Finite Difference Solution to the Ambipolar Transport Equation Under Low Injection The Haynes-Shockley Experiment

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Finite Difference Solution to the Ambipolar
Transport Equation Under Low InjectionThe
Haynes-Shockley Experiment

• CE402 Case Study 3 (EE)
• Brian Standley
• Nathan Fletcher
• Anthony Oberle

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CE402 Case Study 3 (EE)

• Problem - Charge Flow through a Semiconductor
  Material
• Analysis Method

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CE402 Case Study 3 (EE)

• Background - Semiconductor Characteristics
• Two carrier types
• Doping
• Goal - Model ambipolar transport
• Excess minority carriers

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CE402 Case Study 3 (EE)

• Haynes-Shockley Experiment
• observe excess carrier behavior
• measure material properties
• Components
• Input Pulse
• n-type Semiconductor
• Bias voltage source
• Measurement circuit

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CE402 Case Study 3 (EE)
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CE402 Case Study 3 (EE)

• Derivation

Continuity Equation
Time-Dependant Diffusion Equations
Apply Low Injection Approximation
Ambipolar Transport Equation
Poissons Equation
Current Densities
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CE402 Case Study 3 (EE)

• Semiconductor Material (definitions)

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CE402 Case Study 3 (EE)

• Initial Conditions

• Neumann Boundary Conditions

• Parameters

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CE402 Case Study 3 (EE)

• Description of Explicit Finite Divided Difference
  Method

Discretize
Initialize
Solve for Next Time Level
Iterate
Output
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CE402 Case Study 3 (EE)

• Discretization

x-dx i-1
x0 i0
xdx i1
x2dx i2
xX
xL-dx iM-1
xL iM
xLdx iM1
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CE402 Case Study 3 (EE)

• Derivation

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CE402 Case Study 3 (EE)

• Algorithm Stability - stiff equation
• Minimum resolution in both time and space
• These are strict!

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CE402 Case Study 3 (EE)

• C Program
• for(l 0 l lt L l)
• for(i 0 i lt M i)
• Cl1i
• Km1(i0 ? Cl1 Cli-1)
• K0Cli
• Kp1(iM ? ClM-1 Cli1)
• dtstep(dtl, dxi, t_off, Lg0, Lg1)g0

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CE402 Case Study 3 (EE)

• Convergence

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CE402 Case Study 3 (EE)

• Error v Time

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CE402 Case Study 3 (EE)
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CE402 Case Study 3 (EE)

• Summary
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