Adv. Opt. Photon., Vol. 3, Iss. 1, pp. 53-87; doi: 10.1364/AOP.3.000053

Intersubband approach to silicon based lasers—circumventing the indirect bandgap limitation

Greg Sun

^{100 Morrissey Boulevard, University of Massachusetts Boston,Boston, Massachusetts 02125-3393, U.S.A.}

^{E-mail: greg.sun@umb.edu}

Received February 23, 2010; revised June 23, 2010; accepted July 28, 2010; published September 10, 2010

Silicon-based lasers have long been sought, since they will permit monolithic integration of photonics with high-speed silicon electronics and thereby significantly broaden the reach of silicon technology. Among the various approaches that are currently being pursued to overcome the intrinsic limitations of Si as an efficient light source, intersubband transitions in Si-based quantum well structures offer a rather feasible alternative that conveniently circumvents the indirect-band nature of Si. Various approaches for achieving lasing action based on intersubband transitions within the group-IV materials are reviewed. Relevant theories are presented in detail. Challenges facing the valence band approach, which has not so far been successful, are analyzed, and proposals that bring the intersubband process to the conduction band are discussed.

1. Introduction

Silicon has been the miracle material for the electronics industry, and, for the past twenty years, technology based on Si microelectronics has been the engine driving the digital revolution. For years, the rapid “Moore’s law” miniaturization of device sizes has yielded an ever-increasing density of fast components integrated on Si chips, pushing down feature size close to its ultimate physical limits. At the same time, there has been a parallel effort to broaden the reach of Si technology by expanding its functionalities well beyond electronics. Si is now being increasingly investigated as a platform for building photonic devices. The field of Si photonics has seen impressive growth since the early visions of the 1980s and 1990s [1, 2]. The huge infrastructure of the global Si electronics industry is expected to benefit the fabrication of highly sophisticated Si photonic devices at costs that are lower than those currently required for compound semiconductors. Furthermore, Si-based photonic devices make possible the monolithic integration of photonic devices with high-speed Si electronics, thereby enabling an oncoming Si-based optoelectronic revolution.

Among the many devices that make up a complete set of components for Si photonics are light emitters, amplifiers, photodetectors, waveguides, modulators, couplers, and switches. The most challenging task has been the fabrication of an efficient light source. The difficulty arises from the fact that Si has an indirect bandgap; the minimum of the conduction band and the maximum of the valence band do not occur at the same value of crystal momentum in wave-vector space (Fig. 1). Since the wave vector of a photon is negligibly small compared with that of an electron, the optical transition requires the emission or absorption of a phonon in order to conserve momentum. The radiative recombination of electron–hole pairs in the indirect material is therefore a second-order process, occurring with a lower probability and slower rate than in the direct material. The rather inefficient radiative process must compete with nonradiative processes that take place at much faster rates. As a result, as marvelous as it has been for electronics, bulk Si has not been the material of choice for making light emitting devices, including lasers.

Fig. 1.
Photon emission process in (a) the direct and (b) the indirect bandgap semiconductors.

Nevertheless, driven by its enormous payoff in technology advancement and commercialization, many research groups around the world have been seeking novel approaches to overcome the inherent problem of Si for the development of efficient light sources based on Si. One interesting method is to use small Si nanocrystals dispersed in a dielectric matrix, usually Si O 2 . Such nanoscaled Si clusters are naturally formed in the thermal annealing of a Si-rich oxide thin film. Silicon nanocrystals situated in a much wider bandgap Si O 2 can effectively localize electrons with quantum confinement, which improves the radiative recombination probability, shifts the emission spectrum toward shorter wavelengths, and decreases the free carrier absorption. Optical gain and stimulated emission have been observed in these Si nanocrystals by both optical pumping [3, 4] and electrical injection [5], but the origin of the observed optical gain has not been fully understood, as the experimental results, sensitive to the methods of sample preparation, were not always reproducible. In addition, before Si-nanocrystal-based lasers can be demonstrated, the active medium has to be immersed in a tightly confined optical waveguide or cavity.

Another approach is motivated by the light amplification in Er-doped optical fibers that utilize the radiative transitions in Er ions ( Er 3 + ) [6]. By incorporating Er 3 + in Si, these ions can be excited by energy transfer from electrically injected electron–hole pairs in Si and will subsequently relax by emitting photons at the telecommunication wavelength of 1.55 μ m . However, the degree to which Er 3 + ions can be doped in Si is relatively low, and there is a significant energy backtransfer from the Er 3 + ions to the Si host due to the resonance with a defect level in Si. As a result, both efficiency and maximum power output have been extremely low [7, 8]. To reduce the backtransfer of energy, Si O 2 , with an enlarged bandgap, has been proposed as host to remove the resonance between the defect and the Er 3 + energy levels [9]. Once again, Si-rich oxide is employed to form Si nanocrystals in close proximity to Er 3 + ions. The idea is to excite Er 3 + ions with the energy transfer from the nearby Si nanocrystals. Light emitting diodes (LEDs) with efficiencies of about 10% have been demonstrated [10], on a par with commercial devices made of GaAs, but with power output only in tens of microwatts. While there have been attempts to develop lasers using doped Er in a Si-based dielectric, that goal remains elusive.

The only approach that thus far has led to the demonstration of lasing in Si exploited the effect of stimulated Raman scattering [11, 12, 13], analogous to that produced in fiber Raman amplifiers. With both the optical pumping and the Raman scattering below the bandgap of Si, the indirectness of the Si bandgap becomes irrelevant. Depending on whether it is a Stokes or anti-Stokes process, the Raman scattering either emits or absorbs an optical phonon. Such a nonlinear process requires optical pumping at very high intensities ( 100 MW ∕ cm 2 ) , and the approximately centimeter device lengths are too large to be integrated with other photonic and electronic devices in Si very-large-scale integration-type circuits [14].

Meanwhile, the search for laser devices that can be integrated on Si chips has gone well beyond the monolithic approach, with solutions sought using hybrid integration of III–V compounds with Si. A laser with an AlGaInAs quantum well (QW) active region bonded to a silicon waveguide cavity has been demonstrated [15]. This fabrication technique allows the optical waveguide to be defined by the CMOS compatible Si process while the optical gain is provided by III–V materials. Rare-earth-doped visible-wavelength GaN lasers, fabricated on Si substrates, are also potentially compatible with the Si CMOS process [16]. Another effort has produced InGaAs quantum dot lasers deposited directly on Si substrates with a thin GaAs buffer layer [17]. Although these hybrid approaches offer important alternatives, they do not represent the ultimate achievement of Si-based lasers monolithically integrated with Si electronics.

While progress is being made along these lines, another method has emerged in the past decade that has received a great deal of attention—an approach in which the lasing mechanism is based on intersubband transitions (ISTs) in semiconductor QWs. Such transitions take place between quantum confined states (subbands) of conduction or valence bands and do not cross the semiconductor bandgap. Since carriers remain in the same energy band (either conduction or valence), optical transitions are always direct in momentum space, rendering the indirectness of the Si bandgap irrelevant. Lasers using ISTs therefore provide a promising alternative that completely circumvent the issue of indirectness in the Si bandgap. In addition, this type of laser can be conveniently designed to employ electrical pumping—the so-called quantum cascade laser (QCL). The pursuit of Si-based QCLs may turn out to be a viable approach to achieving electrically pumped Si-based coherent emitters that are suitable for monolithic integration with Si photonic and electronic devices.

In this review, lasing processes based on ISTs in QWs are explained by drawing a comparison to conventional band-to-band lasers. Approaches and results towards SiGe QCLs using ISTs in the valence band are overviewed, and the challenges and limitations of the SiGe valence-band QCLs are discussed with respect to materials and structures. In addition, ideas are proposed to develop conduction-band QCLs, among them a novel QCL structure that expands the material combination to SiGeSn. This is described in detail as a way to potentially overcome the difficulties that are encountered in the development of SiGe QCLs.

2. Lasers Based on Intersubband Transitions

Research on quantum confined structures, including semiconductor QWs and superlattices (SLs), was pioneered by Esaki and Tsu in 1970 [18]. Since then confined structures have been developed as the building blocks for a majority of modern-day semiconductor optoelectronic devices. QWs are formed by depositing a narrow-bandgap semiconductor with a layer thickness less than the deBroglie wavelength of the electron ( ∼ 10 nm ) between two wider-bandgap semiconductors [Fig. 2(a)]. The one-dimensional quantum confinement leads to quantized states (subbands) within both the conduction and the valence bands. The energy position of each subband depends on the band offset ( Δ E C , Δ E V ) and the effective mass of the carriers. In directions perpendicular to the direction of growth z (in-plane), the carriers are unconfined and can thus propagate with an in-plane wave vector k , which gives an energy dispersion for each subband [Fig. 2(b)].

Fig. 2.
(a) Conduction and valence subband formations in a semiconductor QW; (b) in-plane subband dispersions with optical transitions between conduction and valence subbands.

If the band offset is large enough, multiple subbands may be present within either the conduction or the valence band, as shown in Fig. 3, where two subbands are confined within the conduction band. The electron wavefunctions [Fig. 3(a)] and energy dispersions [Fig. 3(b)] are illustrated for the two subbands. The concept of ISTs refers to the physical process of carrier transitions between subbands within either the conduction or valence band, as illustrated in Fig. 3. Carriers originally occupying a higher energy subband can make a radiative transition to a lower subband by emitting a photon. Coherent sources utilizing this type of transition as the origin of light emission are called intersubband lasers.

Fig. 3.
Two subbands formed within the conduction band confined in a QW and their election envelope functions; (b) in-plane energy dispersions of the two subbands. The radiative IST between the two subbands is marked with a green arrow.

