% Slides RIO 2003
% rio1.tex
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%  Lettere greche
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%
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%
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\newcommand{\Hh}{\mathbb{H}}
\newcommand{\R}{\mathbb{R}}

%  definizioni specifiche per Kane

\newcommand{\pmez}{\frac{p}{2}}
\newcommand{\ppt}{\partial}
\newcommand{\wcc}{w_{cc}}
\newcommand{\wcv}{w_{cv}}
\newcommand{\wvc}{w_{vc}}
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%  nuove per KP
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% 
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%
%**********************************************
%Equazione di  Wigner: Momento di Ordine Zero1,4
%**********************************************
\newcommand{\EqnZeroP}[6]
		{
		\displaystyle     \frac{\ppt n_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     \frac{1 }{q}\frac{\ppt J_{{#1}{#2}}}{\ppt x}	
		\displaystyle    \,{#5}\,  i \frac{\hbar P}{2 m_0}\left[ \frac{\ppt n_{{#2}{#3}}}{\ppt x} - \frac{\ppt n_{{#4}{#1}} }{\ppt x}\right]
		\displaystyle    \,{#6}\,  \frac{ P }{q} \left[J_{{#2}{#3}} + J_{{#4}{#1}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Zero1,4
%**********************************************
\newcommand{\EqnZeroPP}[6]
		{
		\displaystyle     \frac{\ppt n_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     \frac{1 }{q}\frac{\ppt J_{{#1}{#2}}}{\ppt x}	
		\displaystyle    \,{#5}\,  i \frac{\hbar P}{2 m_0}\left[ \frac{\ppt n_{{#3}{#2}}}{\ppt x} - \frac{\ppt n_{{#1}{#4}} }{\ppt x}\right]
		\displaystyle    \,{#6}\,  \frac{ P }{q} \left[J_{{#3}{#2}} + J_{{#1}{#4}}\right]
		}
%**********************************************
%Equazione di  Wigner: Momento di Ordine Zero Generiche 2,3
%**********************************************
\newcommand{\EqnZeroG}[6]
		{
		\displaystyle     \frac{\ppt n_{{#1}{#2}} }{\ppt t}  & = & 
		\displaystyle     \frac{1 }{q}\frac{\ppt J_{{#1}{#2}}}{\ppt x} 
		\displaystyle    + \frac{i}{\hbar} \left[ V_{#2}(x,t)-V_{#1}(x,t)\right] n_{{#1}{#2}} %%%\\
	%	&&
		\displaystyle    \,{#5}\, i \frac{\hbar p}{2 m_0}\left[ \frac{\ppt n_{{#1}{#4}} }{\ppt x} + \frac{\ppt n_{{#3}{#2}} }{\ppt x}\right]
		\displaystyle    \, {#6}\,  \frac{p }{q}\left[J_{{#3}{#2}} - J_{{#1}{#4}}\right]
		}
%**********************************************

%
\begin{document}
\pagestyle{empty}
%
\begin{slide}
\begin{center}
{\bf 
International Conference on Transport Theory -
$\mathbf {18^{th}}$ ICTT 
Rio de Janeiro, July 20-25, 2003
}
\vskip 50pt
{\bf \large Quantum hydrodynamic equations
arising from the
Wigner-Kane model}
\end{center}
\vskip 40pt
\vskip 2pt
\begin{center}
{\bf Stefano Biondini} \, and \, {\bf Giovanni Borgioli}\\
Dipartimento di Elettronica e Telecomunicazioni\\
Universit\`a di Firenze - 50139 Firenze
\vskip -3pt
{\bf Giovanni Frosali}\\
Dipartimento di Matematica Applicata ``G.Sansone"\\
Universit\`a di Firenze - 50139 Firenze
\end{center}
\end{slide}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
\begin{slide}

Mathematical modeling for semiconductor devices has relied, from the beginning, on a {\bf fluid dynamical approach}, 
%It means that the set of equations describing the phenomenology are formulated 
in terms of macroscopic quantities (charge density, current, energy, ...).  
 
