@@ -29,7 +29,7 @@ _(based on chapter 3 of the book)_
## The starting point
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 material geometry.
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.
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@@ -39,21 +39,22 @@ Combining all these equations with Ohm's law, we get:
$$
V = I ρ \frac{l}{A} ⇒ E = ρ j,
$$
Even though we only showed it for a specific case, this equation is generally valid for any conductor.
In order to understand how electrons conduct in metals, Drude applied Boltzmann's kinetic theory of gases to electrons.
Three assumptions for the motion of electrons were made:
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 $\tau$. The probability of scattering in a time interval $dt$ is $dt / \tau$
- 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 $⟨\mathbf{p}⟩=0$
- The Lorentz force $\mathbf{F}_L=-e\left(\mathbf{E}+\mathbf{v}×\mathbf{B}\right)$ acts on the electrons in between the scattering events
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 we treat the electrons as classical particles, neglecting all quantum mechanical effects. Later lectures will address this.
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 motion of the electrons is hard to calculate.
This is what electron motion looks like under these assumptions:
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:
```python
%matplotlibinline
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@@ -169,7 +170,7 @@ m\frac{d\mathbf{v}}{dt} = -m\frac{\mathbf{v}}{τ} + 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 is the main result: now we only need to apply it.
This equation equation of motion of the average electron is our main result: now we only need to apply it.
### Consequences of the Drude model
Let us first consider the case without magnetic fields, $\mathbf{B} = 0$, and a constant electric field $\mathbf{E}$.
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@@ -189,13 +190,13 @@ where $\sigma$ is the conductivity, so that $ρ=\frac{1}{\sigma}$.
We now look into the origins of electron scattering.
Drude naively assumed that electrons scatter from all the atoms in a solid.
However, that is not strictly true.
However, that is not true.
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: $τ_0$
The scattering rates $1/τ$ due to different mechanisms add up in the following way:
Because the different scattering mechanisms are independent, the scattering rates $1/τ$ due to different mechanisms add up:
@@ -206,40 +207,41 @@ This explains the empirical *Matthiessen's Rule* (1864).
Here the solid is $ρ(T)$ of a pure crystal, and the dashed of an impure one.
It is interesting to consider an electron's average velocity magnitude.
For example, we consider a meter long copper wire with a 1V voltage drop, $E = 1$ V/m, and a scattering time of $τ∼25$ fs (Cu, $T=300$ K).
$⇒ v=\mu E=\frac{eτ}{m}E=2.5$ mm/s.
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!
The electrons move unexpectedly slow—millimeters per second!
The following question arises: how come you don't wait for hours until the light turns on after you flick a switch?
The answer lies in the meaning of a "steady state".
The moment we hit the switch, the voltage jumps inside the wire, and the current needs time to settle.
We call this the *transient state*.
In contrast, the Drude model assumes that the system had sufficient time to settle in order to reach a steady state.
A proper analysis is outside the scope of this course and relies on a quasi-magnetostatic approximation to Maxwell's equations to find the evolution of the system from the transient state to the steady state.
??? 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.
### Hall effect
We now consider a situation with finite a magnetic $\mathbf{B}$ and electric field $\mathbf{E}$.
Unlike last time, we choose to fix the steady state current density $\mathbf{j}$ and use it to derive the electric field $\mathbf{E}$.
Let us consider a conductive wire with current $\mathbf{j}$ flowing along the x-direction.
A magnetic field $\mathbf{B}$ acts on the wire along the z-direction.
Because of the Lorentz force, the electrons are deflected in a direction perpendicular to $\mathbf{B}$ and $\mathbf{j}$.
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:
The deflection creates a charge imbalance which leads to the electric field $\mathbf{E}_\mathrm{H}$.
The corresponding voltage drop in the direction perpendicular to $\mathbf{B}$ and $\mathbf{j}$ is called the _Hall voltage_.
Let us take a look at the equations of motion again:
The first term is the same as before, while the second is the electric field **perpendicular** to the current flow.
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 electric field relation is linear in $\mathbf{j}$ which allows us to write it in matrix form
