# S-shaped current-voltage characteristics of n+-i-n-n+ graphene field-effect
transistors due to the Coulomb drag of quasi-equilibrium electrons by
ballistic electrons

V. Ryzhii1,2, M. Ryzhii3, V. Mitin4, M. S. Shur5, and T. Otsuji1 1Research
Institute of Electrical Communication, Tohoku University, Sendai 980-8577,
Japan
2Institute of Ultra High Frequency Semiconductor Electronics of RAS, Moscow
117105, Russia
3Department of Computer Science and Engineering, University of Aizu, Aizu-
Wakamatsu 965-8580, Japan
4Department of Electrical Engineering, University at Buffalo, SUNY, Buffalo,
New York 14260 USA
5Department of Electrical, Computer, and Systems Engineering, Rensselaer
Polytechnic Institute, Troy, New York 12180, USA

###### Abstract

Keywords: graphene, field-effect transistor, ballistic electrons, Coulomb
electron drag, S-shape current-voltage characteristics.
We demonstrate that the injection of the ballistic electrons into the two-
dimensional electron plasma in lateral n+-i-n-n+ graphene field-effect
transistors (G-FET) might lead to a substantial Coulomb drag of the quasi-
equilibrium electrons due the violation of the Galilean and Lorentz invariance
in the systems with a linear electron dispersion. This effect can result in
the S-shaped current-voltage characteristics (IVs). The resulting negative
differential conductivity enables the hysteresis effects and current
filamentation that can be used for the implementation of voltage switching
devices. Due to a strong nonlinearity of the IVs, the G-FETs can be used for
an effective frequency multiplication and detection of terahertz radiation.

##  I Introduction

The lateral transport of electrons and holes in the graphene layer (GL)
heterostructures could enable the detection, amplification, and generation of
terahertz radiation [1, 2] and other numerous applications (see, for example,
[3]). In this paper, we analyze the electron transport in the lateral
n+-i-n-n+ graphene field-effect transistors (G-FETs) with the n+ source and
drain contacts and the gated n-region. Figure 1 shows the G-FET structure and
the band diagrams at different source-drain voltage: $V<\hbar\omega_{0}/e$ and
$V>\hbar\omega_{0}/e$, where $\hbar\omega_{0}\simeq 200$ meV is the optical
phonon energy in graphene and $e$ is the electron charge. The n-region is
formed by the positive gate bias $V_{g}$. Similar lateral heterostructure GL
devices including those based on more complex lateral periodical cascade
devices were reported previously [1, 2, 3, 4, 5, 6, 7]. One of the remarkable
advantages of the GLs is the possibility of very high directed velocities of
the electron (hole) ensembles close to the characteristic velocity
$v_{W}\simeq 10^{8}$ cm/s [6, 7] providing the collision-less, i.e., ballistic
electron (BE) motion [8, 9] in relatively long channels. As demonstrated
experimentally, in the graphene encapsulated in hexagonal boron nitride the
ballistic transport is realized in the samples with the length of a few $\mu$m
at room temperature [10] and of 28 $\mu$m at decreased temperatures [11].

As was predicted decades ago [9], the BE motion interrupted by the emission of
the optical phonons can enable the self-excitation of the current oscillations
leading to the radiation emission [12, 13, 14, 15, 16].

Figure 1: (a) Schematic views of the lateral n+-i-n-n+ G-FETs with
electrically induced- n-region and their band diagrams corresponding to (b)
$T<eV<\hbar\omega_{0}$ (intermediate current densities at a pronounced drag)
and (c) to $eV>\hbar\omega_{0}$ monotonic (elevated current densities)
potential distributions. The BEs are injected via the virtual cathode. Arrows
correspond to the BEs injected from the source, the DQEs, i.e., the QEs
dragged by the injected BEs, and the QEs injected from the drain.

Considering the lateral forward-biased n+-i-n-n+ G-FET with the sufficiently
perfect GL, we assume that the transport of the injected electrons from the
emitter n+ region into the i-region ($-l_{i}<x<0$, where $l_{i}$ is the
i-region length) is ballistic. This implies that the BE transit time in the
i-region is much shorter that the characteristic times of their scattering on
the impurities and the acoustic phonons, $\tau_{imp}$ and $\tau_{ac}$. The
impurity and acoustic phonon scattering of the BEs injected into the n-region
($0<x<l_{n}$, where $l_{n}$ stands for the n-region length) is also
insignificant. Thus, $l_{i},l_{n}\ll v_{W}\tau_{imp},v_{W}\tau_{ac}$. We
demonstrate that the Coulomb collisions of the BEs, injected into the
n-region, with the thermalized quasi-equilibrium electrons (QEs) can lead to
the ”conversion” of a fraction of these electrons into the dragged equilibrium
electrons (DQEs) moving toward the n+ collector drain region (analogous to the
mutual electron-hole drag). Such a Coulomb drag in GLs, i.e., in the electron
systems with the linear energy spectrum can be fairly effective. The GL
electron system is neither a Galilean nor a truly Lorentz-invariant system
[17, 18, 19]. The Coulomb drag in question is fundamentally similar to the
drag between spatially separated standard [20, 21] and graphene-based [22, 23,
24, 25, 26, 27, 28, 29] two-dimensional electron-hole systems. This effect was
extensively studied in graphene both theoretically and experimentally (see,
for example, [22, 23, 27]). An essential distinction of the ballistic-
equilibrium drag is the current non-conservation (and possible multiplication)
due to electron-electron collisions. The Coulomb electron drag in the G-FETs
under consideration with the current multiplication might pronouncedly affect
the device characteristics resulting in the S-type current-voltage
characteristics (IVs). The latter can lead to the hysteresis phenomena and the
instability of the uniform current flow (the current filamentation).

Similar phenomenon can take place in the reverse-biased p+-p-i-n-n+ devices
[6, 7] due to the interband tunneling generation [23, 24] of the electron-hole
pairs in the i-region.

The physics behind is the Coulomb drag by the BEs. BEs collisions with QEs
results in the latter contributing to the current. As a result, the current
voltage characteristic is nonlinear even at low applied voltages, since the
voltage increase results in a higher level of the BE injection. At a certain
threshold voltage, this nonlinearity might lead to the infinite differential
conductance. At the threshold voltage, the switching occurs into another
stable branch of the current-voltage characteristic with the dominant
contribution of the BEs. As a consequence, there are two stable branches of
the IV: (a) the low current branch with a relatively few injected BEs and (b)
the high current branch with the dominant BE transport. The switching occurs
at the threshold voltage and the value of current after switching depends on
the load line (i.e. on the load resistance). During the switching, the current
traverses the unstable branch with the negative differential conductance.
Depending on the load resistance and on the applied voltage, the final state
might correspond to the current filamentation when the device cross section is
divided into regions corresponding to the low current and high current
branches, respectively

The paper is organized as follows. In Sec. II, we find the potential
distribution in the G-FET injection region (i-region) and derive the injected
current density as a function of the potential drop across this region.
Section III deals with the analysis of the Coulomb drag of the QEs by the BEs.
In this section, the net current density in the n-region is expressed as a sum
of the BEs, DQEs, and QEs. Using the results of Sec. III, we derive the IVs in
Sec. IV and demonstrate that they can be both monotonic and S-shaped. In Sec.
V, we consider the possibility of the current switching in the G-FETs enabled
by the S-shape of their IVs. Section VI deals with the instability of uniform
current spatial distributions at the fixed terminal current, which, as
indicated, can lead to the formation of the stationary and pulsing current
filaments. Section VII is devoted to the comments associated with the device
model. In Conclusions (Sec. VIII) we summarize the main results of the paper.
Some, mainly intermediate mathematical results, are given in Appendix A and
Appendix B.

