Electrons in metals I

Exercises

Preliminary provocations

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.

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 $b$.

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 $R_H$.

   Hall voltage is measured across the sample width. Hence,

   $$V_H = -\int_{0}^{W} E_ydy$$

   where $E_y = -v_xB$.

   $R_{xy}$ = $-\frac{B}{ne}$, so it does not depend on the sample geometry.

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?

   If hall resistance and magnetic field are known, the charge density is calculated from $R_{xy} = -\frac{B}{ne}$. As $V_x = -\frac{I_x}{ne}B$, a stronger field makes Hall voltages easier to measure.

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?

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

   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.

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.

1. Write down the Newton's equation of motion for the electron, compute $\frac{d\mathbf{v}}{{dt}}$.

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

   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

   $$\frac{dv_x}{dt} = -\frac{e v_y B_z}{m}$$

   $$\frac{dv_y}{dt} = \frac{e v_x B_z}{m}$$

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.

   We can compute $v_x(t)$ and $v_y(t)$ by solving the differential equations in 1.

   From

   $$v_x'' = -\frac{e^2B_z^2}{m^2}v_x$$

   and the initial conditions, we find $v_x(t) = v_0 \cos(\omega_c t)$ with $\omega_c=eB_z/m$. From this we can derive $v_y(t)=v_0\sin(\omega_c t)$.

   We now calculate the particle position using $x(t)=x(0) + \int_0^t v_x(t')dt'$ (and similar for $y(t)$). From this we can find a relation between the $x$- and $y$-coordinates of the particle

   $$(x(t) - x_0)^2 + (y(t) - y_0)^2 = \frac{v_0^2}{\omega_c^2}.$$

   This equation describes a circular motion around the point $x_0=x(0), y_0=y(0)+v_0/\omega$, where the characteristic frequency $\omega_c$ is called the cyclotron frequency. Intuition: $\frac{mv^2}{r} = evB$ (centripetal force = Lorentz force due to magnetic field).

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.

   Due to the applied electric field $\bf E$ in the $x$-direction, the equations of motion acquire an extra term:

   $$m v_x' = -e(E_x + v_yB_z).$$

   Differentiating w.r.t. time leads to the same 2nd-order D.E. for $v_x$ as above. However, for $v_y$ we get

   $$v_y'' = -\omega_c^2(v_d+v_y),$$

   where we defined $v_d=\frac{E_x}{B_z}$. The general solutions are

   $$v_y(t) = c_1\sin(\omega_c t)+ c_2\cos(\omega_c t) -v_d \\ v_x(t) = c_3\sin(\omega_c t)+ c_4\cos(\omega_c t).$$

   Using the initial conditions $v_x(0)=v_0$ and $v_y(0)=0$ and the 1st order D.E. above, we can show

   $$v_y(t) = v_0\sin(\omega_c t)+ v_d\cos(\omega_c t) -v_d \\ v_x(t) = v_d\sin(\omega_c t)+ v_0\cos(\omega_c t).$$

   By integrating the expressions for the velocity we find:

   $$(x(t)-x_0)^2 + (y(t) - y_0 + v_d t))^2 = \frac{v_0^2}{\omega_c^2}.$$

   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

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.

1. Calculate the Drude scattering time $τ$ at room temperature.

   Find electron density from

   $$n_e = \frac{Z n N_A}{W}$$

   where Z is valence of copper atom, n is density, $N_A$ is Avogadro constant and W is atomic weight. Use $\rho = 1/\sigma$ from the lecture notes to calculate scattering time.

   $$\sigma = \frac{n e^2 \tau}{m}$$

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.

   $\lambda = \langle v \rangle\tau$

3. The Drude model assumes that $\lambda$ is independent of temperature. How does the electrical resistivity $ρ$ depend on temperature under this assumption? Sketch $ρ(T)$.

   Scattering time $\tau \propto \frac{1}{\sqrt{T}}$; $\rho \propto \sqrt{T}$

4. The empirical observation known as Matthiessen's Rule states that $ρ(T) \propto T$. Discuss this result with reference to your answer above.

   In general, $\rho \propto T$ as the phonons in the system scales linearly with T (remember high temperature limit of Bose-Einstein factor becomes $\frac{kT}{\hbar\omega}$ leading to $\rho \propto T$). Inability to explain this linear dependence is a failure of the Drude model.

Exercise 4: The Hall conductivity matrix and the Hall coefficient

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:

$$\mathbf{E} = \begin{pmatrix} ρ_{xx} & ρ_{xy} \\ ρ_{yx} & ρ_{yy} \end{pmatrix} \mathbf{j}$$

1. Sketch the expressions for $ρ_{xx}$ and $ρ_{xy}$ derived in the lecture notes as a function of the magnetic field $\bf{B}$.

   $\rho_{xx}$ is independent of B and $\rho_{xy} \propto B$

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 $\mathbf{E}$.

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

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

   This describes a Lorentzian

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Last update: October 30, 2023

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