Heteroepitaxial graphene-on-silicon (GOS) technology

From the technological point of view, synthesis and growth of high-quality graphene is critical. Thermal decomposition of epitaxially grown hexagonal SiC on a SiC substrate is a promising synthesis technology for epitaxial graphene (EG).Reference de Heer, Berger, Wu, First, Conrad, Li, Li, Sprinkle, Hass, Sadowski, Potemski and Martinez¹⁵ (See the articles in this issue by Ruan et al., Nyakiti et al., Bartelt and McCarty, Starke et al., and Wong et al. for more information about epitaxial graphene.) Epitaxial graphene is formed on hexagonal SiC crystals by silicon sublimation during ultrahigh-vacuum annealing. Such an epitaxial growth method can provide a definite graphene crystal orientation, which is mandatory for use in ultrahigh-frequency optoelectronic devices whose operations rely on superior carrier-transport properties.

Because EG synthesis is more amenable than exfoliation to mass production, it is likely to have a greater impact on industrial applications. The drawbacks of EG, however, are the limited diameter of available SiC substrates and their high current prices. For practical use of graphene in electronics, the formation of graphene on silicon substrates, if it proves possible, would be the most promising approach.

With this motivation, we developed a heteroepitaxial graphene-on-silicon (GOS) materials-synthesis technology.Reference Suemitsu, Miyamoto, Handa and Konno^16–Reference Fukidome, Abe, Takahashi, Imaizumi, Inomata, Handa, Saito, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita, Ito and Suemitsu¹⁹ The first step in the GOS process is the heteroepitaxy of SiC on silicon substrates. Instead of the conventional silane/propane two-source system, a single source of monomethylsilane (SiCH₆) was introduced to grow a single-domain SiC film at relatively low temperatures (∼1000°C) under a vacuum of 10^–4 Torr.Reference Suemitsu and Fukidome¹⁷ Cubic 3C-SiC is the only polytype that grows on silicon substrates, not the hexagonal SiC phases that have been extensively explored for epitaxial graphene growth. Fortunately, the surface crystallographic structures of cubic 3C-SiC(111) and hexagonal 6H-SiC(0001) are identical, so that hexagonal graphene can be synthesized by sublimation of silicon from cubic 3C-SiC(111).Reference Suemitsu and Fukidome¹⁷

Further study of GOS growth in terms of chirality and crystal orientation has been carried out.Reference Fukidome, Takahashi, Abe, Imaizumi, Handa, Kang, Karasawa, Suemitsu, Otsuji, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita and Suemitsu^18, Reference Fukidome, Abe, Takahashi, Imaizumi, Inomata, Handa, Saito, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita, Ito and Suemitsu¹⁹ It was found that graphene can be grown on various crystal orientations of the silicon substrate: 3C-SiC(111)/Si(111), 3C-SiC(110)/Si(110), 3C-SiC(100)/Si(110), and 3C-SiC(111)/Si(110). The authors’ findings on this growth method include the influence of the substrate orientation on the graphene–SiC interface structure and on the stacking of multiple graphene layers.

Figure 1 a shows the Raman spectra at the D, G, and G′ bands of graphene grown on SiC(110)/Si(110), SiC(100)/Si(100), and SiC(111)/Si(111). Figure 1b shows the corresponding transmission electron microscope images. The G′ band, in particular, is related to the intrinsic electronic structure of graphene, because it originates from an intervalley double-resonant process involving excitation and de-excitation of an electron and two in-plane transverse optical phonons in the vicinity of the K and K′ points. Interlayer interactions can break the symmetry and split the electronic band, which can readily be detected in the splitting of the G′ band into multiple peaks.

Figure 1.

(a) Raman spectra and (b) transmission electron microscope images of heteroepitaxial graphene on SiC(110)/Si(110) (top), SiC(100)/Si(100) (middle), and SiC(111)/Si(111) (bottom), along with (c) schematic plan-view representations of the differences in their stacking formations. Adapted from References 18 and 19.

