Materials Science and Engineering/Derivations/Semiconductor Devices

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pn Junctino Electrostatics[edit | edit source]

Effect of an Electric Field - Conductivity and Ohm's Law[edit | edit source]



The Built-in Potential (Vbi)[edit | edit source]

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[edit | edit source]

Solution of charge density[edit | edit source]

Solution of electric field[edit | edit source]

Solution of V[edit | edit source]

Solution of xn and xp[edit | edit source]

Depletion Width with Step Junction and VA not equal to zero[edit | edit source]

Pn junction electrostatics.png

pn Junction Diode: I-V Characteristics[edit | edit source]

Assumptions[edit | edit source]

  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[edit | edit source]

Quasineutral Region Consideration[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Ohmic Contacts[edit | edit source]

Depletion Region Edge[edit | edit source]

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[edit | edit source]

  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[edit | edit source]

Derivation 1[edit | edit source]

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

Differentiate the equation of velocity


Derivation 2[edit | edit source]

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[edit | edit source]


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.