More on Conductors

Table of Contents

Current Density (Classical Perspective)

Drift Velocity and Current

When electrons drift in a conductor under an applied electric field $E_x$, they acquire an average drift velocity $v_{dx}$ in the x-direction. The resulting current density is:

$J_x = \frac{\Delta Q}{A\Delta t} = \frac{qNAv_{dx}\Delta t}{A\Delta t} = qNv_{dx}$

where:

• $q$ = electron charge (magnitude)
• $N$ = number density of free electrons
• $v_{dx}$ = drift velocity in x-direction

This assumes constant effective mass $m^*$ due to non-relativistic conditions.

Free Electron Model

In the classical free electron model, kinetic energy increases quadratically with momentum:

$E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m}$

Theoretically extending to infinity, though quantum effects modify this at high energies.

Thermal Effects and Scattering

Thermal Motion

Even without an external electric field, electrons possess significant thermal energy:

$E_{\text{thermal}} \sim k_B T$

At room temperature ($T \approx 300$ K): $k_B T \approx 26 \text{ meV} \approx 4.2 \times 10^{-21} \text{ J}$

This corresponds to thermal velocities via: $\frac{1}{2}mv_{\text{thermal}}^2 \sim k_B T$

Random vs. Drift Motion

• Random thermal motion: $v_{\text{thermal}} \sim 10^6$ m/s
• Drift velocity: $v_{\text{drift}} \sim 10^{-2}$ m/s (cm/s order)

The drift represents a tiny bias superimposed on much larger random thermal motion.

Classical Conduction Theory

Equation of Motion with Scattering

The classical equation of motion for electrons includes a friction term representing scattering:

$m\frac{dv}{dt} + \frac{mv}{\tau} = qE$

where $\tau$ is the relaxation time (average time between scattering events).

Steady-State Solution

At steady state ($dv/dt = 0$), the terminal drift velocity is:

$v_d = \frac{q\tau E}{m} = \mu E$

where $\mu = \frac{q\tau}{m}$ is the mobility.

Time-Dependent Solution

For time-dependent fields, the complete solution is:

$v(t) = v_d\left[1 - e^{-t/\tau}\right]$

The system reaches $\sim 63\%$ of terminal velocity after time $\tau$.

Conductivity

Using Ohm’s law $\mathbf{J} = \sigma \mathbf{E}$:

$\sigma = \frac{nq^2\tau}{m}$

where $n$ is the electron density.

Scattering Mechanisms

Energy-Dependent Scattering

Higher energy electrons have increased scattering probability due to:

• Enhanced interaction cross-sections
• Access to more scattering channels
• Stronger coupling to lattice vibrations (phonons)

Sources of Scattering

1. Phonon scattering: Interaction with lattice vibrations
2. Impurity scattering: Defects and foreign atoms
3. Grain boundary scattering: Polycrystalline interfaces
4. Surface scattering: Important in thin films

The classical model provides the foundation, though quantum mechanical treatments are needed for complete understanding of transport in metals.

Metal Electrons under External Field

$\Delta k = \tau * \frac{E(-q)}{\bar{h}}$

J = qN_fv_f (N_f is the current density of all the electrons near fermi level (unbalanced electrons))…

Density of State

• There was an implicit definiton of infinities (infinitely large crystal / single frequency of momentum vector yielding an infinitely large density of states)
• normalized density of states (g(E)) by energy/volume is a constnant (can be proved…)
• conductivity is j/E = q^2(g(E_f))v_f^2*\tau

Density of state is proportional to square root of energy…

$\Delta E is very small (around 1\%) compared to E_f$.

2D Case…

We know that the real material is three dimensional so we need to look at dispersion in 3D. Since it is not easy to do that, let’s look at the 2D situation…

k is 2D is a vector, so that (kx, ky) yields the direction of propagation.

E_

Fermi Energy vs. Temperature

Fermi Level in metals is fairly stubborn - doesn’t change as a function of temperature…

Integral of sqrt Energy dE/ 1+ exp E does not have a closed form solution…

At 0 degree case, doesn’t matter what fermi energy is, and appriximatino is given by

Fermi energy = hbar^2 / 8effective mass * (3n/pi)^2/3

Fermi energy at a certain temperature = E_f0 * (1-pi^2/12 (kT/E_f0)^2)

Average energy of electrons are aroudnd 3/5 E_fermi

Random dance of electrons, when it gets colder… in perfect cooridinate (less random… at short time scale, still random - vacuum field fluctuation)