Institute for Electrical Communication, Tohoku University, Sendai 980-8577, Japan for Photonics and Infrared Engineering, Bauman Moscow State Technical University, Moscow 105005, Russia 3 Institute for Physics of Microstructures of Russian Academy of Sciences and Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia 4 Department of Computer Science and Engineering, University of Aizu, Aizu-Wakamatsu 965-8580, Japan 5 Department of Electrical, Electronics, and System Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

2 Center

∗ [email protected]

Abstract: We propose and analyze the concept of injection terahertz (THz) lasers based on double-graphene-layer (double-GL) structures utilizing the resonant radiative transitions between GLs. We calculate main characteristics of such double-GL lasers and compare them with the characteristics of the GL lasers with intra-GL interband transitions. We demonstrate that the double-GL THz lasers under consideration can operate in a wide range of THz frequencies and might exhibit advantages associated with the reduced Drude absorption, weaker temperature dependence, voltage tuning of the spectrum, and favorable injection conditions. © 2013 Optical Society of America OCIS codes: (140.2020) Diode lasers; (140.3070) Infrared and far-infrared lasers; (140.5960) Semiconductor lasers; (250.0250) Optoelectronics.

References and links 1. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12,1482–1485 (2012). 2. L. Britnell, R. V. Gorbachev, R. Jalil, B.D . Belle, F. Shedin, A. Mishenko, T. Georgiou, M. I. Katsnelson, L. Eaves, S. V. Morozov, N. M. R. Peres, J. Leist, A. K. Geim, K. S. Novoselov, and L. A. Ponomarenko, “Fieldeffect tunneling transistor based on vertical graphene heterostructures,” Science 335, 947–950 (2012). 3. T. Georgiou, R. Jalil, B. D. Bellee, L. Britnell, R. V. Gorbachev, S. V. Morozov, Y.-J. Kim, A. Cholinia, S. J. Haigh, O. Makarovsky, L. Eaves, L. A. Ponomarenko, A. K. Geim, K. S. Nonoselov, and A. Mishchenko, “Vertical field-effect transistor based on graphene-WS2 heterostructures for flexible and transparent electronics,” Nat. Nanotechnol. 7, 100–103 (2013). 4. L. Britnell, R. V. Gorbachev, A. K. Geim, L. A. Ponomarenko, A. Mishchenko, M. T. Greenaway, T. M. Fromhold, K. S. Novoselov, and L. Eaves, “Resonant tunneling and negative differential conductance in graphene transistors,” Nat. Commun. 4, 1794–1799 (2013). 5. V. Ryzhii, T. Otsuji, M. Ryzhii, V. G. Leiman, S. O. Yurchenko, V. Mitin, and M. S. Shur, “Effect of plasma resonances on dynamic characteristics of double-graphene layer optical modulators,” J. Appl. Phys. 112, 104507 (2012). 6. V. Ryzhii, T. Otsuji, M. Ryzhii, and M. S. Shur, “Double graphene-layer plasma resonances terahertz detector,” J. Phys. D Appl. Phys. 45, 302001 (2012). 7. V. Ryzhii, A. Satou, T. Otsuji, M. Ryzhii, V. Mitin, and M. S. Shur, “Dynamic effects in double-graphenelayer structures with inter-layer resonant-tunneling negative differential conductivity,” J. Phys. D Appl. Phys. 46, 315107 (2013). 8. V. Ryzhii, M. Ryzhii, V. Mitin, M. S. Shur, A. Satou, and T. Otsuji, “Terahertz photomixing using plasma resonances in double-graphene-layer structures,” J. Appl. Phys. 113, 174506 (2013).

