Emission of terahertz plasmons from driven electrons in grated graphene

Chengxiang Zhao; Chengxiang Zhao; Chengxiang Zhao; Yan Liu; Yuan Qie; Fangwei Han; Fangwei Han; Hu Yang; Haiming Dong

doi:10.1364/OE.27.026569

1. Introduction

Graphene is an ideal two-dimensional material with distinctive electronic and optical properties. It has also been found to be a unique electronic material for plasmonic devices owing to the virtue of plasmon oscillations with superior features. These include such as high field confinement, low Ohmic loss, and long wave propagation, highly tunable via electrostatic means [1–5]. More importantly, the frequency of plasmons in graphene ranges from terahertz (THz) to infrared, which implies that graphene could be applied as a candidate material for THz light sources [6,7] and detectors [8]. At present, THz sources based on graphene have attracted intensive research interest both theoretically and experimentally due to their unique characteristics and potential applications as practical devices. The main research effort in this field at present is to develop high-power, tunable THz radiation sources. Thus, the key issue in the effort to realize graphene based plasmonic sources is to find a way to effectively excite the THz plasmons from graphene. Several plasmon excitation mechanisms, involving such as incident plane waves [9,10], injecting an electron stream [11,12], and using strong near-field excitation [2], have been proposed and tested. Recent research outcomes have indicated that the plasmon excitation from graphene can be enhanced via an electrodynamic mechanism by which cerenkov plasmon emission can be achieved using an electron beam driven by a dc electric field [13,14].

In general, from the view of condensed matter theory, the physical essence of plasmon emission by drifting electrons in graphene is that plasmons are excited by the coupling between plasmons and electrons in a graphene system driven by dc electric field. This can refer to the phonon emission by the coupling between phonons and electrons in that plasmons and phonons both involve collective excitation and bosons [15–17]. However, it is difficult to directly excite terahertz plasmons by the coupling between plasmons and electrons in graphene because of the miss-match of energy and momentum of plasmons with electrons in graphene (i.e., the energy of plasmons is outside the range of single particle excitation: SPE) [18](and see Fig. 1(a)). Thus, the plasmons cannot be excited using single particle excitation. As known to us, plasmons also cannot be excited by plasmon-light coupling for their energy and momentum miss-matching. However, a grating is always used for exciting plasmons with light. The reason why a grating allows for the excitation of plasmons can be understood in analogy with the theory of electrons regarding periodic atomic potential. The momentum of an incident photon is conserved to a reciprocal lattice vector. The periodic corrugation can provide the missing momentum needed for light to excite the plasmons [10]. The similar understanding that the electrons can gain additional momentum similar to incident photons, can be applied to the coupling between the plasmons and electrons in a grated graphene system. Thus, the plasmon spectrum will be within the single particle excitation range (see Figs. 1(c) and 1(d)) and plasmons can be excited by drifting electrons. In this work, we theoretically investigated the terahertz plasmon emission from drifting electrons in a grated graphene system driven by a dc electric field by applying the Boltzmann’s equilibrium equation method. The plasmons were excited by plasmon-drifting electron coupling. The effect of the electric field, grating parameters, and electron density on the plasmon emission spectrum was investigated. We hope our work will be helpful for gaining insight into graphene-based plasmonics and that the results will be examined via more experimentation for the application of graphene-based structures as electrically tunable THz plasmonic devices.

Fig. 1. Black solid line is the plasmon spectrum in graphene obtained by random phase approximation (RPA) in (a)-(d). The grey areas below the red lines (depicted by $\gamma q$) are the single particle excitation (SPE) regions for different grating periods.