The original idea of creating light sources based on ISTs was proposed by Kazarinov and Suris [19] in 1971, but the first QCL was not demonstrated until 1994 by a group led by Capasso at Bell Laboratories [20]. In comparison with the conventional band-to-band lasers, lasers based on ISTs require a much more complex design of the active region, which consists of carefully arranged multiple QWs. The reason for added complexity can be appreciated by comparing the very different band dispersions that are involved in these two types of laser. In a conventional band-to-band laser, it appears that the laser states consist of two broad bands. But a closer look at the conduction and valence band dispersions [Fig. 2(b)] reveals a familiar four-level scheme where in addition to the upper laser states | u ⟩ , located near the bottom of the conduction band and the lower laser states | l ⟩ , near the top of the valence band, there are two other participating states—intermediate states | i | and ground states | g ⟩ . The pumping process (either injection or optical) places electrons into the intermediate states, | i ⟩ , from which they quickly relax toward the upper laser states | u ⟩ by inelastic scattering intraband processes. This process is rapid, occurring on a subpicosecond scale. But once they reach states | u ⟩ , they tend to stay there for a much longer time determined by the band-to-band recombination rate, which is of the order of nanoseconds. After undergoing lasing transitions to the lower states | l ⟩ , electrons will quickly scatter into the lower energy states of the valence band—ground states | g ⟩ —by the same fast inelastic intraband processes. (A more conventional view regards the process as the relaxation of holes toward the top of the valence band.) The population inversion between | u ⟩ and | l ⟩ is therefore established primarily by the fundamental differences of the processes determining the lifetimes of upper and lower laser states. As a result, the lasing threshold can be reached when the whole population of the upper conduction band is only a small fraction of that of the lower valence band.

Let us now turn our attention to the IST shown in Fig. 3(b). The in-plane dispersions of the upper | u ⟩ and lower | l ⟩ conduction subbands are almost identical when the band nonparabolicity is neglected. For all practical purposes the subbands can be considered two discrete levels. To achieve population inversion it is necessary to have the whole population of the upper subband exceed that of the lower subband. For this reason, 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 intraband processes, a complex multiple QW structure needs to be designed to engineer the lifetimes of involved subbands.

Despite their structural requirements, intersubband lasers offer advantages in areas where the conventional band-to-band lasers simply cannot compete. In band-to-band lasers, lasing wavelengths are determined mostly by the intrinsic bandgap of the semiconductors. There is very little room for tuning, accomplished by varying the structural parameters such as strain, alloy composition, and layer thickness. Especially for those applications in the mid-IR to far-IR range, there are no suitable semiconductors with the appropriate bandgaps from which such lasers can be made. With the ISTs, we are no longer limited by the availability of semiconductor materials to produce lasers in this long-wavelength region. In addition, for ISTs between conduction subbands with parallel band dispersions, the intersubband lasers should therefore have a much narrower gain spectrum in comparison with the band-to-band lasers in which conduction and valence bands have opposite band curvatures.

A practical design that features a four-level intersubband laser pumped optically was proposed by Sun and Khurgin [21, 22] in the early 1990s. This work laid out a comprehensive analysis of various intersubband processes that affect the lasing operation, including scattering mechanisms that determine subband lifetimes, conditions for population inversion between two subbands, band engineering to achieve it, and optical gain sufficient to compensate for losses under realistic pumping intensity. The QCLs developed soon thereafter significantly expanded the design in order to accommodate electrical pumping by implementing a rather elaborate scheme of current injection with the use of a chirped SL as the injector region placed between the active regions (Fig. 4). The QCL has a periodic structure with each period consisting of an active and an injector region. Both active and injector regions are composed of multiple QWs. By choosing combinations of layer thicknesses and material compositions, three subband levels with proper energy separations and wave function overlaps are obtained in the active region. The injector region, on the other hand, is designed with a sequence of QWs having decreasing well widths (chirped SL) such that they form a miniband under an electric bias that facilitates electron transport. The basic operating principle of a QCL is illustrated in Fig. 4. Electrons are first injected through a barrier into subband 3 (upper laser state) of the active region, they then undergo lasing transitions to subband 2 (lower laser state) by emitting photons, followed by fast depopulation into subband 1 via nonradiative processes. These electrons are subsequently transported through the injector region into the next active region, where in a cascading manner they repeat the process, typically 20–100 times.

Fig. 4.
Schematic band diagram of two periods of a QCL structure with each period consisting of an active and an injector region. Lasing transitions are between the states 3 and 2 in the active regions, with rapid depopulation of lower state 2 into state 1, which couples strongly with the minibands formed in injector regions that transport carriers to state 3 in the next period. The magnitude-squared wavefunctions for the three subbands in active regions are illustrated.

Advances of QCLs since the first demonstration have resulted in dramatic performance improvement in spectral range, power, and temperature. They have become the dominant mid-IR semiconductor laser sources covering the spectral range of 3 < λ < 25 μ m [23, 24, 25], many of them operating in the continuous-wave mode at room temperature and with peak power reaching a few watts [26, 27]. Meanwhile, QCLs have also penetrated deep into the terahertz regime, loosely defined as the spectral region 100 GHz < f < 10 THz or 30 < λ < 3000 μ m , bridging the gap between the far-IR and gigahertz microwaves. At present, spectral coverage from 0.84 – 5.0 THz has been demonstrated with operation in either the pulsed or continuous-wave mode at temperatures well above 100 K [28].

3. Intersubband Theory

To better explain the design considerations of intersubband lasers, it is necessary to introduce some basic physics that underlies the formation of subbands in QWs and their associated intersubband processes. The calculation procedures described here follow the envelope function approach based on the effective-mass approximation [29]. The k ⋅ p method [30] is outlined to obtain in-plane subband dispersions in the valence band. Optical gain for transitions between subbands in conduction and valence bands is derived. Various scattering mechanisms that determine the subband lifetimes are discussed with an emphasis on the carrier-phonon scattering processes.

3.1. Subbands and Dispersions

Let us treat the conduction subbands first. It is well known that in bulk material near the band edge the band dispersion with an isotropic effective mass follows a parabolic relationship. In a QW structure, along the in-plane direction ( k = k x x ̂ + k y y ̂ ) where electrons are unconfined, such curvature is preserved for a given subband i when the nonparabolicity that describes the energy-dependent effective mass m e * is neglected. Then,

E i , k = E i + ℏ 2 k 2 2 m e * ,

(1)

where ℏ is the Planck constant and E i is the minimum energy of subband i in a QW structure. This minimum energy can be calculated as one of the eigenvalues of the Schrödinger equation along the growth direction z ,

[ − ℏ 2 2 d d z 1 m e * ( z ) d d z + V c ( z ) ] φ i ( z ) = E i φ i ( z )

(2)

where the z dependence of m e * allows for different effective masses in different layers and V c ( z ) represents the conduction-band edge along the growth direction. The envelope function of subband i , φ i ( z ) , together with the electron Bloch function u e ( R ) and the plane wave e j k ⋅ r , gives the electron wavefunction in the QW structure as

Φ i ( r , z ) = φ i ( z ) u e ( R ) e j k ⋅ r ,

(3)

where the position vector is decomposed into in-plane and growth directions, R = r + z z ̂ . Since we are treating electron subbands, the Bloch function is approximately the same for all subbands and all k vectors. The electron envelope function can be given as a combination of the forward and backward propagations in a given region l of the QW structure (either a QW or a barrier region), d l < z < d l + 1 ,

φ i ( z ) = A l e j k z ( z − d l ) + B l e − j k z ( z − d l ) ,

(4)

where A l and B l are constants that need to be fixed by imposing the continuity conditions at each of the interfaces z = d l ,

φ i ( z ) and 1 m e * ( z ) d φ i ( z ) d z continous ,

(5)

in conjunction with the relationship between the subband minimum energy E i and the quantized wave vector k z in the z direction,

E i = ℏ 2 k z 2 2 m e * ( z ) + V c ( z ) .

(6)

The wave-vector component k z assumes either real or imaginary values, depending on E i − V c ( z ) . The continuity conditions in Eq. (5) ensure continuous electron distribution and conservation of electron current across the interface.

In the presence of an electric field E applied in the z direction, the potential term V c ( z ) in the Schrödinger equation, Eq. (2), becomes tilted along the z direction according to − e E z . When the Coulomb effect due to the distribution of electrons in the subband is taken into account, the potential in region l of the QW structure with the conduction-band edge V c , l is modified and becomes

V c ( z ) = V c , l − e E z − e ϕ ( z ) ,

(7)

where e ϕ ( z ) is the potential due to electron distributions in all subbands and is obtained by solving the Poisson equation

∂ 2 ∂ z 2 ϕ ( z ) = − e ε 0 ε ( z ) [ ∑ i n i | φ i ( z ) | 2 − N d ( z ) ]

(8)

consistently with Eq. (2). Here, e is the charge of a free electron, ε 0 is the permittivity of free space, ε ( z ) is the z -dependent dielectric constant of the QW structure, n i is the electron density of subband i , and N d ( z ) is the n-type doping profile in the structure.

In comparison with the conduction band, the situation in the valence band is far more complex, mostly because the interactions between subbands of different effective masses produce strong nonparabolicity. The in-plane dispersion of valence subbands and their associated envelope functions can be obtained in the framework of the effective mass approximation by applying the k ⋅ p theory [30] to QWs [31], where, in the most general treatment, an 8 × 8 Hamiltonian matrix is employed to describe the interactions between the conduction, heavy-hole (HH), light-hole (LH), and spin-orbit split off (SO) bands. Usually, for semiconductors in which the conduction band is separated far in energy from the valence band, the coupling of the conduction band can be ignored. For the group-IV semiconductors Si and Ge with indirect bandgaps, this approximation is particularly adequate. In those structures where there is little strain, such as Ga As ∕ Al Ga As , the SO band coupling can also be ignored. The 8 × 8 Hamiltonian matrix can then be reduced to a 4 × 4 matrix. But for systems with appreciable lattice mismatch, strain induces strong coupling between LH and SO bands. For the SiGe system with a large lattice mismatch, the SO band should be included, and a 6 × 6 matrix Hamiltonian equation needs to be solved to obtain the dispersion relations and envelope functions. Such a 6 × 6 Hamiltonian matrix can be solved exactly in multiple QW structures under the bias of an electric field. A procedure based on the Luttinger–Kohn Hamiltonian [32, 33] is outlined as follows: the 6 × 6 Luttinger–Kohn Hamiltonian matrix, including the uniaxial stress along (001), is given in the HH ( | 3 2 , ± 3 2 ⟩ ) , LH ( | 3 2 , ± 1 2 ⟩ ) , and SO ( | 1 2 , ± 1 2 ⟩ ) Bloch function space as

H = − | 3 2 , 3 2 ⟩ | 3 2 , 1 2 ⟩ | 3 2 , − 1 2 ⟩ | 3 2 , − 3 2 ⟩ | 1 2 , 1 2 ⟩ | 1 2 , − 1 2 ⟩ [ P + Q − S R 0 − 1 2 S 2 R − S † P − Q 0 R − 2 Q 3 2 S R † 0 P − Q S 3 2 S † 2 Q 0 R † S † P + Q − 2 R † − 1 2 S † − 1 2 S † − 2 Q 3 2 S − 2 R P + Δ 0 2 R † 3 2 S † 2 Q − 1 2 S 0 P + Δ ] + V v ( z )

(9)

where V v ( z ) is the valence band edge profile (degenerate for HH and LH bands) of the QW structure and

P = ℏ 2 2 m 0 γ 1 ( k x 2 + k y 2 + k z 2 ) − a v ( ϵ x x + ϵ y y + ϵ z z ) ,

Q = ℏ 2 2 m 0 γ 2 ( k x 2 + k y 2 − 2 k z 2 ) − b 2 ( ϵ x x + ϵ y y − 2 ϵ z z ) ,

S = ℏ 2 2 m 0 2 3 γ 3 ( k x − j k y ) k z ,

R = ℏ 2 2 m 0 3 [ − γ 2 ( k x 2 − k y 2 ) + 2 j γ 3 k x k y ] .

