Fluid models for the simulation of charge transport
in submicron and nanometric device structures 
Consolidated research activities
1. Numerical solution of the 2D DriftDiffusion (DD) transport
model using stabilized mixed finite volume methods.
Participants:
Stefano Micheletti, Fausto Saleri  MOX, Politecnico di Milano
Riccardo Sacco  Dipartimento di Matematica, Politecnico di Milano
A stable and accurate discretization has been
devised introducing proper quadrature formulae
into the dualmixed formulation of the decoupled
DD system. This technique yields a mixed finite volume
approach which enjoys the same conservation and accuracy
properties of the mixed method with a considerable
saving of computational effort.
The next figures show the geometry of a 1micron
MOS device with the computed IV characteristics
and the computed distributions of electric
potential and electron concentration (in logarithmic scale).
Authors: R. Sacco, F. Saleri
Authors: R. Sacco, F. Saleri
Relevant publications:
[NMPDE97] ,
[EastWest97] ,
[CWIQuar97] ,
[SISC01]
2.
Numerical solution of the 2D EnergyBalance (EB) transport
model using stabilized mixed finite volume methods.
Participants:
Stefano Micheletti, Fausto Saleri  MOX, Politecnico di Milano
Riccardo Sacco  Dipartimento di Matematica, Politecnico di Milano
The methods introduced in the solution of the DD
equations have been generalized to deal with
the EB model.
The next figures show the computed electron concentration (in logarithmic
scale) and electron temperature in the same device as in
the previous example. Notice the effect of thermal diffusion
around the drain junction and the corresponding
broadening of the inversion layer. Notice also
the considerable carrier heating at the drain end
of the channel.
Authors: S. Micheletti, R. Sacco, F. Saleri
Relevant publications:
[CVS99]
3. Numerical solution of the 1D and 2D Hydrodynamical
(HD) transport model using upwinded finite differences.
Participants:
Stefano Micheletti  MOX, Politecnico di Milano
Riccardo Sacco  Dipartimento di Matematica, Politecnico di Milano
PhD students:
Luca Ballestra
Robust and accurate first and second order discretizations
have been proposed for the simulation of submicron devices
using both full NavierStokes and Euler descriptions of charge transport.
The next figures show the electron velocity and concentration
in a 1D n^{+}nn^{+} structure modeling the conducting
channel of a submicron MOS device with length equal to 0.4 micron.
Notice the formation of a shock wave corresponding to
electron velocity overshoot at the entrance of the channel,
and the considerable carrier heating at the exit of the channel
due to highfield acceleration across the submicronsized region.
Authors: L. Ballestra,
S. Micheletti, R. Sacco, F. Saleri
The next figures show the results of the simulation of a
2D MESFET submicron device, including the geometry of the device and
the computed electron concentration (in log10 scale),
spatial distribution of the Mach number and electron temperature.
Notice the extremely sharp boundary layer
in the neighborhood of the gate contact, the onset
of supersonic regions leading to the formation of truly
2D shock waves in the semiconductor and the considerable carrier
heating around the drain junction.
Authors: L.Ballestra,
S.Micheletti, R.Sacco, F.Saleri
Authors: L.Ballestra,
S.Micheletti, R.Sacco, F.Saleri
Relevant publications:
[CVS01] ,
[PhD02] ,
[CMAME02]
4. Numerical solution of the 1D QuantumDriftDiffusion
(QDD) transport model.
Participants:
Stefano Micheletti  MOX, Politecnico di Milano
Riccardo Sacco  Dipartimento di Matematica, Politecnico di Milano
Degree students:
Paolo Simioni
This is a very recent research area of relevant impact in modern
nanometric semiconductor device structures.
In the study of such devices, it is mandatory to consider
quantum effects to model effectively charge transport.
Here we deal with a perturbation of the standard DD model
including a dispersive relation into the definition
of the current density.
The next figures show the geometry and the potential barrier
profile of a 1D nanometersized heterostructure,
and the computed IV characteristics and electron concentration
(in logarithmic scale) for
a suitably chosen value of the effective electron mass.
Notice the ability of the model in reproducing the negative differential
resistance that is typical in these applications.
Authors: S.Micheletti, R.Sacco, P.Simioni
Authors: S.Micheletti, R.Sacco, P.Simioni
Relevant publications:
[DegThe01] ,
[SCEE02b]
5. Numerical solution of the 3D
HD transport model using stabilized finite elements.
Participants :
Riccardo Sacco  Dipartimento di Matematica, Politecnico di Milano
Degree students:
Carlo de Falco, Giovanni Scrofani
The activity aims at devising efficient and stable
discretizations of the fully 3D NavierStokes/Euler system
for semiconductor device modeling. This step of the
research is absolutely necessary for a stateoftheart
simulation of reallife devices.
The huge computational effort demands for proper
algorithms to manage complex geometries and
requires a multiprocessor environment.
These issues are currently being investigated in collaboration
with the degree students Carlo de Falco and Giovanni Scrofani.
The discretization employs the 3D parallel tetrahedralbased
software for the solution of the NavierStokes/Euler system
developed by
Prof. Carlo L. Bottasso, Dipartimento di
Ingegneria Aerospaziale, Politecnico di Milano. The code
uses the timediscontinuous stabilized finite element
formulation proposed by T.J.R. Hughes and coworkers.
This collaboration is part of the
IPACS project under
the Large Scale Computing (LSC) initiative developed
in the last two years at Politecnico di Milano.
The next figures, concerning with
the study of a n^{+}nn^{+}ndiode structure,
show the doping density, expressed in um^3,
the electron density, expressed in um^3,
the electron velocity, expressed in 10^7 um/s
and the electron temperature, expressed in units of T_0=77 K
(the data of the last three figures
are sampled along the red line of the 3D figure).
Authors: C. de Falco, R. Sacco, G. Scrofani
The next figures,
concerning with the study of a 3D diode structure,
show the space charge density
and the computed electric potential distribution in the device along with
the electron velocity field.
Authors: C. de Falco, R. Sacco, G. Scrofani
The following movies show the electric potential, the electric field and the electric field strength
in different sections of the device (click on the thumbnails to see animations).
Authors: C. de Falco, R. Sacco, G. Scrofani
The last simulated device is a simplified model of an EEPROM memory cell.
The following pictures show the geometry and doping profile of the device, and the electric potential distribution.
