Derivation of the Drift-Diffusion Equationalan.ece.gatech.edu/ECE6451/Lectures/Student...The Method of Moments In the Method of Moments, both sides of a equation are multiplied by

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