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3. Lasers
The word LASER is an acronym for light amplification by stimulated emission of radiation, which sums up the operation of an important optical and electronic device. The laser is a source of highly directional, monochromatic, coherent light, and as such it has revolutionized some longstanding optical problems and has created some new fields of basic and applied optics. The light from a laser, depending on the type, can be a continuous beam of low or medium power, or it can be a short burst of intense light delivering mil lions of watts. Light has always been a primary communications link between humans and the environment, but until the invention of the laser, the light sources available for transmitting information and performing experiments were generally neither monochromatic nor coherent, and were of relatively low intensity. Thus the laser is of great interest in optics; but it is equally important in optoelectronics, particularly in fiber-optic communications. The last three letters in the word laser are intended to imply how the device operates: by the stimulated emission of radiation. In section 2 we discussed the emission of radiation when excited electrons fall to lower energy states; but generally, these processes occur randomly and can therefore be classed as spontaneous emission. This means that the rate at which electrons fall from an upper level of energy E2 to a lower level E1 is at every instant proportional to the number of electrons remaining in E2 (the population of E2). Thus if an initial electron population in E2 were allowed to decay, we would expect an exponential emptying of the electrons to the lower energy level, with a mean decay time describing how much time an average electron spends in the upper level. An electron in a higher or excited state need not wait for spontaneous emission to occur, however; if conditions are right, it can be stimulated to fall to the lower level and emit its photon in a time much shorter than its mean spontaneous decay time. The stimulus is provided by the presence of photons of the proper wavelength. Let us visualize an electron in state E2 waiting to drop spontaneously to E1 with the emission of a photon of energy hn12 = E2 -E1 (FIG. 15). Now we assume that this electron in the upper state is immersed in an intense field of photons, each having energy hn12 = E2 -E1, and in phase with the other photons. The electron is induced to drop in energy from E2 to E1, contributing a photon whose wave is in phase with the radiation field. If this process continues and other electrons are stimulated to emit photons in the same fashion, a large radiation field can build up. This radiation will be monochromatic since each photon will have an energy of precisely hn12 = E2 -E1 and will be coherent, because all the photons released will be in phase and reinforcing. This process of stimulated emission can be described quantum mechanically to relate the probability of emission to the intensity of the radiation field. Without quantum mechanics we can make a few observations here about the relative rates at which the absorption and emission processes occur. Let us assume the instantaneous populations of E1 and E2 to be n1 and n2, respectively. We know from earlier discussions of distributions and the Boltzmann factor that at thermal equilibrium the relative population will be
n2 n1 = e-(E2 -E1)/kT = e-hn12/kT (eqn. 7)
if the two levels contain an equal number of available states.
FIG. 15 Stimulated transition of an electron from an upper state to
a lower state, with accompanying photon emission.
The negative exponent in this equation indicates that n2 V n1 at equilibrium; that is, most electrons are in the lower energy level as expected. If the atoms exist in a radiation field of photons with energy hn12, such that the energy density of the field is t(n12), 3 then stimulated emission can occur along with absorption and spontaneous emission. The rate of stimulated emission is proportional to the instantaneous number of electrons in the upper level n2 and to the energy density of the stimulating field t(n12). Thus we can write the stimulated emission rate as B21n2 t(n12), where B21 is a proportionality factor. The rate at which the electrons in E1 absorb photons should also be proportional to t(n12), and to the electron population in E1. Therefore, the absorption rate is B12n1t(n12), where B12 is a proportionality factor for absorption. Finally, the rate of spontaneous emission is proportional only to the population of the upper level. Introducing still another coefficient, we can write the rate of spontaneous emission as A21n2.
For steady state the two emission rates must balance the rate of absorption to maintain constant populations n1 and n2 (FIG. 16).
(eqn. 8a)
This relation was described by Einstein, and the coefficients B12, A21, B21 are called the Einstein coefficients. We notice from Eq. (eqn. 8a) that no energy density t is required to cause a transition from an upper to a lower state; spontaneous emission occurs without an energy density to drive it. The reverse is not true, however; exciting an electron to a higher state (absorption) requires the application of energy, as we would expect thermodynamically.
At equilibrium, the ratio of the stimulated to spontaneous emission rates is generally very small, and the contribution of stimulated emission is negligible. With a photon field present,
Stimulated emission rate Spontaneous emission rate
(eqn. 8b)
As Eq. (eqn. 8b) indicates, the way to enhance the stimulated emission over spontaneous emission is to have a very large photon field energy density r(n12). In the laser, this is encouraged by providing an optical resonant cavity in which the photon density can build up to a large value through multiple internal reflections at certain frequencies (n).
