Stimulated Emission and Laser Processes

Abstract

The development of lasers has already a relatively long history from first theoretical [17E1] and experimental [28K1] work on stimulated emission over first realizations of lasers [58S1, 60M1] to first semiconductor lasers [62H1, 62M1]. We present here after this short introduction to the topic laser processes in semiconductors (Sects. 22.1 and 22.2), continue with short presentations of cavity lasing (Sect. 22.3) and of random lasing (Sect. 22.4) and give then an introduction to the layout of semiconductor light emitting or laser diodes (Sect. 22.5) including organic light emitting diodes (O-LED) and present research trends.

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The development of lasers has already a relatively long history from first theoretical [17E1] and experimental [28K1] work on stimulated emission over first realizations of lasers [58S1,  60M1] to first semiconductor lasers [62H1,  62M1]. We present here after this short introduction to the topic laser processes in semiconductors (Sects. 22.1 and 22.2), continue with short presentations of cavity lasing (Sect. 22.3) and of random lasing (Sect. 22.4) and give then an introduction to the layout of semiconductor light emitting or laser diodes (Sect. 22.5) including organic light emitting diodes (O-LED) and present research trends.

Stimulated emission from semiconductors is usually identified in experiments by the occurrence of one or several of the following criteria: a strongly superlinear increase of the optical luminescence output I _lum as a function of the pump power I _exc above a certain threshold I _exc ^th with slopes α in I _lum ∝ I _exc ^α of three and more; a simultaneous spectral narrowing of the emission, often accompanied by the appearance of laser modes imposed by some cavity length and a spatially directed emission above I _exc ^th. The increase of the coherence length of the emitted light is also characteristic for laser emission, but is less used to identify laser emission from semiconductors, though semiconductors laser with external, wavelength-selective cavities are developed with extremly narrow spectral emission.

Many of the high excitation effects outlined in Chaps. 19–21 can lead to stimulated emission under suitable conditions; see, e.g., [73B1,  75K1,  81K1,  81S1,  92E1,  93K1,  94C1,  95K1,  97N1,  99M1,  03I1,  03R1,  03R2] and references therein. In this chapter we treat those aspects of the high excitation phenomena of Chaps. 19–21 which are specific for stimulated emission.

Pumpingcan be accomplished by excitation with intense (laser-) light [81K1,  94C1], (I _exc ≳ 104 W∕cm² depending on the material and the pump conditions), with electron beams [64B1,  73B1,  92B1,  92N1] (j ≳ 10 A∕cm², U ≳ 30 keV), by a flash-lamp [83B1] or by a forward biased pn junction [81S1,  92E1,  92N1,  04H1], provided that it is possible to grow the material in the desired highly p- and n-doped versions. Optical pumping is usually the proper choice for scientific investigations of the gain processes, since it allows resonant excitation of some species, e.g., certain exciton or biexciton levels. Pumping by electron beams, on the other hand, is unselective and an energy of about 3 E _g is deposited for the creation of one electron–hole pair. Electron–beam pumping was widely used before high power lasers were available [64B1,  73B1] and more recently for the realization of color projection TV tubes with high brightness [92B1,  92N1].

Pumping by a pn (hetero-) junction biased in the forward direction is obviously the best choice for most technical applications in laser diodes, which are found, for example, in every CD player, laser scanner, laser printer or bar code reader [81S1,  92E1].

Stimulated emission has, until now been almost exclusively limited to direct gap semiconductors. An exception, namely Al_1 − y Ga _y As close to the direct-indirect crossover, was mentioned in Sect. 21.3 and in [96K1] and we shall come back to this later. Our considerations are restricted therefore to direct gap materials where not stated otherwise.

22.1 Excitonic Processes

The processes which lead to stimulated emission can either be of intrinsic nature, i.e., they involve only excitations of the perfect lattice, or of extrinsic nature, involving some impurity or defect states. We shall see in the following examples of both groups.

We use the language of the weak coupling limit for reasons of simplicity, but state, that we are actually dealing with polaritons. We start with intrinsic processes in the intermediate density regime, namely the recombination of an exciton with emission of one or more LO phonons, or with inelastic scatteringby another exciton or a free carrier, or finally the decay of a biexcitoninto a photon and an exciton [76L1]. All these processes were already introduced in Chap. 20. In Fig. 22.1 we summarize these effects again actually drawing them in the strong coupling or polariton picture. For gain due to trion recombination see e.g. [02P2].

Fig. 22.1

Inelastic scattering processes in the intermediate density regime: exciton-LO phonon emission (c), inelastic exciton–exciton scattering (b), inelastic exciton–electron scattering (a) and biexciton decay (d) [81K1]

In all four cases some initial state decays under emission of a photon and leaves behind in the crystal another excitation, for example an exciton in its ground state in the case of biexciton decay, an LO phonon in the case of the phonon replica, or an exciton in an excited state n _B ≳ 2 or a free carrier at higher kinetic energy in the cases of inelastic exciton-excitonor-free carrier scattering, respectively. All these processes have the low threshold for population inversion typical for four-level laser systemsas long as this excited final state in the sample is not (thermally) populated. If there are no excitons thermally excited into n _B = 2 state, or no thermally populated LO phonons, then population inversion is reached in the presence of two or even one exciton, respectively. See also (22.2c). We consider now the rate equations for these processes, assuming that for a given photon energy ℏω there is just one (laser-) mode containing photons with a density N _ph. The general rate equationwhich holds is [81K1]:

$$\frac{\mathrm{d}{N}_{\text{ Ph}}} {\mathrm{d}t} = -2\kappa {N}_{\text{ Ph}} + \sum \frac{2\pi } {\hslash } \delta \left (\Delta E\right ){\left \vert W\right \vert }^{2}Q,$$

(22.1)

where − 2κN _Ph contains all losses of the resonator, e.g., due to finite reflection, diffraction or absorption due to other processes than the ones included in the second term on the r.h.s of (22.1). See e.g. (22.2b). The second term on the right of (22.1) contains a sum over all (scattering) processes contributing to the emission at ℏω,δ(ΔE) stands for the energy conservation, Q is a population factor into which we can integrate the $\bf{k}$-conservation, and | W | ² finally is the transition matrix element. As a first approach to the strong coupling limit one can also incorporate into | W | ² the probabilities that the exciton polariton is exciton-like in the initial state and photon-like in the final state. The next step would involve the use of the polariton dispersion curves of Fig. 22.1 instead that of excitons and photons. See [81K1].