For Si(111) growth, the G′ band exhibits multiple peak components.Reference Fukidome, Takahashi, Abe, Imaizumi, Handa, Kang, Karasawa, Suemitsu, Otsuji, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita and Suemitsu^18, Reference Fukidome, Abe, Takahashi, Imaizumi, Inomata, Handa, Saito, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita, Ito and Suemitsu¹⁹ This shows that the graphene is stacked in a Bernal manner, so that multilayer graphene on this substrate is semiconducting with nonzero carrier masses. In contrast, when a Si(100) or Si(110) substrate is used, the G′ band shows a single peak. This Raman spectrum, in conjunction with the low-energy electron diffraction pattern (not shown), shows that the stacking is non-Bernal and turbostratic (i.e., the stacking is not chiral and the honeycomb lattice is stacked on top with random rotational angle), and the graphene on this surface is metallic.Reference Fukidome, Takahashi, Abe, Imaizumi, Handa, Kang, Karasawa, Suemitsu, Otsuji, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita and Suemitsu^18, Reference Fukidome, Abe, Takahashi, Imaizumi, Inomata, Handa, Saito, Enta, Yoshigoe, Teraoka, Kotsugi, Ohkouchi, Kinoshita, Ito and Suemitsu¹⁹

Even for multilayered graphene, each layer preserves the optoelectronic properties of monolayer graphene. This characteristic provides a fundamental advantage for terahertz photonic device applications, in that the quantum efficiency can be increased in proportion to the number of stacked layers while still preserving the gapless and linear energy spectrum.Reference Ryzhii, Ryzhii, Mitin and Otsuji^11, Reference Ryzhii, Ryzhii, Satou, Otsuji, Dubinov and Aleshkin^13, Reference Ryzhii, Dubinov, Otsuji, Mitin and Shur¹⁴

Carrier relaxation/recombination dynamics and terahertz gain

Ultrafast carrier dynamics in optically or electrically pumped graphene

The unusual optical properties of photoexcited carriers in graphene reflect the structures of both the electronic and vibrational bands. The honeycomb lattice of graphene is formed with strong in-plane covalent bonds of carbons having a low atomic mass, giving rise to extremely high optical-phonon energies, ∼198 meV (1600 cm^–1) at the zone center Γ and ∼163 meV (1315 cm^–1) for the transverse mode at the zone edges K and K′, as shown in Figure 2aReference Suzuura and Ando^20, Reference Maultzsch²¹ (also see the article in this issue by Pop et al.). Carrier–phonon interactions, through intravalley and intervalley as well as intraband and interband processes, as shown in Figure 2b, play a dominant role in cooling photoelectrons and photoholes, which leads to ultrafast nonequilibrium energy relaxation processes in graphene.

Figure 2.

(a) Phonon band diagram of graphene. The identified modes are as follows: LA, longitudinal acoustic; LO, longitudinal optical; TA, transverse acoustic; TO, transverse optical. The red circles show the phonons that interact with carriers. (b) Interaction of optical phonons and carriers through intervalley, intravalley, interband, and intraband transitions. Adapted from Reference Reference Maultzsch21.

Carrier relaxation and recombination dynamics in optically pumped graphene are shown schematically in Figure 3. After the electrons and holes are first photogenerated with high energies (Figure 3a), collective excitations due to carrier–carrier scattering result in ultrafast carrier quasi-equilibration among the electron and hole populations separately, within a time scale of 10–100 fs (Figure 3b).^22–25 At this point, the quasi-Fermi level for electrons (holes) is below (above) the Dirac point (which is the position of the Fermi level in undoped graphene). Then, carriers in the high-energy tails of their distributions emit optical phonons, cooling the entire population and accumulating around the Dirac point (Figure 3c).Reference Dawlaty, Shivaraman, Chandrashekhar, Rana and Spencer^22–Reference Rana, George, Strait, Dawlaty, Shivaraman, Chandrashekhar and Spencer²⁵ Because of the fast intraband relaxation (picosecond or less) and relatively slow interband recombination (≫1 ps) of photoelectrons and holes, large excesses of electrons and holes build up just above and below the Dirac point, respectively. With sufficiently intense excitation, a population inversion can be achieved, as seen in Figure 3c (mathematically indicated by an average electron and hole occupation greater than 0.5), leading to the potential for stimulated emission as in lasers.Reference Satou, Otsuji and Ryzhii²⁶

Figure 3.

