Hall Effect and Magnetoresistance in Semiconductors

Hello! Welcome to our next lesson.

In our previous sessions, we've built a solid understanding of transport phenomena using the Boltzmann Transport Equation (BTE), considering driving forces from electric fields and temperature gradients. This led us to concepts like electrical conductivity, thermal conductivity, and the Seebeck effect.

Today, we'll complete our semi-classical transport picture by incorporating the final term in the electron's equation of motion: the Lorentz force, $-e(\mathbf{v}\times \mathbf{B})$. This will allow us to explore how a magnetic field influences charge transport. We will specifically aim to compute the Hall coefficient and magnetoresistance for single and multi-band systems. These effects are not just theoretically interesting; they form the basis of one of the most powerful and widely used experimental techniques for characterizing charge carriers in materials.

1. Hall Effect and Magnetoresistance: The Single-Band Model

Let's begin by adding the magnetic field to the linearized BTE. Within the relaxation-time approximation, the equation for the deviation from equilibrium, $\delta f$, becomes:

$$-e\mathbf{v}\cdot \mathbf{E}\frac{\partial f^0}{\partial ϵ}-\frac{e}{\hslash c}(\mathbf{v}\times \mathbf{B})\cdot \frac{\partial \delta f}{\partial \mathbf{k}}=-\frac{\delta f}{\tau }$$

The new term, involving $\mathbf{B}$, couples the different components of the electron's momentum, which in turn mixes the components of the current response. This leads to a conductivity (and resistivity) tensor with non-zero off-diagonal elements.

The following reading from the "Boltzmann Transport" notes we've been using provides a formal derivation of the resistivity tensor ${\rho }_{\alpha \beta }$ in the presence of a magnetic field.

As you saw, for a simple isotropic band with effective mass $m^{\ast }$, carrier density $n$, and a magnetic field $\mathbf{B}=B\hat{z}$, the DC ($\omega =0$) resistivity tensor takes the form:

$$\rho =\left(\begin{array}{ccc}{\rho }_{xx}& {\rho }_{xy}& 0\\ {\rho }_{yx}& {\rho }_{yy}& 0\\ 0& 0& {\rho }_{zz}\end{array}\right)=\left(\begin{array}{ccc}{m}^{\ast }/(n{e}^2\tau )& B/(ne)& 0\\ -B/(ne)& m^{\ast }/(n{e}^2\tau )& 0\\ 0& 0& m^{\ast }/(n{e}^2\tau )\end{array}\right)$$

Let's unpack the two key phenomena this reveals:

The Hall Effect

The off-diagonal components, ${\rho }_{xy}=-{\rho }_{yx}$, describe the Hall effect. A current flowing in the x-direction ($j_x$) generates an electric field in the y-direction ($E_y={\rho }_{yx}{j}_x$). This transverse field is the Hall field. The Hall coefficient, $R_H$, is defined to normalize this effect by the magnetic field:

$$R_H=\frac{{\rho }_{yx}}{B}=-\frac{1}{ne}$$

(Note: For positive charge carriers (holes), the charge $e$ becomes $+e$, so $R_H=1/ne$.)

This is a remarkable result. The Hall coefficient gives a direct measurement of two fundamental properties of the charge carriers:

1. Carrier Sign: The sign of $R_H$ tells us whether the dominant charge carriers are electrons (negative $R_H$) or holes (positive $R_H$).
2. Carrier Density: The magnitude of $R_H$ gives the volume density of the carriers, $n$.

Magnetoresistance

The diagonal components, ${\rho }_{xx}$ and ${\rho }_{yy}$, represent the longitudinal resistivity. Magnetoresistance is the change in this resistivity due to the magnetic field:

$$\Delta {\rho }_{xx}={\rho }_{xx}(B)-{\rho }_{xx}(0)$$

From our derived tensor, we find a crucial result for this simple model:

$${\rho }_{xx}(B)=\frac{m^{\ast }}{n{e}^2\tau }={\rho }_{xx}(0)⟹\Delta {\rho }_{xx}=0$$

The single-band model with an energy-independent scattering time predicts zero magnetoresistance. This is because the Lorentz force, which acts perpendicular to velocity, does no work on the charge carriers and, in this simple case, does not change the average time between collisions that impedes their forward motion.

However, experimentally, most metals and semiconductors do exhibit non-zero magnetoresistance. This discrepancy points to the limitations of our simple model and motivates the next step.

2. The Two-Band Model

The most common reason for non-zero magnetoresistance in many materials is the simultaneous presence of more than one type of charge carrier. Let's consider the simplest multi-band case: a two-band model, with a population of electrons and a population of holes.

The key insight is that the total conductivity is the sum of the conductivities of the individual carrier populations, as they conduct in parallel:

$${\sigma }_{\text{total}}={\sigma }_{\text{electron}}+{\sigma }_{\text{hole}}$$

The total resistivity is then the inverse of this total conductivity tensor:

$${\rho }_{\text{total}}=({\sigma }_{\text{electron}}+{\sigma }_{\text{hole}}{)}^{-1}$$

It is this matrix inversion that couples the dynamics of the two carrier types and gives rise to a much richer behavior, including a non-zero magnetoresistance.

