The Physics of Energy, page 101
Bound on Collection Efficiency of PV Cells
Photons with energies below the energy of the band gap cannot excite electrons from the valence to the conduction band in a semiconductor. For photons with energies greater than , the excess beyond is lost as electrons thermalize in the conduction band. Together, these give an upper bound on the collection efficiency of simple photovoltaic cells.
For the case of silicon, where eV the upper bound on the collection efficiency for blackbody radiation at 6000 K is approximately 44%. For the actual spectrum of solar radiation on Earth (e.g. AM1.5), the bound is somewhat higher,
25.5Band Structure of Silicon
In this section we describe the band structure of silicon in some detail. In particular, we discuss the indirect nature of the band gap between the valence band and the conduction band in silicon. This indirect band gap has the consequence that the photon absorption length in silicon is longer than in materials with direct band gaps. As a result, crystalline silicon solar cells must be more than an order of magnitude thicker than so-called thin-film photovoltaics based on materials with direct band gaps (§25.9). The increased absorption length in silicon is at least partially compensated by the correlated suppression of the recombination process.
The band structure of a solid depends upon its crystal structure. Crystal structures can generally be described mathematically in terms of three-dimensional lattices. A lattice in N dimensions is a set of points that are generated through linear combinations with integer coefficients of a set of N linearly independent basis vectors. For example, the simple cubic lattice in three dimensions can be described as the set of points , where are the usual Cartesian unit vectors , and . The cubic lattice can be envisioned through the infinite replication of the unit cell shown in Figure 25.13(a). The face-centered cubic (FCC) lattice is the lattice that includes all the points in the simple cubic lattice, but that also includes the center point of each of the (two-dimensional) faces of the unit cube. This lattice is generated by the basis vectors
(25.12)
Note that the basis vectors of the simple cubic lattice can all be described as linear combinations of ; for example, . A unit cell of the FCC lattice is depicted in Figure 25.13(b). The FCC crystal lattice is described by repeating this cell infinitely in each spatial dimension.
Figure 25.13 Unit cells of (a) a simple cubic lattice with the lattice vectors ; (b) a face-centered cubic (FCC) lattice with lattice vectors ; and (c) the diamond crystal structure of the silicon crystal built from two copies of the FCC lattice. Note that the orange FCC lattice is translated by relative to the blue lattice and that each point in one lattice has four nearest neighbors in the other. The points in the orange lattice that are not contained in the blue unit cell have been faded. Compare Figure 25.1.
The diamond crystal structure, which describes crystalline silicon and is shown in Figure 25.13(c), consists of two copies of the FCC lattice. One copy can be taken to be the lattice generated by the vectors in eq. (25.12), while the second copy is offset from the first by the vector . Once the diamond crystal structure has been defined in this way, it is straightforward to verify that each point on either of the two intertwined FCC lattices has four nearest neighbors on the other FCC lattice at a distance . For example, as shown in Figure 25.18(c), the point v is separated by a distance of from the points and c.
Crystalline silicon is built on the diamond crystal structure. As in one dimension, momentum in a 3D periodic potential is only conserved up to equivalences analogous to eq. (25.5). In a 3D periodic potential the equivalences take the form
(25.13)
where G is a momentum vector with the property that , where n is an integer for all vectors under translation by which the 3D crystal structure is invariant.6 Since the periodicity of the diamond crystal structure is the same as the FCC lattice, the vectors can be taken to be the vectors that generate the FCC lattice.
Using the equivalences (25.13), we can define a region of momentum space such that every momentum vector p can be taken to precisely one point in the region by a translation of the form (25.13). This is the generalization of the first Brillouin zone to three dimensions. For the diamond lattice, this region has the shape of a three-dimensional solid bounded by six squares and eight hexagons (a truncated octahedron), which is depicted in Figure 25.14(a). Just as the energy bands of the particle in a 1D periodic potential can be thought of as functions in the first Brillouin zone (as shown for example in Figure 25.7), we can describe the energy bands of silicon as functions over the first Brillouin zone. These are therefore functions of three variables . A projection of the band structure onto one dimension is shown in Figure 25.14(b).
Figure 25.14 (a) The shape of the Brillouin zone for silicon; (b) a much simplified graph showing the relevant valence (red) and conduction (blue) energy bands of crystalline silicon in a one-dimensional projection over the Brillouin zone. A few of the band boundaries that overlap in this projection are also shown.
Precise computation of the band structure of a 3D solid such as silicon is a rather complicated problem in practice, though in principle the problem is essentially that of solving the Schrödinger equation in three dimensions in a specified potential with periodic boundary conditions. The methods needed for such an analysis are beyond the scope of this book. A key feature, however, of the band structure of silicon is clearly evident in Figure 25.14(b): the highest energy state in the valence band (point A) and the lowest energy state in the conduction band (point B) occur at different lattice momenta in the Brillouin zone. This is an example of an indirect band gap – as opposed to a direct band gap – which occurs when the highest state in the valence band has the same lattice momentum as the lowest state in the conduction band. A direct band gap occurs, for example in the semiconductor cadmium telluride (CdTe) (§25.9). Both types of band gaps are illustrated schematically in Figure 25.15.
