BONDING CHARACTERISTICS
4. Simple models of cohesive properties 1. Introduction
If we know the energy C/(A), for instance expressed through a pairwise potential describing the interaction between the atoms (ions), we can solve for the atomic volume £2a at the energy minimum and also find the corresponding bulk modulus K and binding energy [/bind- In the early days of solid state physics, this was an important field of research.
One was looking for simple mathematical descriptions, in particular for ionic compounds. Modern approaches to cohesive properties, including atomic volumes and bulk moduli, rely on large quantum mechanical calculations of the electronic structure. However, simple mathemati-cal models may serve to give an insight into trends. We now consider such models for ionic compounds, simple (i.e. free-electron-like) and transition metals, and make a brief comment on semiconductors.
4.2. Ionic compounds
The cohesive properties of ionic compounds, in their main features, can be explained in terms of classical physics. This is in contrast to the metals, where quantum mechanics plays a major role. We assume that
two ions, i and j , interact through a potential (e.g. Born and Huang 1954),
<t)(r) = e2ZiZj/r + B/r\ (1.11)
The first term is the Coulomb interaction between charges Z,e and Zje. When summed over the lattice, it gives the Madelung energy EM, expressed through the Madelung constant aR. (The subscript R means that EM ~ &R/R, where R is the nearest-neighbour distance.
One may also define a Madelung constant aa such that EM ~ <xa/a, where a is the lattice parameter, or the cube root of the atomic volume S2a, and yet another type, o?c, is introduced in §4.3.) The last term in eq. (1.11) represents a repulsive interaction that prevents the ions from coming too close to each other. We assume that it acts only between the nearest-neighbours of unlike ions. As an example, consider diatomic compounds (e.g. NaCl, MgO). Let R be the shortest distance between anions and cations. The total energy of a lattice with v nearest unlike neighbours is (per stoichiometric unit, and with |Z,-| = \Zj\ = Z)
Utot = -aR(Ze)2/R + vB/R\ (1.12)
The equilibrium distance R$ is obtained from dUtot/dR = 0;
(Ro)n~l = vBn/[aR(Ze)2]. (1.13)
The binding energy [/bind = — Utot(Ro) depends on three quantities; 7?o, a and vB. If eq. (1.13) is used to eliminate vB, we obtain
H)-0 ^ = 2 ^ , , _ - | . (1.14)
For many ionic compounds, the exponent n ~ 8 to 10, i.e. l — l/n ^ 1.
The electrostatic energy, taken between point charges, is therefore by far the most important contribution to the binding energy of ionic crystals, in line with the arguments in §2. By proceeding as outlined in §2, we obtain for the bulk modulus
„ oiR{Ze)\n - 1) aR(Ze)2(n - 1)
A = -. = 77; . ( l . l j )
18fl04
18^/3
Simple models of cohesive properties 9
Fig. 1.2. The experimental bulk modulus K of alkali halides in the NaCl-type crystal structure, as a function of the volume per atom, £2a.
In the last step we specialised in the NaCl-type lattice, for which Qa = On the basis of the crude model (eq. (1.15)), we can now understand the variation of the bulk modulus of alkali halides crystallising in the NaCl-type structure, as a function of the atomic volume £2a, (fig. 1.2). K is obtained here as (cn + 2cn)/3 with ctj from the Landolt-Bornstein tables (Every and McCurdy 1992). The dashed line in the figure is just a guide to the eye. Its slope corresponds to K ~ Q~l-02, and not ~ ^4 / 3 as suggested by eq. (1.15). Considering the crudeness of the model, for instance the neglect of interactions between next nearest-neighbours, we should not expect a better account of the bulk modulus.
4.3. Free-electron-like metals
The simplest representation of a metal is the jellium model. The ion charges are "smeared out" into a uniform positive background. The distribution of the conduction electrons is also spatially uniform. Since there is a charge neutrality everywhere, this system has no electrostatic energy. Let there be N atoms in a volume V. The only energy that varies
with the atomic volume Qa = V/N in this case is the kinetic energy of the electrons (Appendix B),
<£kin) = 2.210Zr;2 [Ry]. (1.16)
Here, and in the rest of §4, energies refer to an average per atom. The dimensionless parameter rs is a measure of the electron number density, rsao being the radius of a sphere of volume £2a/Z, and a$ the Bohr radius, i.e. (47t/3)(aors)3 = QJZ. Z is the number of valence electrons contributed by each ion, e.g. Z = 1 for the alkali metals, Z = 3 for Al and Z = 4 for Pb, in their free-electron descriptions. The energy is expressed in Rydberg units (1 Ry = me4/(87t€oh2); 1 mRy/atom =1.313 kJ/mol). See Appendices B and H for details.
