We predict the self-excitation of terahertz (THz) oscillations due to the plasma instability in the lateral n^{+}-i-n-n^{+} graphene field-effect transistors (G-FETs). The instability is associated with the Coulomb drag of the quasi-equilibrium electrons in the gated channel by the injected ballistic electrons resulting in a positive feedback between the amplified dragged electrons current and the injected current. The plasma excitations arise when the drag effect is sufficiently strong. The drag efficiency and the plasma frequency are determined by the quasi-equilibrium electron Fermi energy (i.e., by their density). The conditions of the terahertz plasma oscillation self-excitation can be realized in the G-FETs with realistic structural parameters at room temperature enabling the potential G-FET-based radiation sources for THz applications.

The self-excitation of high-frequency plasma oscillations in the field-effect transistor (FET) structures^{1} associated with the plasma instability was predicted a long time ago.^{2,3} The detection and emission of the terahertz (THz) radiation in FETs, primarily attributed to the plasma oscillations excitation, were reported in many theoretical and experimental papers (see Ref. 4 and the references therein). The plasmonic effects in graphene field-effect transistors (G-FETs)^{5} can enable marked advantages of such devices for the THz detection and emission.^{6–14}

The linearity of the carrier energy dispersion law in graphene determines the specifics of the electron–electron (electron–hole and hole–hole) Coulomb interactions. In particular, it enables a pronounced Coulomb carrier drag^{15–21} and can affect the plasma-electron-beam instability.^{22} As recently demonstrated,^{23} in the lateral n^{+}-i-n-n^{+} G-FETs with the structure shown in Figs. 1(a) and 1(b), the two-dimensional ballistic electrons (BEs) injected from the source via the i-region into the gated n-region can effectively drag the quasi-equilibrium electrons (QEs) in this region. Due to the BE-QE scattering, the current of the dragged QEs can exceed the current of the injected BEs (the current amplification, schematically shown in Fig. 1(c) as the current depended current source). The latter can cause the reverse injection of the QEs from the drain and affect the G-FET current–voltage (*I–V*) characteristics resulting in their strong nonlinearity.^{23}

In this paper, we demonstrate that the QE drag by the BEs in the G-FETs can provide the positive feedback between the BE current injected from the source and the QE reverse current. This can enable the instability of the DC flow in the G-FET channel. Since the QE system in the G-FET can play the role of the THz resonant plasmonic cavity,^{3,5} the instability in question can lead to the self-excitation of the THz plasma oscillations feeding an antenna and resulting in the THz radiation. We derive the G-FET source–drain small-signal impedance as a function of the device structural parameters and show that the real part of the G-FET impedance can be negative at the plasma frequency, at which the imaginary part turns zero, that corresponds to the self-excitation of plasma oscillations.^{24}

We assume the following:

The length,

*l*, of the undoped i-region [see Fig. 1(a)] is sufficiently short allowing for the ballistic motion of the injected electrons: $li\u226a\tau i/vW$, where $vW\u2243108$ cm/s,_{i}*τ*is the characteristic time of the BE scattering on the acoustic phonons and impurities. In the graphene encapsulated in hBN, this condition can be realized if $li\u22721\u2009\mu $m at room temperature_{i}^{25}and in even fairly long graphene layers at reduced temperatures;^{26}The characteristic time,

*τ*, of the BE-QE collisions_{ee}*τ*is the shortest scattering time in the n-region: $\tau ee<\tau op\u226a\tau n$, where the latter times are associated with the scattering of the BEs on the optical phonons and other momentum-non-conserving collisions, such as those involving impurities;_{ee}Then, the energy of the BEs injected into the n-region exceeds the energy of optical phonons ($\u210f\omega 0\u2243200$ meV), the pertinent scattering mechanism can substantially dissipate the energy and momentum of the BEs.

