APPENDIX B: Lattice Vibrations
In chapter 2 we treated the crystalline lattice as stationary. This is of course not true. In chapter 3 we discussed the acceleration of electrons by an external electric field and the subsequent scattering of them by their interaction with the vibrating lattice. In this Appendix we give the elements of a quantitative theory of lattice vibrations. As usual we will start with elementary notions and subsequently build on these.
Consider first a periodic linear chain with 1 atom per unit cell. This is shown in figure Bl. The forces on each atom can be considered as Hooke-like (F = —kx) with the bonds between atoms acting as springs. Then the force on each atom will be proportional to the displacements, Щ (t), where l denotes lattice site, of the nearby atoms from their equilibrium position. We assume only nearest neighbour interactions. Then using Newton’s third law and denoting by M the mass of each atom and by к the spring constant of the corresponding force
But the ui must satisfy Bloch’s theorem since they constitute waves in a periodic medium. Then we must have
where q is a wavenumber and a is the distance between nearest neighbour. We get
This is a typical harmonic oscillator equation.
Assuming a harmonic time dependence of the form
FIGURE B1 Notation for the positions of the atoms of a monoatomic chain.
we arrive at the frequencies of oscillation
where in B5 a simple trigonometric formula has been used. At very small со, B5 may be further simplified as follows
The graph of co(q) is shown in figure B2. The range of vibrational frequencies shown in figure B2 is said to constitute the acoustic mode of frequencies of a linear monoatomic chain because it resembles the waves found in a continuous elastic medium. As we will see, additional modes of vibrations exist in a linear chain if it contains more than one atom per unit cell.
Consider then a linear periodic chain with two atoms per unit cell. Let the displacements of the two types of atoms of the unit cell be denoted u] and uf and their respective masses Mt and M2. For Mi we will have assuming a unique spring constant к between Mi and M2, see figure B3.
FIGURE B2 Frequencies of lattice vibrations of a monoatomic chain. These frequencies constitute the acoustic branch.
FIGURE B3 Notation for the position of the atoms of a diatomic chain, and for M2
Assuming now 1) a harmonic variation for both uj and uj exactly as in equation B4 and 2) a Bloch relation for uj and uj separately i.e.
we transform equations B7 and B8 into
For the system of equations B9 and BIO to have a solution for uj and uj their determinant must be zero. Hence
Equation Bll gives a second degree algebraic equation for со2 for each q. The form of the solution is shown in figure B4. It can be seen that in addition to the acoustic mode of vibration (the one going linearly to zero as q —> 0) a second mode of vibration has appeared which is called optical mode of vibration occurring at higher frequencies and hence energies. The reason for the name optical mode comes from the fact that the range of frequencies involved are very near those of the optical spectrum of the electromagnetic radiation. It is exactly the interaction with these type of vibrations that leads to the constancy of the electron drift velocity at high electric fields.
FIGURE B4 The dispersion relation CO(q) of both acoustic and optical frequencies.
Real solids of course cannot be described by linear chains and we therefore need a 3-dimensional generalizations of B7 and B8. We denote by the vectors m| the displacements in 3 dimensions of atom type i in unit cell l (vector). The cartesian components of this vector we call u'j, i.e. j denotes direction in space. The most efficient way of performing this generalization is to assume that the u'j are small enough so that the potential energy V of the lattice can be expanded in a Taylor series in terms of the uj measured from their equilibrium zero values. We will have
where in B12 the symbol |0 means evaluated at the equilibrium values of zero displacements. Now the constant term V„ will give zero force and the first derivatives will be zero at the equilibrium position of the atoms, i.e. that of zero u'j. Hence the only term which will contribute to the interatomic forces will be the last quadratic term. Denoting the second derivatives
the equation of motion can be written as or in matrix form
As there are many indices entering B13 and B14 we offer the following physical explanation of B14: each term on the RHS of B14 is the force exerted on atom type i in the Zth unit cell due to the vector displacement u[ of atom type i’in the f'th cell. The sum of all the forces on atom type i in the /th cell gives the corresponding acceleration. Obviously the number of equations in B14 is tremendous but Bloch’s theorem can be used to reduce them to a small number. This reduction is accomplished by the following intermediate steps.
1. The force constants can only depend on the relative distance between unit cells
2. As already noted the displacements u from waves in a crystalline solid, so they must obey Bloch’s theorem
Substituting the above equations in B14 eliminates the summations over l and we get
Equations B15 form the generalization of B7 and B8. The Oth lattice site is of course only nominal so that u'0 denotes the displacements of any unit cell. Equations B15, however, are much smaller in number compared to B14: The number of atoms per unit cell = N is certainly a small number and so is the number of interactions between the atoms; on the other hand if m separates atoms more than second nearest neighbours the interactions can almost always be ignored. As a general rule the number of vibrational modes is 3N, and of them 3 are acoustic ones so that the optical ones are 3N — 3.
The above presented theory is a classical one. One should have constructed a hamilto- nian and quantized it. This is beyond the scope of the present book but if such a quantization is performed then the frequencies of oscillation would be quantized and the energies would occur in multiples of ^co(q). However, the distinction between acoustic and optical modes would remain and also the fact that it is the optical modes that absorb energy from the electrons and lead to the stabilization of drift velocity.