Optical Transistor for Amplification of Radiation in a Broadband Terahertz Domain
K. H. A. Villegas,1 F. V. Kusmartsev,2,3,* Y. Luo,2,† and I. G. Savenko1,4
1Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
2Micro/Nano Fabrication Laboratory (MNFL), Microsystem and Terahertz Research Center, Chengdu, China
3Physics Department, Loughborough University, Loughborough LE11 3TU, United Kingdom
4A. V. Rzhanov Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia
(Received 30 December 2018; revised manuscript received 13 October 2019;
accepted 23 January 2020; published 26 February 2020)
We propose a new type of optical transistor for a broadband amplification of terahertz radiation. It is
made of a graphene-superconductor hybrid, where electrons and Cooper pairs couple by Coulomb forces.
The transistor operates via the propagation of surface plasmons in both layers, and the origin of
amplification is the quantum capacitance of graphene. It leads to terahertz waves amplification, the negative
power absorption, and as a result, the system yields positive gain, and the hybrid acts like an optical
transistor, operating with the terahertz light. It can, in principle, amplify even a whole spectrum of chaotic
signals (or noise), which is required for numerous biological applications.
DOI: 10.1103/PhysRevLett.124.087701
The growing interest in the terahertz frequency range
(0.3–30 THz) is due to its potential applications in diverse
fields such as nondestructive probing in medicine, allowing
for noninvasive tumor detection, biosecurity, ultrahigh-band-
width wireless communication networks, vehicle control,
atmospheric pollution monitoring, intersatellite communica-
tion, and spectroscopy [1–4]. However, the terahertz range
still remains a challenge for modern technology due to the
lack of a compact, powerful, and scalable solid-state source
[5]. This problem is known as the “terahertz gap.”
To “close” this gap from the lower frequencies, one can
mention electronic devices with negative differential resi-
stance (NDR). For instance, superlattice electronic devices
generate higher harmonics by means of NDR [6] and can
reach a 0.5 THz gap, while the output power is less than
0.5 mW [7]. The radiation power of resonant tunneling
diodes (RTDs) [8] is less than 1 μW, and it further decreases
by 3 orders of magnitude at room temperature. Also, RTDs
suffer from their small electron transition times and parasitic
capacitance, associated with the double-barrier structure.
The use of layered high-temperature superconductors
(HTSCs) with intrinsic Josephson junctions, such as
Bi2Sr2CaCu2O8 (bismuth strontium calcium copper oxide)
[9–11] can produce radiation with Josephson oscillations
generated by an applied bias voltage [12,13]. Here, a
tunable emission, from 1 to 11 THz, has been recently
observed [14]. However, the power output is 1 μW, which
is still inadequate for practical applications.
It can be enhanced with the use of Bose-Einstein
condensates [15,16]. However, such approach requires a
hybridization of several bands with different parity, making
the output power small. Quantum cascade lasers (QCLs)
[17–19] can generate a high-frequency terahertz radiation,
while transistors [20–23], Gunn diodes [24], and frequency
multipliers [25] are approaching the terahertz gap from the
low-frequency side. The latter covers the whole terahertz
range, while having small power. The general fundamental
obstacle of all these terahertz sources is the small emission
rate (on the order of 10 ms). It can be increased with the
Purcell effect when terahertz sources are placed in a cavity
[26,27], however, the quantum efficiency is still about 1%,
and their manufacture is difficult.
Graphene and carbon nanotubes may serve as highly
tunable sources and detectors of terahertz radiation
[28–34], and even in terahertz lasers [35–41]. In the
dual-gate graphene-channel field-effect transistor [42]
embedded into a cavity resonator [43,44], one observes
spontaneous broadband light emission in the 0.1–7.6 THz
range with the maximum radiation power of ∼10 μW at a
temperature of 100 K. There are also emerging sources of
multiple harmonic generation of terahertz radiation in
superlattices [45,46], frequency difference generation in
midinfrared QCLs [47], and terahertz optical combs [48].
Graphene covered with a thin film of colloidal quantum
dots has a strong photoelectric effect, which provides
enormous gain for the photodetection (about 108 electrons
per photon) [49]; graphene grown on SiC has a strong
photoresponse [50], and graphene composites can improve
solar cells efficiency [51]. Note, both graphene and the
superconductor alone are practically insensitive to light
[52]. Here, we show that graphene placed in the vicinity of
a superconductor represents an active media with strong
light-matter coupling. It can operate as an optical transistor
that amplifies broadband electromagnetic radiation. That is
useful for studying chemical and biological processes or in
telecommunications for encryption-decryption procedures,
where it is important to image a whole spectrum.