Similarly, to obtain more stimulated emission than absorption we must have n2 / n1:
(eqn. 9)
Thus if stimulated emission is to dominate over absorption of photons from the radiation field, we must have a way of maintaining more electrons in the upper level than in the lower level. This condition is quite unnatural, since Eq. (eqn. 7) indicates that n2 /n1 is less than unity for any equilibrium case. Because of its unusual nature, the condition n2 / n1 is called population inversion. It is also referred to as a condition of negative temperature. This rather startling terminology emphasizes the non-equilibrium nature of population inversion, and refers to the fact that the ratio n2 /n1 in Eq. (eqn. 7) could be larger than unity only if the temperature were negative. Of course, this manner of speaking does not imply anything about temperature in the usual sense of that word. The fact is that Eq. (eqn. 7) is a thermal equilibrium equation and cannot be applied to the situation of population inversion without invoking the concept of negative temperature.
FIG. 17 Resonant modes within a laser cavity.
In summary, Eqs. (eqn. 8b) and (eqn. 9) indicate that if the photon density is to build up through a predominance of stimulated emission over both spontaneous emission and absorption, two requirements must be met. We must provide (1) an optical resonant cavity to encourage the photon field to build up and (2) a means of obtaining population inversion.
An optical resonant cavity can be obtained using reflecting mirrors to reflect the photons back and forth, allowing the photon energy density to build up. One or both of the end mirrors are constructed to be partially transmitting so that a fraction of the light will "leak out" of the resonant system. This transmitted light is the output of the laser. Of course, in designing such a laser one must choose the amount of transmission to be a small perturbation on the resonant system. The gain in photons per pass between the end plates must be larger than the transmission at the ends, scattering from impurities, absorption, and other losses. The arrangement of parallel plates providing multiple internal reflections is similar to that used in the Fabry-Perot interferometer, thus the reflecting ends of the laser cavity are often referred to as Fabry-Perot faces. As FIG. 17 indicates, light of a par ticular frequency can be reflected back and forth within the resonant cavity in a reinforcing (coherent) manner if an integral number of half-wavelengths fit between the end mirrors. Thus the length of the cavity for stimulated emission must be
L = ml / 2 (eqn. 10)
where m is an integer. In this equation l is the photon wavelength within the laser material. If we wish to use the wavelength l0 of the output light in the atmosphere (often taken as the vacuum value), the index of refraction n of the laser material must be considered
l0 = ln (eqn. 11)
In practice, L W l, and Eq. (eqn. 10) is automatically satisfied over some portion of the mirror. An important exception occurs in the vertical cavity surface-emitting lasers discussed in Section 4.4, for which the cavity length is comparable to the wavelength.
There are ways of obtaining population inversion in the atomic levels of many solids, liquids, and gases, and in the energy bands of semiconduc tors. Thus the possibilities for laser systems with various materials are quite extensive. An early laser system used a ruby rod. In gas lasers, electrons are excited to metastable levels in molecules to achieve population inver sion. These are interesting and useful laser systems, but in view of our empha sis on semiconductor devices in this book, we will move to the description of semiconductor lasers.
4. Semiconductor lasers
The laser became an important part of semiconductor device technology in 1962 when the first p-n junction lasers were built in GaAs (infrared) 5 and GaAsP (visible).
We have already discussed the incoherent light emission from p-n junctions (LEDs), generated by the spontaneous recombination of electrons and holes injected across the junction. In this section we shall concentrate on the requirements for population inversion due to these injected carriers and the nature of the coherent light from p-n junction lasers. These devices differ from solid, gas, and liquid lasers in several important respects. Junction lasers are remarkably small (typically on the order of 0.1 * 0.1 * 0.3 mm), they exhibit high efficiency, and the laser output is easily modulated by controlling the junction current. Semiconductor lasers operate at low power compared, for example, with ruby or CO2 lasers; on the other hand, these junction lasers compete with He-Ne lasers in power output. Thus the function of the semiconductor laser is to provide a portable and easily controlled source of low-power coherent radiation. They are particularly suitable for fiber-optic communication systems (Section 2.2).
FIG. 18 Band diagram of a p-n junction laser under forward bias. The
crosshatched region indicates the inversion region at the junction.
4.1 Population inversion at a junction
If a p-n junction is formed between degenerate materials, the bands under forward bias appear as shown in FIG. 18. If the bias (and thus the current)
is large enough, electrons and holes are injected into and across the transition region in considerable concentrations. As a result, the region about the junction is far from being depleted of carriers. This region contains a large concentration of electrons within the conduction band and a large concentration of holes within the valence band. If these population densities are high enough, a condition of population inversion results, and the region about the junction over which it occurs is called an inversion region.