Because the energy (eq. (1.16)) is lowered if the system is allowed to expand, i.e. if rs increases further, it represents a repulsive term. We need also an attractive term to get a minimum in the total energy, i.e.
cohesion. Its essential physical origin is the fact that the positive charge is not uniformly distributed, but approximately concentrated in positive ionic charges +Ze centred at the lattice points. It can not be described accurately in as simple a form as the kinetic energy. We therefore do not derive a closed-form expression for the binding energy, but turn to the bulk modulus K. Following the approach in §2, we consider only the repulsive term (eq. (1.16)) and get
K = *L < = ( — " ) -109[Nm-2]. (1.17) 47t(4/97T)2^m(rsa0)5 \ rs )
Here rs refers to the actual value for the metal considered. Figure 1.3 shows experimental values for the bulk modulus, versus the parame-ter rs, with K calculated from the single-crystal elastic coefficients ctj (Every and McCurdy 1992) using the methods described in Chapter 18 (§3). For Li, Rb and Cs, low temperature Q; are used to suppress the effects of anharmonic softening. The dashed line is a guide to the eye, and corresponds to K ~ r~3'5. Considering the extreme simplicity of the model, with K arising entirely from the kinetic energy of a free-electron gas, we should not expect a better account of the data in fig.
1.3.
It is seen in fig. 1.3 that rs varies considerably even among free-electron-like metals of the same valency, for instance among the alkali
Simple models of cohesive properties 11
Fig. 1.3. The bulk modulus of some free-electron-like metals as a function of the electron density expressed through the parameter rs, plotted as log K versus log rs.
metals that are all described by Z = 1. Obviously, the atomic volume,
£2a, depends crucially on the "size" of the ions (e.g. the ion Na+, con-sisting of the nucleus and the filled electron shells). However, the energy does not depend much on the precise lattice configuration, for a given ion. This can be illustrated by considering the Coulomb energy Ec (per ion), when ions of charge +Ze interact with a rigid uniform electron gas with a density given by rs. Then
Ec = -acZ5/3Zr;1 [Ry]. (1.18) Here, ac is a Madelung constant given in table 1.2. The fact that ac
depends so weakly on the configuration of the positive point charges is consistent with the experimental and theoretical result that the atomic volume is the same within about 1% for different lattice structures (e.g.
bcc, fee and hep lattices) for a certain free-electron-like metal (Rudman 1965). The atomic volume of alloys is further discussed in Chapter 19, in connection with Vegard's (1921) rule.
Finally, it must be stressed that one cannot tell from the element alone, i.e. without a quantum mechanical calculation, what is the elec-tron structure in the solid. For instance, tin is rather free-elecelec-tron-like in the metallic tetragonal lattice structure (jS-Sn or white tin), but is a semiconductor when crystallising in the diamond-type lattice
struc-Table 1.2
ture (a-Sn or gray tin). The atomic volume of Sn is 27% larger in the semiconducting state.
4.4. Transition metals
The d-electrons play a major role in the transition metals. We will use a simple model (Friedel 1969) that neglects the s- and p-electrons altogether. Let the d-state of an isolated atom have the energy E® rel-ative to some reference level. When the atoms are brought together in a solid, the d-level broadens into a band described by an electron density-of-states Nd(E) (per atom and spin direction). If a metal has nd d-electrons, the cohesive energy (per atom) becomes
Ucoh = ndE»-2 Nd(E)EdE, (1.19) where E' is the bottom of the d-band and £F is the Fermi level. The factor of 2 in the integral comes from summation over the two-spin directions. Friedel (1969) assumed that Nd(E) is rectangular in shape, with a width Wj and a "centre of gravity" shifted from the atomic level E® to Ed (figs. 1.4 and 1.5). The total number of d-states is 10, which fixes the height of Nd(E) to 5/Wd, when Nd(E) refers to one spin direction. Then
Simple models of cohesive properties 13
N(E)
Fig. 1.4. A schematic picture of how an electron d-level shifts from its value E® in an atom and broadens into a band of width Wd in the solid.
Fig. 1.5. The electron density of states N(E) for a real transition metal (bcc W; from Einarsdotter et al. (1997)) and a representation through Friedel's rectangular model
density of states.
C/coh = (E°d - Ed)nd + ( 1 / 2 0 ) ^ ^ ( 1 0 - nd). (1.20) Neglecting the fact that Wd, Ed and E® vary with nd, which is a crude but reasonable approximation in our context, this model predicts that the cohesive energy varies parabolically with nd, i.e. with the d-band filling. Typically, in the 4d-transition metal series, Wd = 0.5 [Ry] and E® - Ed < 0.1 [Ry]. The model neglects any structural dependence of Nd(E) but this is not too serious since f/COh is an integrated quantity of Nd(E) and the most important factor is the width Wd of Nd(E).