##  II Potential distribution and injected current

The injection current density $j_{i}$ is determined by the voltage drop
$\Phi=\varphi|_{x=0}$ across the GL i-region and by the space charge in this
region (the space-charge-limited electron injection [30]). The potential
$\Phi$ is determined by the potential spatial distribution across the entire
G-FET structure corresponding to the boundary conditions
$\varphi_{x=-l_{i}}=0$ and $\varphi_{x=l_{n}}=V$, where $l_{i}$ and $l_{n}$
are the lengths of the i- and n-regions. For the lateral G-FET structure with
the blade-like regions near the i-region edges and for the injected BE density
$\Sigma_{i}=j_{i}/ev_{W}$ (where $v_{W}\simeq 10^{8}$ cm/s is the
characteristic electron velocity in GLs and $e=|e|$ is the electron charge)
the potential distribution across the i-region satisfying the conditions
$\varphi_{x=-l_{i}}=0$ and $\varphi_{x=0}=\Phi$, and $j_{i}$ versus $\Phi$
relation can be found as follows (compare with, for example, [30, 31, 32,
33]):

$\displaystyle\varphi=\frac{2\Phi}{\pi}\cos^{-1}\biggl{(}-\frac{x}{l_{i}}\biggr{)}-\frac{2e\Sigma_{i}}{\kappa}x\ln\biggl{[}\frac{l_{i}-\sqrt{l_{i}^{2}-x^{2}}}{l_{i}+\sqrt{l_{i}^{2}-x^{2}}}\biggr{]},$
(1)

$\displaystyle
E=\biggl{[}\frac{4e\Sigma_{i}l_{i}}{\kappa}-\frac{2\Phi}{\pi}\biggr{]}\frac{1}{\sqrt{l_{i}^{2}-x^{2}}}$
$\displaystyle+\frac{2e\Sigma_{i}}{\kappa}\ln\biggl{[}\frac{l_{i}-\sqrt{l_{i}^{2}-x^{2}}}{l_{i}+\sqrt{l_{i}^{2}-x^{2}}}\biggr{]}.$
(2)

Here $\kappa$ is the dielectric constant of the material surrounding the GL.
To consider the regime of electron injection limited by the electron space
charge near the source n-i-junction, we set the electric field at a point
$x=-l_{i}+0$ (the ”virtual cathode” [30], see Fig. 1) to be equal to zero.
Accounting for Eq. (2), this condition yields

$j_{i}=v_{W}\biggl{(}\frac{\kappa\Phi}{2\pi\,l_{i}}\biggr{)},\qquad\Sigma_{i}=\frac{\kappa\Phi}{2\pi\,el_{i}}.$
(3)

Equations (1) - (3) are valid when $e\Phi,eV>T$, where $T$ is the temperature
in the energy units. Equation (3) is in line with the well known result
obtained for the devices with blade-like injection contacts (but for the
carrier transport with the saturation velocity $v_{S}\ll v_{W}$). However, Eq.
(3) yields different voltage dependence from those found for different bulk
contacts [4].

Hence, according to Eqs. (1) and (2), we obtain

$\displaystyle\varphi=\frac{2\Phi}{\pi}\biggl{[}\cos^{-1}\biggl{(}-\frac{x}{l_{i}}\biggr{)}-\frac{x}{2l_{i}}\ln\biggl{(}\frac{l_{i}-\sqrt{l_{i}^{2}-x^{2}}}{l_{i}+\sqrt{l_{i}^{2}-x^{2}}}\biggr{)}\biggr{]},$
(4)

$\displaystyle
E=\frac{\Phi}{l_{i}}\ln\biggl{[}\frac{l_{i}-\sqrt{l_{i}^{2}-x^{2}}}{l_{i}+\sqrt{l_{i}^{2}-x^{2}}}\biggr{]}.$
(5)

Thus, $E|_{x=-l_{i}}=0$ and $E|_{x=0}\simeq(\Phi/l_{i})\ln(d/2l_{i})^{2}$
$(|E|_{x=0}|\gg\Phi_{i}/l_{i}$). Here, $d$ is the thickness of the gate layer.

##  III Coulomb electron drag

The BEs injected into the n-region have the energy $\varepsilon_{i}=e\Phi$ and
the momentum $p_{i}\simeq e\Phi/v_{W}$. The collisions between the injected
BEs and QEs in the n-region result in the transfer of a part of the ballistic
electron momentum to the QEs. Due to the linearity of the electron spectrum in
GLs, the injected and the BEs scattered by the QEs with small energies
preserve the direction of their movement (in the direction $x$ from the
emitter to the collector), while their directed momentum changes from $p_{i}$
to $p_{i}^{\prime}<p_{i}$. Despite the lost portion of the momentum, the BE
continues its motion toward the collector with the same velocity $v_{W}$. Due
to the collision with the BE, the QE receives the momentum
$p_{s}=p_{i}-p_{i}^{\prime}$. According to the energy and momentum
conservation laws for the linear electron energy dispersion relation, the QEs
also move with the velocity $v_{W}$ in the $x$-direction, so that, in contrast
to both bulk and conventional two-dimensional semiconductor systems, the
momentum conservation at the electron-electron collisions does not lead to the
velocity conservation [17]. In other words, a portion of the QEs becomes
excited with the average directed momentum and velocity upon collisions with
the injected BEs.

Thus, the electron collisions between the injected BEs and the QEs convert QEs
into BEs doubling of the net current carried by the original and ”secondary”
BEs. Hence, the BE current density $j_{n}^{(BE)}$ (associated with the
original BEs, which came from the i-region, and the QEs dragged by the BEs, to
which we refer to as the DQEs) in the n-region ($0<x<l_{n}$) can exceed the BE
current density $j_{i}(\Phi)$ in the i-region. This we interpret as the
amplified QE drag by the injected BEs.

The spatial variation of the BE current density $j_{n}^{(BE)}$ across the
n-region due to the BEs scattering on the QEs and optical phonons is
determined by

$\frac{dj_{n}^{(BE)}}{dx}=-\frac{K}{l_{n}}j_{n}^{(BE)}$ (6)

with $K=K_{ee}+K_{ac}+K_{op},$ where

$K_{ee}=\frac{l_{n}}{v_{W}\tau_{ee}},\qquad
K_{ac}=\frac{l_{n}}{v_{W}\tau_{ac}},$ (7)

$\displaystyle
K_{op}=\frac{l_{n}}{v_{W}\tau_{op}}\frac{(e\Phi-\hbar\omega_{0}+\mu)}{\hbar\omega_{0}}\Theta(e\Phi-\hbar\omega_{0})$
$\displaystyle={\overline{K}}_{op}\frac{(e\Phi-\hbar\omega_{0}+\mu)}{\hbar\omega_{0}}\Theta(e\Phi-\hbar\omega_{0}).$
(8)

Here $\tau_{ee}$, $\tau_{ac}$, and $\tau_{op}=(\rho\hbar\,v_{W}^{2}/D_{0})$
[33, 34, 35, 36] are the characteristic times of the electron-electron (BEs on
QEs) scattering, and the BE scattering on acoustic and optical phonons, $\rho$
and $D_{0}$ are the GL density and the optical deformation potential,
respectively, and $\Theta(e\Phi-\hbar\omega_{0})$ is the unity step function
reflecting the threshold character of the optical phonon emission. To account
for the temperature and electron spectrum smearing of the optical phonon
emission threshold, we set $\Theta(z)=[1+\exp(-2z/T)]^{-1}$ with $T$ being the
QE temperature. The linear factor $\propto(e\Phi-\hbar\omega_{0}+\mu)$ in the
expression for $K_{op}$ is associated with the linearity of the GL density of
states near the Dirac point. The Fermi electron energy in the gated n-region,
$\mu$, appears in the latter function argument to account for the optical
phonon emission with the electron transitions to the states above the Fermi
level. For simplicity we neglect the BE scattering on impurities not only in
the i-region, but in the n-region as well because in the G-FETs under
consideration with the gated n-region the electrons are primarily induced by
the gate voltage (not by ionized impurities). The quantity $K_{ee}$ markedly
exceeds $K_{ac}$. At the electron densities $\Sigma_{n}\simeq
1\times(10^{12}-10^{13})$ cm-2 and room temperature $T$ one can set for the
energy of the BEs injected into the n-region $\varepsilon\sim\hbar\omega_{0}$
$\tau_{ee}^{-1}\simeq(10-50)$ ps-1, $\tau_{ac}^{-1}\simeq 0.5$ ps-1, and
$\tau_{op}^{-1}\simeq(1-2)$ ps-1 [17, 33, 34, 35, 36, 37, 38]. If
$l_{n}=(0.5-1.0)~{}\mu$m, we find $K_{ee}\simeq 5-50$, $K_{ac}\simeq
0.25-0.5$, and ${\overline{K}}_{op}\simeq 0.5-2$. At lower temperatures,
$K_{ac}$ becomes even smaller. For brevity we do not distinguish the different
intra-valley and inter-valley optical phonon modes using for their energies
the common value $\hbar\omega_{0}\simeq 200$ meV and accounting for the
contribution of both modes by choosing the proper scattering time $\tau_{op}$.

