More on Conductors

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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