Coulomb drag and plasmonic effects in graphene field-effect transistors enable resonant terahertz detection

The unique electron properties of graphene enable substantial performance enhancement of different devices.^1,2 This is associated with the possibility of a very high electron (and hole) directed velocity^3,4 close to the graphene characteristic velocity⁵ $vW=108$ cm/s and with the realization of the electron ballistic motion^6–8 in relatively long graphene layer channels.^9,10

The quasi-one-dimensional electron–electron (electron–hole and hole–hole) Coulomb interaction in graphene due to the linearity of carrier energy dispersion laws can lead to the pronounced Coulomb carrier drag.^11–23 Usually, the Coulomb drag effect reveals when the electrons (or holes) propagating along a graphene layer under the influence of the electric field transfer a part of their directed momentum to the holes (or electrons) in the graphene layer located close to the former layer. This happens due to the Coulomb interaction between the carriers belonging to these neighboring layers. Such an interaction leads to the appearance of the drag current²³ (and the references therein). A very similar drag effect can be realized in monopolar graphene lateral structures when the hot (ballistic) electrons accelerated in a part of the structure are injected into the region of this structure with sufficiently dense quasi-equilibrium electron (QE) plasma. In this case, the transfer of the ballistic electrons' (BE) momenta to the QEs results in the drag of the latter and, hence, their drag current. As we showed recently,^24,25 in the lateral n⁺-i-n-n⁺ graphene field-effect transistors (GFETs), the two-dimensional BEs injected from the n⁺-region via the i-region into the gated n-region can drag the QEs toward the n⁺ drain contact. In contrast to the graphene double-layer structures in which the electrons propagating in one graphene layer drag the electrons (or holes) in another,^18–23 in the lateral n⁺-i-n-n⁺ GFETs, the hot BEs accelerated in the depleted i-region are injected into the n-region of the same graphene layer. Colliding with the QEs in this region, the BEs transfer QEs energy and momentum. This momentum is redistributed over the QEs. Due to the specifics of the two-dimensional electrons scattering in graphene, the dragged electrons (DEs) move with a high velocity v_W. Hence, such a Coulomb drag enables the current amplification leading to a strong nonlinearity of the GFET source-drain current–voltage (I–V) characteristics. The resulting I–V characteristics could be either monotonic or S-shaped.²⁴ The Coulomb drag also leads to plasma instability.²⁵ In addition, the Coulomb drag effect could enable the detection of terahertz (THz) radiation based on the rectification of the incoming signals.

In this paper, we analyze the operation of the THz detectors based on the lateral n⁺-i-n-n⁺ GFETs exploiting the transfer of the energy and momentum of the BEs to the QEs in the gated n-region, leading to the Coulomb drag of the QEs toward the drain in the lateral n⁺-i-n-n⁺ GFETs. We account for a strong nonlinearity of the DE current and the resonant plasmonic properties of the gated n-region. The former enables an effective rectification of the ac current stimulated by incoming radiation, while the latter provides an elevated response at the resonant plasmonic frequencies.

We show that these detectors can surpass the THz detectors using other nonlinearity mechanisms of the current rectification such as the thermionic, tunneling, resonant-tunneling mechanisms, and, in particular, the plasmonic effects in the traditional materials^26–36 and graphene structures.^37–43 

Figure 1 shows the structure of the GFET under consideration. The positive dc voltage V_g forms the n-region. Apart from the dc bias component $V¯$⁠, the source-drain voltage comprises the ac signal component $δVω(t)=δVω exp(−iωt)$ produced by incident radiation of frequency ω received by an antenna, where $δVω$ is the component amplitude. As a result, the net source-drain current J includes the dc component $J¯$⁠, the ac component $δJω$ (including the fundamental and higher harmonics), and the rectified component $ΔJω$⁠.

We assume that the conditions of the BE transport across the i-region (its length l_i being sufficiently small) and of the effective drag of the QEs by the BEs (effective electron–electron scattering and weak electron scattering on disorder) are met^24,25 (see also Refs. 9, 10, 44–47).

