Optical Gain and Co-Stimulated Emissions of Photons and Phonons in Indirect Bandgap Semiconductors

Optical Gain and Co-Stimulated Emissions of Photons and Phonons

in Indirect Bandgap Semiconductors

M. J. CHEN, C. S. TSAI1;2 and M. K. WU

Department of Materials Science and Engineering, National Taiwan University, Taipei 106, Taiwan 1_{Department of Electrical Engineering and Computer Science and Institute of Surface and Interface Science,} University of California, Irvine, CA 92687, U.S.A.

2_{Institute of Electrooptics Engineering, National Taiwan University, Taipei 106, Taiwan} (Received February 24, 2006; accepted June 5, 2006; published online August 22, 2006)

A model calculation on optical gain and co-stimulated emission of photons and phonons in indirect bandgap semiconductors such as silicon is presented. An analytical expression for optical gain via phonon-assisted optical transitions in indirect bandgap semiconductors is presented. Population inversion can occur when the difference between the quasi-Fermi levels for electrons and holes is greater than the photon energy. The rate equations and their steady state solutions for electron, photon, and phonon involved in the phonon-assisted optical transitions are presented. It is shown that co-stimulated emissions of photons and phonons will occur when the threshold condition for laser oscillation is satisfied. The magnitude of optical gain in bulk crystalline silicon is calculated and shown to be smaller than the free carrier absorption at room temperature. However, it is shown, for the first time, that the optical gain is greater than the free carrier absorption in bulk crystalline silicon at the temperature below 23 K. Thus, the calculation predicts that the co-stimulated emissions of photons and phonons could take place in bulk crystalline silicon at the low temperature. [DOI:10.1143/JJAP.45.6576]

KEYWORDS: indirect bandgap semiconductors, crystalline silicon, optical gain, co-stimulated emission of photons and phonons, phonon-assisted optical transition, laser oscillation, nanostructured PN junction diode

1. Introduction

Silicon is the dominant semiconductor material in ultra-large-scale-integration (ULSI) electronics, and it is also the desirable substrate material for realization of integrated optoelectronics.1,2) _{However, until the most recent} realiza-tion of Raman-effect based laser,2,3) _{it was generally} believed that silicon’s indirect bandgap would prevent its use as efficient light emitters and lasing media. Numerous earlier approaches,5–7) such as silicon nanocrystals,8,9) Si/ SiO2 supperlattices,10) porous silicon,12) erbium-doped sili-con,13,14) and periodic nanopatterned crystalline silicon15) with emphases on photoluminescence aspects, have been attempted with some progresses toward overcoming this difficulty. For the approaches with emphases on electro-luminescence aspects, high efficiency electroelectro-luminescence from silicon light-emitting diodes (LED), in which the emission wavelength corresponds to the silicon bandgap energy, was reported.16–18) _{Room-temperature} electrolumi-nescence at bandgap energy was also observed in metal– oxide–semiconductor (MOS) tunneling diodes on silicon.19) Recently, stimulated emission at the wavelength correspond-ing to the silicon bandgap energy was observed in a silicon nanostructured pn junction diode using current injection at room temperature.20)Most recently, field-effect electrolumi-nescence from silicon nanocrystals in a floating-gate transistor structure was reported.21)On the theoretical study of the subject, optical gain in materials with indirect transitions was assessed recently based on a simple two-level model.22)

In this paper, we present a considerably more detailed theoretical treatment of optical gain at bandgap energy in indirect bandgap semiconductors, including the calculation of the optical gain in bulk crystalline silicon versus temperature and the derivation of the rate equations of electron, photon, and phonon involved. In §2, an analytical

expression for optical gain via phonon-assisted optical transitions in indirect bandgap semiconductors which takes into account the structures of the conduction and valence band band-edges is presented. We obtain the conditions required for population inversion in indirect bandgap semi-conductors. The rate equations and the steady-state solutions for electron, photon and phonon involved are presented in §3. In §4, calculations of optical gain and free carrier absorption in bulk crystalline silicon at room and low temperatures are provided. The optical gain is shown to be greater than the free carrier absorption at the temperature below 23 K. Finally, discussions and conclusions are provided in §5 and §6, respectively.

2. Optical Gain of Quantum Transition Involving Photons and Phonons

Figure 1 shows a schematic diagram of optical emission and absorption in indirect bandgap semiconductors such as silicon. In order for the crystal momentum to be conserved,

Fig. 1. Schematic diagram of phonon-assisted optical transitions in indirect bandgap semiconductors.

Vol. 45, No. 8B, 2006, pp. 6576–6588 #2006 The Japan Society of Applied Physics

6576

it is necessary that phonons participate in the optical transitions between the conduction and valence band band-edges. The spectral peak of the spontaneous emission spectrum is known to be slightly below the bandgap energy of silicon,16–19) which means that the optical emission is assisted by emission of a phonon, as depicted in Fig. 1. On the other hand, optical absorption at this photon energy is assisted by the absorption of a phonon, also depicted in Fig. 1. As shown in Appendix A, the theoretical analysis has arrived at an expression for optical gain gðh
!Þ via phonon-assisted optical transitions in indirect bandgap semiconduc-tors such as silicon as follows

gðh
!Þ ¼ h 3_c2 8
n2_ðh
!Þ2Rspðh
!Þ
1  nq nqþ1 exp h
! þ h
  F kBT
   ð1Þ

where Rspðh
!Þ is the spontaneous emission rate, nq is the phonon occupation number, h
! and h
 are the photon and phonon energies, h
is the reduced Planck constant, c is the velocity of light in free space, n is the refractive index, F is the difference of the quasi-Fermi levels for electrons and holes, kB is the Boltzmann constant, and T is the temper-ature. From eq. (1) we see that optical gain can take place if the quantity in braces is positive, i.e.,

nqþ1 nq > exp h
! þ h
  F kBT
 : ð2Þ

Equation (2) indicates that for population inversion or optical gain to occur, the populations of phonons, electrons, and holes must satisfy a certain condition. Note that the term in the left hand side of eq. (2) is the ratio of the phonon emission rate to the phonon absorption rate.23) _The term (h
! þ h
) in the right hand side is the energy of the system after stimulated emission of a photon and a phonon, and F is the energy of the system before stimulated emission. Thus eq. (2) states that for optical gain to occur in the phonon-assisted optical transitions, the ratio of the phonon emission rate to the phonon absorption rate must be greater than a Boltzmann factor determined by the energy difference of the system before and after stimulated emission. We see that

nqþ1 nq > 1 > exp h
! þ h
  F kBT
 if F > h
! þ h
 ð3Þ Thus, if the difference between the quasi-Fermi levels for electrons and holes is greater than the sum of photon and phonon energies, as shown schematically in Fig. 1, the stimulated emission rate will exceed the absorption rate and population inversion occurs.

