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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

slide3

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

slide5
Starting with the continuity equation for electrons:

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

slide9
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

slide14
Find FN and FP:

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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