Derivation of the Drift-Diffusion Equationalan.ece.gatech.edu/ECE6451/Lectures/Student...The Method...
Transcript of Derivation of the Drift-Diffusion Equationalan.ece.gatech.edu/ECE6451/Lectures/Student...The Method...
Derivation of the Drift-Diffusion Equation
Lecture Prepared By:Sanjoy Mukherjee
Drift-Diffusion Equation - Applicability
• 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
Basic equations governing transport in semiconductors and semiconductor devices:
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
( ) ( ) ),,(,,,, tkxCtkxBtkxA =+∴
Multiplying by the Moment generating Function Θn(k)( n = order of the moment)
( ) ( ) ( ) ( ) ( ) kdtkxCkkdtkxBkkdtkxAk kkk 333 ),,(,,,, ∫∫∫ Θ=Θ+Θ
Method of Moments Applied to the Boltzmann Transport Equation
τ0fffvfF
tf
xkext −
−=∇⋅+∇⋅+∂∂ vrv
h
v
The Boltzmann Transport Equation with relaxation time approximation:
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
( ) ( ) kdffkdfvkdfFkdtf n
xn
kextnn 30333 1
∫∫∫∫−
Θ−=∇⋅Θ+∇⋅Θ+∂∂
Θτ
vrvv
h
Method of Moments Applied to the Boltzmann Transport Equation
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
If Θ = 1 and n = 1 then:
( ) ( ) kdffkdfvkdfFkdtf
xkext30333 1
∫∫∫∫−
−=∇⋅+∇⋅+∂∂
τvrvv
hSimplifies to
( ) 0=⋅∇+∂∂ vntn
xvv
Carrier Continuity Equation
If Θ = v and n = 1 then:
Simplifies to
nqDFnqJ xnnn ∇+=vv
µ 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
∫∂∂ kfdvt
3r
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
Velocity is time independent
( )vnt
kdtfv vr
∂∂
=∂∂
∫ 3
( )
∫∫
∫
==∴
===
=
==
kdkfkdkfn
dkdN
V
statesofDensitykG
kdkfkGionconcentratcarriern
333
33
3
)(')(4
14
12
21)(
)()(
π
ππ
Drift-Diffusion Equation Derivation – 2nd. Term
( ) ( ) kdFvfkdFfv extkextk33 11
∫∫ ⋅∇−⋅∇vvr
h
vvr
h
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
f is finite and so the surface integral (integral of divergence of fvFext) at infinity vanishes identically
( ) FgFggFIdentityvvvvvv
⋅∇−⋅∇=∇⋅:
( ) ( )[ ]{ }kdvFfFvf kextextk31
∫ ∇⋅−∇rvvvrv
h
( ) ( ) ( )FgGGFgGFgIdentityvvvvvvvvv
⋅∇−⋅∇=∇⋅:
( )∫ ∇⋅− kdvFf kext31 rvv
h
Drift-Diffusion Equation Derivation – 2nd. Term (Continued)
( ) ( ) kdFvfkdvFf extkkext33 11
∫∫ ⋅∇−∇⋅−vvr
h
rvv
h
extx FEngSubstitutivv
=∇−
Substituting:
( ) ( ) kdEvfkdvEf xkkx33 11
∫∫ ∇⋅∇+∇⋅∇vvr
h
rvv
h
vEE xkxxkvv
hvvvv
⋅∇=∇⋅∇=∇⋅∇
( ) kdvvf x3∫ ⋅∇ vvr
Substituting, the second term is finally reduced to:
( ) ( ) ( ) kdvvfkdvEfkdfFv xkxkext333 11
∫∫∫ ⋅∇+∇⋅∇=∇⋅ vvrrvv
h
vvr
h
Drift-Diffusion Equation Derivation – 3rd. Term
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
( ) FgFggFIdentityvvvvvv
⋅∇−⋅∇=∇⋅:
( ) ( )∫ ∫ ⋅∇−⋅∇ kdvfvkdvfv xx33 vvvvvv
( ) ( ) ( )GFgGFgFgGIdentityvvvvvvvvv
∇⋅−⋅∇=⋅∇:
( ) ( )∫ ∫ ∇⋅−⋅∇ kdvvfkdvvf xx33 vvvvvv
( ) ( ) ( ) ( )∫∫ ∫∫ ⋅∇−∇⋅−⋅∇=∇⋅ kdvfvkdvvfkdvvfkdfvv xxxx3333 vvvvvvvvvvrr
Substituting, the third term is finally reduced to:
Drift-Diffusion Equation Derivation – Right Hand Term
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
( ) kdffv 30
1∫ −−r
τ
τ0vvn −
−
nvkfdvandnkfdcall == ∫∫ 33Re v
n = carrier concentration v = average velocity
At equilibrium the ensemble velocity v0 (by definition) = 0
ττvnkdffv −=
−− ∫ 30r
Substituting, the right hand term is finally reduced to:
Drift-Diffusion Equation Derivation – General Form
( ) ( ) kdffvkdfvvkdfFvkdtfv xkext
30333 1∫∫∫∫
−−=∇⋅+∇⋅+
