Optical Transistor for an Amplification of Radiation in a Broadband THz Domain

TL;DR

This paper introduces a novel graphene-superconductor hybrid optical transistor that amplifies broadband THz radiation through surface plasmons and quantum capacitance, enabling amplification of chaotic signals for biological uses.

Contribution

It presents a new hybrid device design that achieves broadband THz amplification using surface plasmons and quantum capacitance effects.

Findings

Achieves positive gain in THz amplification.

Demonstrates amplification of chaotic signals.

Operates as an optical transistor in the THz domain.

Abstract

We propose a new type of optical transistor for a broadband amplification of THz 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 THz 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), that is required for numerous biological applications.

Equations

$$\delta n_{k\omega }$$

$$\delta N_{k\omega }$$

$$P(\omega )=\frac{1}{2}⟨\mathcal{R}e\left[\int d^{2}r\mathbf{J}(\mathbf{r},t)\cdot {\mathbf{E}}^{\ast }(\mathbf{r},t)\right]⟩,$$

$$\displaystyle\mathcal{P}(\omega)=\frac{P(\omega)}{l^{2}E_{0}^{2}}=\frac{1}{2}\frac{e\omega}{kE_{0}}\mathcal{R}e\big{[}\delta n_{k\omega}\big{]}.$$

Full text

The growing interest in terahertz (THz) frequency range (0.3 to 30 THz) is due to its potential applications in diverse fields such as non-destructive probing in medicine, allowing for non-invasive tumor detection, biosecurity, ultra-high bandwidth wireless communication networks, vehicle control, atmospheric pollution monitoring, inter-satellite communication, and spectroscopy  Raab2002 ; Chamberlain2004 ; Eisele2010 ; Mittleman2018 . However, the THz range still remains a challenge for modern technology due to the lack of a compact, powerful, and scalable solid state source RefDragoman . This problem is known as the terahertz gap.

To ‘close’ this gap from the lower frequencies, one can mention electronic devices with negative differential resistance (NDR). For instance, super-lattice electronic devices (SLED) generate higher harmonics by means of NDR Kusmartsev2001 and can reach 0.5 THz gap, while the output power is less than 0.5 mW  Esele2018 . The radiation power of resonant tunneling diodes (RTDs) Asada2016 is less than 1 $\mu W$, and it further decreases by three 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 (HTSC) with intrinsic Josephson junctions, such as BISCCO Ozyuzer2007 ; Welp2013 ; Sun2018 can produce radiation with Josephson oscillations generated by an applied bias voltage Marat1995 ; Marat2000 . Here a tunable emission, from 1 to 11 THz, has been recently observed Krasnov2017 . However, the power output is 1 $\mu$W, which is still inadequate for practical applications.

It can be enhanced with the use of Bose-Einstein condensates RefKavokin ; RefPRLTHz . However, such approach requires a hybridization of several bands with different parity, making the output power small. Quantum cascade lasers (QCL) Kohler-2002 ; Williams-NP-2007 ; Faist2013 can generate a high-frequency THz-radiation, while transistors Hubers-2005 ; Chas-2009 ; Lus-2005 ; Li-2010 , Gunn diodes gunn-diodes , and frequency multipliers freq-mult are approaching the THz-gap from the low-frequency side. The latter covers the whole THz range while having small power. The general fundamental obstacle of all these THz sources is the small emission rate (of the order of 10 ms). It can be increased with the Purcell effect when the THz source is placed in a cavity RefTodorov ; RefChassagneux , however, the quantum efficiency is still about $1\%$, and the manufacture of these devices is difficult.

Graphene and carbon nanotubes may serve as highly tunable sources and detectors of THz radiation Portnoi2006 ; Portnoi2007 ; Portnoi2008 ; Mikhailov2009 ; Portnoi2010 ; Portnoi2014 ; Ryzhii2015 , and even in THz lasers Ryzhii2016 ; Ryzhii2011 ; Ryzhii2012 ; Ryzhii2013 ; Ryzhii2012A ; Li2012 . In the dual-gate graphene-channel field-effect transistor Ryzhii2011A embedded into a cavity resonator Ryzhii2018 ; Hramov2014 , one observes spontaneous broadband light emission in the 0.1-7.6 THz range with the maximum radiation power of $\sim 10~{}\mu$W at the temperature 100 K. There are also emerging sources of THz-radiation that can deliver several frequencies at room temperature, e.g., multiple harmonic generation in superlattices Pereira2017 ; Apostolakis2019 , frequency difference generation in mid infrared QCLs  Razeghi2015 and THz optical combs Forrer2018 .

