Band-to-band optical transitions in semiconductors

This is a continuation from the previous tutorial - radiative recombination in semiconductors.

In the discussion of the characteristics of optical transitions between the energy levels of an individual atom or molecule in the optical transitions for laser amplifiers tutorial, each active atom or molecule is considered a separate system in the sense that it has its own energy levels and it can reside in a particular state independently of the states of other active atoms or molecules.

For the electrons and holes in a semiconductor, the situation is quite different. The states of all of the valence electrons in a semiconductor collectively form energy bands. Because the electron population in the band states is governed by the Fermi-Dirac distribution function, the state of a given electron in a semiconductor is not independent of other electrons.

A band-to-band transition in a semiconductor takes place through the transition of such an electron between a valence band and a conduction band. Consequently, not every concept discussed in the optical transitions for laser amplifiers tutorial regarding optical transitions between the energy levels of an individual atom or molecule is directly applicable to band-to-band optical transitions in a semiconductor.

In particular, the concepts of transition cross section and population inversion have to be modified. When considering a band-to-band optical transition, the characteristics of the band structure have to be considered.

There are two types of band-to-band transitions in a semiconductor. A direct transition takes place when an electron makes an upward or downward transition without the participation of a phonon.

In contrast, when an electron makes an indirect transition, it has to absorb or emit a phonon, thereby exchanging energy and momentum with the crystal lattice, in order to complete the transition.

The transition probability differs significantly between a direct process and an indirect process.

When an electron makes a band-to-band transition between a state \(|1\rangle\) of energy \(E_1\) and wavevector \(\mathbf{k}_1\) in a valance band and a state \(|2\rangle\) of energy \(E_2\) and wavevector \(\mathbf{k}_2\) in a conduction band, both the conservation of energy and the conservation of wavevector have to be satisfied among all parties involved, including any participating photon and phonon.

The magnitude of the electron wavevector in a crystal is of the order of \(2\pi/a\), where the lattice constant \(a\) is smaller than 1 nm, but the wavevector of a photon is \(2\pi/\lambda\), where the wavelength \(\lambda\) is on the order of 1 μm.

Clearly, the photon wavevector is negligibly small in comparison to the electron wavevector. Consequently, the conditions for direct band-to-band transition with the absorption or emission of a photon are

\[\tag{13-9}E_2-E_1=h\nu\qquad\text{and}\qquad\mathbf{k}_2=\mathbf{k}_1+\mathbf{k}_\text{photon}\approx\mathbf{k}_1\]

The requirement of the conservation of wavevector is akin to the requirement of phase matching in the interaction among optical waves and the requirement of the conservation of momentum in the interaction among particles.

The vector quantity \(\hbar\mathbf{k}\) is known as the crystal momentum of an electron in a band state of wavevector \(\mathbf{k}\), but it is not really the momentum of the electron in the usual sense.

As illustrated in Figure 13-3, the conditions for direct transition can be satisfied in a direct-gap semiconductor for transitions between states near the conduction- and valence-band edges.

Band-to-band absorption in a direct-gap semiconductor normally occurs through a direct absorption process for a photon energy of \(h\nu\ge{E}_\text{g}\). As a result, the absorption spectrum of a direct-gap semiconductor shows a sharp edge at \(h\nu=E_\text{g}\) and rises quickly when the photon energy increases above the bandgap.

Band-to-band recombination in a direct-gap semiconductor can also take place through a direct recombination process with emission of a photon of an energy of \(h\nu\ge{E}_\text{g}\). Because the conditions in (13-9) for a direct transition process can be easily satisfied, the probability of radiative recombination in a direct-gap semiconductor is very high, leading to a short radiative lifetime and a high radiative efficiency.

In an indirect-gap semiconductor, the requirement of conservation of wavevector for direct transition cannot be satisfied for transitions between states near the band edges, as illustrated in Figure 13-4(a).

