Non-equilibrium Green's function calculation for GaN-based terahertz-quantum cascade laser structures

Transcription

1 Purdue University Purdue e-pubs Birck and NCN Publications Birck Nanotechnology Center Non-equilibrium Green's function calculation for GaN-based terahertz-quantum cascade laser structures H. Yasuda National Institution of Information & Communication Technology (NICT) T. Kubis Birck Nanotechnology Center, Purdue University I. Hosako National Institution of Information & Communication Technology (NICT) K. Hirakawa Tokyo University Follow this and additional works at: Part of the Nanoscience and Nanotechnology Commons Yasuda, H.; Kubis, T.; Hosako, I.; and Hirakawa, K., "Non-equilibrium Green's function calculation for GaN-based terahertz-quantum cascade laser structures" (2012). Birck and NCN Publications. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

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3 JOURNAL OF APPLIED PHYSICS 111, (2012) Non-equilibrium Green s function calculation for GaN-based terahertz-quantum cascade laser structures H. Yasuda, 1,a) T. Kubis, 2 I. Hosako, 1 and K. Hirakawa 3 1 National Institute of Information and Communications Technology, 4-2-1, Nukui-Kitamachi, Koganei, Tokyo , Japan 2 Network for Computational Nanotechnology, Birk Nanotechnology Center, School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA 3 Institute of Industrial Science and Institute for Nano Quantum Information Electronics, University of Tokyo, Komaba, Meguro-ku, Tokyo , Japan (Received 14 March 2012; accepted 18 March 2012; published online 19 April 2012) We theoretically investigated GaN-based resonant phonon terahertz-quantum cascade laser (QCL) structures for possible high-temperature operation by using the non-equilibrium Green s function method. It was found that the GaN-based THz-QCL structures do not necessarily have a gain sufficient for lasing, even though the thermal backfilling and the thermally activated phonon scattering are effectively suppressed. The main reason for this is the broadening of the subband levels caused by a very strong interaction between electrons and longitudinal optical (LO) phonons in GaN. VC 2012 American Institute of Physics. [ Remarkable progress in terahertz-quantum cascade lasers (THz-QCLs) has been made since their first demonstration in ,2 Low-frequency operation at 1.2 THz at 69 K (Ref. 3) and high operation temperatures of K at 3.22 THz (Ref. 4) and 163 K at 1.8 THz (Ref. 5) have been achieved for THz-QCLs based on GaAs/AlGaAs material systems. However, room-temperature operation of THz-QCLs has not been realized yet. It has been pointed out that to realize high-temperature operation of THz-QCLs, it is important to reduce thermal backfilling and thermally activated longitudinal optical (LO) phonon scattering. 2 The thermal backfilling, by which electrons in the lasing levels are backfilled from the lower level by thermal excitation, can be suppressed by using the double LO phonon depopulation scheme. 6 The remaining task is the reduction of the thermally activated phonon scattering, where electrons in the upper lasing level acquire sufficient in-plane kinetic energy to emit an LO-phonon and relax to the lower lasing level in a nonradiative manner. In this respect, the use of material systems consisting of large LO phonon energies seems promising because large thermal energy is needed to induce the thermally activated phonon scattering. Among such material systems, GaN-based THz-QCL structures, which have an LO phonon energy of about 90 mev, have been considered to be promising for room-temperature operation. Theoretical studies based on rate-equation models and Monte Carlo techniques predicted the improved high-temperature performance of GaN-based QCLs. 7 9 Furthermore, GaN-based THz-QCLs would be key coherent sources in the frequency range of 5 12 THz, at which GaAs-based THz-QCLs cannot be operated due to the Reststrahlen band. However, no clear lasing of a GaN-based QCL has been reported so far. It is certain that the fabrication of GaN-based THz-QCLs is a challenging issue because of the difficulty in growing thick a) Electronic mail: yasuda@nict.go.jp. GaN-based superlattices. However, phenomena affecting the performance of GaN-based THz-QCLs, such as strong LO phonon scattering, that are inherent to the GaN-based material systems need to be more carefully examined in a self-consistent manner. In this work, we analyzed the performance of a GaN-based THz-QCL structure by firstprinciple calculations using the non-equilibrium Green s function (NEGF) method. It was found that the gain of GaNbased THz-QCL structures is not sufficient for lasing, though thermal backfilling and thermally activated phonon scattering are well suppressed. The main reason for this is the broadening of subband levels caused by a strong interaction between electrons and LO phonons in GaN. In our NEGF calculation, the Green s functions G and self-energies R were taken as a function of two spatial coordinates, z and z 0, the lateral electron momentum k jj, and the energy E. Scattering by acoustic phonons, optical phonons, and interface roughness was taken into account. All scattering self-energies were calculated in the self-consistent Born approximation. Anisotropies in the effective mass and the dielectric constants were neglected for simplicity. Details of the calculations were reported elsewhere The calculated GaN-based THz-QCL has a four-well resonant-phonon depopulation structure, where electrons in the lower lasing level are depopulated by LO phonon scattering. To reduce strong built-in electric fields caused by spontaneous and piezoelectric polarizations, we assumed an In 0.04 Al 0.16 Ga 0.8 N/GaN material system, which has a polarization value of C/m 2 at the interface The conduction band discontinuity was set to be 0.25 ev. Since the electron effective mass in GaN is 0.2 m 0, the barriers should be thinner than those typical for GaAs-based THz-QCLs in order to obtain sufficient tunneling currents. Figure 1(a) shows the conduction band profile and the wave functions for the GaN-based THz-QCL at 80 kv/cm solved using the Schrödinger-Poisson equations. The injector, /2012/111(8)/083105/4/$ , VC 2012 American Institute of Physics