Since $j_{n}^{(BE)}|_{x=0}=j_{i}$, where $j_{i}$ is given by Eq. (3), as
follows from Eq. (6), one obtains for the density of the BE current injected
into the n+-contact at $x=l_{n}$

$j_{n}^{(BE)}=j_{i}e^{-K}.$ (9)

The BEs colliding with the QEs in the n-region transfer to the latter the
average (per one QE) momentum equal to

$<p_{x}>=\frac{j_{i}\Phi}{v_{W}^{2}\Sigma_{n}}e^{-K_{ac}-K_{op}}(1-e^{-K_{ee}}),$
(10)

where $\Sigma_{n}$ is the QE density. We have disregarded a weak spatial
nonuniformity of the electron density in this region
$\Sigma_{n}=\Sigma_{d}+\Sigma_{g}\simeq\Sigma_{g}$, where $\Sigma_{d}$ the
density of the ionized donors and
$\Sigma_{g}=[\kappa(V_{g}-\varphi)]/(4\pi\,ed)\simeq\kappa\,V_{g}/(4\pi\,ed)\simeq
const$ is the electron density induced by the gate voltage (the effect of the
quantum capacitance is disregarded for simplicity as well).

The QE drag resulting in the QEs direct momentum induces the QE current, so
that the density of the net current $j_{n}^{>}$ (associated with the injection
of the BEs), into the collector n+-region can be presented as

$j_{n}^{>}=j_{i}e^{-K}+e\Sigma_{n}<v_{x}>.$ (11)

Here $v_{x}$ is the QE average velocity caused by the QE drag. It is related
to $<p_{x}>$ [see the Appendix, Eqs. (A3) and (A5)] as

$<v_{x}>=\frac{<p_{x}>v_{W}^{2}}{T\xi}e^{-K_{ac}}.$ (12)

Here $\xi=\xi(\mu/T)$ is a coefficient determined by the QE statistics, where
$\mu$ is the QE Fermi energy (see, Appendix A).

Table 1: Current-voltage characteristics peculiar points Voltage, $V$ |  $0$ | $V_{min}=\displaystyle V_{0}\frac{(1+\eta-b)}{\eta}$ | $V_{max}=\displaystyle V_{0}\frac{(1+\eta)^{2}}{4b\eta}$
---|---|---|---
Current density, $j$ |  $0$ |  $j_{min}=\displaystyle j_{0}\frac{(1+\eta-b)}{b},\qquad j_{max}^{\infty}\simeq j_{0}$ |  $j_{min}^{\infty}=\displaystyle j_{0}\frac{(1+\eta)}{2b},\qquad j_{max}>j_{max}^{\infty}$

##  IV Characteristics

Figure 2: Qualitative view of the G-FET IV with the Coulomb drag and scheme of
its bistable operation. The S-shaped IV includes three branches: the lower
branch with a monotonic potential distribution $\Phi<V$, the middle branch
with the potential distribution shown in Fig. 1(b), and the upper branch
formed due to the inclusion of the optical phonon emission, which again
corresponds to a monotonic potential distribution seen in Fig. 1(c).

###  IV.1 General equations

Taking into account that the leakage of the QEs from the n-region associated
with the drag is compensated by the conductivity current
$j_{n}^{<}=\sigma_{n}(V-\Phi)/l_{n}$ (so that the net current density in the
n-region $j_{n}^{>}+j_{n}^{<}=j_{i}$), we arrive at the following equation
relating the current density $j=j_{i}$, potential $\Phi$, and applied voltage
$V$:

$\displaystyle
j=je^{-K_{ac}-K_{op}}\biggl{[}e^{-K_{ee}}+\frac{\Phi}{T\xi}(1-e^{-K_{ee}})\biggr{]}$
$\displaystyle+\frac{\sigma_{n}}{l_{n}}(V-\Phi).$ (13)

Considering Eq. (3), Eq. (13) can be presented as a relation between the
potential $\Phi$ and the bias voltage $V$

$\displaystyle\Phi\biggl{(}1+\eta-e^{-K_{ee}-K_{ac}-K_{op}}\biggr{)}$
$\displaystyle-\frac{\Phi^{2}}{T\xi}e^{-K_{ac}-K_{op}}\biggl{(}1-e^{-K_{ee}}\biggr{)}=\eta\,V.$
(14)

Here
$\eta=(2\pi\,\sigma_{n}/\kappa\,v_{W})(l_{i}/l_{n})=\sigma_{n}l_{i}/\sigma_{i}l_{n}$
with $\sigma_{i}=\kappa\,v_{W}/2\pi$ [see Eq. (3)]. The parameter $\eta$ is
actually the ratio of the i-region resistance $r_{i}=l_{i}/\sigma_{i}$ and the
n-region resistance $r_{n}=l_{n}/\sigma_{n}$ (per unit length in the direction
perpendicular to the current flow): $\eta=r_{n}/r_{i}$. At the QE mobility
$\mu_{n}=10^{4}$ cm2/s$\cdot$V, $\Sigma_{n}=5\times 10^{11}$ cm-2, and
$l_{i}/l_{n}=0.1-0.5$, one obtains $\eta\simeq 1.13-5.66$.

Equation (14) yields the following source-drain voltage-current
characteristics

$\displaystyle\frac{j}{j_{0}}\biggl{(}1+\eta-e^{-K_{ee}-K_{ac}-K_{op}}\biggr{)}$
$\displaystyle-b\biggl{(}\frac{j}{j_{0}}\biggr{)}^{2}e^{-K_{op}}\biggl{(}1-e^{-K_{ee}}\biggr{)}\biggr{]}=\eta\frac{V}{V_{0}}.$
(15)

with $j_{0}=v_{W}\kappa\hbar\omega_{0}/2\pi\,el_{i}$,
$V_{0}=\hbar\omega_{0}/e$, and
$b=(\hbar\omega_{0}/T\xi)e^{-K_{ac}}\simeq(\hbar\omega_{0}/\mu)e^{-K_{ac}}$ is
the Coulomb drag parameter (see Appendix A). This parameter determines the
ratio of the current created by the DQEs and the current of the injected BEs,
which is equal to $\xi=b(j_{i}/j_{0})e^{-K_{op}}(1-e^{-K_{ee}})$. When the
potential drop across the i-region $\Phi<V_{0}$, the optical phonon emission
is blocked, i.e., $K_{op}=0$, $\xi\simeq b(j_{i}/j_{0})(1-e^{-K_{ee}})\simeq
b(j_{i}/j_{0})$ (see below). The parameters $\eta$ and $b$ are crucial for the
distribution of the potential drops across the i- and n-regions. Due to the
dependence of the parameter $b$ on the Fermi energy $\mu$, this parameter is
controlled by the gate voltage $V_{g}$: $b\propto\mu^{-1}\propto
V_{g}^{-1/2}$. Setting $\hbar\omega_{0}=200$ meV, $\mu=60$ meV
($\Sigma_{n}\simeq 6\times 10^{11}$ cm-2), and $K_{ac}=0.25$, we obtain
$b\simeq 2.67$. At $\kappa=4$ and $l_{i}=(0.5-1.0)~{}\mu$m, we arrive at the
following estimate: $j_{0}\simeq(1.41-2.82)\times 10^{-4}$ A/$\mu$m.

###  IV.2 Low current densities

In the voltage and current density ranges where $\Phi<V_{0}$, $j<j_{0}$, and
$K_{op}\simeq 0$, so that the optical phonon emission is not involved in the
IV formation, we obtain from Eqs. (14) and (15)

$\displaystyle\frac{j}{j_{0}}\biggl{(}1+\eta-b\frac{j}{j_{0}}\biggr{)}=\eta\frac{V}{V_{0}}.$
(16)

Here and in the following we omit the term $e^{-K_{ee}}\ll 1$.

As follows from Eq. (16), at a certain voltage $V=V_{max}$, where

$\displaystyle V_{max}=V_{0}\frac{(1+\eta)^{2}}{4b\eta},$ (17)

one obtains $dj/dV|_{V=V_{max}}=\infty$. This point corresponds to
$j=j_{min}^{\infty}$

$\displaystyle j_{min}^{\infty}=j_{0}\frac{(1+\eta)}{2b}.$ (18)

Equation (16) describes the IV lower branch in Fig. 2. It also describes the
IV middle branch if the latter exist, that happen if $V_{min}<V_{max}$ and
$j_{min}^{\infty}<j_{max}^{\infty}$ as seen in Fig. 2.

