Psov model polovodicu

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TCAD for Nanostructures
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OUTLINE 1. Introduction - TCAD - what is it,
goals, benefits 2. Device simulation models -
examples, results - quantum (TMM, Wingreen,
NEMO) - semi-classical (MMC - DAMACLES) -
drift-diffuse (Atlas, Dessis, Nextnano) 3.
Conclusion - the comparison of models
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Physical model levels
DRIFT-DIFFUSE
HYDRODYNAMIC
SEMI-CLASSICALL
QUANTUM
local termodynamic equilibrium region
electron wavelength region
electron free-path region
1nm 10nm 100nm 1000nm
critical dimensions
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Semiclassical Transport Models
Drift-Diffusion Model ? Good for devices with
LG0.5 mm ? Cant deal with hot carrier effects
LG 0.5 mm
Hydrodynamic Model ? Hot carrier effects, such
as velocity overshoot, included into the model ?
Overestimates the velocity at high fields
LG ? 0.1 mm
LG Particle-Based Simulation ? Accurate up to
classical limits ? Allows proper treatment of the
discrete impurity effects and e-e and e-i
interactions ? Time consuming
discrete impurity effects, electron-electron
interactions
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Historical Development of Physical Device Modeling

• Closed-form analytical modeling
• Gradual-channel approximation (Schockley, 1952)
• Numerical modeling
• Gummels 1D numerical scheme for BJTs (1964)
• De Mari (1968) 1D numerical model for pn -
  junctions
• Sharfetter and Gummel (1969) 1D simulation for
  Silicon Read (IMPATT) diodes
• Kenedy and OBrien (1970) 2D simulation of
  silicon JFETs
• Slotboom (1973) 2D simulation of BJTs
• Yoshii et al. (1982) 3D modeling for a range of
  semiconductor devices
• Commercial device simulators
• 2D MOS MINMOS, GEMINI, PISCES, CADDET, HFIELDS,
  CURRY
• 3D MOS WATMOS, FIELDAY
• 1D BJT SEDAN, BIPOLE, LUSTRE
• 2D BJT BAMBI, CURRY
• MESFETs CUPID
• Particle-based simulators DAMOCLES
• Quantum transport simulators NEMO

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Physical levels
ACCURACY
Quantum Boltzmann/Wigner Equation
SIMPLICITY
Semiconductor Boltzmann Equation
Hydrodynamic Model
Drift Diffusion Model
Circuit (Spice) Model
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Nanoscale - Microscale
nanotubes
tranzistors
quantum dots
MEMS
atoms
molecules
virus
blood cels
0.1 1 10
100 1000 nm
quantum devices quantum principles of the
functionality
hybrid quantum devices quantum parasitic effects
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Nanostructures

• electron wavelength
• De Broglie ? h / p (cca 10nm) h ...
  Planck constant
• p mv ...electron momentum
• quantum effects
• discrete energetic levels
• (in quantum well...)
• tunneling through the barrier
• (band-to-band...)
• interference of wavefunctions
• (reflection, difraction...)

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Quantum Device Modelling

• STATIONARY
• pure electron states - Schrödinger equation
• (shooting method...)
• selfconsistent Schrödinger-Poisson method
• TRANSPORT
• mixed states
• quantum corrections to classical model
• Transfer matrix method
• Wigner function - Quantum Monte Carlo
• Green function method

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Quantum Device Modelling
Shrödinger equation Wave function y(x,t)
dynamic
kinetic
T-matrix, wave function ay(x,t)
Green function G(x,tx,t)
Density matrix r(x,x,t)
Monte Carlo fW(x,k,t)
Wiegner function fW(x,k,t)
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Quantum Device Modelling

• The evolution of the wavefunction is described by
  the
• time-dependent Schrödinger equation
• If one is considering stationary states, then
• The time-independent Schrödinger equation then
  reduces to
• solution charge density, current density

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Transfer-Matrix Method (TMM)

• the Schrödinger equation is solved in
  steady-state
• the most widely used method of quantum device
  simulation to date
• particles enter and exit the system as continuous
  streams (beams) with amplitudes given by the
  fixed boundary conditions
• particle beams at different energies do not
  interact.
• The result is a state function for each particle
  beam simplicity and the relatively low
  computational requirements.
• can not handle irreversibility (inelastic
  scattering).
• transient simulations are difficult or impossible
  to implement

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Density Matrix Approach

• The density matrix is a single-time function and
  the temporal equation of motion for the density
  matrix is the Liouville equation
• The diagonal terms of the density matrix
  represent the density variation throughout the
  device, whereas the off-diagonal parts represent
  the spatial correlation's that exist in the
  system.
• Since it is defined in position space, it is
  difficult to include scattering in this formalism
  which is best described in momentum space.

