Derivation of the Drift-Diffusion Equation
Lecture Prepared By:
Sanjoy Mukherjee
Drift-Diffusion Equation - Applicability
Basic equations governing transport in semiconductors and
semiconductor devices:
• Instances where Drift-Diffusion Equation cannot be used
– Accelerations during rapidly changing electric fields (transient
effects)
• Non quasi-steady state
• Non-Maxwellian distribution
– Accurate prediction of the distribution or spread of the transport
behavior is required
• Instances when Drift-Diffusion Equation can represent
the trend (or predict the mean behavior of the transport
properties)
– Feature length of the semiconductors smaller than the mean free
path of the carriers
• Instances when Drift-Diffusion equations are accurate
– Quasi-steady state assumption holds (no transient effects)
– Device feature lengths much greater than the mean free paths of
the carrier
The Method of Moments
In the Method of Moments, both sides of a equation are multiplied by a
function (a moment generating function) raised to a integer power, and
then integrated over all space
∴ A( x, k , t ) + B( x, k , t ) = C ( x, k , t )
Multiplying by the Moment
generating Function Θn(k)
( n = order of the moment)
k
3
k
3
k
3
(
)
(
)
(
)
(
)
(
)
Θ
k
A
x
,
k
,
t
d
k
+
Θ
k
B
x
,
k
,
t
d
k
=
Θ
k
C
(
x
,
k
,
t
)
d
k
∫
∫
∫
Method of Moments Applied to the Boltzmann Transport Equation
The Boltzmann Transport Equation with relaxation time approximation:
v
r v
f − f0
∂f Fext v
+
⋅ ∇k f + v ⋅ ∇ x f = −
∂t
τ
h
f = a classical distribution function at nonequilibrium state that represents
the probability of finding a particle at position x, with momentum k and at
time t. The subscript 0 corresponds to the equilibrium state
Multiplying throughout by the moment generating function Θn and
integrating over all k space
v v
v
1
∂f 3
n
3
n r
3
n f − f0 3
∫ Θ ∂t d k + h ∫ Θ (Fext ⋅ ∇ k f )d k + ∫ Θ (v ⋅ ∇ x f )d k = − ∫ Θ τ d k
n
Method of Moments Applied to the Boltzmann Transport Equation
If Θ = 1 and n = 1 then:
r v
1 v v
f − f0 3
∂f 3
3
3
∫ ∂t d k + h ∫ (Fext ⋅ ∇ k f )d k + ∫ (v ⋅ ∇ x f )d k = − ∫ τ d k
Simplifies to
∂n v
v
+ ∇ x ⋅ (nv ) = 0
∂t
Carrier Continuity Equation
If Θ = v and n = 1 then:
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
∫ v ∂t d k + h ∫ v (Fext ⋅ ∇ k f )d k + ∫ v (v ⋅ ∇ x f )d k = − ∫ v τ d k
Simplifies to
v
v
J n = nqµ n F + qDn ∇ x n
Drift-Diffusion Equation
In the subsequent slides we would derive the Drift-Diffusion Equation
from Boltzmann Transport Equation by utilizing this Method of Moments
Drift-Diffusion Equation Derivation – 1st. Term
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
(
)
(
)
∫ v ∂t d k + h ∫ v Fext ⋅ ∇ k f d k + ∫ v v ⋅ ∇ x f d k = − ∫ v τ d k
Velocity is time independent
∂ r 3
v fd k
∫
∂t
n = carrier concentration = ∫ G (k ) f (k )d 3 k
G (k ) = Density of states
1 dN
2
1
=
=
V dk (2π )3 4π 3
1
∴ n = ∫ 3 f (k )d 3k = ∫ f ' (k )d 3k
4π
=
r ∂f 3
∂ v
∫ v ∂t d k = ∂t (nv )
Drift-Diffusion Equation Derivation – 2nd. Term
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
(
)
(
)
d
k
+
v
F
⋅
∇
f
d
k
+
v
v
⋅
∇
f
d
k
=
−
v
d k
v
ext
k
x
∫ ∂t
∫
∫
∫
h
τ
v v
v
v
v v
Identity : F ⋅ ∇g = ∇ ⋅ (gF ) − g∇ ⋅ F
v
1 r v