P = ℏ 2 2 m 0 γ 1 ( k x 2 + k y 2 + k z 2 ) − a v ( ϵ x x + ϵ y y + ϵ z z ) ,

Q = ℏ 2 2 m 0 γ 2 ( k x 2 + k y 2 − 2 k z 2 ) − b 2 ( ϵ x x + ϵ y y − 2 ϵ z z ) ,

S = ℏ 2 2 m 0 2 3 γ 3 ( k x − j k y ) k z ,

R = ℏ 2 2 m 0 3 [ − γ 2 ( k x 2 − k y 2 ) + 2 j γ 3 k x k y ] .

(10)

m 0 is the mass of a free electron, γ 1 , γ 2 , γ 3 are the Luttinger parameters, and a v ′ , b are the deformation potentials [34] with different values in QWs and barriers. The lattice mismatch strain is given by

ϵ x x = ϵ y y = a 0 − a a , ϵ y y = − 2 C 12 C 11 ϵ x x

(11)

with a 0 , a being the lattice constants of the substrate (or buffer) and the layer material, and C 11 and C 12 being the stiffness constants.

The Hamiltonian in Eq. (9) operates on wavefunctions that are combinations of six mutually orthogonal Bloch functions: HH ( | 3 2 , ± 3 2 ⟩ ) , LH ( | 3 2 , ± 1 2 ⟩ ) , and SO ( | 1 2 , ± 1 2 ⟩ ) . The wavefunction is then expressed as

Ψ i ( r , z ) = e j k ⋅ r [ χ 1 ( z ) | 3 2 , 3 2 ⟩ + χ 2 ( z ) | 3 2 , 1 2 ⟩ + χ 3 ( z ) | 3 2 , − 1 2 ⟩ + χ 4 ( z ) | 3 2 , − 3 2 ⟩ + χ 5 ( z ) | 1 2 , − 1 2 ⟩ + χ 6 ( z ) | 1 2 , − 1 2 ⟩ ] ,

(12)

where χ n ( z ) , n = 1 , 2 , … , 6 forms a six-component envelope-function vector χ ( z ) . Each component in a given region l of the QW structure (either QW or barrier), d l < z < d l + 1 , is a superposition of the forward and backward propagations identical to Eq. (4) with constants A n , l and B n , l , n = 1 , 2 , … , 6 that can be fixed by the continuity equations required at each interface z = d l ,

χ ( z ) and [ p + q − s 0 0 − 1 2 s 0 − s † p − q 0 0 − 2 q 3 2 s 0 0 p − q s 3 2 s † 2 q 0 0 s † p + q 0 − 1 2 s † − 1 2 s † − 2 q 3 2 s 0 p 0 0 3 2 s † 2 q − 1 2 s 0 p ] χ ( z ) continuous .

(13)

Here

p = γ 1 ∂ ∂ z ,

q = − 2 γ 2 ∂ ∂ z ,

s = − 3 j γ 3 ( k x − j k y ) .

p = γ 1 ∂ ∂ z ,

q = − 2 γ 2 ∂ ∂ z ,

s = − 3 j γ 3 ( k x − j k y ) .

(14)

It is important to point out that when the above described algorithms are used for the situation where an electric field is applied along the growth direction z , it is necessary to digitize the potential term V c ( z ) and V v ( z ) , since the regions that are used in Eq. (4) are no longer defined by the QW and barrier boundaries; instead, there could be many regions within each QW or barrier, depending on the number of steps used to satisfy the accuracy requirement.

This procedure applied at each wave-vector point ( k = k x x ̂ + k y y ̂ ) produces the in-plane dispersion relation for each subband. An example is illustrated in Fig. 5 for a 70 Å ∕ 50 Å Ga As ∕ Al 0.3 Ga 0.7 As SL [35]. In-plane dispersions of three subbands (two for HH and one for LH) are shown where strong nonparabolicity is demonstrated. It can be seen from Fig. 5 that the band nonparabolicity could be so severe that the LH subband maximum is no longer at the Γ point, which leads to useful valence QCL design applications discussed in Section 4.

Fig. 5.
In-plane dispersions of subbands HH1, LH1, and HH2 for a 70 Å ∕ 50 Å Ga As ∕ Al Ga As SL [35].

3.2. Optical Gain

For lasing to occur between two subbands, it is necessary to induce stimulated emission between them. To sustain such emission of photons, there must be sufficient optical gain to compensate for the various losses in the laser structure. The intersubband optical gain can be obtained by analyzing transition rates between two subbands.

According to the Fermi golden rule, the transition rate between two discrete states 1 and 2 that are coupled by a perturbating electromagnetic (EM) field with a frequency of ω is

g 12 = 2 π ℏ | H m | 2 δ ( E 2 − E 1 − ℏ ω ) ,

(15)

where H m = ⟨ 1 | H ex | 2 ⟩ is the transition matrix element of a perturbation Hamiltonian H ex between the two states with an exact transition energy E 2 − E 1 in the absence of any broadening. In reality, the transition line E 2 − E 1 is not infinitely sharp and is always broadened. As a result, E 2 − E 1 is not exactly known; instead a probability for it to appear in the energy interval E → E + d E is described. In the case of homogeneous broadening, this probability is given by L ( E ) d E , with the Lorentzian line shape centered at some peak transition energy E 0 :

L ( E ) = Γ ∕ 2 π ( E − E 0 ) 2 + Γ 2 ∕ 4 ,

(16)

where Γ is the full width at half-maximum (FWHM) that characterizes the broadening due to various homogeneous processes, including collisions and transitions. Taking this broadening into account, the transition rate in Eq. (15) is modified to an integral in which the δ function in Eq. (15) is replaced with the Lorentzian line shape, Eq. (16):

g 12 = 2 π ℏ ∫ | H m | 2 δ ( E − ℏ ω ) L ( E ) d E = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 0 ) + Γ 2 ∕ 4 .

(17)

In the presence of an EM field with a magnetic vector potential vector A in a medium with isotropic effective mass, the perturbation Hamiltonian H ex that describe the interaction between the field and the electron in isotropic subbands is

H ex = e A ⋅ P m e * ,

(18)

where P is the momentum operator.

From Eq. (18), it is not difficult to see that the selection rules for ISTs in the conduction band are such that only those EM fields that are polarized in the growth direction ( z ) can induce optical transitions. The transition matrix element can then be given as

H m = e A m e * P 12 ,

(19)

where the momentum matrix element

P 12 = ⟨ φ 1 | − j ℏ ∂ ∂ z | φ 2 ⟩

(20)

is evaluated as the envelope function overlap between the two subbands, which is related to the dipole matrix element [36]

z 12 = ⟨ φ 1 | z | φ 2 ⟩ = i ℏ m 0 E 12 P 12

(21)

and to the oscillator strength [37]

f 12 = 2 m 0 m e * 2 E 12 | P 12 | 2 .

(22)

It is not difficult to see from Eq. (17) that the transition rate induced by an EM field between two eigenstates is the same for upward and downward transitions. Now let us apply Eq. (17) to ISTs between the upper subband 2 and lower subband 1 in the conduction band (Fig. 3). Since momenta associated with photons are negligible, all photon-induced transitions are vertical in k space. It is therefore possible to obtain a net downward transition rate (in the units of number of transitions per unit time per unit sample area) between the two subbands by evaluating the following integral:

g net = ∫ g 12 { f 2 ( E 2 , k ) [ 1 − f 1 ( E 1 , k ) ] − f 1 ( E 1 , k ) [ 1 − f 2 ( E 2 , k ) ] } ρ r ( E 2 , k − E 1 , k ) d ( E 2 , k − E 1 , k ) ,

(23)

where f 1 ( E 1 , k ) and f 2 ( E 2 , k ) are the electron occupation probabilities of those states at the same k in subbands 1 and 2, respectively, and ρ r ( E 2 , k − E 1 , k ) is the reduced density of states between E 1 , k and E 2 , k . The density of states of two parallel subbands characterized by the same effective mass m e * are equal, ρ 1 = ρ 2 = m e * ∕ π ℏ 2 . Since the Lorentzian line shape in Eq. (23) should be much broader than the spread of the energy transitions between the two parallel subbands, the latter can be approximated as sharply centered at the subband separation at their energy minima E 12 = E 2 − E 1 . Thus,

g net = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 [ ∫ f 2 ( E 2 , k ) ρ 2 d E 2 , k − ∫ f 1 ( E 1 , k ) ρ 1 d E 1 , k ] = 2 π ℏ | H m | 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,

(24)

where N 1 and N 2 are the total electron densities in subband 1 and 2 per unit area, respectively. The optical gain coefficient γ , describing the increase of the EM field intensity, I , as γ = I − 1 d l ∕ d z , can be defined as the power increase per unit volume divided by the intensity. This in turn can be expressed in terms of the net downward transition rate, Eq. (24), by using the momentum and dipole matrix element relation