Figure 25.15 (a) A direct band gap (as in CdTe), versus (b) an indirect band gap (specifically in silicon). An indirect band gap is more difficult to excite, but decays more slowly.
When a photon excites an electron across a direct band gap, the electron’s lattice momentum does not change significantly. The photon does transfer momentum to the electron, but the photon’s momentum, , is orders of magnitude smaller (see Problem 25.20) than the momentum of an electron with the same energy, and can be neglected. For the same reason, a photon by itself is not sufficient to excite an electron across an indirect band gap because the electron lattice momentum must change. It is possible, in a material with an indirect band gap, for an incoming photon to excite a direct transition from the top of the valence band to the lowest accessible state in the conduction band at the same lattice momentum. This generally requires more energy, however. In silicon, the direct excitation of an electron requires about 3.4 eV. Only a small percentage of the photons in visible light have energy this large (see Problem 25.4).
In order for an electron to cross an indirect band gap, it must absorb momentum from some other source. In §25.1 we mentioned that vibrations of the ionic cores of the atoms that form the lattice could be described by waves that propagate through the crystal lattice. The quanta associated with these waves are known as phonons, in analogy to photons which are the quanta of light waves.7 Phonons also carry lattice momentum, which is conserved along with electron momentum up to equivalences of the form of eq. (25.13). Thus, an electron can be excited across an indirect band gap if a low-energy phonon is emitted or absorbed in the process, thereby conserving lattice momentum. Because it involves an additional process, the quantum amplitude for such an event is smaller than for a comparable excitation of an electron across a direct band gap. As a result, the rate at which electrons are excited across the indirect band gap in silicon is relatively slow. On the other hand, the same physical mechanism also decreases the rate at which an excited electron will drop back from the conduction band into the valence band, giving the charge carriers in indirect band gap materials longer typical lifetimes than in direct band gap materials.
Recall from §23 that light entering a material will be absorbed at a rate , where κ is the absorption coefficient of the material. In general, the absorption coefficient is highly dependent upon the wavelength of the incoming light. For a silicon crystal, the dependence of the absorption length on photon energy is shown in Figure 25.16. From the figure, we see that photons at energies above 3.4 eV will generally be absorbed by exciting an electron across the direct gap within 10 nm or so of entering a silicon crystal. On the other hand, to ensure absorption of most photons with energies in the 1.2–1.5 eV range by excitation of electrons across the indirect band gap, the thickness must be on the order of 100 μm ( nm).
Figure 25.16 Absorption length in silicon for photons at different energies (at 300 K) [152]. Light with energies above 3.4 eV excites electrons from the valence band to the conduction band without changing lattice momentum (direct gap). The absorption length increases dramatically for photons with energy below eV. (Credit: Adapted from data from PVeducation.org)
Direct and Indirect Band Gaps
A direct band gap is one in which the lattice momentum of the highest-energy state in the valence band is the same as the lattice momentum of the lowest-energy state in the conducting band. An indirect band gap is one where these lattice momenta are different. It is harder to excite electrons through an indirect band gap but also harder for the electrons to return to the valence band. In practical photovoltaic cells, the lower absorption in indirect band gap materials is thus mitigated somewhat by their longer carrier lifetimes and diffusion lengths.
25.6p-n Junctions
Of the initial list of challenges assembled in §25.4, it remains to explain how to break the symmetry of the silicon crystal so that, when exposed to sunlight, a voltage difference spontaneously appears across the semiconductor and electrons are encouraged to move in one particular direction to create a useful electric current. In this section we describe how this can be done by doping the silicon using impurities to produce a p-n junction.
25.6.1 Doping
Although we have described silicon as if it were a perfect crystal with the diamond crystal structure, as mentioned in §25.3 any real piece of crystalline silicon contains dislocations and small amounts of impurities. These impurities slightly modify the structure of the lattice, and introduce states in the electron spectrum in the gap region between the valence and conducting bands. It is possible to alter the electron spectrum in the gap by intentionally introducing a small proportion of another element into the silicon. This produces impurities in the crystal structure either through substitution of another element for one of the silicon atoms (substitutional impurities), or by squeezing additional atoms into the crystal structure (interstitial impurities). The introduction of impurities in this fashion is known as doping.
Figure 25.17 Doping a silicon crystal. (a) Undoped: each silicon atom shares four electrons with its nearest neighbors. At the valence band is filled, and the conduction band is empty. At K a few electron–hole pairs are thermally excited. (b) n-type doping: donor impurities have five valence electrons, introducing new filled electron states (donor states, denoted by the solid red line) just below the conduction band . At K most of the donor electrons are thermally excited into the conduction band (leaving behind holes in the donor band) and the Fermi level lies close to (c) p-type doping: acceptor impurities arise from atoms with three valence electrons, introducing new, empty electron states (acceptor states, denoted by the solid blue line) just above the top of the valence band, . At K electrons, thermally excited from the valence band, fill almost all of the acceptor states, creating mobile holes in the valence band. The Fermi level lies close to .
