## Degenerate Semiconductors

In Section 3.4 we gave the name non-degenerate semiconductors to those for which the Fermi level EF is located in the gap at an energy of about 3kT or more away from the band edges. Since for these semiconductors, classical statistics could be applied, we derived simple expressions, Eqs (3.12) and (3.15), for the concentration of electrons and holes, respectively. Under these premises we also derived Eqs (3.24), (3.25), and (3.26) which give the location of the Fermi level for n-type, p-type, and intrinsic semiconductors, respectively.

As the dopant concentration is increased, the Fermi level approaches the band edges and when n or p exceeds Nc or Nv, given by Eqs (3.8) and (3.16), respectively, the Fermi level enters the conduction band in the case of n-type semiconductors or the valence band if the semiconductor is p-type. These heavily doped semiconductors are called degenerate semiconductors and the dopant concentration is usually in the range of 1019-1020 cm-3.

In the case of degenerate semiconductors, the wave functions of electrons in the neighbourhood of impurity atoms overlap and, as it happens in the case of electrons in crystals, the discrete impurity levels form narrow impurity bands as shown in Figure 3.9(a). The impurity bands corresponding to the original donor and acceptor levels overlap with the conduction and valence bands, respectively, becoming part of them. These states which are added to the conduction or valence bands are called bandtail states. Evidently, as a consequence of bandtailing, the phenomenon of bandgap narrowing is produced. Bandgap narrowing has important consequences in the operation of laser diodes (Section 10.3) and in the absorption spectrum of heavily doped semiconductors.

Figure 3.9(b) shows the energy diagram of a degenerate n-type semiconductor. As we know, the energy states below EFn are mostly filled. Therefore most of the electrons have energies in the narrow range between Ec and EFn. The band diagram is somewhat similar to metals and the Fermi level coincides with the highest energy of the electrons in the band. However, if one dopes very heavily, to about 1020 cm-3, a carrier saturation effect appears as a consequence of interaction between dopants. For this very high concentration regime, Eqs (3.22), (3.23), and the law of mass action expressed in Eq. (3.17) do not apply.

oooooooooooooooooooo va

Figure 3.9. (a) Impurity bands and bandtail states in degenerate semiconductors; (b) energy diagram of an n-type degenerate semiconductor.

3.7. OPTICAL PROPERTIES OF SEMICONDUCTORS 3.7.1. Optical processes in semiconductors

When light incides on a semiconductor, we can observe a series of optical phenomena like absorption, transmission, and reflection. All these phenomena induce a series of electronic processes in the semiconductor, which can be studied by recording their respective optical spectra. The absorption spectrum of a typical semiconductor shows several significant features. The dominant absorption process occurs when the energy of the incident photons is equal or larger than the semiconductor gap and therefore electronic transitions from occupied valence band states to empty conduction band states become dominant. These transitions can be either direct or indirect and the absorption coefficient is calculated by means of time-dependent perturbation theory.

At the low energy side of the fundamental edge, exciton absorption can be observed as a series of sharp peaks. An exciton consists of a bound electron-hole pair in which the electron and hole are attracted by the Coulomb interaction and their absorption spectrum is studied in Section 3.7.3. Other absorption processes in semiconductors correspond to electronic transitions between donor levels and the conduction band states, and from the valence band to acceptor levels. The corresponding absorption peaks are located in the infrared ranges, as a consequence of the low values of the donor and acceptor ionization energies (Section 3.3). In heavily doped semiconductors, optical absorption by free carriers can also become significant, since the absorption coefficient is proportional to the carrier concentration. Finally, in ionic crystals, optical phonons can be directly excited by electromagnetic waves, due to the strong electric dipole coupling between photons and transverse optical phonons. These absorption peaks, arising from the lattice vibrations, appear in the infrared energy range.