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Wigner function method
Wigner function (Fourier transformation of
non-diagonal elements of density matrix)

• transport equation formaly similar to the
  semi-classical Boltzmann equation

charge density, current density
QMC - Quantum Monte Carlo (SQUADS - NASA)
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Greens Functions Formalism

• Way of Solving the Many-Body Problem.
• They yield in a direct way the most important
  physical properties of the system - contact,
  scattering - selfenergy S
• transport descibed by the Dyson equation
• They have a simple physical interpretation
  (density of states, charge density, electron
  current)
• They can be calculated in a way that is highly
  systematic and automatic

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WinGreen

• free distributed by FZ Juelich
• based on a nonequilibrium Greens function
  approach to quantum transport in laterally
  extended layered heterostructures.
• incorporate particle scattering.
• A real-space tight binding formulation with a
  multi-orbital basis per lattice site.
• Parallel effective masses are assumed to be
  constant. This leads to extremly sharp resonances
  within the quantum well structures.
• Structure is divided into three regions Two at
  the two ends with a device region in between.
  Each reservoir is in a thermodynamical
  equilibrium state with a given chemical potential
  and temperature.
• The applied voltage corresponds to the difference
  of the two chemical potentials.
• A many-particle mixed state is considered

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WinGreen
http//www.fz-juelich.de/isg/mbe/software.html
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WinGreen
EXAMPLE Resonant Tunneling Diode (RTD)
Current
Conduction band diagrams for different
voltages and the resulting current flow.
Voltage
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The simulated IV characteristics of
GaAs/AlAS/GaAs RTD by Wingreen and the
experimental data
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The electron charge , CB and VB in RTD ( V 0.49
V )
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The transmisivity in RTD ( V 0.49 V )
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The local density of states in RTD ( V 0.49 V )
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NEMO

• available free for US universities
• NEMO 1-D was developed under a NSA/NRO contract
  to Texas Instruments and Raytheon from 93-98
• NEMO 3-D developed at JPL 98-03 under NASA,
  NSA, and ONR funding.
• the physical full band tight binding, full charge
  self-consistent simulation.
• Based on Non-Equilibrium Green function formalism
  NEGF - Datta, Lake, Klimeck.
• represents the state-of-the-art quantum device
  design tool.

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NEMO
http//hpc.jpl.nasa.gov/PEP/gekco/nemo1D/
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NEMO

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Semi-classical Transport Models

• The cornerstone of semiclassical transport theory
  is the Boltzmann transport equation
• This description is based on a physical picture
  in which
• An electron is assumed to be described by a
  wavepacket
• (well defined k and r can be associated to it)
• During free flights, electron momentum changes
  with time
• according to Newton laws
• Electron free-flights are interrupted by
  instantaneous and
• point-like collisions (with impurities, phonons,
  etc.)
• Fermi Golden Rule is commonly used for the
  calculation of the transition probability per
  unit time from state k to state k

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Semi-classical Transport Models

• Boltzmann transport equation
• (stacionary)
• collisional term
• probability of scattering P(k,k) - Fermi golden
  rule

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Monte Carlo Method
semiconductor structures - Ensemble MC
input data
initial conditions
homogenous material - single particle MC
random number generation R (0,1)
free flight
state before collision
R
last flight
0
1
output data
type of collision
AF
OF
IImp
state after collision
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Mikrofyzikální úroven
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Monte Carlo Method
EXAMPLE Superlattices - Bloch Oscillation (BO)
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superlattice Al0,3Ga0,7As/GaAs
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BO by different temperatures - superlattice
9.7/1.7 well/barrier widths
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6
-3
F10 kV/cm N
10
cm
i
10
cm/s)
5
6
0
-5
electron velocity (10
10 K
-10
100 K
-15
300 K
0
0,5
1,0
1,5
2,0
time (ps)
H. Moravcová, J. Voves, Physica E, 17, pp.
307-309, 2003
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Hydrodynamic model

• Derived from Boltzmann equation - momentum
  conservation law
• We expect distribution function as shifted
  Maxwellian -energy conservation law
• Hot carrier effects, such as velocity overshoot,
  included into the model
• Overestimates the velocity at high fields