When the length of the gated n-region $ln>vW\tau ee$, the majority of the BEs injected into the n-region transfer their energy and, what is crucial, the momentum to the QEs. This causes their drag toward the drain forming the current exceeding the injected current since the energy conservation at the electron–electron collisions, the net velocity of the electron system can increase. At the electron densities $\Sigma n\u22431\xd7(1012\u20131013)$ cm^{−2} and temperature $T\u2272300$ K for the energy, $\epsilon BE$, of the BEs injected into the n-region about the optical phonon energy $\u210f\omega 0\u2243200$ meV one can set $\tau ee\u22121\u2243(10\u201350)$ ps^{−1} and $\tau op\u22121=(1\u20132)$ ps^{−1} (Ref. 27) with the QE net scattering time *τ _{n}* on the acoustic phonons and impurities about $\tau n\u223c1\u20132$ ps.

^{28}The minimization of the BE scattering on impurities in the n-region is possible due to primarily electrostatic doping of this region by the gate voltage. Similar situation can occur in the case of the n-region doping by donors sufficiently placed far away from the channel when

*τ*is much shorter than the time of BE collisions with the remote charged impurities.

_{ee}Considering the G-FET with the equivalent circuit shown in Fig. 1(c) and equalizing the BE current across the i-region and the net currents across the n-region, we arrive at the following equation:

Here, *j _{BE}* and

*j*are the densities of the BE and QE currents across the i-region and the gated n-region, accounting for their resistances and kinetic inductivities, $jDP(i)=cid\Phi i/dt$ and $jDP(n)=cnd(V\u2212\Phi i)/2\u2009dt$ are the displacement currents, $\Phi i(t)$ and $V(t)\u2212\Phi i(t)$ are the potential drops across the i- and the n-regions, respectively. The capacitances are given by $Ci/H=ci=(\kappa /2\pi 2)\u2009ln\u2009\lambda $ (accounting for the specific of the structure

_{QE}^{29–32}) and $Cn/H=cn=ln\kappa /4\pi \u2009d$, where

*κ*is the dielectric constant of the material in which graphene is embedded, $ln\u2009\lambda \u2243\u2009ln\u2009(4ln/li)$ is on the order of unity,

*H*is the G-FET width, and

*d*is the gate layer thickness.

The drag current density is given by^{23} $jDQE=bjBE2e\u2212K/j0$, where $j0=vW(\kappa \u210f\omega 0/2\pi \u2009lie)$ is the characteristic current density and $b=(\u210f\omega 0/\mu n)e\u2212Kn$ is the Coulomb drag factor (which describes the drag current multiplication) with $Kn=ln/vW\tau n$. The exponent $K=Kop[(\epsilon BE+\mu n)/\u210f\omega 0\u22121]\xb7\Theta (\epsilon BE/\u210f\omega 0\u22121)$ is the probability of the optical phonon emission by a BE with the energy $\epsilon BE$, which accounts for the Pauli principle and for the threshold character of such an emission process characterized by the step-like function $\Theta (z)$. The function $\Theta (z)=[1+exp\u2009(\u22122z\u210f\omega 0/kBT]\u22121$ describes the temperature smearing of the optical phonon emission threshold. (*k _{B}* is the Boltzmann constant.)

At the dc bias voltage $V=V\xaf$ applied between the source and drain contacts and the dc voltage drop $\Phi \xaf$ across the i-region, the source–drain current density is equal to $j\xafBE=\sigma i\Phi \xafi/li$ and $j\xafQE=\sigma n(V\xaf\u2212\Phi \xaf)/ln$. Here, $\sigma i=vW\kappa /2\pi $ is the i-region dc conductivity in the “virtual cathode” approximation^{29} and $\sigma n=e2\Sigma n\tau n/mn$ is the drift (Drude) conductivity of the n-region, where Σ_{n}, *τ _{n}*, and

*m*are the QE density, scattering time, and fictitious effective mass, respectively. In this case, considering that $\epsilon BE/\u210f\omega 0=e\Phi \xafi/\u210f\omega 0$ (

