Newer
Older
```python tags=["initialize"]
import matplotlib.animation as animation
from IPython.display import HTML
import numpy as np
from common import draw_classic_axes, configure_plotting
configure_plotting()
```
T. van der Sar
committed
_(based on chapter 3 of the book)_
!!! success "Expected prior knowledge"
Before the start of this lecture, you should be able to:
- Write down the force acting on an electron in electric and magnetic fields
- Solve Newton's equations of motion
- Define concepts of voltage, electrical current, and conductivity
!!! summary "Learning goals"
After this lecture you will be able to:
- Discuss the basics of Drude theory, which describes electron motion in metals.
- Use Drude theory to analyze transport of electrons through conductors in electric and magnetic fields.
- Describe concepts such as electron mobility and the Hall resistance.
Ohm's law is an empirical observation of conductors, which states that voltage is proportional to current $V=IR$.
Since we are interested in *material* properties, we would like to rewrite this into a relation that does not depend on the conductor geometry.
We achieve this by expressing the terms in Ohm's law with their microscopic equivalents.
Consider a conducting wire with cross-sectional area $A$ and length $l$.
Such a wire has resistance $R = \rho l / A$ where $\rho$ is the material-dependent resistivity.
An applied voltage $V$ in the wire creates an electric field $E = V/l$.
The resulting current $I$ in the wire is described by the current density $j \equiv I / A$.
Combining all these equations with Ohm's law, we get:
Now that we see that Ohm's law relates local quantities $E$ and $j$, let us try to understand how this relation may arise by considering motion of individual electrons conduct in metals.
In doing so we follow Drude, who applied Boltzmann's kinetic theory of gases to electrons.
We start from the following very reasonable assumptions about how electrons move:
- Electrons scatter randomly at uncorrelated times. The average time between scattering is $\tau$. Therefore, the probability of scattering in a time interval $dt$ is $dt / \tau$
- After each scattering event, the electron's momentum randomizes with a zero average $⟨\mathbf{p}⟩=0$
- Electrons are charged particles with charge $-e$, so that the Lorentz force $\mathbf{F}_L=-e\left(\mathbf{E}+\mathbf{v}×\mathbf{B}\right)$ acts on the electrons in between the scattering events
We are using a convention where $-e$ is the electron charge, so $e>0$.
The first assumption here is the least obvious: why does the time between scattering events not depend on e.g. electron velocity? There is no physical answer to this: the model is only an approximation.
The second assumption can be justified by symmetry: since we expect the electrons to scatter equally to all directions, their average velocity will be zero right after scattering.
Also note that for now we treat the electrons as classical particles, neglecting all quantum mechanical effects.
As we will see in the next lecture, quantum mechanical effects actually help to justify the first assumption.
Even under these simplistic assumptions, the trajectory of the electrons is hard to calculate.
Due to the random scattering, each trajectory is different, and this is how several example trajectories look:
# Use colors from the default color cycle
default_colors = pyplot.rcParams['axes.prop_cycle'].by_key()['color']
blue, red = default_colors[0], default_colors[3]
tau = 1 # relaxation time
gamma = .3 # dissipation strength
a = 1 # acceleration
dt = .1 # infinitesimal
v = np.zeros((2, int(T // dt), walkers), dtype=float) #
# Select random time steps and scattering angles
np.random.seed(1)
scattering_events = np.random.binomial(1, dt/tau, size=v.shape[1:])
angles = np.random.uniform(high=2*np.pi, size=scattering_events.shape) * scattering_events
rotations = np.array(
[[np.cos(angles), np.sin(angles)],
[-np.sin(angles), np.cos(angles)]]
)
for step in range(1, v.shape[1]):
v[:, step] = v[:, step-1]
v[0, step] += a * dt
v[:, step] = np.einsum(
'ijk,jk->ik',
rotations[:, :, step-1, :],
v[:, step, :]
) * (1 - gamma * scattering_events[step-1])
r = np.cumsum(v * dt, axis=1)
scattering_positions = np.copy(r)
scattering_positions[:, ~scattering_events.astype(bool)] = np.nan
scatter_pts = scattering_positions
r_min = np.min(r.reshape(2, -1), axis=1) - 1
r_max = np.max(r.reshape(2, -1), axis=1) + 1
fig = pyplot.figure(figsize=(9, 6))
ax = fig.add_subplot(1, 1, 1)
ax.axis("off")
ax.set(xlim=(r_min[0], r_max[0]), ylim=(r_min[1], r_max[1]))
trajectories = ax.plot([],[], lw=1, color=blue, alpha=0.5)[0]
scatterers = ax.scatter([], [], s=20, c=red)
def frame(i):
concatenated_lines = np.concatenate(
(r[:, :i], np.nan * np.ones((2, 1, walkers))),
axis=1
).transpose(0, 2, 1).reshape(2, -1)
trajectories.set_data(*concatenated_lines)
scatterers.set_offsets(scatter_pts[:, :i].reshape(2, -1).T)
anim = animation.FuncAnimation(fig, frame, interval=100)
### Equations of motion
Each electron with charge $-e$ and velocity $\mathbf{v}$ carries current $-e\mathbf{v}$.
