Patterson, Bailey - Solid State Physics Introduction to theory

276 5 Metals, Alloys, and the Fermi Surface

B

k

Fig. 5.4. Constant-energy surfaces for the de Haas–van Alphen effect

and

where A is the area between the two Fermi surfaces in the plane perpendicular to B. This result was first obtained by Onsager in 1952 [5.20].

Recall that we have already found that the energy levels of an electron in a magnetic field (in the z direction) are given by (3.201)

This equation tells us that the difference in energy between different orbits with the same kz is ωc. Let us identify the E in the equations of the preceding figure with the energy differences of ωc. This tells us that the area (perpendicular to B) between adjacent quantized orbits in k-space is given by

The above may be interesting, but it is not yet clear what it has to do with the Fermi surface or with the de Haas–van Alphen effect. The effect of the magnetic field along the z-axis is to cause the quantization in k-space to be along energy

278 5 Metals, Alloys, and the Fermi Surface

If A0 is the area of an extremal (where one gets the dominant response, see Ziman [5.35, p 322]) cross-sectional area (perpendicular to B) of the Fermi surface and if B1 and B2 are the two magnetic fields that make adjacent tubes equal in area to A0, then

(1/B) is the change in the reciprocal of the magnetic field necessary to induce one fluctuation of the magnetic susceptibility. Thus, experiments combined with the above equation determine A0. For various directions of B, A0 gives considerable information about the Fermi surface.

5.5 Eutectics (MS, ME)

In metals, the study of alloys is very important, and one often encounters phase diagrams as in Fig. 5.6. This is a particularly important technical example as discussed below. The subject of binary mixtures, phase diagrams, and eutectics is well treated in Kittel and Kroemer [5.15].

Alloys that are mixtures of two or more substances with two liquidus branches, as shown in Fig. 5.6, are especially interesting. They are called eutectics and the eutectic mixture is the composition that has the lowest freezing point, which is called the eutectic point (0.3 in Fig. 5.6). At the eutectic, the mixture freezes relatively uniformly (on the large scale) but consists of two separate intermixed phases. In solid-state physics, an important eutectic mixture occurs in the Au1−xSix system. This system occurs when gold contacts are made on Si devices. The resulting freezing point temperature is lowered, as seen in Fig. 5.6.

280 5 Metals, Alloys, and the Fermi Surface

Fig. 5.7. Splitting of energy bands at Fermi wave vector due to distortion

In the case considered (Fig. 5.7), if kF = π/2a, there would be a dimerization of the lattice and the new periodicity would be 2a. Thus, the deformation in the lattice can be approximated by

which is periodic with period π/kF as desired, and c is a constant. As Fig. 5.7 shows, the creation of an energy gap at the Fermi surface leads to a lowering of the electronic energy, but there still is a question as to what electron–lattice interaction drives the distortion. A clue to the answer is obtained from the consideration of screening of charges by free electrons. As (9.167) shows, there is a singularity in the dielectric function at 2kF that causes a long-range screened potential proportional to r−3 cos(2kF r), in 3D. This can relate to the distortion with period 2π/2kF. Of course, the deformation also leads to an increase in the elastic energy, and it is the sum of the elastic and electronic energies that must be minimized.

For the case where k and k′ are near the Brillouin zone boundary at kF = K′/2, we assume, with c1 a constant, that the potential energy due to the distortion is proportional to the distortion, so2

So 2V(K′) ≡ 2V(2kF) = c1c, and in the nearly free-electron model we have shown (by (3.231) to (3.233))

Ek = 12 (Ek0 + Ek0′) ± 12 {4[V (K′)]2 + (Ek0 − Ek0′)2}1/ 2 ,

where

and

2

Ek0′ =V (0) + 2m k + K′ 2 .

2 See e.g. Marder [3.34, p. 277]

We compute an expression relating to the lowering of electron energy due to the gap caused by shifting of lattice ion positions. If we define

As noted by R. Peierls in [5.23], this logarithmic dependence on displacement is important so that this instability not be swamped other effects. If we assume the average elastic energy per unit length is

3 The number of states per unit length with both spins is 2dk/2π and we double as we only integrate from = 0 to kF or –kF to 0. We compute the derivative, as this is all we need in requiring the total energy to be a minimum.

