Unformatted text preview: âˆ‚xi q Ï„F W m*
2
Ï…F Ï„ F W 2 6) Use the balance equation approach and derive a current equation for a semiconductor nanowire. You may assume that only 1 subband is occupied, but do not assume parabolic energy bands. Solution:
The general balance equation is âˆ‚nÏ† = âˆ’âˆ‡ â€¢ FÏ† + GÏ† âˆ’ RÏ† . âˆ‚t
For the 1D current, we use Ï† ( p ) = âˆ’ q Ï… x . () (i) (ii) (iii) (iv) The associated quantity is the electron current 1 nÏ† ( x , t ) = âˆ‘ Ï† ( p ) f x , p, t = I n Lp
The associated flux is 1 2
FÏ† x â‰¡ âˆ‘ âˆ’ qÏ… x Ï… x f x , p, t = âˆ’ qnL Ï… x . Lp
The associated generation rate is ( ECE
Â656 ( ) ) ( ) 14 Fall 2013 Mark Lundstrom âŽ§1
âˆ‚Ï† âŽ«
âŽª
âŽª
GÏ† = âˆ’ qE x âŽ¨ âˆ‘
f âŽ¬ . âŽª L p âˆ‚px âŽª
âŽ©
âŽ
The term inside the sum is âˆ‚Ï† âˆ‚ ( âˆ’ qÏ… x )
âˆ‚Ï…
=
= âˆ’q x
âˆ‚px
âˆ‚px
âˆ‚px
so we find âŽ§
âˆ‚2 E
âŽª1
GÏ† = âˆ’ qE i âŽ¨ âˆ‘ âˆ’ q 2
âˆ‚px
âŽª
âŽ©L p = âˆ’q (v) âˆ‚ ( âˆ‚E px )
âˆ‚2 E
= âˆ’ q 2 âˆ‚px
âˆ‚px (vi) (vii) âŽ«
1
âˆ‚2 E
âˆ‚2 E
âŽª
f âŽ¬ = q 2E i âˆ‘ 2 f = q 2E i nL
2
L p âˆ‚px
âˆ‚px
âŽª
âŽ 2
If we recognize âˆ‚ 2 E âˆ‚px as the inverse effective mass (not necessarily parabolic), then 1
GÏ† = q 2E i nL
m*
Finally, the recombination term is: 0
nÏ† âˆ’ nÏ†
I
RÏ† â‰¡
= n (viii) Ï„I
Ï„Ï†
Now we can put this all together beginning with eqn. (i) and using eqns. (iii), (iv), (vii) and (viii) to find the current equation as âˆ‚I n
I
d
1
2
=âˆ’
âˆ’ qnL Ï… x + q 2 nL
E x âˆ’ n . (ix) *
âˆ‚t
dx
m
Ï„I ( ) Now letâ€™s simplify this equation. Assume slow variations in time::
1
d
2
In = q2 Ï„ I
nLE x + Ï„ I
qnL Ï… x
*
dx
m
and define the mobility as
1
Âµn â‰¡ q Ï„ I
.
m*
With these assumptions, the current equation becomes ( I n = nL qÂµ nE x + q Ï„ I ( d
2
nL Ï… x
dx ) ) 2
If the quantity Ï… x is independent of position, then I n = nL qÂµ nE x + qDn dnL
, dx where ECE
Â656 15 Fall 2013 Mark Lundstrom 2
Ï… x Dn â‰¡ Ï„ I To summarize, under appropriate simplifying conditions, the 1D current equation for a
material with an arbitrary bandstructure can be written as
I n = nL qÂµ nE x + qDn Âµn â‰¡ q Ï„ I
Dn â‰¡ Ï„ I dnL
dx 1
m*
2
Ï…x Question: Does this work for a metallic carbon nanotube?
7) When deriving the momentum balance equation in 3D, a tensor, 1 Ï…i p j Wij = âˆ‘
f ( r , p, t ) Î©p 2
occurs. This problem asks you to write a balance equation for Wxx. You may assume parabolic energy bands and an electric field in the x
Âdirection. Solution: The general balance equation is âˆ‚nÏ† = âˆ’âˆ‡ â€¢ FÏ† + GÏ† âˆ’ RÏ† . âˆ‚t (i) (ii) (iii) (iv) For Wxx , we use Ï… x px 1 * 2
= m Ï… x . 2
2
The associated quantity is Wxx Ï† ( p) = 1 âˆ‘ Ï† ( p) f x, p, t = Wxx . Î©p
The associated flux is 1 âŽ›1
1 2âŽ...
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 Fall '14

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