Naturally, in the absence of the Coulomb drag ($b=0$), such a voltage point
does not exist ($V_{max}\propto 1/b$ tends to infinity). Equation (16) also
corresponds to $dj/dV<0$, i.e., the negative differential conductivity, when
$j\gtrsim j_{min}^{\infty}$.

###  IV.3  High current densities

When $\Phi\gtrsim V_{0}$, $j\gtrsim j_{0}$, and $K_{op}\geq 0$. In this case,
the optical phonon emission starts to play a substantial role. Accounting for
such an emission, from Eq. (14) we obtain the following generalization of Eq.
(16):

$\displaystyle\frac{j}{j_{0}}(1+\eta)$
$\displaystyle-b\biggl{(}\frac{j}{j_{0}}\biggr{)}^{2}\exp\biggl{[}-\overline{K}_{op}\biggl{(}\frac{j}{j_{0}}-1+F\biggr{)}\Theta\biggl{(}\frac{j}{j_{0}}-1\biggr{)}\biggr{]}$
$\displaystyle=\eta\frac{V}{V_{0}},$ (19)

where we have introduced the normalized electron Fermi energy
$F=\mu/\hbar\omega_{0}$. In particular, Eq. (19) describes the IV upper branch
with $V_{0}\lesssim\Phi<V$, i.e., characterized by a monotonic potential
distribution shown in Fig. 1(c).

The IV governed by Eq. (19) exhibits the point near the threshold of the
optical phonon emission, where $V=V_{min}$, $j=j_{max}^{\infty}$,
corresponding to $dj/dV|_{V=V_{min}}=\infty$, for relatively small $F$, are
close to

$\displaystyle V_{min}\simeq V_{0}\frac{(1+\eta-b)}{\eta},$ (20)

$\displaystyle j_{max}^{\infty}=j_{0}.$ (21)

If $V\gg V_{0}$, one can expect that $\Phi$ markedly exceeds $V_{0}$, so that
$K_{op}\gg 1$, and the drag effect is suppressed by the relaxation of the BE
momentum due to the optical phonon emission. In such a limit, the high-voltage
section of the IV becomes monotonically rising.

###  IV.4 IV peculiar points

As follows from the above analysis, the IVs exhibit the following peculiar
points (for $K_{ee}\gg 1$), see also Table I:

(a) $V=0$ and $j=0$;

(b) $V=V_{min}=\displaystyle V_{0}\frac{1+\eta-b}{\eta}$ and

$j=j_{min}=\displaystyle j_{0}\frac{1+\eta-b}{b}$;

(c) $V=V_{max}=\displaystyle V_{0}\frac{(1+\eta)^{2}}{4b\eta}$ and
$j=j_{min}^{\infty}=~{}\displaystyle j_{0}\frac{1+\eta}{2b}$,

with $dj/dV|_{V=V_{max}}=\infty$;

(d) $V=V_{max}=\displaystyle V_{0}\frac{(1+\eta)^{2}}{4b\eta}$ and
$j=j_{max}>j_{max}^{\infty}$;

(e) $V=V_{min}\simeq\displaystyle V_{0}\frac{(1+\eta-b)}{\eta}$ and
$j=j_{max}^{\infty}\simeq j_{0}$

with $dj/dV|_{V=V_{min}}=\infty$.

Figure 3: G-FET normalized IVs ($j/j_{0}$ versus $eV/\hbar\omega_{0}$) for the
Fermi energy $\mu=60$ meV and different parameters $\eta$.

The net current is a monotonic function of the bias voltage if
$j_{min}^{\infty}\geq j_{0}$, i.e., if $(1+\eta)>2b$. In the opposite case
$j_{min}^{\infty}<j_{0}$, i.e., when $(1+\eta)<2b$, the IVs are of the
S-shaped form with a region of the negative differential conductivity $dj/dV$.
The latter corresponds to the voltage range $V_{min}<V<V_{max}$.

Figure 2 shows the schematic view of the G-FET S-shaped IV (analogous to those
in the following Figs. 3 - 5) at the parameters $\eta$ and $b$ corresponding
to $(1+\eta)<2b$ with the indicated peculiar points corresponding to Table I.
In situations when the source-drain voltage is given, the G-FET source-drain
IVs can be of the S-shape.

Figure 4: G-FET normalized IVs ($j/j_{0}$ versus $eV/\hbar\omega_{0}$) for
$\eta=2$ and the Fermi energies $\mu=50$ meV, 75 meV, and 100 meV (the gate
voltages $V_{g}/d\simeq 9.2$ V/$\mu$m, 27.7 V/$\mu$m, and 36.9 V/$\mu$m,
respectively).  Figure 5: The same characteristics as in Fig. 3, but for fixed
Fermi energy $\mu=60$ meV and different temperatures $T$. Inset shows the IVs
details near the point ($V=V_{min}$, $j=j_{0}$).

###  IV.5 Results of numerical calculations

Figure 3 shows the IVs calculated for $K_{ee}=5$, $K_{op}=0.25$, ${\cal
K}_{op}=1$, $b=2.67$, $F=0.3$ ($\mu=60$ meV), $T=10$ meV ($\sim 115$ K) and
different values of other parameters ($K_{ee}$ and $\eta$) demonstrating their
transformation from the monotonic to S-shaped characteristics. As seen from
Fig. 3, an increase in $\eta$ (for example, due to a decrease in the n-region
resistance) leads to a weakening of the S-shape with a shift of $V_{min}$
toward larger values.

As follows from Eq. (19), the IV shape varies with changing parameter $b$,
i.e., with changing the Fermi energy $\mu$, which, in turn, depends on the
gate voltage $V_{g}$. The variation of $\mu$ results in the variation of not
only the parameter $b\propto\mu^{-1}\propto V_{g}^{-1/2}$, characterizing the
drag effect, but the variation of the parameter
$F\propto\mu\propto\sqrt{V_{g}}$, determining the density of electron states
near the threshold of the optical emission, as well. Since in the G-FETs under
consideration, the dominant QE scattering mechanism is associated with the
acoustic phonons (short-range scattering mechanism, which is the same as for
neutral impurities and point defects), the gated n-region conductivity
$\sigma_{n}$ and, therefore, $\eta$ can be set independent of $\mu$ [39, 40].
The change in $\mu$ and, consequently in the QE density affects $K_{ee}$.
However, this can be disregarded until $K_{ee}\gg 1$, i.e., until the QE
density is not too small.

Figure 4 shows the G-FET IVs calculated using Eq. (19) for $\eta=2$ and
different values of the Fermi energy $\mu$. The same other parameters and the
temperature are assumed as for Fig. 3. One can see that the IV shape is fairly
sensitive to the QE Fermi energy in the gated n-region $\mu$, i.e., depends on
the QE density $\Sigma_{n}$ and, hence, on the gate voltage $V_{g}$. An
increase in $\mu$ can result in the transformation from the S-shaped IVs to
the monotonic IVs. This is attributed to a weakening of the drag effect with
increasing $\mu$ (see below).

An increase in the temperature beyond $T=10$ meV leads to the IVs with a less
pronounced S-shape, although such characteristics could be obtained even at
room temperature if the parameters are chosen properly, in particular, by
chosing sufficiently, small $\eta$ and $\mu$ ($\mu\lesssim 75$ meV). Indeed,
choosing $\mu=60$ meV and $\eta=2$ (other parameters are the same as in Figs.
3 and 4), we arrive at the S-shaped IVs shown in Fig. 5, corresponding to the
temperature range from $T=10$ meV to $T=25$ meV. As seen, for the latter set
of the parameters the S-shape can be preserved even at room temperature. The
temperature smearing of the threshold of the optical phonon emission leads to
a small deviation (for moderate values $F$) of the peculiar point positions
from the values given in Table I. The effects of the Fermi energy and the
temperature variations on the IVs are attributed to the Coulomb drag parameter
$b$ versus $\mu$ and $T$ dependences. Figure 6 shows examples of these
dependences calculated using Eq. (A7) in Appendix A. Assuming that
$\tau_{ac}\propto T^{-1}$, in the calculations of $b$ we set
$K_{ac}=0.25(T[{\rm meV}]/25)$. One can see that a decrease in $\mu$ and $T$
provides a rise of $b$ (and, therefore, the IVs with a more pronounced
S-shape).

