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656 9 Fall 2013 Mark Lundstrom Which one is correct? Are both correct? If so, under what conditions is each of them correct? You can address this question using the concepts and approaches that we have been discussing in ECE
656. AFTER you figure out the answer, you may want to consult: [1] P.T. Landsberg, “D grad ν or grad(Dν)?,” J. Appl. Phys., 56, pp. 1119
1122, 1984 [2] P.T. Landsberg and S.A. Hope, “Two Formulations of the Semiconductor Transport Equations,” Solid
State Electronics, 20, pp. 421
429, 1977. Solution: According to Lecture 32, the proper current equation (assuming isotropic transport) is !2!
!
J n = nqµ nE + µ n!W . 3 (i) (ii) (iii) If the temperature is spatially uniform, then we get !
!
!
!
!
J n = nqµ nE + k BTe µ n!n = nqµ nE + Dn!n , (iv) (v) If we ignore the drift energy, then W= 2
k T , 3 Be which, when inserted in (i) gives !
!
!
J n = nqµ nE + µ n! nk BTe . ( ) where Dn = k BTe
µ . qn So equation (A) is correct when the temperature is uniform If the temperature is non
uniform, then we need to use (iii). If the mobility is spatially uniform, then we can write (iii) as !!
!!
!
J n = nqµ nE + ! nk BTe µ n = nqµ nE + ! Dn n , (vi) ( ) ( ) So eqn. (B) is correct when the mobility is spatially uniform. It would be better just to use eqn. (iii), which is correct in both cases. ECE
656 10 Fall 2013 Mark Lundstrom 5) Derive an energy balance equation for graphene, and then simplify it to terminate the balance equation hierarchy. (Consider only electrons in the conduction band with energies above the Dirac point, E = E D = 0 . Solution: The general balance equation is ∂nφ = −∇ • Fφ + Gφ − Rφ . ∂t (i) (ii) u = E p . (iii) (iv) (v) (vi) (vii) (viii) For the energy, φ ( p ) = E p , where E p is the kinetic energy. The associated quantity is ()
() 1 ∑ φ ( p) f x, p, t = W = nS u Ap
The associated flux is 1 Fφ i ≡ ∑ E ( p )υ i f x , p, t ≡ FWi Ap
The associated generation rate is ⎧
⎫
∂φ ⎪1
Gφ = − qE i ⎨ ∑
f ⎬ ⎪
⎪
⎩ A p ∂pi ⎭
and the derivative inside the sum is ∂φ ∂E
=
= υ i , ∂pi ∂pi
so we find ⎧1
⎫
⎪
⎪
Gφ = − qE i ⎨ ∑ υ i f ⎬ = E i J ni = J n iE . ⎪A p
⎪
⎩
⎭ ( nφ ( x , t ) = ( ) ) Finally, the recombination term is: 0
nφ − nφ W − W 0
Rφ ≡
=
τE
τφ () Now we can put this all together beginning with eqn. (i) and using eqns. (iii), (iv), (vii) and (viii) to find: W −W 0
∂...
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