Summary of 5.3.
1 | Induced Electric Field
In a non-uniformly doped semiconductor under thermal equilibrium, an induced electric
field arises to counteract the diffusion of carriers caused by the doping gradient. This field
ensures that no net current flows and maintains equilibrium.
Key Concepts
1. Energy Band Diagram:
o In thermal equilibrium, the Fermi energy level ( E F) remains constant across the
semiconductor.
o The conduction band energy ( Ec ), valence band energy ( E v), and intrinsic Fermi
level ( E F i ) vary spatially due to the graded doping profile but remain parallel to
each other (Figure 5.12).
o As the doping concentration decreases with increasing x , Ec and E v slope upward.
2. Carrier Diffusion and Charge Separation:
o Majority carrier electrons diffuse from regions of high concentration (high
doping) to regions of low concentration (low doping).
o This diffusion leaves behind positively charged donor ions, creating a separation
of charge.
3. Induced Electric Field:
o The separation of charge induces an electric field that opposes further diffusion.
o At equilibrium, this electric field balances the diffusion process, preventing
further charge separation.
4. Quasi-Neutrality:
o In most cases, the mobile carrier concentration (n 0) is approximately equal to the
donor impurity concentration ( N D), ensuring quasi-neutrality.
Key Equations
1. Electric Potential Relation:
o The electric potential (ϕ ) is related to the electron potential energy by:
+1
ϕ= ( E F − E F i ) , ¿ (5.37)
e
where:
ϕ : Electric potential,
e : Elementary charge,
E F : Fermi energy,
E F i : Intrinsic Fermi level.
2. Electric Field Definition:
o The electric field in one dimension is given by:
d ϕ 1 d EFi
E x =− = . ¿ (5.38)
dx e dx
3. Intrinsic Fermi Level Gradient:
o Assuming quasi-neutrality, the electron concentration is approximately:
n 0=ni exp F (
E − EFi
kT
, )
where:
ni : Intrinsic carrier concentration,
k : Boltzmann constant,
T : Absolute temperature.
4. Fermi Level Difference:
o Solving for E F − E F i gives:
¿
5. Intrinsic Fermi Level Gradient:
o Taking the derivative with respect to x :
d EF i 1 d Nd ( x )
=k T . ¿ (5.41)
dx N d (x ) d x
6. Induced Electric Field:
o Combining Equations (5.38) and (5.41), the induced electric field is:
k T 1 d Nd ( x )
E x =− . ¿ (5.42)
e Nd(x) d x
This shows that the electric field depends on the doping gradient.
Physical Interpretation
The induced electric field arises due to charge separation caused by carrier diffusion.
It balances the diffusion current with a drift current, ensuring no net current in thermal
equilibrium.
The strength and direction of the electric field depend on the spatial gradient of the
doping concentration.
Applications
1. Graded Junctions:
o Used in devices like bipolar junction transistors (BJTs) and solar cells to create
built-in electric fields for efficient carrier transport.
2. Energy Band Analysis:
o Understanding how energy bands vary spatially in graded doping profiles helps
optimize device design.
3. Einstein Relation Validation:
o The relationship between mobility and diffusion coefficients is derived using
these principles.
Summary Formulae
Parameter Formula
Electric Potential +1
ϕ= ( E F − E F i )
e
Electric Field 1 d EFi
E x=
e dx
( )
Fermi Level Difference N d (x )
E F − E F i =k T ln
ni
Intrinsic Fermi Level Gradient −d E F i 1 d Nd (x )
=k T
dx Nd ( x ) d x
Induced Electric Field k T 1 d Nd ( x )
E x =−
e Nd(x) d x
In a non-uniformly doped semiconductor under thermal equilibrium, an induced electric field
arises due to carrier diffusion from high to low doping regions, leaving behind positively charged
donor ions (Figure 5.12). This field opposes further diffusion, ensuring no net current flow at
equilibrium. The induced electric field is proportional to the doping gradient and inversely
proportional to the doping concentration (Equation 5.42). These principles are critical for
understanding graded junctions and validating transport relationships like the Einstein relation in
semiconductors.