Figure 2 shows the relationship between the difference between the quasi-Fermi levels F and the phonon occupation number nq required for population inversion involving transverse optical (TO) phonon (h
 ¼ 57:8 meV)-assisted optical transition at photon energy h
! ¼ 1:07 eV in

crystalline silicon at 300 K. The solid curve in Fig. 2 depicts the condition and the region for population inversion based on eq. (2). The phonon occupation number at thermal equilibrium (nq0¼0:12) and the condition F ¼ h
! þ h
 ¼ 1:1278 eV are indicated in dotted lines, respectively. The region above the solid curve in Fig. 2 is the positive optical gain region [gðh
!Þ > 0], while the region below the solid curve is the absorption region [gðh
!Þ < 0]. Since the energy h
! þ h
 is approximately equal to bandgap energy Eg, the condition F > h
! þ h
  Eg corresponds to the situation of electronic population inversion.22) _{It should be} noted in Fig. 2 that there exists a region (the shaded area) where population inversion can occur even though F < h
! þ h
  Eg. This indicates that the positive optical gain [gðh
!Þ > 0] can occurs without an electronic population inversion.22)

Since a phonon is emitted during the emission of a photon, as shown schematically in Fig. 1, a net stimulated emission of phonons will occur when the population inversion is reached. The condition for the net stimulated phonon emission with energy h
 resulting from the net stimulated emission of photons with energy h
! is the same as eq. (2). In other words, the net stimulated emissions of photons and phonons in indirect bandgap semiconductors take place simultaneously when the condition set by eq. (2) is satisfied. The increase in the phonon population due to the net stimulated phonon emission results in a deviation of phonon occupation number nqfrom its value at thermal equilibrium. Figure 2 shows that as the phonon occupation number nq increases from the value at thermal equilibrium nq0, a larger value of F is required to facilitate population inversion.

Similarly, following the procedures similar to those presented in Appendix A, we obtain the expressions for the optical gain associated with emission of multi-phonons as follows gNðh
!Þ ¼ h3_c2 8
n2_ðh
!Þ2RspNðh
!Þ  1  YN i¼1 nqi nqiþ1 exp h
! þ h
1þh
2þ    þh
NF kBT
 " # ð4Þ 0.0 0.2 0.4 0.6 0.8 1.0 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 ∆F=h + =1.1278eV

(

( )

(

Phonon occupation number n_q

∆

F

(eV)

)

(

)

)

Fig. 2. Relationship between F and nqrequired for population inversion in bulk crystalline silicon at 300 K. The phonon energy h
 of 57.8 meV and photon energy h
! of 1.07 eV were used in the calculation.

where N ¼ 1; 2; 3 . . .; gNðh
!Þ is the optical gain coefficient associated with emission of N phonons; nqi is the occupation number of the ith phonon mode and h
i is the corresponding energy. Finally, RspNðh
!Þ is the spontaneous emission rate associated with emission of N phonons. Equation (4) shows that the condition for achieving population inversion associated with N-phonon emission is

YN i¼1 nqiþ1 nqi > exp h
! þ h
1þh
2þ    þh
NF kBT
 : ð5Þ We see that YN i¼1 nqiþ1 nqi > 1 > exp h
! þ h
1þh
2þ    þh
NF kBT
 if F > h
! þPNi¼1h
iEg: ð6Þ Therefore, once the difference between the quasi-Fermi levels F is greater than the sum of photon and phonon energies, which is approximately equal to the bandgap energy Eg, the population inversions associated with emission of 1; 2; . . . ; N phonons can take place.

3. Rate Equations and Co-Stimulated Emissions of Photons and Phonons

In this section, we present the rate equations for the single-mode photon, phonon and carrier in indirect bandgap semiconductors and, then, determine their steady-state solutions. The spontaneous emission rate Rspðh
!Þ, stimulated emission rate Rstðh
!Þ, and absorption rate Rabðh
!Þ of the quantum transition involving a photon with energy h
! and a phonon with energy h
 are derived in Appendices A and B. They are given as follows

Rspðh
!Þ ¼ M  ðnqþ1Þ  NP; ð7aÞ Rstðh
!Þ ¼ M  npðnqþ1Þ  NP; ð7bÞ Rabðh
!Þ ¼ M  npnqexp h
! þ h
  F kBT
 NP ð7cÞ

where the proportional constant M ¼
=8  Acvðh
! þ h
  EgÞ2exp½ðh
! þ h
  EgÞ=kBT, in which Acv is the spontaneous emission coefficient, N and P are the electron and hole concentrations, and np is the photon occupation number which gives the average number of photons per state with the photon energy h
!. Therefore, the net generation rates for photon and phonon are given by

Rstðh
!Þ  Rabðh
!Þ þ Rspðh
!Þ ¼ M  " ðnpþ1Þðnqþ1Þ  npnqexp h
! þ h
  F kBT
# NP: ð8Þ

Note that in eq. (8), the term ðnpþ1Þðnqþ1Þ  NP corre-sponds to the stimulated and spontaneous emission rate of photon and phonon, while the term npnqexp½ðh
! þ h
  FÞ=kBT  NP represents the absorption rate of photon and phonon.

Now, the rate equations for electron concentration N, photon density Np and phonon density Nq in the quantum transitions with a single-optical mode associated with the photon energy h
! and a single-phonon mode associated with the phonon energy h
 are given as follows

dN dt ¼RpRstðh
!Þ þ Rabðh
!Þ  Rspðh
!Þ  N c ; ð9aÞ dNp dt ¼Rstðh
!Þ  Rabðh
!Þ þ Rspðh
!Þ  Np p ; ð9bÞ dNq dt ¼Rstðh
!Þ  Rabðh
!Þ þ Rspðh
!Þ  NqNq0 q ð9cÞ

where Rp is the pumping rate either by current injection or optical excitation,  is spontaneous emission factor which represents the fraction of spontaneous emission entering the optical mode, Nq0 is the phonon density at thermodynamic equilibrium, and c, p and q are the lifetime of carriers, photons and phonons, respectively. Nonradiative Shockley– Read–Hall recombination is the main mechanism that determines the carrier lifetime c. The loss of photon

due to the effects such as optical scattering, free carrier absorption, and output coupling of the resonant cavity, can be characterized by the photon lifetime p. The last term in eq. (9c) represents anharmonic phonon interactions. Due to the anharmonic interactions between phonons,23) _the pho-nons resulting from the phonon-assisted optical transition will decay to other phonon modes. For example, the transverse optical (TO) phonon generated from optical emission in crystalline silicon will decay into the acoustic phonon modes with a typical phonon lifetime in the order of picoseconds.24) _{Therefore the phonon density N}

q will be restored to its thermal equilibrium value Nq0. The photon density Np and the phonon density Nq in eqs. (9b) and (9c), respectively, are related to the photon and phonon occupa-tion numbers np and nq as follows

Np ¼Kpnp; ð10aÞ

Nq ¼Kqnq ð10bÞ

in which Kp and Kq are the density of states for the single-photon and single-phonon modes, respectively. Substituting eqs. (7) and (10) into eq. (9) and assuming that the electron concentration N is equal to the hole concentration P due to charge neutrality, we obtain the following coupled equa-tions:

dN dt ¼RpMnp ðnqþ1Þ  nqexp h
! þ h
  F kBT
   N2 Mðnqþ1ÞN2 N c ; ð11aÞ dnp dt ¼ M Kp np ðnqþ1Þ  nqexp h
! þ h
  F kBT
   N2 þM Kp ðnqþ1ÞN2 np p ; ð11bÞ dnq dt ¼ M Kq np ðnqþ1Þ  nqexp h
! þ h
  F kBT
   N2 þM Kq ðnqþ1ÞN2 nqnq0 q : ð11cÞ

Equations (11a)–(11c) are the rate equations for carrier concentration N, photon occupation number np and phonon occupation number nq, respectively.

We now set d=dt ¼ 0 in eqs. (11a)–(11c) and solve for the steady state solution of the rate equations. From eq. (11b), the steady state photon occupation number np is readily obtained as follows np¼ 1 Kp Mðnqþ1ÞN2 1 p M Kp ðnqþ1Þ  nqexp h
! þ h
  F kBT ! " # N2 : ð12Þ As the denominator in eq. (12) approaches zero, npbecomes very large. Thus, the following equation

M ðnqþ1Þ  nqexp h
! þ h
  F kBT
   N2¼Kp p ð13Þ

is the threshold condition for laser oscillation, meaning that the photon loss of the resonant cavity must be compensated by the optical gain for the on-set of laser oscillation. In the following, we discuss the steady state solutions in two situations: (1) Below threshold, and (2) Above threshold.