Graphene covered with a thin film of colloidal quantum dots has strong photoelectric effect, that provides enormous gain for the photodetection (about $10^{8}$ electrons per photon) Konstantatos2012 ; graphene grown on SiC has strong photoresponse Trabelsi2017 ; and graphene composites can improve solar cells efficiency Bkakri2015 . Note, both graphene and superconductor alone are practically insensitive to light Yung2013 . In this Letter, 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. This new feature allows one to use this device for studying chemical and biological processes or in telecommunications for encryption-decryption procedures, where it is important to image the whole spectrum.

We consider a system consisting of parallel layers of graphene and a superconductor, exposed to an electromagnetic (EM) field incident with the angle $\theta$ and linearly polarized along the x-z plane (p-polarisation), $\mathbf{E}(\mathbf{r},t)=\mathbf{(\sin\theta,0,\cos\theta)}E_{0}e^{-i(k_{\perp}z+\mathbf{k}_{\parallel}\cdot\mathbf{r}+\omega t)}$, where $\mathbf{k}_{\parallel}$, $\omega$, and $\mathbf{r}$ are the in-plane wave vector of the field, frequency, and coordinate, respectively (see Fig. 1). Between graphene layer and superconductor there is a gate voltage that controls its chemical potential and provides AC power. It may pump the energy in the superconductor with AC bias, larger than the superconducting gap exciting quasiparticles and creating NDR. Alternatively, the battery can be replaced by an external laser (pump) with frequency exceeding the superconducting gap (Fig. 1).

The electrons in graphene are coupled by the Coulomb interaction, which has the Fourier image given by $v_{k}=2\pi e^{2}/k$, where $\mathbf{k}$ 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 $u_{k}=2\pi e^{2}\exp(-ak)/k$, where $a$ $\sim 10$ nm is the separation between the layers.

Using the linear response theory for hybrid systems RefOurFano ; RefOurParamag , we can write the electron density fluctuations in the graphene layer $\delta n_{k\omega}$ and Cooper pair density fluctuations in the superconducting layer $\delta N_{k\omega}$ as Villegas2018

[TABLE]

where $\Pi_{k\omega}=\Pi_{k\omega}^{R}+i\Pi_{k\omega}^{I}$ and $P_{k\omega}=P_{k\omega}^{R}+iP_{k\omega}^{I}$ are the complex-valued polarization operators of the graphene and superconductor, respectively, and $W_{k\omega}={eE_{0}}/{ik}$ is the Fourier image of the potential energy due to the external electric field. From Eqs. (Optical Transistor for an Amplification of Radiation in a Broadband THz Domain) we can find the density fluctuations in graphene $n_{k\omega}$ and in superconductor $N_{k\omega}$ as linear functions of applied electric field amplitude $E_{0}$ (see Supplemental Material SM for details). Collective plasmonic hybrid modes in graphene and superconductor can be found from the same system of equations, taking into account the expressions for the polarization loops of the superconductor Villegas2018 and graphene Wunsch2006 ; Hwang2007 . Substituting the expressions for $n_{k\omega}$ and $N_{k\omega}$ into the continuity equations, $kj_{k\omega}=-e\omega\delta n_{k\omega}$ and $kJ_{k\omega}=-2e\omega\delta N_{k\omega}$, for graphene and superconductor, respectively, we can determine the electric currents in each of the layers and their impedances $Z_{G}$ and $Z_{SC}$. 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 $2\Delta=2$ 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 Landau

[TABLE]

where the integration is over the graphene plane, $\langle\cdot\cdot\cdot\rangle$ denotes time-averaging. We normalize the power with the sample area $\int d^{2}=l^{2}$ and the square of the field amplitude $E_{0}^{2}$ to get

[TABLE]

Figure 2(b) shows the dependence of the power absorption on the EM field incidence angle $\theta$ calculated with (3), fixing $\mu=0$ at the Dirac point by gate voltage. All the curves exhibit critical angles at which the power absorption becomes negative, $\alpha<0$. 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 THz frequency range (solid curves and the shaded regions). There is no negative absorption region for isolated graphene, where the power absorption remains positive for any frequency $\nu$ (dashed curves). When the light incidence angle $\theta$ increases, both the maximal intensity (slightly) and the frequency range of the negative light absorption increase, see the shaded area in Fig. 2(b,c). Thus, the angle of light incidence allows us to control the range of light frequencies with the negative absorption.

To understand the $\theta$-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 can not 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 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 $\theta$. If the system is embedded into a cavity resonator, there might even arise lasing similar to one observed in plasmonic lattices remark-X ; Zhou2013 or semiconductor superlattices Hramov2014 .

The mechanism of gain here is similar to one in a waveguide coupled with a superconducting Josephson junction Pedersen2009 . Then, the optical reflectivity of the system reads $\Gamma=(Z_{G}-Z_{SC})^{2}/(Z_{G}+Z_{SC})^{2}$. Near the frequency of the plasmon resonance, there is an area of negative differential resistance of the superconductor, $R_{SC}<0$. If we assume $X_{G}=0$ and $X_{SC}=0$ Pedersen2009 ; remark10 , we find $\Gamma>1$. Note, that graphene can also have NDR (see Sec. III of SM and Kummel1990 ; Hyart2008 ; Hyart2009 ) as in a graphene transistor, which consists of two graphene layers separated by a BN insulator Britnell2013 .