Carrier relaxation and recombination dynamics in optically pumped graphene at room temperature, showing the evolution of f(ε), the fraction of states at energy ε that are occupied. The horizontal axis indicates the Fermi energy, which is at the Dirac point in undoped graphene. The medium blue regions represent photoexcited or thermally excited electrons, and the light blue regions represent holes. (a) Optical excitation creates an electron and a hole, each with one-half of the photon energy, ħΩ. (b) Scattering among carriers redistributes this energy to create hot thermal distributions of electrons and holes. (c) Optical-phonon emission cools the carriers, allowing population inversion in which some states above the Fermi level, with f(ε) > 0.5, contain more electrons than some states, with 1 – f(ε) < 0.5 electron occupancy, below the Fermi energy.

After quasi-equilibration, the energy distributions of both the electrons and holes can be characterized by the usual Fermi–Dirac function, but with a modified temperature and a nonzero quasi-Fermi energy ε_F (relative to the Dirac point). Figure 4a–b shows the simulated time evolution of these quantities after graphene is pumped at t = 0 with a pulse of light with a photon energy of 0.8 eV. In this simulation, all of the scattering mechanisms shown in Figure 3 are taken into account. The calculated carrier energy relaxation times (inverses of the scattering rates) for the various mechanisms range from subpicoseconds to picoseconds, depending on the carrier and optical-phonon energies, the populations, and the initial and final states of carriers.²⁶

Figure 4.

Simulated temporal evolution of the (a) quasi-Fermi energy and (b) carrier temperature in graphene after a pulse of pump light at time t = 0. These two quantities characterize the energy distribution of photoexcited electrons (and also holes, given that, in undoped graphene, the two carriers are symmetrical) once the electron and hole populations have separately come into internal thermal equilibrium. Adapted from Reference Reference Satou, Otsuji and Ryzhii26.

The following trends can be observed: (1) Immediately after the pumping (t ≈ 0 ps), ε_F instantaneously falls due to carrier–carrier scattering (corresponding to Figure 3b). (2) Then, ε_F increases rapidly as a result of carrier cooling due to emission of optical phonons by carriers in the high-energy tails. (3) If the pumping intensity exceeds a certain threshold level, ε_F becomes positive (for both electrons and holes), so the population is inverted (corresponding to Figure 3c). After that, the recombination process continues more slowly (∼10 ps).

Negative terahertz conductivity and amplified stimulated emission

Population inversion is a prerequisite for gain. However, to achieve gain for any particular frequency of electromagnetic radiation, population inversion must overcome Drude absorption by free carriers.

The gain at angular frequency ω is determined by the real part of the net dynamic conductivity, ${\rm{Re \sigma }}_{\rm{\omega }} $, which is proportional to the absorption of photons with that angular frequency. Specifically, negative values of ${\rm{Re \sigma }}_{\rm{\omega }} $ correspond to gain. ${\rm{Re \sigma }}_{\rm{\omega }} $ consists of intraband (${\mathop{\rm Re}\nolimits} {\rm{ \sigma }}_{\rm{\omega }}^{{\mathop{\rm intra}\nolimits} }$) and interband (${\mathop{\rm Re}\nolimits} {\rm{ \sigma }}_{\rm{\omega }}^{{\mathop{\rm inter}\nolimits} }$) contributions and is given byReference Ryzhii, Ryzhii and Otsuji⁹

