Transcript Document

Generalized Monte Carlo Tool for Investigating
Low-Field and High Field Properties of
Materials Using
Non-parabolic Band Structure Model
Raghuraj Hathwar
Advisor : Dr. Dragica Vasileska
Computational Electronics
Outline
• Motivation of modeling different materials
- Strained Silicon
- III-V and II-VI materials
- Silicon Carbide
• The generalized Monte Carlo code
- Free-Flight and drift velocity calculation
• Rappture interfacing
• Results
• Conclusions and future work.
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Technology Trends
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Strained Silicon
•
The four minima of the conduction
band in directions parallel to the
plane of strain are raised. This results
in higher electron mobility.
•
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There is also a splitting of the
light and heavy hole bands
leading to increased hole
mobility.
III-V and II-VI Materials
•
High electron mobility of
compared to silicon.
Gate
Source
n+ cap
L
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AlGaAs/GaAs are lattice
matched.
Drain
n+ cap
Barrier
Barrier / buffer
Substrate
2DEG channel
Channel layer
Gate
n +barrier layer (e.g. Alx Ga1-x As)
•
Undoped channel layer
(e.g. GaAs)
S.I. substrate
(e.g. GaAs)
EF EC
Gate
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AlGaN/GaN interfaces have
spontaneous polarization.
Silicon Carbide (SiC)
•
Very useful in high voltage devices because of its thermal
conductivity, high band gap and high breakdown field.
•
In fact the thermal conductivity of 4H-SiC is greater than
that of copper at room temperature.
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The Monte Carlo Method
• The Boltzmann Transport Equation
• The Chamber-Rees Path Integral
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The Generalized Monte Carlo Flow Chart
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Types of Scattering
•
•
•
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Acoustic Phonon Scattering
Zeroth order Intervalley Scattering
First order Intervalley Scattering
Piezoelectric Scattering
Polar Optical Phonon Scattering
Ionized Impurity Scattering
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Fermi’s Golden Rule and Scattering Rates Calculation
• Calculate the Matrix Element
• Use Fermi’s Golden Rule
• Sum over all k’ states
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Band Structure Model
e.g. GaAs
3 Valley Approximation
Full Band Structure
(equilibrium)
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E-k relation for a General Valley
Here k1 , k2 and k3 are the wave vectors along the three mutually
perpendicular directions that define the valley and m1 , m2 and m3 are the
effective masses of the electrons along those directions
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Conversion from Anisotropic Bands to Isotropic Bands
In order to make the conversion between energy and momentum easy all
anisotropic bands are converted to isotropic bands using
Which gives the following E-k relation
where
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Carrier Free-Flight
From Newton’s 2nd law and Q.M.
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• For simplicity the wave vectors of all electrons are only stored in
the x,y and z coordinate system.
• Therefore before drifting, the wave vectors are transformed from
the x,y,z coordinate system to the 1,2,3 coordinate system
using,
where [a1b1c1], [a2b2c2] and [a3b3c3] are the three mutually
perpendicular directions that define the valley.
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• The electric fields must also be transformed to the directions
along the wave vectors
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The electrons are then drifted and transformed back into the
x,y,z coordinate system.
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Drift Velocity Calculation
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The drift velocities must then be transformed to the x,y,z coordinate
system so that an average can be taken over all electrons.
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Rappture Integration
• The Rappture toolkit provides the basic infrastructure for a large class of
scientific applications, letting scientists focus on their core algorithm
when developing new simulators.
• Instead of inventing your own input/output, you declare the parameters
associated with your tool by describing Rappture objects in the
Extensible Markup Language (XML).
• Create an xml file describing the input structure.
• Integrate the source code with Rappture to read input values and to
output results to the Rappture GUI.
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Material Parameters and Simulation
Parameters
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Valley Parameters
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Scattering Parameters
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Silicon
Drift Velocity vs Electric Field
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Electron Energy vs Electric Field
Gallium Arsenide (GaAs)
Drift Velocity vs Electric Field
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Electron Energy vs Electric Field
Fraction of electrons in the L valley vs Electric Field
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Germanium (Ge)
Drift Velocity vs Electric Field
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Electron Energy vs Electric Field
Rappture GUI Results
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Conclusions and Future Work
• Uses non-parabolic band structure making it as accurate as
possible for an analytic representation of the band structure.
• Interfacing the tool with Rappture enables easy handling of
the parameters and reduces the complexity of using the tool.
• Existing materials band structures can be easily modified to
improve existing results.
• New materials can easily be added to the code.
• The tool can be extended to include impact ionization
scattering to better model high field properties.
• Full band simulation for holes.
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