Chapter 3 Heat Capacity, Entropy, and
3.2 Entropy
3.2.3 Electronic Component of Melting Entropy
As mentioned above, the idea of a ΔSelfollows from an analysis of the perio-dic law of ΔSm (Fig. 3.5) and from the set of experimental data (primarily on electrical properties: the electrical conductivity, the thermal electromotive force emf, and the Hall effect, which reflect changes in the nature of the intera-tomic bonds during melting of a solid). This component essentially reflects the role of changes in the electronic subsystem due to a melting process.
A quantitative calculation of ΔSelfor several semiconductors was carried out on the basis of an analysis of thermoelectric effects at the solid-liquid
in-88 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
terface. Classically, an applied temperature difference causes charged carriers in materials, no matter whether they are electrons (negative charges) or holes (positive charges), to diffuse from the hot side to the cold side, similar to a gas that expands when heated. Mobile charged carriers migrating to the cold side leave behind their oppositely charged and immobile nuclei on the hot side, which give rise to a thermoelectric voltage. Thermoelectric power of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material.
Typically metals have small thermoelectric powers because they have half-filled bands. Electrons and holes both contribute to the induced thermoelec-tric voltage, thus canceling each other’s contribution to that voltage and making it small. By contrast, semiconductors can be doped with an excess amount of electrons or holes and thus can have large positive or negative values of the thermoelectric power depending on the charge of the excess carriers. The sign of the thermoelectric power can determine which charged carriers dominate the electric transport in both metals and semiconductors.
The thermoelectric power also measures the entropy per charge carrier in the material. Superconductors have zero thermoelectric power since the charged carriers carry no entropy. Equivalently, the thermoelectric power is zero be-cause it is impossible to have a finite voltage across a superconductor.
Guided by the concept of the reversibility of thermoelectric effects, which is postulated in the thermodynamics of irreversible processes, investigators es-tablished a relationship between ΔSmand a change in the Seebeck coefficient in the melting transition. The amount of heat δQ evolved at a solid-liquid interface, which is crossed by a certain number (dn) of electrons at Tm for an arbitrarily small current, is given byδQ = ΔS∗Tmdn, or,
ΔS∗=δQ/(Tmdn) (3.45)
where ΔS∗ is the change in entropy per electron. On the other hand, we have,δQ = Δ/cedn where e is the electron charge, and Δ/c is the Peltier heat, which is evolved at the interface due to a change in the Seebeck coefficient upon melting. Δ/c/T = Δaemfwith Δaemf being the change in emf coefficient due to the melting. Comparison of these relations yields
ΔS∗= e(aemf,s− aemf,L). (3.46) If we know the change in the carrier concentration upon melting, Δn, in light of Eq. (3.46),
ΔSel= eΔn(aemf,s− aemf,L). (3.47) With determined ΔSel values of Si, Ge, Sb, Bi, and several semiconductor compounds in terms of Eq. (3.31), Δn can be calculated by Eq. (3.47), which corresponds to measurements from the Hall-effect. This result is convincing evidence that changes in the electronic subsystem contribute substantially to ΔSm, when a substance undergoes a semiconductor-metal melting. Since the mechanism for carrier scattering in the liquid phase has not been finally
3.2 Entropy 89
resolved, and since calculations of the carrier concentration from measure-ments of the Hall coefficient are not completely rigorous, Eq. (3.47) is thus meaningful.
ΔSel can also be calculated by a purely thermodynamic method. The general expression for ΔSm with covalent–metallic interatomic binding is read as
ΔSm= fcΔSmet+ ΔScov (3.48) where subscripts “met” and “cov” denote typical metals (e.g., Cs) and purely covalent crystals (e.g., diamond), and fc is the number of free electrons in a real crystal at its Tm, divided by the total number of electrons. fc has been determined by
fc= exp(−Eg/2kT ) (3.49)
where Eg is the band gap width at Tm. Assuming that all the excess entropy of melting for a purely covalent crystal over that of a good metallic crystal is determined by the difference ΔScov − ΔSmet, Chakraverty [9] suggested calculating it from the Boltzmann equation incorporating the change in the number of bound electrons,
ΔScov− ΔSmet= k ln (4N )!
[(2N )!]2. (3.50)
A covalent crystal is regarded as a solid having 4N electrons, where N is the number of atoms in the crystal. The melting of such a crystal is accompa-nied by a “depairing” of 2N electron pairs and the complete liberation of the electrons. Calculation from Eq. (3.50) yields a value of 23.07 J·mol−1·K−1, in satisfactory agreement with the difference between ΔSm of diamond and ΔSm of cesium: 31.35–7.52 = 23.83 J·mol−1·K−1. A closer look reveals that this difference is essentially the maximal possible ΔSel, which is an obvious characteristic of diamond. A graphic illustration of the conclusion of Chakraverty shows that electron delocalization plays a decisive role in ΔSm. The corresponding relationship is shown in Chakraverty’s plot of ΔSmversus the fraction of electrons in the crystal fc for several simple solids, which is localized at Tm(Fig. 3.7).
