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) structures1 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 drag15–21 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 

FIG. 1.

(a) Cross section and (b) top view of a lateral n+-i-n-n+ G-FET with ballistic i-region and gated n-region using the Coulomb electron drag and (c) G-FET equivalent circuit including an antenna with radiation resistance R.

FIG. 1.

(a) Cross section and (b) top view of a lateral n+-i-n-n+ G-FET with ballistic i-region and gated n-region using the Coulomb electron drag and (c) G-FET equivalent circuit including an antenna with radiation resistance R.

Close modal

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, li, of the undoped i-region [see Fig. 1(a)] is sufficiently short allowing for the ballistic motion of the injected electrons: liτi/vW, where vW108 cm/s, τi 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 li1μm at room temperature25 and in even fairly long graphene layers at reduced temperatures;26 

  • The characteristic time, τee, of the BE-QE collisions τee is the shortest scattering time in the n-region: τee<τopτn, 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;

  • Then, the energy of the BEs injected into the n-region exceeds the energy of optical phonons (ω0200 meV), the pertinent scattering mechanism can substantially dissipate the energy and momentum of the BEs.

When the length of the gated n-region ln>vWτee, 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 Σn1×(10121013) cm−2 and temperature T300 K for the energy, εBE, of the BEs injected into the n-region about the optical phonon energy ω0200 meV one can set τee1(1050) ps−1 and τop1=(12) ps−1 (Ref. 27) with the QE net scattering time τn on the acoustic phonons and impurities about τn12 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 τee is much shorter than the time of BE collisions with the remote charged impurities.

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:

jBE+jDP(i)=jDQE+jQE+jDP(n).
(1)

Here, jBE and jQE 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)=cidΦi/dt and jDP(n)=cnd(VΦi)/2dt are the displacement currents, Φi(t) and V(t)Φi(t) are the potential drops across the i- and the n-regions, respectively. The capacitances are given by Ci/H=ci=(κ/2π2)lnλ (accounting for the specific of the structure29–32) and Cn/H=cn=lnκ/4πd, where κ is the dielectric constant of the material in which graphene is embedded, lnλln(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 by23jDQE=bjBE2eK/j0, where j0=vW(κω0/2πlie) is the characteristic current density and b=(ω0/μn)eKn is the Coulomb drag factor (which describes the drag current multiplication) with Kn=ln/vWτn. The exponent K=Kop[(εBE+μn)/ω01]·Θ(εBE/ω01) is the probability of the optical phonon emission by a BE with the energy εBE, which accounts for the Pauli principle and for the threshold character of such an emission process characterized by the step-like function Θ(z). The function Θ(z)=[1+exp(2zω0/kBT]1 describes the temperature smearing of the optical phonon emission threshold. (kB is the Boltzmann constant.)

At the dc bias voltage V=V¯ applied between the source and drain contacts and the dc voltage drop Φ¯ across the i-region, the source–drain current density is equal to j¯BE=σiΦ¯i/li and j¯QE=σn(V¯Φ¯)/ln. Here, σi=vWκ/2π is the i-region dc conductivity in the “virtual cathode” approximation29 and σn=e2Σnτn/mn is the drift (Drude) conductivity of the n-region, where Σn, τn, and mn are the QE density, scattering time, and fictitious effective mass, respectively. In this case, considering that εBE/ω0=eΦ¯i/ω0 (e is the electron charge) and introducing the normalized current density J¯=j¯BE/j0, from Eq. (1), we arrive at the following equation relating J¯ and V¯:

J¯b(1+η)J¯2eK(J¯)=η(1+η)V¯V0.
(2)

Here, η=σnli/σiln is the ratio of the i- and n-regions resistances ri=li/σiH,rn=ln/σnH, and V0=ω0/e. Equation (2) describes the monotonic and the S-shaped IV characteristics at 2b/(1+η)<1 and 2b/(1+η)>1, respectively.23 

Considering the G-FET dynamic response, we assume that the voltages V and Φi comprise the ac components: V=V¯+δVωexp(iωt) and Φi=Φ¯i+δΦω, where ω is the signal frequency and the normalized source–drain current also includes the pertinent ac contribution δJω=δΦω/V0.

In this case, in the linearized version of Eq. (1), we put δjBE/j0=σiδΦω/lij0(1iωτi)=δΦω/V0(1iωτi) and δjQE/j0=σn(δVωδΦω)/lnj0(1iωτn)=η(δVωδΦω)/V0(1iωτn), where τi and τn determine the pertinent regions kinetic inductance. The scattering time τn 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 cicn. This is justified in the range of frequencies under consideration. The voltage drop across the G-FET δVω can be expressed via the net ac voltage δVω, as δVω=δVωδJ¯ωj0RH, where R is the emitting antenna radiation resistance.

