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: but during the time that the feature size was pushed down towards its ultimate physical limits, there has also been a tremendous 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 early visions in 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, the 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 photonic devices that make up a complete set of necessary components in Si photonics including light emitters, amplifiers, photodetectors, waveguides, modulators, couplers and switches, the most difficult challenge is the lack of an efficient light source. The reason for this striking absence is that bulk Si has an indirect band gap where 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 photons have negligible momentum compared with that of electrons, the recombination of an electron-hole pair will not be able to emit a photon without the simultaneous emission or absorption of a phonon in order to conserve the momentum. Such a radiative recombination is a second-order effect occurring with a small probability, which competes 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.
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 intrinsic problem of Si to develop efficient light sources based on Si. One interesting method is to use small Si nanocrystals dispersed in a dielectric matrix, often times SiO2. Such nano-scaled Si clusters are naturally formed by the thermal annealing of a Si-rich oxide thin film. Silicon nanocrystals situated in a much wider band gap SiO2 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 from these Si nanocrystals by both optical pumping [3,4] and electrical injection , but the origin of the observed optical gain has not been fully understood as the experiments were not always reproducible – results were sensitive to the methods by which the samples were prepared. 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 (Er3+) . By incorporating Er3+ 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 µm. However, the concentration of Er3+ ions that can be doped in Si is relatively low and there is a significant energy back-transfer from the Er3+ 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 back transfer of energy, SiO2 with an enlarged band gap has been proposed as host to remove the resonance between the defect and the Er3+ energy levels . Once again, Si-rich oxide is employed to form Si nanocrystals in close proximity to Er3+ ions. The idea is to excite Er3+ ions with the energy transfer from the nearby Si nanocrystals. Light emitting diodes (LEDs) with efficiencies of about 10% have been demonstrated  on par with commercial devices made of GaAs, but with power output only in tens of µW. While there have been proposals to develop lasers using doped Er in Si-based dielectric, the goal remains elusive.
The only approach so far that has led to the demonstration of lasing in Si exploited the effect of stimulated Raman scattering [11-13], analogous to that produced in fiber Raman amplifiers. With both the optical pumping and the Raman scattering below the band gap of Si, the indirectness of the Si band gap 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 (~100MW/cm2) and the device lengths (~cm) are too large to be integrated with other photonic and electronic devices in any type of Si VLSI-type circuit .
Meanwhile, the search for laser devices that can be integrated on Si chips has gone well beyond the monolithic approach to seek solutions 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 was demonstrated . This fabrication technique allows for 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 . Another effort produced InGaAs quantum dot lasers deposited directly on Si substrates with a thin GaAs buffer layer . 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 and debates continue about which method offers the best promise, yet another approach emerged that has received a great deal of attention in the past decade—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 band gap. 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 band gap irrelevant. Developing lasers using ISTs therefore provides a promising alternative that completely circumvents the issue of indirectness in the Si band gap. 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 might turn out to be a viable path to achieving electrically pumped Si-based coherent emitters that are suitable for monolithic integration with Si photonic and electronic devices.
In this chapter, 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 . 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 narrower band gap semiconductor with a layer thickness thinner than the deBroglie wavelength of the electron (~10nm) between two wider band gap semiconductors (Fig. 2(a)). The one-dimensional quantum confinement leads to quantized states (subbands) in the direction of growth
Obviously, if the band offset is large enough, there could be multiple subbands present within either conduction or 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 a carrier transition between these 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.
The original idea of creating light sources based on ISTs was proposed by Kazarinov and Suris  in 1971, but the first QCL was not demonstrated until 1994 by a group led by Capasso at Bell Laboratories . In comparison with the conventional band-to-band lasers, lasers based on ISTs require much more complex design of the active region which consists of carefully arranged multiple QWs (MQWs). The reason for added complexity can be appreciated by comparing the very different band dispersions that are involved in these two types of lasers. 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
Let us now turn our attention to the intersubband transition shown in Fig. 3(b). The in-plane dispersions of the upper
Still, intersubband lasers offer advantages in areas where the conventional band-to-band lasers simply cannot compete. In band-to-band lasers, lasing wavelengths are mostly determined by the intrinsic band gap 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 band gaps from which such lasers can be made. With the intersubband transitions, 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 to the band-to-band lasers in which conduction and valence bands have opposite band curvatures.
