1. Introduction

Double-graphene-layer (double-GL) structures with narrow inter-GL barrier Boron Nitride (hBN), Tungsten Dioxide (WS₂), and other barrier layers [1–4] have recently been explored for applications in high speed electro-optical modulators [1, 5], transparent electronics [3], terahertz (THz) detectors [6, 7] and photomixers [8]. With the optical or injection pumping, various structures based a single-GL or multiple-GLs (twisted, non-Bernal stacked GLs or GLs separated by relatively thick barrier layers) can exhibit the interband population inversion and lasing of the TE-mode with the in-plane direction of the photon electric field [9 –13]. Due to the gapless energy spectrum of GLs, such a lasing can occur in a wide THz range. The experimental studies of the THz emission from the pumped GLs [14, 15] have validated the concept of GL-based THz lasers (see also the review papers [16, 17]).

Recently [18], in contrast to GL-based lasers using the intra-GL interband radiative transitions [9–13], the use of the resonant radiative inter-GL transitions in double-GL structures with a sufficiently thin barrier between GLs was proposed to realize lasing. Potential advantages unclude much smaller Drude losses of the TM-mode, a weaker temperature sensitivity, the possibility of voltage tuning of the spectrum, and an increased injection efficiency. This can be particularly important at the low end of the THz frequency range (a few THz or less).

In this paper, we develop a device model for the THz laser exploiting the inter-GL radiative transitions in the double-GL structure with a tunneling barrier layer. Such room temperature lasers might compete with the resonant-tunneling diodes [19] and quantum cascade lasers [20–22] at the low end of the THz gap. They can also find applications in the frequency range between 6 and 10 THz, which includes the optical phonon frequencies in A₃B₅ materials, in which the operation of quantum cascade lasers [20] is suppressed.

2. Device structures and principles of operation

We primarily consider the device structure shown in Fig. 1(a). It consists of two GLs chemically doped by donors (n-GL) and acceptors (p-GL), respectively, and separated by a narrow tunneling barrier layer of thickness d (about a few nanometers). The length of each GL is approximately equal to 2L (to the spacing between the side contact). The clad layers between the GLs and the pertinent metal plates are assumed of the same thickness equal to W. One of the edges of each GLs is connected to the side contact while the other one being isolated (independently-contacted GLs). This double-GL structure is similar to those fabricated recently [1–4]. The doping leads to the formation of the two-dimensional electron (2DEG) and the two-dimensional hole (2DHG) gases in the n-GL (the top GL) and the p-GL (the bottom GL), respectively. The bias voltage V between the side contacts induces the extra charges of opposite polarities at GLs and the electric field between GLs. This results in the band structure shown in Fig. 2(a), where μ and Δ are the Fermi energy of 2DEG and 2DHG and the energy separation between the GL Dirac points, respectively, [18]. As seen from Fig. 2(a), the applied bias voltage causes the inter-GL population inversion between the conduction bands and between the valence band (c-c and v-v - transitions accompanied by the photon emission). Thus, the double-GL structure can serve as the laser active region with the lateral injection pumping. The non-radiative inter-GL tunneling and resonant-tunneling were observed in recent experiments [2–4] (see also theoretical papers [23, 24]). The electron and hole lateral injection from the pertinent contacts refills GLs loosing electrons and holes due to nonradiative and radiative inter-GL processes is realized by the electron and hole lateral injection from the pertinent contacts. The device structure is sandwiched by the clad layers (with thickness W ≫ d), which, together with the metal layers, constitute a metal-metal (MM) waveguide for the TM-modes propagating in the direction perpendicular to the x–z plane (see Fig. 1). Such waveguides are successfully used in particular in quantum cascade lasers [21, 22]. Considering this device, we will refer to it as for device ”a”. DC voltages ±V_g (gate voltages) applied between the GL edges and the metal strips of the MM waveguide, can induce sufficient densities of electrons and holes (the electrical induction of the extra electrons and holes by the gate voltages is sometimes dubbed as the ”electrical” doping) even without chemical doping. We will keep in mind this option as well.

 

Fig. 1 Schematic views of double-GL laser cross-sections (a) with the side monopolar injection to each independently contacted GL, tunneling barrier layer, and MM waveguide (device ”a”) and (b) with the injection of electrons and holes from p- and n-contacts to each GL and dielectric waveguide (device ”b”).