_{n}*e*is the electron charge) and introducing the normalized current density $J\xaf=j\xafBE/j0$, from Eq. (1), we arrive at the following equation relating $J\xaf$ and $V\xaf$:

Here, $\eta =\sigma nli/\sigma iln$ is the ratio of the i- and n-regions resistances $ri=li/\sigma iH,\u2009rn=ln/\sigma nH$, and $V0=\u210f\omega 0/e$. Equation (2) describes the monotonic and the S-shaped *I*–*V* characteristics at $2b/(1+\eta )<1$ and $2b/(1+\eta )>1$, respectively.^{23}

Considering the G-FET dynamic response, we assume that the voltages *V* and $\Phi i$ comprise the ac components: $V=V\xaf+\u2009\delta V\omega \u2009exp\u2009(\u2212i\omega t)$ and $\Phi i=\Phi \xafi+\delta \Phi \omega $, where *ω* is the signal frequency and the normalized source–drain current also includes the pertinent ac contribution $\delta J\omega =\delta \Phi \omega /V0$.

In this case, in the linearized version of Eq. (1), we put $\delta jBE/j0=\sigma i\delta \Phi \omega /lij0(1\u2212i\omega \tau i)=\delta \Phi \omega /V0(1\u2212i\omega \tau i)$ and $\delta jQE/j0=\sigma n(\delta V\omega \u2212\delta \Phi \omega )/lnj0(1\u2212i\omega \tau n)=\eta (\delta V\omega \u2212\delta \Phi \omega )/V0(1\u2212i\omega \tau n)$, where *τ _{i}* and

*τ*determine the pertinent regions kinetic inductance. The scattering time

_{n}*τ*coincides with the ratio of the n-region inductance and resistance. Since the transit time of the BEs across the i-region is short, the i-region kinetic ballistic inductance can be disregarded. We also disregard the displacement current across the i-region due to $ci\u226acn$. This is justified in the range of frequencies under consideration. The voltage drop across the G-FET $\delta V\omega $ can be expressed via the net ac voltage $\delta V\omega $, as $\delta V\omega =\delta V\omega \u2212\delta J\xaf\omega j0RH$, where

_{n}*R*is the emitting antenna radiation resistance.

As a result, from Eq. (1) for the ac component of the normalized current $\delta J\xaf\omega $, we obtain the following equation:

Here,

is the net impedance of the loop circuit under consideration. Deriving Eqs. (3) and (4), we have introduced the quantities: $M0=\eta /(1+\eta )$, $M=1\u2212[b/(1+\eta )]d[J2e\u2212K(J)]/dJ|J=J\xaf$, which depend on the parameters *b* and *η*, and the plasma frequency

Here, $\mu n\u2243\u210f\u2009vW\pi \Sigma n$ is the QE Fermi energy. The plasma frequency given by Eq. (5) corresponds to its standard value for the plasma wavelength $\lambda =2\pi \u2009ln\u22434.4ln$. Setting $\mu n=30\u2013100$ meV, *κ* = 4, $d=(5\u201310)\xd710\u22126$ cm, and $ln=10\u22124$ cm, we obtain $\Omega /2\pi \u2243(0.53\u20131.37)$ THz.

In the range of low frequencies $\omega ,\Omega \u226a\tau n\u22121$, Eq. (4) yields $Z\omega \u2243R+ri+rn$ (in the absence of the Coulomb drug, *b* = 0 and *M* = 1) and $Z\omega \u2243R+ri$ (when the drug is pronounced, $M\u226a1$).