γ ( ω ) = g net ℏ ω I L p ,

(25)

where L p is the length of the QW structure, which is equal to the length of one period in the case of QCLs. To relate the EM field intensity l that propagates in plane to the optical potential A polarized along z , a real expression for the potential A has to be used:

A = A 0 cos ( β ⋅ r − ω t ) z ̂ = 1 2 A 0 z ̂ [ e j ( β ⋅ r − ω t ) + e − j ( β ⋅ r − ω t ) ] ,

(26)

where β is the in-plane propagation wave vector of the EM field. Only one of the two terms on the right-hand side of Eq. (26) couples with subbands 1 and 2, E 2 − E 1 = ℏ ω . Thus, the optical potential that participates in the transition matrix Eq. (19), is only half of its real amplitude, A = A 0 ∕ 2 . Since the EM field intensity I is related to the optical potential amplitude A 0 as I = ε 0 c n eff A 0 2 ω 2 ∕ 2 , Eq. (25) can be written as

γ ( ω ) = e 2 | P 12 | 2 2 ε 0 c n eff m e * 2 ω L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) = e 2 m 0 2 ω z 12 2 2 ε 0 c n eff m e * 2 L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,

(27)

where c is the speed of light in free space and n eff is the effective index of refraction of the QCL dielectric medium. The population inversion N 2 − N 1 > 0 is clearly necessary in order to achieve positive gain, which peaks at the frequency ω 0 = E 12 ∕ ℏ with a value of

γ ( ω 0 ) = 2 e 2 m 0 2 ω z 12 2 ε 0 c n eff m e * 2 Γ L p ( N 2 − N 1 ) .

(28)

For transitions between valence subbands with nonparallel dispersions and strong mixing between HH, LH, and SO bands, we have to reexamine the IST rate. Consider the IST in Fig. 5 from the upper state | u ⟩ in subband LH1 to the lower state | l ⟩ in subband HH1; if the spread of IST energies is wide compared with the homogeneous broadening, the Lorentzian line shape in the net downward transition rate, Eq. (23), can be approximated as a δ function, yielding

g net ( v ) = 2 π ℏ | H m ( v ) | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − F HH ( E h ) ] | E l − E h = ℏ ω .

(29)

Here ρ r ( E l − E h ) is the reduced density of states for the transition between subbands LH1 and HH1, f LH ( E l ) and f HH ( E h ) are hole occupation probabilities at states with energy E l and E h in subband LH1 and HH1, respectively, at the same in-plane wave vector k separated by a photon energy ℏ ω . The optical transition matrix element between LH1 and HH1, taking the mixing into account, is given by

H m ( v ) = ∑ n = 1 6 e A m n * ⟨ χ n ( l ) | − j ℏ ∂ ∂ z | χ n ( h ) ⟩ ,

(30)

where χ n ( l ) and χ n ( h ) are, respectively the n th component of the envelope function vectors for subband LH1 and HH1 as defined in Eq. (12), and m n * are the corresponding hole effective mass in the z direction with m 1 , 4 * = m 0 ∕ ( γ 1 − 2 γ 2 ) for HH, m 2 , 3 * = m 0 ∕ ( γ 1 + 2 γ 2 ) for LH, and m 5 , 6 * = m 0 ∕ γ 1 for SO. The optical gain can then be expressed as

γ ( v ) = π e 2 ε 0 c n eff ω L p | ∑ n = 1 6 P n ( l h ) m n * | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − f HH ( E h ) ] | E l − E h = ℏ ω = π e 2 m 0 2 ω ε 0 c n eff L p | ∑ n = 1 6 z n ( l h ) m n * | 2 ρ r ( E l − E h ) | [ f LH ( E l ) − f HH ( E h ) ] | E l − E h = ℏ ω .

(31)

The gain is thus given in terms of momentum matrix element p n ( l h ) = ⟨ χ n ( l ) | − j ℏ ∂ ∕ ∂ z | χ n ( h ) ⟩ as well as dipole matrix element z n ( l h ) = ⟨ χ n ( l ) | z | χ n ( h ) ⟩ between the same n th component of the envelope function vectors of the two valence subbands.

In comparison with the optical gain, Eq. (28), for the conduction subbands, we can see that it is not necessary to have total population inversion, N l − N h > 0 , in order to have positive gain between the valence subbands. Instead, all that is required is local population inversion | [ f LH ( E l ) − f HH ( E h ) ] | E l − E h = ℏ ω > 0 in the region where the IST takes place (those states near | u ⟩ and | l ⟩ in Fig. 5).

3.3. Intersubband Lifetimes

It has been established in Eqs. (27, 28) that the population inversion between the upper (2) and lower (1) subbands, N 2 − N 1 > 0 , is necessary in order to obtain optical gain. But what determines the population inversion? This question is answered with the analysis of the lifetimes of these subband states as a result of various intersubband relaxation mechanisms, including carrier–phonon, carrier–carrier, impurity, and interface roughness scattering processes. Among them, phonon scattering is the dominant process, especially when the energy separation between the two subbands exceeds that of an optical phonon, since the transitions from the upper to the lower subband with the emission of optical phonons are highly efficient. Unlike the optical transitions, the phonon scattering processes do not necessarily occur as vertical transitions in k space. In the case of phonon scattering, the conservation of in-plane momenta can be satisfied by a wide range of phonon momenta, as shown in Fig. 6(a), where intersubband as well as intrasubband transitions due to phonon scattering are illustrated.

Fig. 6.
Intersubband and intrasubband transitions due to electron–phonon scattering. (b) 22 → 11 transition induced by the electron–electron scattering.

Until now, practically all approaches to developing Si-based QCLs are based on materials from group IV, mostly Si, Ge, SiGe alloy, and, more recently, SiGeSn alloy. Different from the polar III–V and II–VI semiconductors, group-IV materials are nonpolar. The carrier scatterings by nonpolar optical phonons are much slower than those due to polar optical phonons [38]. Starting from the Fermi golden rule, Eq. (15), the scattering rate for a carrier in subband 2 with the in-plane wave vector k to subband 1 with k ′ by a phonon with an energy ℏ ω Q and wave vector Q = q + q z z ̂ can be expressed as an integral over all the participating phonon states:

1 τ 12 = 2 π ℏ ∫ | H e p | 2 δ ( E 2 , k − E 1 , k ′ ∓ ℏ ω Q ) d N f ,

(32)

where H e p is the electron–phonon interaction matrix element. The carrier energies E 1 , k ′ and E 2 , k are given by Eq. (1) for conduction subbands, but for valence subbands they need to be obtained by the k ⋅ p method described above. In general, the calculation of phonon scattering of holes and indirect-valley electrons is more complex than that of the Γ -valley electrons. We will proceed with the following approximations: (1) all phonons are treated to be bulklike by neglecting the phonon confinement effect in QW structures; (2) energies of acoustic phonons are negligible, ℏ ω Q ≈ 0 ; and (3) optical phonon energies are taken as a constant, ℏ ω Q ≈ ℏ ω 0 . The matrix element of carrier–phonon interaction for different types of phonon can be written as [39, 40]

| H e p | 2 = { Ξ 2 K B T 2 c L Ω δ q , ± ( k ′ − k ) | G 12 ( q z ) | 2 acoustic phonon ℏ D 2 2 ρ ω 0 Ω δ q , ± ( k ′ − k ) [ n ( ω 0 ) + 1 2 ∓ 1 2 ] | G 12 ( q z ) | 2 nonpolar optical phonon } ,

(33)

where the upper sign is for absorption and the lower for emission of one phonon, K B is the Boltzmann constant, Ω is the volume of the lattice mode cavity, c L is the elastic constant for acoustic mode, Ξ and D are the acoustic and optical deformation potential, respectively, and n ( ω 0 ) is the number of optical phonons at temperature T ,

n ( ω 0 ) = 1 exp ( ℏ ω 0 ∕ K B T ) − 1 .

(34)

The wavefunction interference effect between conduction subbands is

G 12 ( q z ) = ⟨ φ 1 | e j q z z | φ 2 ⟩

(35)

and between valence subbands is

G 12 ( q z ) = ∑ n = 1 6 ⟨ χ n ( 1 ) | e j q z z | χ n ( 2 ) ⟩ .

(36)

The Kronecker symbol δ q , ± ( k ′ − k ) in the matrix element of Eq. (33) represents the in-plane momentum conservation k ′ = k ± q .

Since phonon modes have densities of states Ω ∕ ( 2 π ) 3 , the participating phonon states in the integral of Eq. (32) can be expressed as

d N f = Ω ( 2 π ) 3 q d q d θ d q z ,

(37)

where θ is the angle between k and q . For conduction subbands with the parabolic dispersion given in Eq. (1), the phonon scattering rate, Eq. (32), can be evaluated analytically:

1 τ 12 = { Ξ 2 K B T m e * 4 π c L ℏ 3 ∫ | G 12 ( q z ) | 2 d q z acoustic D 2 m e * [ n ( ω 0 ) + 1 2 ∓ 1 2 ] 4 π ρ ℏ 2 ω 0 ∫ | G 12 ( q z ) | 2 d q z nonpolar optical } .

(38)

For valence subbands where there is a strong nonparabolicity, Eq. (32) can no longer be integrated analytically. However, if we take the wave vector of the initial state in subband 2 to be at the Γ point, k = 0 , then the phonon wave vector is q = k ′ . Equation (38) can then be used to evaluate the phonon scattering rate between valence subbands by replacing the effective mass with some average effective mass in the final subband 1.

The phonon scattering rate in Eq. (38) has been used to compare the lifetimes of two similar three-level systems, Si Ge ∕ Si and Ga As ∕ Al Ga As , as shown in Fig. 7(a)[38]. The lifetime difference between the upper (3) and lower (2) subband is calculated as the function of the transition energy E 3 − E 2 which is varied by changing the barrier width between the two QWs that host the two subbands. The main result, shown in Fig. 7(b), is that the lifetimes in the SiGe system can be an order of magnitude longer than in the Ga As ∕ Al Ga As system because of SiGe’s lack of polar optical phonons. This property can potentially lead to a significantly reduced lasing threshold for the SiGe system. The sudden drops in the lifetimes have to do with the shifting of subband energy separations E 2 − E 1 and E 3 − E 2 to either below or above the optical phonon energy.