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31567

9. V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. 101, 083114 (2007). 10. A. A. Dubinov, V. Ya. Aleshkin. M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabry-Perot resonator,” Appl. Phys. Express 2, 092301 (2009). 11. V. Ryzhii, M. Ryzhii, A. Satou, T. Otsuji, A. A. Dubinov, and V. Ya. Aleshkin, “Feasibility of terahertz lasing in optically pumped expitaxial multiple graphene layer structures,” J. Appl. Phys. 106, 084507 (2009). 12. V. Ryzhii, A. A. Dubinov, T. Otsuji, V. Mitin, and M. S. Shur, “Terahertz lasers based on optically pumped multiple graphene structures with slot-line and dielectric waveguides,” J. Appl. Phys. 107, 054505 (2010). 13. V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110, 094503 (2011). 14. S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012). 15. T. Watanabe, T. Fukushima, Y. Yabe, S. A. Boubanga-Tombet, A. Satou, A. A. Dubinov, V. Ya. Aleshkin, V. Mitin, V. Ryzhii, and T. Otsuji, “The gain enhancement effect of surface plasmon-polaritons on terahertz stimulated emission in optically pumped monolayer graphene,” New J. Phys. 15, 075003 (2013). 16. T. Otsuji, S. A. Boubanga Tombet, A. Satou, M. Ryzhii, and V. Ryzhii, “Terahertz-wave generation using graphene - toward new types of terahertz lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 8400209 (2013). 17. A. Tredicucci and M. S. Vitiello, “Device concepts for graphene-based terahertz photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 8500109 (2014). 18. V. Ryzhii, A. A. Dubinov, V. Ya. Aleshkin, M. Ryzhii, and T. Otsuji, “Injection terahertz laser using the resonant inter-layer radiative transitions in double-graphene-layer structure,” Appl. Phys. Lett. 103, 163507 (2013). 19. H. Kanaya, H. Shibayama, R. Sogabe, S. Suzuki, and M. Asada, “Fundamental oscillation up to 1.31 THz in resonant tunneling diodes with thin well and barriers”, Appl. Phys. Express 5, 124101 (2012). 20. B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photonics 1, 517 (2007). 21. K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, D. L. Sivco, and A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060 (2002). 22. M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008). 23. R. M. Feenstra, D. Jena, and G. Gu, “Single-particle tunneling in doped graphene-insulator-graphene junctions,” J. Appl. Phys 111, 043711 (2012). 24. F. T. Vasko, “Resonant and nondissipative tunneling in independently contacted graphene structures,” Phys. Rev. B 87, 075424 (2013). 25. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56, 281–284 (2007). 26. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007). 27. F. Carosella, C. Ndebeka-Bandou, R. Ferreira, E. Dupont, K. Unterrainer, G. Strasser, A. Wacker, and G. Bastard, “Free carrier absorption in quantum cascade structures,” Phys. Rev. B 85, 085310 (2012). 28. F. T. Vasko, V. V. Mitin, V. Ryzhii, and T. Otsuji, “Interplay of intra- and interband absorption in a disordered graphene,” Phys. Rev. B 86, 235424 (2012) 29. F. T. Vasko and A. V. Kuznetsov, Electronic States and Optical Transitions in Semiconductor Heterostructures (Springer, 1999) 30. K. J. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors (Springer, 1993). 31. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1964). 32. D. M. Hoffman, G. L. Doll, and P. C. Eklund, “Optical properties of pyrolytic boron nitride in the energy range 0.05 - 10 eV,” Phys. Rev. B 30, 6051–6056 (1984). 33. H. Shi, H. Pan, Y.-W. Zhang, and B. Yakobson, “Quasiparticle band structures and optical properties of strained monolayer MoS2 and WS2,” Phys. Rev. B 87, 155304 (2013). 34. V. Ryzhii, I. Semenikhin, M. Ryzhii, D. Svintsov, V. Vyurkov, A. Satou, and T. Otsuji, “Double injection in graphene p-i-n structures,” J. Appl. Phys. 113, 244505 (2013). 35. R. Rengel and M. J. Martin, “Diffusion coefficient, correlation function, and power spectral density of velocity fluctuations in monolayer graphene,” J. Appl. Phys. 114, 143702 (2013). 36. 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–L153 (2007). 37. M. Ryzhii and V. Ryzhii,“”Population inversion in optically and electrically pumped graphene,” Physica E 40, 317–320 (2007).

