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Si-based quantum staircase terahertz lasers
Microelectronics Journal 34 (2003) 391–393 www.elsevier.com/locate/mejo
Si-based quantum staircase terahertz lasers G. Suna,*, Richard A. Sorefb a Department of Physics, University of Massachusetts at Boston, Boston, MA 02125, USA Air Force Research Laboratory, Sensors Directorate, Hanscom Air Force Base, MA 01731, USA
b
Abstract Design results are presented for electrically pumped quantum staircase intersubband p-i-p SiGe/Si strain-balanced superlattice lasers to be operated at 77 K or higher. The wavelength of laser emission will be in the THz range. Two approaches of quantum staircase lasers will be presented, one utilizes the inverted light-hole effective mass, while the other inverted heavy-hole mass. Optical gain on the order of a few 100 cm21 can be achieved for both laser designs. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: SiGe/Si strain balanced superlattice; Quantum staircase laser; THz laser
1. Introduction Recently, there has been tremendous effort directed towards the development of Si-based optoelectronic devices because of their potential for monolithic integration with Si-electronics. We have previously studied some of these structures [1 –4]. The focus of this work is on the feasibility of SiGe/Si laser structures capable of THz emission. In order to obtain , 10 mm-thick superlattices (SLs) not constrained by a critical-thickness limit, the proposed coherently strained epilayer devices would be grown upon a relaxed SiGe buffer on Si. The lasers proposed here use radiative transitions between valence subbands mainly because the valence-band offsets in SiGe/Si quantum wells (QWs) are generally larger than the conduction band offsets. Two designs utilizing inverted hole effective mass produced by strong subband coupling of different types are investigated. One involving inverted light-hole mass uses the intersubband transition between the ground-state light-hole (LH1) to heavy-hole (HH1) in SiGe/Si SLs which allows surface emitting. The other, however, uses HH2 to HH1 transition with inverted heavy-hole mass enabling edge emission.
2. Inverted LH mass laser It is known that the strong interaction between the valence subbands of different kinds causes the LH mass to * Corresponding author. Tel.: þ 1-617-287-6432; fax: þ1-617-287-6053. E-mail address: [email protected] (G. Sun).
be inverted [5]. We have examined the SL consisting of ˚ Si barriers on a (100) ˚ Si0.7Ge0.3 QWs and 50 A 90 A Si0.81Ge0.19 buffer for investigating the feasibility of lasing. The result of in-plane dispersion of valence subbands in the above structure calculated by using the Kane model [6] is shown in Fig. 1. The LH1 subband has a energy valley of ˚ 21, which extends from 0 2 meV deep at kx ¼ ky ¼ 0:014 A 21 ˚ to 0.025 A . The lasing transition is designed to take place between the two lowest subbands HH1 and LH1 in the LH1 valley of the k-space at 50 mm. As marked by numericals 1, 2, 3, 4, we indicate in Fig. 1 how this inverted mass intersubband laser operates. By selective electrical pumping, holes are injected into LH1 at G-point (k ¼ 0) (state 4). Those holes then quickly populate the valley of LH1 (upper laser state 3) through intrasubband relaxation processes, undergo a lasing transition to lower laser state 2 in subband HH1, and finally relax toward state 1 at k ¼ 0 of the HH1 subband. The intrasubband relaxation processes 4– 3 and 2 –1 are very fast compared to the intersubband process 3 –2, which will lead to rapid population of the upper laser state 3 as well as fast depopulation of the lower laser state 2. The upper-state lifetime is long because the intersubband transition energy is below that of optical phonons, allowing only much weaker acoustic phonon scattering between the two different bands. The proposed laser structure is a so-called quantum staircase scheme which is different from the popular quantum cascade design in the sense that it eliminates the need for the chirped SL as the injector region. This structure uses electrical pumping of a set of isolated single Si0.7Ge0.3 QWs with Si barriers. As shown in Fig. 2 where the increase
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00032-6
392
G. Sun, R.A. Soref / Microelectronics Journal 34 (2003) 391–393
Fig. 1. In-plane dispersion of valence subbands showing the laser operating process.