282 5 Metals, Alloys, and the Fermi Surface

The lattice distorts if the quasifree-electron energy is lowered more by the distortions than the elastic energy increases. Now, as defined above,

equals the density (per unit length) of orbitals at the Fermi energy (for free electrons), and we define

as an effective interaction energy. Therefore, the distortion amplitude c is proportional to yF times an exponential;

Our calculation is of course done at absolute zero, but this equation has a formal similarity to the equation for the transition temperature or energy gap as in the superconductivity case. See, e.g., Kittel [23, p 300], and (8.215). Comparison can be made to the Kondo effect (Sect. 7.5.2) where the Kondo temperature is also given by an exponential.

5.6.1 Relation to Charge Density Waves (A)

The Peierls instability in one dimension is related to a mechanism by which charge density waves (CDW) may form in three dimensions. A charge density wave is the modulation of the electron density with an associated modulation of the location of the lattice ions. These are observed in materials that conduct primarily in one (e.g. NbSe3, TaSe3) or two (e.g. NbSe2, TaSe2) dimensions. Limited dimensionality of conduction is due to weak coupling. For example, in one direction the material is composed of weakly coupled chains. The Peierls transitions cause a modulation in the periodicity of the ionic lattice that leads to lowering of the energy. The total effect is of course rather complex. The effect is temperature dependent, and the CDW forms below a transition temperature with the strength p (see as in (5.46)) growing as the temperature is lowered.

284 5 Metals, Alloys, and the Fermi Surface

CeCu2Si2, UBe13, UPt3, CeAl3, UAl2, and CeAl2. In particular, they may show large, low-T electronic specific heat. Some materials show f-band superconductiv- ity—perhaps the so-called “triplet superconductivity” where spins do not pair. The novel results are interpreted in terms of quasiparticle interactions and incompletely filled shells. The heavy fermions represent low-energy excitations in a strongly correlated, many-body state. See Stewart [5.30], Radousky [5.25]. See also Fisk et al [5.8].

5.8 Electromigration (EE, MS)

Electromigration is of great interest because it is an important failure mechanism as aluminum interconnects in integrated circuits are becoming smaller and smaller in very large scale integrated (VLSI) circuits. Simply speaking, if the direct current in the interconnect is large, it can start some ions moving. The motion continues under the “push” of the moving electrons.

More precisely, electromigration is the motion of ions in a conductor due to momentum exchange with flowing electrons and also due to the Coulomb force from the electric field.4 The momentum exchange is dubbed the electron wind and we will assume it is the dominant mechanism for electromigration. Thus, electromigration is diffusion with a driving force that increases with electric current density. It increases with decreasing cross section. The resistance is increased and the heating is larger as are the lattice vibration amplitudes. We will model the inelastic interaction of the electrons with the ion by assuming the ion is in a potential hole, and later simplify even that assumption.

Damage due to electromigration can occur when there is a divergence in the flux of aluminum ions. This can cause the appearance of a void and hence a break in the circuit or a hillock can appear that causes a short circuit. Aluminum is cheaper than gold, but gold has much less electromigration-induced failures when used in interconnects. This is because the ions are much more massive and hence harder to move.

Electromigration is a very complex process and we follow Fermi’s purported advice to use simpler models for complex situations. We do a one-dimensional classical calculation to illustrate how the electron wind force can assist in breaking atoms loose and how it contributes to the steady flow of ions. We let p and P be

4To be even more precise the phenomena and technical importance of electromigration is certainly real. The explanations have tended to be controversial. Our explanation is the simplest and probably has at least some of the truth (See, e.g., Borg and Dienes [5.3].) The basic physics involving momentum transfer was discussed early on by Fiks [5.7] and Huntington and Grove [5.13]. Modern work is discussed by R. S Sorbello as referred to at the end of this section.