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31568

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 A3 B5 materials, in which the operation of quantum cascade lasers [20] is suppressed. +

V

+

V g

W p z

d x

+

V

V g

n -G L W

T u n n e lin g b a rrie r la y e r p -G L 2 L n

p

(a )

i-G L

d 2 L

B a rrie r la y e r i-G L

n

(b )

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”).

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 afew 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 #200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31569

(a )

(b ) c -c h w

m e V

D

v -v h w

m

e V

m m

T u n n e lin g b a rrie r

c -v

D = 0

h w

c -v h w

B a rrie r

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.

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 ±Vg (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. 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

κΔ , (1) 4π e2 d where Σi is the density of donors in the upper GL (n-GL) and acceptors in the lower GL (p-GL), κ is the dielectric constant of the barrier layer, and e = |e| is the electron charge. In the case of ”electrical” doping, Σi is the density of electrons and holes induced by the gate voltages ±Vg applied between the GLs and the metal strips of the MM waveguide: Σi = æVg /4π eW , where Σ = Σi +

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31570

æ is the dielectric constant of the clad layers. In the case of relatively short GLs (such that the diffusion length of electrons and holes Ld L) and strong degeneration of 2DEG and 2DHG ( μ kB T , kB and √ T are the Boltzmann constant and the temperature, respectively), Δ = eV − 2μ and μ h¯ vW π Σ. Here vW 108 cm/s is the characteristic velocity of electrons and holes in GLs. This yields e μ = ( 2VV0 +V02 +Vt2 −V0 ), (2) Δ/e = V +V0 − 2VV0 +V02 +Vt2 , 2 √ 2 κ /2e3 d and V = 2¯ h vW π Σi /e. At V ≤ Vt , Eq. (2) yields Δ ≤ 0, while at where V0 = h¯ 2 vW t V > Vt , Δ > 0. Figure 2(a) shows the band diagram at the bias voltage V > Vt . 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 Σi = 1012 cm−2 , one obtains V0 136 mV and Vt 221 mV. The quantities Δ = 5 − 10 meV correspond to V 229 − 237 mV and μ 112.0 − 113.5 meV. a (ω ) component of the tensor of the double-GL structure dynamic conThe real part of σyy ductivity is given by [25] (see also [11,26–28]) 2 h¯ ω e μ 4γ μ a 2 exp − sinh + , (3) Reσyy (ω ) 2 4¯h kB T 2kB T π (¯h2 ω 2 + γ 2 ) while the real part of σzza (ω ) component can be presented as [18] 2 e 2|zu,l |2 Σi (1 + Δ/Δi )γ h¯ ω Reσzza (ω ) = − , h¯ [¯h2 (ω − ωmax )2 + γ 2 ]

(4)

Here zu,l = ϕu∗ (z)zϕl (z), where ϕu (z) and ϕl (z) are the z-dependent factors of the wave functions in the upper and lower GLs, respectively, is the matrix element for the inter-GL transitions, γ h¯ ν is the relaxation broadening, ν is the collision frequency of electrons and holes, and Δi = 4π e2 dΣi /κ . The right-hand side of Eq. (3) is positive. It comprises the intra-GL interband and intraband (Drude) terms. The former is rather small at μ kB T . The quantity h¯ ωmax in Eq. (4) corresponds to the maximum probability of the inter-GL radiative transitions: 8π e2 |zu,l |2 Σi (1 + Δ/Δi ) = Δ + Δdep , (5) κd The second term in the right-hand side of Eq. (5) describes the depolarization shift Δdep [29]. 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] 2 h¯ ω − 2μ e 4γ μ b tanh + , Reσzzb = 0. (ω ) = 2 (6) Reσxx 4¯h 4kB T π (¯h2 ω 2 + γ 2 ) h¯ ωmax = Δ −

b (ω ) depends on the trade-off of the interband (negative) and intraband The sign of Reσxx b (ω ) < 0. (positive) contributions. In a certain range of frequencies and Fermi energy, Reσxx 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: #200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31571