We consider here as an example the P ₂ line, i.e., the inelastic scattering between two excitons in the n _B = 1 state, from which one is scattered under energy and momentum conservation into a state with n _B = 2, while the other one appears as a photon. In the strong coupling limit one would state, that there occours an inelastic polariton–polariton scatteringprocess between two exciton polaritons, from which one is scattered onto the photonlike (lower) polariton branch and the other into an excited exciton-like polariton state. Consequently, this process and the others depicted in Fig. 22.1 are early examples of polariton lasing, a term presently used for lasing in micro cavities. See e.g. Sects. 20.5 and 22.3 below.

In this case we have

$$\begin{array}{rl} \Delta E =&{E}_{\text{ ex}}\left ({n}_{\text{ B}} = 1,\bf{{k}}_{1}\right ) + {E}_{\text{ ex}}\left ({n}_{\text{ B}} = 1,\bf{{k}}_{2}\right ) \\ & - \hslash \omega - {E}_{\text{ ex}}\left ({n}_{\text{ B}} = 2,\bf{{k}}_{1} +\bf{ {k}}_{2}\right ), \end{array} $$

(22.2a)

and the population factor Q includes spontaneous and stimulated emission, reabsorption and reads

$$\begin{array}{rl} Q =&{N}_{\text{ ex}}\left (\bf{{k}}_{1},{n}_{\text{ B}} = 1\right ) \times {N}_{\text{ ex}}\left (\bf{{k}}_{2},{n}_{\text{ B}} = 1\right ) \\ & \times \left [1 + {N}_{\text{ ex}}\left (\bf{{k}}_{1} +\bf{ {k}}_{2},{n}_{\text{ B}} = 2\right )\right ]\left (1 + {N}_{\text{ Ph}}\right ) \\ & - {N}_{\text{ Ph}}{N}_{\text{ ex}}\left (\bf{{k}}_{1} +\bf{ {k}}_{2},{n}_{\text{ B}} = 2\right )\left [1 + {N}_{\text{ ex}}\left (\bf{{k}}_{1},{n}_{\text{ B}} = 1\right )\right ] \\ & \times \left [1 + {N}_{\text{ ex}}\left (\bf{{k}}_{2},{n}_{\text{ B}} = 1\right )\right ],\end{array} $$

(22.2b)

where we consider both photons and excitons as bosons.

We can now decompose Q into two terms. One, which is independent of N _Ph and describes the spontaneous emission, and another one which is linear in N _Ph and gives the net gain or absorption. For the latter, Q _stim, we find

$$\begin{array}{rl} {Q}_{\text{ stim}} =&{N}_{\text{ Ph}}\left \{{N}_{\text{ ex}}\left (\bf{{k}}_{1},{n}_{\text{ B}} = 1\right ){N}_{\text{ ex}}\left (\bf{{k}}_{2},{n}_{\text{ B}} = 1\right )\right. \\ & - {N}_{\text{ ex}}\left (\bf{{k}}_{1}\bf{{k}}_{2},{n}_{\text{ B}} = 2\right ) \\ &\left.\times \left [1 + {N}_{\text{ ex}}\left (\bf{{k}}_{1},{n}_{\text{ B}} = 1\right ) + {N}_{\text{ ex}}\left (\bf{{k}}_{2},{n}_{\text{ B}} = 1\right )\right ]\right \}.\end{array} $$

(22.2c)

If we assume thermal equilibrium between the excitons in the various states, we see that inversion occurs, i.e., Q _stim > 0 at low temperatures, even if we have only two excitons which collide as mentioned already above.

The luminescence and gain increase roughly quadratically with the exciton density and thus superlinearly with the pump intensity at low temperature until the gain overcomes the losses and stimulated emission sets in. At higher temperatures the n _B = 2 states will be also populated, reabsorptionoccurs and may even overcome the gain in some spectral regions, resulting in excitation-induced absorption. This effect limits laser emission due to exciton–exciton scattering towards higher temperatures together with the increasing thermal dissociation of excitons [76H1,  78K1,  81K1].

The biexcitondecay and the exciton-free carrier scatteringalso have gain values which increase superlinearly with density, while the gain of the ex-n LO process grows essentially linearly with the generation rate until laser emission sets in. For some early examples in ZnO see e.g. [73K1,  75K1,  81K1] and references therein.

If Fig. 22.2 we show the calculated temperature dependences of the laser thresholddensities for three of the above-mentioned processes for a given constant value of κ. A variation of κ will shift the absolute densities or excitation intensities and the various quadratic processes with respect to the linear one. A similar effect results also from a variation of the transition matrix elements and/or the scattering cross section. The increase of the thresholds with increasing temperature comes from the thermal population of the final states as indicated above. The high threshold at low temperatures of the process involving free carriers originates from the assumption of thermal equilibrium. In this case no free carriers (i.e., excitons in the continuum state) are present at low temperatures. The exciton-free carrier-scattering process can be influenced by doping. The fact that the calculated carrier densities are relatively high and may exceed the Mott density(Chap. 21) is due to the use of rather high losses in the calculations. Lower loss values reduce the calculated threshold densities.