Applying the Kirchhoff circuit law to the GFET channel, we arrive at the following equation equating the BE current density J_BE injected into the gated n-region with the sum of the current density, J_QE, associated with the quasi-equilibrium Drude transport and controlled by the gate capacitance and the current density, J_DE, of the dragged electrons

The density of the BE injected current in the depleted i-region and the DE current in the n-region are given by^20,21

Here, $σi=κ vW/2πli$⁠, b is the drag parameter,^20,21$Φ$ is the potential drop across the i-region, $J0=κvWℏω0/2π eli$ is the characteristic current equal to the BE current at $Φ=ℏω0/e$⁠, where $ℏω0$ is the optical phonon energy in graphene, and e is the electron charge. Due to the linearity of the gated channel response, $JQE∝(V−Φ)$⁠. The introduced quantity J₀ is the threshold current at which optical phonon emission starts. Such an emission affects the dc and ac characteristics at the current densities exceeding J₀. In the following, we assume that $JBE≤J0$⁠.

As follows from the dc version of Eq. (1), the GFET dc characteristics (when $V=V¯$⁠) are determined by the drag parameter b and by the DE current nonlinearity parameter N. When $J¯=J¯BE/J0≤1,N=(b/2)[d(dJDE2/dJ)/dJ]|J=J¯=b$ [see Eq. (2)]. At sufficiently large b, the GFET source-drain I–V characteristics can have the S-type shape.^24,47 In the following, we disregard such a case.

Using the linearized ac version of Eq. (1) for the fundamental current harmonic (corresponding to frequency ω), we obtain

Here, $Zω$ is the net GFET circuit impedance accounting for the impedances of the i- and n-regions and the load resistance R_l.

For the GFETs, solving the hydrodynamic equation for the electron liquid in the n-region coupled with the Poisson equation for the ac component of electric potential in the gradual channel approximation⁴⁸ and accounting for the boundary conditions at the edges of the i- and n-regions, we arrive at the following expressions for the impedance:⁴⁹ 

where $ρl=RlH/ri, η=ri/rn$⁠, and $ri=σi−1$ and r_n are the DC resistances of the i- and n-regions (per unit GFET width H), respectively. The quantity

is the plasma frequency in the gated channel, d and κ are the gate layer thickness and dielectric constant, respectively, and ν_n is the frequency of the electron scattering in the n-region.

At small frequencies and b = 0, Eqs. (4)–(6) yield $Zω≃Z0=rn+ri(1+ρl)$⁠. Formally, the low-frequency limit of Eq. (6) might give $Z0<0$⁠, if $2b>1+η(1+ρl)$⁠. However, such a case corresponding to the S-shaped dc I–V characteristics is not under consideration here. At the signal frequencies coinciding with mth resonant plasma frequency $ωm≃(2m−1)Ω$ (provided that $Ω≫νn$⁠), Re $Zω$ exhibits the resonances with the sharpness depending on the resonance quality factor $Qm=4Ω/[νn+α(2m−1)2(π/2ln)2]$⁠. The latter steeply decreases with increasing m, so that the higher resonances are relatively weak, mainly due to viscosity. To account for the viscosity of the electron liquid, we set^49–52 $νn=ν+αk2$⁠, where ν is the scattering frequency on the disorder (acoustic phonons, residual impurities, and defects), α is the viscosity, and k is the wavenumber of the electron density perturbations.

The plasma frequency can be expressed via the gated n-region lumped geometrical capacitance C_g and inductance L_g as $Ω=(CgLg)−1/2$⁠. Equation (4) differs from the analogous equation (obtained previously²⁵ in the framework of the model assuming the lumped element modeling of the resonant plasmonic response) by accounting for the generally nonuniform spatial distributions of the carrier density and potential in the gated n-region (distributed model). As a result, Eq. (4) describes not only the impedance fundamental plasmonic resonance (corresponding to the plasma frequency given in Eq. (5) or the same value obtained using the lumped parameters) but also its harmonics as well.

Averaging Eq. (1) with J_DE given in Eq. (2) over the signal period $2π/ω$⁠, we obtain the following equation relating the ac and rectified currents, $δJω$ and $ΔJω$⁠:

Here, the symbol $⟨⋯⟩$ implies the averaging over the signal period. The output signal is equal to $ΔVω=ΔJωRlH$ (see Fig. 1).