4 Yasuda et al. J. Appl. Phys. 111, (2012) FIG. 2. (a) and (b) show the energy-resolved electron distributions and the spectral functions, respectively, in the injector, active, and ejector regions of the GaN-based THz-QCL at 200 K. FIG. 1. (a) Conduction band profile of a GaN-based resonant-phonon QCL calculated with the Schrödinger-Poisson s equations. One period of the QCL sequence was designed to be 1.0/4.1/1.6/3.1/1.0/2.1/1.6/5.2 nm. The bold and regular numbers represent InAlGaN and GaN layers, respectively. The quantum well, indicated above by the underline, is n-doped with n ¼ cm 3. (b) and (c) show the absorption coefficient a(z,x) and spectral functions, respectively, of the GaN-based THz-QCL calculated using the NEGF program. active, and ejector regions are defined as shown in Fig. 1(a) for convenience. We assumed that the left and right ends of the quantum cascade unit are connected to ideal electrodes. Figure 1(b) shows the absorption coefficient of the QCL structure calculated as a function of position and photon energy at the lattice temperature of 200 K, obtained from the NEGF calculation. Figure 1(c) illustrates the spectral function for one period of the QCL structure; that is, the density of states at in-plane wave vector k jj ¼ 0. The black regions in Fig. 1(b) indicate loss. The local optical gain is observed mainly in the active region. The maximum local optical gain in Fig. 1(b) remains, however, only 1.0 cm 1. When averaged over one period of the QCL structure, the gain vanished. At 40 K, the maximum local gain and the maximum total gain over one period are 4.5 cm 1 and 1.0 cm 1, respectively. The performance of the GaN-based THz-QCL does not exceed that of the GaAs-based THz-QCL even at high temperatures. These results indicate that GaN-based resonant-phonon THz-QCL structures of the conventional resonant phonon depopulation scheme are not suitable for high-temperature operation. Figure 2(a) shows the electron distributions Ð qðz; EÞdz ¼ 2 Ð dzim Ð dkg < ðz; z; k jj ; EÞ=ð2pÞ 3 as a function of energy in the injector, active, and ejector regions for the GaN-based QCL at 200 K. As seen in the electron distribution of the active region in Fig. 2(a), a population inversion between the upper lasing level, denoted as 3, and the lower lasing level, denoted as 2, is well maintained even at 200 K. The thermal backfilling revealed in the electron distribution curve of the ejector region extending from the lowest level, denoted as 1, to the lower lasing level 2 has no influence on the population inversion. Furthermore, the electron distribution curve in the active region has no dip at around 280 mev (Žx LO above the lower lasing level 2, where x LO is the LO phonon frequency). This suggests that the thermally activated phonon scattering is well suppressed in the GaN-based THz-QCL. 11 The distribution peak in the ejector around 140 mev in Fig. 2(a) shows the non-equilibrium accumulation of electrons scattered by LO phonons from the upper lasing level 3. Figure 2(b) shows the spectral functions Ð Aðz; EÞdz as a function of energy in the injector, active, and ejector regions for the GaN-based QCL at 200 K. The spectral functions of the upper lasing level 3 and the lower lasing level 2 in the active region are broadened and overlap in energy. The increased level broadening is one of reasons for the reduction of peak gain. 16,17 The reason for the unexpectedly low optical gain of the GaN-based THz-QCL is attributed to the broadening of the quantized levels. The broadening is caused by the extremely strong LO phonon electron interaction in GaN. The Fröhlich interaction Hamiltonian was used for the self-energy of the LO phonon electron coupling. The retarded self-energy for the LO phonon scattering in the self-consistent Born approximation can be expressed as