We consider a system consisting of parallel layers
of graphene and superconductor, exposed to an electro-
magnetic (EM) field incident with the angle θ and linearly
PHYSICAL REVIEW LETTERS 124, 087701 (2020)
0031-9007=20=124(8)=087701(7) 087701-1 © 2020 American Physical Society
polarized along the x-z plane (p polarization),
Eðr; tÞ ¼ ðsin θ; 0; cos θÞE0e−iðk⊥zþkk·rþωtÞ, where kk, ω,
and r are the in-plane wave vector of the field, frequency,
and coordinate, respectively (see Fig. 1). Between the
graphene layer and superconductor there is a gate voltage
that controls its chemical potential. The energy is supplied
by ac bias and/or by an external laser (pump) with
frequency exceeding the superconducting gap Δ (see
Fig. 1). There, exciting quasiparticles in the superconductor
are creating NDR.
The electrons in graphene are coupled by the Coulomb
interaction, which has the Fourier image given by
vk ¼ 2πe2=k, where k is in-plane momentum (lying in
the x-y plane). The electrons between the two layers are
also Coulomb coupled, and the Fourier image of the
interlayer interaction reads uk ¼ 2πe2 expð−akÞ=k, where
a ¼ 10 nm is the separation between the layers.
Using the linear response theory for hybrid systems
[53,54], we can write the electron density fluctuations in
the graphene layer δnkω and Cooper pair density fluctua-
tions in the superconducting layer δNkω as [55]
δnkω ¼ Πkωðvkδnkω þ ukδNkω þWkωÞ;
δNkω ¼ PkωðvkδNkω þ ukδnkω þWkωÞ; ð1Þ
where Πkω ¼ ΠRkω þ iΠIkω and Pkω ¼ PRkω þ iPIkω are the
complex-valued polarization operators of the graphene and
superconductor, respectively, and Wkω ¼ eE0=ik is the
Fourier image of the potential energy due to the external
electric field. From Eqs. (1) we can find the density
fluctuations in the graphene nkω and in the superconductor
Nkω as linear functions of applied electric field ampli-
tude E0 (see Supplemental Material [56] for details).
Collective plasmonic hybrid modes in the graphene and
superconductor can be found from the same system of
equations, taking into account the expressions for the
polarization loops of the superconductor [55] and graphene
[63,64]. Substituting the expressions for nkω and Nkω into
the continuity equations, kjkω ¼ −eωδnkω and kJkω ¼
−2eωδNkω, for graphene and superconductor, respectively,
we can determine the electric currents in each of the layers
and their impedances ZG and ZSC. The collective modes of
the hybrid system are presented in Fig. 2(a) for the undoped
and doped graphene cases. The upper mode has a gap
2Δ ¼ 2 meV. If in this hybrid a single graphene layer is not
interacting with a superconductor, only one mode exists,
which is due to the superconductor.
The formula for the power absorption or gain reads [65]
PðωÞ ¼ 1
2


Re
Z
d2rJðr; tÞ · Eðr; tÞ

; ð2Þ
where the integration is over the graphene plane, and h  i
denotes time averaging. We normalize the power with the
sample area
R
d2 ¼ l2 and the square of the field amplitude
E20 to get
PðωÞ ¼ PðωÞ
l2E20
¼ 1
2
eω
kE0
Re½δnkω: ð3Þ
Figure 2(b) shows the dependence of the power absorp-
tion on the EM field incidence angle θ calculated with (3),
fixing μ ¼ 0 at the Dirac point by gate voltage. All the
curves exhibit critical angles at which the power absorption
becomes negative, α < 0. This suggests that the incident
angle can be used to switch the amplifier device on or off.
Furthermore, increasing the frequency of the incident EM
wave increases the critical angle.
Figure 2(c) shows the power absorption spectrum. We
see that coupling graphene to the superconductor layer
results in a negative power absorption in the terahertz
frequency range (solid curves and shaded regions). There is
no negative absorption region for isolated graphene, where
the power absorption remains positive for any frequency ν
(dashed curves). When the light incidence angle θ
increases, both the maximal intensity (slightly) and the
frequency range of the negative light absorption increase
[see the shaded area in Figs. 2(b) and 2(c)]. Thus, the angle
of light incidence allows us to control the range of light
frequencies with the negative absorption.