The current responsivity of the GFET operating as a detector of incoming radiation with the frequency ω is estimated as $Rω=ΔJωH/SIω$⁠. Here, $S=λω2g/4π$ is the antenna aperture,⁵³ where $Iω$ and $λω=2π c/ω$ are the intensity and wavelength of incoming radiation, $g∼2$⁠, and c is the speed of light in a vacuum. Considering that $|δVω|2=8πλω2Iω/c$ and using Eq. (8), we arrive at the following expression for the GFET detector responsivity:

Here, $R=32π2137gJ0Hℏω02∝κli$⁠. The parameters of different GFETs under consideration corresponding to room temperature are given in Table I. The drag factor b was calculated accounting for the structural parameters.^20,21 Assuming $κ=4–6, li=0.1 μ$m, $H=10 μ$m, and g = 2, one obtains $J0H≃1.4–2.1$ mA. For the GFET(a), one gets $R≃240$ A/W, while for GFET(b) and GFET(c) we find $R≃160$ A/W. Accounting for the GFET resistance corresponding to these estimates, we obtain for the detector characteristic voltage responsivity $RV=R(ℏω0/eJ0H)≃2.3×104$ V/W. For the peak voltage responsivity $RωV$ corresponding to the data shown in Fig. 2, we obtain $RωV≃(1.24−3.29)×103$ V/W.

According to Eqs. (5) and (6), the real part of the GFET impedance Re $Zω$ can become negative at specific values of $2bJ¯/J0$ and ρ_l. This can lead to plasma instability.^21,46,54 At even more liberal conditions, $|Zω|$ can be close to zero, which leads to the divergence of the expression for $Rω$⁠. However, at small $|Zω|$⁠, the Johnson–Nyquist (J–N) noise can be strong, preventing the detection of incoming signals. Both the plasma instability and the excessive J–N noise can be suppressed by choosing a sufficiently large load resistance R_l, i.e., a large parameter ρ_l. The parameter ρ_l should be properly chosen for the GFET operation as a THz detector. We determine the necessary value of ρ_l from the condition that the J–N noise does not exceed the GFET dark current noise. The mean square values of the J–N noise and the dark current noise are given by⁵⁵ $in,J−N2=4THΔf|ReZω|/|Zω|2$ and by $in,dark2=4eIdarkΔf$⁠, respectively, where $Δf$ is the bandwidth, T is the temperature in energy units, and I_dark is the net source-drain dark current. The maximum of the latter can be estimated as $Idark≃HJ¯≃HJ0$⁠. At the plasmonic resonance $Zω≃ZΩ$⁠, and considering the condition $in,J−N2=4THΔf/|ZΩ|≲in,dark2$⁠, we find $|ZΩ|≳(T/eJ0)=ri(T/ℏω0)$⁠. Accounting for Eqs. (4)–(6), we obtain the following estimate for the dimensionless load resistance $ρLopt$ optimized to prevent the plasma instability and the excessive J–N noise at the current densities $J¯$ up to J₀: $ρl+1=[(2b−1)/η](Ω/νn)2+T/ℏω0$⁠. Equation (9) with Eqs. (4) and (5) yields

Figure 2 shows the responsivity $Rω$ as a function of the incident radiation frequency $ω/2π$ for different normalized bias current $J¯$ calculated for the GFETs (see Table I) using Eqs. (6) and (10). The parameter ρ_l is chosen to be $ρl=8.2$ (larger than the optimized value).

As seen, the responsivity exhibits sharp resonant peaks corresponding to $ω≃Ω$⁠, since a sufficiently large fundamental resonance quality factor $Q1∝Ω/ν1$ was assumed. As for higher resonant peaks (not shown), they are substantially smeared due to the viscosity. According to Fig. 2, the peak responsivity is rather sensitive to the dc current $J¯$⁠. This is because the rectifying effect associated with the carrier drag increases with $J¯$ being large near the point $J¯∼J0$⁠. The fact that the Coulomb drag effect contributes to the GFET operation as a detector is implicitly reflected by the drag factor b in the above equations.

The responsivity of the GFET-based THz detectors can be higher than those in Fig. 2, provided careful optimization. The estimate of the responsivity maximum limited by the J–N noise max $Rω≃R[b/(1+η−2b)](ℏω0/T)2$ can yield max $Rω>R$ and max $RωV>RV$⁠.