5 Yasuda et al. J. Appl. Phys. 111, (2012) R ret LO ðz; z0 ; k jj ; EÞ ¼ cp ð 1 ð ð2pþ 3 dq z dl jj 1 q 2 jj þ q2 z ðq 2 jj þ q2 z þ n 2 Þ 2 eiqzðz z0 Þ ð1 þ N 0 ÞG ret ðz; z 0 ; l jj ; E hx LO Þ þ N 0 G ret ðz; z 0 ; l jj ; E þ hx LO Þ þ 1 2 G< ðz; z 0 ; l jj ; E hx LO Þ 1 2 G< ðz; z 0 ; l jj ; E þ hx LO Þ ð de 0 G < ðz; z 0 ; l jj ; E 0 Þ þ ip 2p E E 0 hx LO G< ðz; z 0 ; l jj ; E 0 Þ E E 0 ; (1) þ hx LO where G ret and G < are the retarded and correlation Green s functions, respectively, q is the phonon momentum, N 0 is the phonon number, n is the Debye screening length, l jj is the electron momentum defined by k jj q jj, and P is the principal value. 12 The Fröhlich coupling constant c is given by c ¼ e2 hx LO 1 1 ; (2) 2e 0 e 1 e r where e r and e 1 are the low- and high-frequency relative dielectric constants, respectively. The values of e r and e 1 are 9.7 and 5.28, respectively, for GaN, and and 10.89, respectively, for GaAs; the Fröhlich coupling constant c for GaN is 15 times larger than that in GaAs (c GaAs ). Therefore, the self-energy that describes the interaction between the electrons and the LO phonons in GaN is extremely strong. The real and imaginary parts of the retarded self-energy relate to the energy shift and the energy broadening of the electron state, respectively. 18 Thus, the broadening of the subband levels is very significant in GaN-based QCL structures. To investigate the influence of the Fröhlich coupling constant on gains of THz-QCLs, we performed NEGF calculations for a typical GaAs-based resonant-phonon THz-QCL with hypothetically changing values of e 1, while keeping other material parameters fixed to those of GaAs. The lattice temperature was set to 40 K because thermally activated phonon scattering is negligible at this temperature. Figures 3(a), 3(b), and 3(c) show the optical gains, the electron distributions, and the spectral functions in the active regions, respectively, for c ¼ c GaAs,3c GaAs, and 9c GaAs. The optical gain decreased dramatically with an increase in c. The subband levels are misaligned and the electrons are not efficiently injected to the upper lasing level for c ¼ 9c GaAs due to the large broadenings and energy shifts of the subband levels. Comparing the results for c ¼ c GaAs and c ¼ 3c GaAs, the optical gain decreased from 94 cm 1 to 33 cm 1 at 13.9 mev. The population inversion in the active region decreased from FIG. 3. The NEGF results of a GaAs-based THz-QCL at 40 K for various Fröhlich coupling constants. One period of the GaAs-based resonant-phonon QCL sequence, modified structure of Ref. 19, is 5.1/9.6/2.3/7.3/4.0/15.8 nm. The bold and regular numbers represent Al 0.15 Ga 0.85 As and GaAs layers, respectively. The quantum well, indicated above by the underline, is n- doped with n ¼ cm 3. The electric field is 14.0 kv/cm. (a) Gain averaged over one period of the QCL. (b) Electron distributions in the active region. (c) Spectral functions in the active region cm 2 to cm 2. The full width at half maximum of the spectral function of the upper lasing level increased from 4.6 mev to 12.9 mev. Thus, an increase in the Fröhlich coupling constant significantly degrades the performance of THz-QCLs. In conclusion, we investigated the performance of a GaN-based THz-QCL by using with the NEGF method for possible high-temperature operation. The calculated results at 200 K showed that the GaN-based THz-QCL does not have a larger THz gain than the GaAs-based THz-QCLs, even though the thermal backfilling and the thermallyactivated phonon scattering are well suppressed. The main reason for this is the broadening of the subband levels caused by strong interaction between electrons and LO phonons in GaN. Novel methods to minimize the influence of the level broadening will be needed to make use of the material properties of GaN and realize the high-temperature operation of GaN-based THz-QCLs. We thank P. Vogl and G. Klimeck for fruitful discussions. This work was partly supported by the Grant-in-Aid for Scientific Research from JSPS (# ) and the Specially Coordinated Fund for Promoting Science and Technology from MEXT (NanoQuine).

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