Fig. 22.2

The calculated threshold density N _theory ^th for various gain processes in the intermediate density regime of CdS as a function of temperature, and the experimentally observed excitation intensity at threshold I _exc ^th as a function of the lattice temperature [81K1]

On the other hand this finding shows that stimulated emission occurs frequently in the transition region between intermediate and high densities. The exciton binding energy decreases with increasing density, i.e., the excitons become “soft” and inelastic scattering processes between carriers may take place in an EHP or the emission of phonons or plasmons in the recombination process of an electron–hole pair. See Chap. 21 and below for ZnO. While scientists tried to separate these various processes in the 1970s and 1980s it is now, at least from the theoretical side, the aim to present calculations that are valid continuously from low densities to the EHP as already mentioned in Chap. 21 where some references are also given. A technical term to describe these effects is gain and loss in a “strongly Coulomb correlated electron–hole plasma”. For some more experimental and theoretical references see, apart from the references given in Chap. 21, e.g. [77O1,  93J1,  96G1,  96H1,  96K1,  97P1,  98M1,  98P1,  99B1,  00S1,  02T2,  11G1]. In connection with the recent research boom on ZnO, triggered by the hope to obtain a material for blue/near UV optoelectronics similar to GaN, one can frequently read the claim of excitonic lasing processes in ZnO around room temperature and even above, which should have additionally very low laser thresholds. See e.g. for some of these references the ones given in Fig. 20.3a or in Chap. 11 of [10K3]. We show in the next Sect. 22.2 that this is in most cases an erroneous assignment. Instead the stimulated emission arises from an EHP.

For the purpose of didactics and clarity, we still separate these processes here, but one should be aware of the continuous transition from the intermediate to the high density regime, especially in direct-gap semiconductors.

In Fig. 22.2 we give experimental data for the excitation intensity at threshold for CdS which show the trends predicted by theory and dicussed above. The low value of the threshold around 80 K comes from a cooperative effect, since various processes are spectrally degenerate in CdS at this temperature; see Fig. 20.3. Similar data are known for other semiconductors like ZnO. See [81K1] and reference therein. For more recent data on GaN and ZnO see [99B1,  08F1,  10K3,  11G1], the references therein or Sect. 22.3.

In Fig. 22.3 we show an example of the optical input–output curves, showing clearly the laser threshold. As an example, the exciton-LO process in ZnO has been chosen [80W1].

Fig. 22.3

The luminescence intensity of the first two LO phonon replicas of ZnO as a function of the generation rate G, showing the laser threshold [ 75K1,  80W1]

Similar input–output curves are obtained for the light output as a function of the forward current (density) in laser diodes. Often one defines a differential or slope efficiency which is the (maximum) of the derivative of curves like Fig. 22.3. It must be mentioned that theses slope efficencies may reach or even exceed unity but they say nothing about the absolute internal or external luminescence yield of the device.

Recent publications also stressing (bi-) excitonic gain processes in quantum structuresare, e.g., [93J1,  95K1,  95K2,  97C1,  01L1,  01R1], which are even partly claimed to coexist with EHP phenomena.

Inelastic scattering processes between two excitons or free carriers are treated in quantum structures, e.g., in [87F1,  89F1,  93J1,  94C2,  96R1,  97F1].

We come back to the laser emission in structures of reduced dimensionality below and in Sect. 22.3.

Stimulated emission due to inelastic scattering between heavy and light exciton in CuI thin films is reported in [02T2] and stimulated THz emission due to intraexcitonic transitions in Cu₂O [06H1,  06H2]. See in this context also Sect. 13.3.

Another group of intrinsic laser processes, still in the intermediate density regime, involves only interaction processes of virtually and coherently created particles. As an example we take the two-photon or hyper-Raman scatteringalready introduced in Sects. 20.3 and 13.1.4. There we presented a process where a biexciton is created virtually by two quanta and decays under energy and momentum conservation.

If we now send an additional third quantum ℏω_s into the sample with momentum and energy coinciding with those of a possible decay channel, this quantum can stimulate the decay of the virtually excited biexciton into another photon ℏω_s, $\bf{{k}}_{\text{ s}}$. A second photon ℏω_f, $\bf{{k}}_{\text{ f}}$ must then necessarily be simultaneously emitted according to (20.7). See [76L1,  85H1].

If the third quantum lies energetically below the two pump quanta, ℏω_f is necessarily located above, and the whole process is an example of electronic CARS(coherent anti-Stokes Raman scattering). This latter process also occurs with optical phonons. The sample in this case is again illuminated with some pump photon and with quanta corresponding to the Stokes emission, resulting in stimulated emission of the anti-Stokes line.

Alternatively these processes can be named non-degenerate four-wave mixing (NDFWM)and can also be considered as diffraction from a moving laser-induced grating written by the interference of one pump and one stimulating quantum, ℏω_p and ℏω_s, respectively, and read by one of the two incident beams resulting in a Doppler-shifted diffracted signal. After the investigation of the above mentioned inelatic scattering or coherent NDFWM processes with respect to their laser properties in bulk material and in quantum structures (see above or [01L1]), we see now that analogous phenomena are investigated involving cavity polaritons,as outlined already in Sect. 20.2. The experiments are beautiful and the introduction of a new language in connection with the rediscovery continues. Stimulated emission due to polariton–polariton scatteringin a microcavity in analogy to Fig. 20.1 or 22.1b is named polariton laser or PLASER,and the low threshold well known for four level laser systems mentioned above comes now partly under names like thesholdless lasing or lasing without inversion. See e.g. the Ref. [00B1,  00T1,  02B4,  02B5,  02S5] of Chap. 20 or for more recent considerations of microcavity lasers [02K3,  02Z1] and Sect. 20.3. The close connection between stimulated emission and Bose–Einstein condensationof cavity polaritons has already been mentioned in Sect. 20.5.4. Further references are e.g. [10D1,  01O1,  02L2].