Fig. 7.
HH valence band diagram of one period of a Si Ge ∕ Si SL with a hole energy increase in the upward direction. (b) Comparison of lifetime difference ( τ 3 − τ 2 ) between the Si Ge ∕ Si and Ga As ∕ Al Ga As SL (in a similar three-level scheme) as a function of the transition energy ( E 3 − E 2 ) [38].

Among the different phonon scattering processes—emission and absorption of acoustic and optical phonons—the emission of an optical phonon is by far the fastest process. But in far-IR QCLs where the subband energy separation is less than the optical phonon energy and the emission of an optical phonon is forbidden, phonon scattering may no longer be the dominant relaxation mechanism.

Other scattering mechanisms need to be taken into consideration, such as the carrier–carrier [41], impurity [42], and interface roughness scatterings [43], all of which are elastic processes. The carrier–carrier scattering is a two-carrier process that is particularly important when carrier concentration is high, increasing the probability of two carriers interacting with each other. There are many possible outcomes as a result of this interaction in inducing intersubband as well as intrasubband transitions. Among them, the 22 → 11 process, in which both carriers originally in subband 2 end up in subband 1, is the most efficient one in terms of inducing ISTs [Fig. 6(b)]. It has been reported that IST times of the order of tens of picoseconds have been measured for carrier densities of 10 9 – 10 11 ∕ cm 2 in Ga As ∕ Al Ga As QWs [44]. In QCLs, in which doping is introduced mostly away from the active region where optical transitions take place, impurity scattering does not seem to play a major role in determining the lifetimes of laser subbands. However, its influence on carrier transport in the injection region can be rather important. Interface roughness depends strongly on the process of structural growth, so its effect on scattering should be more significant in narrow QWs, particularly for those involving transitions between two wavefunctions that are localized in multiple QWs spanning several interfaces.

4. Valence Band SiGe Quantum Cascade Lasers

Until now, all of the demonstrated QCLs have been based on epitaxially grown type-III–V semiconductor heterostructures such as Ga In As ∕ Al In As , Ga As ∕ Al Ga As , and In As ∕ Al Sb , using electron subbands in the conduction band. With the promise of circumventing the indirectness of the Si bandgap, a Si Ge ∕ Si laser based on ISTs was first proposed in 1995 [38] by Sun et al., who conducted a comparative study between the Si Ge ∕ Si and Ga As ∕ Al Ga As systems. Since then there has been a series of theoretical and experimental investigations aimed at producing Si-based QCLs. A natural choice of the material system is SiGe because Si and Ge are both group-IV elements. SiGe alloys have been routinely deposited on Si to produce heterojunction bipolar transistors or to form a strain-inducing layer for CMOS transistors [45].

While QCLs based on SiGe alloys can be monolithically integrated on Si, there are significant challenges associated with this material system. First, there is a 4% lattice mismatch between Si and Ge. Layers of Si 1 − x Ge x alloys deposited on Si substrates induce strain that can be rather significant in QCLs, in which a working structure typically consists of at least hundreds of layers. The total thickness may easily exceed the critical thickness above which the built-in strain simply relaxes to develop defects in the structure. In dealing with the issue of strain in Si Ge ∕ Si quantum cascade structures, one popular approach is to use strain-balanced growth where the compressively strained Si 1 − x Ge x and tensile strained Si are alternately stacked on a relaxed Si 1 − y Ge y buffer deposited on a Si substrate. The buffer composition ( y < x ) is chosen to produce strains in Si 1 − x Ge x and Si that compensate for each other, so that the entire structure maintains a neutral strain profile [46, 47]. Strain-balanced structures have effectively eliminated the limitations of critical thickness and produced high-quality Si Ge ∕ Si structures consisting of nearly 5000 layers ( 15 μ m ) by chemical vapor deposition [48].

Second, the band offsets between compressively strained SiGe and tensile strained Si or between SiGe of different alloy compositions is such that the conduction-band QWs are shallow, and nearly all band offsets are in the valence band. Practically all of the investigations of SiGe QCLs are focused on ISTs in the valence band. But the valence subband structure is much more complex than that of the conduction subband because of the mixing between the HH, LH, and SO bands. Their associated subbands are closely intertwined in energy, making the design of valence QCLs extremely challenging. Third, any valence QCL design inevitably involves HH subbands, since they occupy lower energies relative to LH subbands because of their large effective mass. In SiGe, the HH effective mass is high ( ∼ 0.2 m 0 ) , which leads to small IST oscillator strengths between the laser states and poor carrier transport behavior associated with their low mobilities.

The challenge presented by the valence-band mixing also creates an opportunity to engineer desirable subband dispersions such that total population inversion between the subbands becomes unnecessary, in a manner analogous to the situation in conventional band-to-band lasers discussed in Section 2. It has been reported that in QCLs the population inversion was established only locally in k space, in the large k -vector region of the conduction subbands, because the interactions between the subbands produced nonparallel in-plane dispersions [49]. Band nonparabolicity is found to be much pronounced in the valence band than in the conduction band [31]. As a matter of fact, in the valence band of most diamond and zinc blende semiconductors, LH and HH subbands usually anticross, and near the point of anticrossing, the LH subband in-plane dispersion becomes electronlike. Thus, an earlier design was accomplished to effectively tailor the dispersions of two valence subbands in a Ga As ∕ Al Ga As QW (Fig. 5) similar to those of the conduction and valence bands, in which one of the subbands is electronlike and the other holelike; i.e., one of the subbands will have its effective mass inverted [35]. 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 in Fig. 5), we can see that the arrangement closely resembles the one in the conventional band-to-band semiconductor laser. The upper and lower laser states can now be populated and depopulated through fast intrasubband processes, while the lifetime of the upper laser states is determined by a much slower intersubband process between subbands LH1 and HH1. Such a large lifetime difference between the upper and lower laser states is certainly favorable for achieving population inversion between them.

The inverted mass approach was later applied to the SiGe system [50, 51]. Two slightly different schemes were developed, one utilizing the inverted LH effective mass [50], and the other the inverted HH mass [51]. In both cases, the effective mass inversion is the result of strong interaction between the valence subbands. The inverted-effective-mass feature requires the coupled subbands to be closely spaced in energy, typically less than all the optical phonon energies in the SiGe material system ( 37 meV for the Ge–Ge mode, 64 meV for the Si–Si mode, and 51 meV for the Si–Ge mode [52]), suppressing the nonradiative ISTs due to the optical phonon scattering, but also limiting the optical transitions in the terahertz regime. The structures under investigation were strain balanced with compressively strained Si 1 − x Ge x QW layers and the tensile strained Si barrier layers deposited on a relaxed Si 1 − y Ge y buffer layer ( 0 < y < x ) on Si. The in-plane dispersions of the inverted LH scheme are shown in Fig. 8 for a 90 Å ∕ 50 Å Si 0.7 Ge 0.3 ∕ Si SL. The three lowest subbands are shown. The numbers 1, 2, 3, and 4 and the arrows indicate how this inverted mass intersubband laser mimics the operation of a conventional band-to-band laser. The lifetime of the upper laser state 3 is long because the IST energy at 6 THz ( ∼ 50 μ m ) is below that of optical phonons, allowing only weak acoustic phonon scattering between the two subbands. Calculation results have shown that optical gain in excess of 150 ∕ cm can be achieved without total population inversion’s being established between the LH and HH subbands.

Fig. 8.
Dispersions of subbands HH1, LH1, and HH2 in a 90 Å ∕ 50 Å Si 0.7 Ge 0.3 ∕ Si SL strained balanced on a Si 0.81 Ge 0.19 buffer, obtained with a 6 × 6 valence band matrix, taking into account HH, LH, and SO interactions and strain effect [50].

The inverted LH effective mass approach utilizes optical transitions between the LH and HH subband. It can be argued from the component overlap of the envelope functions in Eq. (30) that the optical transition matrix elements between subbands of different types are always smaller than those between subbands of the same type. We therefore tried to engineer the same inverted-effective-mass feature between two HH subbands. The challenge is to lift the LH subband above the HH2 subband. Once again, a strain-balanced SL structure is considered but with different SiGe alloy compositions and layer thicknesses [51]. The band structure for a 90 Å ∕ 35 Å Si 0.8 Ge 0.2 ∕ Si SL, under an electric bias of 30 kV ∕ cm , is shown in Fig. 9(a), where each QW has two active doublets formed by bringing HH1 and HH2 subbands in the neighboring QWs into resonance under the bias. There is a 3 meV energy split within the doublet. The resulting in-plane dispersions for the two doublets are shown in Fig. 9(b). Simulation results have shown that optical gain of 450 ∕ cm at 7.3 THz can be achieved at a pumping current density of 1.5 kA ∕ cm 2 at 77 K .

Fig. 9.
Band diagram of the Si 0.8 Ge 0.2 ∕ Si SL under an electric bias of 30 kV ∕ cm . The labels ( n − 1 , n , n + 1 , … ) represent the QWs in which the wave functions are localized [51]. (b) Dispersions of the four levels (two doublets) in a QW.

Electroluminescence (EL) from a Si Ge ∕ Si quantum cascade emitter was first demonstrated in a Si Ge ∕ Si quantum cascade emitter using HH to HH transitions in the mid-IR range in 2000 [53]. Since then, several groups have observed EL from the same material system with different structures. EL emissions have been attributed to various optical transitions, including HH to HH [54], LH to HH [55], and HH to LH [56], with emission spectra ranging from mid-IR to terahertz ( 8 – 250 μ m ) . However, lasing has not been observed. Improvements in the QCL design have been made. One of the most successful III–V QCL designs has utilized the approach of bound to continuum, where the lower laser state sitting at the top of a miniband is delocalized over several QWs while the upper laser state is a bound state in the minigap, as illustrated in Fig. 10[57, 58]. Electrons injected into the bound upper state 2 are prevented from escaping the bound state by the minigap and undergo lasing transitions to the lower state 1. The depopulation of the lower state 1 is accelerated through the efficient miniband carrier transport. Such a design has led to improved performance in terms of operating temperature as well as output power for III–V QCLs. A similar bound-to-continuum design has been implemented in SiGe with both bound and continuum formed by HH states, but once again showing just EL with no lasing [59]. It is believed that in this structure LH states are mixed within the HH states. Although the effect of this intermixing has not been fully understood, these LH states can in principle present additional channels for carriers to relax from the upper laser state reducing its lifetime. An improved version has been sought that utilizes strain to lift the LH states out of all involved HH states for bound and continuum. As a result, a clear intersubband TM polarized EL has been obtained, suggesting that LH states have been pushed away from the HH radiative transitions [60].