To understand the θ dependence, note that the wave
vector of the plasma wave is related to the projection of the
incident light wave vector on the plane of the sample. Both
the angular dependence of the absorption and gain are
related to the amplitude of this wave propagating on the
surface. The light incident perpendicular to the graphene
surface cannot excite such plasma waves and, therefore, in
this case, we do not have the gain. However, at large
incident angle, there is a reflection of the incident radiation
FIG. 1. System schematic. Graphene coupled with a two-
dimensional superconductor by the Coulomb force and
connected to an electrical pump source (a battery). Figure also
shows the pump-probe configuration for the terahertz radiation
amplification: The hybrid system is exposed to an external laser
(pump, depicted by red arrow) and broadband EM field at
incidence angle θ (probe, depicted by yellow arrows). The
frequency of the pump (probe) should be above (below) the
superconducting gap. Both the optical and electrical pump can
provide energy for the amplification.
PHYSICAL REVIEW LETTERS 124, 087701 (2020)
087701-2
due to the difference in the refractive index of the hybrid
and air. Thus, we conclude that the most optimal effect will
be observed at small but nonzero θ. If the system is
embedded into a cavity resonator, there might even arise
lasing similar to one observed in plasmonic lattices [66,67]
or semiconductor superlattices [44].
The mechanism of gain here is similar to one in a
waveguide coupled with a superconducting Josephson
junction [68]. Then, the optical reflectivity of the system
reads Γ ¼ ðZG − ZSCÞ2=ðZG þ ZSCÞ2. Near the frequency
of the plasmon resonance, there is an area of negative
differential resistance of the superconductor, RSC < 0. If
we assume XG ¼ 0 and XSC ¼ 0 [68,69], we find Γ > 1.
Note that a graphene transistor [70] can also have NDR (see
Sec. III of [56] and [71–73]).
The graphene-superconductor junction (Fig. 1) has a
large tunneling resistance. An electron in graphene with
energy below the superconducting gap can tunnel into the
superconductor only due to the Andreev scattering [74].
The probability of such tunneling is small, since all
electrons are paired. Therefore, the resistance of the
junction is high. With applied bias voltage above the
gap, quasiparticles appear and they can tunnel. As a result,
the resistance decreases and NDR arises. The latter can
appear even at zero bias when we pump the superconductor
with external light with the frequency above the gap. The
light excites electron and hole quasiparticles coexisting
with superconducting fluctuations on the surface of the
superconductor [75]. Then, in addition to the Andreev
scattering, there starts normal tunneling of quasiparticles
into the superconductor. The resistance of the junction
decreases and the NDR arises. Such a mechanism of NDR
can exist only in a highly nonequilibrium excited state
created by the pump [76].
Graphene separated by a dielectric layer (e.g., made of
BN, SiO2, or Ta2O5) from the superconductor (Nb, Pb, or
HTSCs) together form a parallel plate capacitor, in which
the capacitance C is given by C ¼ Cplate þ Cq, where Cplate
is the classical capacitance Cplate ¼ ϵ0A=a, A is the area of
the sample, and ϵo is the dielectric constant (e.g., ϵo ¼ 3.9
for SiO2). The quantum capacitance Cq of graphene
emerges due to its conical energy-momentum relation,
and it has the form Cq ¼ 2Ae2jEFj=πℏ2v2F [77–79]. The
component of incident light parallel to the superconducting
surface induces the fluctuation of charge density δns, which
is associated with a traveling plasmon wave with the
amplitude Es ∼ δns, where δns ¼ δns0 cosðkxþ ωtÞ
(Fig. 3). This charge density wave (CDW) on the surface
of the superconductor generates a mirror CDW of the
opposite sign in the neutral graphene layer, being of
the same order as the charge fluctuations in the super-
conducting layer, i.e., δnG ∼ δns. These plasmons have a
(a) (b) (c)
FIG. 2. (a) Hybrid collective plasmonic modes for undoped (blue curves) and doped (red curves, μ ¼ 3.0 meV) graphene layer. Black
dashed curves show the modes of isolated doped graphene layer. (Inset) Enlargement of the lower energy modes. (b) Power absorption
as a function of the angle of incidence θ (see also Fig. 1) for graphene-superconductor hybrid for frequencies ν ¼ 0.5 (red), 1.0 (green),
and 2.0 THz (blue). (c) Graphene power absorption spectrum for the angles of incidence θ ¼ 1.0° (red), θ ¼ 1.5° (green), and θ ¼ 3.0°
(blue). Dashed curves show the data corresponding to the isolated graphene. In (b) and (c) μ ¼ 0. (Insets) The effect of temperature
T ¼ 0 (red), T ¼ 0.5Tc (green), and T ¼ Tc (blue) is shown. The graphene-superconductor separation is a ¼ 10 nm.
EG
ESC
EF
EF
+
superconductor
graphene
-
++
++
+--
--
-
FIG. 3. Schematic of the mechanism of terahertz amplification.
The incident light induces a collective hybrid plasmon mode. The
interplay between this mode and the quantum capacitance of
graphene amplifies the incident electromagnetic field (see text for
details).