The absorption of light by a semiconductor can be described macroscopically in terms of the absorption coefficient. If light of intensity 10 penetrates the surface of a solid, then the intensity 1(z) at a distance z from the surface varies as

where a is a material property called the absorption coefficient, which depends on the light wavelength and is given in units of cm-1. The parameter 1/a is called the penetration depth. Evidently, the higher the absorption coefficient, the smaller the depth at which light can penetrate inside the solid. For semiconductors like GaAs, a increases very sharply when the photon energy surpasses Eg, since the optical transitions are direct (Section 3.7.2). On the contrary, for indirect semiconductors like Si or Ge, the increase of a is slower since the optical transitions require the participation of phonons. Therefore, the increase of a is not as sharp as in the case of direct semiconductors. In addition, the onset of absorption does not occur exactly when hv = Eg as for direct transitions, but in an interval of the order of the energy of the phonons around Eg.

### 3.7.2. Interband absorption

Interband absorption across the semiconductor gap is strongly dependent on the band structure of the solid, especially on the direct or indirect character of the gap. In this section we are mainly going to review the case of interband optical transitions in direct gap semiconductors, since most of the optoelectronic devices of interest in light emission (Chapter 10) are based on this type of materials.

For the calculation of the optical absorption coefficient we have to make use of the quantum mechanical transition rate, Wif, between electrons in an initial state f i which are excited to a final state f This rate is given by the Fermi Golden rule of Eq. (2.26) of Chapter 2:

In this expression, the matrix element Hf corresponds to the optical external perturbation on the electrons and p(E) is the density of states function for differences in energy E between final and initial states equal to the excitation photon energy

The perturbation Hamiltonian H' (Section 2.2.4) associated to electromagnetic waves acting on electrons of position vector r is given by

and corresponds to the energy of the electric dipole of the electron -er under the action of the wave electric field F. Since the electronic states are described by Bloch functions (Section 2.4), we have to calculate matrix elements of the form

The integral in Eq. (3.55) is extended to the volume of the unit cell, since the integral over the whole crystal can be decomposed as a sum over the unit cells. The functions ui(r) and uf (r) in Eq. (3.55) have the periodicity of the lattice, according to the Bloch theorem.

The density of states function p(E) that appears in Eq. (3.53) has to be calculated at the energy hv of the incident photons, since the final and initial states, which lie in different energy bands, should be separated in energy by hv. For this reason the function p(E) is usually called the optical joint density of states function, and for its calculation one has to know the structure of the bands. Another condition which has to be fulfilled is that, for direct gap semiconductors, the electron wave vector of the final state should be the same as the one of the initial state, i.e.

since the momentum associated to the photon can be considered negligible.

Optical transitions around k = 0 in direct III-V semiconductors like GaAs involve the valence band of p-like atomic orbitals and the conduction band originated from s-like orbitals. It is also known from the electric dipole selection rules that transitions from p-states to s-states are allowed, and therefore a strong optical absorption should be expected.

Let us now calculate the optical DOS. We can observe that the conduction band as well as the three valence bands (heavy hole, light hole, and split-off band) are all parabolic close to the r point (Figure 3.2(a)). The direct gap Eg equals the energy difference between the minimum of the conduction band and the maxima of the heavy and light hole bands, which are degenerated at r. For these transitions, conservation of energy requires h2k2 h2k2 h2k2

where ^ is the reduced mass of the electron-hole system, and we have taken into account only one of the two degenerated hole bands for simplicity; the split-off hole band would show the absorption transition at higher photon energies. Considering the expression found in Section 2.3, Eq. (2.36), for the DOS function, we should have the absorption coefficient a for hv > Eg:

Taking into account the expression for the rate of optical transitions Wif given by Eq. (3.53) and that the optical absorption coefficient a is proportional to Wif, then, for hv > Eg, we find that a should be also proportional to the square root of the photon energy minus the bandgap. Therefore, a plot of a2 as a function of hv should yield a straight line which intercepts the horizontal axis (a = 0) at a value of the energy equal to the semiconductor gap.

As an example, we show in Figure 3.10, a2 as a function of the photon energy for PbS which shows the linear behaviour just discussed [2]. Some other direct gap semiconductors do not verify this relation so exactly, since often not all the assumptions that we have considered in the derivation of Eq. (3.58) are fulfilled. For instance, in the case of GaAs at low temperatures, the region around the onset of this absorption can be overshadowed by the exciton absorption that will be studied in the next section. Also, Eq. (3.58) was only strictly valid for values of k close to k = 0, but, as the photon energy increases, this condition does not hold anymore.