We see an inverse proportionality. As fc in the solid phase increases, the possible destruction of homeopolar bonds in the course of melting becomes progressively less significant for ΔSm. Clearly not conforming to this behavior is Se. This circumstance indicates that the nature of the interatomic bond remains the same as this substance goes into the liquid phase. Thus, there is essentially no delocalization of electrons when Se melts. This effect was used to calculate the number of electrons in the liquid phase. Taking account of ϕc
in the melt of semiconductors, which goes into a metallic state upon melting, ΔSel can be determined by
ΔSel= k ln [4N (1− ϕc)(1− fc)]!
{[2N(1 − ϕc)(1− fc)]!}2. (3.51)
90 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
Fig. 3.7 ΔSmof semiconductors and of semimetals versus the change in the relative number of bound electrons at a melting transition.
ΔSel values of five substances in terms of Eq. (3.51) are shown in Fig.3.8 as a function of the relative number of paired electrons, which become free in the molten state. The origin of coordinates here corresponds to an ideal metal, with fc = 1 and ΔSel = 0. The uppermost point on this plot is the largest possible ΔSel, characteristic of a hypothetical covalent crystal.
Diamond can apparently be regarded as such a substance. The data of the five semi-conductors of Si, Ge, AlSb, GaSb, and InSb conform almost perfectly to the straight line connecting these two extreme points.
Fig. 3.8 ΔSel of certain semiconductors versus the relative number of electrons which are delocalized at the time of melting.
All the approaches discussed above ΔSel values produce similar results.
The proposition that there is a significant ΔSel passing from melting in a semiconductor-metal fashion can be thus regarded as a demonstrated fact:
physical reality.
3.2 Entropy 91
Let us look at yet another approach for calculating ΔSel. This approach is based on the argument that a semiconductor–metal transition observed during the melting of the most important semiconductors forces us to switch from MB statistics to FD statistics in describing the behavior of the electrons in the liquid phase.
We now examine the entropy of a closed, equilibrium system, which can be written as S = R lnΩ. The increment in entropy due to the change in the carrier energy spectrum upon melting can thus be described by
ΔSel= Sel(L)− Sel(s)= R ln(ΩL/Ωs). (3.52) For a metallic melt, we write the density of states in the form for a system of a degenerate electron gas,
Ωel(ε) = (2π/h2P)(2m)3/2√
εe. (3.53)
Here εeis the energy of the electron. The density of states for an electron gas in a crystal in the premelting region can be written in a way, which reflects the circumstance that a semiconductor at such temperature behaves as if it had an intrinsic conductivity,
Ωs(εn, εp) = [Ω(εn)Ω(εp)]1/2 (3.54) where Ω(εn) and Ω(εp) are the densities of one-particle states for electrons and holes, respectively. energy of the electrons and holes, respectively. Using Eqs. (3.52)–(3.54) and the above equations, there is
The total energy of the electrons and holes in an intrinsic semiconductor with an Eg can be written as where the first integral corresponds to electrons, and the second to holes.
Replacing f0(ε) and f0(ε) by the corresponding distribution functions, and replacing Ω(ε) and Ω(ε) by the above mentioned equations, there is
ε =
92 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
Using the value of the chemical potential of an electron for a semiconductor, μ =−Eg
2 +3
4kT lnm∗p
m∗n, (3.58)
and integrating Eq. (3.57), there is
ε = (2π In the one-particle approximation, we have
εn = 3kT /2, (3.60)
and
εp= (3kT + ΔE)/2. (3.61)
In the liquid state, εn can be found as the energy a degenerate electron gas (i.e., introducing an FD distribution),
An analysis of Eq. (3.55) led us to conclude that a calculation of ΔSel re-quires knowledge of not only certain physical constants of a crystal, but also the concentration of free carriers in the melt. The latter can be determined from measurements of the Hall coefficient. Accordingly, some precise experi-mental studies were carried out for the temperature dependence of the Hall coefficients of Ge, Si and III-Sb compounds in the solid and liquid states with error being less than 4%. The abrupt change observed in the Hall coefficient at Tm corresponds to a metallization of the bonds of these substances with melting.
The so calculated ΔSel values of some substances are shown in the last column of Table 3.4 while ΔSel values obtained by other methods are also shown, based on measurements of the thermal emf at Tm (ΔSel(I)), the dif-ference between ΔSm and the sum of ΔSvib and ΔSpos (ΔSel(II)), and a calculation of the change in the configurational entropy of the binding elec-trons (ΔSel(III)). The obtained results are similar to errors within 10%.
Table 3.4 ΔSelof Ge, Si, and III-Sb compounds calculated in various ways. The meanings in parentheses for ΔSelsee the text
Substance Entropy units