Fig. 3. Optical gain as a function of the tunneling time with several hole densities.
of hole energy is chosen upward, holes from the lower subband HH1 in the previous lasing period tunnel through the Si-barrier to the upper subband LH1 in the next one. At each end, there is a region doped pþ: a SiGe emitter for current injection (gap wider than QW gap) and SiGe hole collector (gap equal to QW gap). The tunneling time, tt ; between subbands HH1 and LH1 can be controlled by tuning the width of the Si-barrier separating the QWs. Also shown in Fig. 2 is the competing process of acoustic phonon scattering which leaks holes directly to the lower subband HH1 in the next laser period without contributing to the lasing process, and therefore resulting in current loss. We performed detailed calculation of the optical gain based on our previous result of intersubband lifetime in SiGe/Si SLs [7]. Fig. 3 shows the optical gain as a function of the tunneling time for several densities of hole population. Positive optical gain ranging from , 100 cm21 to over 1000 cm21 is obtained when the tunneling time tt is shorter than that of upper laser state estimated at tu ¼ 2:0 ns. This is obviously expected because it corresponds to the case of total population inversion between the two subbands LH1 and HH1. It is interesting to
point out that even when tt . tu ; the situation where there is no total population inversion between the two subbands, fairly large optical gain on the order of a few 100 cm21 can still be achieved.
Fig. 2. A schematic of the inverted-mass intersubband laser with a quantum staircase scheme.
Fig. 4. Band diagram of the proposed inverted HH mass quantum staircase laser under bias.
3. Inverted HH laser A similar inverted mass feature can also be engineered for the HH subband in SiGe/Si SLs. We analyzed a model system consisting of coupled QWs under electric field bias, ˚ Si0.8Ge0.2 and as shown in Fig. 4 where the wells are 90 A ˚ Si. The operating field is chosen to be the barriers are 35 A 30 kV/cm just enough to produce a 3 meV spacing between the HH1 and HH2 subbands from two neighboring wells. The resulting quantum staircase has two active doublets per QW forming a four-level system. The laser transitions are indicated by the vertical arrows. The fast transitions within the doublets are responsible for the population of the upper laser state and depopulation of the lower laser state. The wavefunctions found at this bias is also shown in Fig. 4. We have calculated the in-plane dispersion of
G. Sun, R.A. Soref / Microelectronics Journal 34 (2003) 391–393
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the valence subbands in such a laser structure. The result of the two doublets in one QW is shown in Fig. 5 where the inverted HH effective mass in demonstrated. We have also estimated the gain as a function of pump current density at 77 K as presented in Fig. 6. Peak gain of 450 cm21 at 41 mm is expected to be larger than the cavity loss such as free carrier absorption.
4. Conclusion
Fig. 5. Dispersion of the four active levels.
We have designed and simulated two SiGe/Si SL intersubband lasers pumped with the quantum staircase scheme. One involves inverted LH mass, and the other inverted HH mass, making it possible to achieve population inversion locally in k-space, eliminating the need of total population inversion between the two laser subbands. In both cases, optical gain on the order of a few 100 cm21 can be obtained in the THz range.
References
Fig. 6. Gain as a function of the pump current density.
[1] L. Friedman, R.A. Soref, G. Sun, Y. Lu, IEEE J. Sel. Top. Quantum Electron. 4 (1998) 1029. [2] R.A. Soref, L. Friedman, L.C. Lew Yan Voon, L.R. Ram-Mohan, G. Sun, J. Vac. Sci. Technol., B 16 (1998) 1525. [3] L. Friedman, G. Sun, R.A. Soref, Appl. Phys. Lett. 78 (2000) 401. [4] R.A. Soref, G. Sun, Appl. Phys. Lett. 79 (2001) 3639. [5] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Les Ulis), 1998, Chapter 3. [6] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [7] G. Sun, L. Friedman, R.A. Soref, Phys. Rev. B 62 (2000) 8114.