We now give two examples of stimulated emission in disordered systems, see Sect. 14.4, which still needs for lasing beyond amplified spontaneous emission (ASE) some type of cavity in contrast to lasing in random media treated shortly in Sect. 22.4 below.The tail of localized statesin CdS_1 − x Se _x . typically has a maximum tailing parameter E ₀(x) of about 5 meV and contains roughly 1018 states per cm³. So the lower portion of this tail can be easily filled by optical pumping at low temperatures where thermal excitation into the extended states with a much higher density of states is prevented. If the gain value, i.e., the inversion, of the filled states is sufficiently large, laser emission sets in.

We show schematically in Fig. 22.4 (left-hand side) the density of states with the mobility edgeME and the chemical potential, μ, which indicates the energy up to which the states are filled at the highest excitation intensity. The right-hand side gives observed emission spectra showing the spikes of laser modes slightly above threshold.

Fig. 22.4

Left: The density of localized and extended states in CdS_1 − x Se _x (schematic). Right: Observed emission spectra below and above threshold [ 87M1]

Another aspect of disordered systems has been exploited recently to observe stimulated emission in indirect gap Al_1 − y Ga _y As and similar materials [90C1,  94C1,  94K1,  94W1,  96K1] as mentioned briefly in Chap. 21 and references therein. GaAs is a direct-gap material with E _g ≈ 1. 4 eV at 4 K and AlAs an indirect one with E _g ≈ 2. 2 eV. There exist alloys of all compositions y. For y < y _c = 0. 57, the minimum in the conduction band at the X point in the first Brillouin zone becomes lower than that at Γ. Under high (optical) pumping most of the electrons therefore sit in the X minimum. However, the alloy disordercouples the states and Γ and at X so that the electron–hole pairs can recombine without participation of phonons. This fact enhances the transition rate so strongly that stimulated emission has been reached for y < 0. 57 at a wavelength of 620 nm corresponding to 2. 2 eV, i.e., already in the orange part of the spectrum with gain values up to 200 cm⁻¹ [94W1].

The coupling between Γ and X states can also be presented in another way. The alloying destroys to some extent the translational invariance of the lattice. As a consequence the $\bf{k}$-conservation, which is based on this translational invariance as shown in Sect. 3.1.3, is partly relaxed, allowing recombination processes which violate a strict $\bf{k}$-conservation rule.

We should like to mention that some recombination processes which involve impurity centers also lead to optical gain. In fact, one of the first theoretical considerations of stimulated emission from semiconductors started with the recombination of bound-exciton complexes(BEC) like excitons bound to neutral acceptors or donors in CdS [62T1], because of their small spectral width. However, direct stimulated emission from BEC is observed only very rarely, but a slightly more complex mechanism gives optical gain. This is the recombination of a BEC under emission of a photon and an acoustic phonon, the so-called acoustic sideband. See Sect. 14.1 and [81K1]. This process also becomes possible because of the disturbed translational invariance. It is again a four-level process with low threshold as long as the acoustic phonon states are not (thermally) populated.

The two-electron transition explained in connection with Fig. 14.5 also yields gain as shown in [75K1] for ZnO.

Stimulated emission from trapped excitons in SnO₂ nano wires is reported in [07L1], although this material has a dipole forbidden, direct gap. See also Sect. 13.2.1.2.

Stimulated emission in the IR is reported for Si due to 2p → 1s transitions within a donor [00P1]. Stimulated Raman scattering and other processes for Si-based structures are discussed in [05R1].

One expects some advantages for semiconductor lasers in structures of reduced dimensionality. We come back to this aspect in Sect. 22.5. As a consequence of these expectations, gain processes have been widely investigated in structures of reduced dimensionality. We already mentioned a few examples above. Often the parameters of the gain are provided, such as its spectral position, spectral width, maximum value, temperature dependence and the pump power necessary to observe it, but no detailed information is given about the recombination process. We give in the following a further, rather limited selection of references to gain measurements, allowing the reader to enter deeper into the field. Apart from [92E1,  93J1,  94C1,  99K1,  01L1] we mention the following.

The gain in GaN-based structures or more generally group III-nitrides is investigated, e.g., in [97N1,  98O1,  99B1,  99M1,  03R1]. These structures resulted in commercially available light-emitting and laser diodes for the short wavelength part of the visible spectrum including the near-UV.

For the green spectral range, gain processes in Zn_1 − y Cd _y Se-based structures have been investigated [92N1,  93J1,  95K1,  95U1,  96G1,  97C1,  97P1,  98M1,  02G1] as well as ZnTe-based structures [94M1,  01C2,  01S3]. Laser diodes based on this material combination are still awaiting their commercial application, because the device lifetime is still limited to unacceptably low values by the creation of dark line defectsunder operation.

For laser emission in the IR based on lead salts or Ga_1 − y In _y N_1 − x As _x see[01L1,  03I1] and [02K4,  04K1,  04P2,  04S1], respectively.

Some examples for the investigation of gain and lasing in quantum wiresof III–V and II–VI materials can be found, e.g., in [90C1,  94C1,  94C2,  94K1,  99K1,  01H1].