##  V Current switching by the voltage pulses

The S-shaped IVs with hysteresis can enable the bistable operation controlled
by the source-drain voltage. At the fixed source-drain voltage
${\overline{V}}$ in the voltage range $V_{min}<{\overline{V}}<V_{max}$, there
are two branches of the stable states: the ”low” stable with the current
densities $0<j^{(low)}<j_{min}~{}{\infty}$ and the ”high” stable with
$j_{max}^{\infty}<j^{(high)}<\infty$. The stability of these states is due the
positive differential conductivities $\sigma_{D}^{(low)}=dj^{(low)}/dV$ and
$\sigma_{D}^{(high)}=dj^{(high)}/dV$ at the pertinent branches. In contrast,
the states in the intermediate branch
$j_{min}^{\infty}<j^{(int)}<j_{max}^{\infty}$ are unstable (see below).

The transition from the low state to the high state requires the voltage pulse
$\Delta V>V_{max}-{\overline{V}}>0$. The reverse transition can be realized by
applying the voltage pulse $\Delta V<{\overline{V}}-V_{min}<0$. The pulse
duration should be sufficiently longer than the characteristic time of the
temporal relaxation of the electron system $\tau_{rc}$. This time, as is
estimated in the next section, can be an order of a few ps.

Hence, the G-FETs with the S-shaped IVs can be used for the frequency
multiplication of the incoming signals. A strong IV nonlinearity at certain
applied voltages can be also used for the signal rectification and, therefore,
for the signal detection.

Figure 6: The Coulomb drag parameter $b$ versus Fermi energy at different
temperatures. Dashed line corresponds to the dependence calculated
disregarding a small effect of the quasi-equilibrium holes.

##  VI Aperiodic instability of uniform current flow

In the devices with the S-shaped IVs the current tends to filamentation under
the condition when the net terminal current is fixed. This is due to the
instability of the uniform state of the electron plasma toward the spatial
perturbations in the in-plane $y$-direction, perpendicular to the current flow
(in the $x$-direction).

Let us consider the dynamic behavior of the electron system. Introducing the
normalized average current density ${\overline{J}}=I/Lj_{0}$, where $I$ is the
net current through the G-FET (which is maintained to be fixed) chosen to be
such that ${\overline{J}}$ is in the range of the negative differential
conductivity), $L$ is the width in the $y$-direction, and introducing

$\displaystyle\tau_{rc}=\frac{c_{n}}{(\sigma_{i}/l_{i}+\sigma_{n}/l_{n})}=\biggl{(}\frac{l_{i}l_{n}}{2dv_{W}}\biggr{)}\frac{1}{(1+\eta)},$
(22) $\displaystyle{\cal
L}=\frac{l_{i}l_{n}}{2(\sigma_{i}/\sigma_{n}+l_{i}/l_{n})}=l_{n}\sqrt{\frac{\eta}{2(1+\eta)}},$
(23)

we present an equation governing the spatio-temporal variations in the gated
n-region given in Appendix B [Eq. (B1)] in the following form:

$\displaystyle-\tau_{rc}\frac{\partial}{\partial
t}\biggl{(}\frac{j}{j_{0}}\biggr{)}+{\cal L}^{2}\frac{\partial^{2}}{\partial
y^{2}}\biggl{(}\frac{j}{j_{0}}\biggr{)}=-\biggl{[}{\biggl{(}\frac{j}{j_{0}}\biggr{)}-\overline{J}}\biggr{]}$
$\displaystyle-\frac{b}{(1+\eta)}\biggl{[}\biggl{(}\frac{j}{j_{0}}\biggr{)}^{2}-{\overline{J}}^{2}\biggr{]}.$
(24)

The quantity $\tau_{rc}$ is a product of the gated n-region capacitance
$c_{n}$ and the G-FET source-drain resistance
$r=(\sigma_{i}/l_{i}+\sigma_{n}/l_{n})^{-1}$.

Now we focus on the stability of the uniform current flow with
$J={\overline{J}}$. Assuming that the potential $j={\overline{J}}j_{0}+\delta
je^{i(qy-\omega t)}$, where $q$ and $\omega$ are the wave number and the
frequency of the perturbation, respectively, we obtain from Eq. (23) the
following dispersion equation for the transit potential perturbations:

$\displaystyle(i\omega\tau_{rc}-q^{2}{\cal L}^{2})\delta
j=\biggl{(}1-\frac{2b{\overline{J}}}{1+\eta}\biggr{)}\delta j.$ (25)

When

$\displaystyle{\overline{J}}>\frac{(1+\eta)}{2b},$ (26)

the right-hand side of Eq. (25) is negative. This corresponds to
${\overline{J}}$ falling to the current range
$j_{min}^{\infty}/j_{0}<{\overline{J}}<1$, in which, as mentioned above, the
differential conductivity is negative. In this current range, Eq. (25) for the
electron plasma perturbations increment (the grows rate) yields

$\displaystyle{\rm Im}~{}\omega=\frac{2b{\overline{J}}-1-\eta-q^{2}{\cal
L}^{2}}{\tau_{rc}},$ (27)

which is positive for sufficiently small wave numbers $q$, i.e., for
sufficiently long perturbations. However, the perturbation length is limited
by the device size, $L$, in the $y$-direction.

Since, according to Eq. (25), Re $\omega=0$, inequality (26) corresponds to a
temporal aperiodic rise of the plasma perturbations (aperiodic plasma
instability). Hence, the temporal variation of the electron system out of
equilibrium, including the transformation of the current spatial distribution
and the duration of the switching process is characterized by the time
$\tau_{rc}$ given by Eq. (22). Setting, for example, $\eta=2$,
$l_{i}=1~{}\mu$m, $l_{n}=1~{}\mu$m, and $d=(0.05-0.10)~{}\mu$m, for the
characteristic time, $\tau_{rc}$, determining the time scale of the dynamic
processes in the G-FETs, we obtain $\tau_{rc}\simeq 1.67-3.33$ ps.

Setting ${\overline{J}}=0.875$ (that corresponds to the dc potential at $x=0$
equal to ${\overline{\Phi}}_{0}=175$ mV, i.e.,
${\overline{\Phi}}_{0}<V_{0}=200$ mV), $T=10$ meV, and $\mu=60$ meV
($b=2.67$), from Eq. (27) we find that the plasma instability in question can
occur if $\eta\leq 3.66$.

The current instability associated with the S-type IVs is akin to those
predicted by B. K. Ridley (see, for example, [41, 42, 43, 44, 45]), although
it is caused by a different mechanism, namely, by the electron drag.

As can be concluded from Eq. (27), the spatial scale of the current filaments
is determined by the characteristic length ${\cal L}$ given by Eq. (23).
Hence, the filamentation is possible when the width of the G-FET in the
y-direction $L\gg{\cal L}$. Setting $l_{n}=(1-2)~{}\mu$m and $\eta=2$, we find
${\cal L}\simeq(0.57-1.15)~{}\mu$m. Depending on the boundary conditions at
the G-FET edges (in the $y$-direction, $y=0$ and $y=L$), the arising filaments
cam be either stationary or pulsating. The formation of the nonlinear filament
structure might change the source-drain voltage drop at the fixed net current.
In the case of the pulsating filaments, the source-drain voltage can comprise
an ac component.

##  VII Comments

###  VII.1 Origin of the S-shaped IVs

As shown above, at sufficiently strong drag effect (large $b$), the IVs can be
of the S-shape. This is associated with the following two reasons: First, if
the potential drop across the i-region $\Phi$ is smaller than the voltage
corresponding to the optical phonon emission, there the IV ambiguity with two
possible values of the current: a relatively low with a small contribution of
the DQEs and rather large with a marked contribution of the DQEs [this
ambiguity is described by Eq. (16)]. In the first case, the potential
difference $V-\Phi>0$ removes the injected BEs that have accumulated in the
n-region. Such a low IV branch corresponds to an elevated source-drain voltage
$V$ [see Fig. 1(c)]. In contrast, in the second case, the DQE current through
the n-region is compensated by the reverse current injected from the drain.
The latter requires $V-\Phi<0$, i.e., a lowered voltage $V$ as seen from Fig.
1(b). Second, when $\Phi$ is sufficient for the emission of optical phonons by
the injected BEs (near the point separating the i- and n-region), the drag
suppressed, and the transport become normal, i.e., with the monotonic
potential distribution. The latter situation corresponds to the upper branch
of the S-shaped IV. In the less probable case of too large parameter $b$, the
upper branch can appear because of reflection of the DQEs by a strong braking
electric field $(\Phi-V)/l_{n}$.

