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Lecture 6. OUTLINE Semiconductor Fundamentals (cont’d) Continuity equations Minority carrier diffusion equations Minority carrier diffusion length Quasi-Fermi levels Poisson’s Equation Reading : Pierret 3.4-3.5, 5.1.2; Hu 4.7, 4.1.3. Derivation of Continuity Equation.

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Lecture 6
Lecture 6

OUTLINE

  • Semiconductor Fundamentals (cont’d)

    • Continuity equations

    • Minority carrier diffusion equations

    • Minority carrier diffusion length

    • Quasi-Fermi levels

    • Poisson’s Equation

      Reading: Pierret 3.4-3.5, 5.1.2; Hu 4.7, 4.1.3


Derivation of continuity equation
Derivation of Continuity Equation

  • Consider carrier-flux into/out-of an infinitesimal volume:

Area A, volume Adx

Jn(x)

Jn(x+dx)

dx

EE130/230A Fall 2013

Lecture 6, Slide 2


Continuity

Equations:

EE130/230A Fall 2013

Lecture 6, Slide 3


Derivation of minority carrier diffusion equation
Derivation of Minority Carrier Diffusion Equation

  • The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers.

  • Simplifying assumptions:

    1. The electric field is small, such that

    in p-type material

    in n-type material

    2. n0 and p0 are independent of x (i.e. uniform doping)

    3. low-level injection conditions prevail

EE130/230A Fall 2013

Lecture 6, Slide 4


EE130/230A Fall 2013

Lecture 6, Slide 5


Carrier concentration notation
Carrier Concentration Notation

  • The subscript “n” or “p” is used to explicitly denote n-type or p-type material, e.g.

    pn is the hole (minority-carrier) concentration in n-type mat’l

    np is the electron (minority-carrier) concentration in n-type mat’l

  • Thus the minority carrier diffusion equations are

EE130/230A Fall 2013

Lecture 6, Slide 6


Simplifications special cases
Simplifications (Special Cases)

  • Steady state:

  • No diffusion current:

  • No R-G:

  • No light:

EE130/230A Fall 2013

Lecture 6, Slide 7


Example
Example

  • Consider an n-type Si sample illuminated at one end:

    • constant minority-carrier injection at x = 0

    • steady state; no light absorption for x > 0

Lp is the hole diffusion length:

EE130/230A Fall 2013

Lecture 6, Slide 8


The general solution to the equation

is

where A,B are constants determined by boundary conditions:

Therefore, the solution is

EE130/230A Fall 2013

Lecture 6, Slide 9


Minority carrier diffusion length
Minority Carrier Diffusion Length

  • Physically, Lp and Ln represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated.

  • Example: ND = 1016 cm-3; tp = 10-6 s

EE130/230A Fall 2013

Lecture 6, Slide 10


Summary continuity equations
Summary: Continuity Equations

  • The continuity equations are established based on conservation of carriers, and therefore hold generally:

  • The minority carrier diffusion equations are derived from the continuity equations, specifically for minority carriers under certain conditions (small E-field, low-level injection, uniform doping profile):

EE130/230A Fall 2013

Lecture 6, Slide 11


Quasi fermi levels
Quasi-Fermi Levels

  • WheneverDn = Dp  0, np  ni2. However, we would like to preserve and use the relations:

  • These equations imply np = ni2, however.The solution is to introduce twoquasi-Fermi levels FNand FPsuch that

EE130/230A Fall 2013

Lecture 6, Slide 12


Example quasi fermi levels
Example: Quasi-Fermi Levels

Consider a Si sample with ND = 1017 cm-3 and Dn = Dp = 1014 cm-3.

What are p and n ?

What is the np product ?

EE130/230A Fall 2013

Lecture 6, Slide 13


EE130/230A Fall 2013

Lecture 6, Slide 14


Poisson s equation
Poisson’s Equation

area A

Gauss’ Law:

E(x)

E(x+Dx)

Dx

s :permittivity (F/cm)

 :charge density (C/cm3)

EE130/230A Fall 2013

Lecture 6, Slide 15


Charge density in a semiconductor
Charge Density in a Semiconductor

  • Assuming the dopants are completely ionized:

    r = q (p – n + ND – NA)

EE130/230A Fall 2013

Lecture 6, Slide 16


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