A further reduction of the dimensionality leads to gain and lasing from quantum dotsincluding dots in insulators or self-assembled islands in semiconductors [92N1,  93F1,  93H1,  93M1,  95G1,  95W1,  99E1,  99I1,  99K1,  00K1,  01B1,  01E1,  02E1,  02S1,  03C1,  03H1,  05C1], in [97W1] of Chap. 1 and [95W1,  95W2,  96G1,  96W1] of Chap. 20. Other materials for gain and laser emission in the blue and near-UV, which have become fashionable again, are ZnO and ZnO-based quantum structures including nanorods (see, e.g., [75K1,  81K1,  02K3,  02Z1,  04L1,  04P1,  05K1,  06H1,  07K1,  07K2,  07Z1,  08Z2,  10K1] and references therein or [97S1,  01H1,  02K4] of Chap. 15) allowing, similarly to GaN, partial lasing well above room temperature [98B1,  00O1,  04P1].

ZnO powders are also used to investigate lasing in “active random media”,a process that involves enhanced backscattering of light at the ZnO grains [99C1,  00S1,  01C1,  01M1,  01S1,  04H2,  04Q1]. This process has been discussed in Sects. 8.15, 9.6 and 14.4 in connection with localization of free carriers and/or excitons and we shall come back to this aspect in Sect. 22.4.

22.2 Electron–Hole Plasmas

The laser process which has presently the highest importance with respect to technical applications in laser diodes is the stimulated emission from a degenerate electron–hole plasma as outlined in Sect. 21.4. Population inversion occurs when the chemical potential μ of the electron–hole pair system is located above the reduced gap E′ _g. See Fig. 21.1 or (21.16). It should be mentioned that some of the inelastic scattering processes or recombination under emission of a phonon or of a plasmon–phonon mixed state quasiparticle mentioned above may also occur in a plasma, contributing to the long-wavelength part of the gain spectra, or at densities below those fulfilling (21.16).

Population inversion can be achieved much more easily in indirect gap semiconductors like Si or Ge due to the longer carrier lifetimes, but the indirect nature of the transition makes the gain values so small that impracticably large volumes have to be pumped. However, far-infrared lasing has been reported in p-Ge between the heavy and light hole valence bands, or in a magnetic field between hole Landau (or cyclotron) levels [91G1]. Light emission and possibly gain from (partly doped) Si nanostructures are discussed in [01D1,  01K2,  02L1,  03D1] and the references given therein. On the other hand, gain values of up to 104 cm⁻¹ can be reached with direct gap semiconductors, so that devices with a length of the active material of about 100 μm can be pumped in a forward-biased pn junction to give power densities in the 105 W∕cm² range at the surface of the laser. Differential internal quantum efficienciesdeduced from the slope of the light power output versus electrical current input in excess of 50% have been reported.

We come now to one of the reasons why structures of reduced dimensionality are in principle favorable for the use as active materials in laser diodes.

In Fig. 22.5 we show calculated gain spectra resulting from an EHP for idealized In_. 53Ga_. 47As/InP quantum structures of various quasi-dimensions d from 3 down to 0 as indicated in the figure.

Fig. 22.5

Calculated gain spectra of idealized In_1 − y GaAs/InP quantum structures for various quasi-dimensions but with constant electron–hole pair density ${n}_{\text{ P}} = 3 \times \! 1018 \mathrm{c{m}^{-3}}$ [86A1,  99K1]

The density is kept constant and the confining linear lengths are always 10 nm. It is obvious that the absolute gain values increase and that the widths of the gain spectra decrease with decreasing d. This effect would allow smaller active volumes and/or lower threshold currents for decreasing d.

We stress that these calculations are valid for idealized structures. The calculations include, e.g., only homogeneous broadening.

In reality, inhomogeneous broadening due to alloying and fluctuations of the width in the confinement direction(s) tend to increase the width and to decrease the height of the gain spectra with decreasing d. Furthermore the confining potentials are sometimes rather moderate, e.g., for self-assembled islandsso that carriers can escape at elevated temperatures. Consequently, commercially available laser diodes are still based on double-heterostructures or quantum wells. The future will show if the various types of wires, e.g., etched, T-shaped or V-grooves (see Sects. 8.11–8.13) or even quantum dots or islands will make it into application.

For some examples of EHP gain spectra of bulk semiconductors and in quantum wells see Chap. 21.

Some references for stimulated emission in nano structures like quantum wells, ∼ wires and ∼ dots in addition to the ones cited already above in Chaps. 20 and 21 are [92D1,  93H1,  95W1,  96W2,  00O1,  04Z1,  05C1,  05C2,  06R1,  06R2,  09S1,  09W1,  09Z1,  10Z1].

As mentioned already above, the stimulated emission of ZnO is again investigated by many research groups world wide. Frequently the densities, at which stimulated and/or laser emission are observed, are in the range from a few times 10¹⁷ cm^− 3 to a few times 10¹⁸ cm^− 3. The use of (19.20), which overestimates the Mott density considerably as we learned e.g. with Table 21.1, lead many authors to the conclusion, that they are at the above densities still in a regime where excitons are good quasi-particles, and consequently they attributed the stimulated emission to excitonic processes, mainly to inelastic exciton–exciton scattering, the so-called P-bands, connected with the claim of very low threshold at room temperature.

