@@ -106,15 +106,15 @@ We first consider an electron in free space, moving in a plane perpendicular to
3. The Drude model assumes that $\lambda$ is independent of temperature. How does the electrical resistivity $\rho$ depend on temperature under this assumption? Sketch $\rho(T)$.
5. Compare your sketch of $\rho(T)$ with that in the lecture notes. In what respect do they differ? Discuss possible reasons for differences.
### Exercise 3: Hall resistivity and Hall coefficient
We apply a magnetic field $\bf B$ perpendicular to a current carrying 2D sample. In this situation, the electric field $\mathbf{E}$ is related to the current density $\mathbf}J}$ by the resistivity matrix:
### Exercise 3: The Hall conductivity matrix and the Hall coefficient
We apply a magnetic field $\bf B$ perpendicular to a current carrying 2D sample. In this situation, the electric field $\mathbf{E}$ is related to the current density $\mathbf{J}$ by the resistivity matrix:
1. Sketch $\rho_{xx}$ and $\rho_{xy}$ as a function of the magnetic field $\bf B$.
2. Invert the resistivity matrix to obtain the conductivity matrix $\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\\sigma_{yx} & \sigma_{yy} \end{pmatrix} $ and express $\mathbf{J}$ as a function of $\mathbf{E}$.
3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$.
4.Define the Hall coefficient. What does the sign of the Hall coefficient signify?
2. Invert the resistivity matrix to obtain the conductivity matrix $$\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\\sigma_{yx} & \sigma_{yy} \end{pmatrix} $$ and express $\mathbf{J}$ as a function of $\mathbf{E}$.
3. Sketch $\sigma_{xx}$ and $\sigma_{xy}$ as a function of the magnetic field $\bf B$. Calculate the value of $\sigma_{xx}$ for $\rho_{xx}=0$. Discuss what is going on here.
4.Give the definition of the Hall coefficient. What does the sign of the Hall coefficient indicate?
### Exercise 4: Drude model of thermal and electrical conductivity
