Solid State Physics
Yuan Ming Huang
4.2.1 Dispersion relation
We start with the simplest case of one-atomic linear chain with nearest neighbor interaction (see Fig. 4).
Figure 4. Vibrations of a linear one-atomic chain (displacements).
The nth lattice point is connected with (n-1)th lattice point on the left side and the (n+1)th lattice point on the left side. If one expands the energy near the equilibrium point for the nth atom and use quasi elastic approximation, he comes to the following equations. The force exerted by the (n-1)th lattice point can be described by the force f1
The force exerted by the (n+1)th lattice point can be described by the force f2
Applying the Newtonian equation, one gets the following equation
To solve this infinite set of equations let us take into account that the equation does not change if we shift the system as a whole by the quantity a times an integer n. We can fulfill this condition automatically by searching the solution as
, where .
It is just a plane wave but for the discrete co-ordinate na. Immediately we get (see Problem 2.1)
, where .
The expression (9) is called the dispersion law. It differs from the dispersion relation for a homogeneous string, ¦Ø = sq. Another important feature is that if we replace the wave number q as
where g is an integer, the solution (8) does not change (because exp (2¦Ð i¡Á integer) = 1). Consequently, it is impossible to discriminate between q and q' and it is natural to choose the region
to represent the dispersion law in the whole q-space. So, for a linear chain, the wave number q takes L discrete values in the interval (¨C¦Ð/a, ¦Ð/a). Note that this interval is just the same as the Wigner-Zeitz cell of the one-dimensional reciprocal lattice. This law is shown in Fig. 4.
Figure 4. Vibrations of a linear one-atomic chain (spectrum).
Note that there is the maximal frequency ¦Øm that corresponds to the minimal wave length ¦Ëmin = 2¦Ð/qmax = 2a. The maximal frequency is a typical feature of discrete systems vibrations.
Because of the discrete character of the vibration states one can calculate the number of states, z, with different q in the frequency interval ¦Ø, ¦Ø+d¦Ø. One easily obtains (see Problem 2.2)
This function is called the density of states (DOS). It is plotted in Fig. 5.
Figure 5. Density of states for a linear one-atomic chain.
We shall see that density of states is strongly dependent on the dimensionality of the structure.
Now we discuss the properties of long wave vibrations. At small q we get from Eq. (9)
is the sound velocity in a homogeneous elastic medium. In a general case, the sound velocity becomes q-dependent, i. e. there is the dispersion of the waves. One can discriminate between the phase (up) and group (ug) velocities. The first is responsible for the propagation of the equal phase planes while the last one describes the energy transfer. We have
At the boundaries of the interval we get up = (2/¦Ð)u while ug = 0 (boundary modes cannot transfer energy).