(1)$$\displaylines{ {\rm{Re \sigma }}_{\rm{\omega }} = {\rm{Re \sigma }}_{\rm{\omega }}^{{\mathop{\rm inter}\nolimits} } + {\rm{Re \sigma }}_{\rm{\omega }}^{{\mathop{\rm intra}\nolimits} } \cr \approx {{e^2 } \over {4\hbar }}(1 - 2f_{\hbar {\rm{\omega }}} ) + {{(\ln 2 + {\rm{\varepsilon }}_{\rm{F}} /2k_{\rm{B}} T)e^2 } \over {{\rm{\pi }}\hbar }}{{k_{\rm{B}} T{\rm{\tau }}_{\rm{m}} } \over {\hbar (1 + {\rm{\omega }}^2 {\rm{\tau }}_{\rm{m}} ^2 )}}{\rm{,}} \cr} $$

where e is the elementary charge, ħ is the reduced Planck constant, f_{ħ ω} is the Fermi distribution function at energy ħω, k _B is the Boltzmann constant, T is the temperature, and τ _m is the momentum relaxation time of carriers. The intraband contribution, ${\mathop{\rm Re}\nolimits} {\rm{ \sigma }}_{\rm{\omega }}^{{\mathop{\rm intra}\nolimits} }$, corresponds to the Drude absorption and is always positive, contributing as a loss, although this contribution becomes less important for ω > 1/τ_m.

Typical simulated results for ${\rm{Re \sigma }}_{\rm{\omega }} $ are shown in Figure 5 as a function of time and frequency at a fixed pump fluence of 8 μJ/cm² (intensity of 1 × 10⁸ W/cm² and pulse duration of 80 fs [full width at half-maximum]) with a momentum relaxation time of τ _m = 3.3 ps.Reference Satou, Boubanga Tombet, Otsuji and Ryzhii²⁷ It can be clearly seen that, even at room temperature, a broad terahertz gain bandwidth from ∼2.2 THz to 10 THz is obtained on the picosecond time scale after pumping.

Figure 5.

Simulated time evolution of the terahertz dynamic conductivity of graphene after pulsed pumping at a fixed pump fluence of 8 μJ/cm² (intensity of 1 × 10⁸ W/cm² and pulse duration of 80 fs [full width at half-maximum]). The pump-photon energy and carrier momentum relaxation time τ_m were assumed to be 0.8 eV and 3.3 ps, respectively. The conductivity values were normalized to the quantum conductivity [e ²/(4ħ)]. The red color encompasses positive values as well as small negative values. After the initial carrier relaxation, a spectral window temporarily appears, at frequencies high enough to avoid free-carrier absorption, that has negative conductivity (not red) and thus potential power gain.

We used time-domain spectroscopic studies to show that a graphene sheet can amplify an incoming terahertz field.Reference Otsuji, Boubanga-Tombet, Chan, Satou and Ryzhii^28–Reference Karasawa, Komori, Watanabe, Satou, Fukidome, Suemitsu, Ryzhii and Otsuji³⁰ The novel pump/probe geometry is shown in Figure 6. In this setup, a single 80-fs, 1550-nm, 4-mW, 20-MHz-repetition-rate beam from a fiber laser is split into two parts: one for optical pumping and generation of the terahertz probe beam in a CdTe crystal and the other for optical probing of the terahertz field. The delay of the optical probe relative to the arrival of the pump pulse can be swept through the time domain of interest. The terahertz probe pulse double-reflects to stimulate terahertz emission in graphene, which is detected as a second, delayed terahertz signal.

Figure 6.

Geometry for the observation of amplified terahertz emission from optically pumped graphene. When the pump (thick red arrow) enters CdTe, it generates a terahertz wave (blue arrow labeled “1”) that is measured by electro-optic sampling using an optical probe (thin red arrow). Some of the terahertz wave is reflected and then reflected again from the substrate (blue arrow labeled “2”). This second terahertz pulse is delayed by the round-trip transit time through the CdTe, allowing it to be probed separately. Adapted from Reference Reference Boubanga-Tombet, Chan, Watanabe, Satou, Ryzhii and Otsuji29.

