## Abstract

We propose a novel intersubband laser based on transitions between subbands in the **k**-space region where the light-hole effective mass is inverted. We consider GaAs/AlGaAs quantum well (QW) structures which give rise to several confined subbands including ground state heavy-hole subband (HH1) and light-hole subband (LH1). The inverted-effective-mass feature in subband LH1 emerges only in wide QWs which produce closely spaced subbands in energy that are strongly coupled. Laser wavelength determined by this energy separation is typically in the THz range. The laser structure is designed to facilitate electrical pumping with a quantum cascade scheme consisting of isolated single GaAs/AlGaAs QWs. Our calculation shows that with only a small fraction of the carrier population in the upper subband (LH1), it is still possible to achieve population inversion locally in **k**-space between the two subbands where the light-hole effective mass is inverted. Our result indicates that optical gain in excess of 150/cm can be achieved with a relatively small pump current density on the order of 100A/cm^{2}.

© Optical Society of America

## 1. Introduction

In less than four decades that have elapsed since their inception, radiation sources based on stimulated emission have extended their spectral range from far infrared nearly to the soft X-ray region. At the same time, development of microelectronics has pushed the frontier of electronic devices beyond the 100GHz range. Conspicuously, the wide frequency gap between electronics and optics, specifically in the 100GHz~10THz range, remains largely uncovered by the coherent sources of radiation. On one hand, for solid-state microelectronic devices to reach the THz range, it would require miniaturization on the nanometer scale, which is not only difficult but also impractical since the power output is expected to be extremely small, due to the 1/*f*
^{2} power degradation resulting from the transit time limitation. On the other hand, for solid state quantum transition devices, the THz range is on the same order of magnitude as their line broadening and the thermal energy. Currently, significant research effort has been directed towards meeting the challenge of developing submillimeter (or THz) lasers spanning the gap between the far infrared and GHz microwaves (50 ~ 1000*μ*m), which are in great demand for applications in spectroscopy, communication, bio-medical imaging and so on.

Rapid advance of epitaxial growth techniques has opened the possibility for the development of a fundamentally new type of the semiconductor devices based on transitions between the subbands in QWs and superlattices. Using the method of band-gap engineering, it is possible to adjust the energy of intersubband transition over a wide range. The first demonstration of a conduction-band quantum-cascade laser (QCL)[1] based on intersubband transitions has produced lasing around 5*μ*m. Since then, the QCL has been refined and its operating range has been extended toward far-infrared[2]–[4]. But, it is unlikely that this type of QCLs can operate in the THz range. This is because the momentum scattering time in semiconductors is less than 1ps, the operation of 1THz intersubband lasers would require obtaining stimulated emission from a line with a *Q*-factor less than one, which seems to be highly improbable. Here we propose a new kind of intersubband lasers based on the inverted-effective-mass feature of the subband in-plane dispersion. Such laser sources will potentially cover the THz range bridging the gap between the far infrared and GHz microwaves. The proposed lasers can be constructed with a much simplified design of isolated single QWs.

## 2. Main features of the inverted mass scheme

It appears that the laser states in a conventional band-to-band semiconductor laser consist of two broad bands. But a closer look at Fig.1(a) reveals a familiar four-level scheme. Clearly, the upper laser states, |*u* >, are all located near the bottom of the conduction band, while the lower states, |*l* >, are all near the top of the valence band. The lifetime of the upper laser states is determined by the interband recombination rate which can be as long as nanoseconds. From the lower laser states , |*l* >, electrons scatter into the lower energy states of the valence band, |*g* >, by inelastic intraband processes (a more conventional way is to look at it as relaxation of holes towards the top of the valence band.) This process is very fast, occurring on sub-picosecond scale. Pumping (injection or optical), on the other hand, places electrons at the intermediate states, |*i* >, above the quasi-Fermi level of the conduction band from which they quickly relax toward the upper laser states, |*u* >. Therefore, the lasing threshold can be reached when the whole population of the upper conduction band is only a tiny fraction of
that of the lower valence band. The population inversion is attained mostly due to the fundamental difference between the processes determining the lifetimes of upper and lower laser states.

