Nonequilibrium plasmons with gain in graphene

Graphene supports strongly confined transverse-magnetic sheet plasmons whose spectral characteristics depend on the energetic distribution of Dirac particles. The question arises whether plasmons can become amplified when graphene is pumped into a state of inversion. In establishing a theory for the dynamic nonequilibrium polarizability, we are able to determine the exact complex-frequency plasmon dispersion of photoinverted graphene and study the impact of doping, collision loss, and temperature on the plasmon gain spectrum. We calculate the spontaneous emission spectra and carrier recombination rates self-consistently and compare the results with approximations based on Fermi's golden rule. Our results show that amplification of plasmons is possible under realistic conditions but inevitably competes with ultrafast spontaneous emission, which, for intrinsic graphene, is a factor of 5 faster than previously estimated. This work casts new light on the nature of nonequilibrium plasmons and may aid the experimental realization of active plasmonic devices based on graphene.

Figure 1

Plasmon decay rate ${\gamma }_{\mathrm{pl}}$ over plasmon energy $\hslash {\omega }_{\mathrm{pl}}$ for inverted intrinsic graphene. At energies below $2\mu$ (where $\mu$ is the chemical potential of particles and holes) plasmons undergo stimulated emission processes, accompanied by interband recombination of carriers. Above energies of $2\mu$ stimulated absorption processes associated with inter- and intraband recombination of particle/hole pairs dominate.

Figure 2

The dynamic polarizability $\Pi \left[n\right]$ of an arbitrary nonequilibrium carrier distribution $n\left(\epsilon \right)$ can be represented as the sum of the intrinsic zero-temperature polarizability ${\Pi (q,\omega )|}_{\mu =0}^{T=0}$ and the polarizabilities of the particle and hole plasmas ${\Pi }^{\left(e\right)}\left[n\right]$ and ${\Pi }^{\left(h\right)}\left[\overline{n}\right]$, where $\overline{n}\left(\epsilon \right)=1-n(-\epsilon )$.

Figure 3

Plasmon frequency dispersion (top) and decay rate (bottom) for (a) the equilibrium case (${\mu }_e=-{\mu }_h=\overline{\mu }$) and (b) the photoinverted intrinsic case (${\mu }_e={\mu }_h=\overline{\mu }/2$), using dimensionless variables for frequency ${\tilde{\omega }}_{\mathrm{pl}}=\hslash {\omega }_{\mathrm{pl}}/\overline{\mu }$, decay rate ${\tilde{\gamma }}_{\mathrm{pl}}=\hslash {\gamma }_{\mathrm{pl}}/\overline{\mu }$, and wave vector $\tilde{q}=\hslash v_{F}q/\overline{\mu }$. The exact complex-frequency dispersion [solid (red) lines] is shown together with the dispersion in the real-frequency approximation (dashed black lines), in the classical Drude limit (dotted black lines), and using the optical conductivity (dash-dotted black lines). The complex-frequency dispersion [solid (red) lines] crosses through regions of no loss (II and V; white), regions of interband (III; yellow) and intraband (IV; blue) damping, and, in case (b), a region of amplification (I; green), characterized by a negative decay rate.

Figure 4

Complex-frequency plasmon dispersion of inverted extrinsic graphene for a varying carrier imbalance $m=({\mu }_e-{\mu }_h)/\overline{\mu }$ with $\overline{\mu }={\mu }_e+{\mu }_h$. (a) Frequency-dispersion curves for $m=0\cdots 1$. Insets: Two representative dispersion curves passing the branch singularity [filled (magenta) circle] at the intersection of regions I–III from above (top left; black line) and below [bottom right; gray (red) line]. (b) Loss dispersion over plasmon frequency. Dashed curves in (a) and (b) mark solutions with $m$ values just above and below $m_c$. (c) Decay rate as a function of ${\omega }_{\mathrm{pl}}$ and $m$, depicting gain regions (red), inter-/intraband loss regions (blue), and regions of no loss/gain (gray). The discontinuity (dashed line) originates in the branch singularity [filled (magenta) circle]. The dimensionless frequency, loss, and wave vector are defined by ${\tilde{\omega }}_{\mathrm{pl}}=\hslash {\omega }_{\mathrm{pl}}/\overline{\mu }$, ${\tilde{\gamma }}_{\mathrm{pl}}=\hslash {\gamma }_{\mathrm{pl}}/\overline{\mu }$, and $\tilde{q}=\hslash v_{F}q/\overline{\mu }$.

Figure 5

Impact of temperature and collision loss on plasmon decay rate and frequency dispersion. (a) Plasmon decay rate at $m=0.5$ for collision loss rates ranging from ${\tilde{\tau }}^{-1}=0$ to ${\tilde{\tau }}^{-1}=0.08$, where ${\tilde{\tau }}^{-1}=\hslash {\tau }^{-1}/\overline{\mu }$. (b) Plasmon decay rate at $m=0.5$ for temperatures ranging from $\tilde{T}=0$ (red) lines] to $\tilde{T}=0.1$ (black lines), where $\tilde{T}=k_{B}T/\overline{\mu }$. Insets: Corresponding frequency dispersion curves for $m=0.5$ (red lines) and $m=0.6$ (blue lines).

Figure 6

(a) Spontaneous plasmon emission spectra ${\tilde{G}}_{\mathrm{pl}}\left(\omega \right)$ and (b) electron recombination rates ${\tilde{\Gamma }}_{\mathrm{pl}}^{\left(e\right)}$ at zero temperature and dependent on the carrier imbalance $m$. The respective dimensionless quantities are defined as ${\tilde{G}}_{\mathrm{pl}}={\left(\hslash v_F\right)}^2{G}_{\mathrm{pl}}/{\overline{\mu }}^2$ and ${\tilde{\Gamma }}_{\mathrm{pl}}^{\left(e\right)}=\hslash {\Gamma }_{\mathrm{pl}}^{\left(e\right)}/\overline{\mu }$ (see Appendix pp2). Exact results (i) derived from the exact complex-$\omega$ plasmon dispersion curves, together with approximative results obtained from Fermi's golden rule in conjunction with (ii) the classical (Drude) dispersion, (iii) the complex-$\omega$ dispersion, and (iv) the real-$\omega$ dispersion.

Figure 7

Impact of collision loss and temperature on the plasmon gain spectra. (a) Plasmon gain spectra ${\tilde{G}}_{\mathrm{pl}}\left(\tilde{\omega }\right)$ for $m=0$ and collision loss rates ranging from ${\tilde{\tau }}^{-1}=0$ to ${\tilde{\tau }}^{-1}=0.08$. (b) Plasmon gain spectra for $m=0$ and temperatures ranging from $\tilde{T}=0$ to $\tilde{T}=0.1$. (c) Particle recombination rates over carrier imbalance parameter $m$ for temperatures from $\tilde{T}=0$ to $\tilde{T}=0.1$. The dimensionless spectral gain and carrier recombination rates are defined as ${\tilde{G}}_{\mathrm{pl}}={\left(\hslash v_F\right)}^2{G}_{\mathrm{pl}}/{\overline{\mu }}^2$ and ${\tilde{\Gamma }}_{\mathrm{pl}}^{\left(e\right)}=\hslash {\Gamma }_{\mathrm{pl}}^{\left(e\right)}/\overline{\mu }$.