An indirect optical transition between two such states is possible, however, if the process is assisted by the absorption or emission of a phonon of an energy \(\hbar\Omega\) and a wavevector \(\mathbf{K}\) that satisfy the following conditions:

\[\tag{13-10}E_2-E_1=h\nu\pm\hbar\Omega\qquad\text{and}\qquad\mathbf{k}_2=\mathbf{k}_1+\mathbf{k}_\text{photon}\pm\mathbf{K}\approx\mathbf{k}_1\pm\mathbf{K}\]

An indirect transition process has a much lower probability than a direct transition process because, in comparison to a direct process that involves only a photon and an electron, an indirect process is a high-order process that requires the participation of a phonon.

Near the band edges of an indirect-gap semiconductor, both optical absorption and radiative carrier recombination can take place only through an indirect transition process.

Direct optical transition between a state near the valence-band edge and a state high above the conduction-band edge and that between a state well below the valence-band edge and a state near the conduction-band edge are possible in an indirect-gap semiconductor, as illustrated in Figure 13-4(b). If the photon energy is sufficiently larger than \(E_\text{g}\), direct optical absorption occurs readily in an indirect-gap semiconductor.

Therefore, the absorption coefficient of an indirect-gap semiconductor first increases gradually as the photon energy increases just above the bandgap where only indirect absorption takes place. It then has a sharp increase when the photon energy reaches the threshold for direct absorption to occur.

Carrier recombination through a direct optical transition process in an indirect-gap semiconductor is highly unlikely, however, because a state high above the conduction-band edge is normally not occupied by an electron while a state deep down below the valence-band edge is generally occupied.

Consequently, band-to-band carrier recombination in an indirect-gap semiconductor is generally an indirect process, which has a low radiative recombination probability and a long radiative lifetime.

Because of this long radiative lifetime, competing nonradiative recombination processes can easily take place, resulting in a low radiative efficiency for an indirect-gap semiconductor. This is the reason why the important semiconductors Si and Ge are not useful for fabricating lasers and LEDs, though they are good for making photodetectors.

Direct Transition Rates

To evaluate the transition rates of direct band-to-band optical transitions in a semiconductor, a few conditions imposed by the band structure have to be considered.

As a result, the formulation for the direct transition rates of a semiconductor is different from the transition rates obtained in the optical transitions for laser amplifiers tutorial for individual atoms or molecules.

First, the conditions in (13-9) that dictate the conservation of energy and momentum for a direct optical transition have to be satisfied. By taking \(|\mathbf{k}_2|\approx|\mathbf{k}_1|=k\) for momentum conservation and considering the fact that the electron and hole energies vary with the value of \(k\) quadratically near the band edges, we have

\[\tag{13-11}E_2=E_\text{c}+\frac{\hbar^2k^2}{2m_\text{e}^*}\]

\[\tag{13-12}E_1=E_\text{v}-\frac{\hbar^2k^2}{2m_\text{h}^*}\]

By applying the condition \(E_2-E_1=h\nu\) for energy conservation and using the relation of \(E_\text{c}-E_\text{v}=E_\text{g}\), (13-11) and (13-12) can be used to find \(E_2\) and \(E_1\) in terms of the photon energy as

\[\tag{13-13}E_2=E_\text{c}+\frac{m_\text{r}^*}{m_\text{e}^*}(h\nu-E_\text{g})\]

\[\tag{13-14}E_1=E_\text{v}-\frac{m_\text{r}^*}{m_\text{h}^*}(h\nu-E_\text{g})\]

where \(m_\text{r}^*\) is the reduced effective mass defined as

\[\tag{13-15}m_\text{r}^*=\frac{m_\text{e}^*m_\text{h}^*}{m_\text{e}^*+m_\text{h}^*}\]

To satisfy the conservation of energy and momentum simultaneously, a band-to-band optical transition associated with a photon of an energy \(h\nu\) can occur only between a conduction-band state of energy \(E_2\) given by (13-13) and a valence-band state of energy \(E_1\) given by (13-14).