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2. Theoretical considerations and approaches

Here we consider a grated graphene system as depicted in Fig. 2. A 20 nm-thick Al$_2$O$_3$ film is introduced between graphene and the grating to serve as the top gate dielectric layer. Graphene covered with Al$_2$O$_3$ film is placed on a 200 nm SiO$_2$ wafer that can be produced on a Si substrate. Here, $d$ and $w$ represent, respectively, the period and the width of the gold strips. The applied grating is used not only to excite plasmons via plasmon-electron coupling, but also to tune the electron density in the graphene by applying a gate voltage. S and D is, respectively, the source and drain electrode, to which can be applied a source-to-drain electric field F$_x$ (taken along the x-direction of the graphene sheet). In graphene, the frequency of intra-conduction band plasmon ranges from terahertz to infrared band and the frequency of inter-band plasmon is generally higher than the intra-band plasmon frequency [18]. Thus, here we consider that plasmons in graphene are induced by electrons in the conduction band because terahertz plasmons are induced by intra-conduction band electron transition in graphene [18]. We take the energy balance equation, derived from the semi-classic Boltzmann equation (BE), as governing transport equation to study the plasmon emission by drifting electrons in graphene. The BE in non-degenerate statistics takes the form

(1)$$-\frac{e F_x}{\hbar}\frac{\partial f(\textbf{k})}{\partial k_x}=g_sg_v\sum_\textbf{k'}[F(\textbf{k}',\textbf{k})-F(\textbf{k},\textbf{k}')],$$

where g$_s$=2 and g$_v$=2 are respectively the spin and valley degeneracy; $F(\textbf {k},\textbf {k}')=f(\textbf {k})[1-f(\textbf {k}')]W(\textbf {k},\textbf {k}')$, and $W(\textbf {k},\textbf {k}')$ is the electronic transition rate induced by interactions with scattering centers such as impurities, phonons, and plasmons. Here, $f(\textbf {k})$ is the momentum-distribution function, for an electron in state $\mid \textbf {k}>$ with momentum $\textbf {k} =(k_x,k_y)$, and can be described approximately by a drifted energy-distribution function [19] when an electron gas is subjected to a driving electric field. For graphene, we can take $f(\textbf {k})\simeq f(E(\textbf {k}^*))$, where $\textbf {k}^*=(k_x-k_Fv_x/v_F,k_y)$ is the momentum drifted by electron velocity $v_x$, $k_F$ is the Fermi wavevector for an electron in graphene, $v_F=10^8$ cm/s is the Fermi-velocity, and $E(\textbf {k})=\hbar v_F k$ is the energy of an electron in conduction band in graphene. Here we take the drifted Fermi-Dirac function with electron temperature, $T_e$, as the distribution function for electrons in graphene. Specifically, we take $f(x)= (e^{(x-\mu )/k_B T_e}+1)^{-1}$ with $\mu$ being the Fermi-energy (or chemical potential) for the system, which can be determined by the condition of electron number conservation [20].

Fig. 2. Device model that combines a gold-grated graphene with a dielectric substrate. From top to bottom is the gold grating, Al$_2$O$_3$ dielectric medium, graphene, and SiO$_2$/Si. S and D are, respectively, the source and drain electrode.

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For electron interactions with charged impurities, phonons, and plasmons in graphene, the electronic transition rate can be obtained from Fermi’s golden rule,

(2)$$W(\textbf{k},\textbf{k}')=\sum_j{\frac{2\pi}{\hbar}}|U^j(\textbf{q})|^2\delta[E(\textbf{k})-E(\textbf{k}')\pm\hbar\omega_j],$$

which measures the probability of scattering of an electron from state $|\textbf {k} >$ to state $|\textbf {k}'>$. The square of the matrix elements for electron interactions with impurities: $|U^{im}(\textbf {q})|^2$, acoustic-phonons via deformation potential coupling: $|U^{ap}(\textbf {q})|^2$, and with optic-phonons under the usual valence-force-field approximation: $|U^{op}(\textbf {q})|^2$, have been documented previously [15]. Here, $\textbf {q}=(q_x,q_y)$ is the change in the electron wavevector during a scattering event. For plasmon-electron (p-e) (plasmon-phonon) scattering, $\textbf {q}$ is also the plasmon (phonon) wavevector and the upper (lower) case refers to absorption (emission) of plasmon (phonon) with energy $\omega _j=\omega _p$. It should be noted that a momentum complication is caused by applying the grating so that the momentum is conservative during the coupling between plasmons and electrons.