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Fig. 2 Band diagrams of laser structures with (a) inter-GL and (b) intra-GL radiative transitions. Wavy arrows indicate the inter-GL and intra-GL radiative c − c, v − v, and c − v transitions in devices ”a” and ”b”, respectively.

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For comparison, we also consider the double GL-structure laser with the lateral injection of electrons and holes into each GL (and propagating toward each other along GL) and employing the inter-band transitions under the conditions of population inversion (see, for example, [9–12]). The device structure (device ”b”), which, for the adequate comparison, is assumed to consist of two undoped GLs with the side p- and n-contacts forming a p-i-n-structure. One of the option to create a laser with such an active region is to use a dielectric waveguide. This device (device ”b”) and its band diagram are shown in Fig. 1(b) and Fig. 2(b), respectively.

3. Inter-GL and intra-GL dynamic conductivities

The electron and hole density in the pertinent GLs is given by

The inter-GL transitions assisted by the emission of photons with the polarization corresponding to the photon electric field perpendicular to GLs (along the axis z) conserve the electron momentum and, hence, do not involve scattering (resonant-tunneling photon-assisted transitions). Assuming κ = 4, d = 4 nm, and Σ_i = 10¹² cm⁻², one obtains V₀ ≃ 136 mV and V_t ≃ 221 mV. The quantities Δ = 5 – 10 meV correspond to V ≃ 229 – 237 mV and μ ≃ 112.0 – 113.5 meV.

The real part of ${\sigma }_{yy}^a(\omega )$ component of the tensor of the double-GL structure dynamic conductivity is given by [25] (see also [11,26–28])

For the GL-structure with the intraband population inversion caused by the injection of both electrons and holes in each GL and with the suppressed the inter-GL transitions (device ”b”), the conductivity tensor components are given by [9]

The sign of $\text{Re}{\sigma }_{xx}^b(\omega )$ depends on the trade-off of the interband (negative) and intraband (positive) contributions. In a certain range of frequencies and Fermi energy, $\text{Re}{\sigma }_{xx}^b(\omega )<0$. This is to be exploited in the THz GL-based lasers.

4. Terahertz gain and gain-overlap factors

The THz gain for the TM-mode in device ”a” and the TE-mode in device ”b” is given by the following equation:

The solution of the Maxwell equations with the pertinent complex permittivity of the waveguide and metal strips yields the spatial distributions of the electric and magnetic fields, E_x, E_y, and E_z in the propagating modes and consequently, the gain-overlap factor. These equations were solved numerically using the effective index and transfer matrix methods (see, for instance [30, 31]). It was assumed that the cladding waveguide layers (above and below the double-GL structure and the inter-GL layers were made of hBN and WS₂, respectively. The hBN complex permittivity was extracted from [32] (æ ≃ ε(x, 0, 0) ≃ 4). It was also assumed that the metal strips in the MM waveguide are made of Au.

Figures 3(a) and 3(b) show the examples of the spatial distributions of the photon electric field components $\left|E_z^a(x,z,\omega )\right|$ and $\left|E_y^a(x,z,\omega )\right|$ in the TM mode of a MM waveguide with the sizes L = W = 5 μm in device ”a”. Figure 3(c) demonstrates the spatial distribution $\left|E_x^b(x,z,\omega )\right|$ in the dielectric waveguide. The obtained dependences correspond to ω/2π = 8 THz. Figure 3(d) shows the spatial distributions of the electric field components $\left|E_z^a(x,0,\omega )\right|$, $\left|E_y^a(x,0,\omega )\right|$, and $\left|E_x^b(x,0,\omega )\right|$ at the double-GL structure plane.

 

Fig. 3 Spatial distributions of the photon electric field components: (a) $\left|E_z^a(x,z,\omega )\right|$, (b) $\left|E_y^a(x,z,\omega )\right|$ in the TM mode in MM waveguide (in device ”a”), (c) $E_x^b(x,z,\omega )|$ in the TE mode (in device ”b”), and (d) amplitudes of electric field components at GL plane (z = 0) for ω/2π = 8 THz). White horizontal strips correspond to GLs with the barrier layer in between.

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As follows from Fig. 3(c), the TM-mode component E_y is small in comparison with the E_z component of this mode [compare the solid and dashed lines in Fig. 3(d)].