Si-based quantum staircase terahertz lasers G. Suna,*, Richard A. Sorefb a Department of Physics, University of Massachusetts at Boston, Boston, MA 02125, USA Air Force Research Laboratory, Sensors Directorate, Hanscom Air Force Base, MA 01731, USA
b
Abstract Design results are presented for electrically pumped quantum staircase intersubband p-i-p SiGe/Si strain-balanced superlattice lasers to be operated at 77 K or higher. The wavelength of laser emission will be in the THz range. Two approaches of quantum staircase lasers will be presented, one utilizes the inverted light-hole effective mass, while the other inverted heavy-hole mass. Optical gain on the order of a few 100 cm21 can be achieved for both laser designs. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: SiGe/Si strain balanced superlattice; Quantum staircase laser; THz laser
1. Introduction Recently, there has been tremendous effort directed towards the development of Si-based optoelectronic devices because of their potential for monolithic integration with Si-electronics. We have previously studied some of these structures [1 –4]. The focus of this work is on the feasibility of SiGe/Si laser structures capable of THz emission. In order to obtain , 10 mm-thick superlattices (SLs) not constrained by a critical-thickness limit, the proposed coherently strained epilayer devices would be grown upon a relaxed SiGe buffer on Si. The lasers proposed here use radiative transitions between valence subbands mainly because the valence-band offsets in SiGe/Si quantum wells (QWs) are generally larger than the conduction band offsets. Two designs utilizing inverted hole effective mass produced by strong subband coupling of different types are investigated. One involving inverted light-hole mass uses the intersubband transition between the ground-state light-hole (LH1) to heavy-hole (HH1) in SiGe/Si SLs which allows surface emitting. The other, however, uses HH2 to HH1 transition with inverted heavy-hole mass enabling edge emission.
2. Inverted LH mass laser It is known that the strong interaction between the valence subbands of different kinds causes the LH mass to * Corresponding author. Tel.: þ 1-617-287-6432; fax: þ1-617-287-6053. E-mail address: [email protected] (G. Sun).
be inverted [5]. We have examined the SL consisting of ˚ Si barriers on a (100) ˚ Si0.7Ge0.3 QWs and 50 A 90 A Si0.81Ge0.19 buffer for investigating the feasibility of lasing. The result of in-plane dispersion of valence subbands in the above structure calculated by using the Kane model [6] is shown in Fig. 1. The LH1 subband has a energy valley of ˚ 21, which extends from 0 2 meV deep at kx ¼ ky ¼ 0:014 A 21 ˚ to 0.025 A . The lasing transition is designed to take place between the two lowest subbands HH1 and LH1 in the LH1 valley of the k-space at 50 mm. As marked by numericals 1, 2, 3, 4, we indicate in Fig. 1 how this inverted mass intersubband laser operates. By selective electrical pumping, holes are injected into LH1 at G-point (k ¼ 0) (state 4). Those holes then quickly populate the valley of LH1 (upper laser state 3) through intrasubband relaxation processes, undergo a lasing transition to lower laser state 2 in subband HH1, and finally relax toward state 1 at k ¼ 0 of the HH1 subband. The intrasubband relaxation processes 4– 3 and 2 –1 are very fast compared to the intersubband process 3 –2, which will lead to rapid population of the upper laser state 3 as well as fast depopulation of the lower laser state 2. The upper-state lifetime is long because the intersubband transition energy is below that of optical phonons, allowing only much weaker acoustic phonon scattering between the two different bands. The proposed laser structure is a so-called quantum staircase scheme which is different from the popular quantum cascade design in the sense that it eliminates the need for the chirped SL as the injector region. This structure uses electrical pumping of a set of isolated single Si0.7Ge0.3 QWs with Si barriers. As shown in Fig. 2 where the increase
0026-2692/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0026-2692(03)00032-6
392
G. Sun, R.A. Soref / Microelectronics Journal 34 (2003) 391–393
Fig. 1. In-plane dispersion of valence subbands showing the laser operating process.