###  VII.2 Electron injection and transit-time delay

In the case of the lateral n+-contact, the BE injection is limited by the two-
dimensional space-charge in the i-region. The G-FETs with the BE tunneling
injection through the Schottky contact can exhibit a similar behavior.
However, in the latter case, the $j_{i}$ versus $\Phi$ relation can be
different [a nonlinear in contrast to Eq. (3)]. This can lead to a
modification of the IVs in comparison with derived above.

At AC voltage, the density of the BE current injected into the n-region
exhibits a delay due to the finite transit time $\tau_{tr}=l_{i}/v_{W}$ of the
BEs across the i-region. Such a BE transit delay can, in principle, affect the
transient processes in the G-FETs under consideration, in particular, the
dynamic of the instability considered above.

According to the Shockley-Ramo theorem [46, 47], one needs to replace the
quantity $j_{i}=\sigma_{i}\Phi/l_{i}$ (which constitutes the quasi-stationary
current density) in the right-hand side of Eq. (9) by the current density of
the BEs propagating across the i-region. Therefore, the ac component of the
induced current density can be presented as [6, 7]

$\displaystyle j_{ind}=\frac{2}{\pi}j_{i}\int_{0}^{1}\frac{ds\,\displaystyle
e^{i\omega\tau_{tr}s}}{\sqrt{1-s^{2}}}\simeq
j_{i}\biggl{[}J_{0}(\omega\tau_{tr})+\frac{2i}{\pi}\omega\tau_{tr}\biggr{]},$
(28)

where $J_{0}(s)$ is the Bessel function and the factor $2/\pi\sqrt{1-s^{2}}$
under the integral appears due to the electric field created by the BEs in the
case of the ”blade-shaped” highly conductive n+\- and gated n-regions [48].

According to Eq. (28), the relative role of the transit-time effect is weak in
comparison with the effect of the gated n-region RC-recharging is
characterized by the ratio $2\tau_{tr}/\pi\tau_{rc}$. Taking into account Eq.
(22), we find $2\tau_{tr}/\pi\tau_{rc}=4(1+\eta)d/\pi\,l_{n}$.

For $\eta=2$, $l_{n}=1~{}\mu$m, $d=0.05-0.10~{}\mu$m, one obtains
$2\tau_{tr}/\pi\tau_{rc}\simeq 0.19-0.38<1$. Since the latter inequality is
normally satisfied for the G-FETs with realistic parameters, we disregarded
the BE transit delay, although this effect can lead to a moderate decrease of
the instability increment.

###  VII.3 Plasmonic resonance effects

The two-dimensional electron system in the gated n-region of the GL channel
can exhibit the plasmonic resonances corresponding to the plasma oscillation
frequencies $\Omega\propto d^{1/4}V_{g}^{1/4}/l_{n}$ and its harmonics [49].
The excitation of the plasma oscillations is possible when the source-drain
voltage $V$ comprises the ac component with the frequency
$\omega\simeq\Omega$. This component can be associated with the incident
radiation received by an antenna. This effect combined with the pronounced IV
nonlinearity should lead to a resonantly large rectified current, which can be
used for the detection of the incoming radiation. According to the estimate of
the plasma frequency $\Omega$, it can be in the terahertz frequency (THz)
range. Due to the positive feedback between the currents injected to the
i-region from the source and the reverse current injected to the n-region from
the drain, one might expect the plasma instability of the net steady-state
source-drain current resulting in the self-excitation of the THz plasma
oscillations [50]. However, the analysis of such effects is beyond the scope
of the present paper.

###  VII.4 Technological aspects

The crystallographic quality of graphene synthesized by a popular engineering
method of the thermal decomposition from the SiC substrate is now approaching
the high end of those for exfoliated graphene [51], in particular, exhibiting
the BE transport [10, 11]. The G-FET (similar to that under consideration in
this paper) process technology is getting matured for both semiconductor
integrated device processes based on e-beam lithography and gate stack
formation with the plasma chemical vapor deposition (CVD) [52] or the atomic
layer deposition (ALD) and exfoliation and dry-transfer in hBN/graphene/hBN
van der Waals hetero-stacking for the gate stack [1]. The processed GL
channels in those G-FET devices with sub-micrometer dimensions exhibit field-
effect mobilities beyond 100,000 cm2/Vs [51, 52].

##  VIII Conclusions

We proposed and evaluated the characteristics of a lateral n+-i-n-n+ G-FET
with the ballistic transport of the electrons injected from the source
n+-region into the i-region. We demonstrated that the ballistic electrons
entering the n-region can effectively drag the quasi-equilibrium electrons
toward the drain if the electron-electron scattering in the gated n-region
prevails over the impurity and acoustic phonon scattering. The Coulomb
ballistic-equilibrium electron drag in question with the electron current
multiplication is associated with the linearity of the electron energy
dispersion law in graphene. The drag effect can result in non monotonous
potential distributions in the G-FET channel and the strongly nonlinear S-type
source-drain IVs. The S-type IVs might lead to the filamentation of the
current in the G-FET channel (with the stationary or pulsating filaments) and
to the hysteresis phenomena, enabling the switching between different current
states. Apart from this application, the plasmonic phenomena in the G-FETs
under consideration can be used for the THz radiation detection, generation,
and signal frequency-multiplication. The latter applications require a
separate consideration.

## Acknowledgments

The work at RIEC and UoA was supported by Japan Society for Promotion of
Science, KAKENHI Grant Nos. 21H04546, 20K20349, Japan; RIEC Nation-Wide
Collaborative research Project No. H31/A01, Japan; The work at RPI was
supported by Office of Naval Research (N000141712976, Project Monitor Dr. Paul
Maki). The authors are grateful to D. Svintsov for very useful discussions.
One of the authors (V.R.) is also thankful to Yu. G. Gurevich for helpful
information.

## Appendix A. QE Coulomb drag parameter

The average momentum transferring from BEs to QEs (per one QE) can be
presented as:

$\displaystyle<p_{x}>=\frac{j_{i}\Phi}{v_{W}^{2}\Sigma_{n}}e^{-K_{ac}-K_{op}}(1-e^{-K_{ee}}).$
(A1)

The QE distribution function

$\displaystyle f=\biggl{[}1+\exp\biggl{(}\frac{v_{W}p+\mu-
p_{x}<v_{x}>}{T}\biggr{)}\biggr{]}^{-1},$ (A2)

whee $<v_{x}>$ is the average drift velocity obtained by QEs due to the
collisions with the BE flux, $T$ is temperature and $\mu$ is the electron
Fermi energy: $\mu\simeq\hbar\,v_{W}\sqrt{\kappa\,V_{g}/4ed}$. The latter
yields the relation between $<p_{x}>$ and $<v_{x}>$:

$\displaystyle<p_{x}>=\frac{\displaystyle\int\frac{dp_{y}dp_{x}p_{x}}{\biggl{[}1+\displaystyle\exp\biggl{(}\frac{v_{W}p+\mu-
p_{x}<v_{x}>}{T}\biggr{)}\biggr{]}}}{\displaystyle\int\frac{dp_{x}dp_{y}}{\biggl{[}1+\displaystyle\exp\biggl{(}\frac{v_{W}p+\mu}{T}\biggr{)}\biggr{]}}}$
$\displaystyle\simeq\frac{<v_{x}>T}{v_{W}^{2}}\xi(\mu/T).$ (A3)

where

$\displaystyle\xi(\mu/T)=\displaystyle\frac{3}{2}\frac{{\cal
F}_{2}(\mu/T)}{{\cal F}_{1}(\mu/T)}.$ (A4)

Here ${\cal F}_{n}(\eta)=\int_{0}^{\infty}duu^{n}[1+\exp(u-\eta)]^{-1}$ is the
Fermi-Dirac integral. At $\mu/T\gg 1$, $\xi(\mu/T)\simeq\mu/T$.

Hence,

$\displaystyle j_{n}^{>}=e\Sigma_{n}<v_{x}>\simeq j_{i}e^{-K}$
$\displaystyle+\frac{j_{i}\Phi}{T\xi}e^{-K_{ac}-K_{op}}(1-e^{-K_{ee}}).$ (A5)

Using Eq. (A4), the quantity
$b=(\hbar\omega_{0}/T\xi)e^{-K_{ac}}(1-e^{-K_{ee}})\simeq(\hbar\omega_{0}/T\xi)e^{-K_{ac}}$,
which we call as the Coulomb drag parameter, is presented as

$\displaystyle b=\frac{3\hbar\omega_{0}}{2T}e^{-K_{ac}}\frac{{\cal
F}_{1}(\mu/T)}{{\cal F}_{2}(\mu/T)}.$ (A6)

When $\mu>T$,
$b\simeq(\hbar\omega_{0}/\mu)e^{-K_{ac}}\simeq(\hbar\omega_{0}/\mu)$. Since
Fermi energy $\mu$ depends on the QE density $\Sigma_{n}$, $\mu$ and,
therefore, $b$ are controlled by the gate voltage $V_{g}$.