Figure 4(a) demonstrates the dependences of the inter-GL matrix element |z_u,l|² calculated using the Schrodinger equation for the wave functions accounting for the delta-function-like dependence in the z–direction of potentials of GLs separated by the barrier of width d. Following [32], we assume that the conduction band offset between GLs ans WS₂ barrier layer and the effective electron mass in the latter are equal to ΔE_g = 0.4 eV and m = 0.27m₀ (m₀ is the mass of free electron) [33]. Figure 4(b) shows the dependence of the depolarization shift Δ_dep on the energy separation between the Dirac points Δ calculated using the obtained values |z_u,l|² and Eq. (5). Due to the inter-GL population inversion, Δ_dep < 0, and its value varies in a wide energy range with varying Δ

 

Fig. 4 Dependences of (a) the inter-GL matrix element |z_u,l|² and (b) depolarization shift Δ_dep on the energy separation between the Dirac points Δ calculated for different values of spacing between GLs d.

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As follows from Fig. 4(a), sufficiently large values of |z_u,l|² can be achieved only in the double-GL structures with rather thin barrier layers (d about a few nm).

Figure 5 shows the frequency dependences of the gain-overlap factors and the waveguide absorption coefficients calculated using Eq. (9) for different waveguide lateral sizes. As seen from Fig. 5, ${\mathrm{\Gamma }}_{yy}^a\ll {\mathrm{\Gamma }}_{zz}^a$. This is a consequence of a weak E_y component [see Fig. 3(d)]. Hence, we can disregard the first term in right-hand side of Eq. (10) and reduce this equation to

 

Fig. 5 Frequency dependences of gain-overlap factors ${\mathrm{\Gamma }}_{jj}^{a,b}(\omega )$ (solid lines) and waveguide absorption coefficients α^a,b(ω) (dashed lines): (a) L = 5 μm and W = 5 μm and (b) L = 15 μm and W = 5 μm.

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As for device ”b”, taking into account Eq. (6), the THz gain can be calculated using the following equation:

The obtained data for the MM waveguide absorption coefficients are in line with those found previously [22]. A significant increase in the absorption coefficient of the TM mode at relatively low frequencies seen in Fig. 5 is attributed to relatively small spacing between the metal strips in the MM waveguides under consideration compared to the radiation wavelength λ = 2πc/ω.

5. Frequency and voltage dependences of the THz gain

Figure 6 shows the frequency dependences of the THz gain g^a(ω) a for double-GL structure (device ”a”) with Σ_i = 1 × 10¹² cm⁻², different values of the energy separation between the Dirac points Δ (i.e., different voltages V), and different waveguide geometrical parameters. The frequency dependences g^a(ω) shown in Fig. 6 correspond to the relaxation broadening γ = 1 meV (ν = 1.6 × 10¹² s⁻¹)and doping level Σ_i = 1 × 10¹² cm⁻², i.e., ${\mu }_i=\overline{h}{v}_W\sqrt{\pi {\mathrm{\Sigma }}_i}\simeq 110\hspace{0.17em}\text{meV}$. The actual value of the Fermi energy μ > μ_i.

 

Fig. 6 THz gain g^a(ω) versus frequency dependences (solid lines) for different values of energy separation between the Dirac points Δ and THz gain g^b(ω) (dashed lines) for different Fermi energies in GLs: (a) L = 5 μm and W = 5 μm and (b) L = 15 μm and W = 5 μm.

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As follows from Fig. 6 (see solid lines), the THz gain as a function of the frequency exhibits a resonant behavior with the peak at ω = ω_max. In this peak, g^a(ω) is positive and fairly large. The peak position is determined by Δ, in line with Eq. (5). The THz gain g^a(ω) outside the resonance is negative, practically coinciding with the absolute value of the MM waveguide absorption coefficient. Since, as follows from Eq. (2), Δ is a function of the bias voltage V, the position of the THz gain resonant peak, ω_max, of the THz gain is voltage tunable. The quantity Δ and, consequently, ω_max depend on Σ_i. In device ”a” with the ”electrical” doping Σ_i ∝ V_g, and, hence, in such a device, the position of the resonance can be also tuned by the gate voltage, V_g. Figure 7 demonstrates how the frequency ω_max at which the THz gain g^a(ω) achieves a maximum value with with varying bias voltage V and gate voltage V_g.

 

Fig. 7 Tuning of frequency ω_max corresponding to peak of the THz gain g^a(ω) by (a) bias voltage V and (b) gate voltage V_g.