3.1 Below threshold

Below the threshold condition for laser oscillation, the photon density is low and the net stimulated emission rate Rstðh
!Þ  Rabðh
!Þ in the rate equations can be neglected. Then the rate equations become

dN dt ¼RpMðnqþ1ÞN 2_N c ¼0; ð14aÞ dnp dt ¼ M Kp ðnqþ1ÞN2 np p ¼0; ð14bÞ dnq dt ¼ M Kq ðnqþ1ÞN2 nqnq0 q ¼0: ð14cÞ

We shall now determine the typical values of the two terms in eq. (14c) using the typical data shown in Table I. Accordingly, the first term in eq. (14c) is approximately equal to 104
_ðn

qþ1Þ s1, while the second term ðnq nq0Þ=q 1012
ðnqnq0Þs1. Consequently, the follow-ing approximation is valid

nq nq0 ð15Þ

Substituting eq. (15) into eq. (14a) we obtain the following solution for the carrier concentration

N ¼ 1 c þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 c þ4Mðnq0þ1ÞRp s 2Mðnq0þ1Þ : ð16Þ

Again, using the typical values shown in Table I, we see that 1=_c2 is much larger than 4Mðnq0þ1ÞRp. Thus

N  1 c þ 1 c 1 þ1 24M 2 cðnq0þ1ÞRp ! 2Mðnq0þ1Þ ¼cRp: ð17Þ

Finally, substituting eqs. (15) and (17) into eq. (14b), the photon density Np is obtained

NppMc2ðnq0þ1ÞR2p: ð18Þ In summary, at below threshold condition the carrier concentration N increases linearly with the pumping rate Rp, the phonon occupation number nq remains close to its thermal equilibrium value nq0, and the photon density Np increases with the square of the pumping rate Rp.

3.2 Above threshold

Equation (3) shows that F > h
! þ h
 is the necessary condition for the occurrence of population inversion. It is usually the case that F  h
! þ h
 when the threshold condition eq. (13) is reached. As a result, the factor exp½ðh
! þ h
  FÞ=kBT  1 can be neglected and the threshold condition eq. (13) is simplified as follows:

Mðnqþ1ÞN2¼ Kp

p

: ð19Þ

Thus the rate equations eq. (11) are reduced to the following forms dN dt ¼RpMnpðnqþ1ÞN 2_Mðn qþ1ÞN2 N c ; ð20aÞ dnp dt ¼ M Kp npðnqþ1ÞN2þ M Kp ðnqþ1ÞN2 np p ; ð20bÞ dnq dt ¼ M Kq npðnqþ1ÞN2þ M Kq ðnqþ1ÞN2 nqnq0 q : ð20cÞ Table I. Typical values of relevant parameters.

Parameter Typical value Reference

Rp(cm3s1) 1021–1025 34 M (cm3_s1_{) 10}15 _41,42 N (cm3_{) 10}15_–1020 _41,42 Kp(cm3) 1011 34 Kq(cm3) 1023 23 c(s1) 106 42 p(s1) 1012 34 q(s1) 1012 24 nq0 0.12 23,36

Substituting the threshold condition of eq. (19) into eqs. (20a) and (20c) and seting d=dt ¼ 0, the following equation is obtained n3_qþ ð1  2A1Þn2qþ ðA 2 12A1Þnqþ ðA21A2Þ ¼0 ð21Þ where A1¼nq0þ qRp Kq ; A2 ¼ Kpq2 pMc2Kq2 :

Using the typical data listed in Table I, the value of A2₁ is found to be much greater than A2, so the A2term in eq. (21) can be neglected. Thus the corresponding steady state solution for phonon occupation number is obtained as follows

nqA1¼nq0þ q Kq

Rp: ð22aÞ

The phonon occupation number nq is seen to increase linearly with the pumping rate Rp. For convenience, using eq. (10b), eq. (22a) is now written in the following form

Nq Nq0þqRp: ð22bÞ

After substituting eq. (22a) into the laser oscillation thresh-old condition of eq. (19), the following carrier concentration N at steady state is obtained,

N  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kp Mp nq0þ1 þ q Kq Rp ! v u u u u t : ð23aÞ

Again, using the typical values of parameters listed in Table I, it is seen that the situation qRp=Kq  ðnq0þ1Þ holds. Thus, the following approximate carrier concentration N is obtained N  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kp Mpðnq0þ1Þ s Nth: ð23bÞ

Equation (23b) shows that the carrier concentration remains nearly constant at Nth. Furthermore, from eq. (20a) the solution for photon density Np is then given as follows

Npp Rp Mðnq0þ1ÞNth2 þ Nth c
   ¼pðRpRpthÞ ð24Þ where Rpth¼Mðnq0þ1ÞNth2 þ Nth c : ð25Þ

is the pumping rate at threshold.

As described above, although the phonon occupation number nq increases linearly with the pumping rate Rp, the amount of increase is small and so remains very close to its thermodynamic equilibrium value nq0. Thus the expressions for optical gain associated with emission of one phonon and N phonons, i.e., eqs. (1) and (4), reduce to eqs. (26a) and (26b), respectively: g1ðh
!Þ ¼ h3_c2 8
n2_ðh
!Þ2Rspðh
!Þ  1  exp h
!  F kBT
   ; ð26aÞ gNðh
!Þ ¼ h3_c2 8
n2_ðh
!Þ2RspNðh
!Þ  1  exp h
!  F kBT
   ; ð26bÞ and gðh
!Þ > 0 if F > h
!: ð27Þ In comparison with eqs. (1), (2), and (4), it is seen that the factors involving phonons, such as nq and h
, disappear in eqs. (26a), (26b), and (27) as a result of nqnq0. This means that even though it is necessary that phonons participate in the optical transitions for the conservation of crystal momentum, the population of phonon does not influence the magnitude of optical gain significantly since nq is close to nq0 in practice. Equations (26a) and (26b) take the same form as the optical gain in direct bandgap semiconductors, and eq. (27) is also similar to the Bernard– Duraffourg condition for the occurrence of population inversion in direct bandgap semiconductors.25) Thus, it is seen that population inversion can be accomplished in indirect bandgap semiconductors provided that the differ-ence between the quasi-Fermi levels for electrons and holes, F, is greater than the photon energy. It should be noted that for the direct bandgap semiconductors, since the energy of photon emitted is equal to the bandgap energy, F has to be greater than the bandgap energy for the achievement of population inversion. However, as shown in Figs. 1 and 2, for the indirect bandgap semiconductors the energy of photon emitted is slightly smaller than the bandgap energy, it is not necessary for F to be greater than the bandgap energy for the occurrence of population inversion. This is the difference for the condition of population inversion between the direct and indirect bandgap semiconductors.

Equations (26a) and (26b) show that the signs of optical gain coefficients gðh
!Þ and gNðh
!Þ are determined by the difference between the quasi-Fermi levels (F), and their magnitudes are proportional to the spontaneous emission rate Rspðh
!Þ and RspNðh
!Þ. Also, as indicated in eq. (A·7), the magnitude of F strongly influences the spontaneous emission rate Rspðh
!Þ. A PN junction is well-known to be a convenient way for injection of electrons and holes with adequate concentrations to achieve the condition for pop-ulation inversion represented by eq. (27). The gain coef-ficient can be enhanced by increasing the spontaneous emission rate Rspðh
!Þ. Furthermore, it has been well recognized that the spatial confinement caused by the carrier localization structures leads to an increase in the sponta-neous emission rate and, thus, the optical gain coefficient. Consequently, a PN junction with carrier localization structures such as a nanostructured silicon PN junction diode20)is proposed for this purpose.