For our experiments, a high-quality exfoliated monolayer-graphene/SiO₂/silicon sample, having a surface potential variance of 4.05 meV and a Dirac voltage close to zero, indicating very little doping,²⁸was placed on the stage. A 120-μm-thick CdTe(100) crystal placed on the sample acted as both a terahertz probe pulse emitter and an electro-optic sensor.

Figure 7a shows the measured temporal response.Reference Otsuji, Boubanga-Tombet, Chan, Satou and Ryzhii²⁸ The second pulse, the doubly reflected terahertz signal, is more intense with graphene present than without it. This indicates that graphene acts as an amplifying medium in the terahertz frequency range. As shown in the inset of Figure 7b, the emission spectra exhibit threshold-like behavior as a function of laser pumping intensity, attesting to the occurrence of negative conductivity and population inversion in graphene.²⁹A normal dispersion, in which the phase changes sign across the gain peak, testifies to the occurrence of amplification attributed to stimulated emission of photocarriers in the inverted state.

Figure 7.

(a) Temporal responses to an optical pulse,Reference Otsuji, Boubanga-Tombet, Chan, Satou and Ryzhii²⁸ measured using electro-optic sampling (EOS), for samples with and without graphene (GR) at the substrate–CdTe interface. Graphene has little effect on the terahertz pulse generated by the first passage of the pump (“1,” see Figure 6), but the delayed, doubly reflected pulse (“2”) samples the graphene after pump-excited carriers have equilibrated, and its increased amplitude demonstrates the gain of photoexcited graphene in the terahertz spectral region. (b) Gain spectra obtained by Fourier transforming the temporal responses of the second pulse, normalized to the frequency response without graphene. The inset shows that power gain appears only for sufficient pump power.Reference Boubanga-Tombet, Chan, Watanabe, Satou, Ryzhii and Otsuji²⁹ (c) Comparison of the measured gain spectra for terahertz-probe delays of 2.0 ps (with a thin CdTe crystal) and 3.5 ps (with a thick CdTe crystal). The shorter delay in the thinner crystal allows the reflected teraherz pulse to arrive earlier, when there are more unrecombined (but already quasi-equilibrated) photoexcited carriers to create gain.

The observed voltage gain exceeded the calculated maximum value of 1.011 (=$\sqrt {1.023}$) determined by the absorbance of monolayer graphene (2.3%). Part of this discrepancy could result from experimental artifacts. For example, the ∼70% reflection of the terahertz probe pulse at the SiO₂–silicon interface will also interact with graphene and be amplified. Because the round-trip delay in the 300-nm-thick SiO₂ layer is ∼4 fs, almost no distortion is seen in the temporal response. This multipath reflection effect can at most double the gain but cannot fully explain the observed large discrepancy in the gain. Another effect arises from the pumping-induced increase in substrate reflectivity due to photocarrier-dependent Drude conductivity. For a graphene/SiO₂/silicon sample, whose silicon substrate has a low resistivity of 0.005 Ω cm, the reflectivity increase is as small as 1–2% and quite monotonic over the entire terahertz spectral range of interest. Thus, a small percentage of the increase in reflectivity should be subtracted from the obtained photon echo signal, but this would barely affect the gain spectral profile.Reference Boubanga-Tombet, Chan, Watanabe, Satou, Ryzhii and Otsuji²⁹

We also conducted a similar experiment with a thinner (80-μm-thick) CdTe crystal to shorten the terahertz-probe delay time from the original 3.5 ps to 2.0 ps, which is the timing giving the largest negative conductivity and, hence, the greatest gain spectral width, according to Figure 5.Reference Boubanga-Tombet, Chan, Watanabe, Satou, Ryzhii and Otsuji²⁹ The gain spectra of these two cases are plotted in Figure 7c. The emission spectra are clearly broader (in particular, at the higher-frequency end) for a terahertz-probe delay time of 2 ps. This tendency coincides with the simulated results in Figure 5, providing clear evidence of the nonequilibrium relaxation dynamics of the quasi-Fermi energy reflecting a population inversion.

