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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### Exercise 4: Drude model of thermal and electrical conductivity
Metals are good conductors of heat and electricity. Heat transfer from hot to cold medium of a conductor generates thermal current whereas, electrical transport through a thermoelectric material results in heat transfer. Hence, the thermal ($\kappa$) and electrical conductivity ($\sigma$) of a metal are proportional to each other and it's ratio $\frac{\kappa}{\sigma}$ is given by the _Wiedemann Franz law_.
1. Given the thermal conductivity $\kappa=\frac{1}{3}v^2\tau c_v$ in the framework of Drude model, derive an expression for the ratio $\frac{\kappa}{\sigma}$ as a function of Temperature T. (Hint: Apply free electron gas model for the heat capacity $c_v$)
2. The ratio $\frac{\kappa}{\sigma T}$ is called the _Lorenz_ number $L$. From (1), calculate the value of it.
3. _Peltier_ coefficient $\pi$ is an intrinsic material property that defines the amount of heat carried per unit charge. In contrast, the _Seebeck_ coefficient $S$ determines the voltage induced in response to the temperature difference across the material. Given that $\pi = -\frac{c_vT}{3e}$ and $S=\frac{\pi}{T}$, calculate the _Seebeck_ coefficient of copper at room temperature.
4. Compare your results of _Lorenz_ number and _Seebeck_ coefficient against the experimental values ($L_{cu} = 2.20\cdot10^{-8}Watt Ohm/K^2$; $S_{cu} = 1.8\cdot10^{-6}V/K$). By applying ideal gas theory for electrons in a metal, how does the Drude model perform in the calculations of $L$ and $S$.