In summary, the steady state phonon density Nq, carrier concentration N, and the photon density Np are given by eqs. (22b), (23b), and (24), respectively. When the threshold condition is reached, while the carrier concentration remains approximately at Nth, both the photon and phonon densities increase linearly with the pumping rate Rp.

It is of interest to depict the results for above threshold, i.e., eqs. (22b), (23b), and (24), with those for below threshold, namely, eqs. (15), (17), and (18), as shown in Fig. 3. Below the threshold, the photon density Npincreases 6580

rather slowly. However, once the threshold is reached, the photon density Np increases rapidly and linearly with the

pumping rate Rp. As to the phonon, its density Nq remains practically constant before threshold, but increases linearly with the pumping rate Rponce the threshold is reached. Both the photon and phonon densities grow rapidly after the threshold, which indicates the co-stimulated emission of photons and phonons. On the other hand, the carrier concentration N increases linearly with the pumping rate Rp before the threshold is reached, but clamps at its threshold value Nth after the threshold is reached. In other words, the stimulated emission uses up almost all of the carriers generated by the external pumping after the thresh-old is reached. Any further increase in the pumping rate Rp does not increase the carrier concentration N but contributes to the increase in photon and phonon densities.

4. Temperature Dependence of Optical Gain and Free Carrier Absorption

Equation (27) indicates that population inversion in indirect bandgap semiconductors can occur provided that F > h
!. However, in order to achieve optical amplifica-tion, and eventually lasing, the magnitude of optical gain has to be large enough to overcome the optical losses resulting from the silicon itself and the optical cavity. A major source for material losses is free carrier absorption.26)We shall now make an estimate on the magnitudes of gain coefficient and free carrier absorption in bulk crystalline silicon.

Equations (26a) and (26b) show that the magnitude of gain coefficient gðh
!Þ is proportional to the spontaneous emission rate Rspðh
!Þ. Thus the magnitude of optical gain in bulk crystalline silicon can be estimated based on the spontaneous emission coefficient Acv, which is determined here using the optical absorption data in bulk crystalline silicon.27)_{Appendix B shows that the absorption coefficient} 1ðh
!Þ associated with the one-phonon assisted optical transition is given by 1ðh
!Þ ¼ C1 1 ðh
!Þ2 1 ½1  expðh
1=kBTÞ 
8ðh
! þ h
1EgÞ 2_ exp½ðh
! þ h
1Þ=kBT½expðh
!=kBTÞ  1 ð28Þ

where C1is a constant which is proportional to the spontaneous emission coefficient Acvand is independent of temperature.28) Using the optical absorption coefficient associated with one TO phonon (h
1¼57:8 meV) in bulk crystalline silicon,27)the value of C1is estimated to be 2:2
104cm1. As shown in Appendix A, by rewriting eq. (26a) in terms of the constant C1, the magnitude of optical gain associated with the one-phonon assisted process can be determined using the following equation g1ðh
!Þ ¼ C1 1 ðh
!Þ2  1 ½1  expðh
1=kBTÞ  f1  exp½ðh
!  FÞ=kBTg
Z Ec Z Ev

½Ec ðEgEgÞ1=2ðEvÞ1=2fcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
1ÞdEvdEc ð29Þ

where fcand (1  fv) are the Fermi–Dirac distributions with respective quasi-Fermi levels Fe and Fh for electrons and holes, and the integration variables Ec and Ev are the energies of the conduction band and valence band states, respectively.

The calculated gain spectra of bulk crystalline silicon for various values of F at 300 K are plotted in Fig. 4. In this calculation, the effect of bandgap narrowing Eg due to high carrier density was taken into account using the following relationship29)

Eg¼14  lnðN=1:4
1017Þ meV ð30Þ where N is the carrier density in cm3_{. Also, the relationship} between the quasi-Fermi levels, Fe and Fh, and the carrier density was calculated using the Joyce–Dixon approxima-tion.30)_{As shown in Fig. 4, when the value of F increases,} the magnitude of optical gain coefficient is enhanced and the range of photon energy corresponding to positive optical gain also increases. The gain coefficient changes sign to become absorption when the photon energy is greater than

(

)

(

)

Fig. 3. Phonon, photon, and carrier densities versus the pumping rate below and above threshold.

F. The peak in the gain spectra near photon energy 1.07 eV is associated with one TO phonon-assisted radiative recom-bination,31,32) _{and the shoulder in the gain spectra around} photon energy 0.98 eV is attributed to two-phonon ðTO þ OÞ-assisted radiative recombination.31,32) _{According to the} measured photoluminescence spectra of bulk crystalline silicon at 26 K,31)_{the ratio of the spontaneous emission rate} of the two-phonon process Rsp2ðh
!Þ to the one-phonon process Rsp1ðh
!Þ is 0.07, and this value was adopted in this calculation.

Figure 5 shows the plot of gain coefficients (in logarith-mic scale) of bulk crystalline silicon versus the difference between the quasi-Fermi levels F for one- and two-phonon-assisted processes at 300 K. The optical gains are negative for both the one- and two-phonon processes when F < 0:98 eV. For the range 0:98 < F < 1:07 eV, posi-tive optical gain only occurs for the two-phonon process while the one-phonon process still results in absorption. Population inversion for both one- and two-phonon proc-esses takes place simultaneously when F > 1:07 eV. Figure 5 indicates that two-phonon assisted radiative re-combination requires a smaller value of F to facilitate population inversion, as compared to the one-phonon process.

Figures 4 and 5 show that the maximum gain coefficient in bulk crystalline silicon can be as large as 22.5 cm1when F ¼ 1:15 eV. However, optical amplification in bulk crystalline silicon around the bandgap energy Eg has never been observed at room temperature. This fact may be attributed to the excessive free carrier absorption.26) _The magnitude of free carrier absorption fcat room temperature in units of cm1 _{is given by}28,33)

fc¼ ð1:01
1020N þ 0:51
1020PÞ2T ð31Þ where N and P are again the carrier densities in units of cm3_{for electrons and holes, and  is the wavelength in mm,} and T is the temperature in K. The plots in Fig. 6 show the magnitudes of optical gain and free carrier absorption in bulk crystalline silicon versus the carrier density at photon energy h
! of 1.07 eV at 300 K. In the calculation, the electron concentration N is assumed to be equal to the hole concentration P due to charge neutrality. The gain coef-ficient increases linearly with the carrier density, and the transparent carrier density is 3
1019_cm3 _{in bulk} crystal-line silicon, which is about one order of magnitude greater than that in bulk GaAs.34) _{Note that the transparent carrier} density is defined as the carrier density where stimulated emission rate Rstðh
!Þ equals absorption rate Rabðh
!Þ.34) Figure 6 clearly demonstrates that the free carrier absorption is larger than the optical gain in bulk crystalline silicon. This conclusion is identical to that of Dumke.26)_{As a result, even} though the condition for population inversion eq. (27) is met, it is not possible to accomplish optical gain or optical amplification in bulk crystalline silicon at room temperature. In Fig. 7, we show the calculated gain spectra of bulk crystalline silicon for the carrier concentrations of 8:8
1019cm3 at the temperatures of 10, 150, and 300 K. In this calculation, the temperature dependence of the bandgap energy is described by the Varshni’s equation35)

EgðTÞ ¼ Egð0Þ  AT2

T þ B ð32Þ

where Egð0Þ ¼ 1:17 eV, A ¼ 4:9
104eV/K, and B ¼ 655 K. We use the inverse first-order Sommerfeld approx-imation to calculate the relationship between the

quasi-0.95 1.00 1.05 1.10 1.15 10-3 10-2 10-1 100 101 102

∆

F (eV)

Gain coefficient (cm

)

Fig. 5. Calculated gain coefficients of one- and two-phonon-assisted processes in bulk crystalline silicon versus F at 300 K.

