Drude Model

This simulation models 2D electron trajectories under the influence of electric and magnetic fields.

A particle with charge $q$ in an electric field, $\mathbf{E}$, and magnetic field, $\mathbf{B}$, will experience the Lorenz force, $\mathbf{F} = q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$, where $v$ is the velocity of the particle. For an electron, the charge is $e = -1.602*10^{-19} C$ and the mass is $m_e = 9.109*10^{-31} kg$.

If we consider first the case of only a magnetic field, we can see that the force acting a particle is

$\mathbf{F} = e\mathbf{v}\times \mathbf{B}$

which acts perpindicular to the motion of the particle and does no work. This causes the particle to have circular motion and, balancing the Lorenz force and the centrifugal force $\mathbf{F_{lor}} = \mathbf{F_{cent}}$ we find

$mv^2/r = evB$
$mv = erB$
$mr\omega = erB$ (where $v = r\omega$)
$\omega_c = eB/m^{*}$

This frequency $\omega_c$ is known as the cyclotron frequency.

The Drude model makes the assumption that a metal or semiconductor serves as a host for free electrons, and their net motion gives electrical current. We can introduce $\tau'$ as the mean scattering time, the average time an electron travels between collisions in this medium. These collisions randomize the particle's momentum, a process which is known as diffuse scattering. Between one scattering event and the next, the electron is influenced by the Lorentz force of applied electric and magnetic fields. For example, consider a uniform electric field applied in the $\hat{x}$ direction. The force on electrons within the field will be

$F_x\hat{x} = eE_x\hat{x}$

Which can be used to solve for acceleration $a$

$a = F_x/m = eE_x/m$

At some time $t=\tau$ after a collision ($t=0$) before the next scattering event the electron will reach a velocity of

$v = eE\tau/m^{*}$

and an average velocity of electrons, or drift velocity,

$v_D = eE\tau/{2m^{*}}$

Now let us examine what happens when the electron is under the influence of both electric and magnetic fields. For example let us take examine the case of an electric field in the $\hat{x}$ direction and a magnetic field in the $\hat{z}$ direction. The Lorenz force law gives

$F_x = eE_x + e v_y B \qquad F_y = -e v_x B$

Combined with $\mathbf{F} = m\mathbf{a}$, this produces a system of two ordinary differential equations

$m\frac{d^2x}{dt^2} = qE_x + qB\frac{dy}{dt} \qquad m\frac{d^2y}{dt^2} = qB\frac{dx}{dt}$

which can be solved to show that the electron will travel in unifom circular motion combined with a uniform drift velocity $v_d$ in the $\hat{y}$ direction where

$v_d = E/B$

Drude Model Simulator

Refresh page to reset
25
5
10
0
0
0

Questions

To clearly see effects, use low $E_F$ and the Step button and Zoom functionality of the simulator.

1. After each scattering event, what happens to the speed of each electron when $E = 0$?
Hint: Track a single electron in both real-space and velocity-space graphs. Use cursor tooltip to trace correct particle between plots.

2. Following a single electron in real space, approximate its average distance traveled for $\tau = 10, 20$ $ps$.
Hint: This is easier to see for faster moving electrons. They will also come closer to the mean free path, which is calculated using the Fermi velocity.

3. For $E_x = 10^5$ $V/m$ and $B_z = -1$ $T$, what is the drift velocity?

4. Set $B$ to $-1$ $T$. Determine the approximate cyclotron resonance frequency.
Note: The time of a single step is $0.5$ $ps$.