3.7.3. Excitonic effects

As we have seen, photons of energy larger than the semiconductor gap can create electron-hole pairs. Usually the created electron and hole move independently of each other, but in some cases, due to the Coulomb interaction between them, the electron and hole can remain together forming a new neutral particle called exciton. Since excitons have no charge, they cannot contribute to electrical conduction. Exciton formation is very much facilitated in quantum well structures (Sections 1.5 and 4.10), because of the confinement effects which enlarge the overlapping of the electron and hole wave functions.

The simplest picture of an exciton consists of an electron and a hole orbiting inside the lattice around their centre of mass, as a consequence of the Coulombic attraction between them (Figure 3.11). There are two basic types of excitons: (a) Excitons for which the wave function of the electron and hole have only a slight overlap, i.e. the exciton radius encompasses many crystal atoms. These excitons are called Wannier-Mott excitons and are usually detected in semiconductors. (b) Other excitons, mainly observed in insulators, have a small radius of the order of the lattice constant and are called Frenkel excitons.

Wannier-Mott excitons can be described according to a model similar to the hydrogen atom. Considering the exciton as a hydrogenic system, the energies of the bound states should be given by an expression similar to Eq. (2.15) of Chapter 2, with the proper corrections, in a similar manner as we did with ionization donor and acceptor impurity levels (Section 3.3) in extrinsic semiconductors. Evidently, the mass in this expression should be now the reduced mass ^ of the system formed by the electron and hole effective

In addition, we have to consider that the electron and the hole are immersed within a medium of dielectric constant ere0, where er is the high frequency relative dielectric constant of the medium. The bound states of the excitons are, therefore, given by:

where RH is the Rydberg constant for the hydrogen atom and Rex is called the exciton Rydberg constant.

Figure 3.12 shows the excitonic bound states given by Eq. (3.60) and the exciton ionization energy EI. The energy needed for a photon to create an exciton is smaller than the one needed to create an independent electron-hole pair, since we can think of this process as creating first the exciton and, later, separating the electron from the hole by providing an amount of energy equal to the exciton binding energy. Therefore, as shown in Figure 3.12, the exciton bound states, given by Eq. (3.60), should be located within the gap just below the edge of the conduction band.

Figure 3.13 shows the absorption spectrum of GaAs for photon energies close to the gap [3]. It is seen that the first three peaks predicted by Eq. (3.60), with Rex = 4.2 meV, are well resolved. This is because the spectrum was taken at very low temperatures, the spectrometer had a high resolution, and the sample was ultrapure. The advantage of conduction band

valence band

Figure 3.12. Excitonic states located in the gap, close to the conduction band edge.

valence band

Figure 3.12. Excitonic states located in the gap, close to the conduction band edge.

Photon energy (eV)

Figure 3.13. Exciton absorption spectrum of GaAs at 1.2 K. After [3].

Photon energy (eV)

Figure 3.13. Exciton absorption spectrum of GaAs at 1.2 K. After [3].

working at low temperatures is double. On the one hand, excitons are less likely to be destroyed by phonons and, on the other, the thermal broadening of the absorption lines is reduced. Excitons are much better observed in intrinsic semiconductors than in doped ones, where the free charge carriers partially screen the Coulombic interaction between the electron and the hole.

A closer look to the band structure of semiconductors allows predicting which regions in £-space are favourable for the formation of excitons. Since the exciton is composed of a bound electron-hole pair, the velocity vectors of both particles should be the same, and therefore, according to Eq. (2.52) of Chapter 2, their respective conduction and valence bands should be parallel. This is evidently the case in the vicinity of the point k = 0 in GaAs, i.e. around the spectral region corresponding to the direct gap.

If the intensity of the light that creates excitons is high enough, their density increases so much that they start to interact among themselves and with the free carriers. In this high density regime, biexcitons consisting of two excitons can be created. Biexcitons have been detected in bulk semiconductors as well as in quantum wells and dots. Biexcitons consist of two electrons and two holes and, similarly to the way followed to study excitons in terms of hydrogenic atoms, they can be compared to hydrogen molecules. In addition to biexcitons, trions consisting of an exciton plus either a hole or an electron have also been experimentally detected in several nanostructures, among them, III-V quantum wells and superlattices.