4π a ga (ω ) = − √ [Reσyy (ω )Γayy (ω ) + Reσzza (ω )Γazz (ω )] − α a (ω ), c κ

(7)

4π b (ω )Γbxx (ω ) − α b (ω ), gb (ω ) = − √ Reσxx c κ

(8)

where

L

Γa,b j j (ω )

a,b 2 −L dx|E j (x, 0, ω )| a,b 2 −L −W dxdz|E j (x, z, ω )|

= L W

( j = x.y, z),

(9)

are the gain-overlap factors, Exb (x, z, ω ), Eya (x, z) and Eza (x, z, ω ) are the components of the electric field in the pertinent modes, and c is the speed of light in vacuum, and α a.b (ω ) = 2Im ky (ω ) is the absorption coefficient of the propagating mode due to the losses in the pertinent waveguide. Considering Eqs. (3) and (4), Eq. (7) yields h¯ ω 4γ μ 2π e2 μ a sinh + Γayy (ω ) − 2 exp − g (ω ) = √ kB T 2kB T h¯ c κ π (¯h2 ω 2 + γ 2 ) 4|zu,l |2 Σi (1 + Δ/Δi )γ h¯ ω a Γzz (ω ) − α a (ω ), (10) + [¯h2 (ω − ωmax )2 + γ 2 ] 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, Ex , Ey , and Ez 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 |Eza (x, z, ω )| and |Eya (x, z, ω )| in the TM mode of a MM waveguide with the sizes L = W = 5 μ m in device ”a”. Figure 3(c) demonstrates the spatial distribution |Exb (x, z, ω )| 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 |Eza (x, 0, ω )|, |Eya (x, 0, ω )|, and |Exb (x, 0, ω )| at the double-GL structure plane. As follows from Fig. 3(c), the TM-mode component Ey is small in comparison with the Ez 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 |zu,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 ΔEg = 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 Δdep on the energy separation between the Dirac points Δ calculated using the obtained values |zu,l |2 and Eq. (5). Due to the inter-GL population inversion, Δdep < 0, and its value varies in a wide energy range with varying Δ As follows from Fig. 4(a), sufficiently large values of |zu,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

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31572

Fig. 3. Spatial distributions of the photon electric field components: (a) |Eza (x, z, ω )|, (b) |Eya (x, z, ω )| in the TM mode in MM waveguide (in device ”a”), (c) Exb (x, z, ω )| 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.

from Fig. 5, Γayy Γazz . This is a consequence of a weak Ey component [see Fig. 3(d)]. Hence, we can disregard the first term in right-hand side of Eq. (10)and reduce this equation to 8π e2 |zu,l |2 Σi (1 + Δ/Δi )γ h¯ ω a √ Γ (ω ) − α a (ω ), (11) ga (ω ) = h¯ c κ [¯h2 (ω − ωmax )2 + γ 2 ] zz This, in particular, implies that the Drude absorption of the TM mode in device ”a” is insignificant. As for device ”b”, taking into account Eq. (6), the THz gain can be calculated using the following equation: 2μ − h¯ ω 2π e2 4γ μ √ − Γbxx (ω ) − α b (ω ). tanh (12) gb ( ω ) = 4kB T h¯ c κ π (¯h2 ω 2 + γ 2 ) 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 = 2π c/ω . 5.