A practical design that featured 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 in 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 MQWs. 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 they repeat the process in a cascading manner, typically 20 to 100 times.
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. Intersubband theory
In order 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 follows the envelope function approach based on the effective-mass approximation . The
3.1. Subbands and dispersions
Let us treat the conduction subbands first. It is well known in bulk material that 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 (
where the position vector is decomposed into in-plane and growth directions
in conjunction with the relationship between the subband minimum energy
In the presence of an electric field
consistently with Eq.(2), where
In comparison with the conduction band, the situation in the valence band is far more complex mostly because of the interactions between subbands of different effective masses that 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
The 6×6 Luttinger-Kohn Hamiltonian matrix including the uniaxial stress along (001) is given in the HH (
The Hamiltonian in Eq.(9) operates on wavefunctions that are combinations of six mutually orthogonal HH (
to maintain undisruptive carrier distribution and current across the interface.
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
This procedure applied at each wave vector point (
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 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 perturbation electro-magnetic (EM) field with a frequency of
essentially replacing the
In the presence of an EM field with an optical potential vector
From Eq.(18), it is not difficult to see that the selection rules for intersubband transitions in the conduction band are such that only those EM fields that are polarized in the growth direction (
where the momentum matrix element
is evaluated as the envelope function overlap between the two subbands, which is related to the dipole matrix element 
and to the oscillator strength 
It is not difficult to see from Eq.(17) that the transition rate induced by an EM field between two eigen states is the same for upward and downward transitions. Now let us apply Eq.(17) to intersubband transitions between the upper subband 2 and lower subband 1 in the conduction band (Fig. 3). Since momentums associated with photons are negligible, all photon-induced transitions are vertical in
The optical gain coefficient
For transitions between valence subbands with nonparallel dispersions and strong mixing between HH, LH, and SO bands, we have to re-examine the intersubband transition rate. Consider the intersubband transition in Fig. 5 from the upper state
in terms of momentum matrix
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,
3.3. Intersubband lifetimes
It has been established in Eqs.(27) and (28) that the population inversion between the upper (2) and lower (1) subbands,
Up to now, practically all approaches in 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 . Starting from Fermi Gold rule Eq.(15), the scattering rate for a carrier in subband 2 with the in-plane wave vector
where the upper sign is for absorption and lower for emission of one phonon,
The wavefunction interference effect between conduction subbands is
and between valence subbands is
The Kronecker symbol
Since phonon modes have density of states
But 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
The phonon scattering rate in Eq.(38) has been used to compare the lifetimes of two similar three-level systems, SiGe/Si and GaAs/AlGaAs, as shown in Fig. 7(a) . The lifetime difference between the upper (3) and lower (2) subband is calculated as the function of the transition energy
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 when 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 , impurity , and interface roughness scatterings , all of which are elastic processes. The carrier-carrier scattering is a two-carrier process that is particularly important when carrier concentration is high which increases the possibility 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 where both carriers originally in subband 2 end up in subband 1 is the most efficient one in terms of inducing intersubband transitions (Fig. 6(b)). It has been reported in experiment that the intersubband transition times on the order of tens of ps have been observed for carrier densities of
4. Valence band SiGe QCLs
Up to now, all of the demonstrated QCLs are based on epitaxially grown III-V semiconductor heterostructures such as GaInAs/AlInAs, GaAs/AlGaAs, and InAs/AlSb, using electron subbands in conduction band. With the promise of circumventing the indirectness of Si band gap, a SiGe/Si laser based on intersubband transitions was first proposed by Sun et al in 1995  where a comparative study was performed between the SiGe/Si and GaAs/AlGaAs 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 as a strain-inducing layer for CMOS transistors .