At practical electron densities in the n-region when $μn≫T$⁠, the drag parameter could be estimated as^25,47

The formal dependence of b on $ℏω0$ is associated with the voltage normalization by the optical phonon energy $ℏω0≃0.2$ eV. At room and lower temperatures and the electron densities in the n-region up to $Σn=1011–1012$ cm^–2, the frequency of electron scattering on residual impurities, acoustic phonons, and defects can be about $ν=0.5–1.0$ ps^–1 or smaller.^44–47 For $ln=0.5−0.75 μ$m and $μn=37.5–50$ meV, one obtains $b≃1.89–4.15$⁠. The latter interval comprises the values of b used in the calculations (see Table I). This implies that the GFET detectors with sufficiently perfect GL channels using the electron drag mechanism can operate at room temperature. Lowering temperature can lead to a markedly larger parameter b and, hence, higher values of the detector responsivity and detectivity. (Both characteristics are proportional to b.) One needs to mention that an increase in the collision frequency ν_n results not only in the plasma resonances broadening but also in a decrease in the drag parameter b.^25,47 This might impair or even suppress the radiation THz detection mechanism under consideration in GFETs with insufficiently perfect channels.

One of the crucial parameters is the electron density in the n-region Σ_n, which determines the plasma frequency $Ω∝μn1/2∝Σn1/4$ and the drag parameter $b∝μn−1∝Σn−1/2$⁠. As demonstrated, at $μn=37.5–50$ meV (the electron densities $Σn=(1–2)×1011$ cm^–2), the GFETs under consideration (with other parameters given in Table I) can exhibit a pronounced resonant plasmonic response in the frequency range $ω/2π≃0.8−1.5$ THz.

The quantity $b/(1+η)$ plays the role of the drag nonlinearity parameter. The latter (being about $0.13−0.24$⁠, as in the above estimates) might exceed the nonlinearity parameter characterizing the response of the FET-based plasmonic photodetectors.^26–36 In this regard, it is instructive to compare the ratio, $γ(drag)$⁠, of the drag nonlinear and linear current components $ΔJω$ and $⟨|δJω|2⟩$ using Eq. (8), and the ratio, $γ(plasm)$⁠, of such components for the plasma nonlinearity detection mechanism (see, for example, Refs. 26–28). Taking into account Eq. (11), for $J¯≲J0$⁠, one can find that

For the plasmonic nonlinearity (see, for example, Ref. 56),

Here, $m=μn/vW$ and s are the fictitious electron mass in graphene and the plasma wave velocity, respectively (in GFETs, $s∝Ω$⁠). Considering this, we obtain

In the gated graphene structures, $(s/vW)2$ can be fairly large.

Using Eq. (10), one can estimate the GFET photodetector ultimate dark-current limited detectivity (at T = 300 K) $Dω*=Rω/2eIdark/Hli=Rω/2eJ0/li$⁠.⁵⁷ Setting $li=0.1 μ$ m, κ = 4, so that $J0=1.4$ A/cm, and $Rω≃R=160$ A/W (assuming the load resistance optimization), we arrive at the estimate max $Dω*≃8×108$ cm Hz^1/2/W. The latter estimate corresponds to the noise equivalent power NEP $≃1.25×10−9$ W/Hz^1/2.

Lowering the temperature leads to a decrease in the scattering frequencies ν and ν_n. The latter provides higher values of the drag factor and sharpening of the plasma resonances. At lower temperatures, the J–N noise weakens that softens the pertinent limitation. All this might result in a significant increase in the GFET detector responsivity and detectivity (a decrease in the NEP).

In conclusion, we proposed using the nonlinearity of the Coulomb drag dependence on the injected BEs for the rectification of the incoming THz signals and their detection. The plasmonic resonance properties of the GFET channel might result in a substantial enhancement of the detector responsivity in the THz range, which can markedly surpass that of the existing uncooled detectors. Further enhancement of the GFETs based THz detectors can be achieved in the arrays of multiple GFETs similar to the standard plasmonic FET arrays.^30,34,58

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 via Project No. H31/A01, Japan. The work at RPI was supported by the Office of Naval Research (No. N000141712976, Project Monitor Dr. Paul Maki).