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Spreadsheet Modeling of Electron Distributions in Solids Wingfield V. Glassey Department of Chemistry, The College of Wooster, Wooster, OH 44691;
[email protected] This article describes a series of spreadsheet modeling exercises constructed as part of a new upper-level elective course on Solid State Materials and Surface Chemistry at The College of Wooster. The spreadsheet exercises were designed to accompany discussion of the electronic structure of materials. The concepts introduced in this portion of the course are subsequently employed in discussion of the electronic and optical properties of semiconductors and semiconductor devices such as photoconductive materials and light emitting diodes (LEDs). In its present form the course also incorporates discussion of structure–property relationships in solidstate materials, X-ray methods for structure determination, reaction chemistry on surfaces, and the use of spectroscopic methods to probe the structure and binding of atoms and molecules on surfaces. The spreadsheet modeling exercises are intended to afford students the opportunity to directly manipulate and explore the properties of the mathematical expressions used to model electron distributions in solids. In the process students gain familiarity with the conceptual framework used to describe the behavior of electrons in solids. Students are asked to construct mathematical models describing the electron distribution in solids and the effect of temperature on the electron distribution. In the final exercise, students demonstrate their understanding by constructing a model to demonstrate how experimental measurement of the electrical conductivity of a semiconductor (a macroscopic property) can be used to graphically determine the electronic-band gap in the material (a microscopic property). The exercises demonstrate that the electrical properties of materials derive from an entire electron distribution and, as such, should not be attributed to the behavior of individual electrons. The exercises also serve to demonstrate how basic concepts from quantum and statistical mechanics, both microscopic theories, can be used to model the macroscopic properties of materials. Accordingly the exercises could be used as a bridge into discussions of the application of quantum and statistical ideas in materials science. Topical examples include the use of the particle-ina-box model of quantum mechanics to describe the optoelectronic properties of nanoscale semiconductor particles, so-called quantum dots (1), and color centers, atom defects common in alkali and alkaline earth halides (2, 3). The presentation offered in this article assumes no prior knowledge of quantum and statistical mechanics. As such, the exercises are suitable for students at all levels of the college chemistry curriculum.
the magnitude of the band gap, Eg, the energy gap between the highest occupied and lowest unoccupied electron states in the material. The characteristic features of electron distributions in metals, semiconductors, and insulators are illustrated in Figure 1. As shown in Figure 1A, the distribution of electron states in a metal is continuous and there are no band gaps. At 0 K, the electron states are completely filled up to an energy EF and the electron states above EF are empty. The energy EF is known as the Fermi energy or Fermi level. In contrast, semiconductors (Figure 1B) are characterized by band gaps on the order of 1–3 eV. At 0 K, the electron states below the band gap are completely filled and those above the band gap are empty. The band of filled energy levels is occupied by the valence electrons and is, as such, referred to as the valence band (VB). The bonding between the atoms in a semiconductor results from the filling of electron states in the valence band. The electrical conductivity of semiconductors results from the thermal excitation of electrons from the valence band to the band of empty states above the band gap. The presence of electrons in the band of energy levels above the band gap is associated with electrical conductivity in the material. As such, the empty states above the band gap are collectively referred to as the conduction band (CB). The magnitude of the band gap in semiconductors is typically sufficient to render the electrical conductivity of semiconductors lower than that of metals by a factor of 106–1010. As illustrated in Figure 1C, band gaps in insulators are larger than those in semiconductors. As a result, very few electrons are excited to the conduction band and the electrical con-
CB
Eg EF
E VB
Background
Metals, Semiconductors, and Insulators Materials can often be conveniently classified as metals, semiconductors, or insulators (nonmetals) on the basis of their ability to conduct electricity. In general terms, the electrical conductivity of a material is a function of temperature and
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A
B
C
Figure 1. Band structure of (A) a metal, (B) a semiconductor, and (C) an insulator at 0 K. Filled energy levels are shaded.
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ductivity of insulators is frequently lower than that of metals by a factor of 1015 or more.
Electron Distributions in Metals In contrast to the localized nature of the valence d electrons in transition metals, the valence electrons in simple (sp) metals such as Na and Al are effectively delocalized. As such, simple metals such as Na and Al can be considered examples of free-electron metals (FEMs) and their valence electrons described by free-electron theories (4). In a free-electron metal the behavior of individual (valence) electrons is effectively described using the particle-in-
a-box model of quantum mechanics (4) and the distribution of electron energy levels visualized in the form of a density of states (DOS) plot. In condensed-matter theory, the term DOS refers to the number of electron states per unit energy. As such, the DOS function serves as a practical tool for visualizing the large number of electron states in nonmolecular materials (4, 5). For the purpose of this study, it is sufficient to note that the DOS for a FEM, D(E ) ∝ √E (Figure 2). A detailed derivation of the DOS function for a FEM can be found in most texts dealing with aspects of materials science (4, 6). In a FEM, the electrons occupy orbitals two at a time, in accord with the Pauli principle. At 0 K, each filled state contains exactly two electrons. As the temperature is increased, electrons are thermally excited and begin to occupy previously unoccupied energy levels. The thermal distribution of electrons among the energy levels at a temperature, T, is determined by the Fermi–Dirac distribution function (4):
D(E ) ⬀ E
f (E ) =
EF
EF − E k BT
E
1 + e
D(E ) Figure 2. Density of states (DOS) for a free-electron metal (FEM).