PHYSICAL REVIEW LETTERS 124, 087701 (2020)
087701-3
wavelength larger than a micrometer, and therefore the
charge density for each half wavelength can be viewed as a
local temporary graphene doping, moving with the wave.
During this half period, the charge fluctuation corresponds
to the local change in the chemical potential or the Fermi
energy EF, which is directly related to the amplitude of the
plasmon wave propagating in graphene EG. Because of the
quantum capacitance of graphene, described by the relation
EF ∼
ffiffiffiffiffi
nG
p
, the waves amplitude EG ∼ EF is significantly
enhanced, and it is different from the amplitude of
the plasmon wave, propagating on the surface of the
superconductor.
In order to stimulate radiation of electromagnetic waves
in the terahertz range (similar to photoconductive antennas
[80]), one has to add a supply of energy. It can be done by
introducing a pump-probe setup, where the energy required
for the amplification of the probe comes from the pump.
For the pump, we have to expose graphene to an external
laser with the frequency above the superconducting gap. Or
we can apply ac bias voltage with the amplitude larger than
the gap. Because of the pump, there forms a state, in which
energy is accumulated in electronic excitations. To have the
pump-probe configuration, we expose the junction to an
additional external light (probe) with the frequency lower
than the gap. Then, there on the surface will emerge
superconducting charge density fluctuations. Effectively,
they represent coherent waves traveling inside the capacitor
formed by the graphene and superconductor. Because of
graphene quantum capacitance [77–79], the amplitude of
these electromagnetic waves is amplified. The energy
is provided by the pump, resulting in negative light
absorption for a probe (see Fig. 2) or a reflection coefficient
larger than 1.
Note that the graphene-superconductor system possesses
many benefits. Both superconducting and graphene layers
have giant mobility and small resistivity, giving rise to
minute losses. With increasing temperature, the super-
conducting gap decreases, while graphene mobility
changes a little [81,82]. Since the radiation amplification
occurs in graphene, the temperature has a weak effect on
the operation of the terahertz transistor [see insets in
Figs. 2(b) and 2(c)]. The working temperature range is
similar to one for the stack of the Josephson junctions made
of HTSCs [9,10], and it is limited by the critical temper-
ature Tc.
The radiation power reads Pr ¼ hVgIi, where the voltage
Vg and the transverse current I are periodically changing in
time around the Dirac point [83]. Then, assuming simple
periodic behavior for both IðtÞ and VgðtÞ and the graphene-
superconductor separation a ¼ 10 nm [84], the maximal
outcome power reads hI × Vgi ∼ 200–250 μW=cm2 [85].
Evidently, it can be increased for larger areas of the surface
or employing multilayer hybrid structures.
Conclusions.—We have shown that, in a hybrid graphene-
superconductor system exposed to an electromagnetic field
of light, the absorption coefficient can become negative in a
certain range of frequencies and at a nonzero angle of
incidence. We suggest that the system can serve as an
amplifier of terahertz radiation. The essence of the ampli-
fication is the quantum capacitance of graphene, which
provides the conversion of the charge density wave induced
by incident light into emitted radiation with much stronger
intensity. That is also related to the negative differential
conductivity of the hybrid, where there is a strong Coulomb
coupling of graphene and superconductor.
Such devices are now in strong demand and may be
complementary to quantum cascade lasers. Moreover, the
use of high-temperature superconductors extends the range
of temperatures required for their operation. The existence
of Dirac or Weyl cones in graphene, topological insulators,
and Weyl semimetals brings in a new physical concept
called quantum capacitance. Its essence is in strong
dependence of the Fermi energy on the charge doping.
A weak charge density wave can induce a strong electric
field, allowing us to achieve the amplification of incident
electromagnetic radiation.
The situation is somewhat similar to lasers, where the
pumping results in the population inversion. The difference
is that, here, the amplification can occur in a broad
frequency range simultaneously, while in lasers it is pinned
to a specific resonant frequency. Such amplification of the
broadband spectrum, e.g., for chaotic or noise radiation,
opens exciting opportunities of new types of molecular and
biological noise spectroscopy, where the response of the
system can be measured in a broad frequency range,
providing new opportunities in molecular and biological
noise spectroscopy [86–88].
We thank Vadim Kovalev and Gennadii Sergienko for
fruitful discussions. K. H. V. and I. G. S. acknowledge the
support of the Institute for Basic Science in Korea (Project
No. IBS-R024-D1) and the Russian Foundation for Basic
Research (Project No. 18-29-20033).
*Corresponding author.
F.Kusmartsev@lboro.ac.uk
†luoyi@mtrc.ac.cn
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PHYSICAL REVIEW LETTERS 124, 087701 (2020)
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