If the value of $\mu$ approaches to the Dirac point, the drag of the quasi-
equilibrium holes (QHs) can become crucial. This is because the QHs are
dragged by the BEs to the same direction partially neutralizing the current of
the dragged QEs.

Considering that the QH Fermi energy is equal to $-\mu$, one can obtain the
expression for the drag parameter $b$ replacing Eq. (A6):

$\displaystyle b=\frac{2\hbar\omega_{0}}{3T}e^{-K_{ac}}\biggl{[}\frac{{\cal
F}^{2}_{1}(\mu/T)}{{\cal F}_{2}(\mu/T)}-\frac{{\cal F}^{2}_{1}(-\mu/T)}{{\cal
F}_{2}(-\mu/T)}\biggr{]}$ $\displaystyle\times[{\cal F}_{1}(\mu/T)+{\cal
F}_{1}(-\mu/T)]^{-1}.$ (A7)

Equation (A7) does not account for the mutual electron-hole drag [18]. The
latter should lead to a somewhat smaller value of $\xi(\mu/T)$ in comparison
with Eq. (A6). Thus the QHs weaken the drag current multiplication. Although
for the G-FETs with the parameters used in the main text, this is negligible.

For different devices with the two-dimensional carriers but with the quadratic
dispersion (having the 2D channels in the heterostructures made of the
standard materials and the graphene bilayer heterostructures),
$<v_{z}>=<p_{z}>/m$, where $m$ is the carrier effective mass) and
$j_{n}^{>}=j_{i}$, so that there is no electron current multiplication.

## Appendix B. Spatio-temporal variations of electron system in the gated
region

The electron charge in the diode active region ($0<x<l_{n}$) $Q=-c_{n}\Phi$,
where $c_{n}=\kappa\,l_{n}/4\pi\,d$ is the gated n-region capacitance, obeys
the following equation:

$\displaystyle\frac{\partial Q}{\partial
t}+\frac{\sigma_{n}l_{n}}{2}\frac{\partial^{2}\Phi}{\partial
y^{2}}=\frac{\sigma_{i}\Phi}{l_{i}}\biggl{(}1-\frac{e\Phi}{T\xi}e^{-K_{ac}-K_{op}}\biggr{)}$
$\displaystyle-\frac{\sigma_{n}}{l_{n}}(V-\Phi).$ (B1)

The factor 1/2 in the second term in the left-hand side of Eq. (B1), appears
because the potential in the n-region varies between $\varphi|_{x=0}=\Phi$ and
$\varphi|_{x=l_{n}}=V$ (approximately linearly). Equation (B1) can be
presented as

$\displaystyle\frac{c_{n}l_{i}}{\sigma_{i}}\frac{\partial}{\partial
t}\biggl{(}\frac{j}{j_{0}}\biggr{)}+\frac{\sigma_{n}}{\sigma_{i}}\frac{l_{n}l_{i}}{2}\frac{\partial^{2}}{\partial
y^{2}}\biggl{(}\frac{j}{j_{0}}\biggr{)}=\frac{j}{j_{0}}(1+\eta)$
$\displaystyle-b\biggl{(}\frac{j}{j_{0}}\biggr{)}^{2}\exp\biggl{[}-\overline{K}_{op}\biggl{(}\frac{j}{j_{0}}-1+F\biggr{)}\Theta\biggl{(}\frac{j}{j_{0}}-1\biggr{)}\biggr{]}$
$\displaystyle-\eta\frac{V}{V_{0}}.$ (B2)

In the case of the steady-state uniform current flow with $j<j_{0}$ when
$K_{op}=0$, Eq. (B1) is reduced to Eq. (24). If the average current density
through the G-FET $I/L$ and its normalized value $I/Lj_{0}$ are given, Eq.
(B2) can also be presented in the following form:

$\displaystyle-\tau_{rc}\frac{\partial}{\partial
t}\biggl{(}\frac{j}{j_{0}}\biggr{)}+{\cal L}^{2}\frac{\partial^{2}}{\partial
y^{2}}\biggl{(}\frac{j}{j_{0}}\biggr{)}$
$\displaystyle=\biggl{(}\frac{j}{j_{0}}\biggr{)}-\frac{b}{(1+\eta)}\biggl{[}\biggl{(}\frac{j}{j_{0}}\biggr{)}^{2}-{\overline{J}}^{2}\biggr{]}.$
(B3)

Here $\tau_{rc}=c_{n}/(\sigma_{i}/l_{i}+\sigma_{n}/l_{n})$ and ${\cal
L}^{2}=l_{i}l_{n}/2(\sigma_{i}/\sigma_{n}+l_{i}/l_{n})$.