While QCLs based on SiGe alloys can be monolithically integrated on Si if successfully developed, there are significant challenges associated with this material system. First, there is a 4% lattice mismatch between Si and Ge. Layers of Si1-xGex alloys deposited on Si substrates induce strain which can be rather significant in QCLs because a working structure typically consists of at least hundreds of layers with a total thickness that easily exceeds 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 SiGe/Si QC structures, one popular approach is to use strain balanced growth where the compressively strained Si1-xGex and tensile strained Si are alternately stacked on a relaxed Si1-yGey buffer deposited on a Si substrate where the buffer composition (
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 valence band. Practically all of the investigations of SiGe QCLs are focused on intersubband transitions in the valence band. But the valence subband structure is much more complex in comparison with the conduction subbands 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 in general has to inevitably involve 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
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 way analogous to the situation in conventional band-to-band lasers discussed in section II. It was reported in QCLs that the population inversion was only established locally in
The inverted mass approach was later on applied to the SiGe system [50,51]. Two slightly different schemes were developed, one utilized the inverted LH effective mass , and the other inverted HH mass . 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 (37meV for the Ge-Ge mode, 64meV for the Si-Si mode, and 51meV for the Si-Ge mode ), which suppresses the nonradiative intersubband transitions due to the optical phonon scattering, but also limits the optical transitions in the THz regime. The structures under investigation were strain balanced with compressively strained Si1-xGex QW layers and the tensile strained Si barrier layers deposited on a relaxed Si1-yGey buffer layer (
intersubband laser mimics the operation of a conventional band-to-band laser. The lifetime of the upper laser state 3 is long because the intersubband transition energy at 6THz (~50μm) is below that of optical phonons, allowing only much weaker 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 being established between the LH and HH subbands.
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 between subbands of different types is always smaller than that 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 . The band structure for a 90Å/35Å Si0.8Ge0.2/Si SL under an electric bias of 30kV/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 3meV energy split within the doublet. The resulting in-plane dispersions for the two doublets are shown in Fig. 9(b). Simulation results showed that optical gain of 450/cm at 7.3THz can be achieved at a pumping current density of 1.5KA/cm2 at 77K.
Electroluminescence (EL) from a SiGe/Si quantum cascade emitter was first demonstrated in a SiGe/Si quantum cascade emitter using HH to HH transitions in the mid-IR range in 2000 . 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 , LH-to-HH , and HH-to-LH , with emission spectra ranging from mid IR to THz (8 ~ 250μm). But lasing has not been observed. Improvement on the QCL design has been to be made. One of the most successful III-V QCL designs has been 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 that are injected into the bound upper state 2 are prevented from escaping the bound state by the minigap, and then undergo lasing transitions to a 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, once again showing just EL with no lasing . It is believed that in this structure LH states are mixed within the HH states. Although the impact 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 after by lifting the LH states out of all involved HH states for bound and continuum with the use of strain, as a result, a clear intersubband TM polarized EL is shown suggesting that LH states have been pushed away from the HH radiative transitions .
Nearly a decade has passed since the first experimental demonstration of EL from a SiGe/Si quantum cascade emitter . During this period III-V QCLs have been improved dramatically 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 propelled some researchers back into the conduction band to look for solutions.