1 r v v
3
3
(
)
(
)
v
∇
⋅
f
F
d
k
−
f
v
∇
⋅
F
d
k
k
ext
k
ext
∫
∫
h
h
v v v v
vv vv
v
Identity : (gF ⋅ ∇ )G = ∇ ⋅ (gFG ) − G∇ ⋅ (gF )
1
h
{∫ [
] }
v rv
v v r 3
∇ k ( fv Fext ) − ( fFext ⋅ ∇ k )v d k
f is finite and so the surface integral (integral of
divergence of fvFext) at infinity vanishes identically
v v r 3
1
− ∫ ( fFext ⋅ ∇ k )v d k
h
Drift-Diffusion Equation Derivation – 2nd. Term (Continued)
Substituting:
v v r 3
1
1 r v v
− ∫ ( fFext ⋅ ∇ k )v d k − ∫ fv (∇ k ⋅ Fext )d 3 k
h
h
v
v
Substituting − ∇ x E = Fext
v
v r 3
1
1 r v v
3
(
)
(
)
f
∇
E
⋅
∇
v
d
k
+
f
v
∇
⋅
∇
E
d
k
x
k
k
x
∫
∫
h
h
v v
v v
v v
∇ k ⋅ ∇ x E = ∇ x ⋅ ∇ k E = h∇ x ⋅ v
∫
r v v 3
fv (∇ x ⋅ v )d k
Substituting, the second term is finally reduced to:
v
v r 3
r v v 3
1 r v v
1
3
v (Fext ⋅ ∇ k f )d k = ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ fv (∇ x ⋅ v )d k
∫
h
h
Drift-Diffusion Equation Derivation – 3rd. Term
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
∫ v ∂t d k + h ∫ v (Fext ⋅ ∇ k f )d k + ∫ v (v ⋅ ∇ x f )d k = − ∫ v τ d k
v v
v
v
v v
Identity : F ⋅ ∇g = ∇ ⋅ (gF ) − g∇ ⋅ F
vv
v 3
v v v 3
∫ v ∇ x ⋅ ( fv )d k − ∫ v f (∇ x ⋅ v )d k
vv
v v
vv
v v v
Identity : G∇ ⋅ (gF ) = ∇ ⋅ (gFG ) − (gF ⋅ ∇ )G
v
vv 3
v v v 3
∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k
Substituting, the third term is finally reduced to:
v
rr v
vv 3
v v v 3
v v v 3
3
∫ v (v ⋅ ∇ x f )d k = ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k − ∫ v f (∇ x ⋅ v )d k
Drift-Diffusion Equation Derivation – Right Hand Term
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
∫ v ∂t d k + h ∫ v (Fext ⋅ ∇ k f )d k + ∫ v (v ⋅ ∇ x f )d k = − ∫ v τ d k
−
Re call
∫
1 r
3
(
)
v
f
−
f
d
k
0
∫
τ
v
fd 3 k = n and ∫ v fd 3 k = v n
n = carrier concentration
v = average velocity
−n
v − v0
τ
At equilibrium the ensemble
velocity v0 (by definition) = 0
Substituting, the right hand term is finally reduced to:
r f − f0 3
v
d k = −n
−∫v
τ
τ
Drift-Diffusion Equation Derivation – General Form
r ∂f 3
rr v
r f − f0 3
1 r v v
3
3
∫ v ∂t d k + h ∫ v (Fext ⋅ ∇ k f )d k + ∫ v (v ⋅ ∇ x f )d k = − ∫ v τ d k
v
v r 3
r v v 3
∂ (nv ) 1
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ fv (∇ x ⋅ v )d k +
∂t
h
v
v
vv 3
v v v 3
v v v 3
∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k − ∫ v f (∇ x ⋅ v )d k = −n
τ
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
h
τ
Standard Drift-Diffusion Equation for Electrons/Holes
The general Drift-Diffusion derived in the previous slides may be further
simplified with the help of certain assumptions
• Assumptions
h 2k 2
– The energy of the carriers, E =
2m
– Mass is isotropic and constant
1 2
∴ E x = E y = E z = Ei = mvi
2
– Material is isotropic, and so the spatial temperature
v
gradient is zero
∇ x Ei = 0
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
h
τ
v
v
Substituting − ∇ x E = Fext
v v r 3
1
− ∫ fFext ⋅ ∇ k v d k
h
v
2
v v 1 v2
v
h k
h k kv = mvv v 1 v
⇒ ∇k E =
→ v = ∇ k E ⇒ ∇ k v = ∇ k E
E=
2m
h
h
m
v2
h 2k 2
h2
⇒ ∇k E =
again E =
2m
m
v v h
∴ ∇k v =
m
2
v
Fext
−
m
∫
fd 3 k
2
recall
∫
fd 3 k = n
v
Fext
−
n
m
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
h
τ
Assuming the mass is isotropic and constant and therefore:
v x2