## References

  * [1] S. Boubanga-Tombet, W. Knap, D. Yadav, A. Satou, D. B. But, V. V. Popov, I. V. Gorbenko, V. Kachorovskii, and T. Otsuji, “Room temperature amplification of terahertz radiation by grating-gate graphene structures,”Phys. Rev. X 10, 031004-1-19 (2020).
  * [2] J. A. Delgado-Notario, V. Clericò, E. Diez, J. E. Velazquez-Perez, T. Taniguchi, K. Watanabe, T. Otsuji, and Y. M. Meziani, “Asymmetric dual grating gates graphene FET for detection of terahertz radiations,”APL Photon. 5, 066102-1-8 (2020).
  * [3] V. Ryzhii, T. Otsuji, and M. S. Shur, “Graphene based plasma-wave devices for terahertz applications,”Appl. Phys. Lett.116, 140501-1-6 (2020).
  * [4] M. Ryzhii and V. Ryzhii, “Injection and population inversion in electrically induced p–n junction in graphene with split gates, ”Jpn. J. Appl. Phys. 46, L151 (2007).
  * [5] M. Ryzhii, V. Ryzhii, T. Otsuji, V. Mitin, and M. S. Shur, “Electrically induced n-i-p junctions in multiple graphene layer structures,”Phys. Rev. B 82, 075419 (2010).
  * [6] V. Ryzhii, M. Ryzhii, V. Mitin, and M. S. Shur, “Graphene tunnelinhg transit-time terahertz oscillator based on electrically induced p-i-n junctions,”Appl. Phys. Express 2, 034503 (2009).
  * [7] V. L. Semenenko, V. G. Leiman, A. V. Arsening, V. Mitin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Effect of self-consistent electric field on characteristics of graphene p-i-n tunneling transit-time diodes,”J. Appl. Phys. 113, 024503 (2013).
  * [8] M. S. Shur and L. F. Eastman, “Ballistic transport in semiconductor at low temperatures for low-power high-speed logic,”IEEE Trans. Electron Devices 26, 1677-1683 (1979).
  * [9] V. I. Ryzhii, N. A. Bannov, and V. A. Fedirko, “Ballistic and quasi ballistic transport in semiconductor structures (review)”, Sov. Phys. Semicond. 18, 481 (1984).
  * [10] A. S. Mayorov, R. V. Gorbachev, S. V. Morozov, L. Britnell, R. Jalil, L. A. Ponomarenko, P. Blake, K. S. Novoselov, K. Watanabe, T. Taniguchi, and A. K. Geim, “Micrometer-scale ballistic transport in encapsulated graphene at room temperature,”Nano Lett. 11, 2396-2399 (2011).
  * [11] L. Banszerus, M. Schmitz, S. Engels, M. Goldsche, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ballistic transport exceeding 28 $\mu$m in CVD grown graphene,”Nano Lett. 16, 1387-1391 (2016).
  * [12] A. A. Andronov and V. A. Kozlov, “Low-temperature negative differential microwave conductivity in semiconductors following elastic scattering of electrons,”JETP Lett. 17, 87 (1973).
  * [13] V. L. Kustov, V. I. Ryzhii, and Yu. S. Sigov, “Nonlinear plasma instabilities in semiconductors subjected to strong electric fields in the case of inelastic scattering of electrons by optical phonons,”Sov. Phys. JEPT 99 (1980).
  * [14] L. E. Vorob’ev, S. N. Danilov, V. N. Tulupenko, and D. A. Firsov, “Generation of millimeter radiation due to electric-field-induced electron-transit-time resonance in indium phosphide,”JEPT Lett. 73, 219 (2001).
  * [15] S. Sekwao and J. P. Leburton, “Terahertz harmonic generation in graphene,”Appl. Phys. Lett. 106, 063109 (2015).
  * [16] A. A. Andronov and V. I. Pozdniakova, “Terahertz dispersion and amplification under electron streaming in graphene at 300 K, ”Semiconductors 54, 1078-1085 (2020).
  * [17] X. Li, E. A. Barry, J. M. Zavada, M. Buongiorno Nardelli, and K. W. Kim, “Influence of electron-electron scattering on transport characteristics in monolayer graphene,”Appl. Phys. Lett. 97, 082101 (2010).
  * [18] D. Svintsov, V. Vyurkov, S. Yurchenko, T. Otsuji, and V. Ryzhii, “Hydrodynamic model for electron-hole plasma in graphene,”J. Appl. Phys. 111, 083715 (2012).
  * [19] D. Svintsov, V. Ryzhii, A. Satou, T. Otsuji, and V. Vyurkov, “Carrier-carrier scattering and negative dynamic conductivity in pumped graphene,”Opt. Express 22, 19873 (2014).
  * [20] T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, “Mutual friction between parallel two-dimensional electron systems,”Phys. Rev. Lett. 66, 1216 (1991).
  * [21] U. Sivan, P. M. Solomon, and H. Shtrikman, “Coupled electron-hole transport,”Phys. Rev. Lett. 68, 1196 (1992).
  * [22] M. Schütt, P. M. Ostrovsky, M. Titov, I. V. Gornyi, B. N. Narozhny, and A. D. Mirlin, “Coulomb drag in graphene near the Dirac point,”Phys. Rev. Lett. 110, 026601 (2013).
  * [23] R. V. Gorbachev, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. Tudorovskiy, I. V. Grigorieva, A. H. MacDonald, S. V. Morozov, K. Watanabe, T. Taniguchi, and L. A. Ponomarenko, “Strong Coulomb drag and broken symmetry in double-layer graphene,”Nat. Phys. 8, 896 (2012).
  * [24] J. C. Song, D. A. Abanin, and L. S. Levitov, “Coulomb drag mechanisms in graphene,”Nano Lett. 13, 3631 (2013).
  * [25] D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Hydrodynamic electron transport and nonlinear waves in graphene, ”Phys. Rev. B 88, 245444 (2013).
  * [26] S. H. Abendinpour, G. Vignale, A. Principi, M. Polini, W.-K. Tse, and A. H. MacDonald, “Drude weight, plasmon dispersion, and ac conductivity in doped graphene sheets,”Phys. Rev. B 84, 045429 (2011).
  * [27] J. Li, T. Taniguchi, K. Watanabe, J. Hone, A. Levchenko, and C. R. Dean, “Negative Coulomb drag in double bilayer graphene,”Phys. Rev. Lett. 117, 046802 (2016).
  * [28] Y. Nam, D. K. Ki, D. Soler-Delgado, and A. F. Morpurgo, “Electron-hole collision limited transport in charge-neutral bilayer graphene,”Nat. Phys. 13, 1207 (2017).
  * [29] D. Svintsov, “Fate of an electron beam in graphene: Coulomb relaxation or plasma instability,”Phys. Rev. B 101, 235440 (2020).
  * [30] A. Grinberg, S. Luryi, M. Pinto, and N. Schryer, “Space-charge-limited current in a film,”IEEE Trans. Electron Devices 36, 1162 (1989).
  * [31] S. G. Petrosyan and A. Ya. Shik, “Contact phenomena in low-dimensional electron systems,”Sov. Phys. JETP 69, 1261 (1989).
  * [32] B. Gelmont, M. Shur, and C. Moglestue, “Theory of junction between two-dimensional electron gas and p-type semiconductor,”IEEE Trans. Electron Devices 39, 1216 (1992).
  * [33] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, “Electrostatics of edge channels,”Phys. Rev. B 46, 4026 (1992).
  * [34] M. V. Beznogov and R. A. Suris, “Theory of space charge limited ballistic currents in nanostructures of different dimensionalities,”Semiconductors 47, 514 (2013).
  * [35] R. S. Shishir and D. K. Ferry, “Intrinsic mobility in graphene,”J. Phys.: Cond. Mat. 21, 344201 (2009).
  * [36] E. H. Hwangand and S. Das Sarma, “Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic graphene,”Phys. Rev. B 77, 115449 (2008).
  * [37] K. M. Borysenko, J. T. Mullen, E. A. Barry, S. Paul, Y. G. Semenov, J. M. Zavada, M. B. Nardelli, and K. W. Kim, “First-principles analysis of electron-phonon interactions in graphene,”Phys. Rev. B 81, 121412(R) (2010).
  * [38] M. V Fischetti, J. Kim, S. Narayanan, Zh.-Y. Ong, C. Sachs, D. K. Ferry, and S. J. Aboud, “Pseudopotential-based studies of electron transport in graphene and graphene nanoribbons,”J. Phys: Cond. Mat. 25, 473202 (2013).
  * [39] F. T. Vasko and V. I. Ryzhii, “Voltage and temperature dependences of conductivity in gated graphene heterostructures,”Phys. Rev. B 76, 233404 (2007).
  * [40] V. Ryzhii, D. S. Ponomarev, M. Ryzhii, V. Mitin, M. S. Shur, and T. Otsuji, “Negative and positive terahertz and infrared photoconductivity in uncooled graphene,”Opt. Mat. Express 9, 585 (2019).
  * [41] B. K. Ridley, “Specific negative resistance in solids,”Proc. Phys. Soc. 82, 954 (1963).
  * [42] A. Blicher, Field-Effect and Bipolar Power Transistor Physics, (Ac. Press, New York, 1981).
  * [43] A. F. Volkov and Sh. M. Kogan, “Nonuniform current distribution in semiconductors with negative differential conductivity,”Sov. Phys. JETP 25, 1095 (1967).
  * [44] F. G. Bass, V. S. Bochkov, and Yu. G. Gurevich, “Influence of sample size on the current-voltage characteristic in media with an ambiguous dependence of electron temperature on field strength,”Sov. Phys. JETP 31, 972 (1970).
  * [45] F. G. Bass, Yu. G. Gurevich, S. A. Kostylev, and N. A. Terent’eva, “Dynamics of electrical instabilities in a medium with an S-type negative differential conductance,”Sov. Phys. Semicond. 17, 808 (1983).
  * [46] W. Shockley, “Currents to conductors induced by a moving point charge,”J. Appl. Phys. 9, 635 (1938).
  * [47] S. Ramo, “Currents induced by electron motion,”Proc. IRE 27, 584 (1939).
  * [48] V. Ryzhii and G. Khrenov, “High-frequency operation of lateral hot-electron transistor,”IEEE Trans. Electron Devices 42, 166 (1995).
  * [49] V. Ryzhii, A. Satou, and T. Otsuji, “Plasma waves in two-dimensional electron-hole system in gated graphene heterostructures,”J. Appl. Phys. 101, 024509 (2007).
  * [50] V. Ryzhii, A. Satou, I. Khmyrova, M. Ryzhii, T. Otsuji, V. Mitin, and M. S. Shur, “Plasma effects in lateral Schottky junction tunneling transit-time terahertz oscillator,”J. Phys.: Conf. Ser. 38, 228 (2006).
  * [51] T. Someya, H. Fukidome, H. Watanabe, T. Yamamoto, M. Okada, H. Suzuki, Yu Ogawa, T. Iimori, N. Ishii, T. Kanai, K. Tashima, B. Feng, S. Yamamoto, J. Itatani, F. Komori, K. Okazaki, Sh. Shin, and I. Matsuda, “Suppression of supercollision carrier cooling in high mobility graphene on SiC(0001), ”Phys. Rev. B 95, 165303 (2017).
  * [52] D. Yadav, G. Tamamushi, T. Watanabe, J. Mitsushio, Y. Tobah, K. Sugawara, A. A. Dubinov, A. Satou, M. Ryzhii, V. Ryzhii, and T. Otsuji, “Terahertz light-emitting graphene-channel transistor toward single-mode lasing,”Nanophotonics 7, 741 (2018). 7(4)