This claim is already surprising, since the threshold of this process goes with increasing temperature through a minimum and increases at room temperature already significantly due to reabsorption by the inverse process as shown in Fig. 22.2 for CdS or calculated for ZnO in [81K1]. See e.g. Sect. 20.2. In fact, the inspection of Fig. 21.1 or Table 21.1 shows clearly that the Mott density is reached for ZnO around 10¹⁸ cm^− 3. The finite homogeneous width at room temperature makes it even difficult to speak about excitons as good quasi-particles for densities approaching 10¹⁷ cm^− 3. Definitely (20.3) with the low density value of the exciton binding energy is by no means any longer adequate. Since, on the other hand, the densities at which lasing is observed experimentally at room temperature by several groups (see e.g. the references in Fig. 20.3a) are too low to reach population inversion of the band-to-band transitions in the Coulomb-correlated EHP, alternative recombination processes have been suggested in [07K1,  07K3,  10K3] which may result in optical gain and lasing in a Coulomb-correlated EHP without population inversion of the band-to-band transition. These are radiative recombination processes of an electron–hole pair in the EHP under emission of one or more plasmon–phonon mixed state quanta or under scattering with a third carrier. Indeed the experimental data in Fig. 20.3a extrapolate nicely the data for the X-2LO and the X-el processes from temperatures below room temperature to beyond 500 K.

The fact, that lasing in ZnO around and above room temperature is due to radiative recombination in an EHP has been confirmed recently independently in [11V1].

22.3 Cavity Lasing

Apart from random lasing discussed below in Sect. 22.4, a laser needs a gain medium and some cavity. Without the latter one is limited to amplified spontaneous emission. The gain medium can be e.g. a gas, a dye, some ions in solid state matrices or – and this is the topic of this chapter – a semiconductor. The cavity may consist e.g. of external mirrors and/or gratings (for a few examples out of the many given in literature see [80W2,  92E1,  09K1]), of as grown or cleaved facets of the semiconductor itself [92E1] or of nano rods, which may form with their end facets longitudinal resonators or which may support whispering gallery modes [10K3].

We concentrate here on micro cavities, i.e. cavities with dimensions in at least one dimension of the order of one or several half wave lengths of the emitted light. A few examples e.g. for nano rods, ∼ ribbons, ∼ rings or ∼ tetrapods are [04Z1,  05P1,  06H1,  06P1,  07Z1,  08F1,  08Z1,  08Z2,  09U2,  09W1,  09Z1,  10E1,  10K2,  10K3,  10Z1,  11V1], for micro spheres and ∼ pyramids [05C1,  07H1], for micro discs acting as whispering galleries [92M1,  93H1,  94J1,  98B3,  98L1,  00C1,  00L1,  02K5,  03R1,  05S1,  06C1,  06M1,  06R1], for Bragg mirror cavities (partly in the form of micro pillars or VCSEL see also Sect. 20.5) [91B2,  97F2,  04L2,  06L1,  06R2,  07C1,  08K1,  09S1,  11G1] or photonic crystals [06S1].

The semiconductors forming the active material in the cavity may be in the form of thin layers, quantum wells, ∼ wires or ∼ dots or they may form the nano rods or ∼ belts themselves.

The gain processes may be all of the ones discussed in the preceding Chaps. 19–21 with some emphasis on the EHP.

We discussed already the lasing of cavity polaritons in Sect. 20.5 and found that the laser emission itself is not a proof for a Bose-Einstein condensation and may result even from an EHP [03D1,  10D1].

22.4 Random Lasing

We learned in Sect. 8.15 that disorder may lead to localization and outlined various models to describe this phenomenon. This concept can be applied to all wave-like species like free carriers, excitons (Sect. 14.4) or light waves.

If the light is confined in a disordered medium, which exhibits optical gain due to some pump process, laser emission may result from these localized modes. This phenomenon is usually called shortly “random lasing”. The localization length of these localized random lasing modes can be small (i.e. extending only over a few scattering centres) or more extended [01V1,  07V1]. We give here only a few out of the many references [99C1,  03C3,  04Q1,  08W1,  09U1,  09Y1,  10L2] and present examples for random lasing from optically pumped ZnO powder samples [09F1,  10K4].

In Fig. 22.6a1 we show how the ZnO powder with an average grain size of a few 100 nm was filled into a depression of a few μm depth and a few ten μm length lateral dimensions. This depression was etched into GaAs and GaAs was chosen because it absorbes at the emission wave length of ZnO. The inner lateral surface of the depression is rough and its shape is irregular to avoid lateral resonator effects.

Fig. 22.6

SEM image of one ensemble of ZnO nano crystals filled into a micro depression in GaAs (a1) and spatially resolved photoluminescence at room temperature under excitation with 2.5 MW/cm² (According to [10K4])

Figure 22.6a2 shows on a grey scale (colour online) the spatially resolved and spectrally integrated near edge emission under optical pumping with a frequency tripled Nd:YAG laser (ℏω_{{ e}xc}  = 3.49 eV, τ_{{ e}xc}  = 5 ns). The spectral resolution shows for these small samples reproducible and well defined localized laser modes, while larger powder samples have a very high spectral density of modes, where the occurrence of individual modes is more random [09F1,  10K4]. With high spatial and spectral resolution is was possible to observe emission spectra showing reproducibly only a few laser modes. See Fig. 22.7.

Fig. 22.7

Spectrally and spatially resolved luminescence (middle panel) taken from the marked, vertically extended and horizontally limited (3.5 μm) μm spatial region (see insert). The lower spectrum in the upper panels is vertically integrated and the upper one is a cut along the dotted line in the middle panel (According to [10K4])

Some of the random laser modes in Fig. 22.7 extend over the whole height (Y-direction) of the powder sample; others are limited to around of 1 μm, essentially limited by the spatial resolution in Y-direction. By shifting the detection window in X-direction it was confirmed that the spatial extension of the latter modes is also limited in this direction.