When the quantum aspects become not negligible or even predominant,
it is useful to formulate semi-classical models in terms of macroscopic variables
{\bf ( quantum hydrodynamic models )}.


% Such models are generally generated by a hierarchy of coupled moment equations and referred to 
% as {\bf quantum hydrodynamic models.}


Interesting results are present in literature (Gardner, J\"ungel, Markowich, Ringhofer,  Unterreiter, etc.). 
Published results are generally devoted to single-band problems.

{\bf In this work} we face the problem of a fluid dynamical formulation  for a 
semiconductor device characterized by a {\bf two--band (i.e. conduction and 
valence band) structure, where the carriers dynamics is driven by interband tunneling. 
}
\end{slide}
%
%
\begin{slide}
{\bf Physical  description of the Kane model.}
\setcounter{equation}{0} %           SECTION   222222222222222222222222222222222222%
%
%
%\Large
Let $\psi_c(x,t)$ be the conduction band electron wave (envelope) function and 
$\psi_v(x,t)$ be the valence band electron wave (envelope) function. 

In the one-dimensional case,  {\bf the Kane model} reads as  follows [kane56]:
\begin{equation}
\left\{
\begin{array}{rcl}
\displaystyle
i \hbar \frac{\ppt \psi_c}{\ppt t} &=&\left[
\displaystyle
-\frac{\hbar^2}{2m}\frac{\ppt^2}{\ppt x^2} 
+ V_c  \right]
\psi_c -
\displaystyle  \frac{\hbar^2}{m} P \frac{\ppt \psi_v}{\ppt x} 
\\
%\, \rule{0mm}{5mm}
\\
\displaystyle
i \hbar \frac{\ppt \psi_v}{\ppt t} &=&\left[
\displaystyle
- \frac{\hbar^2}{2m}\frac{\ppt^2}{\ppt x^2}
+ V_v  \right]
\psi_v + 
\displaystyle 
\frac{\hbar^2}{m} P \frac{\ppt \psi_c}{\ppt x} 
\end{array}
     \right. \, .
                                            \nn \label{kane1z}
\end{equation}
%
%

%\Large
% where $i$ is the imaginary unit, $\hbar$ is the Planck constant scaled by $2 \pi$,
where $m$ is the bare  mass of the carriers,
$V_c$ and $V_v$ are the minimum of the conduction band energy and maximum of 
the valence band energy respectively,
and $P$ is the {\bf coupling coefficient}   given by
\begin{equation}
P = \frac{m}{\hbar} \sqrt{\frac{m-m^*}{2 m m^*} E_g}                           \nn %   \label{coefp}
\end{equation}
which is obtained through the energy dispersion relation and where
$m^*$ is the effective electron mass depending, through the $x-$coordinate,
on the layer composition, but otherwise isotropic, and $E_g = V_c - V_v$ 
is the $x-$dependent gap energy.
%

According to  {\bf the envelope function theory} approach it is sufficient to study the evolution 
of  the so-called envelope function because it is not 
necessary to find the exact evolution of the full wave equation. 

The {\bf envelope function} is a smooth function which can be obtained by replacing the wave
function by its average in each primitive cell.

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 3

\begin{slide}
%
{\bf Formulation of the Kane model in terms of Wigner functions}
\setcounter{equation}{0}
 
We define {\bf the density matrix}
$
\rho_{ij}(r,s,t) = \bar\psi_i(r,t) \psi_j(s,t) \,\,\,\,\, i,j = c,v \,.
$

Taking formally the derivatives with respect to time $t$ in the density matrix element and 
using Kane eqs. lead to a {\bf set of four coupled evolution equations} for the density
matrix elements $\rho_{ij}$.