Let us now turn our attention to the intersubband transitions shown in Fig.1(b). The in-plane dispersions of the upper |*u* > and lower |*l* > conduction subbands are almost identical when the band nonparabolicity can be neglected. For all practical purposes they can be considered as two discrete levels. Then, in order to achieve population inversion it is necessary to have the whole population of the upper subband exceed that of the lower subband, *N _{u}* >

*N*. For this reason, as pointed out in our earlier work[5], a three- or four-subband scheme becomes necessary to reach the lasing threshold. Even then, since the relaxation rates between different subbands are determined by the same physical processes, a complex multiple QW structure needs to be designed to engineer the lifetimes of involved subbands.

_{l}Fortunately, the band nonparabolicity resulting from the interaction between the subbands produces nonparallel in-plane dispersions, especially in the large k-vector region of the conduction subbands. As a result, the total intersubband population inversion becomes unnecessary, as reported recently in a QCL[6] where population inversion is established only locally in **k**-space for laser operation. In comparison with the conduction-band nonparabolicity, this effect is known to be much stronger in the valence band. Hence, we propose a scheme of valence intersubband lasers in which we can effectively engineer the dispersion of two valence subbands in a QW similar to that of conduction and valence bands. In order to achieve this, one of the subbands shall be electron-like and the other hole-like, i.e., one of the subbands shall have its effective mass inverted. Such inversion is the result of interaction between the subbands, which is much stronger in the valence band. Indeed, in the valence band of most diamond and zinc-blende semiconductors, light- and heavy-hole subbands usually anti-cross, and near the point of anti-crossing, the light-hole subband in-plane dispersion becomes electron-like. The calculated in-plane dispersions of a single GaAs/AlGaAs QW with a well width of 70 Åfor subbands HH1, LH1, and HH2 are shown in Fig.2. The inverted-effective-mass feature in the light-hole subband LH1 is clearly demonstrated.

If we now designate states near the Γ-point of subband LH1 as the intermediate states, |*i* >, states near the valley (inverted-effective-mass region) of subband LH1 as the upper laser states, |*u* >, states in subband HH1 vertically below the valley of subband LH1 as the lower laser states, |*l* >, and states near the Γ-point of subband HH1 as the ground states, *g* > (counting the hole energy downward), we can see that the situation closely resembles the one in the conventional band-to-band laser. The upper and lower laser states can be quickly populated and depopulated through fast intrasubband processes, while the lifetime of upper laser states is determined by a much slower intersubband process between subbands LH1 and HH1. In order to produce the inverted-effective-mass feature in the in-plane dispersion of subband LH1, the QW width needs to be relatively wide (*d* > 60Å) so that light- and heavy-hole subbands are closely spaced in energy. In GaAs/AlGaAs QWs, this energy separation is typically less than the optical phonon energy (36meV), which suppresses the nonradiative intersubband transitions due to the optical phonon scattering. Thus, near the inverted-effective-mass region in **k**-space, the lifetime of upper laser states can be as long as a few nanoseconds, much longer than that of lower laser states on the order of picoseconds, *τ _{u}* >

*τ*, a necessary condition for achieving population inversion between the laser states.

_{l}## 3. Analysis and results

The in-plane dispersion of valence subbands can be calculated using the Kane model [7] in QW structures. Since in GaAs-based materials the Γ_{8} coupling with Γ_{6} bands is weak, the 8 × 8 Hamiltonian matrix in the **k** · **p** theory reduces to a 4 × 4 valence band matrix taking into account the coupling between heavy-hole and light-hole bands [8]. The inverted-effective-mass feature produced by the anti-crossing between subbands LH1 and HH2 is clearly demonstrated in Fig.2 showing a shallow energy valley in the in-plane dispersion of subband LH1 for a well width of 70Å.

