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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 4\break
\parindent=20pt                                                                    
\bigskip 
Exercises E.2.3, E.2.4 from Datta's book.
Note that the Hall coefficient according to Drude model 
$$R_H={E_y\over J_xB_z}=-{1\over \vert e\vert n_s}$$
is negative for electrons. Is it also in E.2.3?
\bigskip                                                                        
3. Some ambiguities in the discussion of screening in Datta's book (pages 72-74) can
be rectified as follows. Instead of (1.2.2) we write the 2D Schr\"odinger equation
$$[E_0+{(i\hbar\nabla+e{\bf A})^2\over 2m}+U]\Psi(x,y)=E\Psi(x,y),$$ 
with constant $E_0$ and the band-edge energy $E_s=E_0+U$. Otherwise use the same
formulas as Datta: the potential is related to electric field by 
$e{\bf E}=-\nabla U$, the 2D electron density $n_s=N_s(E_f-E_s)$ and the Gauss' law
$$\nabla\cdot{\bf E}={\rho\over\epsilon}={e\delta n_s\over \epsilon d}$$ where $d$ is
the thickness of the 2D electron gas. Derive from these that $E_s$ is determined by
the equation
$$(\nabla^2-\beta^2)\delta E_s=-\beta^2\delta E_f$$
Solve this for the idealized case of one dimension and a step change of
$E_f$, and show that $\beta^{-1}=\sqrt{\pi\hbar^2\epsilon d/e^2m}$ represents
the length where the space charge is screened. 
Note that a similar analysis applies also to the density discontinuity (but $E_f=$
constant) at the junction of a narrow conductor, which was studied in exercises
E.1.3 and E.1.4. 

This simple theory of screening is known as Thomas-Fermi theory. In more accurate
theory the Schr\"odinger equation has to be solved for a spatially varying $U(x)$.
Another fact that was neglected here is that ${\bf E}$ really is 3 dimensional.
\bigskip                                                                        


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