0 _2x1019 4x1019 6x1019 8x1019 1x1020 0 50 100 150 200 250 300 350 400 450

Carrier density (cm

)

free carrier absorption

optical gain of bulk crystalline silicon

Gain/Absorption coefficient (cm

-1 )

Fig. 6. Calculated optical gain and free carrier absorption coefficients in bulk crystalline silicon versus the carrier density at photon energy h
! ¼ 1:07 eV at 300 K. 0.9 1.0 1.1 1.2 -10 0 10 20 30 0.98eV 1.075 eV 1.10 eV 1.125 eV ∆F = 1.15 eV

Gain coefficient (cm

)

Photon energy (eV)

1.07eV

Fig. 4. Calculated gain spectra of bulk crystalline silicon for various values of F at 300 K.

Fermi levels and the carrier density when the temperature is below 100 K.34)_{Figure 7 shows that the gain spectra is} blue-shifted due to the increase in the bandgap energy when the temperature is lowered. The magnitude of optical gain is also shown to increase with decreasing temperature. On the other hand, it should be noted from eq. (31) that as the temperature decreases, the magnitude of free carrier absorp-tion also decreases. Such a temperature dependence is associated with the acoustic phonon scattering.33) Thus it suggests that the optical gain may be larger than the free carrier absorption at a sufficiently low temperature. For this reason we calculate the magnitude of optical gain and free carrier absorption at low temperature. The calculation shows that the optical gain is greater than the free carrier absorption when the temperature is below 23 K. For example, Fig. 8 shows the calculated gain coefficient and free carrier absorption in the bulk crystalline silicon versus the carrier density at photon energy h
! ¼ 1:1 eV at the temperature of 10 K. It is seen that at the temperature of 10 K, the optical gain greatly exceeds the free carrier absorption when the carrier density is greater than 4
1019_cm3_{. Therefore, the}

optical amplification, as well as the co-stimulated emissions of photons and phonons, can be achieved in bulk crystalline silicon at the low temperature below 23 K.

5. Discussion

The recent paper by Trupke et al. presented a reassess-ment of optical gain in indirect bandgap semiconductors based on a simple two-level model.22)The paper points out that the magnitude of optical gain can be orders of magnitude greater than the absorption coefficient and, thus, suggests that the possibility of laser operation in bulk crystalline silicon should be readdressed. In comparison with ref. 22, the theoretical treatment presented here is more specific as it takes into account three important aspects: (1) the structures of the conduction and valence band band-edges, (2) the temperature dependence, and (3) the effect of bandgap narrowing. Using the expressions obtained for the optical gain, the gain spectrum in bulk crystalline silicon is calculated and the magnitude of optical gain is obtained. Furthermore, this treatment explicitly shows that the magnitude of the gain coefficient in bulk crystalline silicon is smaller than the free carrier absorption at room temper-ature. Thus optical amplification is not possible for bulk crystalline silicon at room temperature. However, at the low temperature below 23 K, the optical gain is shown, for the first time, to be greater than the free carrier absorption. This result indicates that it is possible to achieve optical amplification in bulk crystalline silicon at the temperature below 23 K. In addition, the rate equations and their steady state solutions for electron, photon, and phonon involved in the phonon-assisted optical transitions are obtained. It is shown that once the threshold condition for laser oscillation is reached, co-stimulated emissions of photons and phonons will take place in indirect bandgap semiconductors.

On the other hand it should also be noted that some of the theoretical findings presented in this paper differ from that of Dumke.26) He estimated the magnitude of optical gain in indirect semiconductors and concluded that optical amplifi-cation in germanium and silicon is impossible because the optical gain is too small to overcome the free carrier absorption. Dumke’s calculations are based on the assump-tion that the absolute value of maximum optical gain equals the absorption coefficient measured under thermal equi-librium. However, this assumption is questionable for the phonon-assisted optical transitions in indirect bandgap semiconductors.22)_{As shown in Figs. 4–6, we find that the} gain coefficient around the photon energy of 1.07 eV in bulk crystalline silicon at room temperature can be as large as 22.5 cm1, which is at least two orders of magnitude greater than the absorption coefficient (0.58 cm1).27) Also, as shown in Appendix C, such a large disparity between the optical gain and absorption coefficients results from the involvement of phonons in optical transitions and the effect of bandgap narrowing at high carrier density. At a given photon energy, the stimulated emission of photons is assisted by emission of phonons, while the absorption of photons is also assisted by absorption of phonons. The difference in the phonon emission and absorption rates results in the large disparity between the maximum optical gain and absorption coefficient in bulk crystalline silicon. For a given photon energy, the bandgap narrowing at high carrier density results

0 _2x1019 4x1019 6x1019 8x1019 1x1020 0 10 20 30 40 T = 10K =1.10 eV hω

free carrier absorption optical gain for

bulk crystalline silicon

Gain/Absorption coefficient (cm

-1 )

Carrier density (cm

)

Fig. 8. Calculated optical gain and the free carrier absorption coefficients vs the carrier density in bulk crystalline silicon at photon energy h
! ¼ 1:10 eV at 10 K. 0.9 1.0 1.1 1.2 -20 0 20 40 60 19 –3 8.8×10 N = P = cm 300K 150K Τ=10Κ

Gain coefficient (cm

)

Photon energy (eV)

Fig. 7. Calculated gain spectra of bulk crystalline silicon for the carrier concentration of 8:8
1019_cm3 _{at the temperatures of 10, 150, and} 300 K.

in the increase in the density of conduction and valence band states involved in the optical transition, and thus causes the gain coefficient to increase.

6. Conclusions

A theoretical study on optical gain at bandgap energy in indirect bandgap semiconductors, such as silicon, is presented in this paper. The expressions for optical gain associated with one and multi-phonon emission in indirect bandgap semiconductors are obtained. Population inversion can be achieved in indirect bandgap semiconductors when the difference between the quasi-Fermi levels for electrons and holes, F, is greater than the photon energy. It is to be emphasized that two-phonon process requires a lower value of F to facilitate population inversion than the one-phonon process. In contrast to the direct bandgap semiconductors, population inversion in the indirect bandgap semiconductors can be achieved even if F is smaller than the bandgap energy.

The rate equations for the single-mode photon, phonon and carrier in indirect bandgap semiconductor are also presented. The condition required for laser oscillation and the steady state solutions below and above threshold are obtained. Below the threshold the photon and phonon densities increase slowly with the pumping rate. However, once the threshold is reached, both the photon and phonon densities increase rapidly and linearly with the pumping rate. The simultaneous increase of the photon and phonon densities can be easily understood since a phonon is emitted during the emission of a photon, a stimulated emission of phonons will take place due to the stimulated emission of photons. Thus co-stimulated emissions of photons and phonons will take place when the threshold condition for laser oscillation is satisfied. As for the carrier concentration, it increases linearly with the pumping rate before the threshold is reached, but remains practically at its threshold value after the threshold is reached.

