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Matalien l\"amp\"otilojen fysiikan teoria (Kyl-0.104)\hfil Autumn 2000\break
Electronic transport in mesoscopic systems\hfil Exercise set 2\break
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Exercises E.1.3, E.1.4 from Datta's book.
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3. Show that the average drift of all particles with velocity ${\bf v}_d$ is equivalent
to the distribution function
$f_0(p^2/2m-{\bf v}_d\cdot{\bf p})$ (to first order in $v_d$), where
$f_0(\epsilon)=\{\exp[\beta(\epsilon-\mu)]+1\}^{-1}$ is the equilibrium Fermi
distribution. From this calculate the electric current (in 2 dimensions)
$${\bf j}=2e{1\over L^2}\sum_{\bf k}{\hbar {\bf k}\over m}f({\bf k})$$
in such a way that does not
use the energy states outside the neighborhood ($\pm$ a few $k_BT$) of the
Fermi surface (assuming
$k_BT\ll E_f-E_s$).   
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4. In the lectures we derived the Einstein relation $\sigma=e^2N_sD$ for degenerate
electron gas. Derive the corresponding relation
$$\mu={\vert e\vert D\over kT}$$
for nondegenerate electrons.
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