Therefore if the electron density is $n$, the *average* current they carry is $-ne⟨\mathbf{v}⟩$.
The key idea is that although the motion of an individual electron is hard to calculate, the *average* motion of the electrons is much simpler.
For convenience from now on, we will omit the average brackets, and write $\mathbf{v}$ instead of $⟨\mathbf{v}⟩$.
This also applies to $F$.
We derive an equation of motion for the "average" electron in the following way:
Consider everything that happens in an (infinitesimal) time interval $dt$.
A fraction $dt/τ$ of the electrons scatters, and their average velocity becomes zero.
$$
m\mathbf{v}(t + dt) = 0
The rest of the electrons $(1 - dt/τ)$ are accelerated by the Lorentz force $F$, so their velocity becomes
m\mathbf{v}(t + dt) = m\mathbf{v}(t) + \mathbf{F}⋅dt.
To find the average velocity, we take a weighted average of these two groups of particles:
m\mathbf{v}(t+dt) &= [m\mathbf{v}(t) + F dt]\left(1 - \frac{dt}{\tau}\right) + 0⋅\frac{dt}{\tau} \\
&= [m\mathbf{v}(t) + \mathbf{F} dt] \left(1 - \frac{dt}{\tau}\right) \\
&= m\mathbf{v}(t) + dt [\mathbf{F} - m\frac{\mathbf{v(t)}}{\tau}] - \frac{\mathbf{F}}{\tau} dt^2
We now neglect the term proportional to $dt^2$ (it vanishes when $dt → 0$).
Finally, we recognize that $\left[\mathbf{v}(t+dt) - \mathbf{v}(t)\right]/dt = d\mathbf{v}(t)/dt$, which results in
m\frac{d\mathbf{v}}{dt} = -m\frac{\mathbf{v}}{τ} + \mathbf{F}.
Observe that the first term on the right-hand side has the same form as a drag force: it always decelerates the electrons.
This equation equation of motion of the average electron is our main result: now we only need to apply it.
Let us first consider the case without magnetic fields, $\mathbf{B} = 0$, and a constant electric field $\mathbf{E}$.
After a long enough time, we expect the average electron velocity to become constant, $d\mathbf{v}/dt = 0$, and we immediately get a steady-state solution:
\mathbf{v}=\frac{-eτ}{m}\mathbf{E}=-|μ|\mathbf{E}.
In the above equation, we define the *mobility* $μ\equiv |e|τ/m$ to be a ratio between the electron *drift velocity* and the electric field.
Electron mobility describes the response of the electron to the electric field: for a given field, a higher $\mu$ means the electrons will (on average) move faster.
We substitute the steady-state velocity into the definition of current density:
\mathbf{j}=-en\mathbf{v}=\frac{n e^2τ}{m}\mathbf{E}=\sigma\mathbf{E},\quad \sigma=\frac{ne^2τ}{m}=ne\mu,
where $\sigma$ is the conductivity, so that $ρ=\frac{1}{\sigma}$.
#### Origins of scattering
We now look into the origins of electron scattering.
Drude model assumes that electrons scatter from all the atoms in a solid.
That, however, is not true (the next lecture explains why).
