The Drude Model: Deriving Ohm's Law from Classical Mechanics
The Drude model provides a classical explanation for the conductivity of metals by treating electrons as a gas of pinballs bouncing off a lattice of heavy, stationary ions. By applying Newton's laws and simple probability to these collisions, we can surprisingly derive Ohm's Law and understand why metals resist electric current to varying degrees.
1. Momentum Evolution over a Short Time Interval
To understand how current flows, we must look at the average momentum \(\mathbf{p}(t)\) of an ensemble of electrons. We consider the evolution of this momentum over an infinitesimal time step \(dt\).
The Probability of Collision
We define \(\tau\) (tau) as the relaxation time or mean free time—the average time an electron travels between collisions. The probability \(P_{\text{coll}}\) that an electron undergoes a collision in time \(dt\) is: \[ P_{\text{coll}} = \frac{dt}{\tau} \] Conversely, the probability that an electron survives without a collision is \(1 - \frac{dt}{\tau}\).
Calculating Momentum at \(t + dt\)
The momentum of the electron gas at time \(t + dt\) comes from two contributions:
- Fraction that did not collide: These electrons retain their old momentum \(\mathbf{p}(t)\), plus any momentum added by external forces (like an electric field) during \(dt\).
- Fraction that collided: Drude assumed that after a collision, the electron emerges with random direction. Therefore, the average momentum of the collided fraction is zero.
Mathematically, we write the ensemble average momentum as:
\[ \mathbf{p}(t+dt) = \left(1 - \frac{dt}{\tau}\right) \left[ \mathbf{p}(t) + \mathbf{F}_{\text{ext}} dt \right] + \left(\frac{dt}{\tau}\right) \times \mathbf{0} \]Expanding this and ignoring second-order terms (terms containing \(dt^2\)):
\[ \mathbf{p}(t+dt) = \mathbf{p}(t) - \frac{\mathbf{p}(t)}{\tau}dt + \mathbf{F}_{\text{ext}} dt \]Rearranging to form a derivative:
\[ \mathbf{p}(t+dt) - \mathbf{p}(t) = \mathbf{F}_{\text{ext}} dt - \frac{\mathbf{p}(t)}{\tau} dt \] \[ \frac{\mathbf{p}(t+dt) - \mathbf{p}(t)}{dt} = \mathbf{F}_{\text{ext}} - \frac{\mathbf{p}(t)}{\tau} \]Taking the limit as \(dt \to 0\), we arrive at the fundamental equation of motion in the Drude model:
2. Physical Meaning of the Equation
This differential equation has a clear physical interpretation:
- First Term (\(\mathbf{F}_{\text{ext}}\)): This represents Newton’s Second Law. The external field accelerates the electrons.
- Second Term (\(-\mathbf{p}/\tau\)): This represents a "frictional" drag force. It is not viscous drag in real space, but rather a relaxation in momentum space. The collisions continuously try to randomize the momentum, causing the net drift to decay exponentially.
3. Case Studies: Free Decay and Steady State
Case A: Free Decay (No Electric Field)
If we switch off the external field (\(\mathbf{F}_{\text{ext}} = 0\)), how long does the current last? \[ \frac{d\mathbf{p}}{dt} = -\frac{\mathbf{p}}{\tau} \]
The solution to this first-order differential equation is an exponential decay: \[ \mathbf{p}(t) = \mathbf{p}_0 e^{-t/\tau} \] This implies that without an electric field to sustain it, the directed momentum (current) vanishes within a few relaxation times (\(\tau \approx 10^{-14}\) s).
Case B: Steady State (Constant Electric Field)
In a DC circuit, the current is constant. This means the rate of change of momentum is zero (\(d\mathbf{p}/dt = 0\)). The force on an electron is \(\mathbf{F}_{\text{ext}} = -e\mathbf{E}\).
\[ 0 = -e\mathbf{E} - \frac{\mathbf{p}}{\tau} \]Solving for the steady-state momentum \(\mathbf{p}\): \[ \mathbf{p} = -e\mathbf{E}\tau \]
Since \(\mathbf{p} = m\mathbf{v}_d\) (where \(\mathbf{v}_d\) is the drift velocity), we get:
4. Deriving Ohm's Law
Current density \(\mathbf{j}\) is defined as the charge passing through a unit area per unit time. For electron density \(n\): \[ \mathbf{j} = -ne\mathbf{v}_d \]
Substituting our expression for drift velocity \(\mathbf{v}_d\): \[ \mathbf{j} = -ne \left( -\frac{e\tau}{m}\mathbf{E} \right) \] \[ \mathbf{j} = \left( \frac{ne^2\tau}{m} \right) \mathbf{E} \]
This matches the vector form of Ohm's Law, \(\mathbf{j} = \sigma \mathbf{E}\), where the conductivity \(\sigma\) is:
Consequently, the resistivity \(\rho = 1/\sigma\) is: \[ \rho = \frac{m}{ne^2\tau} \]
5. Velocity Scales: Drift vs. Thermal
A crucial insight of the Drude model is the distinction between the random thermal speed of electrons and their slow collective drift.
Thermal Velocity (\(v_{th}\))
Using the equipartition theorem from classical thermodynamics: \[ \frac{1}{2}mv_{th}^2 = \frac{3}{2}k_B T \] \[ v_{th} = \sqrt{\frac{3k_B T}{m}} \] At room temperature, \(v_{th} \approx 10^5 \text{ m/s}\). This is incredibly fast.
Drift Velocity (\(v_d\))
Under typical applied voltages, the drift velocity is roughly: \[ v_d \approx 10^{-4} \text{ m/s} \]
6. Mean Free Path and Limitations
The mean free path \(\ell\) is the average distance traveled between collisions: \[ \ell = v_{th} \tau \] Using \(\tau \approx 10^{-14}\) s, we get \(\ell \approx 10^{-9}\) m (10 Å), which roughly matches interatomic spacing.
Where Drude Fails
- Temperature Dependence: Drude predicts \(\rho \propto \sqrt{T}\) (because \(v_{th} \propto \sqrt{T}\)). Experimentally, for metals, \(\rho \propto T\).
- Electronic Heat Capacity: The model predicts a large heat capacity for electrons (\(\frac{3}{2}nk_B\)), but the actual value is roughly 100 times smaller.
- Hall Effect: It cannot explain positive Hall coefficients (holes).
These failures arise because electrons follow Fermi-Dirac statistics (quantum mechanics), not Maxwell-Boltzmann statistics. This was later corrected by the Sommerfeld model.
Summary Diagrams
1. Force Balance
Visual Description: An electron experiencing two opposing vectors. A constant vector -eE pointing left (driving force) and a velocity-dependent vector -p/τ pointing right (resistive drag). In steady state, these arrows are equal length.
2. Velocity Magnitude
Visual Description: A bar chart comparing velocities.
Bar 1: Thermal Velocity (~100,000 m/s) - Fills the chart.
Bar 2: Drift Velocity (~0.0001 m/s) - Invisible line at the bottom.
Illustrates that current is a slow drift superimposed on fast random motion.