Fig. 10.
Illustration of two periods of a bound-to-continuum QCL. Lasing transition occurs between an isolated bound upper state 2 (formed in the minigap) and a delocalized lower state 1 (sitting on top of a miniband).

Nearly a decade has passed since the first experimental demonstration of EL from a Si Ge ∕ Si quantum cascade emitter [53]. During this period, III–V QCLs have been dramatically improved to allow for commercialization and system integration for various applications. However, there are still no SiGe QCLs. The seemingly inherent difficulties with the valence QCL approach have impelled some researchers to return to the conduction band to look for solutions.

5. Conduction Band Si-Based Quantum Cascade Lasers

Before QCLs can be designed by using conduction subbands, there must be sufficient conduction-band offset. Contrary to the situation in compressively strained Si 1 − x Ge x , tensile strained Si 1 − x Ge x can have larger conduction-band offsets, but the conduction-band minima occur at the two Δ 2 valleys whose effective mass (longitudinal) along the growth direction is very heavy ( m l ∼ 0.9 m 0 ) . This results in small oscillator strengths and poor transport behavior, possibly even worse than the HHs. Any approach to developing Si-based QCLs based on transitions between conduction subbands necessarily needs to go beyond conventional methods of selecting the material system and growth technique.

Prospects of developing such Si-based QCLs have been investigated theoretically. One approach stayed with the Si-rich Si Ge ∕ Si material system, but instead of the conventional growth direction (100), the structural orientation has been rotated to the [111] crystal plane [61]. Conduction band offset was calculated to be 160 meV at the conduction-band minima consisting of six-degenerate Δ -valleys, sufficient for designing far-IR QCLs. The effective mass along the (111) direction can be obtained as the geometric average of the longitudinal and transverse effective masses of the Δ valley, ∼ 0.26 m 0 , lower than that of longitudinal m l ∼ 0.9 m 0 in the (100) structure. Another design, relying on the Ge-rich Ge ∕ Si Ge material system, has been proposed to construct conduction-band QCLs by using compressively strained Ge QWs and tensile strained Si 0.22 Ge 0.78 alloy barriers grown on a relaxed [100] Si 1 − y Ge y buffer [62]. The ISTs in this design are within the L valleys, which are the conduction-band minima in Ge QWs whose effective mass along the (100) direction has been determined to be ∼ 0.12 m 0 . Since Si 1 − x Ge x alloys with x < 0.85 are similar to Si in that the conduction-band minima appear in the Δ valleys, the conduction-band lineup in the Ge ∕ Si 0.22 Ge 0.78 structure is rather complex, with conduction-band minima in Ge at L valleys but in Si 0.22 Ge 0.78 at Δ 2 valleys along the (100) growth direction. Although the band offset at the L valleys is estimated to be as high as 138 meV , the overall band offset between the absolute conduction-band minima in Ge and Si 0.22 Ge 0.78 is only 41 meV . Although the quantum confinement effect helps to lift those electron subbands at Δ 2 valleys, the two Δ 2 valleys are inevitably entangled with the L valleys in the conduction band, leading to design complexity and potentially creating additional nonradiative decay channels for the upper laser state.

Recently, a new group-IV material system that expands beyond the Si 1 − x Ge x alloys has been successfully demonstrated with the incorporation of Sn. These new ternary Ge 1 − x − y Si x Sn y alloys have been studied for the possibility of forming direct bandgap semiconductors [63, 64, 65, 66]. Since the first successful growth of this alloy [67], device-quality epilayers with a wide range of alloy contents have been deposited [68, 69]. Incorporation of Sn provides the opportunity to engineer separately the strain and band structure since the Si (x) and Sn (y) compositions can be varied independently. Certain alloy compositions of this material system offer three advantages: (1) the possibility of a cleaner conduction-band lineup in which the L valleys in both well and barrier sit below other valleys ( Γ , Δ ) , (2) an electron effective mass along the (001) growth direction that is much lower than the HH mass, and (3) a strain-free structure that is lattice matched to Ge. In addition, recent advances in the direct growth of a Ge layer on Si provide a relaxed matching buffer layer on a Si substrate upon which the strain-free Ge ∕ Ge 1 − x − y Si x Sn y is grown [70]. Based on this material system, a strain-free QCL operating in the conduction L valleys has been proposed [71].

Since band offsets between ternary Sn-containing alloys and Si or Ge are not known experimentally, we have calculated the conduction-band minima for a lattice-matched heterostructure, consisting of Ge and ternary Ge 1 − x − y Si x Sn y , based on Jaros’s band offset theory [72], which is in good agreement with experiment for many heterojunction systems. For example, this theory predicts an average valence band offset Δ E v , av = 0.48 eV for a Ge ∕ Si heterostructure (higher energy on the Ge side), close to the accepted value of Δ E v , av = 0.5 eV . The basic ingredients of our calculation are the average (between HH, LH, and SO bands) valence band offset between the two materials and the compositional dependence of the band structure of the ternary alloy. For the Ge ∕ α - Sn interface, Jaros’ theory predicts Δ E v , av = 0.69 eV (higher energy on the Sn side). For the Ge 1 − x − y Si x Sn y ∕ Ge interface we have used the customary approach for alloy semiconductors, interpolating the average valence band offsets for the elementary heterojunctions Ge ∕ Si and Ge ∕ α - Sn . Thus we used (in electron volts)

Δ E v , av ( x , y ) = E v , av ( Ge Si Sn ) − E v , av ( Ge ) = 0.48 x + 0.69 y .

(39)

Once the average valence band offset is determined, the energies of individual conduction-band edges in the Ge 1 − x − y Si x Sn y alloy can be calculated relative to those in Ge from the compositional dependence of the spin-orbit splitting of the top valence band states and the compositional dependence of the energy separations between those conduction-band edges and the top of the valence band in the alloy [73]. We have assumed that all required alloy energies can be interpolated between the known values for Si, Ge, and α - Sn as

E Ge Si Sn ( x , y ) = E Ge Si Sn ( 1 − x − y ) + E Si x + E Sn y − b Ge Si ( 1 − x − y ) x − b Ge Sn ( 1 − x − y ) y − b Si Sn x y .

(40)

The bowing parameters b Ge Si , b Ge Sn , and b Si Sn have been discussed in [74, 75]. Finally, for the indirect conduction-band minimum near the X point, Weber and Alonso find

E x = 0.931 + 0.018 x + 0.206 x 2

(41)

(in electron volts) for Ge 1 − x Si x alloys [76]. On the other hand, the empirical pseudopotential calculations of Chelikovsky and Cohen place this minimum at 0.90 eV in α - Sn , virtually the same as its value in pure Ge [77]. We thus assume that the position of this minimum in ternary Ge 1 − x − y Si x Sn y alloys is independent of the Sn concentration y . The conduction-band minima results are shown in Fig. 11 for Sn concentrations 0 < y < 0.1 . The Si concentration x was calculated by using Vegard’s law in such a way that the ternary Ge 1 − x − y Si x Sn y is exactly lattice matched with Ge.

Fig. 11.
Conduction band minima at L , Γ , X points of Ge 1 − x − y Si x Sn y that is lattice matched to Ge [71].

It can be seen from Fig. 11 that a conduction-band offset of 150 meV at L valleys can be obtained between lattice-matched Ge and Ge 0.76 Si 0.19 Sn 0.05 alloy while all other conduction-band valleys ( Γ , X , etc) are above the L -valley band edge of the Ge 0.76 Si 0.19 Sn 0.05 barrier. This band alignment presents a desirable alloy composition from which a QCL operating at L valleys can be designed by using Ge as QWs and Ge 0.76 Si 0.19 Sn 0.05 as barriers without the complexity arising from other energy valleys.

Figure 12 shows the QCL structure based on Ge ∕ Ge 0.76 Si 0.19 Sn 0.05 QWs. Only L -valley conduction-band lineups are shown in the potential diagram under an applied electric field of 10 kV ∕ cm . To solve the Schrödinger equation to yield subbands and their associated envelope functions, it is necessary to determine the effective mass m z * along the (001) growth direction ( z ) within the constant-energy ellipsoids at the L valleys along the (111) direction, which is tilted with respect to (100). Using the L -valley principal transverse effective mass m t * = 0.08 m 0 and the longitudinal effective mass m l * = 1.60 m 0 for Ge, we obtain m z * = ( 2 ∕ 3 m t * + 1 ∕ 3 m l * ) − 1 = 0.12 m 0 . The squared magnitudes of all envelope functions are plotted at energy positions of their associated subbands. As shown in Fig. 12, each period of the QCL has an active region for lasing emission and an injector region for carrier transport. These two regions are separated by a 30 Å barrier. The active region is constructed with three coupled Ge QWs that give rise to three subbands marked 1, 2, and 3. The lasing transition at the wavelength of 49 μ m is between the upper laser state 3 and the lower laser state 2. The injector region consists of four Ge QWs of decreasing well widths, all separated by 20 Å Ge 0.76 Si 0.19 Sn 0.05 barriers. The depopulation of lower state 2 is through scattering to state 1 and to the miniband downstream formed in the injector region. These scattering processes are rather fast because of the strong overlap between the involved states. Another miniband in the injector region formed of quasi-bound states is situated 45 meV above the upper laser state 3, effectively preventing escape of electrons from upper laser state 3 into the injector region.

Fig. 12.
L -valley conduction-band profile and squared envelope functions under an electric field of 10 kV ∕ cm . Layer thicknesses in ångströms are marked with bold numbers for Ge QWs and regular numbers for GeSiSn barriers. The array marks the injection barrier [71].

