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\vsize=25truecm                                                             
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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 3\break
\parindent=20pt                                                                    
\bigskip 
Exercises E.2.1, E.2.2 from Datta's book.
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3. In the lectures the Landau levels were solved for an ideal conductor. In the 
presence of impurities the Landau levels will be broadened. Show this by calculating
in the first order perturbation theory the effect of a single circularly symmetric
impurity potential
$$U(r,\phi)=\cases{ U_0& for $r<r_0$\cr 0&for $r>r_0$\cr}$$
on the energies in the lowest Landau level. Calculate the energies by assuming
$r_0$ small compared to the cyclotron radius of lowest Landau level, $r_0\ll
\sqrt{\hbar/m\omega_c}$. (Note that because all the states in the same Landau level
have initially equal energies, we have to deal with degenerate perturbation theory in
general. However, for a proper choice of basis functions the perturbation forms a
diagonal matrix $\langle\psi_i\vert U\vert\psi_j\rangle=c_i \delta_{ij}$, and one can
effectively use simple (nondegenerate) perturbation theory. In the present case this
is achieved by using the symmetric gauge centered at the same position as the
impurity, and using circularly symmetric eigenstates, which were studied in exercise
5 of set 1.)  
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4. (a) Use the Landauer formula to derive the contact resistance (Sharvin resistance) 
$${1\over R_c}={e^2k_F^2S\over 4\pi^2\hbar}\eqno(1)$$
for a channel between two bulk (3D) conductors. Assume that the transverse cross
section $S$ of the channel is not too small: $S\gg k_f^{-2}$. (Hint: consider first a
rectangular channel. How this case can be generalized to different shapes?)

(b) The Sharvin resistance can also be derived semiclassically by considering
electrons as point particles that move with velocity $v_f$ along straight trajectories
except when hitting a wall. In three dimensions their density of states is
$N_v=mk_f/\pi^2\hbar^2$ (at the Fermi surface, including 2 for spin).   
Derive the Sharvin resistance (1) in this way. In order to calculate contact
resistance only, it has to be assumed that the reflections of the electrons at the
channel walls are specular (mirror like). (Hint: start from similar formula as in
exercise 3 of set 2 but in 3D).



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