Instead, scattering is the result of deviations from a perfect crystal:
- Phonons: $τ_\mathrm{ph}(T)$ ($τ_\mathrm{ph}\rightarrow\infty$ as $T\rightarrow 0$)
- Impurities/vacancies or other crystalline defects: $τ_0$
Because the different scattering mechanisms are independent, the scattering rates $1/τ$ due to different mechanisms add up:
\frac{1}{τ}=\frac{1}{τ_\mathrm{ph}(T)}+\frac{1}{τ_0}\ \Rightarrow\ ρ=\frac{1}{\sigma}=\frac{m}{ne^2}\left( \frac{1}{τ_\mathrm{ph}(T)}+\frac{1}{τ_0} \right)\equiv ρ_\mathrm{ph}(T)+ρ_0
This approximate empirical observation is known as *Matthiessen's Rule* (1864).
It allows to separate the impurity scattering rate from the electron-phonon scattering rate by comparing temperature dependences of a clean and dirty metal.
The difference between two resistances is due to the impurity scattering.
Let us apply Drude model to compute a typical drift velocity of electrons in a metal.
For example, consider a one meter long copper wire with a 1V voltage applied to it, $E = 1$ V/m.
Taking a scattering time of $τ∼25$ fs (valid in Cu at $T=300$ K), we obtain $v=\mu E=\frac{eτ}{m}E=2.5$ mm/s.
The electrons move unexpectedly slow—millimeters per second!
??? question "How come you do not have to wait for minutes until the light in your room turns on after you flick a switch?"
How fast the light turns on is controlled by the speed at which the electric field develops inside the wire.
Electric field propagates much faster than electrons—in vacuum it moves at the speed of light after all.
Conduction properties become much more interesting once we turn on both and electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$.
Let us consider a conductive wire with current $\mathbf{j}$ flowing along the x-direction (notice how we are specifying the direction of the current and not of the electric field).
A magnetic field $\mathbf{B}$ acts on the wire along the z-direction, like shown in the figure:
Let us take a look at the equations of motion again:
m\frac{d\mathbf{v}}{dt} = -m\frac{\mathbf{v}}{τ} - e(\mathbf{E} + \mathbf{v}\times\mathbf{B})
Once again, we consider the steady state $d\mathbf{v}/dt = 0$.
After substituting $\mathbf{v} = -\mathbf{j}/ne$, we arrive to
\mathbf{E}=\frac{m}{ne^2τ}\mathbf{j} + \frac{1}{ne}\mathbf{j}\times\mathbf{B}.
The first term is the same as before and describes the electric field parallel to the current, while the second is the electric field **perpendicular** to the current flow.
In other words, if we send a current through a sample and apply a magnetic field, a voltage develops in the direction perpendicular to the current—this is called *Hall effect*, the voltage is called *Hall voltage*, and the proportionality coefficient $B/ne$ the *Hall resistivity*.
Because of the Lorentz force, the electrons are deflected in a direction perpendicular to $\mathbf{B}$ and $\mathbf{j}$.
The deflection creates a charge imbalance, which in turn creates the electric field $\mathbf{E}_\mathrm{H}$ compensating the Lorentz force.
The above relation between the electric field and the current is linear, which allows us to write it in matrix form
with $a, b ∈ \{x, y, z\}$, and $ρ$ the *resistivity matrix*.
Its diagonal elements are $ρ_{xx}=ρ_{yy}=ρ_{zz}=m/ne^2τ$—the same as without magnetic field.
The only nonzero off-diagonal elements when $\mathbf{B}$ points in the $z$-direction are
ρ_{xy}=-ρ_{yx}=\frac{B}{ne}\equiv -R_H B,
where $R_H=-1/ne$ is the *Hall coefficient*.
So by measuring the Hall voltage and knowing the electron charge, we can determine the density of free electrons in a material.
While most materials have $R_H < 0$, interestingly some materials are found to have $R_H > 0$. This would imply that the charge of the carriers is positive. We will see later in the course how to interpret this.
1. Drude theory is a microscopic justification of the Ohm's law.
2. We can calculate the resitivity from the characteristic scattering time $\tau$.
3. The Lorentz force leads to the Hall voltage that is perpendicular to the direction of electric current pushed through a material and the magnetic field.
### Warm-up questions
1. How does the resistance of a purely 2D material depend on its size?
2. Check that the units of mobility and the Hall coefficient are correct.
(As you should always do!)
3. Explain why the scattering times due to different types of scattering events add up in a reciprocal way.
### Exercise 1: Extracting quantities from basic Hall measurements
We apply a magnetic field $\bf B$ along the $z$-direction to a planar (two-dimensional) sample that sits in the $xy$ plane. The sample has width $W$ in the $y$-direction, length $L$ in the $x$-direction and we apply a current $I$ along the $x$-direction.