The nonradiative transition rates between different subbands in such a low-doped nonpolar material system with low injection current should be dominated by deformation-potential scattering of nonpolar optical and acoustic phonons. For this Ge-rich structure, we have used bulk-Ge phonons for the calculation of the scattering rate to yield lifetimes for the upper laser state τ 3 and the lower laser state τ 2 , as well as the 3 → 2 scattering time τ 32 [78]. The results obtained from Eq. (38) are shown in Fig. 13 as a function of operating temperature. These lifetimes are at least 1 order of magnitude longer than those of III–V QCLs owing to the nonpolar nature of GeSiSn alloys. The necessary condition for population inversion τ 32 > τ 2 is satisfied throughout the temperature range. Using these predetermined lifetimes in the population rate equation under current injection,

∂ N 3 ∂ t = η J e − N 3 − N ¯ 3 τ 3 ,

∂ N 2 ∂ t = N 3 − N ¯ 3 τ 32 − N 2 − N ¯ 2 τ 2 ,

∂ N 3 ∂ t = η J e − N 3 − N ¯ 3 τ 3 ,

∂ N 2 ∂ t = N 3 − N ¯ 3 τ 32 − N 2 − N ¯ 2 τ 2 ,

(42)

where N i ( i = 2 , 3 ) is the area carrier density per period in subband i under injected current density J with an injection efficiency η , and N ¯ i is the area carrier density per period due to thermal population. Solving the above rate equation at steady state yields the population inversion

N 3 − N 2 = τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ,

(43)

which can then be used to evaluate the optical gain of the TM-polarized mode, following Eq. (28), at the lasing transition energy ℏ ω 0 = E 3 − E 2 = 25 meV , as

γ ( ω 0 ) = 2 e 2 m 0 2 ω 0 z 23 2 ε 0 c n eff m z * 2 Γ L p [ τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ] .

(44)

For the QCL structure in Fig. 12, the following parameters are used: index of refraction n eff = 3.97 , lasing transition FWHM Γ = 10 meV , length of one period of the QCL L p = 532 Å , area doping density per period of 10 10 cm 2 , and unit injection efficiency η = 1 .

Fig. 13.
Upper-state lifetime τ 3 , lower-state lifetime τ 2 , and scattering time τ 32 between them as a function of temperature.

Since the relatively small conduction-band offset limits the lasing wavelength to the far-IR or terahertz regime (roughly 30 μ m and beyond), the waveguide design can no longer rely on that of conventional dielectric waveguides such as those used in laser diodes and mid-IR QCLs. This is mainly because the thickness required for the dielectric waveguide would exceed what can be realized with the epitaxial techniques employed to grow the laser structures. One solution is to place the QCL active structure between two metal layers to form a so-called plasmon waveguide [79, 80]. While the deposition of the top metal is trivial, placing bottom metal requires many processing steps such as substrate removal, metal deposition, and subsequent wafer bonding. The QCL waveguides are typically patterned into ridges, as shown in Fig. 14.

Fig. 14.
Schematic of a ridge plasmon waveguide with the Ge ∕ Ge Si Sn QCL sandwiched between two metal layers.

This plasmon waveguide supports only the TM polarized EM mode that is highly confined within the QCL region, − d ∕ 2 < z < d ∕ 2 . We assume the Drude model to describe the metal dielectric function

ε M = 1 − ω p 2 ω 2 + j γ m ω ,

(45)

where ω p is the metal plasmon frequency, and γ m is the metal loss ( ℏ ω p = 8.11 eV , ℏ γ m = 65.8 meV for Au [81]), and ε D = n eff 2 ≈ 16 for the Ge-rich Ge ∕ Ge 0.76 Si 0.19 Sn 0.05 QCL active region. Consider the EM wave propagating along the x direction as shown in Fig. 14; its electric field can be obtained as

E = { E 0 ε cosh ( k d 2 ) ( j β z ̂ + q x ̂ ) e − q ( z − d ∕ 2 ) e j ( β x − ω t ) z > d 2 E 0 [ j β cosh ( k z ) z ̂ − k sinh ( k z ) x ̂ ] e j ( β x − ω t ) | z | < d 2 E 0 ε cosh ( k d 2 ) ( j β z ̂ − q x ̂ ) e q ( z + d ∕ 2 ) e j ( β x − ω t ) z < − d 2 } ,

(46)

where ε = ε M ∕ ε D and E 0 is a constant. The complex propagation constant β = β ′ + j β ″ follows the relations β 2 − k 2 = k D 2 , β 2 − q 2 = ε k D 2 with k D = ε D ω ∕ c . It is easy to see that the continuity of the normal component of the electric displacement is satisfied at the boundaries z = ± d ∕ 2 ; the requirement of continuity of tangential electric field leads to

k 2 [ ε 2 tanh 2 ( k d 2 ) − 1 ] = k D 2 ( 1 − ε ) ,

(47)

which determines the TM modes that can propagate in this plasmon waveguide. The waveguide loss a w is dominated by the metal loss, which can be determined from the imaginary part of the propagation constant, as a w = 2 β ″ . As a superposition of two surface plasmon modes bound to the two metal–dielectric interfaces at z = ± d ∕ 2 , this TM mode decays exponentially into the metal, providing an excellent optical confinement factor defined as Γ w = ∫ − d ∕ 2 d ∕ 2 | E | 2 d z ∕ ∫ − ∞ ∞ | E | 2 d z . We have simulated the TM-polarized mode in a QCL structure of 40 periods ( d = 2.13 μ m ) that is confined by a double-Au-plasmon waveguide and obtained near-unity optical confinement Γ w ≈ 1.0 and waveguide loss α w = 110 ∕ cm . Assuming a mirror loss α m = 10 ∕ cm for a typical cavity length of 1 mm , the threshold current density J th can be calculated from the balancing relationship, Γ w γ th = α w + α m , where γ th is the optical gain, Eq. (44), obtained at the threshold J th . The result is shown in Fig. 15 for J th that ranges from 22 A ∕ cm 2 at 5 K to 550 A ∕ cm 2 at 300 K . These threshold values are lower than those of III–V QCLs as a result of the longer scattering times due to nonpolar optical phonons.

Fig. 15.
Simulated threshold current density of the Ge ∕ Ge Si Sn QCL as a function of temperature.

While GeSiSn epilayers with alloy compositions suitable for this QCL design have been grown with metal organic chemical vapor deposition [68, 69], implementation of Ge ∕ Ge Si Sn QCLs is currently challenged by the structural growth of the large number of heterolayers in the QCL structure with very fine control of layer thicknesses and alloy compositions. Nevertheless, progress is being made towards experimental demonstration.

6. Summary

Si-based lasers have been long sought after for the possibility of monolithic integration of photonics with high-speed Si electronics. Many parallel approaches are currently taken to reach this goal. Among them, Si nanocrystals and Er-doped Si have been investigated rather extensively. While EL has been demonstrated, lasing has not been observed. The only reported lasing in Si so far has been achieved by using stimulated Raman scattering, which requires optical pumping at very high intensity on a device of large scale—impractical for integration with Si electronics. The QCLs that have been successfully developed in III–V semiconductors offer an important alternative for the development of Si-based lasers. The salient feature of QCLs is that lasing transitions take place between subbands that are within the conduction band without crossing the bandgap. Such a scheme makes the indirect nature of the Si bandgap irrelevant. To assist appreciation of the QCL designs, some theoretical background underlying the basic operating principles has been introduced here. In particular, subband formation and energy dispersion in semiconductor QWs are described in the framework of envelope functions with the effective-mass approximation for both conduction and valence band, taking into account mixing between HH, LH, and SO bands. Optical gain based on ISTs is derived, and intersubband lifetimes are discussed with a more detailed treatment of carrier-phonon scattering.

The development of Si-based QCLs has been primarily focused on ISTs between valence subbands in the Si-rich Si Ge ∕ Si material system. Such a material system has been routinely used in CMOS-compatible processes. There are two reasons for using holes instead of electrons. One is that compressively strained Si 1 − x Ge x with tensile strained Si grown on relaxed Si 1 − y Ge y has very small conduction-band offset; QWs are too shallow to allow elaborate QCLs. Tensile strained Si 1 − x Ge x , on the other hand, can have a larger conduction-band offset, but the conduction-band minima occur at the two Δ 2 valleys whose effective mass (longitudinal) along the growth direction is heavy ( m l ∼ 0.9 m 0 ) , resulting in small oscillator strength and poor transport behavior such as reduced tunneling probabilities. It is generally believed that SiGe QCLs have to be pursued within the valence band as a p-type device. But the situation in the valence band also presents challenges from several perspectives. First, the strong mixing of HH, LH, and SO bands makes the QCL design exceedingly cumbersome, albeit the opportunities presented by the strong nonparabolicity in valence subbands to take advantage of schemes such as the inverted effective mass where the total population inversion between subbands may not be needed. Second, there is a great deal of uncertainty in various material parameters for the SiGe alloy. Oftentimes, approximations have to be made to linearly extrapolate parameters from those of Si and Ge, reducing the accuracy and increasing the ambiguity of the designs. Third, any valence QCLs have no choice but to deal with HH subbands; their large effective mass hinders carrier injection efficiency and leads to small IST oscillator strengths between laser states. Fourth, for any significant band offset needed to implement QCLs, lattice-mismatch-induced strain in SiGe QWs and Si barriers, even in strain-balanced structures, is significant, presenting a challenge in structural growth and device processing. While EL was demonstrated from a valence-band Si Ge ∕ Si quantum cascade emitter nearly a decade ago, lasing remains elusive.