(3)$$\hbar \textbf{q}=\hbar \textbf{q}'.$$

The left side of the equation is the momentum of a plasmon and the right is the momentum change of an electron with an additional momentum complication with $\textbf {q}'=(q'_x+2\pi /d,q'_y)$. That is, the change of an electron wavevector during a scattering event $\textbf {q}'=(q'_x,q'_y)$ will be added to a wavevector $2\pi /d$ in the direction of the grating arrangement. In Fig. 1 the black solid line is the plasmon spectrum in graphene obtained by random phase approximation (RPA). From Fig. 1(a), we can see that the plasmon spectrum (black solid line) is outside the SPE region below the red line (i.e., $0<\hbar \omega _p<\gamma q$ without the grating). In this situation the plasmons cannot be excited by electron transition during the plasmon-electron coupling. However, from Figs. 1(b)–1(d) we can see that the spectrum of a plasmon can be within the SPE region (i.e., $0\,<\,\hbar \omega _p\,<\,\gamma q'$) with tuning of the period of the applied grating. As observed from Fig. 1, the SPE region is enlarged with decrease in the period of the grating. The plasmon spectrum will fall within the region of SPE when the period of the grating is proper, and the plasmons can couple with electrons. The value of a suitable period is determined by the electron density $n_e$ because the plasmon spectrum is in proportion to ${n_e}^{1/4}$ in RPA (i.e., $\omega _p=(2e^2\gamma q/\hbar ^2 \varepsilon _0)^{1/ 2}(\pi n_e)^{1/4}$) [18,21].

The square of the matrix elements for plasmon-electron coupling can be written as [22]

(4)$$|U^{p-e}(\textbf{k},\textbf{q})|^2=\hbar V(q)[\frac{\partial\varepsilon(\textbf{q},\omega)}{\partial \omega}]^{{-}1}_{\omega_p}{\frac{1}{|\textbf{q}|^2}}(<\Psi_\textbf{k}|e^{i\textbf{q}\cdot \textbf{r}}|\Psi_\textbf{k}'>)^2,$$

where $V(q)=2\pi e^2/q\varepsilon _0$ is the two-dimensional Fourier transform of the Coulomb potential with $\varepsilon _0$ being the effective dielectric constant. $\Psi _\textbf {k}$ and $\Psi _\textbf {k}'$ are, respectively, the initial and final state. Applying the random phase approximation (RPA) [21,23], the dynamical dielectric function can be written as

(5)$$\varepsilon(\textbf{q},\omega)=1-V(q)\sum_\textbf{k} \Pi(\textbf{k},\textbf{q},\omega),$$

where

(6)$$\Pi(\textbf{k}, \textbf{q}, \omega)=g_sg_v\frac{{F}[E(\textbf{k}+\textbf{q})]-{ F}[E(\textbf{k})]}{ \hbar\omega+E(\textbf{k}+\textbf{q})-E(\textbf{k})+i\delta},$$

is the pair bubble or density-density correlation function in the absence of e-e screening, with ${ F}[E(\textbf {k})]$ being the energy-distribution function for electrons in conduction band. In Eq. (6), an infinitesimal quantity $i\delta$ has been introduced to make the integral converge when Fourier transforming from time representation to spectrum-representation. Then combine Eq. (3), Eq. (5) and Eq. (6) and the plasmon-electron coupling matrix can be written as

(7)$$|U(\textbf{k},\textbf{q})|^2=\frac{\hbar^3 \omega_p^3 \pi}{q^2 E_F}\frac{(1+\cos{\varphi})^2}{ 4}$$

with $\varphi$ being the angle between $\textbf {k}$ and k'.

The BE with the electronic transition rate given by Eq. (1) cannot be solved easily. Here we employ the usual balance-equation approach to solve the problem. For the first and second moments, the momentum-balance equation and the energy-balance equation can be derived by multiplying, respectively, $g_sg_v \sum _\textbf {k} k_x$ and $\sum _\textbf {k} E(\textbf {k})$ to both sides of the BE, which reads

(8)$$\frac{eF_x}{\hbar}=\frac{16}{ n_e}\sum_{\textbf{k'},\textbf{k}}(k_x-k_x')F(\textbf{k}',\textbf{k}),$$

and $P_t=(4eF_x/\hbar )\sum _\textbf {k}E(\textbf {k}){(\partial f(\textbf {k})/ \partial k_x)}$ is the energy transfer rate induced by electron-plasmon coupling with