Population inversion at a junction is best described by the use of the concept of quasi-Fermi levels (Section 4.3.3). Since the forward-biased condition of FIG. 18 is a distinctly nonequilibrium state, the equilibrium equations defining the Fermi level are not applicable. In particular, the concentration of electrons in the inversion region (and for several diffusion lengths into the p material) is larger than equilibrium statistics would imply; the same is also true for the injected holes in the n material. We can use Eq. ( 4-15) to describe the carrier concentrations in terms of the quasi-Fermi levels for electrons and holes in steady state. Thus
(eqn. 12a)
(eqn. 12b)
Using Eqs. (eqn. 12a) and (b), we can draw Fn and Fp on any band dia gram for which we know the electron and hole distributions. For example, in FIG. 18, Fn in the neutral n region is essentially the same as the equilibrium Fermi level EFn. This is true to the extent that the electron concentration on the n side is equal to its equilibrium value. However, since large numbers of electrons are injected across the junction, the electron concentration begins at a high value near the junction and decays exponentially to its equilibrium value np deep in the p material. Therefore, Fn drops from EFn as shown in FIG. 18. We notice that, deep in the neutral regions, the quasi-Fermi levels are essentially equal. The separation of Fn and Fp at any point is a measure of the departure from equilibrium at that point. Obviously, this departure is considerable in the inversion region, since Fn and Fp are separated by an energy greater than the band gap (FIG. 19).
Unlike the case of the two-level system discussed in Section 3, the condition for population inversion in semiconductors must take into account the distribution of energies available for transitions between the bands. The basic definition of population inversion holds-for dominance of stimu lated emission between two energy levels separated by energy hn, the electron population of the upper level must be greater than that of the lower level. The unusual aspect of a semiconductor is that bands of levels are avail able for such transitions. Population inversion obviously exists for transitions between the bottom of the conduction band Ec and the top of the valence band Ev in FIG. 19. In fact, transitions between levels in the conduction band up to Fn and levels in the valence band down to Fp take place under conditions of population inversion. For any given transition energy hn in a semiconductor, population inversion exists when
(eqn. 13a)
For band-to-band transitions, the minimum requirement for population inversion occurs for photons with hv = Ec -Ev = Eg
(Fn -Fp) / Eg (eqn. 13b)
FIG. 19 Expanded view of the inversion region.
FIG. 20 Variation of the inversion-region width with forward bias:
V(a) 6 V(b).
When Fn and Fp lie within their respective bands (as in FIG. 19), stimulated emission can dominate over a range of transitions, from hv = (Fn -Fp) to hv = Eg. As we shall see below, the dominant transitions for laser action are determined largely by the resonant cavity and the strong recombination radiation occurring near hv = Eg.
In choosing a material for junction laser fabrication, it is necessary that electron-hole recombination occur directly, rather than through trap ping processes such as are dominant in Si or Ge. Gallium arsenide is an example of such a "direct" semiconductor. Furthermore, we must be able to dope the material n-type or p-type to form a junction. If an appropriate resonant cavity can be constructed in the junction region, a laser results in which population inversion is accomplished by the bias current applied to the junction (FIG. 20).
4.2 Emission spectra for p-n junction lasers
Under forward bias, an inversion layer can be obtained along the plane of the junction, where a large population of electrons exists at the same location as a large hole population. A second look at FIG. 19 indicates that spontaneous emission of photons can occur due to direct recombination of electrons and holes, releasing energies ranging from approximately Fn -Fp to Eg.
That is, an electron can recombine over an energy from Fn to Fp, yielding a photon of energy hv = Fn -Fp, or an electron can recombine from the bottom of the conduction band to the top of the valence band, releasing a photon with h n = Ec -Ev = Eg. These two energies serve as the approximate outside limits of the laser spectra.
FIG. 21 Light intensity vs. photon energy hv for a junction laser:
(a) incoherent emission below threshold; (b) laser modes at threshold; (c)
dominant laser mode above threshold. The intensity scales are greatly compressed
from (a) to (b) to (c).
The photon wavelengths which participate in stimulated emission are determined by the length of the resonant cavity as in Eq. (eqn. 10). FIG. 21 illustrates a typical plot of emission intensity vs. photon energy for a semiconductor laser. At low current levels (FIG. 21a), a spontaneous emission spectrum containing energies in the range Eg 6 h n 6 (Fn -Fp) is obtained. As the current is increased to the point that significant population inversion exists, stimulated emission occurs at frequencies corresponding to the cavity modes as shown in FIG. 21b. These modes correspond to successive numbers of integral half-wavelengths fitted within the cavity, as described by Eq. (eqn. 10). Finally, at a still higher current level, a most preferred mode or set of modes will dominate the spectral output (FIG. 21c). This very intense mode represents the main laser output of the device; the output light will be composed of almost monochromatic radiation superimposed on a relatively weak radiation background, due primarily to spontaneous emission.
The separation of the modes in FIG. 21b is complicated by the fact that the index of refraction n for GaAs depends on wavelength l. From
Eq. (eqn. 10) we have
(eqn. 14)
If m (the number of half-wavelengths in L) is large, we can use the derivative to find its rate of change with lambda_0:
(eqn. 15)
Now reverting to discrete changes in m and l0, we can write
(eqn. 16)
If we let delta_m = -1, we can calculate the change in wavelength ?l0 between adjacent modes (i.e., between modes m and m -1).