0.8
Fig. 1.6. The cohesive energy [/coh (solid curve) calculated from eq. (1.20) when E® = Ed a nd Wrf = 0 . 5 Ry, and plotted versus the position of the metal in the Periodic Table. Filled circles are experimental values and open circles are results from an early and very influential ab initio electron structure calculation (Moruzzi et al. 1977, 1978).
As is seen in fig. 1.6, the cohesive energy calculated in this approx-imate manner is in qualitative agreement with experiments. Since the model makes no reference to how C/COh changes with volume, we cannot estimate the atomic volume. Such considerations should use the fact that Wd increases with decreasing volume that corresponds to attractive forces between the atoms. This is balanced by the repulsive force arising when the conduction electrons (s- and p -electrons) are forced into the ion cores on compression. Thus, the s- and /^-electrons are important in determining the atomic volume and the bulk modulus, but not for the cohesive energy, again in line with the arguments in §2. There, the balance between the repulsive and attractive forces also was found to imply a covariation between K and C/C0h. Such a connection holds also for transition metals, as seen by comparing fig. 1.6 and 1.7. However, there is no simple correlation between the bulk modulus K and the atomic volume J2a in contrast to the behaviour for simple metals, ionic and covalent solids.
Example: relative stability of fee and bee structures. In pioneering work by Pettifor (1970), Skriver (1985) and others, the difference in cohesive energy between the fee, hep and bec lattice structures of transition met-als was obtained from electron structure calculations. Such theoretical
15
0.08
0.06 g
0.04 >r
CO
0.02
Sr Y Zr Nb Mo Tc Ru Rh Pd Ag
Fig. 1.7. The experimental bulk modulus K (filled symbols, left scale), obtained as in Chapter 18 with c/y from Every and McCurdy (1992), plotted versus the position of the metal in the Periodic Table. The figure also shows the experimental volume per
atom (open symbols, right scale), from Rudman (1965).
results were first thought to be less accurate because for some elements they seemed not to agree with the semiempirical results (cf. fig. 1.8). It is now well established that ab initio electron structure calculations can give very reliable results for the cohesive energy of transition metals in various hypothetical structures. The difference between such data and the semiempirical values that are derived, e.g. from the fitting of thermodynamic functions to alloy phase diagrams, is not physically significant. Discrepancies between the two approaches to the cohesive energy of metastable structures may arise when a metastable structure is in fact dynamically unstable (see also Chapter 4 (§3), and a review by Grimvall 1998).
It was noted above that for free-electron-like metals, the bulk mod-ulus K varied significantly with the atomic volume Qa when different elements were compared, but £2a did not vary much for different metal-lic structures of the same element. As seen in fig. 1.7, there is no corresponding close relation between K and Qa for the transition met-als. However, like the case of simple metals (§4.3) it is a good rule of thumb for the transition metals, that Qa does not depend much on the lattice structure as long as the electronic structure is not much changed.
This is further dealt with in Chapter 19 (§2).
Simple models of cohesive properties
400
16
Zr Nb Mo Tc Ru Rh Pd Ag
Fig. 1.8. The difference in cohesive energy between the fee and bec lattice structures of 4d-transition metals (solid line; after Skriver 1985) and the same quantity obtained by the often used semiempirical estimate using Miedema's approach (symbols; Miedema and Niessen 1983). Where the two sets of data disagree significantly, they do in fact not
reflect the same physical quantity.
2.00
1.90 h
CO
O 1 8 0
|-5 9
1.70
\-1.60
ll-VI compounds
0.15 0.20 0.25 0.30 0.35 log(fta/A3)
0.40 0.45
Fig. 1.9. The experimental bulk modulus K = (c\\ + 2c\2)/3, with CJJ from Every and McCurdy (1992), as a function of the atomic volume Q&; plotted as log K versus
log fia
-Simple models of cohesive properties 17
4.5. Semiconductors
The bonding in semiconductors cannot be described quantitatively by the simple approaches we have discussed for ionic solids, free-electron-like metals and transition metals. However, one can argue (Cohen 1985,
1988) that the bulk modulus varies approximately inversely with a power of the atomic volume. Consider the Ansatz
K*
K-g. (1.2!) Figure 1.9 shows that this gives a good fit when the exponent p « 7/6.
Hence, K varies as a~35 within a class of materials with the same lat-tice structure, where a is a latlat-tice parameter. The parameter K* takes different values for group IV, III-V and II-VI compounds.