vv
fv v = f  0
0

0
0
0 
 fE x
1
 2
fE y
0  =  0
0  taking E x = E y = E z = Ei = mvi2
m
2
 0
fE z 
0 v z2 
0
v
v
2  ∂fE x ∂fE y ∂fE z  2 v
vv
now ∇ x ⋅ ( fv v ) = 
f∇ x Ei + Ei ∇ x f
+
+
=
m  ∂x
∂z  m
∂y
v
v
2 v
vv 3
∴ ∫ ∇ x ⋅ ( fv v )d k = n∇ x Ei + Ei ∇ x n
m
0
v y2
[
[
]
]
Assuming the material is isotropic i.e. temperature or energy is spatially independent
0
1
v
v
v
v
Ei = E
2
2
vv 3
3
∴ ∫ ∇ x ⋅ ( fv v )d k =
n ∇ x E i + E i ∇ x n  →
E ∇ xn
3m
m
[
]
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
h
τ
v
v
∂ (nv ) Fext
2 v
n+
E ∇ x n = −n
−
m
∂t
3m
τ
Notice that this
term is completely
ignored in the text
v
v
v
Fext = −qF where q = electronic ch arg e, F = field
E=
3
k BT , where T = temperature
2
v
v
∂ (− qn v )
q τF
qτ
τ
+ (− qn v ) =
n+
k BT ∇ x n
∂t
m
m
2
density = J n = − qn v
electron
current
electron
mobility = µ n =
qτ
m
v
v
∂J n
τ
+ J n = nq µ n F + µ n k BT∇ x n
∂t
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
τ
h
vv
v v
vv
v v v
Identity : G∇ ⋅ (gF ) = ∇ ⋅ (gFG ) − (gF ⋅ ∇ )G
vv
v 3
(
v
∇
⋅
f
v
∫ x )d k
v
v
v v
v v
Identity : ∇ ⋅ (gF ) = F ⋅ ∇g + g∇ ⋅ F
[
]
[
]
v v v 3
vv v
3
v
f
∇
⋅
v
d
k
+
v
v
⋅
∇
f
d
k
x
∫ x
∫
v v 1v v
1v v
Re call ∇ x ⋅ v = ∇ x ⋅ ∇ k E = ∇ k ⋅ ∇ x E
h
h
0
v 1 v v  3
vv v
3
v
f
∇
⋅
∇
E
d
k
v
v
f
d
k
+
⋅
∇
x
∫  h k x 
∫
[
Assuming the material is isotropic
i.e. energy is spatially independent
v
∇xE = 0
]
Next Slide
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
Previous Slide
vv v
3
v
v
⋅
∇
fd
k
x
∫
Assuming the mass is isotropic and constant and therefore:
v x2
vv 
vv =  0
0

0
0
 Ex 0
1
 2
0  =  0 E y 0  taking E x = E y = E z = Ei = mvi2
2
m
 0
0 v z2 
0 E z 
v
2 Ei  ∂f ∂f ∂f  2
vv v
+ +
=
now v v ⋅ ∇ x f =
Ei ∇ x f
m  ∂x ∂y ∂z  m
v
2
vv v
∴ ∫ v v ⋅ ∇ x f d 3k =
Ei ∇ x n
m
0
v y2
[
[
]
]
1
v
v
v
E
E
=
2
2
i
vv
3
3
∴ ∫ v v ⋅ ∇ x fd k =
E i ∇ x n  →
E ∇ xn
3m
m
[
]
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
v
v r 3
v
v
∂ (nv ) 1
vv 3
v v v 3
+ ∫ ( f∇ x E ⋅ ∇ k )v d k + ∫ ∇ x ⋅ ( fv v )d k − ∫ ( fv ⋅ ∇ x )v d k = − n
∂t
τ
h
v
v
∂ (nv ) Fext
2 v
n+
E ∇ x n = −n
−
m
∂t
3m
τ
As before in the
text version
v
v
∂J n
τ
+ J n = nq µ n F + µ n k BT∇ x n
∂t
ALTHOUGH BOTH THE TEXT VERSION AND MY
VERSION ENDS UP WITH THE SAME ANSWER MY
APPROACH IS ACCURATE SINCE IT ACCOUNTS FOR
ALL THE TERMS IN THE GENERAL DRIFTDUFFUSION EQUATION.
Drift-Diffusion Equation for Electron and Holes – And Finally
v
v
∂J n
Taking τ
+ J n = nq µ n F + µ n k BT∇ x n
∂t
If the above equation is restricted to only zero order in Jn, then
∂J n
~0
∂t
v
v
∴ J n = nqµ n F + µ n k BT∇ x n
k T


Diffusion coeff  Einstein Re lation  Dn = µ n B
q


v
v
∴ J n = nqµ n F + qDn ∇ x n.......... .......... ......(1)
Similarly, for holes (moves in opposite direction);
v
v
∴ J p = npµ p F − pD p ∇ x p.......... .......... .....( 2)
Equation (1) and (2) are the Drift-Diffusion Equations for
Electrons and Holes respectively
Resources
•
Books
– The Physics of Semiconductors, Kevin F. Brennan, Cambridge University Press,
New York (1999)
– Introduction to Modern Statistical Mechanics, David Chandler, Oxford University
Press, New York (1987)
– Introduction to Statistical Thermodynamics, Terrell L. Hill, Dover Publications
Inc., New York (1986)
•
Websites
– A great site hosted by the UIUC, Some great 1-D derivations in statistical
mechanics
• http://www-ncce.ceg.uiuc.edu/tutorials/bte_dd/html/bte_dd.html
– A good site with introductory derivations on statistical mechanics and some
classical physics derivations, hosted by James Graham in UC-Berkeley
• http://astron.berkeley.edu/~jrg/ay202/lectures.html
– The Mathworld® site. I find it one of the most helpful to check out theorems and
formulae (I checked out the divergence theorem for this derivation)
• http://mathworld.wolfram.com/
End of Lecture