Next, we have to consider the density of states in the conduction and valence bands that satisfy the conservation of energy and momentum for the optical transition. This can be done by considering the states in the conduction and valence bands that satisfy (13-13) and (13-14), respectively.

The density of states for band-to-band optical transitions corresponding to optical frequencies in the range from \(\nu\) to \(\nu+\text{d}\nu\) can be evaluated as

\[\tag{13-16}\rho(\nu)\text{d}\nu=\rho_\text{c}(E_2)\text{d}E_2=-\rho_\text{v}(E_1)\text{d}E_1\]

where the minus sign in front of \(\rho_\text{v}(E_1)\) is introduced because the sign of \(\text{d}E_1\) is opposite to that of \(\text{d}\nu\), as can be seen in (13-14).

Using (12-16) [refer to the electron and hole concentrations tutorial] for \(\rho_\text{c}(E)\text{d}E\) and (13-13) for \(E_2\), or (12-16) for \(\rho_\text{v}(E)\text{d}E\) and (13-14) for \(E_1\), we find that

\[\tag{13-17}\rho(\nu)\text{d}\nu=\frac{4\pi(2m_\text{r}^*)^{3/2}}{h^2}(h\nu-E_\text{g})^{1/2}\text{d}\nu\qquad(\text{m}^{-3})\]

for direct band-to-band optical transitions associated with absorption or emission of photons in the frequency range between \(\nu\) and \(\nu+\text{d}\nu\).

Finally, the probabilities of occupancy for the states that are involved in an optical transition have to be considered.

For an optical transition from a valence-band state \(|1\rangle\) of energy \(E_1\) to a conduction-band state \(|2\rangle\) of energy \(E_2\), state \(|1\rangle\) has to be occupied and state \(|2\rangle\) has to be empty before the transition takes place. Therefore, the probability of the transition associated with optical absorption is \(f_\text{v}(E_1)(1-f_\text{c}(E_2))\).

For optical emission, the probability is \(f_\text{c}(E_2)(1-f_\text{v}(E_1))\) because it involves the transition from an occupied conduction-band state \(|2\rangle\) to an empty valence-band state \(|1\rangle\).

Based on the above discussions, we can easily write down the transition rates for direct band-to-band optical transitions in a semiconductor by following the line of reasoning employed to derive the atomic transition rates in the optical transitions for laser amplifiers tutorial.

In the presence of an optical radiation field that has a spectral energy density of \(u(\nu)\), the induced transition rates per unit volume of the semiconductor in the spectral range between \(\nu\) and \(\nu+\text{d}\nu\) are

\[\tag{13-18}R_\text{a}(\nu)\text{d}\nu=B_{12}u(\nu)f_\text{v}(E_1)[1-f_\text{c}(E_2)]\rho(\nu)\text{d}\nu\qquad(\text{ m}^{-3}\text{ s}^{-1})\]

for optical absorption associated with upward transitions of electrons from the valence band to the conduction band and

\[\tag{13-19}R_\text{e}(\nu)\text{d}\nu=B_{21}u(\nu)f_\text{c}(E_2)[1-f_\text{v}(E_1)]\rho(\nu)\text{d}\nu\qquad(\text{ m}^{-3}\text{ s}^{-1})\]

for stimulated emission resulting from downward transitions of electrons from the conduction band to the valence band.

The spontaneous emission rate is independent of \(u(\nu)\) and can be expressed as

\[\tag{13-20}R_\text{sp}(\nu)\text{d}\nu=A_{21}f_\text{c}(E_2)[1-f_\text{v}(E_1)]\rho(\nu)\text{d}\nu\qquad(\text{ m}^{-3}\text{ s}^{-1})\]

The \(A\) and \(B\) coefficients in (13-18) - (13-20) are the Einstein \(A\) and \(B\) coefficients, which are evaluated in the following through a procedure similar to that used in the optical transitions for laser amplifiers tutorial. 

We consider a semiconductor in thermal equilibrium at a temperature \(T\) with blackbody radiation, which has a spectral energy density of \(u(\nu)\) given by (10-20) [refer to the optical transitions for laser amplifiers tutorial].

