diff --git a/src/3_drude_model.md b/src/3_drude_model.md index 05644aad3504531d83eaa6fc4e4520e535b8bac2..9bb975bfd13a22643876ce4aed34debc528c4e09 100644 --- a/src/3_drude_model.md +++ b/src/3_drude_model.md @@ -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: $$\mathbf{E} = \begin{pmatrix} \rho_{xx} & \rho_{xy} \\ \rho_{yx} & \rho_{yy} \end{pmatrix} \mathbf{J}$$ 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