Consequently it can be stated, that closely localized and spatially more extended modes may coexist in random lasing, answering thus a question raised in [01V1,  07V1]. The occurrence of one or of the other type depends among others on the value of the optical gain [09F1,  09W3,  10K4].

22.5 Basic Concepts of Laser Diodes and Present Research Trends

Though this is not a textbook on semiconductor technology or devices, we outline in the first part of this section some developments and trends in the design of light emitting and laser diodes. For direct gap semiconductors we concentrate mainly on laser diodes (LD), for indirect ones and for organic semiconductors on light emitting diodes (LED) and on organic LED (O-LED), since electrically pumped LD are still difficult to realize for the two latter groups of materials. A detailed discussion of this topic is, however, beyond the scope of this book and may be found, e.g., in [81S1,  92E1,  94C1,  99K1,  12B1] and references therein. In the second part we list some of the present research trends in the field of semiconductors lasers.

The simplest way to build a laser diode would be to strongly bias a pn junction in the forward direction. In order to fulfill the condition μ > E′ _g for simple band-to-band recombination at least one (better both) of the n and p doped layers had to be doped so highly, that the Fermi level is in the band, i.e., that the population is degenerate (see Fig. 22.8a).

Fig. 22.8

Schematic drawings of a simple pn junction laser biased in the forward direction (a) of a double-heterostructure (b) of the effect of gain guiding (c) of a GRINSCH structure (d) and of a VCSEL (e)

This design has the disadvantage that neither the carriers nor the photons are confined or guided in any direction. Consequently, the threshold current density of such structures is extremely high so that such devices could only be operated in a pulsed mode and at reduced temperatures. This means that they were essential to prove that the concept of laser diodes works [62H1], but they were only of very limited practical use.

A major breakthrough was the invention of double-heterostructuresembedded in the intrinsic region of a pin diode (Fig. 22.8b). An undoped GaAs layer is, e.g., grown between n- and p-doped Al_1 − y Ga _y As layers. The reduced band gap of this layer allows one to reduce the doping levels, as can be seen from a comparison between Figs. 22.8a, b. Furthermore the carriers are confined and the light quanta are already guided in one dimension since the larger band gap material has a smaller refractive index. Lateral guiding of the light can be achieved, e.g., by gain guiding(Fig. 22.8c) using a strip-like contact. Consequently, the current is injected along this line and photons experience maximum amplification along the direction of this line, which coincides trivially with the axis of the resonator, which in turn consists generally in the cases shown in Fig. 22.8a–d of cleaved semiconductor surfaces, possibly coated on one side with highly reflecting (Bragg-) mirror. Photons emitted under an oblique angle with respect to the resonator axis are lost for the laser, but are only amplified over a shorter distance. A lateral wave guiding can be achieved, e.g., by etching a ridge under the upper contact, possibly followed by a coating or overgrowth with high-band-gap (i.e., low-index) material to reduce surface recombination. This design already allows cw room temperature operation.

Since the threshold current in the forward-biased pin laser diode depends, among other things, on the volume in which inversion has to be reached, a more advanced design has been realized in which the electrons and holes are confined in one (or a few) quantum wells with a typical width of about 10 nm, while the light quanta are guided in a structure with a width comparable to their wavelength, i.e., about 1 μm [81S1,  92E1,  96K1]. These graded index separate confinement heterostructure (GRINSCH)structures are shown in Fig. 22.8d. The funnel-shaped bandstructure of the optical waveguide helps to collect the injected carriers in the quantum well. Minimum threshold currents reached with these and similar structures are in the 1 mA range. A certain drawback of these structures is the limited spatial overlap of light-field and electron wavefunction.

Another currently very lively field of research concerns the development of surface-emitting laser diodes which can be arranged in one- and two-dimensional arrays and addressed individually. These arrays are very important ingredients for high brightness displays and in parallel electro–optic data handling (Sect. 24.2).

Among the most promising concepts are so-called vertical cavity surface emitting lasers (VCSELs)shown in Fig. 22.8e. These are monolithic devices, where two stacks of Bragg mirrors (BM) are grown epitaxially together with the cavity, which contains one or a few quantum wells or self-organized quantum islands at the positions of the antinodes of the cavity (also see Sect. 17.1). The current is injected, e.g., through n ⁺ and p ⁺ doped Bragg reflectors forming the cavity. Ideally two-dimensional arrays of VCSELs are fabricated, in which every single laser structure can be addressed individually to allow, for high-brightness two-dimensional displays. The concept of VCESL’s is closely related to lasing involving cavity polaritons and their interaction processes. See Sects. 20.5, 22.1 or 22.3 and the references given there or [04K1,  10D1,  12B1].

Laser diodes have developed into various directions. Linear arrays are optimized towards maximum output to pump, e.g., Nd-solid-state lasers. Others are optimized towards minimum threshold currents or towards maximum modulation band width for data transfer. A basically simple way to modulate the output power is via current modulation. Another possibility has been realized by sending short optical pulses from the laser [98H1] with sufficient excess energy to heat the carrier gas to temperatures that reduce μ below E′ _g. As a consequence, the laser switches off, although the electron–hole pair density is increased. After cooling the carriers the laser starts operating again.

Concerning the emission wavelength there are the two windows in glass fiber communication at 1.55 and 1. 3 μm that can be covered by Ga_1 − y In _y As_1 − x P _x -based structures and more recently by Ga_1 − y In _y N _x As_1 − x -based ones. Ga_1 − y Al _y As-based structures emit in the (nearinfra-) red. See e.g. [00E1,  01S2,  04W2,  07D1,  07H1,  11M1] and references therein.