D (E )
T
T EF (at T = 0)
T
E
T
E
e
EF − E k BT
In practice, the Fermi level, EF is often used to refer to the energy of the highest filled (or partially filled) state. Strictly speaking this definition is only appropriate in the limit T → 0. When using eq 1 to describe the electron distribution at a temperature above 0 K, a more rigorous, but less chemically intuitive definition of the Fermi level must be used. The Fermi level is (rigorously) defined to be the energy at which the available energy levels are half-occupied. Accordingly, the Fermi level corresponds to the energy at which the Fermi– Dirac distribution function (eq 1) takes the value 0.5. This can be confirmed by setting E = EF in eq 1. Both the density of states, D(E ), and the Fermi–Dirac distribution, f (E ), play integral roles in determining the electron distribution among the available energy levels in a material. To calculate the number of electrons with energies in the range E → E + dE, N(E ), we must first determine both the number of electron states available at this energy, the DOS, D(E ), and the probability that each will be occupied. The latter being determined by the Fermi–Dirac function, f (E ). On noting that N(E ) is proportional to the number of available energy levels, D(E ), and the probability that each is occupied, f (E ), the electron distribution is given by
N (E ) dE = D(E ) f (E ) dE
0
1
N (E )
f (E )
Figure 3. Temperature dependence of the electron distribution in a free-electron metal. The electron distribution at 0 K is indicated by a solid line. The effect of increasing temperature on the electron distribution is shown by a progression of dashed lines.
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(1)
(2)
The form of the Fermi–Dirac distribution (eq 1) and the resulting electron distribution (eq 2) is illustrated in Figure 3 for a FEM at 0 K (solid line). Figure 3 also demonstrates the effect of increasing the temperature on both functions. As the temperature is increased, an increasing number of electrons are excited to higher energy levels and the electron distribution is increasingly “smeared” towards higher energies. The smearing of the electron distribution results from the increasing asymmetry of the Fermi–Dirac distribution as indicated in Figure 3.
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Electrical Conductivity in Semiconductors
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CB
T
EC
EF (at T = 0)
E
For a material to be an electrical conductor, the application of an electric field must result in an electrical current within a material. For this to occur, the applied field must be able to accelerate some of the electrons in the material. In order to be accelerated, electrons must be able to acquire additional kinetic energy and move to a higher energy level. Thus, if a material is to be an electrical conductor, it must have accessible empty states immediately above the Fermi level. Further, if the material is to be a good conductor it must also have a significant number of filled states at the Fermi level to serve as a source of conduction electrons. The electron distribution for a FEM shown in Figure 2 demonstrates that (free-electron) metals effectively meet both criteria and are thus, not unexpectedly, good electrical conductors. The situation is somewhat different in a semiconductor. In a semiconducting material at 0 K, the valence band is completely filled and there are no empty states immediately above the Fermi level (Figure 1B). However, as temperature is increased, an increasing number of electrons are thermally excited into the conduction band. Once in the conduction band, electrons can be accelerated by the application of an external electric field and contribute to the electrical conductivity of the material. The vacancies or “holes” generated in the valence band also contribute to the electrical conductivity of semiconductors by acting as acceptor states for valenceband electrons accelerated in the applied electric field. As a result, the electrical conductivity of a semiconductor is expressed as the sum of electron and hole conductivities, σe and σh. Both contributions are proportional to the density of charge carriers (electrons and holes, respectively) in the material. Thus, to effectively model the electrical conductivity of a semiconductor, it is necessary to develop a model capable of predicting the extent of electron–hole pair formation as a function of temperature. By comparison with thermal energies (kBT ≈ 0.025 eV at room temperature) the band gap in a semiconductor is large. As a result, few electrons are thermally excited to the conduction band and the density of charge carriers is low. For example, in silicon, which has a band gap of ∼1.1 eV, there are approximately 1010 electrons per cm3 in the conduction band; small by comparison with a conduction band DOS on the order of 1019 states per cm3 (6). The effect of temperature on the electron distribution for a semiconductor (eq 2) is illustrated schematically in Figure 4. In a semiconductor, the number of electrons at the bottom of the conduction band, N(EC ), serves as an effective measure of the total number of electrons in the conduction band, NC (6). Accordingly, N(EC ) serves as a measure of the number of electron–hole pairs generated by thermal excitation of electrons from the valence band and, as such, the electrical conductivity (σ). In order to estimate N(EC ) it is first necessary to determine the probability that electron states at the bottom of the conduction band are occupied. The average occupation of the electron states at the bottom of the conduction band is given by the value of the Fermi– Dirac distribution function (eq 1) for an energy E = EC ,
T VB
Tⴝ0
0
1
f (E )
T
Figure 4. Temperature dependence of the electron distribution in a semiconductor.
that is,
e
f (E C) =
EF − EC kBT
1 + e
(3)
EF − EC k BT
At 0 K, the Fermi level is located at the center of the band gap. As the temperature is increased, few electrons are thermally excited to the conduction band and the Fermi level remains close to the center of the band gap, that is, EC − EF ≈ Eg兾2 . Further, on noting that the thermal energies of the electrons are small by comparison with the band gap, that is, kBT