The temperature dependence of optical gain and free carrier absorption in bulk crystalline silicon is calculated for the first time. It is shown that the optical gain is not sufficiently large to overcome the free carrier absorption at room temperature. Accordingly, it is impossible to observe optical amplification in bulk crystalline silicon at room temperature. However, at the temperature below 23 K, the gain coefficient exceeds the free carrier absorption. It suggests that the co-stimulated emissions of photons and phonons as well as the optical amplification can take place in bulk crystalline silicon at the temperature below 23 K. Acknowledgement

This work was initiated at and supported by the Academia Sinica and was subsequently supported in part by the National Science Council, Taiwan, R.O.C under Contract No. NSC94-2112-M-002-019 (M. J. Chen and M. K. Wu), and the Nicholas Foundation Prize, University of California at Irvine, California, U.S.A. (C. S. Tsai).

1) R. A. Soref: Proc. IEEE 81 (1993) 1687. 2) R. A. Soref: Thin Solid Films 294 (1997) 325. 3) O. Boyraz and B. Jalali: Opt. Express 12 (2004) 5269.

4) H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang and

M. Paniccia: Nature 433 (2005) 725.

5) See, for example, D. J. Lockwood: Light emission in Silicon: from Physics to Devices (Academic, New York, 1998).

6) L. Pavesi, S. V. Gaponenko and L. Dal Negro: Towards the First Silicon Lasers (Kluwer Academic, Dordrecht, 2003) NATO Science Series.

7) L. Pavesi: Mat. Today 8 (2005) 18.

8) L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzo and F. Priolo: Nature 408 (2000) 440.

9) L. Tsybeskov, K. L. Moore, D. G. Hall and P. M. Fauchet: Phys. Rev. B 54 (1996) R8361.

10) Z. H. Lu, D. J. Lockwood and J. M. Baribeau: Nature 378 (1995) 258. 11) J. Ruan, P. M. Fauchet, L. Dal Negro, M. Cazzanelli and L. Pavesi:

Appl. Phys. Lett. 83 (2003) 5479.

12) L. T. Canham: Appl. Phys. Lett. 57 (1990) 1046.

13) J. Palm, F. Gan, B. Zheng, J. Michel and L. C. Kimerling: Phys. Rev. B 54 (1996) 17603.

14) P. G. Kik, A. Polman, S. Libertino and S. Coffa: J. Lightwave Technol. 20(2002) 862.

15) S. G. Cloutier, P. A. Kossyrev and J. Xu: Nat. Mater. 4 (2005) 887. 16) L. Tsybeskov, K. L. Moore, S. P. Duttagupta, K. D. Hirschman, D. G.

Hall and P. M. Fauchet: Appl. Phys. Lett. 69 (1996) 3411.

17) W. L. Ng, M. A. Louren, R. M. Gwilliam, S. Ledain, G. Shao and K. P. Homewood: Nature 410 (2001) 192.

18) M. A. Green, J. Zhao, A. Wang, P. J. Reece and M. Gal: Nature 412 (2001) 805.

19) M. J. Chen, J. F. Chang, J. L. Yen, C. S. Tsai, E. Z. Liang, C. F. Lin and C. W. Liu: J. Appl. Phys. 93 (2003) 4253.

20) M. J. Chen, J. L. Yen, J. Y. Li, J. F. Chang, S. C. Tsai and C. S. Tsai: Appl. Phys. Lett. 84 (2004) 2163.

21) H. A. Atwater, G. I. Bourianoff and R. J. Walters: Nat. Mater. 4 (2005) 143.

22) T. Trupke, M. A. Green and P. Wu¨rfel: J. Appl. Phys. 93 (2003) 9058. 23) N. W. Ashcroft and N. D. Mermin: Solid State Physics (Brooks Cole,

1976).

24) J. Shah: Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer-Verlag, Heidelberg, 1999) 2nd ed. 25) M. G. A. Bernard and G. Duraffourg: Phys. Status Solidi 1 (1961) 699. 26) W. P. Dumke: Phys. Rev. 127 (1962) 1559.

27) G. G. Macfarlane, T. P. McLean, J. E. Quarrington and V. Roberts: Phys. Rev. 111 (1958) 1245.

28) J. C. Sturm and C. M. Reaves: IEEE Trans. Electron Devices 39 (1992) 81.

29) A. Cuevasa, P. A. Basoreb, G. Giroult-Matlakowski and C. Dubois: J. Appl. Phys. 80 (1996) 3370.

30) W. B. Joyce and R. W. Dixon: Appl. Phys. Lett. 31 (1977) 354. 31) P. J. Dean, J. R. Haynes and W. F. Flood: Phys. Rev. 161 (1967) 711. 32) G. Davies: Phys. Rep. 176 (1989) 83.

33) K. G. Svantesson and N. G. Nilsson: J. Phys. C 12 (1979) 3837. 34) L. A. Coldren and S. W. Corzine: Diode Lasers and Photonic

Integrated Circuit (Wiley, New York, 1995).

35) V. Alex, S. Finkbeiner and J. Weber: J. Appl. Phys. 79 (1996) 6943. 36) B. K. Ridley: Quantum Processes in Semiconductors (Oxford

University Press, Oxford, 1999) 3rd ed.

37) See, for example, A. E. Siegman: Lasers (University Science Books, Mill Vallet, 1986).

38) A. Yariv: Quantum Electronics (Wiley, New York, 1989) 3rd ed. 39) W. van Roosbroeck and W. Shockley: Phys. Rev. 94 (1954) 1558. 40) H. Schlangenotto, H. Maeder and W. Gerlach: Phys. Status Solidi A 21

(1974) 357.

41) G. Augustine, A. Rohatgi and N. M. Jokerst: IEEE Trans. Electron Devices 39 (1992) 2395.

42) D. K. Schroder: Semiconductor Material and Device Characterization (Wiley, New York, 1998) 2nd ed.

Appendix A

A.1 Stimulated emission rate Rstðh
!Þ

The stimulated emission rate Rstðh
!Þ per unit volume of the quantum transition involving the emission of a photon with energy h
! and a phonon with energy h
 is given by23)

Rstðh
!Þ ¼ Bcvphðh
!Þ
Z Ec Z Ev

ðnqþ1Þ  DcðEcÞDvðEvÞfcðEcÞ

½1  fvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc ðA:1Þ where Bcvis the stimulated emission coefficient, and phðh
!Þ is the photon density per energy interval. In eq. (A·1), the integration must cover those states in both the conduction and valence bands that may participate in the transition with photon energy h
! because the momentum in the initial state and final state are not the same. The integral variables Ecand Evare the energies of the conduction band and valence band states, respectively. The delta function in the integral restricts Ec and Ev to the energy states which contribute to the quantum transitions involving the emission of a photon with energy h
! and a phonon with energy h
. DcðEcÞand DvðEvÞ are the well-known densities of states for electrons and holes in the conduction and valence bands

DcðEcÞ ¼NcðEcEgÞ1=2; Nc¼ ð1=2
2Þ  ð2m e=h
2_Þ3=2_; ðA:2aÞ DvðEvÞ ¼NvðEvÞ1=2; Nv ¼ ð1=2
2Þ  ð2m h=h
2_Þ3=2 ðA:2bÞ

where h
 h=2
is the reduced Planck constant, h is Planck constant, Eg is the indirect bandgap energy, and m e and m

h are the effective masses of electron and hole, respectively. In our calculation, the energy of the top of the valence band is taken as zero, as depicted in Fig. 1. The distribution functions fcand (1  fv) for electrons and holes are given by the Fermi–Dirac distribution with respective quasi-Fermi levels Feand Fh. When the quasi-Fermi levels for electrons and holes locate in the bandgap and far away from the conduction and valence band edges, respectively, the distribution functions fcand (1  fv) may be approximated by the Boltzmann distribution

fcðEcÞ ¼

1

1 þ exp½ðEcFeÞ=kBT

exp½ðEcFeÞ=kBT if ðEcFeÞ=kBT  1; ðA:3aÞ

1  fvðEvÞ ¼1  1 1 þ exp½ðEvFhÞ=kBT ¼ 1 1 þ exp½ðFhEvÞ=kBT exp½ðFhEvÞ=kBT if ðFhEvÞ=kBT  1 ðA:3bÞ

where kBis Boltzmann constant and T is absolute temperature.

Finally, the factor nqin eq. (A·1) is the phonon occupation number, and the probability of phonon emission is proportional to (nqþ1).36)Since the phonons participating in the compensation of the momentum mismatch lie near the Brillouin zone edge, the phonon energy h
 can be considered independent of its wavevector.36)_{At thermodynamic equilibrium, the phonon} occupation number nqbecomes nq0 and is given by Bose–Einstein statistics36)

nq¼nq0¼

1 expðh
=kBTÞ  1

ðA:4Þ

Thus nq is also independent of the wavevector as well as Ecand Ev. As a result, the (nqþ1) term may be taken out of the integral in eq. (A·1). Using eqs. (A·2) and (A·3), eq. (A·1) is reduced to

Rstðh
!Þ ¼ Bcvphðh
!Þ  ðnqþ1Þ  NcNv
Z Ec Z Ev

ðEcEgÞ1=2ðEvÞ1=2exp½ðEcEvFÞ=kBT  ðh
!  EcþEvþh
Þ dEvdEc

¼Bcvphðh
!Þ  ðnqþ1Þ  NcNvexp½ðh
! þ h
  FÞ=kBT
Z_h
!þh
 Eg ðEcEgÞ1=2ðh
!  Ecþh
Þ1=2dEc ¼BcvNcNvphðh
!Þ  ðnqþ1Þ 
8ðh
! þ h
  EgÞ 2_ exp½ðh
! þ h
  FÞ=kBT ðA:5Þ

where F ¼ FeFh is the difference between the quasi-Fermi levels for electrons and holes.

A.2 Spontaneous emission rate Rspðh
!Þ

Similarly, the spontaneous emission rate Rspðh
!Þ per unit volume of the quantum transition involving the emission of a photon with energy h
! and a phonon with energy h
 is written as follows36)

Rspðh
!Þ ¼ Acv Z

Ec

Z Ev

ðnqþ1Þ  DcðEcÞDvðEvÞfcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc ðA:6Þ

where Acvis the spontaneous emission coefficient. If we again apply Boltzmann approximation to the distribution functions fc and (1  fv), the spontaneous emission rate Rspðh
!Þ is reduced to the following simplified form

Rspðh
!Þ ¼ AcvNcNv ðnqþ1Þ 

8ðh
! þ h
  EgÞ 2_

A.3 Absorption rate Rabðh
!Þ

Lastly, the absorption rate Rabðh
!Þ per unit volume of the quantum transition involving the absorption of a photon with energy h
! and a phonon with energy h
 takes the following form36)

Rabðh
!Þ ¼ Bvcphðh
!Þ  Z

Ec

Z Ev

nqDcðEcÞDvðEvÞ½1  fcðEcÞfvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc ðA:8Þ

where Bvcis the coefficient for absorption. The nqterm is associated with the probability of phonon absorption.36)The delta function in the integral again restricts the energy (EcEv) to be equal to (h
! þ h
) due to the absorption of a photon and a phonon. Again, when the quasi-Fermi levels for electrons and holes locate in the bandgap far away from the conduction and valence band edges, the distribution functions (1  fc) and fv can be approximated by

1  fcðEcÞ ¼ 1 1 þ exp½ðFeEcÞ=kBT 1; ðA:9aÞ fvðEvÞ ¼ 1 1 þ exp½ðEvFhÞ=kBT 1: ðA:9bÞ

Thus, eq. (A·8) is reduced to the following simple form:

Rabðh
!Þ ¼ Bvcphðh
!Þ  nqNcNv Z Ec Z Ev ðEcEgÞ1=2ðEvÞ1=2ðh
!  EcþEvþh
Þ dEvdEc ¼BvcNcNvphðh
!Þ  nq
8ðh
! þ h
  EgÞ 2_: ðA:10Þ

A.4 Optical gain gðh
!Þ

At thermodynamic equilibrium there is only one common Fermi level, i.e., F ¼ 0, and the Fermi level is located far away from conduction and valence band edges. As a result, the Boltzmann distribution represents valid approximation for the distribution functions fcand fv. Now, the principle of detailed balance requires that the upward transition rate must equal the downward transition rate at thermodynamic equilibrium

Rabðh
!Þ ¼ Rspðh
!Þ þ Rstðh
!Þ ðA:11Þ

Using eqs. (A·5), (A·7), and (A·10), the above equation yields

Bvcphðh
!Þnq¼ ½Bcvphðh
!Þ þ Acv  ðnqþ1Þ  exp 

h
! þ h
 kBT



ðA:12Þ

Since nqis given by Bose–Einstein distribution, i.e., eq. (A·4), at thermodynamic equilibrium the common factors involving phonons such as nq and h
 in eq. (A·12) cancel out and the following simple expression is obtained

phðh
!Þ ¼ Acv Bvcexpðh
!=kBTÞ  Bcv ¼Dpnp¼ 8
n2_n gðh
!Þ2 h3_c3 1 expðh
!=kBTÞ  1 ðA:13Þ

where the photon density per energy interval phðh
!Þ is given by Plank’s law, Dp¼8
n2ngðh
!Þ2=h3c3is the density of states of photon per energy interval, np¼ ½expðh
!=kBTÞ  11is the photon occupation number which gives the average number of photons per state with the photon energy h
! at thermodynamic equilibrium, c is the velocity of light in free space, and n and ng are the refractive index and the group index of the materials, respectively. We see that

Bvc¼Bcv; ðA:14aÞ

Acv¼DpBcv: ðA:14bÞ

Clearly, eqs. (A·14a) and (A·14b) are the same as the well-known results of Einstein’s treatment for a simple two-level system. Thus the relationship between Acv, Bcv, and Bvc in the phonon-assisted optical transitions are the same as that in a simple two-level system.