#### Question 1.
Suppose we measure a Hall voltage $V_H$. Express the Hall resistance $R_{xy} = V_H/I$ as a function of magnetic field. Does $R_{xy}$ depend on the geometry of the sample? Also express $R_{xy}$ in terms of the Hall coefficient $R_H$.
??? question "What is the relation between the electric field and the electric potential?"
$V_b - V_a = -\int_{\Gamma} \mathbf{E} \cdot d\mathbf{\ell}$ if $\Gamma$ is a path from $a$ to $b$.
#### Question 2.
Assuming we control the magnetic field $\mathbf{B}$, what quantity can we extract from a measurement of the Hall resistance? Would a large or a small magnetic field give a Hall voltage that is easier to measure?
#### Question 3.
Express the longitudinal resistance $R=V/I$, where $V$ is the voltage difference over the sample along the $x$ direction, in terms of the longitudinal resistivity $ρ_{xx}$. Suppose we extracted $n$ from a measurement of the Hall resistance, what quantity can we extract from a measurement of the longitudinal resistance? Does the result depend on the geometry of the sample?
### Exercise 2: Motion of an electron in a magnetic and an electric field.
Consider an electron in free space experiencing a magnetic field $\mathbf{B}$ along the $z$-direction.
%Assume that the electron starts at the origin with a velocity $v_0$ along the $x$-direction.
#### Question 1.
Write down the Newton's equation of motion for the electron, compute $\frac{d\mathbf{v}}{{dt}}$.
#### Question 2.
What is the shape of the motion of the electron? Calculate the characteristic frequency and time-period $T_c$ of this motion for $B=1$ Tesla.
#### Question 3.
Now we accelerate the electron by adding an electric field $\mathbf{E} = E \hat{x}$. Adjust the differential equation for $\frac{d\mathbf{v}}{{dt}}$ found in (1) to include $\mathbf{E}$. Sketch the motion of the electron.
#### Question 4.
Consider now an ensemble of electrons in a metal. Include the Drude scattering time $τ$ into the differential equation for the velocity you formulated in 3.
Note that the differential equation now describes the *average* velocity of the electrons in the ensemble.
### Exercise 3: Temperature dependence of resistance in the Drude model
We consider copper, which has a density of 8960 kg/m$^3$, an atomic weight of 63.55 g/mol, and a room-temperature resistivity of $ρ=1.68\cdot 10^{-8}$ $\Omega$m. Each copper atom provides one free electron.
#### Question 1.
Calculate the Drude scattering time $τ$ at room temperature.
#### Question 2.
Assuming that electrons move with the thermal velocity $\langle v \rangle = \sqrt{\frac{8k_BT}{\pi m}}$, calculate the electron mean free path $\lambda$, defined as the average distance an electron travels in between scattering events.
#### Question 3.
The Drude model assumes that $\lambda$ is independent of temperature. How does the electrical resistivity $ρ$ depend on temperature under this assumption? Sketch $ρ(T)$.
#### Question 5.
Compare your sketch of $ρ(T)$ with that in the lecture notes. In what respect do they differ? Discuss possible reasons for differences.
### Exercise 4: The Hall conductivity matrix and the Hall coefficient
We apply a magnetic field $\bf B$ along the $z$-direction to a current carrying 2D sample in the xy plane. In this situation, the electric field $\mathbf{E}$ is related to the current density $\mathbf{j}$ by the resistivity matrix:
$$
\mathbf{E} = \begin{pmatrix} ρ_{xx} & ρ_{xy} \\ ρ_{yx} & ρ_{yy} \end{pmatrix} \mathbf{j}
$$
#### Question 1.
Sketch the expressions for $ρ_{xx}$ and $ρ_{xy}$ derived in the lecture notes as a function of the magnetic field $\bf{B}$.
#### Question 2.
Invert the resistivity matrix to obtain the conductivity matrix,
$$
\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{yx} & \sigma_{yy} \end{pmatrix}
$$
allowing you to express $\mathbf{j}$ as a function of $\mathbf{E}$.
#### Question 3.
Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$.
#### Question 4.
Give the definition of the Hall coefficient. What does the sign of the Hall coefficient indicate?