For full color displays,diodes in the yellow and green (e.g., those based on Cd_1 − y Zn _y Se) and in the blue (based on Al _y1Ga${}_{1-y1-y2}$In _y2N or on Cd _y1Mg _y2 Zn${}_{1-y1-y2}$O) are under investigation. References for these materials have been given above, but some can be added, e.g., [92N2,  96R1,  96R2] for ZnO (though this is definitely not the “first report on lasing in ZnO by optical pumping” as can be seen from [75K1,  81K1] and the references given therein) e.g.[04L1,  07K1,  07T1,  08Z1,  09W1,  10E1,  10K1,  10K3] for some further and partly more recent papers and references an ZnO, [01C2,  01S3] for ZnTe-based materials and [00F1,  00T1,  09N1] for Zn_1 − y Cd _y Se-based structures or [02A1] for CdS-based ones and [75R1,  98K1,  00K2,  00L2,  07T1,  09M1,  09N1,  09O1,  09Q1,  10L1,  11P1,  11W1] for group III-nitride structures.

Shorter emission wavelengths also allow one to increase the density of bits per unit area n _b, e.g., on CDs. There is a rough scaling law n _b ∼ λ^− 2. This means that a transition from the near-IR to the near-UV allows one to increase n _b by a factor often.

Organic semiconductorsor phosphors are also being investigated for the purpose of (full color) displays. Presently these are mainly in the shape of light emitting diodes or electroluminescent devices, i.e., devices that operate below the onset of stimulated emission [87T1,  02C1,  02C2,  02T1,  03M1,  04H1,  04Q1,  04W1,  04W3,  06C2,  06K1,  09W2,  11S1]. In this sense, one also investigates Si nanocrystals as they occur in porous Si (see, e.g., [01K2] and references therein).

Now we give a few references on some further present research trends in semiconductor lasers.

The incorporation of CdSe-based quantum islands as active materials might help to improve the lifetime of Cd_1 − y Zn _y S_1 − x Se _x -based II–VI lasers [01K1,  02K1].

Inter-subband or inter-miniband transitionslike the ones described in Sect. 21.5 can be inverted to give laser emission. Examples for quantum structures can be found, e.g., in [96W1,  02K2,  04W2]. We already mentioned an example for bulk p-Ge [91G1].

Efficient THz radiation can be obtained e.g. by operating a VCSEL (see above) simultaneously on two spectrally close modes and by producing the difference frequency of them in a nonlinear crystal [04H3].

Quantum cascade lasersare unipolar devices that use, e.g., an electron several times for the emission of photons. See, e.g., [96F1,  02K2,  03C3] and references therein. The basic idea is roughly the following. Electrons are injected in a ${p}^{+}{in}^{+}$ structure, which contains a periodic array of superlattices and quantum wells in the intrinsic region. The electron reaches the n _z = 2 state in the quantum well through the miniband states of the SL and performs a ${n}_{\text{ z}} = 2 \rightarrow {n}_{\text{ z}} = 1$ transition under emission of a photon, e.g., in the mid-IR, and tunnels from the n _z = 1 state into the minibandof the next SL, where the process repeats itself. For the incorporation of quantum cascade lasers in photonic crystalssee [03C1].

Another topic are lasers without inversion,treated, e.g., in [89H1,  96F1,  01B2] and references therein.

Concerning the resonators, semiconductor laser resonators are made in the simplest case by cleaved surfaces. They may also contain Bragg mirrors like the cavities in Sect. 17.1 or the VCSEL structures mentioned above or in [01E1,  02P1], or other distributed feedback(DFB) structures [81S1,  92E1].

For various designs of micro-cavity lasers see Sect. 22.3 or [12B1].

For quantum kinetics, spatio-temporal dynamics and quantum fluctuations in semiconductor lasers see, e.g., [96H1,  03G1] for lasing without inversion or threshold less lasing see e.g. [94I1].

Another point we want to address is the temperature dependence of the laser threshold. As seen, e.g., from Fig. 22.2 there may be various types of dependencies. For laser diodes one often finds for the forward current at threshold an empirical relation

$${I}_{\text{ th}}\left (T\right ) = {I}_{0}\exp \left \{T/{T}_{0}\right \}.$$

(22.3)

The art of device development and materials engineering is currently to make I ₀ as small as possible, e.g., by reducing losses including the density dependent Auger-recombination and to make T ₀ large so that the properties of the laser do not change much with varying operating temperature.

Recently one tried also to obtain LD from group IV element-semiconductor based structures, including diamond for the UV. For a few examples see e.g. [00P1,  01K3,  04D1,  05R1,  08L1].

Organic semiconductors can be optically pumped to show laser emission [98B2]. However, it is still difficult to pump O-LED electrically beyond laser threshold. A few references for O-LED are [87T1,  98B2,  02C1,  02C2,  02T1,  03M1,  04H1,  04Q1,  04W1,  04W3,  04X1,  06C2,  06K1,  09W2,  11S1] where also the various organic compounds are found, which are used in O-LED.

With this small excursion into more application-oriented topics in semiconductor laser research and development we close this chapter.

22.6 Problems

• 1. How would you expect the curves of Fig. 22.2 a to shift with respect to each other if the total losses of the cavity increases or decreases by a factor $\sqrt{10}$?

• 2. Why is lasing via a degenerate EHP at room temperature more likely in standard III–V than in II–VI compounds? To answer this question calculate the effective density of states for electrons and holes in various 2d and 3d materials. Why should you do so?

• 3. Calculate the gain spectra for a 3d degenerate EHP of a direct and an indirect-gap semiconductor with otherwise identical parameters. Do the same for a quasi-2d direct gap material.

• 4. Why does lasing generally occur not at the maxima of the gain spectra but rather on their low-energy sides?