% RIO 2003,   da lucidi biondini ....19 sett. 2002
%  14 luglio 2003
%  stampare in DIPartimento 
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%  definizioni specifiche per Kane

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%introdotte da Biondini
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% 

\date{}
\begin{document}
%\newcommand{\df}{\buildrel\rm def\over =}
% 
\title{\vskip55pt\bf
Quantum hydrodynamical model for the
two-band Kane system in a tunneling diode }

\pagestyle{empty}


\begin{slide}

\begin{center}
\bf
DERIVATION of  \\
Quantum Hydrodynamic Models \\ from the 
Wigner equations
\end{center}
%%%%% {\bf QUANTUM HYDRODYNAMIC MODEL}
%%%%% { \bf  from the Wigner equations}

The equations for the moments of the Wigner functions can be derived by multiplying the equations of  Wigner 
system, %introduced in \cite{kane}, 
by $1$, $v$ and $v^2$ and by integrating over the velocity space.


 {\bf Definition of macroscopic quantities}

% Now we define density of particles, density of current, energy and third moment, 
% in the {\bf 1-D} dimension.

{\bf Quantum mechanical probability densities} $\ncc$ and $\nvv$ for the positions of electrons in 
band of conduction and valence. The {\bf interband terms} $\ncv$ and $\nvc$ are complex functions
such that $\ncv = \overline{\nvc}$.
\begin{equation*}
	\begin{array}{rclr}
	%particelle
	n_{ij}(x,t) & =&  \displaystyle\inti \wij(x,v,t)dv \,, \quad i,j=c,v\,. 
	\end{array} \nn
\end{equation*}
Similarly we define the {\bf current densities} by the first moments of the Wigner functions, with the classical
meaning of the conduction (valence) current densities $J_{cc}$ ($J_{vv}$).
$$
J_{i j} = - q \inti v w_{i j}(x,v,t) dv \,, \quad i,j=c,v\,.
$$
\end{slide}

\begin{slide}
For $ i,j=c,v $, the set of moments is 
\vskip -18pt
\begin{eqnarray*}
	%particelle
	n_{ij}(x,t) & =&  \displaystyle\inti \wij(x,v,t)dv \,, 
\\
J_{i j}(x,t) &=& - q \inti v w_{i j}(x,v,t) dv \,, 
\\
	\Ener_{ij}(x,t)  &=&  \displaystyle\frac{m}{2}\inti v^2 \wij(x,v,t)dv, \\
	%Momento Terzo
	M^{(3)}_{ij}(x,t)  &=&  \displaystyle\inti v^3 \wij(x,v,t)dv\\
	%Momento Quarto
	%M^{(4)}_{ij}(x,t)  &=&  \displaystyle\inti v^4 \wij(x,v,t)dv
\\
\dots \qquad & \dots & \qquad \ldots
\end{eqnarray*}
%***********************************************


\end{slide}
\begin{slide}
\begin{center}
{\bf The total particle density}
\end{center}
For the two-band Schr\"odinger-like Kane model, {\bf the total density} is
$$
{\bf n_{tot}} (x,t) = |\psi|^2 = |\psi_c|^2+ |\psi_v|^2 = n_{cc}(x,t) + n_{vv}(x,t) \,.
$$
\vskip -4pt
In general we know that {\bf the quantum continuity equation} reads as follows
$$
\displaystyle
\frac{\pt |\psi|^2}{\pt t} = - \div J
$$
where
$$
J(x) = \frac{i \hbar}{2 m} (\psi \nabla \psibar - \psibar \nabla \psi) \,\,.
$$
\vskip -4pt
For the {\bf two-band model}  the continuity equation takes the form
\begin{center}
\framebox{  
$ \displaystyle
\frac{\pt \ {\bf (\ncc + \nvv)} }{\pt t} \, = \, \div \left( \bf J_{tot} \right)
$
}
\end{center}
where the total {\bf current density} for the two-band system is
\begin{eqnarray*}
{\bf J_{tot}}(x, t) &=& \jcc + \jvv -  \frac{\hbar P}{m} \Re {\bf n_{cv}}
\end{eqnarray*}

\end{slide}


%


\begin{slide}

The equations for the current (first moments of equations) contain the second order moments,
$ \Ener_{ij}(x,t) $,
 interpreted as {\bf energy terms.}

A {\bf simple quantum hydrodynamical  model} is obtained directly from the first two moments, by
manipulating  the energy term appearing under the divergence symbol and
taking into account that %equations for the current densities $J_{ij}$.
% 
\\[-6mm]
\begin{eqnarray*}
\frac{\pt \Ener_{ij} }{\pt x}
&=& \frac{m}{2 q^2} \frac{\pt}{\pt x} \left( \frac{J_{ij}^2 }{n_{ij}}  \right) 
\displaystyle -
 \frac{\hbar^2}{4 m} 
n_{ij} \frac{\pt}{\pt x}
\left[ \frac{1}{\sqrt{n_{ij}} }
\displaystyle
 \frac{\pt^2 {\sqrt{n_{ij}}} }
{\pt x^2}  \right]
  \\[3mm]
&+&  \frac{\hbar^2}{2 m} \frac{\pt}{\pt x} \left( n_{ij}  T_{ij} \right) \,,  \nn
\end{eqnarray*}
\\[-2mm]
with $i,j =c,v$.

The quantum correction term $ \frac{1}{\sqrt{n_{ij}} }
\displaystyle
 \frac{\pt^2 {\sqrt{n_{ij}}} }
{\pt x^2}  
$
can be interpreted as internal self-potential, the so-called {\bf Bohm potential.}
\end{slide}

\begin{slide}
The {\bf `'temperatures" } $T_{ij}$ are defined by the operators
$
\displaystyle P_\pm = \displaystyle \frac{1}{2} \frac{\pt}{\pt x} \pm i \frac{m v}{\hbar} $ \,,
as
\vskip -30pt
\begin{equation}
T_{ij} =\displaystyle \frac{<P_+ P_- w_{ij}> }{<w_{ij}>} 
-  \frac{<P_+ w_{ij}> < P_- w_{ij}>}{<w_{ij}>^2} \,,  \nn
\end{equation}
\vskip -10pt
where $< \cdot > $ means average.

The closure of the previous equations system can be done using the  {\bf isothermal assumption.}

The closure can be performed also 
by using the  {\bf thermal equilibrium Wigner functions} for the two-band semiconductor
model, in the case of the Kane  Hamiltonian, [barletti2003]. 

\end{slide}
\begin{slide}
%
\begin{center}
{\bf FINAL COMMENTS}
\end{center}
In this contribution, we present only the first preliminary results. The future 
research is oriented towards the closure %of the moment equations 
and  the numerical validation of the model.
\vskip -30pt
\begin{itemize}
\item The set of Madelung-type equations is closed. The system obtained from Wigner is {\bf not closed}.
\item  {\bf Closure conditions}.
\begin{itemize}
\item Thermodynamic equilibrium
\item Chapman-Enskog expansion
\item MEP (Maximum entropy principle)
\end{itemize}
\item {\bf Non-smooth} potentials.
\item {\bf Numerical validation} of  QHD models
\item {\bf Comparison} with other models and with experiments.
\end{itemize}
\end{slide}
\end{document}