Toward terahertz graphene injection lasers

When installed in a proper terahertz cavity as a gain medium, graphene is expected to lase. Because the electronic transitions arise from the π-electron sheets, the electric field vector of the emitted terahertz photons must lie within the graphene plane. This permits gain for terahertz photons propagating either perpendicular or parallel to the graphene plane. Therefore, structures with both verticalReference Dubinov, Aleshkin, Ryzhii, Otsuji and Ryzhii^12, Reference Ryzhii, Ryzhii, Satou, Otsuji, Dubinov and Aleshkin¹³ and waveguide-typeReference Ryzhii, Dubinov, Otsuji, Mitin and Shur¹⁴ cavities can be used to construct a graphene terahertz laser. From the viewpoint of higher quantum efficiency and hence lower pumping threshold, a waveguide cavity is superior because it can have a larger volume of gain overlap, distributed along the length of the cavity in which the terahertz photons propagate.

Figure 8a–b depicts simulated room-temperature gain spectra in different geometries of a slot-line waveguide cavity structure that includes a 20-layer non-Bernal-stacked graphene sample.Reference Ryzhii, Dubinov, Otsuji, Mitin and Shur¹⁴ The cross-sectional electric field distribution is also shown for one geometry. The width of the slot line was fixed at 15 μm, whereas the depth of the cavity was treated as a parameter. The quasi-Fermi energy that defines the level of population inversion was set in the calculations at a feasible level of 30 meV.

Figure 8.

(a) The bottom image shows the simulated distribution of the terahertz electric field, E _ω, in a slot-line waveguide laser structure with multiple graphene layers (GLs) on top, for a slot width of D = 15 μm and a substrate (cavity) thickness of W = 25 μm. In the envisioned operation, GLs in the slot are illuminated from the top with photons of energy ħΩ to provide gain to a terahertz wave traveling in the plane of the image. (b) Calculated frequency dependence of the terahertz gain coefficient in laser structures consisting of K = 20 GLs and the having quasi-Fermi energy of the topmost GL ${\rm{\varepsilon }}_{\rm{F}}^{\rm{T}} = 30{\rm{ meV}}$, for different cavity thicknesses. Adapted from Reference Reference Ryzhii, Dubinov, Otsuji, Mitin and Shur14.

Depending on the cavity thickness in the range of 12–25 μm, the gain coefficient reached a peak value ranging from 2 to 8 cm^–1at 2–3 THz, whereas its lower (higher) cutoff frequencies fell in the range of 1.5–2.2 THz (3.5–7.6 THz). If the number of graphene layers could be increased to ∼100, the gain factor would increase by one order of magnitude.Reference Ryzhii, Dubinov, Otsuji, Mitin and Shur¹⁴

To assess the practicality of the graphene terahertz laser structure, consider a distributed-feedback laser-cavity structure similar to that demonstrated for a terahertz quantum-cascade laser,Reference Williams, Kumar, Hu and Reno³¹ in which the gain coefficient of the slot-line waveguide is spatially and periodically modulated. We assume front and back mirrors having a feasible reflectance of ∼85% and a waveguide loss, including radiative loss and free-carrier absorption, of ∼10 cm^–1. A 1-mm-long laser cavity can then provide a net gain of ∼5 (∼7 dB), which is sufficient to realize lasing.

Optical pumping with rather high photon energy, on the order of a few electronvolts, significantly heats the carriers, as shown in Figure 3b, which dramatically increases the pumping threshold required to achieve population inversion.Reference Ryzhii, Ryzhii, Mitin and Otsuji^11, Reference Dubinov, Aleshkin, Ryzhii, Otsuji and Ryzhii^12, Reference Satou, Otsuji and Ryzhii²⁶ However, decreasing the pump-photon energy suppresses carrier heating. For example, for a decrease in pump-photon energy from 0.8 eV to 0.2 eV (such as for CO₂gas lasers or terahertz quantum-cascade lasers), the maximum carrier temperature is suppressed from ∼1400 K to below ∼450 K. Population inversion then occurs from the earliest time after pumping at deeper levels, reducing the pumping threshold.