4.3 The basic semiconductor laser
To build a p-n junction laser, we need to form a junction in a highly doped, direct semiconductor (GaAs, for example), construct a resonant cavity in the proper geometrical relationship to the junction, and make contact to the junction in a mounting which allows for efficient heat transfer. The first lasers were built as shown in FIG. 22. Beginning with a degenerate n-type sample, a p region is formed on one side, for example by diffusing Zn into the n-type GaAs. Since Zn is in column II of the periodic table and is introduced substitutionally on Ga sites, it serves as an acceptor in GaAs; therefore, the heavily doped Zn diffused layer forms a p+ region (FIG. 22b). At this point we have a large-area planar p-n junction. Next, grooves are cut or etched along the length of the sample as in FIG. 22c, leaving a series of long p regions isolated from each other. These p-n junctions can be cut or broken apart (FIG. 22d) and then cleaved into devices of the desired length.
At this point in the fabrication process, the very important requirements of a resonant cavity must be considered. It is necessary that the front and back faces (FIG. 22e) be flat and parallel. This can be accomplished by cleaving. If the sample has been oriented so that the long junctions of FIG. 22d are perpendicular to a crystal plane of the material, it is possible to cleave the sample along this plane into laser devices, letting the crystal structure itself provide the parallel faces. The device is then mounted on a suitable header, and contact is made to the p region. Various techniques are used to provide adequate heat sinking of the device for large forward current levels.
FIG. 22 Fabrication of a simple junction laser: (a) degenerate n-type sample;
(b) diffused p layer; (c) isolation of junctions by cutting or etching;
(d) individual junction to be cut or cleaved into devices; (e) mounted laser
structure.
FIG. 23 Use of a single heterojunction for carrier confinement in laser
diodes: (a) AlGaAs heterojunction grown on the thin p-type GaAs layer; (b)
band diagrams for the structure of (a), showing the confinement of electrons
to the thin p region under bias.
4.4 Heterojunction lasers
The device described above was the first type used in the early development of semiconductor lasers. Since the device contains only one junction in a single type of material, it is referred to as a homojunction laser. To obtain more efficient lasers, and particularly to build lasers that operate at room temperature, it is necessary to use multiple layers in the laser structure. Such devices, called heterojunction lasers, can be made to operate continuously at room temperature to satisfy the requirements of optical communications. An example of a heterojunction laser is shown in FIG. 23. In this structure the injected carriers are confined to a narrow region so that population inversion can be built up at lower current levels. The result is a lowering of the thresh old current at which laser action begins. Carrier confinement is obtained in this single-heterojunction laser by the layer of AlGaAs grown epitaxially on the GaAs.
In GaAs the laser action occurs primarily on the p side of the junction due to a higher efficiency for electron injection than for hole injection. In a normal p-n junction the injected electrons diffuse into the p material such that population inversion occurs for only part of the electron distribution near the junction. However, if the p material is narrow and terminated in a barrier, the injected electrons can be confined near the junction. In FIG. 23a, an epitaxial layer of p-type AlGaAs (Eg ? 2eV) is grown on top of the thin p-type GaAs region. The wider band gap of AlGaAs effectively terminates the p-type GaAs layer, since injected electrons do not surmount the barrier at the GaAs-AlGaAs heterojunction (FIG. 23b). As a result of the confinement of injected electrons, laser action begins at a substantially lower current than for simple p-n junctions. In addition to the effects of carrier confinement, the change of refractive index at the heterojunction provides a waveguide effect for optical confinement of the photons.
A further improvement can be obtained by sandwiching the active GaAs layer between two AlGaAs layers (FIG. 24). This double-heterojunction structure further confines injected carriers to the active region, and the change in refractive index at the GaAs-AlGaAs boundaries helps to confine the generated light waves. In the double-heterojunction laser shown in FIG. 24b the injected current is restricted to a narrow stripe along the lasing direction, to reduce the total current required to drive the device. This type of laser was a major step forward in the development of lasers for fiber-optic communications.
FIG. 24 A double-heterojunction laser structure: (a) multiple layers used
to confine injected carriers and provide waveguiding for the light; (b)
a stripe geometry designed to restrict the current injection to a narrow
stripe along the lasing direction. One of many methods for obtaining the
stripe geometry, this example is obtained by proton bombardment of the unshaded
regions in (b), which converts the GaAs and AlGaAs to a semi-insulating
form.
FIG. 25 Separate confinement of carriers and waveguiding: (a) use of
separate changes in AlGaAs alloy composition to confine carriers in the
region (d) of the smallest band gap and to obtain waveguiding (w) at the
larger step in the refractive index; (b) grading the alloy composition,
and therefore the refractive index, for better waveguiding and carrier confinement.