Now, rewriting the expressions for stimulated emission rate Rstðh
!Þ and absorption rate Rabðh
!Þ in terms of spontaneous emission rate Rspðh
!Þ yields

Rstðh
!Þ ¼ Bcv Acv phðh
!Þ  Rspðh
!Þ; ðA:15Þ Rabðh
!Þ ¼ Bvcphðh
!Þ  nq
Z Ec Z Ev

DcðEcÞDvðEvÞ½1  fcðEcÞfvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc

¼ ðBvc=AcvÞ phðh
!Þ  nq=ðnqþ1Þ  Acv ðnqþ1Þ
Z Ec Z Ev

DcðEcÞDvðEvÞfcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
Þ 

½1  fcðEcÞfvðEvÞ fcðEcÞ½1  fvðEvÞ

dEvdEc

¼ ðBvc=AcvÞ phðh
!Þ  nq=ðnqþ1Þ  Acv ðnqþ1Þ
Z Ec Z Ev

DcðEcÞDvðEvÞfcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
Þ  exp½ðEcEvFÞ=kBT dEvdEc

¼ ðBvc=AcvÞ phðh
!Þ  nq=ðnqþ1Þ  exp½ðh
! þ h
  FÞ=kBT  Acv ðnqþ1Þ
Z Ec Z Ev

DcðEcÞDvðEvÞfcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc

¼Bvc Acv phðh
!Þ  nq nqþ1 exp h
! þ h
  F kBT
 Rspðh
!Þ: ðA:16Þ

Although the relationships between Acv, Bcv, and Bvc in eq. (A·14) are established under the condition of thermodynamic equilibrium, we assume that they remain valid even under the non-equilibrium conditions to derive optical gain.37,38)_Using eqs. (A·14)–(A·16), the optical gain gðh
!Þ of the quantum transition involving the photon energy h
! and the phonon energy h
 is obtained as follows gðh
!Þ Rstðh
!Þ  Rabðh
!Þ phðh
!Þ  ðc=ngÞ ¼ h 3_c2 8
n2_ðh
!Þ2Rspðh
!Þ  1  nq nqþ1 exp h
! þ h
  F kBT
   : ðA:17Þ

It should be noted that at thermodynamic equilibrium, F ¼ 0 and nq is given by the Bose–Einstein statistics.36)As a result, the optical gain becomes negative, i.e., absorption, and eq. (A·17) is reduced to the well-known van Roosbroeck– Shockley relation39) Rspðh
!Þ ¼ ðh
!Þ  8
n2ðh
!Þ2 h3_c2 1 expðh
!=kBTÞ  1 ðA:18Þ

where ðh
!Þ is the absorption coefficient. Clearly, eq. (A·18) provides a direct link between the absorption coefficient and the spontaneous emission rate under thermodynamic equilibrium situations.

After substituting eq. (A·7) into eq. (26a) and set F ¼ 0, the absorption coefficient 1ðh
!Þ associated with the one-phonon assisted optical transition is obtained as follows:

1ðh
!Þ ¼ h3_c2 8
n2_ðh
!Þ2Acv1NcNv ðnq1þ1Þ 
8ðh
! þ h
1EgÞ 2 exp½ðh
! þ h
1Þ=kBT½expðh
!=kBTÞ  1 ¼C1 1 ðh
!Þ2 1 1  expðh
1=kBTÞ 
8ðh
! þ h
1EgÞ 2 exp½ðh
! þ h
1Þ=kBT½expðh
!=kBTÞ  1 ðA:19Þ

where the proportional constant C1¼ ðh3c2=8
n2Þ Acv1NcNv. According to eqs. (26a) and (A·6), the optical gain associated with the one-phonon assisted process can be rewritten in terms of the constant C1 as follows

g1ðh
!Þ ¼ C1 1 ðh
!Þ2 1 1  expðh
1=kBTÞ  f1  exp½ðh
!  FÞ=kBTg
Z Ec Z Ev

½Ec ðEgEgÞ1=2ðEvÞ1=2fcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
1ÞdEvdEc:

ðA:20Þ

Appendix B

It is necessary to relate the rates of stimulated emission, spontaneous emission and absorption to the electron and hole concentrations. The spontaneous emission rate as defined in eq. (A·6) is related to the concentrations of electrons and holes as follows: Rspðh
!Þ ¼ Bðh
!Þ  NP ¼Acv ðnqþ1Þ  NcNv Z Ec Z Ev

ðEcEgÞ1=2ðEvÞ1=2fcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
Þ dEvdEc

¼Acv ðnqþ1Þ  NcNvI

ðB:1Þ

where Bðh
!Þ is the bimolecular radiative recombination coefficient,34)_{N and P are the electron and hole concentrations, and} the integral in eq. (B·1) is denoted by I, respectively. Accordingly,

Bðh
!Þ ¼ Acv ðnqþ1Þ  NcNv I

NP: ðB:2Þ

Since Bðh
!Þ is independent of F,40) _{we can calculate the value of I=NP in eq. (B·2) in the region where Boltzmann} approximation is valid. Using eqs. (A·2), (A·3), and (A·5), we obtain

I NP¼
8ðh
! þ h
  EgÞ 2_{exp }h
! þ h
  F kBT ! NcNvexp  EgF kBT ! ¼ 1 NcNv
8ðh
! þ h
  EgÞ 2_{exp }h
! þ h
  Eg kBT
 : ðB:3Þ

Thus the bimolecular radiative recombination coefficient Bðh
!Þ and the spontaneous emission rate Rspðh
!Þ are given by

Bðh
!Þ ¼
8Acvðnqþ1Þðh
! þ h
  EgÞ 2_{exp }h
! þ h
  Eg kBT
 ¼M  ðnqþ1Þ; ðB:4Þ Rspðh
!Þ ¼
8Acvðnqþ1Þðh
! þ h
  EgÞ 2_{exp }h
! þ h
  Eg kBT
 NP ¼ M  ðnqþ1Þ  NP ðB:5Þ

where the proportional constant M ¼
=8  Acvðh
! þ h
  EgÞ2exp½ðh
! þ h
  EgÞ=kBT. Similarly, for the stimulated emission rate eq. (A·15) and the absorption rate eq. (A·16),

Rstðh
!Þ ¼ Bcv Acv phðh
!Þ  Rspðh
!Þ ¼ 1 Dp DpnpMðnqþ1ÞNP ¼M  npðnqþ1Þ  NP; ðB:6Þ Rabðh
!Þ ¼ Bvc Acv phðh
!Þ  nq nqþ1 exp h
! þ h
  F kBT
 Rspðh
!Þ ¼ 1 Dp Dpnp nq nqþ1 exp h
! þ h
  F kBT
 Mðnqþ1ÞNP ðB:7Þ ¼M  npnqexp h
! þ h
  F kBT
 NP:

Note that Dp and np have been defined in connection with eq. (A·13). Appendix C

By using eqs. (A·20), (26a), (28), and (29), the ratio of the optical gain coefficient to the absorption coefficient for one-phonon assisted optical transition is given by

g1ðh
!Þ 1ðh
!Þ ¼Rsp1ðh
!; F 6¼ 0Þ Rsp1ðh
!; F ¼ 0Þ f1  exp½ðh
!  FÞ=kBTg ½expðh
!=kBTÞ  1 ¼ Z Ec Z Ev

½Ec ðEgEgÞ1=2ðEvÞ1=2fcðEcÞ½1  fvðEvÞ ðh
!  EcþEvþh
1ÞdEvdEc

=8  ðh
! þ h
1EgÞ2exp½ðh
! þ h
1Þ=kBT

f1  exp½ðh
!  FÞ=kBTg ½expðh
!=kBTÞ  1

:

ðC:1Þ

Under the condition of high carrier injection, i.e., F  h
!, the distribution functions fcðEcÞ 1 and ½1  fvðEvÞ 1, and so eq. (C·1) reduces to the following simple form:

g1ðh
!Þ 1ðh
!Þ ðh
! þ h
1EgþEgÞ 2 ðh
! þ h
1EgÞ2 exp½h
1=kBT ¼ 1 þ Eg h
! þ h
1Eg
2 exp½h
1=kBT: ðC:2Þ

The first factor in eq. (C·2) represents the effect of bandgap narrowing, while the second factor results from the involvement of phonons in optical transitions.22) Equation (C·2) clearly shows that both bandgap narrowing and phonon participation contribute to the large disparity between the maximum optical gain and the absorption coefficient in bulk crystalline silicon.