Current injection is a better solution because such electrical pumping can provide any pumping energy, even below a few millielectronvolts, thus reducing heating. In traditional semiconductor laser diodes,Reference Ryzhii and Ryzhii^10, Reference Ryzhii, Ryzhii, Mitin and Otsuji¹¹ a narrow-bandgap intrinsic (“i”) active region is sandwiched between wide-bandgap p- and n-type regions. Forward biasing the p–i–n diode injects carriers into the i region at the fixed energy levels of the band edges, leading to population inversion between the conduction and valence bands. Similarly, in quantum-cascade laser diodes, forward bias injects carriers into the upper subbands of multiple quantum wells in a superlattice heterostructure, leading to population inversion between the subbands. In both cases, the population inversion occurs between fixed energy levels that are defined by the bandgap or subbandgap energies, and majority carriers (electrons in n-type regions or holes in p-type regions) are originally introduced by donor or acceptor impurity doping.

In contrast, the lack of a bandgap in graphene means that the energy levels that contain the inverted carrier populations are not fixed. Moreover, so far, no mature impurity doping technique has been developed for graphene. One must therefore consider how to define the energy levels at which the carrier populations are inverted and how to realize a p–i–n junction.

We have proposed a device with a dual-gate structure that can solve these issues.Reference Ryzhii and Ryzhii^10, Reference Ryzhii, Ryzhii, Mitin and Otsuji¹¹Figure 9 a shows the structure and band diagram of a dual-gate graphene-channel field-effect transistor (FET).Reference Ryzhii, Ryzhii, Mitin and Otsuji¹¹ Applying opposite-polarity biases to the dual gates (i.e., V _G1 = –V _G, V _G2 = +V _G) attracts holes (electrons) beneath gate G1 (G2), making a p–i–n junction in the graphene channel.

Figure 9.

(a) Schematic drawings of proposed dual-gate field-effect transistor structure for electrically pumping graphene to create lasers at terahertz frequencies. Opposite biases on the two gates, G1 and G2, create an effective p-type/intrinsic/n-type diode. A small bias between drain D and source S allows holes and electrons to flow from under the gates into the intrinsic region between the gates. Sufficient injection will create population inversion in this region and thus the possibility of optical gain for photons of energy ħω₀. The band diagram on the bottom shows how the dual gate biases, with the same magnitude but opposite polarity, attract electrons and holes and thereby generate equal quasi-Fermi levels μ_i in the n-type and p-type regions. Both types are injected into the intrinsic section by the drain–source bias, which is split equally on the two sides, resulting in equal quasi-Fermi energies μ in the intrinsic section. As shown, a positive drain–source bias shifts the electron quasi-Fermi energy up by μ and the hole quasi-Fermi energy down by μ, resulting in population inversion. (b) Simulated real parts of the terahertz conductivities (negative values corresponding to gain) of monolayer graphene at 300 K. The axis unit is normalized to the quantum conductivity. The parameter ${\rm{\eta }}_{\rm{o}}^{{\rm{decay}}}$is the optical-phonon decay factor described in the text. Adapted from Reference Reference Ryzhii, Ryzhii, Mitin and Otsuji11.

In the intrinsic section, both electrons and holes are injected, as follows: If a slight dc drain-to-source (DS) bias is applied, V _DS = +V _D (V _D > 0), the quasi-Fermi levels of electrons and holes in the intrinsic section will split by V _D, resulting in population inversion. Injected electrons and holes can recombine through optical-phonon-assisted interband indirect transitions, Auger processes, and/or radiative direct transitions, contributing to spontaneous terahertz photon emission when V _Dis on the order of millivolts to tens of millivolts, with little of the extra heating that occurs during optical pumping.

