ABSTRACT

Bloch electrons in a perfect periodic potential can sustain an electric current even in the absence of an external electric field. This infinite conductivity is limited by the imperfections of the crystals, which lead to deviations from a perfect periodicity. The most important deviation is the atomic thermal vibration from the equilibrium position in the lattice; however, electric perturbations can also promote this type of vibration. A quantitative treatment of the external electric perturbation of a crystal, therefore, starts with the observation of the change in the lattice vibrations [1]:

Uper ¼ X R

f(r R) (6:1)

where f(r) is the electrostatic energy resulting from the attraction (/r1) balanced against the kinetic potential (/r2)

r is the atomic position

In the case of a square-wave potential perturbation,

f(r, t)per ¼ f1(r)

f2(r)

(6:2)

where f1(r) and f2(r) are the electric potentials for odd and even N half-cycle values, respectively. Considering the generalized term for a periodic potential energy,

U(r) ¼ X R

f(r R u(R)) ¼ Uper X R

u(R) rf(r R)þ (6:3)

where u(r) is the atomic displacement in the lattice. The difference between the two forms is the perturbation that causes the transitions between the

Bloch levels that initiates the lower conductivity. This process can be considered as a transition of the emitting electrons to absorbing phonons due to a change in their energy by the phonon wave vector. The result is the electron scattering by the lattice vibrations. In most cases, the crystal conduction is dominated by electronic transitions of a single phonon absorption or emission. The energy and crystal momentum conservation laws require that the phonon energy satisfy

Ek ¼ Ekþq hv[k (k þ q)] (6:4)

where the positive sign obeys the phonon emission of the q wave vector between the transitions of the electron with the k wave vector. It has to be remembered that the surface of the allowed q for a given k vector will be very close to the set of vectors connecting k to all other points of the constant energy: EkEkþq. Considering that the static model works for the platinum electrode, the total potential energy of the crystal will be

U(r) ¼ 1 2

f[r(R) r(R0)] ¼ 1 2

f[R R0 þ u(R) u(R0)] (6:5)

The following Hamiltonian governs the resolution of the dynamic system:

H ¼ X R

P2(R)

2M þ U (6:6)

where P(R) is the linear momentum M is the mass R is the position of the crystal atom

For a pair potential, f, of the form (1=r2 1=r), it is not difficult to obtain clear information from the Hamiltonian, but with a periodic electric potential the problem gets more complicated. We will consider that the platinum atoms will not deviate much from the equilibrium positions, that is, small u(r) values. Thus, the Taylor expansion of the term U(r) is of the following expression:

U(r) ¼ 1 2

X R

f(R)þ 1 4

[u(R) u(R0)] rf(R R0)þ 1 8

([u(R)

u(R0)] r)2f(R R0)þ (6:7)

When no net force is applied to the crystal atom, the linear term vanishes because the coefficient u(R) isrf(R R0). The latter is minus the force exerted by other atoms in the equilibrium position. The non-vanishing term is the quadratic contribution, which is called the harmonic approximation.