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Drift-Diffusion Model
zeroth and first moment of BTE Variable
s n, p, and V are solved simultaneously on a
mesh. Transport is localy described by
phenomenological mobility v?(E)E and diffusion
coefficient D(E)kT/q ?(E)
Continuity equations
Current density
Poissons equation
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Drift-Diffusion Model

• Good for devices with LG0.5 mm
• Cant deal with hot carrier effects
• Temporal variations occur in a time-scale much
  longer than the momentum relaxation time.
• The drift component of the kinetic energy was
  neglected, thus removing all thermal effects.
• Thermoelectric effects associated with the
  temperature gradients in the device are
  neglected, i.e.
• The spatial variation of the external forces is
  neglected, which implies slowly varying fields.
• Parabolic energy band model was assumed, i.e.
  degenerate materials can not be treated properly.

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SILVACO International
Device simulation ATLAS structure and modules
SILVACO/ATLAS User Manual, 2003
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SILVACO/ATLAS

• program includes the self-consistent
  Schrodinger-Poisson model. It solves the one
  dimensional Schrodingers equation along a
  vertical slices. The Eigen energies and Eigen
  functions can also be visualized.
• The quantum transport model. This model is based
  on moments of the Wigner function. The
  equations-of-motion consist of quantum correction
  to the carrier temperatures in the carrier
  current and energy flux added to the standard
  drift diffussion current continuity equation.
• The quantum mechanical correction as given by
  Hansch et. al. is suitable for accurate
  simulation of the effects of quantum mechanical
  confinement near the gate oxide interface in
  MOSFETs.
• The Van Dort model - the effects of the quantum
  confinement are modeled by an effective
  broadening of the bandgap near the surface as a
  function of a perpendicular electric field and
  distance from the surface.

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SILVACO/ATLAS
EXAMPLE HEMT simulation mesh
structure, doping
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SILVACO/ATLAS
EXAMPLE HEMT simulation electron
concentration IV characteristics
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SILVACO/ATLAS
EXAMPLE HEMT simulation 2DEG energy
levels electron wavefunctions
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SILVACO/ATLAS
EXAMPLE RTD simulation energy
levels electron
wavefunctions
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SYNOPSYS TCAD - models

• Transport models
• Drift-diffuse, Hydrodynamic, Thermodynamic,
• Quantum models
• Metal conductivity, Trap kinetics
• Tunneling models (FowlerNordheim, Band-to-band)
• Gate current and interface models
• Mechanical stress models
• Full band Monte Carlo
• Optoelectronic models, Laser models
• The main drawback of these models is the problem
  with realistic boundary conditions and very
  approximate approach.

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SYNOPSYS/DESSIS User Manual, 2005
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SYNOPSYS/DESSIS

• Program implements three quantization models
• The van Dort quantum correction model is a
  numerically robust and fast. It is only suited to
  MOSFET simulations. It does not give the correct
  density distribution in the channel.
• The 1D Schrödinger equation is the most
  physically sophisticated quantization model in
  DESSIS. Simulations with this model tend to be
  slow and often lead to convergence problems,
  which restrict its use to situations with small
  current flow. Therefore it is used mainly for the
  validation and calibration of other quantization
  models.
• The density gradient model is numerically robust,
  but significantly slower than the van Dort model.
  It can be applied to MOSFETs, quantum wells and
  SOI structures, and gives a reasonable
  description of terminal characteristics and
  charge distribution inside a device. It can
  describe 2D and 3D quantization effects.

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SYNOPSYS/DESSIS
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Nextnano

• developed at TU Munich - free available
• calculating the realistic electronic structure of
  three-dimensional heterostructure quantum devices
  under bias and its current density close to
  equilibrium.
• The electronic structure is calculated fully
  quantum mechanically, whereas the current is
  determined by employing a semiclassical concept
  of local Fermi levels
• considers only stationary pure single-particle
  states by solving the Schrödinger equation for
  envelope functions.
• Multi-band SchrödingerPoisson equation
• includes strain
• piezo and pyroelectric charges
• Si/Ge and III-V materials

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Nextnano
http//www.nextnano.de/
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Conclusions

• The quantum models are not yet completely
  included in the commercial TCAD tools.
• These tools can be used for approximate
  description of nanostructures. The stationary
  energy levels and wavefunctions can be determined
  by Silvaco/Atlas or by Synopsys/Dessis.
• Free available simulators can serve as a good
  help in this lack.
• For the precise quantum transport simulation we
  need models based on the Wigner or Green function
  formalism.
• Green function based method looks to be the most
  realistic for the quantum transport simulation