Generalized Monte Carlo Tool for Investigating Low-Field and High Field Properties of Materials Usin...
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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. Outline. Motivation of modeling different materials - Strained Silicon

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Raghuraj Hathwar Advisor : Dr. Dragica Vasileska

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Raghuraj hathwar advisor dr dragica vasileska

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


Outline

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.


Raghuraj hathwar advisor dr dragica vasileska

Technology Trends


Raghuraj hathwar advisor dr dragica vasileska

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.

  • There is also a splitting of the light and heavy hole bands leading to increased hole mobility.


Raghuraj hathwar advisor dr dragica vasileska

III-V and II-VI Materials

  • High electron mobility of compared to silicon.

  • AlGaAs/GaAs are lattice matched.

  • AlGaN/GaN interfaces have spontaneous polarization.


Raghuraj hathwar advisor dr dragica vasileska

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.


Raghuraj hathwar advisor dr dragica vasileska

The Monte Carlo Method

  • The Boltzmann Transport Equation

  • The Chamber-Rees Path Integral


Raghuraj hathwar advisor dr dragica vasileska

The Generalized Monte Carlo Flow Chart


Types of scattering

Types of Scattering

  • Acoustic Phonon Scattering

  • Zeroth order Intervalley Scattering

  • First order Intervalley Scattering

  • Piezoelectric Scattering

  • Polar Optical Phonon Scattering

  • Ionized Impurity Scattering


Raghuraj hathwar advisor dr dragica vasileska

Fermi’s Golden Rule and Scattering Rates Calculation

  • Calculate the Matrix Element

  • Use Fermi’s Golden Rule

  • Sum over all k’ states


Raghuraj hathwar advisor dr dragica vasileska

Band Structure Model

e.g. GaAs

3 Valley Approximation

Full Band Structure

(equilibrium)


Raghuraj hathwar advisor dr dragica vasileska

E-k relation for a General Valley

Here k1 , k2and k3are the wave vectors along the three mutually perpendicular directions that define the valley and m1 , m2and m3 are the effective masses of the electrons along those directions


Raghuraj hathwar advisor dr dragica vasileska

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


Raghuraj hathwar advisor dr dragica vasileska

Carrier Free-Flight

From Newton’s 2nd law and Q.M.


Raghuraj hathwar advisor dr dragica vasileska

  • 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.


Raghuraj hathwar advisor dr dragica vasileska

  • The electric fields must also be transformed to the directions along the wave vectors

  • The electrons are then drifted and transformed back into the x,y,z coordinate system.


Raghuraj hathwar advisor dr dragica vasileska

Drift Velocity Calculation


Raghuraj hathwar advisor dr dragica vasileska

The drift velocities must then be transformed to the x,y,z coordinate system so that an average can be taken over all electrons.


Rappture integration

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.


Raghuraj hathwar advisor dr dragica vasileska

Material Parameters and Simulation Parameters


Raghuraj hathwar advisor dr dragica vasileska

Valley Parameters


Raghuraj hathwar advisor dr dragica vasileska

Scattering Parameters


Raghuraj hathwar advisor dr dragica vasileska

Silicon

Electron Energy vs Electric Field

Drift Velocity vs Electric Field


Raghuraj hathwar advisor dr dragica vasileska

Gallium Arsenide (GaAs)

Electron Energy vs Electric Field

Drift Velocity vs Electric Field


Raghuraj hathwar advisor dr dragica vasileska

Fraction of electrons in the L valley vs Electric Field


Raghuraj hathwar advisor dr dragica vasileska

Germanium (Ge)

Drift Velocity vs Electric Field

Electron Energy vs Electric Field


Raghuraj hathwar advisor dr dragica vasileska

Rappture GUI Results


Conclusions and future work

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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