Materials Science and Engineering/Derivations/Semiconductor Devices

pn Junctino Electrostatics

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Effect of an Electric Field - Conductivity and Ohm's Law

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The Built-in Potential (Vbi)

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The electric field is the derivative of the potential with position

 

Integrate across the depletion region

 

The sum of the drift and the diffusion at equilibrium is equal to zero

 

Use the Einstein relationship:

 

 

  and   are the n- and p-side doping concentrations.

 

 

  

Depletion Width

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Solution of charge density

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Solution of electric field

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Solution of V

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Solution of xn and xp

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Depletion Width with Step Junction and VA not equal to zero

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pn Junction Diode: I-V Characteristics

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Assumptions

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  1. Diode operated in steady state
  2. Doping profile modeled by nondegenerately doped step junction
  3. Diode is one-dimensional
  4. In quasineutral region the low-level injection prevails
  5. The only processes are drift, diffusion, and thermal recombination-generation

General Relationships

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Quasineutral Region Consideration

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The quasineutral p-region and n-region are adjacent to the depletion region

 

 

The electric field is zero and the derivative of the electron and hole concentration is zero in the quasineutral regions.

 

 

Depletion Region

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Continuity equations:

 

 

Assume that the thermal recombination-generation is zero throughout depletion region. Sum the   and   solutions.

 

Strategy to find the minority carrier current density in the quasineutral regions:

  • Evaluate current densities at the depletion region edges
  • Add edge current densities
  • Multiply by A to find the current

Boundary Conditions

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Ohmic Contacts

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Depletion Region Edge

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Establish boundary conditions at the edges of the depletion region.

Multiply defining equations of the electron quasi-Fermi level,  , and the hole quasi-Fermi level,  .

 

Monotonic variation in levels

 

"Law of Junction"

 

Evaluate the "law of junction" at the depletion region edges to find the boundary conditions.

 

 

  

 

 

  

Derivation

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  1. Solve minority carrier diffusion equations with regard to the boundary conditions to determine value of   and   in quasineutral region.
  2. Determine the minority carrier current densities in quasineutral region
  3. Evaluate quasineutral region solutions of   and  . Multiply result by area

Solve the equation below with regard to two boundary conditions.

 

 

 

General solution:

 

 

  
  
  
  

Evaluate at the depletion region edges

 

 

Multiply the current density by the area:

 

Ideal diode equation:

  
  

The Effective Mass

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Derivation 1

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The velocity of an electron in a one-dimensional lattice is in terms of the group velocity

 

 

 

Differentiate the equation of velocity

 

 

 

 

  

Derivation 2

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In the free electron model, the electronic wave function can be in the form of  . For a wave packet the group velocity is given by:

  =  

In presence of an electric field E, the energy change is:

 

Now we can say:

 

where p is the electron's momentum. Just put previous results in this last equation and we get:

 

From this follows the definition of effective mass:

 

The Zimman Model

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Wavefunction:

 

Strong disturbance when individual reflections add in phase

 

 

 

 

The potential energy is from the actual potential   weighted by the probability function  

 

 

Average over one period

 

 

 

 

The kinetic energy is the same in the case of both wave functions

 

The total energy is the kinetic energy plus the potential energy

 

The energy of an electron cannot be between the lower and higher value.