As a result, from Eq. (1) for the ac component of the normalized current δJ¯ω, we obtain the following equation:

δJω=riZωδVωV0.
(3)

Here,

Zω=ri[(MM0)M0(1+ω2τn2)1+iωτn(Ω2ω2)τn21Ω2τn2+1+Rri],
(4)

is the net impedance of the loop circuit under consideration. Deriving Eqs. (3) and (4), we have introduced the quantities: M0=η/(1+η), M=1[b/(1+η)]d[J2eK(J)]/dJ|J=J¯, which depend on the parameters b and η, and the plasma frequency

Ω=8πe2Σndκmnln2=eln8μndκμnln.
(5)

Here, μnvWπΣn is the QE Fermi energy. The plasma frequency given by Eq. (5) corresponds to its standard value for the plasma wavelength λ=2πln4.4ln. Setting μn=30100 meV, κ = 4, d=(510)×106 cm, and ln=104 cm, we obtain Ω/2π(0.531.37) THz.

In the range of low frequencies ω,Ωτn1, Eq. (4) yields ZωR+ri+rn (in the absence of the Coulomb drug, b = 0 and M = 1) and ZωR+ri (when the drug is pronounced, M1).

At the plasmonic resonance ω=Ω2τn2, the impedance imaginary part becomes zero, and Eq. (4) yields

ZωR=R+ri+ri(MM0)M0Ω2τn2.
(6)

Equation (6) yields the condition ZωR<0 in the following forms:

MM0<1(R+ri)riΩ2τn2,b>1+(R+ri)rn1Ω2τn2d[J2eK(J)]/dJ|J=J¯.
(7)

Considering that η/Ω2τn2=(liln/4πdvWτn)=(li/4πd)Kn, inequality (7) can be presented as

ω0μn>[1+(Rri+1)(li4πd)Kn]d[J2eK(J)]/dJ|J=J¯eKn.
(8)

The latter condition is valid at not too small μn (μn>kBT). As follows from Eqs. (7) and (8), the instability criteria primarily requires a sufficiently large value b=(ω0/μn)eKn, i.e., not too large Kn. This implies a relatively strong Coulomb electron drag.

Figures 2 and 3 show the real part Re Zω/ri and the imaginary part Im Zω/ri of normalized impedance vs signal frequency ω/2π calculated for different values of the normalized bias current J¯ using Eq. (4) and the M vs J¯ dependence shown in the inset in Fig. 2. The resonant impedance ZωR/ri as a function of the normalized bias current J¯ 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=105 cm, ln=104 cm, d=105 cm, Kn = 1, Kop = 1, and κ = 4. For these parameters, assuming that the G-FET width H=103 cm, we obtain ri140 Ω. One can see that at selected structural parameters and the bias current (bias voltage), Re Zω<0 in the THz range. Just in the range, where Re Zω<0, Im Zω changes its sign turning zero at the plasmonic resonance. This corresponds to the self-excitation of high-frequency oscillations24—the plasma oscillations in our case, followed by the radiation emission from the antenna. When J¯1, i.e., V¯V0200 mV, in a G-FET with the above parameters the dc current J¯j0H1.4 mA. According to Figs. 2 and 3, the swing of the normalized current, at which Zω<0, is about of ΔJ¯0.5J¯, so that Δj=ΔJ¯j0H0.7 mA. Setting |Zω|=2ri=280 Ω, so that |Zω| matches the standard THz antenna radiation resistance (see, for example, Ref. 33), for the characteristic emitted THz power, we obtain Pω140μ W, or Pω/H14 mm/mW.

FIG. 2.

The real part Re Zω/ri (a) and imaginary part Im Zω/ri (b) of the G-FET impedance vs signal frequency f=ω/2π for different values of normalized bias current J¯: Ω/2π=0.96 THz, τn=1 ps, μn=50 meV, b = 1.47, η=9.2, and R/ri=1. Inset shows parameter M vs bias current J¯ for different values of 2b/(1+η): 1—2b/(1+η)=0.1, 2—0.5, and 3—1.0.

FIG. 2.

The real part Re Zω/ri (a) and imaginary part Im Zω/ri (b) of the G-FET impedance vs signal frequency f=ω/2π for different values of normalized bias current J¯: Ω/2π=0.96 THz, τn=1 ps, μn=50 meV, b = 1.47, η=9.2, and R/ri=1. Inset shows parameter M vs bias current J¯ for different values of 2b/(1+η): 1—2b/(1+η)=0.1, 2—0.5, and 3—1.0.

Close modal
FIG. 3.

The same as in Figs. 2(a) and 2(b), but for Ω/2π=0.68 THz, τn=1 ps, μn=30 meV, b = 2.25, η=5.52, and R/ri=1. Inset shows the resonant impedance ZωR/ri vs normalized bias current J¯ for different values of the plasma frequency: Ω/2π=0.96 THz—solid line, and Ω/2π=0.68 THz—dashed line (other parameters are the same as for Figs. 2 and 3, respectively).

FIG. 3.

The same as in Figs. 2(a) and 2(b), but for Ω/2π=0.68 THz, τn=1 ps, μn=30 meV, b = 2.25, η=5.52, and R/ri=1. Inset shows the resonant impedance ZωR/ri vs normalized bias current J¯ for different values of the plasma frequency: Ω/2π=0.96 THz—solid line, and Ω/2π=0.68 THz—dashed line (other parameters are the same as for Figs. 2 and 3, respectively).

Close modal

The plasma instability under consideration caused by the negativity of the impedance real part in a certain frequency range can occur at the monotonic IV 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, σn/(1+ω2τn2), 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).

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

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