(9)$$P_t={-}16\sum_\textbf{k',{\textbf k}}\hbar\omega_p F(\textbf{k},\textbf{k}').$$

From the energy-balance equation the net energy transfer rate induced by electron-plasmon coupling is given as $P_p=\int _0^\infty d\omega _p \int _0^{2\pi } d\theta P(\omega _p,\theta )$. Here $P(\omega _p,\theta )=P^-(\omega _p,\theta )-P^+(\omega _p,\theta )$ is the distribution function of plasmon emission strength with $\theta$ being the plasmon emission angle measured from the x-axis, which is the difference between plasmon emission and plasmon absorption. Combining Eq. (9) with Eqs. (3)–(7),

(10)$$\begin{aligned}P^\pm(\omega_p,\theta)&=\frac{\hbar^3\omega^3_p (N_q+1/ 2\mp 1/2)}{ 2\varepsilon_0E_F\pi^2}\int dk\int d\phi f(\textbf{k})\times[1 \nonumber \\ & \quad -f(\textbf{k}')] k (1+\cos{\varphi})^2\delta[E(\textbf{k})-E(\textbf{k'})\pm\hbar\omega_p] \end{aligned}$$

with $\phi$ being the angle between $\textbf {k}$ and the x-axis. Here we treat plasmons as bosons so that $N_q=(e^{\hbar \omega _q/k_BT}-1)^{-1}$ becomes the plasmon occupation number. In the fully screened case, the graphene is completely covered by metal, and the effective dielectric constant [24] is $\varepsilon _{screened}=[\varepsilon _1+\varepsilon _2 \coth (qh)]/2$, where $\varepsilon _1=4$ and $\varepsilon _2=3.2$ are, respectively, the dielectric constant of the SiO$_2$ layer and Al$_2$O$_3$ layer [25], and $h$ is the thickness of the Al$_2$O$_3$ layer. In the unscreened/open case, the graphene is opened in vacuum, and the effective dielectric constant is [24]$\varepsilon _{open}={\bigg (}\varepsilon _1+\varepsilon _2[1+\varepsilon _2\tanh (qh)]/[\varepsilon _2+\tanh (qh)]{\bigg )}/2$. For partial shielding by the grating, the effective dielectric constant is $\varepsilon _0= [w\times \varepsilon _{screened}+(d-w)\times \varepsilon _{open}]/2d$.

In this study, we were able to solve the momentum-balance equation (Eq. (8)) and the energy-balance equation (Eq. (9)) self-consistently for a given electric field $F_x$. Thus, we can obtain the drift velocity $v_x$ and temperature $T_e$ for an electron in graphene in the presence of $F_x$. Putting $v_x$ and $T_e$ into Eq. (9), we can calculate the frequency and angular distribution $P(\omega _p,\theta )$ of the plasmon emission generated electrically from graphene. However, it is intricate work to solve self-consistently the two equations with the grating applied on graphene. This can be considered in later work. In the present work, we used the previous results ($v_x$, $T_e$) without applying the grating on graphene to investigate quasi-quantificationally the plasmons emission in such a grated graphene system. All the parameters (i.e., $n_e$, $F_x$, $v_x$, and $T_e$) used in the calculations have been documented in previous work that investigated the transport properties of graphene in a driving electric field [15,20].

