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\section*{Wigner-path approach to nonequilibrium quantum transport}

Carlo Jacoboni, Paolo Bordone, and Rossella Brunetti\\

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INFM - National Research Center on nanoStructures and \\
bioSystem at Surfaces (S$^3$)\\
Dipartimento di Fisica, Universit\`a di Modena e Reggio Emilia,\\
Via Campi 213/A, 41100 Modena, Italy.\\
e-mail: bordone@unimo.it

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The Wigner-function (WF) approach has proved to be quite useful to deal 
with quantum electron transport in mesoscopic systems.
It explicitely refers to variables defined in an $({\bf r},{\bf p})$ phase 
space. Furthermore when the phase 
coherence length of the electrons tends to vanish, the WF reduces to the 
classical distribution function, while when the electron wavefunction
extends to a finite size, the corresponding dynamics of the WF can be 
interpreted as the motion of representative points in phase 
space.
Such motion is identical to the motion of classical perticles as long as 
the potential does not change more than quadratically in the region 
occupied by the wavefunction. Thus in many cases the dynamics of each single 
electron in terms of the WF can still be interpreted as an ensemble of 
classical particles. Such situations allows to understand why the Boltzmann 
equation (BE), based on semiclassical approximations, works often so well 
even for mesoscopic systems, and to identify the special situations where 
quantum effects are relevant.
Wigner paths (WP) in the above mentioned phase space can be defined that
provide a pictorial representation of the quantum evolution of the 
system of interest and constitute a useful tool for the development of
Monte Carlo simulative algorithms,
in strict analogy with the ones used in semiclassical transport theory
to solve the BE.

Applications of the WP mothod for the study of quantum 
electron transport in nonequilibrium conditions are presented in this work. 
Scattering with the potential profile and with phonons is included in the 
paths in a way that takes automatically into accounts all quantum effects, 
such as intracollisional field effect and collisional broadening.

We have studied a typical quantum device as the resonant-tunneling 
diode. Results will be presented for different configurations
of the potential profile and for different values of the applied bias. 
Correlation effects due to a fully quantum particle simulation are
clearly visible. 

The WP method can also be extended to the case of a momentum and 
frequency-dependent WF, $f_w({\bf p},\omega)$, defined starting from a 
two-time Green function, related to the electron spectral density
$A({\bf p},\omega)$ and used to analyse the dynamics of polaron 
formation in compound
semiconductors crystals. Numerical calculations have been perfomed
for electrons in GaAs and CdTe and compared, when possible, with the 
results predicted by the semiclassical polaron theory in the weak-coupling
limit. From the WF the spectral density can be obtained
and then it is possible to estimate both the quantum electron mass
renormalization and the polaronic energy shift, even though at present,
due to the heavy computational burden required, our analysis
must be restrictec to times of the order of two, three times
the average phonon scattering time.



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