At the plasmonic resonance $\omega =\Omega 2\u2212\tau n\u22122$, the impedance imaginary part becomes zero, and Eq. (4) yields

Equation (6) yields the condition $Z\omega R<0$ in the following forms:

Considering that $\eta /\Omega 2\tau n2=(liln/4\pi \u2009dvW\tau n)=(li/4\pi \u2009d)Kn$, inequality (7) can be presented as

The latter condition is valid at not too small *μ _{n}* ($\mu n>kBT$). As follows from Eqs. (7) and (8), the instability criteria primarily requires a sufficiently large value $b=(\u210f\omega 0/\mu n)e\u2212Kn$, i.e., not too large

*K*. This implies a relatively strong Coulomb electron drag.

_{n}Figures 2 and 3 show the real part Re $Z\omega /ri$ and the imaginary part Im $Z\omega /ri$ of normalized impedance vs signal frequency $\omega /2\pi $ calculated for different values of the normalized bias current $J\xaf$ using Eq. (4) and the *M* vs $J\xaf$ dependence shown in the inset in Fig. 2. The resonant impedance $Z\omega R/ri$ as a function of the normalized bias current $J\xaf$ calculated using Eq. (6) is shown in the inset in Fig. 3. The structural parameters used for Figs. 2 and 3 correspond to realistic values: $li=10\u22125$ cm, $ln=10\u22124$ cm, $d=10\u22125$ cm, *K _{n}* = 1,

*K*= 1, and

_{op}*κ*= 4. For these parameters, assuming that the G-FET width $H=10\u22123$ cm, we obtain $ri\u2243140$ Ω. One can see that at selected structural parameters and the bias current (bias voltage), Re $Z\omega <0$ in the THz range. Just in the range, where Re $Z\omega <0$, Im $Z\omega $ changes its sign turning zero at the plasmonic resonance. This corresponds to the self-excitation of high-frequency oscillations

^{24}—the plasma oscillations in our case, followed by the radiation emission from the antenna. When $J\xaf\u223c1$, i.e., $V\xaf\u223cV0\u2243200$ mV, in a G-FET with the above parameters the dc current $J\xafj0H\u22431.4$ mA. According to Figs. 2 and 3, the swing of the normalized current, at which $Z\omega <0$, is about of $\Delta J\xaf\u22430.5J\xaf$, so that $\Delta j=\Delta J\xafj0H\u22430.7$ mA. Setting $|Z\omega |=2ri=280$ Ω, so that $|Z\omega |$ matches the standard THz antenna radiation resistance (see, for example, Ref. 33), for the characteristic emitted THz power, we obtain $P\omega \u2243140\u2009\mu $ W, or $P\omega /H\u224314$ mm/mW.

The plasma instability under consideration caused by the negativity of the impedance real part in a certain frequency range can occur at the monotonic *I*–*V* characteristics. Its origin is associated with the dynamic feedback between the injected source current and the reverse current injected from the drain with a marked delay of the latter due to a small dynamic conductivity, $\sigma n/(1+\omega 2\tau n2)$, of the n-region at elevated frequencies. In this regard, the mechanism in question is, to some extent, akin to those explaining the current-driven plasma instabilities previously reported (see, for example, Refs. 2, 3, 34, and 35).

In conclusions, we predicted the possibility of the current driven plasma instability in the lateral G-FETs with the BE injection into the gated n-region region and the Coulomb drag of the QE by the BEs. The plasma instability and the pertinent self-excitation of the THz oscillation are associated with the amplification of the current due to the transfer of the BE momentum to the QEs. The plasma oscillations self-excitation can lead to the THz radiation emission using the proper antenna. The G-FETs under consideration can be connected in series forming a periodic lateral structure (like in Refs. 11, and 34–36) that can enhance the THz emission.

The work at RIEC and UoA was supported by the Japan Society for Promotion of Science (KAKENHI Nos. 21H04546 and 20K20349), Japan; and the RIEC Nation-Wide Collaborative research Project No. H31/A01, Japan. The work at RPI was supported by the Office of Naval Research (No. N000141712976, Project Monitor Dr. Paul Maki).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

^{+}-i-n-n

^{+}graphene field-effect transistors due to the Coulomb drag of quasi-equilibrium electrons by ballistic electrons