In thermal equilibrium, the electrons in both conduction and valence bands follow the same distribution function \(f(E)\) given by (12-1) [refer to the introduction to semiconductors tutorial] that is characterized by a single Fermi level, \(E_\text{F}\). Therefore, \(f_\text{c}(E_2)=f(E_2)\) and \(f_\text{v}(E_1)=f(E_1)\).

For the semiconductor to maintain thermal equilibrium with blackbody radiation, the total absorption rate in any given frequency range has to be equal to the total emission rate in the same frequency range:

\[\tag{13-21}R_\text{a}(\nu)\text{d}\nu=R_\text{e}(\nu)\text{d}\nu+R_\text{sp}(\nu)\text{d}\nu\]

By substituting (13-18), (13-19), and (13-20) in (13-21) and using the fact that \(f_\text{c}(E_2)=f(E_2)\) and \(f_\text{v}(E_1)=f(E_1)\) in this situation, we have

\[\tag{13-22}\frac{B_{12}u(\nu)}{B_{21}u(\nu)+A_{21}}=\frac{f(E_2)[1-f(E_1)]}{f(E_1)[1-f(E_2)]}=\text{e}^{-(E_2-E_1)/k_\text{B}T}=\text{e}^{-h\nu/k_\text{B}T}\]

This result can be rearranged to yield the following relation:

\[\tag{13-23}u(\nu)=\frac{A_{21}/B_{21}}{(B_{12}/B_{21})\text{e}^{h\nu/k_\text{B}T}-1}\]

Similar to what is done in (10-30) [refer to the optical transitions for laser amplifiers tutorial], the coefficient \(A_{21}\) can be expressed in terms of a spontaneous time constant as

\[\tag{13-24}A_{21}=\frac{1}{\tau_\text{sp}}\]

Note, however, that \(\tau_\text{sp}\) is not the same as the radiative carrier lifetime \(\tau_\text{rad}\) or the total spontaneous carrier recombination lifetime \(\tau_\text{s}\) defined in the radiative recombination tutorial.

The physical meaning and the characteristics of \(\tau_\text{sp}\) are further discussed in the next tutorial.

By identifying \(u(\nu)\) in (13-23) with the spectral energy density given in (10-20) [refer to the optical transitions for laser amplifiers tutorial] for blackbody radiation, we find that

\[\tag{13-25}B_{12}=B_{21}=\frac{c^3}{8\pi{n}^3h\nu^3\tau_\text{sp}}\]

where \(n\) is the refractive index of the semiconductor.

Though the relation in (13-25) for the coefficients \(B_{12}\) and \(B_{21}\) was obtained by considering the interaction of a semiconductor with blackbody radiation in thermal equilibrium, it is an intrinsic property of the semiconductor material that is independent of the source and characteristics of the optical radiation.

Using the results obtained above and the relation given by (10-15) [refer to the optical transitions for laser amplifiers tutorial] between the spectral intensity \(I(\nu)\) and the spectral energy density \(u(\nu)\) of an optical field at a frequency \(\nu\), we obtain the following relations for direct band-to-band optical transitions:

\[\tag{13-26}\begin{align}R_\text{a}(\nu)&=\frac{c^3}{8\pi{n}^3h\nu^3\tau_\text{sp}}u(\nu)f_\text{v}(E_1)[1-f_\text{c}(E_2)]\rho(\nu)\\&=\frac{c^2}{8\pi{n}^2h\nu^3\tau_\text{sp}}I(\nu)f_\text{v}(E_1)[1-f_\text{c}(E_2)]\rho(\nu)\qquad(\text{ m}^{-3})\end{align}\]

for optical absorption,

\[\tag{13-27}\begin{align}R_\text{e}(\nu)&=\frac{c^3}{8\pi{n}^3h\nu^3\tau_\text{sp}}u(\nu)f_\text{c}(E_2)[1-f_\text{v}(E_1)]\rho(\nu)\\&=\frac{c^2}{8\pi{n}^2h\nu^3\tau_\text{sp}}I(\nu)f_\text{c}(E_2)[1-f_\text{v}(E_1)]\rho(\nu)\qquad(\text{ m}^{-3})\end{align}\]

for stimulated emission, and

\[\tag{13-28}R_\text{sp}(\nu)=\frac{1}{\tau_\text{sp}}f_\text{c}(E_2)[1-f_\text{v}(E_1)]\rho(\nu)\qquad(\text{m}^{-3})\]

for spontaneous emission.

The validity of these relations is quite general.

When the carriers in the conduction and valence bands of a semiconductor are in thermal equilibrium, they are governed by the same Fermi-Dirac distribution with \(f_\text{c}\) and \(f_\text{v}\) characterized by the same Fermi level \(E_\text{F}\).

When the carriers are in quasi-equilibrium, \(f_\text{c}\) and \(f_\text{v}\) are characterized by different quasi-Fermi levels, \(E_\text{Fc}\) and \(E_\text{Fv}\), respectively.

In either situation, the relations in (13-26) - (13-28) are valid.

The relations in (13-26) and (13-27) for the transitions induced by a radiation field are also valid regardless of whether the interaction optical field is a coherent field like a laser field or an incoherent field like blackbody radiation.

Note that \(R_\text{a}(\nu)\text{d}\nu\), \(R_\text{e}(\nu)\text{d}\nu\), and \(R_\text{sp}(\nu)\text{d}\nu\) given in (13-18), (13-19), and (13-20), respectively, represent the transition rates per unit volume of a semiconductor and thus have units of cubic meters per second.

In contrast, \(W_{12}(\nu)\text{d}\nu\), \(W_{21}(\nu)\text{d}\nu\), and \(W_\text{sp}(\nu)\text{d}\nu\) given in (10-17), (10-18), and (10-19), respectively, [refer to the optical transitions for laser amplifiers tutorial], represent the transition rates of a single atom or molecule and are measured in units per second.

As mentioned in the beginning of this tutorial, it is not possible to consider the transition rates of each individual electron separately from other electrons in the band structure of a semiconductor.

Consequently, the transition rates obtained in this section for the band-to-band transitions in a semiconductor already account for the distribution and density of the carriers in the energy bands of a semiconductor.

The concept of transition cross section defined in the optical transitions for laser amplifiers tutorial is not directly applicable to the band-to-band transitions in a semiconductor though an equivalent gain cross section can be obtained, as defined in a later tutorial.

Instead, \(R_\text{a}(\nu)\), \(R_\text{e}(\nu)\), and \(R_\text{sp}(\nu)\) for the band-to-band transitions in a semiconductor are respectively equivalent to \(N_1W_{12}(\nu)\), \(N_2W_{21}(\nu)\), and \(N_2W_\text{sp}(\nu)\) for the transitions between the energy levels of active atoms or molecules in a material.

Example 13-2

In this example, we consider direct band-to-band optical transitions in GaAs at \(\lambda=850\text{ nm}\) wavelength at \(300\text{ K}\).

(a) Find the reduced effective mass \(m_\text{r}^*\) for GaAs.

(b) Find the energy levels, \(E_2\) and \(E_1\), for the optical transitions at this wavelength.

(c) Calculate the value of the density of states \(\rho(\nu)\) for these transitions.

(d) By taking \(\tau_\text{sp}=500\text{ ps}\), find the spontaneous emission rate \(R_\text{sp}(\nu)\) for intrinsic GaAs at this optical wavelength.