### 3.7.4. Emission spectrum

In Section 3.7.2, we have considered transitions of electrons from the valence band to the conduction band in semiconductors caused by photon absorption. In the inverse process, light can be emitted when an excited electron drops to a state in a lower energy band. If in this process light is emitted, we have photoluminescence due to a radiative transition. The emitted photons have in general a different frequency than the previously absorbed, and the emission spectrum is usually much narrower than the absorption spectrum. In effect, suppose, as in Figure 3.14, that a photon of energy hv > Eg is absorbed and, as a consequence, an electron-hole pair is produced. In this process, the electron and/or the hole can get an energy higher than the one corresponding to thermal equilibrium. Subsequently, the electrons (the same applies to holes) lose the extra kinetic energy very rapidly by the emission of phonons (mainly optical phonons) occupying states close to the bottom of the band. The time taken in this step is as short as 10-13s due to the strong electron-phonon coupling. On the other hand, the lifetime of the radiative process by which electrons drop to the valence band, by the emission of photons of hv ~ Eg, is several orders of magnitude longer (of the order of nanoseconds). Therefore, the emission spectrum should range between Eg and Eg plus an energy of the order of kT, since the electrons have enough time to get thermalized at the bottom of the conduction band. At present, these phenomena can be studied very nicely by means of very fast time-resolved photoluminescence spectroscopy, using ultra short laser pulses. Figure 3.15 shows the spectra of bulk GaAs at 77 K, after having been excited by 14 fs laser pulses [4]. The curves are shown for three different carrier concentrations of increasing values from top to bottom. The fourth curve in the figure represents the autocorrelation (AC) of the laser pulse.

light absorption light emission light emission light absorption

Figure 3.14. De-excitation of an electron by first emitting an optical phonon and subsequently a photon with energy approximately equal to the gap energy.

Figure 3.14. De-excitation of an electron by first emitting an optical phonon and subsequently a photon with energy approximately equal to the gap energy.

Time delay (fs)

Figure 3.15. Spectra of GaAs obtained by time-resolved photoluminescence spectroscopy.

Time delay (fs)

Figure 3.15. Spectra of GaAs obtained by time-resolved photoluminescence spectroscopy.

For many semiconductors, recombination of electrons and holes is mainly non-radiative, that is, instead of emitting photons, the recombination process occurs via recombination centres with energy levels located within the gap. In this case, the energy lost by the electrons is transferred as heat to the lattice. We have already seen in Section 3.3 that impurity states in semiconductors are located within the gap, close to the band edges, but other defects, such as vacancies or metal impurities, can have their levels much deeper within the gap. Only for quantum wells of very high quality, the number of emitted photons divided by the number of excited electron-hole pairs reaches values in the range 0.1-1. Even in direct gap bulk semiconductors, the values of the luminescence yield are very low, between 10-3 and 10-1. Impurities and in general any kind of defects can act as recombination centres by first capturing an electron or a hole and subsequently the oppositely charged carrier. The defects which make possible electron-hole non-radiative recombination are colloquially called traps and in them the recombination centre re-emits the first captured carrier before capturing the second carrier. The recombination centres are called either fast or slow depending on the time that the first carrier remains at the centre before the second carrier is captured.

### 3.7.5. Stimulated emission

Suppose a simple electron system (Figure 3.16) of just two energy levels E1 and E2 (E2 > Ei). Electrons in the ground state E1 can jump to the excited state E2 if they absorb photons of energy E2 - E1. On the contrary, photons of energy E2 - E1 are emitted when the electron drops from E2 to E1. In general, the emission of light by a transition from the excited state E2 to the ground state E1, is proportional to the population of

stimulating absorption

Figure 3.16. (a) Absorption; (b) spontaneous emission; (c) stimulated emission processes under steady state conditions. (The absorption should be equal to the sum of spontaneous and stimulated emission processes.)

stimulating absorption

Figure 3.16. (a) Absorption; (b) spontaneous emission; (c) stimulated emission processes under steady state conditions. (The absorption should be equal to the sum of spontaneous and stimulated emission processes.)