Frequency and voltage dependences of the THz gain

Figure 6 shows the frequency dependences of the THz gain ga (ω ) a for double-GL structure (device ”a”) with Σi = 1 × 1012 cm−2 , different values of the energy separation between the #200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31573

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

Dirac points Δ (i.e., different voltages V ), and different waveguide geometrical parameters. The frequency dependences ga (ω ) shown in Fig. 6 correspond to the relaxation broadening √ γ = 1 meV (ν = 1.6 × 1012 s−1 )and doping level Σi = 1 × 1012 cm−2 , i.e., μi = h¯ vW π Σi 110 meV. The actual value of the Fermi energy μ > μi . 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 = ωmax . In this peak, ga (ω ) is positive and fairly large. The peak position is determined by Δ, in line with Eq. (5). The THz gain ga (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, ωmax , of the THz gain is voltage tunable. The quantity Δ and, consequently, ωmax depend on Σi . In device ”a” with the ”electrical” doping Σi ∝ Vg , and, hence, in such a device, the position of the resonance can be also tuned by the gate voltage, Vg . Figure 7 demonstrates how the frequency ωmax at which the THz gain ga (ω ) achieves a maximum value with with varying bias voltage V and gate voltage Vg . The height of the resonances maxga (ω ) varies with increasing Δ. This can be attributed to the trade-off in a decrease in |zu,l |2 [see Fig. 4(a)] and in an increase of factor (1 + Δ/Δi ) 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.

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31574

Fig. 5. Frequency dependences of gain-overlap factors Γa,b j j (ω ) (solid lines) and waveguide a,b absorption coefficients α (ω ) (dashed lines): (a) L = 5 μ m and W = 5 μ m and (b) L = 15 μ m and W = 5 μ m.

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

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31575

Fig. 7. Tuning of frequency ωmax corresponding to peak of the THz gain ga (ω ) by (a) bias voltage V and (b) gate voltage Vg .

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 ≤ Li , Leh [34]. Here Li ∝ (Di τRinter )1/2 and −1 are the diffusion coLeh ∝ (Deh τRinter )1/2 are the diffusion lengths, Di ∝ νi−1 and Deh ∝ νeh efficients, where νi and νeh are the electron and hole collision frequencies with impurities and with each other, respectively (νi + νeh = ν ), and τRinter is the recombination time associated with the inter-GL processes. As shown [35], and references therein), the diffusion coefficient Di 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 ≤ Li , Leh is satisfied at rather large values of L (tens on μ m). Indeed, setting Deh ∼ Di = 40, 000cm2 /s [35] and τRinter ∼ 10−10 s, one obtains L ≤ Li ∼ Leh ∼ 20 μ m. The implementation of the ”electrical” doping in the double-GL structures is beneficial because in such structures νi 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)/kB T ]} 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 #200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31576

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: (i) The realization of THz lasing in the devices with the double-GL structure and WS2 and hBN layers exploiting the inter-GL radiative transitions is feasible because of the possibility to achieve sufficiently high values of the THz gain; (ii) Due to resonant-tunneling nature of the radiative transitions in the THz laser under consideration, the spectrum of the THz laser radiation can exhibit sharp maxima in the range from a few THz to a dozen THz and can be tuned by the applied voltages; (iii)The double-GL inter-GL THz laser is rather insensitive to the temperature; (iv) The proposed structures should exhibit advantages over the lasers utilizing the intra-GL transitions due to weaker temperature sensitivity, practically absent Drude absorption (crucial at the low end of the THz range), and possibly, higher injection efficiency; (v) Both inter-GL and intra-GL THz lasers can be useful to cover the frequency range of several THz, in which the operation of A3 B5 quantum cascade lasers is hampered by optical phonons. Acknowledgments This work was supported by the Japan Society for promotion of Science (Grant-in-Aid for Specially Promoting Research #23000008), Japan. The work by A.D. was also supported by the Russian Foundation of Basic Research and the Dynasty Foundation, Russia. The work at RPI was supported by the US Army Cooperative Research Agreement (Program Manager Dr. Meredith Reed). V.R. is grateful to M. J. Martin for sending recent Ref. [35].

#200238 - $15.00 USD Received 28 Oct 2013; revised 30 Nov 2013; accepted 6 Dec 2013; published 13 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031567 | OPTICS EXPRESS 31577