The laser structure employs the quantum cascade scheme in the active region to facilitate tunneling of carriers from the lower subband HH1 in the previous lasing cycle to the upper subband LH1 in the next one as shown in Fig.3. The structure is doped with *p*+ regions on both ends for current injection. We have estimated the tunneling time between subbands HH1 and LH1 to be *τ*
_{tun} = 5ns for a barrier width of 50Åtaking into account the small tunneling probability between the heavy- and light-hole subbands. In competing with the tunneling process, the acoustic phonon scattering leaks carriers directly to the lower subband HH1 in the next laser cycle, resulting in current loss. The acoustic phonon process can have a much shorter scattering time of *τ*
_{ph} = 0.1ns at the liquid-nitrogen operating temperature of the laser.

Since the intrasubband process is significantly faster than the intersubband process, we can use quasi-Fermi levels (*E _{Fl}*,

*E*) to describe light-hole and heavy-hole distributions in their respective subbands. Under the assumption of isotropic in-plane dispersion, the light- and heavy-hole populations at a given in-plane wavevector (${k}_{i}=\sqrt{{k}_{i,x}^{2}+{k}_{i,y}^{2}}$) within the interval of Δ

_{Fh}*k*can be described by

_{i}where *d* is the well width, *k _{i}*/

*πd*is the density of states in

**k**-space, and

*f*(

_{l,h}*k*) is the Fermi-Dirac distribution for holes in subbands LH1 and HH1, respectively,

_{i}where *k _{B}* is the Boltzmann constant,

*T*is the temperature, and

*E*(

_{l,h}*k*) is the in-plane dispersion of subbands LH1 and HH1, respectively. The total hole population

_{i}*N*injected by the pump current is distributed between subbands HH1 (

*N*) and LH1(

_{h}*N*) as

_{l}$$\phantom{\rule{.2em}{0ex}}=\sum _{{k}_{i}}\Delta {N}_{l}\left({k}_{i}\right)+\sum _{{k}_{i}}\Delta {N}_{h}\left({k}_{i}\right)$$

where the summation is over all *k _{i}* states.

A rate equation can be established for the population *N _{l}* of subband LH1,

taking into account contributions from carrier tunneling, induced and spontaneous emissions. Since the energy separation between subbands HH1 and LH1 is below the optical phonon energy (36meV) and the laser is designed to operate at the temperature of liquid nitrogen, the contributions from nonradiative intersubband phonon scattering and Auger processes between subbands LH1 and HH1 are neglected in Eq.(4). Also, the carrier-carrier scattering is not expected to be strong in this type of lasers since the Bloch wavefunctions of light-hole and heavy-hole are very different from each other in two-dimensional QWs and there is little overlap between them. The spontaneous emission rate is given by

where *ħ* is the Planck constant, *m _{o}* and

*e*are the free-electron mass and charge,

*∊*and

_{o}*c*are the permittivity and light velocity in vacuum,

*ñ*is the index of refraction,

*E*(

*k*) =

_{i}*E*(

_{l}*k*) -

_{i}*E*(

_{h}*k*) and

_{i}*M*(

_{p}*k*) are the transition energy and momentum matrix element between subbands LH1 and HH1 at

_{i}*k*, respectively. The induced transition rate is related to the spontaneous rate by the Einstein relation,

_{i}*W*

_{ind}(

*k*) =

_{i}*n*(

*E*)

*W*

_{spon}(

*k*), where

_{i}*n*(

*E*) = 1/[exp(

*E*/

*k*) - 1] is the number of photons with energy

_{B}T*E*at temperature

*T*.