The Wigner function is defined by the {\bf inverse Fourier transformation}
$$w_{i,j} (x,v,t) =
{\cal F}^{-1} 
\rho_{i,j} \left(x +\displaystyle  \frac{\hbar}{2 m} \eta,
x -\displaystyle  \frac{\hbar}{2 m} \eta \right)  \,\,.
$$
%
%
\begin{center}
$\bf \Downarrow$ 
\end{center}
Then we obtain the following system, which is {\bf the Wigner function formulation 
of the Kane model} 
\end{slide}

\begin{slide}
%%%%%%%%%%%%
% \hskip-20pt
{
\begin{equation}
\left\{
\begin{array}{rcl}
\displaystyle     i \hbar \frac{\ppt \wcc }{\ppt t} \hskip-2mm &\!=\!&\hskip-2mm 
\displaystyle   \! -  i  \hbar v \frac{\ppt w_{cc}}{\ppt x} + \Theta_{cc} w_{cc} 
\displaystyle    +\frac{ \hbar}{2 m}  P \left[  \frac{\ppt w_{vc}}{\ppt x} 
- \frac{\ppt w_{cv}}{\ppt x} \right]
\displaystyle    - i \hbar P v \left[ w_{vc} +  w_{cv} \right] 
\\
\\
%%%   2 2 2 2 2 2 2  2 
\displaystyle      i \hbar \frac{\ppt \wcv }{\ppt t} \hskip-2mm&\!=\!& \hskip-2mm
\displaystyle    - i  \hbar v     \frac{\ppt \wcv}{\ppt x} 
% -  2 m \hbar v^2 \wcv 
%  - \frac{\hbar^2}{m^*} \ktra \wcv
 +   \Theta_{vc} \wcv 
\displaystyle    +  \frac{\hbar^2}{2 m} P \left[ \frac{\ppt \wvv}{\ppt x} 
+
\frac{\ppt \wcc}{\ppt x} \right]
\displaystyle    - i \hbar P v \left[ w_{vv} - w_{cc} \right] 
\\
\\
%%%%%%%%  33333333333333333333
%\displaystyle     i \hbar \frac{\ppt \wvc }{\ppt t} \hskip-2mm&\!=\!&\hskip-2mm 
%\displaystyle    -  \frac{\hbar^2}{2 m} \frac{\ppt^2 \wvc}{\ppt x^2} 
% +  2 m \hbar v^2 \wvc
% +  \frac{\hbar^2}{m^*} \ktra \wvc 
% +  \Theta_{cv} \wvc 
%\\ \\
%& & \phantom{------}
%\displaystyle    - i \hbar \frac{p}{2}\frac{\ppt \wcc}{\ppt x} 
%\displaystyle    + m p v \wcc 
%\displaystyle    - i \hbar \frac{p}{2}\frac{\ppt \wvv}{\ppt x} 
%\displaystyle    -  m p v \wvv 
%\\ \\
%%%%%%%%  44444444444444444444
\displaystyle   i \hbar v \frac{\ppt \wvv }{\ppt t} \hskip-2mm&\!=\!& \hskip-4mm
\displaystyle   - i  \hbar v \frac{\ppt \wvv}{\ppt x} + \Theta_{vv} \wvv
%% 
\displaystyle   \! - \frac{ \hbar^2}{2 m} P \left[ \frac{\ppt \wcv}{\ppt x} 
  -  \frac{\ppt \wvc}{\ppt x} \right]
\displaystyle  + i \hbar P v \left[ \wcv + \wvc \right] 
\end{array}
    \right.
                                          \nn  %\label{sipro1}
\end{equation}
}

where %%%%%%%%%%%%%%%%%%%%%%$i$ is the imaginary unit, and
{\small
\begin{eqnarray*}
\Theta_{ij} w_{ij}  &=& 
\frac{1}{2\pi} \inti \! \inti
 \left[    V_j\left(x -\displaystyle  \frac{\hbar}{2 m} \eta,t\right) 
            - V_i\left(x +\displaystyle  \frac{\hbar}{2 m} \eta ,t\right) 
\right]   w_{ij}(x,v',t) \cdot
%\\
%&& \phantom{-------------xxxxxx} \cdot
\displaystyle {\rm e}^{i (v-v') \eta} d\eta dv' \nn \\
           \label{tetas}
\end{eqnarray*}
}
with $i,j = c,v$. % etc. etc.
%
\end{slide}

%

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}