Separate confinement and Graded index channels. One of the disadvantages of the double-heterostructure laser shown in FIG. 24 is the fact that the carrier confinement and the optical waveguiding both depend on the same heterojunctions. It is much better to optimize these two functions by using a narrow confinement region for keeping the carriers in a region of high recombination, and a somewhat wider optical waveguide region. In FIG. 25a we show a separate confinement laser in which the width of the optical waveguiding region (w) is optimized by using the refractive index step at a separate heterojunction from that used to confine the carriers. For example, in the GaAs-AlxGa1-xAs system the optical confinement (waveguiding) occurs at a boundary with much larger composition x (and therefore smaller refractive index) than is the case for the carrier confinement barrier. By grading the composition of the AlGaAs it is possible to obtain even better waveguiding. For example, in FIG. 25b a parabolic grading of the refractive index leads to a waveguide within the laser analogous to that shown in FIG. 12 for a fiber. This graded index separate confinement heterostructure (GRINSCH) laser also provides built-in fields for better electron confinement.
Vertical cavity surface-emitting lasers (vcsels). There are advantages to laser structures in which light is emitted normal to the surface, including ease of device testing on the wafer before packaging. An interesting approach is the VCSEL, in which the cavity mirrors are replaced by DBRs, which use many partial reflectors spaced to reflect light constructively. DBRs can be grown by MBE or OMVPE. In FIG. 26 the bottom DBR mirror of a VCSEL is composed of many alternating layers of AlAs and GaAs with thickness one-quarter of a wavelength in each material. The top mirror is composed of deposited dielectric layers (alternating ZnSe and MgF). Current is funneled into the active region from the top contact by using an oxide layer achieved by laterally oxidizing an AlGaAs layer to form an aluminum oxide. The active region of the laser employs InGaAs-GaAs quantum wells, and the GaAs cavity between the two DBRs is one wavelength long. The VCSEL can be made with much shorter cavity length than other structures, and as a result of Eq. (eqn. 16) the laser modes are widely separated in wavelength. Thus single-mode laser operation is more easily achieved with the VCSEL. Lasing can be achieved at very low current (6 50 uA) with this device.
FIG. 26 Schematic cross section of oxide-confined vertical cavity surface-emitting
laser diode. [D. G. Deppe et al., IEEE J. Selected Topics in Quantum Elec.,
3(3) (June 1997): 893-904.]
4.5 Materials for semiconductor lasers
We have discussed the properties of the junction laser largely in terms of GaAs and AlGaAs. However, as discussed in Section 2.2, the InGaAsP/InP system is particularly well suited for the type of lasers used in fiber-optic communication systems. Lattice-matching (Section 1.4.1) is important in creating heterostructures by epitaxial growth. The fact that the AlGaAs band gap can be varied by choice of composition on the column III sublattice allows the formation of barriers and confining layers such as those shown in Section 4.4. The quaternary alloy InGaAsP is particularly versatile in the fabrication of laser diodes, allowing considerable choice of wavelength and flexibility in lattice-matching. By choice of composition, lasers can be made in the infrared range 1.3-1.55 mm required for fiber optics. Since four components can be varied in choosing an alloy composition, InGaAsP allows simultaneous choice of energy gap (and therefore emission wavelength) and lattice constant (for lattice-matched growth on convenient substrates). In many applications, however, other wavelength ranges are required for laser output. For example, the use of lasers in pollution diagnostics requires wavelengths farther in the infrared than are available from InGaAsP and AlGaAs. In this application the ternary alloy PbSnTe provides laser out put wavelengths from about 7 om to more than 30 um at low temperatures, depending on the material composition. For intermediate wavelengths, the InGaSb system can be used.
Materials chosen for the fabrication of semiconductor lasers must be efficient light emitters and also be amenable to the formation of p-n junctions and in most cases the formation of heterojunction barri ers. These requirements eliminate some materials from practical use in laser diodes. For example, semiconductors with indirect band gaps are not sufficiently efficient light emitters for practical laser fabrication. The II-VI compounds, on the other hand, are generally very efficient at emitting light but junctions are difficult to form. By modern crystal growth techniques such as MBE and MOVPE it is possible to grow junctions in ZnS, ZnSe, ZnTe, and alloys of these materials, using N as the acceptor. Lasers can be made in these materials which emit in the green and blue-green regions of the spectrum.
In recent years much progress has been made in the growth of large band gap semiconductors using GaN, and its alloys with InN and AlN. The InAlGaN system has direct band gaps over the entire alloy composition range, and hence offers very efficient light emission. Band gaps range from about 2 eV for InN to 3.4 eV for GaN and 5 eV for AlN. This covers the wavelength range from about 620 nm to about 248 nm, which is from blue to UV. The resurgence of interest in this field was triggered by the work of Nakamura at Nichia Corporation in Japan who demonstrated very high-efficiency blue light emitting diodes (LEDs) in GaN.
Two of the problems that had stymied progress in this field since pio neering work by Pankove in the 1970s was the absence of a suitable sub strate having sufficient lattice match with GaN, and the inability to achieve p-type doping in this semiconductor. GaN bulk crystals cannot be grown easily because of the high vapor pressure of the nitrogen-bearing precursor (generally ammonia). This requires growth at high temperature and pressure. This precludes using bulk GaN wafers as substrates for epitaxial growth. However, epitaxial layers can be grown on other substrates with reasonable success, in spite of the lattice mismatch.
GaN exists in the cubic zinc blende form (which is the preferred structure) as well as the hexagonal wurtzite form. It was demonstrated recently that cubic GaN could be grown heteroepitaxially on sapphire, even though it is not lattice-matched to GaN. In fact, sapphire does not even have a cubic crystal structure-it is hexagonal. The lattice constant of GaN is about 4.5 Å, while that of sapphire is 4.8 Å, which is a huge lattice mismatch. Contrary to what would normally be expected, however, high-quality epitaxial GaN films can be grown on sapphire by MOCVD using ammonia and tri-methyl gallium as the precursors. One possible reason for the high quality of the films, as evidenced by blue LEDs and short wavelength lasers fabricated in these nitrides, is that these large-band-gap semiconductors have very high chemical bond strengths. As discussed in Section 2.1, the addition of In to GaN results in lower bandgap In-rich regions which provide sites for radiative recombination, minimizing the loss of carriers at dislocations. Another breakthrough required in the nitrides was the ability to achieve high p-type doping so that p-n junctions could be formed. It has been demonstrated that Mg (which is a column II element) doping of MOCVD films, followed by high temperature annealing can be used to achieve high acceptor concentrations in these systems.
Short wavelength emitters such as UV/blue semiconductor lasers are important for storage applications such as digital versatile discs (DVDs), which are higher density versions of compact discs (CDs). The storage density on these discs is inversely proportional to the square of the laser wavelength that is used to read the information. Thus reducing the laser wavelength by a factor of two leads to a four-fold increase of storage density. Such an increased storage capacity opens up entirely new applications for DVDs that were not possible previously with conventional CDs, for example, the storage of full-length movies. A recent example of success in this rapidly progressing field is a 417-nm semiconductor laser made with InGaN multi-quantum-well heterostructures.
FIG. 27 Quantum cascade laser based on three-level system in conduction
band. Photons are emitted each time there is a transition from level 3 to
2 in this "staircase" of stages.
Carriers are injected from level 1 of one stage to level 3 of the next
stage through a superlattice (SL) miniband.
4.6 Quantum cascade lasers.
The lasers described above are bipolar devices involving interband transitions. Electrons in the conduction band recombine with holes in the valence band, and photons are emitted at an energy determined by the bandgap. This limits laser operation to direct bandgap materials for reasons discussed in section 4.
The quantum cascade laser (QCL), on the other hand, involves unipolar operation based on inter-subband transitions in a series of coupled quantum wells, separated by superlattices (FIG. 27). As discussed in section 1 and 3, quantum wells and superlattices are made of semicond ductor heterostructures with layer thicknesses comparable to or smaller than the deBroglie wavelengths of the electrons.
In quantum wells, one gets confinement in one direction, but plane wave states in the other two directions. This leads to sub-band levels shown in FIG. 27 . By carefully designing the quantum well structure in a QCL, one can design a three-state system (FIG. 27). By biasing the QCL, car riers are injected from the left into state 3, creating a population inversion between states 3 and 2. There is a relatively thick tunnel barrier between states 3 and 2 in order to allow the population inversion. However, the barrier must be narrow enough to allow overlap of the wavefunctions in states 3 and 2, and allow transitions between these levels and allow lasing. Photons in the QCL are emitted upon transition to state 2. Carriers in state 2 are closely coupled by a thin tunnel barrier to state 1, and are immediately removed to this state, so that the population inversion between states 3 and 2 can be maintained. Carriers in state 1 are subsequently injected to state 3 in the next stage of the QCL through the digitally graded superlattice (SL), and the whole process is repeated, until the current is extracted on the right-hand side of the QCL. It is clear that the number of photons emitted per electron injected depends on the number of these energy staircases, 1, 2, 3.
Since the QCL involves electronic transitions entirely within conduc tion inter-subband quantum well states, it does not require a direct bandgap semiconductor, and the photon energy is not related to the bandgaps of the semiconductors used in the superlattice. Instead, the photon energies are much lower, and are determined by the quantum well design and the sub-band energies. This enables one to generate THz radiation in a very important portion of the electromagnetic spectrum between micro/sub-mm waves and far infrared (300 GHz -3 THz) where there is a paucity of good sources, leading to the so-called THz gap. This frequency range is very useful, for example, for biomolecule and chemical species detection because many molecular transitions are in this frequency range.
Summary
1. The emission and absorption of light by semiconductors gives rise to useful optoelectronic devices. Emission typically occurs in direct band gap materi als. Indirect semiconductors are less optically efficient because they require the involvement of both photons (which carry energy, but little momentum) and phonons (which carry momentum, but little energy).
2. In some optoelectronic devices, electron-hole pairs (EHPs) are generated by the absorption of light. These carriers can be collected as electrical energy in solar cells or photodetectors. Generated EHPs add to the thermally generated reverse current in diodes (Eq. 2), from which one can obtain the open-circuit voltage or short-circuit current in solar cells, a potential renewable source of energy.
3. In photodetectors, there is a trade-off between speed (which requires a short absorption region for a short transit time) and sensitivity (which requires long absorption regions). Clever structures to get both include waveguide photo detectors. One can achieve gain by using APDs, in which impact ionization can increase the number of EHPs generated for a certain number of absorbed photons. However, this can be at the expense of increased impact ionization noise in photodiodes.
4. Photodiodes, in conjunction with light emitters, are used with optical fibers for fiber-optic communication, which is the backbone of the Internet and world wide communication today. Optical fibers are waveguides that work on the basis of the total internal reflection of light from a low-refractive-index outer layer to a high-refractive-index medium (glass).
5. Semiconductors can be used not only to detect photons, but also to emit pho tons by the recombination of EHPs in direct band gap semiconductors. Light emitters can be non-coherent, such as LEDs, or coherent, such as lasers. The color of light depends on the semiconductor, because the photon frequency is proportional to the band gap (Planck relationship).
6. Lasers are more difficult to make than LEDs, because they require not just EHPs, but population inversion (i.e., more carriers in the higher energy levels than in the lower ones). This enables stimulated emission of phase-coherent radiation (governed by the Einstein B coefficient) to dominate over phase-incoherent spontaneous radiation (determined by the Einstein A coefficient). Lasers also require an optical cavity to allow the photon field to build up. Light can come out from the side of the cavity, as in edge-emitting lasers, or from the top surface, as in vertical cavity surface-emitting lasers (VCSELs). Double-heterostructure lasers are useful because they allow both carrier and photon confinement.
Exercises
1. For the p-i-n photodiode of FIG. 7 , (a) explain why this detector does not have gain; (b) explain how making the device more sensitive to low light levels degrades its speed; (c) if this device is to be used to detect light with l = 0.6 um, what material would you use and what substrate would you grow it on?
2. We make a quantum well by sandwiching a 60 Å layer of GaAs (Eg = 1.43 eV) between AlAs with a bandgap of 2.18 eV. We can assume two-thirds of the bandgap difference appear as conduction band discontinuity, and the rest in the valence band. The electron and hole effective masses in GaAs are 0.067 m0 and 0.5 m0, respectively. If we make an LED out of this heterostructure, what is the lowest energy photon that can be emitted from this GaAs layer? If we make a photodetector out of this, what is the longest wavelength that can be detected? You can assume the infinite potential well approximation for this calculation. How many confined states can you have in the conduction and valence bands in the GaAs layer? Qualitatively sketch the electron and hole probability density functions for these states. How far from the hetero-interfaces is the hole most likely to be in the second excited state?
3. For a Si solar cell, the dark saturation current is 2 µA and the short circuit cur rent is 150 mA. When it is optically illuminated, the optically generated current is 0.1 mA. Find the corresponding voltage at current of 100 mA.
4. For a Si photoconductor of length 5 um, doped n-type at 10^15 cm^-3, calculate the change in current density when we shine light on the photoconductor under the following circumstances: We create 10^20 electron-hole pairs/cm^3 @s and carrier-recombination lifetimes, t = 0.1 os. The applied voltage is 2.5 V across the photoconductor's length. How about if we increase the voltage to 2500 V? The electron and hole mobilities are 1500 cm^2 /V - s and 500 cm^2 /V - s, respectively, in the ohmic region for electric fields below 10^4 V/cm. For higher fields, electrons and holes have a saturation velocity of 10^7 cm/s.
5. A Si solar cell with a dark saturation current Ith of 5 nA is illuminated such that the short-circuit current is 200 mA. Plot the I-V curve for the cell as in FIG. 6. (Remember that I is negative, but is plotted positive as Ir.)
6. What is the main limiting factor in increasing the efficiency of a solar cell?
7. A major problem with solar cells is internal resistance, generally in the thin region at the surface, which must be only partially contacted, as in FIG. 5.
Assume that the cell of Exc. 5 has a series resistance of 1 ohm, so that the cell voltage is reduced by the IR drop. Re-plot the I-V curve for this case and com pare with the cell of Exc. 5.
8. (a) Why must a solar cell be operated in the fourth quadrant of the junction I-V characteristic?
(b) What is the advantage of a quaternary alloy in fabricating LEDs for fiber optics?
(c) Why is a reverse-biased GaAs p-n junction not a good photodetector for light of l = 1 um?
9. A Si solar cell 2 cm * 2 cm with Ith = 32 nA has an optical generation rate of 1018 EHP/cm^3 - s within Lp = Ln = 2 mm of the junction. If the depletion width is 1 um, calculate the short-circuit current and the open-circuit voltage for this cell.
10. The maximum power delivered by a solar cell can be found by maximizing the I-V product.
(a) Show that maximizing the power leads to the expression a1 + q kT Vmp b eqVmp/kT = 1 + Isc Ith where Vmp is the voltage for maximum power, Isc is the magnitude of the short-circuit current, and Ith is the thermally induced reverse saturation current.
(b) Write this equation in the form ln x = C -x for the case Isc ? Ith, and Vmp ? kT/q.
(c) Assume a Si solar cell with a dark saturation current Ith of 1.5 nA is illumi nated such that the short-circuit current is Isc = 100 mA. Use a graphical solution to obtain the voltage Vmp at maximum delivered power.
(d) What is the maximum power output of the cell at this illumination?
11. For a solar cell, Eq. (eqn. 2) can be rewritten V = kT q ln a1 + Isc + I Ith b Given the cell parameters of Exc. 10, plot the I-V curve as in FIG. 6 and draw the maximum power rectangle. Remember that I is a negative number but is plotted positive as Ir in the figure. Ith and Isc are positive magnitudes in the equation.
12. Solar cells are severely degraded by unwanted series resistance. For the cell described in Exc. 7, include a series resistance R, which reduces the cell volt age by the amount IR. Calculate and plot the fill factor for a series resistance R from 0 to 5?, and comment on the effect of R on cell efficiency.
13. During the absorption spectra measurement of an unknown semiconductor material, the peak of the spectra appears around 364 nm. What is the band gap of the device and what material is this? If the majority carrier in that material will be able to move with 5 * 10^5 cm/sec velocity, what will be the carrier mass for that material?
14. A semiconductor material of band gap 1.2 eV is used to make a device. At what wavelength it will emit radiation? Is the emitted radiation in the optical range? From these, can you predict the band gap nature of the semiconductor material under observation?
15. The degenerate occupation of bands shown in FIG. 19 helps maintain the laser requirement that emission must overcome absorption. Explain how the degeneracy prevents band-to-band absorption at the emission wavelength.
16. Assume that the system described by Eq. (eqn. 7) is in thermal equilibrium at an extremely high temperature such that the energy density r(n12) is essentially infinite. Show that B12 = B21.
17. The system described by Eq. (eqn. 7) interacts with a blackbody radiation field whose energy density per unit frequency at n12 is r(v12) = 8phv3 12 c3 [ehv12/kT -1] -1 from Planck's radiation law. Given the result of Exc. 16, find the value of the ratio A21 /B12.
18. In a compound AlxGa1-x As material, calculate the energy band gap for the AL concentration of 0.40 using linear interpolation method at 300 K temperature.
Calculate the minimum carrier concentration n = p for population inversion in AlGaAs at 300 K if the intrinsic carrier concentration is 2.1 * 10^3 cm^3.
19. In a Si base long p+ n diode, the excess hole distribution due to optical excitation becomes
Suppose for such a device acceptor concentration is 10^18/cm^3 , diffusion coefficient for holes is 10.36 cm^2 /sec, mobility for holes is 400 cm^2 /V-sec, carrier lifetime = 10 ns, and applied bias is 0.7 V. Due to uniform illumination, 10^20 /cm^3
EHP is generated. Now calculate the excess hole distribution for this long diode at a distance of twice the diffusion length from the junction at the n side.
Quiz
Question 1:
Consider the following band diagram of a simple LED. Assume essentially all re combination is direct and results in light emission. The forward-bias current consists of holes injected from a contact on the left and electrons injected from a contact on the right.
(a) In which region would you expect the optical recombination rate to be the greatest? Select one.
Region(s) A / Region(s) B / Region C
(b) What is the approximate energy of the emitted photons in eV? (c) For a steady state current I = 10 mA, assuming all photons escape, what is the optical output power consistent with your answer to part (b)? _____ W (hint) Watts = Amps #
eV/q = Amps #
Volts (d) If the voltage drop across the depletion region is 0.4 V, what is the separation of the quasi-Fermi levels in Region C? How can this be less than the total forward bias of 1.4 V? (e) What is the electrical power consumed? _______ W (f) In terms of the ratio of optical power out (c) to electrical power in (e), how efficient is this LED? _______%
Question 2:
A solar cell has a short- circuit current of 50 mA and an open- circuit voltage of 0.7 V under full illumination. What is the maximum power delivered by this cell if the fill factor is 0.8?
Question 3:
If one makes an LED in a semiconductor with a direct band gap of 2.5 eV, what wavelength light will it emit? Can you use it to detect photons of wavelength 0.9 um? 0.1 mm?
Question 4:
What is most attractive about solar cells as a global energy source? Why haven't they been adopted more widely so far?