3. Results and discussion

In Fig. 3(a) we show the plasmon emission distribution for different driving electric fields with fixed grating period and electron density. As we can see, net plasmon emissions ranging from terahertz to infrared can be observed for this grating period. As depicted in the Fig. 3, there are almost no plasmon emission by electrons when the electric field is absent or very small ($F_x <0.2$ kV/cm). However, an obvious enhancement of plasmon emission strength can be observed when the electric field $F_x>1.0$ kV/cm. It is known that the electron velocity and temperature increase with the strength of the driving electric field [15]. Consequently, the coupling strength of electrons with plasmons will be enhanced with increase in the driving electric field. Thus, the plasmon emission strength increases with the driving electric field strength. Moreover, the width of the distribution spectra are enlarged and, along with a blue-shift, can be seen with increase in the driving electric field. This is because more electrons with higher energy can couple with plasmons with higher frequency with the increasing driving electric field. These results suggest that the plasmon emission spectrum can be tuned electrically by managing the applied driving electric field. Figures 3(b)–3(d) show the angular and frequency dependence of plasmon emission for three different fields with fixed electron density. We noticed the following features. i) The overall intensity of plasmon emission increases significantly with the strength of the driving electric field. ii) The plasmon emission can be observed overall $\theta$ angles and the intensity of the plasmon emission decreases with decreasing $\theta$ angle. iii) The strongest plasmon emission can be observed at angle $\theta =180^\circ$. This implies that in such a graphene system, the generated plasmon waves travel mainly along the direction of a moving electron.

Fig. 3. (a) Plasmon emission distribution in the direction of $\theta =0^\circ$ for different driving electric fields at a fixed grating period and electron density. The drifting electron velocity v$_x$ and temperature T$_e$ for the driving electric fields of 15 kV/cm, 10 kV/cm, 5.0 kV/cm, 1.0 kV/cm, and 0.2 kV/cm are, respectively, 1.76$\times$10$^7$ cm/s, 1.49$\times$10$^7$ cm/s, 1.06$\times$10$^7$ cm/s, 3.35$\times$10$^6$ cm/s, and 0.72$\times$10$^6$ cm/s, at 712.01 K, 602.70K, 474.78 K, 324.25 K, and 301.26 K. (b)-(d) Angular and frequency dependence of plasmon emission for different electric fields $F_x=15,\ 10$ and $5.0$ kV/cm at a fixed electron density $n_e=1.0\times 10^{12}$ cm$^{-2}$.

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The plasmon emission distributions are different when different grating periods are applied. The emission distributions shown in Fig. 4 are different from the grating periods in Fig. 3. The plasmon spectrum is near the boundary of the SPE for the former, as shown in Fig. 1(c). In this situation, we can see that there is not only the net plasmon emission, but also plasmon absorption. Moreover, the net plasmon emission occurs at low frequency (terahertz) while the net plasmon absorption can be observed at high Frequency (infrared). This means that electrons with low energy tend to emit plasmons, while electrons with high energy tend to absorb plasmons through the landau damping of plasmons. The strength of plasmon absorption at high frequency is much more than the plasmon emission strength at low frequency, which indicates that the coupling strength between plasmons and electrons with high energy are more than that with low energy. Moreover, not only the plasmon emission, but also the plasmon absorption, increase with increase in the electric field strength and an obvious blue shift in the absorption can be seen with increase in the driving electric field. These results suggest that the plasmon emission spectrum can be tuned by the applied driving electric field or by changing the period of the applied grating. It is exciting to compare the results shown in Fig. 3 and Fig. 4, which indicate that it is desirable (and possible) to electrically obtain terahertz plasmons by applying a grating with proper period. This makes production and tuning of terahertz radiation possible using electrical methods.

Fig. 4. Plasmon emission distribution for different driving electric fields at a fixed grating period and electron density. The drifting electron velocity v$_x$ and temperature T$_e$ for the different driving electric fields are the same as in Fig. 3.

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The inset in Fig. 5 is similar to Fig. 1(b) except that $q$ is extended to a larger value so that the plasmon spectrum can go into the SPE region at high frequency. The plasmon frequencies in the SPE region shown in the inset is in line with the x-axis of Fig. 5. Thus, the plasmons with high frequency can couple with electrons under these conditions. However, as we can see, net plasmon emission is absent while there is a net absorption for the high plasmon frequencies. Thus, we cannot obtain net plasmon emission for this period of grating. Moreover, from the results shown in Fig. 5 one can know that the plasmon spectrum region in high frequencies can be in the SPE region despite that net plasmon emission cannot be obtained without application of the grating. Thus, it is important to choose an applied grating with suitable period. This choice is determined by the electron density because the plasmon spectrum depends on the electron density.

Fig. 5. Plasmon emission distribution for different driving electric fields at high frequency for the grating period $L=100$ nm. The inset is similar to Fig. 1(b) except that $q$ is extended to a larger value.