Fig. 3. Optical gain as a function of the tunneling time with several hole densities.
of hole energy is chosen upward, holes from the lower subband HH1 in the previous lasing period tunnel through the Si-barrier to the upper subband LH1 in the next one. At each end, there is a region doped pþ: a SiGe emitter for current injection (gap wider than QW gap) and SiGe hole collector (gap equal to QW gap). The tunneling time, tt ; between subbands HH1 and LH1 can be controlled by tuning the width of the Si-barrier separating the QWs. Also shown in Fig. 2 is the competing process of acoustic phonon scattering which leaks holes directly to the lower subband HH1 in the next laser period without contributing to the lasing process, and therefore resulting in current loss. We performed detailed calculation of the optical gain based on our previous result of intersubband lifetime in SiGe/Si SLs [7]. Fig. 3 shows the optical gain as a function of the tunneling time for several densities of hole population. Positive optical gain ranging from , 100 cm21 to over 1000 cm21 is obtained when the tunneling time tt is shorter than that of upper laser state estimated at tu ¼ 2:0 ns. This is obviously expected because it corresponds to the case of total population inversion between the two subbands LH1 and HH1. It is interesting to
point out that even when tt . tu ; the situation where there is no total population inversion between the two subbands, fairly large optical gain on the order of a few 100 cm21 can still be achieved.
Fig. 2. A schematic of the inverted-mass intersubband laser with a quantum staircase scheme.
Fig. 4. Band diagram of the proposed inverted HH mass quantum staircase laser under bias.
3. Inverted HH laser A similar inverted mass feature can also be engineered for the HH subband in SiGe/Si SLs. We analyzed a model system consisting of coupled QWs under electric field bias, ˚ Si0.8Ge0.2 and as shown in Fig. 4 where the wells are 90 A ˚ Si. The operating field is chosen to be the barriers are 35 A 30 kV/cm just enough to produce a 3 meV spacing between the HH1 and HH2 subbands from two neighboring wells. The resulting quantum staircase has two active doublets per QW forming a four-level system. The laser transitions are indicated by the vertical arrows. The fast transitions within the doublets are responsible for the population of the upper laser state and depopulation of the lower laser state. The wavefunctions found at this bias is also shown in Fig. 4. We have calculated the in-plane dispersion of
G. Sun, R.A. Soref / Microelectronics Journal 34 (2003) 391–393
393
the valence subbands in such a laser structure. The result of the two doublets in one QW is shown in Fig. 5 where the inverted HH effective mass in demonstrated. We have also estimated the gain as a function of pump current density at 77 K as presented in Fig. 6. Peak gain of 450 cm21 at 41 mm is expected to be larger than the cavity loss such as free carrier absorption.
4. Conclusion
Fig. 5. Dispersion of the four active levels.
We have designed and simulated two SiGe/Si SL intersubband lasers pumped with the quantum staircase scheme. One involves inverted LH mass, and the other inverted HH mass, making it possible to achieve population inversion locally in k-space, eliminating the need of total population inversion between the two laser subbands. In both cases, optical gain on the order of a few 100 cm21 can be obtained in the THz range.
References
Fig. 6. Gain as a function of the pump current density.
[1] L. Friedman, R.A. Soref, G. Sun, Y. Lu, IEEE J. Sel. Top. Quantum Electron. 4 (1998) 1029. [2] R.A. Soref, L. Friedman, L.C. Lew Yan Voon, L.R. Ram-Mohan, G. Sun, J. Vac. Sci. Technol., B 16 (1998) 1525. [3] L. Friedman, G. Sun, R.A. Soref, Appl. Phys. Lett. 78 (2000) 401. [4] R.A. Soref, G. Sun, Appl. Phys. Lett. 79 (2001) 3639. [5] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Les Ulis), 1998, Chapter 3. [6] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [7] G. Sun, L. Friedman, R.A. Soref, Phys. Rev. B 62 (2000) 8114.