The Band Theory of Solids

There are several ways of explaining the existence of energy bands in crystalline solids. The simplest picture is to consider a single atom with its set of discrete energy levels for its electrons. The electrons occupy quantum states with quantum numbers n,l, m and s denoting the energy level, orbital and spin state of the electrons. Now if a number N of identical atoms are brought together in very close proximity as in a crystal, there is some degree of spatial overlap of the outer electron orbitals. This means that there is a chance that Semiconductor Physics 3 any pair of these outer electrons from adjacent atoms could trade places. The Pauli exclusion principle, however, requires that each electron occupy a unique energy state. Satisfying the Pauli exclusion principle becomes an issue because electrons that trade places effectively occupy new, spatially extended energy states. The two electrons apparently occupy the same spatially extended energy state.

In fact, since outer electrons from all adjacent atoms may trade places, outer electrons from all the atoms may effectively trade places with each other and therefore a set of outermost electrons from the N atoms all appear to share a spatially extended energy state that extends through the entire crystal. The Pauli exclusion principle can only be satisfied if these electrons occupy a set of distinct, spatially extended energy states. This leads to a set of slightly different energy levels for the electrons that all originated from the same atomic orbital. We say that the atomic orbital splits into an energy band containing a set of electron states having a set of closely spaced energy levels. Additional energy bands will exist if there is some degree of spatial overlap of the atomic electrons in lower-lying atomic orbitals. This results in a set of energy bands in the crystal. Electrons in the lowest-lying atomic orbitals will remain virtually unaltered since there is virtually no spatial overlap of these electrons in the crystal.

The picture we have presented is conceptually a very useful one and it suggests that electrical conductivity may arise in a crystal due to the formation of spatially extended electron states. It does not directly allow us to quantify and understand important details of the behaviour of these electrons, however.

Fermi Energy and Holes

Of particular interest is the existence in semiconductors, at moderate temperatures such as room temperature, of the two energy bands that are partly filled. The higher of these two bands is mostly empty but a number of electrons exist near the bottom of the band, and the band is named the conduction band because a net electron flux or flow may be obtained in this band. The lower band is almost full; however, because there are empty states near the top of this band, it also exhibits conduction and is named the valence band. The electrons that occupy it are valence electrons, which form covalent bonds in a semiconductor such as silicon. Figure 1.7 shows the room temperature picture of a semiconductor in thermal equilibrium. An imaginary horizontal line at energy Ef, called the Fermi energy, represents an energy above which the probability of electron states being filled is under 50%, and below which the probability of electron states being filled is over 50%. We call the empty states in the valence band holes. Both valence band holes and conduction band electrons contribute to conductivity.

In a semiconductor we can illustrate the valence band using Figure 1.8, which shows a simplified two-dimensional view of silicon atoms bonded covalently. Each covalent bond requires two electrons. The electrons in each bond are not unique to a given bond, and are shared between all the covalent bonds in the crystal, which means that the electron wavefunctions extend spatially throughout the crystal as described in the Kronig–Penney model. A valence electron can be thermally or optically excited and may leave a bond to form an electron-hole pair (EHP). The energy required for this is the bandgap energy of the

Molecular Excitons

In inorganic semiconductors electrons and holes exist as distributed wavefunctions, which prevents the formation of stable excitons at room temperature. In contrast to this, holes and electrons are localized within a given molecule in organic semiconductors, and the molecular exciton is thereby both stabilized and bound within a molecule of the organic semiconductor. In organic semiconductors, which are composed of molecules, excitons are clearly evident at room temperature and also at higher operational device temperatures. An exciton in an organic semiconductor is an excited state of the molecule. A molecule contains a series of electron energy levels associated with a series of molecular orbitals that are complicated to calculate directly from Schrodinger’s equation due to the complex ̈ shapes of molecules. These molecular orbitals may be occupied or unoccupied. When a molecule absorbs a quantum of energy that corresponds to a transition from one molecular orbital to another higher energy molecular orbital, the resulting electronic excited state of the molecule is a molecular exciton comprising an electron and a hole within the molecule. An electron is said to be found in the lowest unoccupied molecular orbital and a hole in the highest occupied molecular orbital, and since they are both contained within the