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It is known that the electron density in graphene can easily be tuned from the order of 10$^{11}$ cm$^{-2}$ to 10$^{13}$ cm$^{-2}$ by managing the applied gate voltage [26]. In Fig. 6 we show the plasmon emission distribution with plasmon frequencies for different electron densities at fixed driving electric field strength and grating period. The results indicate that the effect of electron density on the plasmon emission spectrum is complex. The plasmon emission strength and width of the plasmon spectrum increase with increasing electron density for low electron densities (1.0$\times 10^{11}$ cm$^{-2}$-1.0 $\times 10^{13}$ cm$^{-2}$), as depicted by the dashed lines and solid purple line. However, the plasmon emission strength and the width of the plasmon emission spectrum decrease with the subsequently increasing electron density (1.0 $\times 10^{12}$ cm$^{-2}$-5.0 $\times 10^{13}$ cm$^{-2}$) depicted by the solid lines. The reason that the plasmon emission strength first increases then decreases with the subsequently increasing electron density, is because the electron density and electron temperature (velocity) have a competitive effect on the plasmon emission strength. High electron density means more electrons will couple with plasmons and that the plasmons emission strength will be enhanced. However, as shown elsewhere [15], the electron temperature and velocity decrease with increasing electron density for a certain driving electric field, which leads to a decrease in the electron-plasmon coupling strength. On the other hand, as can be seen, there is a blue shift of the plasmon emission spectrum with increasing electron density for low electron densities, while a red shift occurs with the subsequently increasing electron density for high electron densities. Moreover, net plasmon emission exists with low electron densities while net plasmon absorption occurs in the high frequency region for high electron densities. The energy (frequency) of plasmons in graphene is in proportion to ${n_e}^{1/4}$, so that electrons tend to couple with high frequency plasmons for high electron densities. This leads to a plasmon emission spectrum that exhibits a blue shift with increasing electron density. As indicted above, plasmons with high energy tend to be absorbed by electrons, which results in net plasmon absorption at high frequency for high electron densities.

Fig. 6. Plasmon emission spectrum for different electron densities with a fixed electric field and grating parameters (i.e., $L=20$ nm and $w=10$ nm). The electron temperature (423.72 K, 457.89 K, 474.78 K, 514.38 K, and 556.12 K) is dependent on the electron density with decreasing electron density.

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4. Conclusions

In this work we studied theoretically the electrodynamic plasmon emission in a graphene system driven by a dc electric field by applying the Boltzmann’s equilibrium equation method. The conclusions from this work include: a plasmon spectrum can be shifted to within the single particle region by applying a grating on the graphene system, thereby allowing electrons to couple with plasmons. Plasmons with frequencies from terahertz to infrared can be obtained by applying a grating with proper period on the graphene system. Generally, electrons tend to absorb plasmons when coupled with high frequency plasmons, but emit plasmons when coupled with low frequency plasmons. Thus, net emission of low frequency plasmons exists with low electron densities while net absorption of high frequency plasmons exists with high electron densities. The plasmon emission strength increases with increase in the driving electric field while the effect of electron density on plasmon emission is complicated by the competition between the electron density and electron temperature (velocity), which are determined by the electron density. The plasmon emission strength first increases with electron density; then decreases with the subsequently increasing electron density. These results suggest that terahertz plasmons can be generated and tuned by changing the parameters of the grating or by control of the electron density using the applied gate voltage. More importantly, generation and tuning of terahertz radiation can be achieve by electrical methods based on plasmon emission from graphene.

Funding

National Natural Science Foundation of China (11604192, 11847029, 11604380); Applied Basic Research Project of Shanxi Province (201801D221113); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (201804018); Doctoral Start-up Funding from Shanxi Normal University (0505/02070351); Doctoral Scientific Research of Jining Medical University (2018ZJYQD06).

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Emission of terahertz plasmons from driven electrons in grated graphene

Abstract

1. Introduction

2. Theoretical considerations and approaches

3. Results and discussion

4. Conclusions

Funding

References

Cited By

Figures (6)

Equations (10)

Optics Express