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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

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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.

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

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Substrate

Barrier / buffer

Channel layer

Drain

BarrierL

n+ cap

Gate

n+ cap

2DEG channel

Source

III-V and II-VI Materials

• High electron mobility of compared to silicon.

• AlGaAs/GaAs are lattice matched.

• AlGaN/GaN interfaces have spontaneous polarization.

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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 Boltzmann Transport Equation

• The Chamber-Rees Path Integral

The Monte Carlo Method

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The Generalized Monte Carlo Flow Chart

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Types of Scattering

• 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

Full Band Structure(equilibrium)

3 Valley Approximation

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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

• 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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Drift Velocity vs Electric Field Electron Energy vs Electric Field

Silicon

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Gallium Arsenide (GaAs)

Drift Velocity vs Electric Field Electron Energy vs Electric Field

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Fraction of electrons in the L valley vs Electric Field

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Drift Velocity vs Electric Field Electron Energy vs Electric Field

Germanium (Ge)

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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.