A steady-state distribution can be obtained by setting Eq.(4) to be zero, then solved simultaneously with Eq.(3). Solving both equations self-consistently using the Monte Carlo method, we are able to obtain quasi-Fermi levels for subbands HH1 and LH1 with a given hole concentration *N* injected by a pump current density, *J _{p}* =

*eNd*(1/

*τ*

_{tun}+ 1/

*τ*

_{ph}). The result of quasi-Fermi levels for the structure with well and barrier widths of 70Åand 50Å, respectively, is shown in Fig.2 for a pump current density of

*J*= 90A/cm

_{p}^{2}providing a total hole population of 6 × 10

^{17}/cm

^{3}. The positions of quasi-Fermi levels relative to their respective subbands suggest that the population inversion can be established locally near the inverted-effective-mass region of subband LH1 even though the overall hole population in subband LH1 is less than that in subband HH1.

The expression for optical gain at a photon energy *E* can be given as

where *η* = 1/*∊ _{o}cñ* = 377Ω/ñ is the impedance of the medium,

*ρ*(

_{r}*E*-

_{l}*E*) is the reduced density of states for the l-h transition, and Γ is the broadening determined by all the dephasing processes including both intrasubband and intersubband scattering, but mostly by the much faster intrasubband process. Eq.(6) is integrated over the range of

_{h}*E*-

_{l}*E*where the population inversion is established. If this region is wide enough compared to the broadening, the Lorentzian shape can then be approximated by

_{h}*πδ*[

*E*- (

*E*-

_{l}*E*)], and Eq.(6) reduces to

_{h}In case that the broadening is comparable to the energy range of population inversion, this approximation may lead to an overestimate of optical gain by about a factor of √2.

The optical gain in a laser structure with a well width of 70Åand a barrier width of 50Åas a function of photon energy is shown in Fig.4 for several injected hole concentrations in the 5 × 10^{17} ~ 1 × 10^{18}/cm^{3} range. It should be pointed out that the optical gain is calculated under the conditions of liquid-nitrogen operating temperature and a relatively long tunneling time of 5ns. It can be seen from Fig.4 that a maximum optical gain of 170/cm at the laser wavelength of 67*μ*m can be achieved for the hole concentration of 7 × 10^{17}/cm^{3}. Optical gain of this magnitude shall be large enough to compensate loss in GaAs/AlGaAs QWs if TM-polarization is employed for the laser operation to avoid the internal loss due to the strong free-carrier absorption of TE-polarized radiation.

Fig.5 shows maximum gain in the laser structure with a well width of 70Åand a barrier width of 50Åas a function of pump current density. It can be seen from Fig.5 that when the current density is small (< 15A/cm^{2}) the population inversion is not established in the inverted-mass region producing no optical gain. As the injection current increases, the population inversion is achieved and optical gain increases rapidly. Further increase of the current intensity, however, results in optical gain saturation. This is because more of the lower laser states in subband HH1 are occupied by holes due to the high carrier concentration injected by the large pump current through the undesirable phonon scattering process. The same behavior of optical gain saturation is also shown in Fig.4 where the maximum optical gain of the highest hole concentration of 1 × 10^{18}/cm^{3} is reduced compared to those with lower hole concentrations.

## 4. Summary

We have studied a novel intersubband laser based on the inverted-effective-mass feature in the in-plane dispersion of the light-hole valence subband in GaAs/AlGaAs QWs. The intersubband laser transition occurs between the light- and heavy-hole subbands in the region of inverted-effective-mass in the in-plane dispersion. The advantage of designing intersubband lasers based on this feature is that the population inversion needs to be achieved only locally in **k**-space within the inverted-effective-mass region, while the total population in the upper light-hole subband is only a small fraction of the population in the lower heavy-hole subband. We have theoretically investigated simple laser structures of quantum cascade scheme which consists of multiple isolated single QWs. Carrier tunneling has been utilized as the mechanism for the population of the upper light-hole subband. Our results indicate that without establishing total population inversion between the two subbands LH1 and HH1, it is possible to achieve local population inversion in **k**-space due to this inverted-effective-mass feature and obtain an optical gain in excess of 150/cm at a pump current density of 100A/cm^{2}.

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