(a)

From Table 12-2 [refer to the current density in semiconductors tutorial], we have \(m_\text{e}^*=0.067m_0\) and \(m_\text{h}^*=0.52m_0\) for GaAs. We then find from (13-15) that

\[m_\text{r}^*=\frac{0.067\times0.52}{0.067+0.52}m_0=0.0594m_0\]

(b)

The photon energy for \(\lambda=850\text{ nm}=0.85\text{ μm}\) is

\[h\nu=\frac{1.2398}{0.85}\text{ eV}=1.459\text{ eV}\]

At \(300\text{ K}\), the bandgap of GaAs is \(E_\text{g}=1.424\text{ eV}\). We find by using (13-13) and (13-14) that

\[E_2=E_\text{c}+\frac{0.0594}{0.067}\times(1.459-1.424)\text{ eV}=E_\text{c}+31\text{ meV}\]

\[E_1=E_\text{v}-\frac{0.0594}{0.52}\times(1.459-1.424)\text{ eV}=E_\text{v}-4\text{ meV}\]

Therefore, the direct band-to-band absorption or emission of a photon at 850 nm wavelength in GaAs at \(300\text{ K}\) takes place between a conduction-band state that is located at 31 meV above the conduction-band edge and a valence-band state that is located at 4 meV below the valence-band edge, as shown in Figure 13-5.

(c)

The density of states can be calculated directly using (13-17):

\[\begin{align}\rho(\nu)&=\frac{4\pi\times(2\times0.0594\times9.11\times10^{-31})^{3/2}}{(6.626\times10^{-34})^2}\times[(1.459-1.424)\times1.6\times10^{-19}]^{1/2}\text{ m}^{-3}\text{ Hz}^{-1}\\&=7.63\times10^{10}\text{ m}^{-3}\text{ Hz}^{-1}\end{align}\]

(d)

To find \(R_\text{sp}(\nu)\) from (13-28), we have to calculate \(f_\text{c}(E_2)(1-f_\text{v}(E_1))\).

It can be calculated by plugging the values of \(E_2\) and \(E_1\) found above into the Fermi-Dirac distribution function given in (12-1) [refer to the introduction to semiconductors tutorial] with \(E_\text{F}=E_\text{Fi}\) for the intrinsic GaAs found in Example 12-2 [refer to the electron and hole concentrations tutorial].

Alternatively, we know from Example 12-2 [refer to the electron and hole concentrations tutorial] that \((E_\text{c}-E_\text{Fi})/k_\text{B}T\approx(E_\text{Fi}-E_\text{v})/k_\text{B}T\approx{E}_\text{g}/(2k_\text{B}T)=27.49\) for GaAs at \(T=300\text{ K}\).

Because \(E_2\gt{E}_\text{c}\) and \(E_\text{v}\gt{E}_1\), we have

\[f_\text{c}(E_2)\approx\text{e}^{(E_\text{Fi}-E_2)/k_\text{B}T}\qquad\text{and}\qquad1-f_\text{v}(E_1)\approx\text{e}^{(E_1-E_\text{Fi})/k_\text{B}T}\]

Therefore, for intrinsic GaAs at \(300\text{ K}\), we have

\[\tag{13-29}f_\text{c}(E_2)[1-f_\text{v}(E_1)]\approx\text{e}^{(E_1-E_2)/k_\text{B}T}=\text{e}^{-h\nu/k_\text{B}T}\]

Note that this relation is valid only for an unexcited intrinsic semiconductor with \(E_\text{g}\gg{k}_\text{B}T\) and with \(E_\text{Fi}\) located near the center of tis bandgap.

It is not valid when the bandgap is small or when the Fermi level lies close to one of the band edges.

With this relation and with \(\tau_\text{sp}=500\text{ ps}\), we can calculate \(R_\text{sp}(\nu)\), using (13-28), as

\[R_\text{sp}(\nu)=\frac{1}{500\times10^{-12}}\times\text{e}^{-1.459/0.0259}\times7.63\times10^{10}\text{ m}^{-3}=5.23\times10^{-5}\text{ m}^{-3}\]

This is the spontaneous emission rate per unit spectral bandwidth of unexcited intrinsic GaAs in the thermal equilibrium state at \(300\text{ K}\) at \(h\nu=1.459\text{ eV}\) for \(\lambda=850\text{ nm}\).

The next tutorial covers the topic of optical gain in semiconductors.