electrons n2 at the level E2. This is called spontaneous emission and the proportionality coefficient is called the Einstein A21 coefficient, while the corresponding one for the absorption process is called the Einstein B12 coefficient. As Einstein observed, electrons can also drop from E2 to Ei if they are stimulated by photons of energy hv = E2 - E1. This process is therefore called stimulated emission and is governed by the Einstein B21 coefficient. Since stimulated emission is proportional to the density p(v) of photons, in order to have a high rate of stimulated emission, in comparison to the spontaneous one, the radiation energy density should be very high. Evidently the above three Einstein coefficients are related to each other, since in the steady state the rate of upward and downward transitions shown in Figure 3.16 should be equal.

One very interesting aspect of stimulated emission is that the emitted photons are in phase with the stimulating ones. Precisely, the operation of lasers is based on the process of stimulated emission. Semiconductor lasers, which will be studied in Chapter 10, produce monochromatic and coherent light. The rate of stimulated emission should be proportional to n2p(v) and therefore, in order to dominate over absorption (proportional to n1) we should have n2 > n1. This condition is known as population inversion, since in thermal equilibrium, according to the Boltzmann distribution, we have n1 < n2. Observe also that since stimulated emission is proportional to the radiation energy density, lasers need to make use of resonant cavities in which the photon concentration is largely increased by multiple internal optical reflections.

Population inversion in semiconductor lasers is obtained by the injection of carriers (electrons and holes) across p+-n+ junctions of degenerate direct gap semiconductors, operated under forward bias. Figure 3.17(a) shows a non-biased p-n junction and Figure 3.17(b) shows the junction when it is polarized under a forward bias. In this situation, a region around the interface between the p+ and n+ materials, called the active region, is formed, in which the condition of population inversion is accomplished.

From Figure 3.18(a), it can be deduced the range of energies of incoming photons which are able to produce a rate of stimulated emission larger than the absorption rate, therefore resulting in an optical gain. Taking into account the density in energy of electrons through density of states functions for the conduction and valence bands, one can deduce qualitatively the dependence of optical gain as a function of the energy of the incident photons. Evidently, as indicated in Figure 3.18(b), the photons that induce stimulated emission should have energies larger than Eg and lower than EFn - EFp. At higher temperatures, the Fermi-Dirac distribution broadens around the Fermi levels and as a result there is a diminution in optical gain.

cb cb cb cb

Figure 3.17. (a) Energy diagram for a p-n junction made of degenerate semiconductors with no bias; (b) idem, with a forward bias, high enough to produce population inversion in the active region.

Figure 3.17. (a) Energy diagram for a p-n junction made of degenerate semiconductors with no bias; (b) idem, with a forward bias, high enough to produce population inversion in the active region.

Optical gain

Figure 3.18. (a) Density of states of electrons and holes in the conduction and the valence bands, respectively; (b) optical gain as a function of photon energy.

REFERENCES

[1] McKelvey, J.P. (1966) Solid-State and Semiconductor Physics (Harper and Row, New York).

[2] Schoolar, R.B. & Dixon, J.R. (1965) Phys. Rev., 137, A667.

[3] Fehrenbach, G.W., Schäfer, W. & Ulbrich, R.G. (1985) J. Luminescence, 30, 154.

[4] Banyai, L., Tran Thoai, D.B., Reitsamer, E., Haug, H., Steinbach, D., Wehner, M.U., Wegener, M., Marschner, T. & Stolz, W. (1995) Phys. Rev. Lett, 75, 2188.

FURTHER READING

Seeger, K. (1999) Semiconductor Physics (Springer, Berlin).

Singh, J. (2003) Electronic and Optoelectronic Properties of Semiconductor Structures

(Cambridge University Press, Cambridge). Yu, P. & Cardona, M. (1996) Fundamentals of Semiconductors (Springer, Berlin).

### PROBLEMS

1. Electron mean free path. Find the electron mean path in GaAs, at room temperature and T = 77 K knowing that the respective mobilities are approximately 3 x 105 and 104 cm2/Vs, respectively.

