parts of the RPA dielectric function $ε (q, ω)$ vs $ω$ for a 2DEG (parabolic dispersion) system. The zero of the dielectric function marked by the small circle corresponds to the spin plasmon mode in 2DEG [10]. (b) displays the real and imaginary parts $ε (q, ω)$ vs $ω$ for graphene. Note that there is no spin plasmon mode in graphene. Other parameters are chosen to be $P = 0.5, q / k_{F} = 0.1$ , and $r_{s} = 2.5$ for both panels. The thin vertical lines in both panels mark the location of the known plasmon mode in the long-wavelength limit, i.e., in graphene $ω_{pl} / ɛ_{F} = \sqrt{2 r_{s} q / k_{F}}$ and in 2DEG $ω_{pl} / ɛ_{F} = \sqrt{2^{3 / 2} r_{s} q / k_{F}}$ .
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Figure 2
The color plot of the loss function, $- ℑ m [1 / ε (q, ω)]$ , along with the numerically evaluated plasmon dispersion (solid black line), and the approximate analytical dispersion (green circles) as per Eq. (10). Panels (a), (b), (c), and (d) correspond to $P = 0, 0.25, 0.5, and 0.75$ , respectively. Here, $r_{s} = 0.5$ in all the panels.
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Figure 3
(a) $Q^{- 1}$ vs $q$ for various values of $P$ in the clean limit. A sharp rise occurs when the charge plasmon enters the interband e-h continuum of the minority spin species and is approximately given by Eq. (11) and marked by thin vertical lines. A second cusp occurs when the plasmon enters the interband e-h continuum of the majority spin species, whose approximate location is specified by Eq. (11) with $P \to - P$ , and becomes visible for large values of $P$ . (b) shows the critical $q_{c}$ beyond which the plasmon mode enters the interband e-h continuum. The $P$ dependence of the critical $q_{c}$ arises almost entirely from the shift in the boundary of the interband continuum: this is shown by the curve labeled “interband” in which the polarization dependence of the plasmon dispersion has been neglected. Further, the maximum deviation of Eq. (11) from the exactly calculated critical wave vector occurs for $P \to 0$ , and is less that $6.7 %$ . We have taken $r_{s} = 0.5$ in both the panels.
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Figure 4
(a) and (b) display $ω_{pl}$ and $Q^{- 1}$ , respectively, as functions of $q$ for different disorder strengths, for $P = 0.5$ . The shaded region around $ω_{pl}$ indicates the decay rate $γ_{pl}$ , and the plasmon curves for different disorder strength are shifted vertically (by $0.4$ ) for better visibility. (c) and (d) plot $Q^{- 1}$ for different polarization strengths, with $1 / τ = 0.02$ , for large- and small $q$ , respectively. In (c), the cusps in the inverse quality factor arise for the critical wave vectors where the plasmon mode enters the interband e-h continuum of the minority (first cusp – marked by vertical lines), and then the majority spin carriers (second cusp). (d) highlights the existence of the $P$ -dependent critical wave vector of Eq. (18), below which the plasmon modes are unstable to diffusion. Notice that we plot $Q$ rather than $Q^{- 1}$ and the different curves are shifted vertically (by $0.2$ ) for better visibility. The symbols indicate the exact numerical result, which are in excellent agreement with the approximate analytical result of Eq. (19) indicated by the solid black line. In all panels, we have taken $r_{s} = 0.5$ .
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