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The height of the resonances maxg^a(ω) varies with increasing Δ. This can be attributed to the trade-off in a decrease in |z_u,l|² [see Fig. 4(a)] and in an increase of factor (1 + Δ/Δ_i) in Eq. (11) with increasing Δ. The width of the resonant peaks is determined by parameter γ, i.e., by the collision frequencies of electrons and holes ν. At small ν the peaks in question can be very high, but large ν they are smeared.

It worth noting that in the case of relatively large L, device ”a” can exhibit rather high THz gain in a few THz range corresponding to the left peak in Fig. 6(b). This is due to negligible Drude absorption in such a device structure.

For comparison, in Fig. 6 we also show the THz gain versus frequency calculated for device ”b” with the same broadening parameter γ but different Fermi energies μ. In this device, the THz gain can be positive in a wide frequency range provided sufficiently large μ, i.e., sufficiently strong pumping [9–13]. However, as seen from Fig. 6, the THz gain markedly decreases (and becomes negative) when the frequency approaches the low end of the THz range. This is because of an increasing role of the Drude absorption with decreasing frequency of the TE mode.

6. Discussion

The normal operation of the laser under consideration assumes sufficient densities of electrons in GLs. This is achieved by chemical or ”electron” doping of GLs and the injection of electrons to the upper GL and holes to the lover GL. Reaching sufficient population in the entire GLs requires satisfying the condition L ≤ ℒ_i, ℒ_eh [34]. Here ${ℒ}_i\propto {\left(D_i{\tau }_R^{\mathit{inter}}\right)}^{1/2}$ and ${ℒ}_{eh}\propto {\left(D_{eh}{\tau }_R^{\mathit{inter}}\right)}^{1/2}$ are the diffusion lengths, $D_i\propto {\nu }_i^{-1}$ and $D_{eh}\propto {\nu }_{eh}^{-1}$ are the diffusion coefficients, where ν_i and ν_eh are the electron and hole collision frequencies with impurities and with each other, respectively (ν_i + ν_eh = ν), and ${\tau }_R^{\mathit{inter}}$ is the recombination time associated with the inter-GL processes. As shown [35], and references therein), the diffusion coefficient D_i in GLs can be very large. Considering that the electron-hole scattering and recombination in the double-GL structures might be weak due to the spatial separation of GLs with electrons and holes, one can expect that the condition L ≤ ℒ_i, ℒ_eh is satisfied at rather large values of L (tens on μm). Indeed, setting D_eh ∼ D_i = 40, 000cm²/s [35] and ${\tau }_R^{\mathit{inter}}~{10}^{-10}\hspace{0.17em}\text{s}$, one obtains L ≤ ℒ_i ∼ ℒ_eh ∼ 20 μm. The implementation of the ”electrical” doping in the double-GL structures is beneficial because in such structures ν_i can be markedly reduced.

In deriving Eq. (11), neglecting the processes of reabsorption of the emitted photons at the reverse transitions, we disregarded the deviation of factor {1 − exp[(2μ + Δ)/k_BT]} from unity. This factor is really very close to unity at room and lower temperatures. Under these circumstances, the temperature dependence of the THz gain in device ”a” is determined solely by the broadening factor γ. The value of the latter assumed above is realistic at room temperatures providing sufficient quality of GLs. At lower temperatures, this factor can be much smaller. This can leads to substantially higher and narrower peaks of the THz gain. The temperature dependence of the THz gain in device ”b”, is stronger because, apart from a marked change in the Drude absorption with the temperature variation of γ, the temperature variation also affects the population inversion [the first term in the right-hand side of Eq. (12)].

7. Conclusion

We have developed a device model for the proposed injection THz laser with an active region consisting of a double-GL structure with a barrier layer placed within a MM waveguide. The operation of this laser is associated with the inter-GL (intraband) tunneling photon-assisted transitions. Applying the model to the device structure with WS₂ inter-GL layer and hBN clad layers of the MM waveguide, we have derived general formulas for the dynamic conductivity of the double-GL structure and for the THz gain, estimated the values of the matrix element of the inter-GL radiative transitions for different thicknesses of the barrier layer, found the spatial distributions of the photon electric field in the TM mode propagating along the MM waveguide, and calculated the frequency dependences of the gain-overlap factor, the waveguide absorption coefficient, and the THz gain. For comparison, we analyzed the characteristics of a THz laser based on two GLs with simultaneous injection in each GL using the intra-GL (interband) radiative transitions (proposed and studied previously [9–13,36,37]). The results show that: