Einstein s Approach to Gain and Spontaneous Emission

Transcription

1 APPENDIX SIX Einstein s Approach to Gain and Spontaneous Emission A6.1 INTRODUCTION Equation (2.14) gives us the relationship between gain and the stimulated recombination rate, R st = v g gn p. (A6.1) As shown in Fig. 1.6, the net stimulated rate R st = R 21 R 12, is the stimulated emission less the stimulated absorption of photons. Thus, we wish to calculate R st, from which the gain, g, can be obtained. As also suggested in Chapters 1 and 2, the stimulated emission and absorption rates depend on the number of available electronic states and their probability of occupation in both the conduction and valence bands for the transitions to occur. The unknowns are the multiplicative rate constants. Once these are determined, we can calculate R st and the gain, g. Einstein gave us a technique to calculate these rates without delving too deeply into the details of the stimulated emission physics. His technique is to determine the desired rate constants under a particular set of boundary conditions. Once obtained, however, these constants are generally applicable to other situations. As shown in Fig. A6.1, the medium of interest is placed in a closed cavity, which has neither inputs nor outputs, and held under thermal equilibrium. Then, a dynamic balance equation can be set up that expresses the desired rates in terms of the equilibrium Diode Lasers and Photonic Integrated Circuits, Second Edition. Larry A. Coldren, Scott W. Corzine, and Milan L. Mašanović John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 579

2 580 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION Infinitesimal sampling window FIGURE A6.1: Schematic of an arbitrary closed cavity with negligible energy in or out. Contents must be in thermal equilibrium. optical energy density, W(ν). Since this is known (from Appendix 4), the rate constants can be determined. In this situation we must include all carrier recombination mechanisms in writing a dynamic balance equation. The nonradiative rates generate heat, which is naturally taken into account in this closed system. By one means or another, they must saturate in equilibrium. Thus, in equilibrium, for a pair of energies, E 2 and E 1,in the conduction and valence bands, respectively, we have dn p dt = 0 = R 21 + R sp,21 R 12, (A6.2) where the first two terms represent electrons recombining by either stimulated or spontaneous processes, respectively, and the last term represents electrons being generated by stimulated absorption. The 21 subscript on R sp distinguishes this two-level spontaneous rate from the net recombination between two bands in a semiconductor as we have considered elsewhere. We could summarize Eq. (A6.2) by saying that the downward transition rate (i.e., conduction to valence band) must equal the upward transition rate. The equilibrium occupation probability at some temperature is given by the Fermi function introduced in Appendix 2 f i = 1 e (E i E F )/kt + 1, (A6.3) where i = 1 or 2 for the involved transition energies in the valence or conduction bands, respectively. That is, f 2 fraction of states filled at E 2, and f 1 fraction of states filled at E 1. Figure A6.2 illustrates the various energy levels for reference.

3 A6.1 INTRODUCTION 581 E E Fc (nonequilibrium) E 2 E c E F (equilibrium) E v E 1 E Fv (nonequilibrium) k FIGURE A6.2: Energy versus momentum schematic illustrating a transition between two energy states in the conduction and valence bands, respectively. Quasi-Fermi levels for both equilibrium and nonequilibrium carrier densities are also illustrated. In Appendix 1, we also defined a density of states, ρ(e), to describe the distribution of states in a band. The number of states are equally distributed in k-space, but by integrating ρ(e) over some energy range, the number of states in that range is obtained. As shown in Figure A6.2, the radiative recombination of an electron and hole involves states in the conduction and valence bands with the same k-vector. That is, both energy (E 21 = hν 21 ) and momentum ( k-electron k-hole) conservation must be satisfied. (As discussed in Chapter 4, the photon momentum is negligible.) Thus, we can consider the density of state pairs, or a reduced density of states, ρ r (E 21 ), in calculating emission at E 21 from electron-hole recombination. We shall explicitly derive ρ r (E 21 ) for parabolic bands to obtain an analytic expression, however, the concept is entirely general. With reference to Fig. A6.2 and using the results of Appendix 1, we can express the transition energy difference, E 2 E 1 = E 21 as E 21 = E g + 2 k 2 2m c + 2 k 2 2m v = E g + E m v + m c m v, (A6.4) where E = 2 k 2 /2mc for parabolic bands. For relatively thick active regions (bulk), we found from Eq. (A1.39) in Appendix 1 that ρ(e ) de = 1 [ 2m ] 3/2 c 2π 2 2 (E ) 1/2 de. (A6.5)

4 582 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION Solving for E from Eq. (A6.4), and forming de, we have m v E = mc + (E 21 E g ), m v m v de = mc + m v de 21. (A6.6) Finally, plugging Eq. (A6.6) into Eq. (A6.5), we obtain the desired reduced density of states, ρ r (E 21 ) = 1 [ 2m ] 3/2 r 2π 2 2 (E 21 E g ) 1/2, (A6.7) where m r = m c m v /(m c + m v ). A6.2 EINSTEIN A AND B COEFFICIENTS The general approach of Einstein was to assign rate constants to the three radiative processes appearing in Eq. (A6.2), with the assumption that these rates must be proportional to the carrier density. These were written empirically as R sp,21 = A 21 N 2, R 21 = B 21 W(ν)N 2, and R 12 = B 12 W(ν)N 1. Generally speaking, the A rate constant is associated with spontaneous processes, whereas the B rate constants are associated with stimulated processes and hence are weighted by the radiation spectral density, W(ν), which was introduced in Appendix 4. In Einstein s day, most radiative transitions of interest took place between atoms with very isolated, sharp energy levels. The carrier densities therefore referred to the density of atoms with electrons in either energy level 1 or 2. In the current context, we must interpret these definitions somewhat differently because in semiconductors, the energy levels are neither isolated nor sharp. To deal with the continuous nature of energy states in semiconductors, we restrict our attention to a differential population of state pairs existing between E 21 and E 21 + de 21. Using the reduced density of states function derived earlier, the differential population available for producing downward transitions becomes dn 2 = f 2 (1 f 1 ) ρ r (E 21 ) de 21, (A6.8) where dn 2 number of state pairs per unit volume between E 21 and E 21 + de 21 available to interact with photons near E 21, in which the upper state is full and the lower state is empty. Similarly, the differential population available for producing upward transitions becomes dn 1 = f 1 (1 f 2 ) ρ r (E 21 ) de 21, (A6.9)

5 A6.2 EINSTEIN A AND B COEFFICIENTS 583 where dn 1 number of state pairs per unit volume between E 21 and E 21 + de 21 available to interact with photons near E 21, in which the lower state is full and the upper state is empty. Another factor we must consider in semiconductors is that the states in ρ r (E 21 ) have a finite lifetime due to collisions with other electrons and phonons. As a result, a given differential population can actually appear over a small range of energies and thus interact with photons spanning some narrow energy range, E 21. The probability of finding this population at energies away from E 21 is characterized by some lineshape function, L (E E 21 ), which has a full-width half-maximum (FWHM) E 21 and is centered at E 21 as shown in Fig. A6.3. The longer the lifetime of a given state, the narrower the spread in energy E 21, and hence, the more chance there is of finding the state pair at E 21. A direct consequence of the finite state lifetime is that when we consider interactions between our differential population and photons of a given energy, hν, we must somehow account for the fact that the population only spends a fraction of its time at that photon energy. In other words, we must weight our differential population by the probability of finding that population between photon energies hν and hν + hdν, which is given by L (hν E 21 ) hdν. The differential population seen by photons at energy, hν, therefore becomes dn 2 dn 2 L (hν E 21 ) hdν, dn 1 dn 1 L (hν E 21 ) hdν. (A6.10) These are the forms for N 1 and N 2 that we must use when analyzing semiconductors. With our differential populations defined, we can now make use of Einstein s A and B coefficients. The differential downward transition rates created by our differential dn 2 population can be written as dr 21 + dr sp,21 = [dn 2 W(ν)B 21 + dn 2 A 21 ]L (hν E 21 ) hdν. (A6.11) The integral over photon frequency is necessary to include contributions that W(ν) makes over the full range of energies that dn 2 is spread over. (E E 21 ) (state with longer lifetime) 2 pδe 21 ΔE 21 E 21 E FIGURE A6.3: Plot of lineshape function versus transition energy. We require that it be normalized so that its area equals one (the state must exist somewhere!). (The peak probability shown assumes a Lorentzian distribution.)

6 584 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION The differential transition rates on the left-hand side of (A6.11) have units of [volume 1 time 1 ]. The population density, dn 2, has units of [volume 1 ]. Therefore, the spontaneous recombination rate constant, A 21, has units of [time 1 ] and is often expressed as an inverse spontaneous lifetime, 1/τ sp. In atomic systems, A 21 does represent the inverse of the spontaneous lifetime of energy level 2. However, in the present context, A 21 is associated with the spontaneous lifetime (which we will denote as τsp 21) of only the differential population, dn 2, and is not equal to the entire band-to-band spontaneous lifetime, τ sp. Later we will determine the relationship between the two-level lifetime τsp 21 and τ sp. For the stimulated term, we note that W(ν) represents the optical energy per unit volume per unit frequency. Therefore, the units of the stimulated recombination rate constant, B 21, are [(volume frequency)/(energy time)]. The differential upward transition rate created by our differential dn 1 population can be written as dr 12 = [dn 1 W(ν)B 12 ]L (hν E 21 ) hdν, (A6.12) where B 12 is the stimulated generation rate constant and has the same units as B 21. A6.3 THERMAL EQUILIBRIUM Einstein s approach allows us to relate the three rate constants appearing in Eqs. (A6.11) and (A6.12) in a straightforward manner. Under thermal equilibrium, we can set (A6.11) equal to (A6.12) according to (A6.2), which we assume holds for differential rates as well as the integrated rates. To remove the integrals over photon frequency, we note that equilibrium blackbody radiation is broadband and varies little over typical linewidths, E 21, associated with the lineshape function. This allows us to treat the lineshape as a delta function, or hl (hν E 21 ) δ(ν ν 21 ). In other words, we simply evaluate all terms under the integral at ν 21. The balance equation then reduces to dn 2 W(ν 21 )B 21 + dn 2 A 21 = dn 1 W(ν 21 )B 12. (A6.13) Rearranging, we obtain dn 2 W(ν 21 )B 12 =. dn 1 W(ν 21 )B 21 + A 21 (A6.14) Alternatively, from Eqs. (A6.8) and (A6.9), using (A6.3) and E 21 = hν 21,wefind the simple result dn 2 = f 2(1 f 1 ) dn 1 f 1 (1 f 2 ) = e hν21/kt. (A6.15)

7 A6.4 CALCULATION OF GAIN 585 Setting Eq. (A6.15) equal to (A6.14) and solving for W(ν 21 ), we obtain W(ν 21 ) = A 21 /B 21 (B 12 /B 21 )e hν 21/kT 1. (A6.16) From Appendix 4, the blackbody radiation formula that defines the spectral density of photons under thermal equilibrium is given by W(ν 21 ) = ρ 0(ν 21 )hν 21 e hν 21/kT 1. (A6.17) Comparing Eq. (A6.16) with Eq. (A6.17), we see that both can be true for all temperatures only if the following two equalities hold: B 12 = B 21, A 21 = ρ 0 (ν 21 )hν 21 B 21. (A6.18) (A6.19) Equation (A6.18) reveals that stimulated emission and stimulated absorption are truly complementary processes associated with the same rate constant. Perhaps more significantly, Einstein s approach establishes a fundamental link between stimulated and spontaneous emission processes through Eq. (A6.19). Thus, by analyzing the system under thermal equilibrium, Einstein s approach allows us to reduce the three differential rate constants to one independent constant, B 21.We could have designated A 21 as the independent constant, however, too often this leads to the incorrect conclusion that B 21 is inversely dependent on the density of optical modes, ρ 0 (ν 21 ). More correctly, we should view B 21 as the rate constant of a single optical mode, and A 21 as this rate constant multiplied by the equivalent spectral density that induces spontaneous emission into the full density of optical modes near ν 21. It is interesting to note that the equivalent spectral density, ρ 0 (ν 21 )hν 21, implies one photon per optical mode. In Chapter 4, more insight into this observation is gained through a quantum mechanical perspective. A6.4 CALCULATION OF GAIN Now that we have established the connection between the three rate constants, we can leave the closed-system, thermal-equilibrium environment and proceed to calculate the gain for a monochromatic radiation field under nonequilibrium conditions. Under forward bias, the Fermi level in the active region splits into two quasi-fermi levels to reflect the nonequilibrium electron and hole densities. The splitting corresponds roughly to the applied voltage. The nonequilibrium carrier densities are then calculated from N = ρ c (E)f c (E) de P = ρ v (E)[1 f v (E)] de, (A6.20)

8 586 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION where we have assumed N P because of negligible doping in the active region. The factors, ρ c (E) and ρ v (E), refer to the densities of states in the conduction and valence bands, respectively. Also, f c (E) and f v (E) are the Fermi functions, in which E F is replaced by the quasi-fermi levels, E Fc and E Fv, for the conduction and valence bands, respectively. To calculate the gain we turn to Eq. (A6.1) which allows us to relate the gain to the net stimulated emission rate. However, we must keep in mind that contributions to the gain at a particular frequency of radiation, ν 0, will come from many differential populations distributed roughly over energies comparable to the lineshape width, E 21. The gain contributed by each population can be written as dg(hν 0 ) = dr st v g N p = 1 v g N p W(ν)B 21 [dn 2 dn 1 ]L (hν E 21 ) hdν. (A6.21) The second equality is obtained by setting dr st = dr 21 dr 12 and using Eqs. (A6.11) and (A6.12) with B 12 = B 21. For a monochromatic field, W(ν) hν 0 N p δ(ν ν 0 ), where the strength of the delta function is equal to the energy density of the field and ν 0 is the frequency of the wave. With this substitution, the integral reduces to evaluating all photon frequency-dependent factors at ν 0, and we are left with dg(hν 0 ) = hν 0 v g hb 21 [dn 2 dn 1 ]L (hν 0 E 21 ). (A6.22) The appearance of the lineshape function reminds us that the further away the differential population is in transition energy from the photon energy, the less contribution it makes to the gain at that frequency. The total gain at hν 0 is found by integrating dg over all existing populations of state pairs which might possibly interact with the field. Expanding the differential populations in Eq. (A6.22) using Eqs. (A6.8) and (A6.9), simplifying the Fermi factors, and integrating over all possible transition energies, we obtain g(hν 0 ) = hν 0 v g h B 21 ρ r (E 21 )(f 2 f 1 )L (hν 0 E 21 ) de 21. (A6.23) If the energy-dependent factors, B 21 ρ r (E 21 )(f 2 f 1 ), are slowly varying compared to the lineshape function, then we can set L (hν 0 E 21 ) δ(hν 0 E 21 ) and the gain expression reduces to g 21 hν 21 v g hb 21 ρ r (E 21 )(f 2 f 1 ), ( E 21 0) (A6.24) where E 21 is evaluated at the photon energy of interest (i.e., g 21 g(hν 0 = E 21 )).

9 A6.4 CALCULATION OF GAIN 587 Equation (A6.23) is the central result of this section. It reveals that the gain is directly proportional to the rate constant B 21, the reduced density of states, and the Fermi probability factors. It is immediately apparent from Eq. (A6.24) that to achieve positive gain, we must create enough electrons and holes to allow f 2 > f 1. This places a condition on the quasi-fermi levels, which reduces to the requirement that the quasi-fermi level separation be larger than the incident photon energy, or E F > hν 0. Chapter 4 considers these issues in more detail. To fully evaluate the gain, we still need to determine the rate constant, B 21. In lasers that use atomic transitions, a measurement of the spontaneous emission linewidth of a given transition can allow us to estimate A 21 if the broadening of the line is dominated by the spontaneous emission lifetime, τ sp = 1/A 21. With this information, B 21 can readily be determined using Eq. (A6.19). Thus, with Einstein s approach and this one simple measurement, the description of gain and spontaneous emission in atomic systems is completely self-contained. Unfortunately the situation is not so simple in semiconductors since the spontaneous emission spectrum represents a superposition of transitions from all of our differential populations. The resulting broad emission spectrum prevents us from isolating the linewidth of just one differential population, and therefore prevents us from evaluating A 21 (and hence, B 21 ) via direct experiment. The approach that must be followed in semiconductors is to estimate the transition rates using other more in-depth theories and then relate the resulting expressions back to B 21 and A 21. Chapter 4 details the theory required to evaluate the transition rates explicitly from a more fundamental quantum mechanical analysis. An explicit expression for B 21 will be presented there. We can alternatively express the gain in terms of the spontaneous rate constant, or the two-level lifetime, τsp 21 = 1/A 21. Using Eq. (A6.19) in Eq. (A6.24), the gain takes the form g 21 = A 21 h ρ r (E 21 )(f 2 f 1 ) ρ 0 (ν 21 ) v g = λ 2 0 8πn 2 τ 21 sp hρ r (E 21 )(f 2 f 1 ), (A6.25) where we have set ρ 0 (ν 21 ) = 8πn 2 /λ 2 0 v g according to Eq. (A4.5), with λ 0 c/ν 21. Although this expression is equivalent to Eq. (A6.24), it is very deceptive for two reasons. First of all, a common mistake in the literature is to equate τsp 21 with the band-toband spontaneous lifetime, τ sp, incorrectly linking the gain with the carrier lifetime. In fact, some go even further by linking A 21 with the overall spontaneous emission bandwidth (analogous to atomic transitions) a procedure that is completely misguided in semiconductors, but nevertheless encouraged by writing the gain in terms of A 21. The second problem is that when written in this way, one might conclude that the gain varies inversely with the optical mode density. Only on more careful inspection

10 588 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION does one realize that implicit in the two-level lifetime, τsp 21, is an inverse dependence on the mode density (i.e., the higher the mode density, the shorter the lifetime). As a result, the product ρ 0 (ν 21 )τsp 21 (and hence the gain) becomes independent of the optical mode density a conclusion that is obvious from Eq. (A6.24). Reduced optical mode densities possible in very small VCSEL structures (or microcavities ) have lead some researchers to conclude using Eq. (A6.25) that the gain is increased as a result again, a conclusion which is misguided. For these reasons, Eq. (A6.24), which more appropriately defines the gain in terms of the single-mode stimulated rate constant, B 21, is preferable. Another issue we need to resolve is whether we should use Eq. (A6.24) or the more complex Eq. (A6.23) to evaluate the gain. The time between collisions for electrons in typical semiconductors is on the order of 0.1 ps, which leads to a FWHM of E mev (assuming a Lorentzian lineshape). At room temperature, this bandwidth is small enough that we can assume f 2 f 1 to be roughly constant. Furthermore, B 21 does not have a strong energy dependence. Therefore, our main concern lies with ρ r (E 21 ). In bulk active regions, ρ r (E 21 ) varies as E 1/2 and the rate of change can be neglected in comparison to the bandwidth of the lineshape function. In other words, at room temperature, Eq. (A6.24) can generally be used with bulk active regions. For quantum wells, the reduced density of states can be found by using the reduced mass in Eq. (A1.45). It is zero up to the first allowed energy states in the conduction and valence bands where n z = 1, i.e., (E c1 E h1 ). There it abruptly increases to ρ r (hν 21 ) = m r π 2 d, E 21 >(E c1 E h1 ), (A6.26) where it remains constant up to the point where n z = 2. There it again increases by the same amount. Thus, it violates the assumptions about being slowly varying over the bandwidth E 21 made between Eq. (A6.23) and Eq. (A6.24). Therefore, we must use Eq. (A6.23). Unfortunately, the actual lineshape function is not well established. A Lorentzian is often used, but the results are somewhat nonphysical; so other, more complex functions have been developed to better fit the experimental data. The simple Lorentzian with an FWHM of E 21 takes the form 2/π E 21 L (E E 21 ) = 1 + 4(E E 21 ) 2 / E21 2. (A6.27) The integration of Eq. (A6.23) with Eq. (A6.27) will smooth the discontinuities in ρ r (E 21 ) that exist in quantum-well (as well as quantum-wire and box) lasers. However, the plateau gain levels obtained by inserting Eq. (A6.26) in Eq. (A6.24) will be correct, as long as we are > E 21 away from a step edge. The numerical gain calculations in Chapter 4 illustrate this behavior more quantitatively. A common feature of all active materials (regardless of the lineshape broadening or reduced dimensionality) is that the gain increases from an initial unpumped absorption level given by g 21 (f 2 = 0, f 1 = 1), to a transparency gain value given

11 A6.5 CALCULATION OF SPONTANEOUS EMISSION RATE 589 f 2 = 1, f 1 = 0 g 21 Approx N tr ( f 2 = f 1 ) N f 1 = 1, f 2 = 0 FIGURE A6.4: Illustration of gain versus carrier density. Straight-line approximation valid over limited ranges. by g 21 (f 2 = f 1 ), finally approaching a saturation level of g 21 (f 2 = 1, f 1 = 0) (equal in magnitude to the unpumped absorption level) as more and more carriers are injected into the active region. In Fig. A6.4 we schematically illustrate this characteristic. In Chapter 4 we shall find that this characteristic can be well approximated by a logarithmic function; however, for many situations only a small portion of the curve near and somewhat above the transparency point is of interest. In these cases, a straight-line approximation is often very useful. That is, g 21 = a(n N tr ), where a is the differential gain, g/ N, and N tr density. (A6.28) is the transparency carrier A6.5 CALCULATION OF SPONTANEOUS EMISSION RATE In creating a given amount of gain in the semiconductor by increasing dn 2 relative to dn 1, we unfortunately end up creating a large amount of spontaneous emission over a relatively broad range of frequencies. This section deals with developing the relation for the spectrum of spontaneously emitted photons, which when integrated allows us to determine the total number of spontaneous photons being generated per second.

12 590 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION We start by defining the spontaneous emission within a small energy interval to be Rsp hν (hν)hdν, where Rhν sp is the emission rate per unit energy per unit volume occurring at hν. As with the gain, we need to sum over all differential populations to determine the emission rate at a single frequency. The probability of dn 2 appearing at hν is given by L (hν E 21 ) hdν. Weighting dn 2 by this factor, multiplying by the spontaneous rate constant, A 21, and integrating over all state pairs, we obtain Rsp hν (hν)hdν = A 21 [L (hν E 21 ) hdν] dn 2. (A6.29) Canceling out hdν on both sides, and expanding dn 2, we arrive at the desired result: Rsp hν (hν) = A 21 ρ r (E 21 )f 2 (1 f 1 )L (hν E 21 ) de 21. (A6.30) Equation (A6.30) reveals that the spontaneous emission spectrum is smoothed in the same manner as the gain spectrum discussed earlier. It is interesting to note that while we must have E 21 > E g to have a nonzero reduced density of states, it is possible for hν <E g since L (hν E 21 ) can be nonzero for hν E 21 < 0. Thus, spontaneous emission can actually be observed at energies E 21 /2 below the bandgap. This reflects the uncertainty in the energy of states at the band edge, which results from the finite lifetimes of electrons in those states. Again if A 21 ρ r (E 21 )f 2 (1 f 1 ) is slowly varying compared to the lineshape function, then we can set L (hν E 21 ) δ(hν E 21 ), and Eq. (A6.30) simplifies to R 21 sp A 21ρ r (E 21 )f 2 (1 f 1 ), ( E 21 0) (A6.31) where we have defined Rsp 21 Rhν sp (hν = E 21) in analogy with g 21. Comparing Rsp 21 to g 21 given in Eq. (A6.24), it is interesting to note that the two are quite similar. In fact, we can express the spontaneous emission at E 21 in terms of the gain at E 21 as follows: Rsp 21 = v g A 21 f 2 (1 f 1 ) h 2 ν 21 B 21 (f 2 f 1 ) g 21 = 1 h ρ 0(ν 21 ) v g n sp g 21, (A6.32) where we have made use of the relation between the rate constants expressed in Eq. (A6.19). We have also introduced the population inversion factor, which is defined as n sp = f 2(1 f 1 ) (f 2 f 1 ) = 1 1 e (hν 21 E F )/kt. (A6.33) The popular usage of the sp subscript originates from the fact that n sp was initially referred to as the spontaneous emission factor, but was later changed to eliminate conflict with β sp, which is also defined as the spontaneous emission factor. We now refer to n sp as the population inversion factor because it is the semiconductor

13 A6.5 CALCULATION OF SPONTANEOUS EMISSION RATE 591 laser equivalent to the ratio N 2 /(N 2 N 1 ) encountered in atomic laser physics. In atomic systems, when N 2 N 1 > 0, the population is said to be inverted, the ratio N 2 /(N 2 N 1 ) is positive, and optical gain is achieved. Similarly, when n sp > 0, a population inversion is established in the semiconductor, indicating a net optical gain. If the quasi-fermi level separation in the semiconductor is known, then using Eq. (A6.32), we can determine the spontaneous emission rate if we know the gain, or we can determine the gain if the spontaneous emission rate is known. In Chapter 4 a more thorough investigation of this fundamental relationship between spontaneous emission and gain is provided. Now we wish to determine the total spontaneous emission rate, R sp, by integrating (A6.30) over all photon energies: R sp = Rsp hν (hν)hdν = A 21 ρ r (E 21 )f 2 (1 f 1 )L (hν E 21 )de 21 hdν [ ] A 21 ρ r (E 21 )f 2 (1 f 1 ) L (hν E 21 ) hdν de 21 = A 21 ρ r (E 21 )f 2 (1 f 1 ) de 21 = Rsp 21 de 21. (A6.34) The third equality is found by inverting the order of integration and pulling out all terms not dependent on the photon frequency. The approximate sign is used here because A 21 is not completely independent of photon frequency (A 21 ν, from Eq. (A6.19) with ρ 0 (ν) ν 2 and B 21 ν 2 ; see Chapter 4). However, in comparison to L (hν E 21 ), this dependence can be neglected. The integral in brackets then reduces to unity, which leads to the fourth equality. In other words, the lineshape broadening has no effect on the total spontaneous emission rate, and we can simply integrate over the simplified Rsp 21 defined in (A6.31). Setting A 21 = 1/τsp 21 and R sp = N /τ sp in (A6.34), we can define the total spontaneous lifetime in terms of the local spontaneous lifetime through the following nontrivial relation: [ ] 1 1 τ sp N τsp 21 ρ r (E 21 )f 2 (1 f 1 ) de 21 (A6.35) Generally speaking, the term in brackets will tend to go as N 2 due to the double dependence on the quasi-fermi levels (f 2 related to N and 1 f 1 related to P). As a result, the total spontaneous lifetime typically follows a 1/N dependence as assumed in Chapter 2. In performing actual calculations of the total spontaneous emission rate, it is useful to replace A 21 with the single-mode rate constant, B 21. Doing this, Eq. (A6.34)

14 592 EINSTEIN S APPROACH TO GAIN AND SPONTANEOUS EMISSION becomes R sp = B 21 hν 21 ρ 0 (ν 21 )ρ r (E 21 )f 2 (1 f 1 ) de 21. (A6.36) Thus, we see that the spontaneous emission rate includes both optical and electronic density of states functions. By reducing the cavity size to dimensions on the order of the emission wavelength in the material, it is in principle possible to significantly alter the optical mode density, which allows us to actually alter the spontaneous emission rate. An active field of research that studies these microcavity effects is attempting to reduce the spontaneous emission rate substantially. The motivation for this lies in the following relation: R sp = η i η r 1 qv. (A6.37) That is, the total spontaneous emission rate represents the number of carriers lost to spontaneous recombination per second and can therefore be equated with the radiative portion of the injected current. By minimizing R sp, researchers hope to minimize the threshold current of certain types of lasers. In particular, VCSELs represent excellent candidates for such experiments due to their scalable geometry. In concluding this appendix, it is useful to appreciate that the carrier density, the optical gain, and the radiative current in the active region can all be calculated from the quasi-fermi levels using Eqs. (A6.20), (A6.23), (A6.36), and (A6.37). Thus, E Fc and E Fv completely determine all relevant parameters under nonequilibrium conditions. Furthermore, by invoking charge neutrality, we can find E Fc for a given E Fv using (A6.20), reducing the entire problem to one independent parameter. In other words, we can obtain gain versus current, gain versus carrier density, or current versus carrier density, by scanning E Fv over the appropriate ranges. The linear relationship of gain to carrier density, and the quadratic relationship of current to carrier density discussed in Chapter 2 represent approximations to the more rigorous nontrivial relationships derived in this appendix. READING LIST Verdeyen JT. Laser electronics. 2nd ed. Englewood Cliffs, NJ: Prentice Hall; Chapters 7 and 11.