The most common types of doping used for crystalline silicon are n-type doping and p-type doping, implemented by substituting an atom from column V or column III of the periodic table (see Figure D.1) for a silicon atom in the crystal structure. Cartoons of the resulting crystal structure in the vicinity of the impurity are depicted at the top of Figure 25.17. In the case of n-type doping, a silicon atom is replaced by an atom from column V, such as phosphorus or arsenic, which has one more electron in its outer orbital than silicon. This extra electron does not participate in any of the bonds to the four neighbors in the crystal, and is only weakly bound to the nucleus of the atom.8 It therefore takes very little energy to boost this extra electron out of its loosely bound state and into the conduction band. In fact, the energy required to lift such an electron to the conduction band is roughly 0.04–0.07 eV for n-type impurities from column V atoms (compared to the band gap of eV). To a good approximation, n-type doping can be modeled by adding a large number of electron energy levels just below the bottom of the conduction band. At all these levels are filled – the extra electrons remain bound to the impurity atoms. At room temperature, however, there is enough thermal energy available to promote most of these electrons into the conduction band, and the Fermi level moves up. The situation is illustrated schematically at the bottom of Figure 25.17(b).
Similarly, substituting into the silicon lattice an atom from column III, such as aluminum or gallium, with one fewer electron than silicon in the outer orbital, has the opposite effect on the Fermi level. The atom from column III does not have enough electrons to fully bond with all four of the adjacent atoms in the crystal structure. Therefore, one of its neighbors can be thought of as having a hole into which an electron in the valence band can easily be excited. Much like an n-type impurity, the energy difference between the top of the valence band and the new, unfilled state created by the substitution of a group III atom is small. So p-type doping has the effect of moving the Fermi level down towards the top of the valence band (see Figure 25.17(c)).
Thus, by adding n-type and p-type impurities to the silicon crystal the Fermi level can be moved in either direction. Semiconductors that have been doped in this way are known as n-type and p-type materials. At room temperature, electrons in an n-type or p-type material readily move between the Fermi level and the conducting or valence bands. In an n-type material, electrons are easily removed from their loosely bound states into the conducting band, becoming mobile charge carriers, while in a p-type material, electrons easily jump from the valence band into the outer orbitals of the group III atoms that have been substituted in the lattice, producing mobile positive charge carriers (holes) in the valence band. These mobile charge carriers – electrons in an n-type semiconductor and holes in a p-type semiconductor – are known as majority carriers, while the opposite charges in each type of semiconductor are minority carriers.
Doping
The band structure of a crystalline solid can be subtly modified by doping with atoms of different elements. Silicon lies in group IV of the periodic table, with four valence electrons per atom. n-doping is achieved by inserting atoms of an element from column V that have an extra electron. This raises the Fermi level of the semiconductor to near the bottom of the conduction band. p-doping is accomplished by inserting atoms of an element from column III, and lowers the Fermi level close to the top of the valence band.
Figure 25.18 Dynamics at a pn-junction: (a) Before the n-type and p-type materials are brought into contact. Note that for simplicity no excitations across the band gap are shown in this figure. When brought into contact, mobile electrons (holes) will diffuse from n to p (p to n) regions. This sets up a diffusion current in the direction of the n-type material. (b) Electrons diffused from the n-type material fill holes in the p-type material near the junction, leaving fixed charge densities that create an electric field and a potential difference across the junction. drives a field current in the direction of the p-type material. The system comes into a dynamic equilibrium when the two currents balance and the Fermi levels of the two pieces of matter become equal. In the depletion zone near the interface, there are few mobile charge carriers and an electric field points from the n-type material toward the p-type material.
These two different ways of modifying the silicon crystal lattice can be combined to generate an asymmetry in the spatial configuration of the material, so that electrons travel in a preferred direction, and can drive an external circuit. The basic mechanism for doing this is the p-n junction, a fundamental piece of semiconductor technology.
25.6.2 p-n Junction Diodes
A p-n junction is formed when a piece of semiconductor with n-type doping is placed adjacent to another piece of semiconductor with p-type doping. When these materials are connected (see Figure 25.18(a)), the loose negative charge carriers (electrons) in the n-type material diffuse through random thermal motion into the p-type material, while the loose positive charge carriers (holes) in the p-type material diffuse into the n-type material. This diffusion of majority carriers results in a net flow of positive charge from the p-type material into the n-type material, known as the diffusion current, .
As charge flows via the diffusion current, depletion regions build up on both sides of the interface, in which the density of majority carriers is very small. These regions have opposite net charge density, positive in the n-type material and negative in the p-type. Like a parallel plate capacitor, these charges generate an electric field pointing from the n- to the p-type material, resulting in a potential difference between the two materials. This potential difference has two effects: first, it suppresses the diffusion current by the Boltzmann factor , when the system is at temperature T. The Boltzmann factor appears because the diffusion current arises from random thermal motion of electrons from the n-type region into the p-type region (and vice versa for holes), involving an increase in energy of . And second, the electric field between the regions drives a second current , known as the field current (also known as the drift current), which opposes the diffusion current. The voltage difference between the two regions increases until the diffusion current and field current balance and the system reaches a dynamic equilibrium with
