Solutions for Drude model exercises¶

Exercise 1: Extracting quantities from basic Hall measurements¶

Hall voltage is measured across the sample width. Hence,

$V_{H} = - \int_{0}^{W} E_{y} d y$

where $E_{y} = - v_{x} B$ .

$R_{x y}$ = $- \frac{B}{n e}$ , so it does not depend on the sample geometry.
If hall resistance and magnetic field are known, the charge density is calculated from $R_{x y} = - \frac{B}{n e}$ .

As $V_{x} = - \frac{I_{x}}{n e} B$ , a stronger field makes Hall voltages easier to measure.

3.

$$
R_{xx} = \frac{\rho_{xx}L}{W}
$$

where $\rho_{xx} = \frac{m_e}{ne^2\tau}$. Therefore, scattering time ($\tau$) is known and $R_{xx}$ depend upon the sample geometry.

Exercise 2: Motion of an electron in a magnetic and an electric field¶

1.

$$
m\frac{d\bf v}{dt} = -e(\bf v \times \bf B)
$$

Magnetic field affects only the velocities along x and y, i.e., $v_x(t)$ and $v_y(t)$ as they are perpendicular to it. Therefore, the equations of motion for the electron are

$$
\frac{dv_x}{dt} = -\frac{ev_yB_z}{m}
$$

$$
\frac{dv_y}{dt} = \frac{ev_xB_z}{m}
$$

We can compute $v_{x} (t)$ and $v_{y} (t)$ by solving the differential equations in 1.

From

$v_{x}^{″} = - \frac{e^{2} B_{z}^{2}}{m^{2}} v_{x}$

and the initial conditions, we find $v_{x} (t) = v_{0} \cos (ω_{c} t)$ with $ω_{c} = e B_{z} / m$ . From this we can derive $v_{y} (t) = v_{0} \sin (ω_{c} t)$ .

We now calculate the particle position using $x (t) = x (0) + \int_{0}^{t} v_{x} (t^{'}) d t^{'}$ (and similar for $y (t)$ ). From this we can find a relation between the $x$ - and $y$ -coordinates of the particle

$(x (t) - x_{0})^{2} + (y (t) - y_{0})^{2} = \frac{v_{0}^{2}}{ω_{c}^{2}} .$

This equation describes a circular motion around the point $x_{0} = x (0), y_{0} = y (0) + v_{0} / ω$ , where the characteristic frequency $ω_{c}$ is called the cyclotron frequency. Intuition: $\frac{m v^{2}}{r} = e v B$ (centripetal force = Lorentz force due to magnetic field).
Due to the applied electric field $E$ in the $x$ -direction, the equations of motion acquire an extra term:

$m v_{x}^{'} = - e (E_{x} + v_{y} B_{z}) .$

Differentiating w.r.t. time leads to the same 2nd-order D.E. for $v_{x}$ as above. However, for $v_{y}$ we get

$v_{y}^{″} = - ω_{c}^{2} (v_{d} + v_{y}),$

where we defined $v_{d} = \frac{E_{x}}{B_{z}}$ . The general solutions are

$v_{y} (t) = c_{1} \sin (ω_{c} t) + c_{2} \cos (ω_{c} t) - v_{d} v_{x} (t) = c_{3} \sin (ω_{c} t) + c_{4} \cos (ω_{c} t) .$

Using the initial conditions $v_{x} (0) = v_{0}$ and $v_{y} (0) = 0$ and the 1st order D.E. above, we can show

$v_{y} (t) = v_{0} \sin (ω_{c} t) + v_{d} \cos (ω_{c} t) - v_{d} v_{x} (t) = v_{d} \sin (ω_{c} t) + v_{0} \cos (ω_{c} t) .$

By integrating the expressions for the velocity we find:

$(x (t) - x_{0})^{2} + (y (t) - y_{0} + v_{d} t))^{2} = \frac{v_{0}^{2}}{ω_{c}^{2}} .$

This represents a cycloid: a circular motion around a point that moves in the $y$ -direction with velocity $v_{d} = \frac{E_{x}}{B_{z}}$ .

Exercise 3: Temperature dependence of resistance in the Drude model¶

Find electron density from $n_e = \frac{ZnN_A}{W} $

where Z is valence of copper atom, n is density, $N_{A}$ is Avogadro constant and W is atomic weight. Use $ρ$ from the lecture notes to calculate scattering time.
$λ = ⟨ v ⟩ τ$
Scattering time $τ \propto \frac{1}{\sqrt{T}}$ ; $ρ \propto \sqrt{T}$
In general, $ρ \propto T$ as the phonons in the system scales linearly with T (remember high temperature limit of Bose-Einstein factor becomes $\frac{k T}{ℏ ω}$ leading to $ρ \propto T$ ). Inability to explain this linear dependence is a failure of the Drude model.

Exercise 4: The Hall conductivity matrix and the Hall coefficient¶

$ρ_{x x}$ is independent of B and $ρ_{x y} \propto B$
4 (Refer to the lecture notes).

3.

$$
\sigma_{xx} = \frac{\rho_{xx}}{\rho_{xx}^2 + \rho_{xy}^2}
$$

$$
\sigma_{xy} = \frac{-\rho_{yx}}{\rho_{xx}^2 + \rho_{xy}^2}
$$

This describes a [Lorentzian](https://en.wikipedia.org/wiki/Spectral_line_shape#Lorentzian).