∂∂
τrvrrvvr
h
r
( ) ( ) ( )
( ) ( ) ( )τvnkdvfvkdvvfkdvvf
kdvvfkdvEftvn
xxx
xkx
−=⋅∇−∇⋅−⋅∇
+⋅∇+∇⋅∇+∂
∂
∫∫ ∫
∫∫333
331
vvvvvvvvv
vvrrvv
h
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
Standard Drift-Diffusion Equation for Electrons/Holes
• Assumptions– The energy of the carriers,
– Mass is isotropic and constant
– Material is isotropic, and so the spatial temperature gradient is zero
The general Drift-Diffusion derived in the previous slides may be further simplified with the help of certain assumptions
mkE
2
22h=
2
21
iizyx mvEEEE ====∴
0=∇ ixEv
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
∫ ∇⋅− kdvFf kext31 rvv
h
extx FEngSubstitutivv
=∇−
mv
mE
mkEagain
EvEvmkE
mkE
k
k
kkkvmk
k
hvv
hvh
v
h
vvv
h
vv
hvh vv
=∇∴
=∇⇒=
∇=∇⇒∇= →=∇⇒= =
22
22
2222
2
112
∫− kfdmFext 3
v nkfdrecall =∫ 3
nmFextv
−
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
( ) [ ]
( ) [ ]nEEnm
kdvvf
fEEfmz
fEyfE
xfE
mvvfnow
mvEEEEtakingfE
fEfE
mv
vv
fvvf
xiixx
xiixzyx
x
iizyx
z
y
x
z
y
x
∇+∇=⋅∇∴
∇+∇=
∂∂
+∂
∂+
∂∂
=⋅∇
====
=
=
∫vvvvv
vvvvv
vv
2
22
21
000000
2
000000
3
2
2
2
2
Assuming the mass is isotropic and constant and therefore:
Assuming the material is isotropic i.e. temperature or energy is spatially independent
( ) [ ] nEm
nEEnm
kdvvf x
EE
xiixxi
∇ →∇+∇=⋅∇∴=
∫vvvvvv
322 3
13
0
Standard Drift-Diffusion Equation for Electrons/Holes-Text Version
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
( )τvnnE
mn
mF
tvn
xext −=∇+−
∂∂ v
v
32
etemperaturTwhereTkE
fieldFechelectronicqwhereFqF
B
ext
==
==−=
,23
,argvvv
( ) ( ) nTkmqn
mFqvqn
tvqn
xB ∇+=−+∂−∂ v
vτττ
2
mqmobilityelectron
vqnJdensitycurrentelectron
n
n
τµ ==
−==
nTkFnqJtJ
xBnnnn ∇+=+
∂∂ vv
µµτ
Notice that this term is completely ignored in the text
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
( )∫ ⋅∇ kdvfv x3vvv
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
( ) ( ) ( )GFgGFgFgGIdentityvvvvvvvvv
∇⋅−⋅∇=⋅∇:
( ) FggFFgIdentityvvvvvv
⋅∇+∇⋅=⋅∇:
[ ] [ ]∫∫ ∇⋅+⋅∇ kdfvvkdvfv xx33 vvvvvv
EEvcall xkkxx ∇⋅∇=∇⋅∇=⋅∇vv
h
vv
h
vv 11Re
[ ]∫∫ ∇⋅+
∇⋅∇ kdfvvkdEfv xxk
331 vvvvv
h
v
Assuming the material is isotropic i.e. energy is spatially independent
0
0=∇ Exv
Next Slide
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
∫ ∇⋅ kfdvv x3vvv
Assuming the mass is isotropic and constant and therefore:
[ ]
[ ]nEm
kdfvv
fEmz
fyf
xf
mEfvvnow
mvEEEEtakingE
EE
mv
vv
vv
xix
xii
x
iizyx
z
y
x
z
y
x
∇=∇⋅∴
∇=
∂∂
+∂∂
+∂∂
=∇⋅
====
=
=
∫vvvv
vvvv
vv
2
22
21
000000
2
000000
3
2
2
2
2
Previous Slide
[ ] nEm
nEm
kfdvv x
EE
xixi
∇ →∇=∇⋅∴=
∫vvvvv
322 3
13
Standard Drift-Diffusion Equation for Electrons/Holes-My Version
( ) ( ) ( ) ( )τvnkdvvfkdvvfkdvEf
tvn
xxkx −=∇⋅−⋅∇+∇⋅∇+∂
∂∫ ∫∫ 3331 vvvvvvrvv
h
( )τvnnE
mn
mF
tvn
xext −=∇+−
∂∂ v
v
32
nTkFnqJtJ
xBnnnn ∇+=+
∂∂ vv
µµτ
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 DRIFT-DUFFUSION EQUATION.
As before in the text version
Drift-Diffusion Equation for Electron and Holes – And Finally
nTkFnqJtJTaking xBnnnn ∇+=+
∂∂ vv
µµτ
If the above equation is restricted to only zero order in Jn, then
nTkFnqJ xBnnn ∇+=∴vv
µµ
0~tJ n∂∂
Similarly, for holes (moves in opposite direction);
qTkDlationEinsteincoeffDiffusion B
nn µ=
Re
( )1..........................nqDFnqJ xnnn ∇+=∴vv
µ
)2.....(....................ppDFnpJ xppp ∇−=∴vv
µ
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