Teoría de Piaget

Consider a metal sample of electrical conductivity σ subjected to an electric field E. The effect of this field is to exert a force equal to −eE on each free electron in the metal. As a result of this force, the electron delta will be accelerated. The acceleration produced is given by

ax _{= −(e/m)E} (3.5)

assuming that the electric field is acting along x-axis and m is the mass of the elec- tron. If vx represents the velocity component along x-axis, we may write the above

equation as δv x δt  f ield = −(e/m)E (3.6) The partial time derivative of velocity and the subscript ‘field’ indicate that the accel- eration mentioned here is due to the effect of the applied electric field. Contribution to the acceleration due to other possible effects will be considered later. Considering a large group of electrons, the average velocity at any instant may be written as

< vx >= 1 N N X i=1 v_xi (3.7)

where N is the number of electrons considered. Hence, we may write equation (3.6) as _{δ < v} x > δt  f ield = −(e/m)E (3.8) Considering unit area of cross section of the sample and counting the total number of carriers crossing the unit area along the x-axis as shown in Fig. 3.1 in unit time, we find that all the carriers present in an elemental volume of unit area of cross-section and upto a length equal to vx will cross the reference point at origin.

1 1 vx X Y Z

Figure 3.1 Schematic representation of electron flow for calculation of current density.

In other words, all the electrons present in the elemental volume vx will contribute to the current density. If there are n conduction electrons per unit volume in the given sample, the number of electrons contributing to current density is equal to nvx. Since each electron is associated with a charge (−e), the current density at any instant may be written as

According to Ohm’s law, we have

J = σE

This means that as long as the applied field is held constant, the current density will remain constant.

i.e., dJ

dt = 0 Substituting this in equation (3.9), we get

dJ dt = ne.d < vx > dt = 0 or d < vx > dt = 0 (3.10)

Comparing equation (3.8) with equation (3.10), since the time invariance of current density is an experimentally confirmed fact, we conclude that equation (3.8) does not give the total time rate change of velocity and that there must be other processes which make the total time rate change of velocity equal to zero.

i.e.,d < vx > dt = δ < v x > δt  f ield + δ < v x > δt  other = 0 (3.11)

The other process responsible for the time rate change of velocity of conduction elec- trons has been identified as electron- lattice interaction. Since the electrons are mov- ing in the crystal lattice of ion cores, such an interaction is a possibility. An experi- mental observation that a sample tends to warm up when an electric current is passed indicates possible transfer of energy (momentum) from electrons to the lattice.

In order to understand the nature of lattice - electron interaction, let us consider a sample subjected to an electric field E resulting in a current density J and an associated average electron velocity < vx >. If the applied electric field is switched off at any instant, say t = 0, the electron flow will stop indicating that the velocity < vx > goes to zero. This decrease in the velocity is attributed to the interaction of electrons with the lattice. The average velocity is assumed to decay exponentially with time (Fig.3.2) in accordance with the equation

< vx>

Figure 3.2 Exponential decay of the average velocity of electrons when the applied field is removed.

where < vx >t is the instantaneous value of average velocity at time t, < vx >o is the average velocity when the electric field is turned off. The quantity τ is called the relaxation time of conduction electrons. Differentiating equation (3.12)

_{δ < v} x > δt  coll= − < vx > τ (3.13)

In other words, when the applied electric field is turned off, the velocity component will reduce to zero due to electron-lattice collisions and the second component of ac- celeration mentioned as due to other processes in equation (3.11) is now identified as due to collisions. Substituting for the two components of acceleration from equations (3.6) and (3.13), equation (3.11) becomes

d < vx > dt = − eE m − < vx > τ = 0 or < vx > = − eτE m (3.14)

The steady state average velocity of electrons in presence of an applied electric field is called the drift velocity of the carriers. This is proportional to the applied electric field and the constant of proportionality is called the mobility of the carriers. Hence, the mobility of carriers is defined as the drift velocity per unit applied field and is given by

µ = eτ

The current density J can be calculated from equation (3.9) as

J = −ne < vx >= ne 2_τE

m (3.16)

Comparing equation (3.16) with equation (3.3), we have the electrical conductivity of a metal given by the expression

σ = ne 2_τ

m = ne µ (3.17)

This expression for the conductivity of a metal predicts the dependence of conductivity on the total number of valence electrons per unit volume and their mobility.

Drude made use of the kinetic theory of gases to evaluate τ. Considering electrons as equivalent to molecules in a gas, he evaluated the kinetic energy of the electrons as

1 2mv

2₌ 3 2kT

where v is the root mean square velocity of electrons, which can be calculated as

v = (3kT/m)1/2 (3.18)

At room temperature, the thermal velocity of electrons calculated using the above equa- tion is 1.15 × 105_ms−1_{. The drift velocity acquired by the electron due to an applied}

field is much smaller than the thermal velocity. Hence, the relaxation time may be written in terms of the mean free path as

τ = λ/v = λ(m/3kT )1/2

Hence, by substituting this value of τ in equation (3.17), the conductivity of a metal may be expressed as σ = ne 2_τ m = ne2_λ (3mkT )1/2 (3.19)

This equation suggests that the conductivity of a metal must be inversely proportional to the square root of temperature. This is in contrast to the experimental observation that the conductivity is inversely proportional to the temperature. This discrepancy is due to the fact that λ is assumed to be independent of temperature. In fact, λ is a function of temperature which is not explained or accounted for by the classical free electron theory.