Recently, several ideas for developing Si-based conduction-band QCLs have emerged to circumvent the hurdles in the Si Ge ∕ Si valence-band approach. The proposals offer ways to increase the conduction-band offset and to reduce the effective mass along the growth direction. One scheme proposes to orient the structural growth along the (111) direction; another relies on ISTs in the L valleys of the conduction band in a Ge-rich Ge ∕ Si Ge material system. The former has led to advances in increasing the conduction-band offset, and the latter in reducing the effective mass. A third approach that expands the material system beyond SiGe to GeSiSn has been discussed in detail. A Ge ∕ Ge 0.76 Si 0.19 Sn 0.05 QCL that operates at L valleys of the conduction band was designed. According to our estimation of the band lineup, this particular alloy composition gives a clean conduction-band offset of 150 meV at L valleys with all other energy valleys conveniently out of the way. All QCL layers are lattice matched to a Ge buffer layer on a Si substrate, and the entire structure is therefore strain free. The electron effective mass along the growth direction is much lighter than that of heavy holes, bringing a significant improvement in tunneling rates and oscillator strengths. The lasing wavelength of this device is 49 μ m . With different GeSiSn alloy compositions that are lattice matched to Ge, QCLs can be tuned to lase at other desired wavelengths. Lifetimes determined from the deformation potential scattering of nonpolar optical and acoustic phonons are at least an order of magnitude longer than those in III–V QCLs with polar optical phonons, leading to a reduction in threshold current density and the possibility of room temperature operation. While there are considerable challenges in material growth of this QCL design, advances in fine control of structural parameters including layer thicknesses and alloy compositions are moving towards implementation of conduction-band QCLs in the GeSiSn system.

When are we going to realize Si-based lasers that can be integrated with Si electronics? Clearly, breakthroughs in material science and device innovation are necessary before that can happen, but with the variety of approaches that are being pursued—driven by the potential payoff in commercialization—the prospect is promising.

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[79] B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum cascade laser at λ ∼ 100 μ m using metal waveguide for mode confinement,” Appl. Phys. Lett. 83, 2124–2126 (2003). [CrossRef]

[80] K. Unterrainer, R. Colombelli, C. Gmachl, F. Capasso, H. Y. Hwang, A. M. Sergent, D. L. Sivco, and A. Y. Cho, “Quantum cascade lasers with double metal-semiconductor waveguide resonators,” Appl. Phys. Lett. 80, 3060–3062 (2002). [CrossRef]

[81] P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

Greg Sun graduated with B.S. from Peking University in 1984, M.S. from Marquette University in 1988, and Ph.D. from Johns Hopkins University in 1993, all in Electrical Engineering. In the early 1990s, he also worked as a research fellow at Philips Laboratories in Briarcliff Manor, New York, where he participated in the development of II–VI blue lasers. He joined the University of Massachusetts Boston in 1993 as an assistant professor and was promoted to full professor in 2004. He holds joint appointment in Physics and Engineering, and serves as the Director of the Engineering Program at the University of Massachusetts Boston. His research interests lie in the areas of semiconductor quantum structures and their associated physical processes, optoelectronic devices such as detectors and emitters including silicon-based lasers, and plasmon-enhanced phenomena and devices.

[ − ℏ 2 2 d d z 1 m e * ( z ) d d z + V c ( z ) ] φ i ( z ) = E i φ i ( z )
	(2)

φ i ( z ) = A l e j k z ( z − d l ) + B l e − j k z ( z − d l ) ,
	(4)

φ i ( z ) and 1 m e * ( z ) d φ i ( z ) d z continous ,
	(5)

∂ 2 ∂ z 2 ϕ ( z ) = − e ε 0 ε ( z ) [ ∑ i n i \| φ i ( z ) \| 2 − N d ( z ) ]
	(8)

H = − \| 3 2 , 3 2 ⟩ \| 3 2 , 1 2 ⟩ \| 3 2 , − 1 2 ⟩ \| 3 2 , − 3 2 ⟩ \| 1 2 , 1 2 ⟩ \| 1 2 , − 1 2 ⟩ [ P + Q − S R 0 − 1 2 S 2 R − S † P − Q 0 R − 2 Q 3 2 S R † 0 P − Q S 3 2 S † 2 Q 0 R † S † P + Q − 2 R † − 1 2 S † − 1 2 S † − 2 Q 3 2 S − 2 R P + Δ 0 2 R † 3 2 S † 2 Q − 1 2 S 0 P + Δ ] + V v ( z )
	(9)

P = ℏ 2 2 m 0 γ 1 ( k x 2 + k y 2 + k z 2 ) − a v ( ϵ x x + ϵ y y + ϵ z z ) ,

Q = ℏ 2 2 m 0 γ 2 ( k x 2 + k y 2 − 2 k z 2 ) − b 2 ( ϵ x x + ϵ y y − 2 ϵ z z ) ,

R = ℏ 2 2 m 0 3 [ − γ 2 ( k x 2 − k y 2 ) + 2 j γ 3 k x k y ] .

ϵ x x = ϵ y y = a 0 − a a , ϵ y y = − 2 C 12 C 11 ϵ x x
	(11)

Ψ i ( r , z ) = e j k ⋅ r [ χ 1 ( z ) \| 3 2 , 3 2 ⟩ + χ 2 ( z ) \| 3 2 , 1 2 ⟩ + χ 3 ( z ) \| 3 2 , − 1 2 ⟩ + χ 4 ( z ) \| 3 2 , − 3 2 ⟩ + χ 5 ( z ) \| 1 2 , − 1 2 ⟩ + χ 6 ( z ) \| 1 2 , − 1 2 ⟩ ] ,
	(12)

χ ( z ) and [ p + q − s 0 0 − 1 2 s 0 − s † p − q 0 0 − 2 q 3 2 s 0 0 p − q s 3 2 s † 2 q 0 0 s † p + q 0 − 1 2 s † − 1 2 s † − 2 q 3 2 s 0 p 0 0 3 2 s † 2 q − 1 2 s 0 p ] χ ( z ) continuous .
	(13)

g 12 = 2 π ℏ \| H m \| 2 δ ( E 2 − E 1 − ℏ ω ) ,
	(15)

g 12 = 2 π ℏ ∫ \| H m \| 2 δ ( E − ℏ ω ) L ( E ) d E = 2 π ℏ \| H m \| 2 Γ ∕ 2 π ( ℏ ω − E 0 ) + Γ 2 ∕ 4 .
	(17)

g net = ∫ g 12 { f 2 ( E 2 , k ) [ 1 − f 1 ( E 1 , k ) ] − f 1 ( E 1 , k ) [ 1 − f 2 ( E 2 , k ) ] } ρ r ( E 2 , k − E 1 , k ) d ( E 2 , k − E 1 , k ) ,
	(23)

g net = 2 π ℏ \| H m \| 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 [ ∫ f 2 ( E 2 , k ) ρ 2 d E 2 , k − ∫ f 1 ( E 1 , k ) ρ 1 d E 1 , k ] = 2 π ℏ \| H m \| 2 Γ ∕ 2 π ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,
	(24)

A = A 0 cos ( β ⋅ r − ω t ) z ̂ = 1 2 A 0 z ̂ [ e j ( β ⋅ r − ω t ) + e − j ( β ⋅ r − ω t ) ] ,
	(26)

γ ( ω ) = e 2 \| P 12 \| 2 2 ε 0 c n eff m e * 2 ω L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) = e 2 m 0 2 ω z 12 2 2 ε 0 c n eff m e * 2 L p Γ ( ℏ ω − E 12 ) 2 + Γ 2 ∕ 4 ( N 2 − N 1 ) ,
	(27)

γ ( ω 0 ) = 2 e 2 m 0 2 ω z 12 2 ε 0 c n eff m e * 2 Γ L p ( N 2 − N 1 ) .
	(28)

g net ( v ) = 2 π ℏ \| H m ( v ) \| 2 ρ r ( E l − E h ) \| [ f LH ( E l ) − F HH ( E h ) ] \| E l − E h = ℏ ω .
	(29)

H m ( v ) = ∑ n = 1 6 e A m n * ⟨ χ n ( l ) \| − j ℏ ∂ ∂ z \| χ n ( h ) ⟩ ,
	(30)

γ ( v ) = π e 2 ε 0 c n eff ω L p \| ∑ n = 1 6 P n ( l h ) m n * \| 2 ρ r ( E l − E h ) \| [ f LH ( E l ) − f HH ( E h ) ] \| E l − E h = ℏ ω = π e 2 m 0 2 ω ε 0 c n eff L p \| ∑ n = 1 6 z n ( l h ) m n * \| 2 ρ r ( E l − E h ) \| [ f LH ( E l ) − f HH ( E h ) ] \| E l − E h = ℏ ω .
	(31)

1 τ 12 = 2 π ℏ ∫ \| H e p \| 2 δ ( E 2 , k − E 1 , k ′ ∓ ℏ ω Q ) d N f ,
	(32)

\| H e p \| 2 = { Ξ 2 K B T 2 c L Ω δ q , ± ( k ′ − k ) \| G 12 ( q z ) \| 2 acoustic phonon ℏ D 2 2 ρ ω 0 Ω δ q , ± ( k ′ − k ) [ n ( ω 0 ) + 1 2 ∓ 1 2 ] \| G 12 ( q z ) \| 2 nonpolar optical phonon } ,
	(33)

G 12 ( q z ) = ∑ n = 1 6 ⟨ χ n ( 1 ) \| e j q z z \| χ n ( 2 ) ⟩ .
	(36)

1 τ 12 = { Ξ 2 K B T m e * 4 π c L ℏ 3 ∫ \| G 12 ( q z ) \| 2 d q z acoustic D 2 m e * [ n ( ω 0 ) + 1 2 ∓ 1 2 ] 4 π ρ ℏ 2 ω 0 ∫ \| G 12 ( q z ) \| 2 d q z nonpolar optical } .
	(38)

Δ E v , av ( x , y ) = E v , av ( Ge Si Sn ) − E v , av ( Ge ) = 0.48 x + 0.69 y .
	(39)

E Ge Si Sn ( x , y ) = E Ge Si Sn ( 1 − x − y ) + E Si x + E Sn y − b Ge Si ( 1 − x − y ) x − b Ge Sn ( 1 − x − y ) y − b Si Sn x y .
	(40)

∂ N 2 ∂ t = N 3 − N ¯ 3 τ 32 − N 2 − N ¯ 2 τ 2 ,

N 3 − N 2 = τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ,
	(43)

γ ( ω 0 ) = 2 e 2 m 0 2 ω 0 z 23 2 ε 0 c n eff m z * 2 Γ L p [ τ 3 ( 1 − τ 2 τ 32 ) η J e − ( N ¯ 2 − N ¯ 3 ) ] .
	(44)