2. Semiconductor doping. In order to make a p-n diode, a sample of silicon of type n is doped with 5 x 1015 phosphorous atoms per cm3. Part of the sample is additionally doped, type p, with 1017 boron atoms. (a) Calculate the position of the Fermi levels at T = 300 K, in both sides of the p-n junction. (b) What is the contact potential?

3. Carrier concentrations in germanium. Determine the free electron and hole concentration in a Ge sample at room temperature, given a donor concentration of 2.5 x 1014 cm-3 and acceptor concentration of 3.5 x 1014 cm-3, assuming that all impurities are ionized. Determine its n or p character if the intrinsic carrier concentration of Ge at room temperature is ni = 2.5 x 1013 cm-3.

4. Carrier concentrations in silicon. A semiconducting silicon bar is doped with a concentration of 4 x 1014 cm-3 n-type impurities and 6 x 1014 cm-3 of p-type impurities. Assuming that the density of states is constant with increasing temperature and that electron mobility is twice that of holes, determine the carrier concentration and the conducting type at 300 and 600 K. The bandgap energy of silicon is Eg = 1. 1eV and ni (300K) = 1.5 x 1010 cm-3.

5. Diffusion currents in semiconductors. The electron density in an n-type GaAs crystal varies following the relationship n(x) = Aexp(-x/L), for x > 0, being

A = 8 x 1015cm-3 and L = 900 nm. Calculate the diffusion current density at x = 0 if the electron diffusion coefficient equals 190cm2s-1.

6. Diffusion length. In a p-type GaAs sample electrons are injected from a contact. Considering the mobility of minority carriers to be 3700cm2V-1s-1, calculate the electron diffusion length at room temperature if the recombination time is Tn = 0.6 ns.

7. Carrier dynamics in semiconductors. The electron energy close to the top of the valence band in a semiconductor can be described by the relationship E(k) = —9 x 10—36k2(/), where k is the wave vector. If an electron is removed from the state k = 2 x 109ku, being ku the unity vector in the x-direction, calculate for the resulting hole: (a) its effective mass, (b) its energy, (c) its momentum, and (d) its velocity. Hint: suppose that we are dealing with states close to the maximum of the valence band or to the minimum of the conduction band, thus being the parabolic dispersion relationship a good approximation.

8. Energy bands in semiconductors. The conduction band in a particular semiconductor can be described by the relationship Ecb(k) = E1 — E2 cos(ka) and the valence band by Evb(k) = E3 — E4 sin2(ka/2) where E3 < E1 — E2 and —n/a < k < +7r/a. Determine: (a) the bandgap, (b) the variation between the extrema (Emax — Emin) for the conduction and valence bands, (c) the effective mass for the electrons and holes in the bottom of the conduction band and the top of the valence band, respectively. Sketch the band structure.

9. Excess carriers. An n-type Ge bar is illuminated, causing the hole concentration to be multiplied by five. Determine the time necessary for the hole density to fall to 1011 cm"-3 if Th = 2.5 ms. Assume the intrinsic carrier concentration to be 1013 cm—3 and the donor density 8 x 1015 cm—3.

10. Optical absorption in semiconductors. A 700 nm thick silicon sample is illuminated with a 40 W monochromatic red light (X = 600 nm) source. Determine: (a) power absorbed by the semiconductor, (b) power dissipated as heat, and (c) number of photons emitted per second in recombination processes originated by the light source. The absorption coefficient at X = 600 nm equals a = 7 x 104 cm—1 and the bandgap of Si is 1.12eV.

11. Excitons in GaAs. (For the following problem take ^ = 0.05m0 as the exciton reduced mass and er = 13 as the relative dielectric constant.) (a) Calculate the Rydberg energy RH. What is the largest binding energy? (b) Calculate the exciton radius aex in the ground level, following Bohr's theory. (c) Calculate the number of unit cells of GaAs (a0 = 0.56 nm) inside the exciton volume. (d) Supposing that the exciton is in its ground sate, up to what temperature is the exciton stable?

12. Excitons in a magnetic field. Suppose that a magnetic field B is applied to a gallium arsenide sample. Find the value of B for which the exciton cyclotron energy is equal to the exciton Rydberg energy of Eq. (3.60). Take e = 13 for the value of the dielectric constant.

## Post a comment