As the reading shows, the total longitudinal resistivity ${\rho }_{xx}$ in the two-band model is no longer independent of the magnetic field. In the low-field DC limit, it takes the form:

$${\rho }_{xx}(B)\approx {\rho }_{xx}(0)+C\cdot B^2$$

where $C$ is a positive constant that depends on the densities and mobilities of both electrons and holes. This gives a positive, quadratic magnetoresistance at low fields, which is commonly observed in experiments.

Furthermore, the Hall coefficient is no longer a simple measure of the total carrier density. For low fields, it becomes a weighted average:

$$R_H\approx \frac{n_h{\mu }_h^2-n_e{\mu }_e^2}{e(n_h{\mu }_h+n_e{\mu }_e{)}^2}$$

where ${\mu }_e=e{\tau }_e/m_e^{\ast }$ and ${\mu }_h=e{\tau }_h/m_h^{\ast }$ are the electron and hole mobilities. This shows that the measured $R_H$ is highly sensitive to the carrier mobilities. A material with equal numbers of electrons and holes ($n_e=n_h$) will show a negative Hall coefficient if electrons are more mobile (${\mu }_e>{\mu }_h$), and a positive one if holes are more mobile.

3. Application: Analyzing Experimental Data

Your interest in experimental work makes this next part particularly relevant. The two-band model is not just a theoretical construct; it is a workhorse for analyzing real transport data. By measuring ${\rho }_{xx}(B)$ and ${\rho }_{xy}(B)$ simultaneously, we can often untangle the contributions of different carrier types.

The following resource, a bachelor's thesis, provides a clear, practical guide to how this analysis is performed.

The strategy outlined in the thesis is powerful:

1. Measure ${\rho }_{xx}$ and ${\rho }_{xy}$ as a function of a varying magnetic field $H$.
2. Fit the low-field data to extract key parameters:
   • The zero-field resistivity, ${\rho }_{xx}(0)$.
   • The low-field Hall coefficient, $R_H=\frac{d{\rho }_{xy}}{dH}{|}_{H=0}$.
   • The curvature of the magnetoresistance, $S=\frac{d{\rho }_{xx}}{d(H^2)}{|}_{H=0}$.
3. These three experimental numbers can then be used to solve for the underlying physical parameters of the two-band model, such as the carrier densities $n_e,n_h$ and mobilities ${\mu }_e,{\mu }_h$.

The image below shows a beautiful example of this process applied to a real semiconductor device.

This direct connection between the theoretical model and the extraction of microscopic parameters from macroscopic measurements is a cornerstone of experimental condensed matter physics. Given your background in Python and data analysis, you could implement these fitting routines yourself.

Test your understanding!

In a two-band model, the low-field Hall coefficient is given by $R_H\approx \frac{n_h{\mu }_h^2-n_e{\mu }_e^2}{e(n_h{\mu }_h+n_e{\mu }_e{)}^2}$. Consider an undoped semiconductor where temperature is high enough that intrinsic carriers dominate, so the electron and hole densities are nearly equal: $n_e\approx n_h$. However, due to the band structure, the electron mobility is significantly higher than the hole mobility (${\mu }_e\gg {\mu }_h$). What is the sign of the Hall coefficient you would expect to measure?

Show answer

Even though the carrier densities are equal, the Hall coefficient is weighted by the square of the mobilities. Since ${\mu }_e\gg {\mu }_h$, the term $-n_e{\mu }_e^2$ will dominate the numerator. Therefore, the Hall coefficient $R_H$ will be negative. This is a common and important situation where a material with equal numbers of electrons and holes (a "compensated semimetal" or intrinsic semiconductor) can appear to be n-type in a Hall measurement simply because its electrons are more mobile.

Conclusion

We have now seen how the inclusion of a magnetic field in the Boltzmann transport equation gives rise to the Hall effect and magnetoresistance.

Key Takeaways:

• The Hall effect describes the appearance of a transverse electric field in a current-carrying conductor in a magnetic field. The Hall coefficient $R_H$ measures the carrier density and sign.
• In a simple single-band model, $R_H=\pm 1/ne$ and the magnetoresistance is zero.
• A two-band model (e.g., electrons and holes) is necessary to explain the positive, quadratic magnetoresistance commonly observed in real materials.
• In a multi-band system, the measured Hall coefficient is a weighted average of all carrier types, depending on both their densities and mobilities.
• Simultaneous measurement and fitting of ${\rho }_{xx}(B)$ and ${\rho }_{xy}(B)$ is a powerful experimental method to determine the properties of individual carrier populations.

Preview of Next Lesson:
This lesson concludes our module on semi-classical electron transport. We have treated scattering phenomenologically through the relaxation time $\tau$. A major source of scattering in real crystals comes from the vibrations of the ionic lattice. In the next module, we will quantize these lattice vibrations, which will lead us to the concept of the phonon. Understanding phonons is the first step toward building a microscopic theory of electron-phonon scattering and calculating $\tau$ from first principles.