The graphene-superconductor junction (Fig. 1) has large tunneling resistance. An electron in graphene with energy below the superconducting gap can tunnel into superconductor only due to the Andreev scattering Andreev1964 . The probability of such tunnelling 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 Aslamazov1968 . 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 mechanism of NDR can exist only in a highly nonequilibrium excited state created by the pump.

Scattering and de-phasing mechanisms are limiting the gain bandwidth and can flat and eliminate the gain Pereira . Here the plasmon scattering within each and between two graphene and superconducting layers can not only broaden the width of the optical transition, but also enable optical gain and absorption to coexist, constituting the Wacker-Pereira mechanism of optical gain Wacker2007 ; Wacker2012 ; Pereira , as observed in QCLs Terazzi2007 ; Revin2008 . It provides one of the explanations, why $\alpha(\omega)=1-\Gamma(\omega)$ is negative below the plasmon resonance.

From another perspective, graphene separated by dielectric layer (e.g., made of BN, SiO2 or Ta2O5) from the superconductor (Nb, Pb, or HTSC) together form a parallel plate capacitor, which capacitance $C$ is given by ${C}={C_{plate}}+{C_{q}}$, where $C_{plate}$ is the classical capacitance $C_{plate}=\epsilon_{0}A/a$, $A$ is the area of the sample and $\epsilon_{o}$ is the dielectric constant (e.g., $\epsilon_{o}=3.9$ for SiO2). The quantum capacitance $C_{q}$ of graphene emerges due to its conical energy-momentum relation, and it has the form $C_{q}={2Ae^{2}|E_{F}|}/{\pi\hbar^{2}v^{2}_{F}}$ Yu ; Trabelsi2014 ; Trabelsi2016 .

This parallel plate capacitor is connected to a power supply that charges the capacitor and provides the energy for the amplification of incident radiation. The incident light (in particular, its component parallel to the superconducting surface) induces the fluctuation of charge density $\delta n_{s}$, which is associated with a travelling plasmon wave with the amplitude $E_{s}\sim\delta n_{s}$, where $\delta n_{s}=\delta n_{s0}\cos({kx+\omega t})$ (Fig. 3). This charge density wave on the surface of the superconductor generates a mirror charge wave of the opposite sign in the neutral graphene layer, being of the same order as the charge fluctuations in superconducting layer, i.e. $\delta n_{G}\sim\delta n_{s}$. Note that the sign of these charge density fluctuations changes each half-wavelength of the plasmon wave. These plasmons have the wavelength larger than a micrometer, and therefore the charge density for each half-wavelength can be viewed at 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 $E_{F}$, which is directly related to the amplitude of the plasmon wave propagating in graphene $E_{G}$. Due to the quantum capacitance of graphene, described by the relation $E_{F}\sim\sqrt{n_{G}}$, the waves amplitude, $E_{G}\sim E_{F}$, 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 THz range (similar to photo-conductive antennas Burford2017 ), 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. Due to 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 graphene and superconductor. Due to the graphene quantum capacitance Yu ; Trabelsi2014 ; Trabelsi2016 , the amplitude of these electromagnetic waves becomes amplified (by pump), resulting in reflection coefficient larger than 1 (as shown above).

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 superconducting gap decreases, while graphene mobility changes a little. Since the radiation amplification occurs in graphene the temperature has a weak effect on the operation of THz transistor (see insets in Fig. 2(b,c). The working temperature range is similar to one for the stack of the Josephson junctions made of HTSC Ozyuzer2007 ; Welp2013 and it is limited by the critical temperature $T_{c}$.

The radiation power reads $P_{r}=\langle V_{g}I\rangle$, where the voltage $V_{g}$ and the transverse current $I$ are periodically changing in time around the Dirac point. Then, assuming simple periodic behavior for both $I(t)$ and $V_{g}(t)$ and the graphene-superconductor separation, $a$=10 nm remark-distance , the maximal outcome power reads $\langle I\times V_{g}\rangle\sim$ 200-250 $\mu$W/cm2 remark-power . 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 non-zero angle of incidence. We suggest that the system can serve as an amplifier of THz radiation. The essence of the amplification 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 usage 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 a strong dependence of the Fermi energy on the charge doping. A weak charge density wave can induce a strong electric field in these materials, 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 opening new opportunities in molecular and biological noise spectroscopy Koshelets-IEEE-2015 ; Gulevich-2017 .

Acknowledgements. 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 grant RFBR 18-29-20033 mk. The work of F.V.K. was supported by the Government of the Russian Federation through the ITMO Fellowship and Professorship Program.