Electrons (holes) in the intrinsic section are confined by the built-in potential between the p and i (n and i) regions. Carriers can escape either by thermal excitation over this barrier (thermionic current) or by interband quantum-mechanical tunneling through the barrier, which reflects the gapless band structure of graphene.Reference Ryzhii, Ryzhii and Otsuji³² Thermionic and tunneling currents are the major factors that prevent population inversion and reduce the injection efficiency. When the space between the dual-gate electrodes is wide enough, the tunneling current can be suppressed far below the thermionic current,Reference Ryzhii and Ryzhii^10, Reference Ryzhii, Ryzhii, Mitin and Otsuji¹¹ which is a key to maximizing the injection efficiency and, hence, minimizing the threshold lasing current.

Figure 9b shows typical simulated dynamic conductivity of a dual-gate structure having a dual-gate spacing of 20 μm for different bias conditions at room temperature.Reference Ryzhii, Ryzhii, Mitin and Otsuji¹¹ In this simulation, a carrier injection with a density of 10¹² cm^–2, an interband optical-phonon emission rate of 10 ps, and a carrier momentum relaxation time of 3.3 ps was assumed. The optical-phonon decay factor, ${\rm{\eta }}_{\rm{o}}^{{\rm{decay}}}$, which affects the carrier cooling speed,Reference Ryzhii, Ryzhii, Mitin, Satou and Otsuji³³ and the drain bias, V _D, were taken as parameters. The value ${\rm{\eta }}_{\rm{o}}^{{\rm{decay}}} = 2$ occurs when graphene is attached to a conductive substrate with a high thermal conductivity, whereas ${\rm{\eta }}_{\rm{o}}^{{\rm{decay}}} = 10$ applies when graphene is suspended in free space. It can be clearly seen that, for ${\rm{\eta }}_{\rm{o}}^{{\rm{decay}}} = 2$ and V _D= 0.075 V (75 mV), negative dynamic conductivity can be obtained over a very wide frequency range from 2 THz to ∼18 THz. These simulation results are encouraging with respect to the creation of graphene terahertz injection lasers that operate at room temperature.

New aspects of graphene terahertz science and technology

Summary

Recent advances in graphene-based materials and devices for terahertz science and technology were reviewed in this article. Owing to its unique carrier-transport and optical properties, including massless and gapless energy spectra, graphene is anticipated to overcome many technological limitations on conventional electronic and photonic devices operating in the terahertz frequency range. One of the most important aspects arising from the unique ultrafast carrier relaxation/recombination dynamics of graphene is the occurrence of negative dynamic conductivity over a wide terahertz spectral range even at room temperature, which would break through the limit on operating temperatures of state-of-the-art semiconductor lasers. Theoretical calculations suggest that current injection is best suited to preventing carrier heating, hence minimizing the pumping threshold level.

Compared to the existing state-of-the-art semiconductor laser technologies, graphene injection lasers have substantial advantages including relatively high quantum efficiency (∼1), practically high output power (on the order of milliwatts), and room-temperature operation. These advantages arise from (1) the symmetric, gapless, linear band structure that achieves carrier depopulation without the special structures that are mandatory in quantum-cascade lasers; (2) extremely high optical-phonon energy, resulting in ultrafast carrier energy relaxation and efficient population inversion; and (3) extremely long carrier momentum relaxation times that help minimize the intraband Drude conductivity (loss factor) and hence maximize the terahertz gain. Because of the operating principle of interband transitions within the gapless and continuum linear band structure, the lasing frequency and linewidth are determined mainly by the laser cavity profiles (resonance frequencies and finesse), which should be demonstrated when lasing operation is achieved.

The unique terahertz dynamics of two-dimensional plasmons in graphene is another important advantage of graphene materials and devices that helps reduce the propagation loss of the generated terahertz waves. More experiments are needed to verify these exciting predictions, especially in the more practical, epitaxially grown material.

In addition to coherent terahertz light sources, graphene can realize various other functionalities, including amplification, detection, modulation, and frequency multiplication over the terahertz frequency range. In fact, in all of these aspects, graphene could substantially outperform any existing materials. Continued study is also expected toward the creation of graphene terahertz lasers and other novel devices. Design and implementation of high-performance device structures and improvement of synthesis technology that can reproducibly provide the highest quality of large-area graphene will be the key issues.