5. Conduction band Si-based QCLs
Before QCLs can be designed using conduction subbands, there must be sufficient conduction band offset. Contrary to the situation in compressively strained Si1-xGex, tensile strained Si1-xGex can have larger conduction band offset, but the conduction band minima occur at the two
Prospects of developing such Si-based QCLs have been investigated theoretically. One approach stayed with the Si-rich SiGe/Si material system but instead of the conventional growth direction in (100), the structural orientation has been rotated to the  crystal plane . Conduction band offset was calculated to be 160meV at the conduction band minima consisting of six-degenerate Δ-valleys, sufficient for designing far IR QCLs. The effective mass along (111) direction can be obtained as the geometric average of the longitudinal and transverse effective masses of the Δ -valley,
Recently, a new group-IV material system that expands beyond the Si1-xGex alloys has been successfully demonstrated with the incorporation of Sn. These new ternary Ge1-x-ySixSny alloys have been studied for the possibility of forming direct band gap semiconductors[63-66]. Since the first successful growth of this alloy , device- quality epilayers with a wide range of alloy contents have been achieved [68,69]. Incorporation of Sn provides the opportunity to engineer separately the strain and band structure since we can vary the Si (x) and Sn (y) compositions 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 (
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 a ternary Ge1-x-ySixSny based on Jaros' band offset theory  which is in good agreement with experiment for many heterojunction systems. For example, this theory predicts an average valence band offset,
Once the average valence band offset is determined, the energies of individual conduction band edges in the Ge1-x-ySixSny 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 . We have assumed that all required alloy energies can be interpolated between the known values for Si, Ge, and α-Sn as
(in eV) for Ge1-xSix alloys . On the other hand, the empirical pseudo-potential calculations of Chelikovsky and Cohen place this minimum at 0.90 eV in -Sn, virtually the same as its value in pure Ge . We thus assume that the position of this minimum in ternary Ge1-x-ySixSny alloys is independent of the Sn concentration
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 Ge0.76Si0.19Sn0.05 alloy while all other conduction-band valleys (Γ, X, etc) are above the L-valley band edge of the Ge0.76Si0.19Sn0.05 barrier. This band alignment presents a desirable alloy composition from which a QCL operating at L-valleys can be designed using Ge as QWs and Ge0.76Si0.19Sn0.05 as barriers without the complexity arising from other energy valleys.
Figure 12 shows the QCL structure based upon Ge/ Ge0.76Si0.19Sn0.05 QWs. Only L-valley conduction-band lineups are shown in the potential diagram under an applied electric field of 10 kV/cm. In order to solve the Schrödinger equation to yield subbands and their associated envelope functions, it is necessary to determine the effective mass
separated by 20Å Ge0.76Si0.19Sn0.05 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.
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 calculation of the scattering rate to yield lifetimes for the upper laser state
which can then be used to evaluate the optical gain of the TM polarized mode following Eq.(28) at the lasing transition energy
For the QCL structure in Fig. 12, the following parameters are used: index of refraction
Since the relatively small conduction band offset limits the lasing wavelength to the far-IR or THz regime (roughly 30 µm 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 have exceeded 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 the so-called plasmon waveguide [79,80]. While the deposition of 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.
This plasmon waveguide supports only TM polarized EM mode that is highly confined within the QCL region,
which determines the TM modes that can propagate in this plasmon waveguide. The waveguide loss
While GeSiSn epilayers with alloy compositions suitable for this QCL design have been grown with MOCVD [68,69], implementation of Ge/GeSiSn QCLs is currently challenged by the structural growth of the large number of hetero-layers in the QCL structure with very fine control of layer thicknesses and alloy compositions. Nevertheless, progress is being made towards experimental demonstration.
Silicon-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 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 band gap. Such a scheme makes the indirect nature of the Si band gap irrelevant. In order to appreciate 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 SiGe/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 the compressively strained Si1-xGex with tensile strained Si grown on a relaxed Si1-yGey has very small conduction band offset QWs are too shallow to allow for elaborate QCLs. Tensile strained Si1-xGex, on the other hand, can have larger conduction band offset, but the conduction band minima occur at the two
Recently, several ideas of developing Si-based conduction-band QCLs have emerged to circumvent the hurdles in the SiGe/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, and another relies on ISTs in the
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 happens, but with the variety of approaches that are being pursued--driven by the potential pay-off in commercialization--the prospect is promising.
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