same molecule the electron-hole state is said to be bound. A bound exciton results, which is spatially localized to a given molecule in an organic semiconductor. Organic molecule energy levels are discussed in more detail in Chapter 6. These molecular excitons can be classified as in the case of excited states of the helium atom, and either singlet or triplet excited states in molecules are possible. The results from Section 3.6 are relevant to these molecular excitons and the same concepts involving electron spin, the Pauli exclusion principle and indistinguishability are relevant because the molecule contains two or more electrons. If a molecule in its unexcited state absorbs a photon of light it may be excited forming an exciton in a singlet state with spin s = 0. These excited molecules typically have characteristic lifetimes on the order of nanoseconds, after which the excitation energy may be released in the form of a photon and the molecule undergoes fluorescence by a dipole-allowed process returning to its ground state. It is also possible for the molecule to be excited to form an exciton by electrical means rather than by the absorption of a photon. This will be described in detail in Chapter 6. Under electrical excitation the exciton may be in a singlet or a triplet state since electrical excitation, unlike photon absorption, does not require the total spin change to be zero. There is a 75% probability of a triplet exciton and 25% probability of a singlet exciton, as described in Table 3.2. The probability of fluorescence is therefore reduced under electrical excitation to 25% because the decay of triplet excitons is not dipole-allowed. Another process may take place, however. Triplet excitons have a spin state with s = 1 and these spin states can frequently be coupled with the orbital angular momentum of molecular electrons, which influences the effective magnetic moment of a molecular exciton. The restriction on dipole radiation can be partly removed by this coupling, and light emission over relatively long characteristic radiation lifetimes is observed in specific molecules. These longer lifetimes from triplet states are generally on the order of milliseconds and the process is called phosphorescence, in contrast with the shorter lifetime fluorescence from singlet states. Since excited triplet states have slightly lower energy levels than excited singlet states, triplet phosphorescence has a longer wavelength than singlet fluorescence in a given molecule. In addition there are other ways that a molecular exciton can lose energy. There are three possible energy loss processes that involve energy transfer from one molecule to another molecule. One important process is known as Forster resonance energy transfer ̈ . Here a molecular exciton in one molecule is established but a neighbouring molecule is not initially excited. The excited molecule will establish an oscillating dipole moment as its exciton starts to decay in energy as a superposition state. The radiation field from this dipole is experienced by the neighbouring molecule as an oscillating field and a superposition state in the neighbouring molecule is also established. The originally excited molecule loses energy through this resonance energy transfer process to the neighbouring molecule and finally energy is conserved since the initial excitation energy is transferred to the neighbouring molecule without the formation of a photon. This is not the same process as photon generation and absorption since a complete photon is never created; however, only dipole-allowed transitions from excited singlet states can participate in Forster resonance ̈ energy transfer. Forster energy transfer depends strongly on the intermolecular spacing, and the rate ̈ of energy transfer falls off as 1 R6 where R is the distance between the two molecules. A

Band-to-Band Transitions

In inorganic semiconductors the recombination between an electron and a hole occurs to yield a photon, or conversely the absorption of a photon yields a hole-electron pair. The electron is in the conduction band and the hole is in the valence band. It is very useful to analyse these processes in the context of band theory from Chapter 1. Consider the direct-gap semiconductor having approximately parabolic conduction and valence bands near the bottom and top of these bands respectively, as in Figure 3.9. Parabolic bands were introduced in Section 1.5. Two possible transition energies, E1 and E2, are shown, which produce two photons having two different wavelengths. Due to the very small momentum of a photon, the recombination of an electron and a hole occurs almost vertically in this diagram to satisfy conservation of momentum. The x-axis represents the wave-number k, which is proportional to momentum. See Section 1.12. Conduction band electrons have energy Ee = Ec + 2k2 2m∗ e and for holes we have Eh = Ev − 2k2 2m∗ h In order to determine the emission/absorption spectrum of a direct-gap semiconductor we need to find the probability of a recombination taking place as a function of energy E. This

transition probability depends on an appropriate density of states function multiplied by probability functions that describe whether or not the states are occupied. We will first determine the appropriate density of states function. Any transition in Figure 3.9 takes place at a fixed value of reciprocal space where k is constant. The same set of points located in reciprocal space or k-space gives rise to states both in the valence band and in the conduction band. In the Kronig–Penny model presented in Chapter 1, a given position on the k-axis intersects all the energy bands including the valence and conduction bands. There is therefore a state in the conduction band corresponding to a state in the valence band at a specific value of k. Reciprocal space and k-space are equivalent except for a constant factor of π, as explained in Section 1.9. See Figures 1.11 and 1.12, which depict reciprocal space. Therefore in order to determine the photon emission rate over a specific range of photon energies we need to find the appropriate density of states function for a transition between a group of states in the conduction band and the corresponding group of states in the valence band. This means we need to determine the number of states in reciprocal space or k-space that give rise to the corresponding set of transition energies that can occur over a small radiation energy range E centred at some transition energy in Figure 3.9. For example, the appropriate number of states can be found at E2 in Figure 3.9b by considering a small range of k-states k that correspond to small differential energy ranges Ec and Ev and then finding the total number of band states that fall within the range E. The emission energy from these states will be centred at E2 and will have an emission energy range E = Ec + Ev producing a portion of the observed emission spectrum. The density of transitions is determined by the density of states in the joint dispersion relation, which will now be introduced: The available energy for any transition is given by E(k) = hv = Ee(k) − Eh(k) and upon substitution we can obtain the joint dispersion relation, which adds the dispersion relations from both the valence and conduction bands. We can express this transition energy E and determine the joint dispersion relation from Figure 3.9a as E(k) = hv = Ec − Ev + 2k2 2m∗ e + 2k2 2m∗ h = Eg + 2k2 2μ (3.17) where 1 μ = 1 m∗ e + 1 m∗ h Note that a range of k-states k will result in an energy range E = Ec + Ev in the joint dispersion relation because the joint dispersion relation provides the sum of the relevant ranges of energy in the two bands as required. The smallest possible value of transition energy E in the joint dispersion relation occurs at k = 0 where E = Eg from Equation 3.17, which is consistent with Figure 3.9. If we can determine the density of states in the joint dispersion relation we will therefore have the density of possible photon emission transitions available in a certain range of energies. The density of states function for an energy band was derived in Section 1.9. As originally derived, the form of the density of states function was valid for a box having V = 0 inside the box. In an energy band, however, the density of states function was modified. We replaced the free electron mass with the reduced mass relevant to the specific energy band and we replaced m in Equation 1.23 by m* to obtain Equation 1.31a. This is valid because rather than the parabolic E versus k dispersion relation for free electrons in which the electron mass is m, we used the parabolic E versus k dispersion relation for an electron in an energy band as illustrated in Figure 1.7, which may be approximated as parabolic for small values of k with the appropriate effective mass. The slope of the E versus k dispersion relation is controlled by the effective mass, and this slope determines the density of states

Aurther

ADRIAN KITAI

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