changed exercise 3 to derive resistivity/conductivity tensor

src/3_drude_model.md

@@ -106,13 +106,23 @@ 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: Classical Hall effect
+### Exercise 3: Hall resistivity and Hall coefficient
 
-  1. In the hall [conductor](https://en.wikipedia.org/wiki/Hall_effect#/media/File:Hall_Effect_Measurement_Setup_for_Electrons.png), consider a magnetic field $B_z$ applied along z and current I flows along x-direction. Having in mind the electron motion under electric and magentic fields, can you formulate a steady state condition at which the net force on the charges due to these two fields is zero along the y-direction.
-  2. If the Hall voltage $V_H$ and current I are measured in an experiment, calculate the magnetic field. Provide a potential commercial application for the hall effect.
-  3. Given the hall coefficient and electrical conductivity of copper, calculate the carrier mobility.
-  4. From the hall voltage (or hall coefficient) can you qualitatively determine the type of charge carriers. Is there a case at which the hall effect based on Drude model doesn’t make sense?
+When applying a magnetic field $\bf B$ perpendicular to a current carrying 2D sample, the electrons experiences Lorentz force which prevents them from moving in line with the applied electric field. As a result, the resistance of the sample must be expressed in tensor form as follows,
 
+$$\begin{pmatrix} \rho_{xx} & \rho_{xy} \\ \rho_{yx} & \rho_{yy} \end{pmatrix}$$
+
+  1. Given the x and y components of the electric field as follows
+
+$$ E_x = \frac{1}{\sigma_0}j_x + \frac{\omega_c\tau}{\sigma_0}j_y$$
+$$ E_y = -\frac{\omega_c\tau}{\sigma_0}j_x + \frac{1}{\sigma_0}j_y $$
+
+with $\sigma_0$ - drude conductivity, $\omega_c$ - cyclotron frequency, $\tau$ - mean free time, $j_{x, y}$ - current density. Find the longitudinal and Hall resistivities of the 2D sample.
+  2. Derive elements of the conductivity tensor  
+  3. Express the conductivities in terms of resistivities
+  4. Sketch $\rho_{xx}$ and $\rho_{xy}$ as a function of the magnetic field $\bf B$
+  5. Define the Hall coefficient. What does the sign of the Hall coefficient signify??
+  
 ## Exercise 4: Shortcomings of Drude model
   1. Calculate the Seebeck coefficient of Sodium and Copper using Drude model at room temperature.
   2. Compare it against the experimental values (Sodium = $-5*10^-6 V/K$, Copper = $-1.6*10^-6 V/K$) and discuss the vulnerabilities of Drude model. Please justify your opinions.
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