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Week13HWSolutionsV2

7 when deriving the momentum balance equation in 3d a

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Unformatted text preview: ∂xi q τF W m* 2 υF τ 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 ∂nφ = −∇ • Fφ + Gφ − Rφ . ∂t For the 1D current, we use φ ( p ) = − q υ x . () (i) (ii) (iii) (iv) The associated quantity is the electron current 1 nφ ( x , t ) = ∑ φ ( p ) f x , p, t = I n Lp The associated flux is 1 2 Fφ x ≡ ∑ − qυ x υ x f x , p, t = − qnL υ x . Lp The associated generation rate is ( ECE ­656 ( ) ) ( ) 14 Fall 2013 Mark Lundstrom ⎧1 ∂φ ⎫ ⎪ ⎪ Gφ = − qE x ⎨ ∑ f ⎬ . ⎪ L p ∂px ⎪ ⎩ ⎭ The term inside the sum is ∂φ ∂ ( − qυ x ) ∂υ = = −q x ∂px ∂px ∂px so we find ⎧ ∂2 E ⎪1 Gφ = − qE i ⎨ ∑ − q 2 ∂px ⎪ ⎩L p = −q (v) ∂ ( ∂E px ) ∂2 E = − q 2 ∂px ∂px (vi) (vii) ⎫ 1 ∂2 E ∂2 E ⎪ f ⎬ = q 2E i ∑ 2 f = q 2E i nL 2 L p ∂px ∂px ⎪ ⎭ 2 If we recognize ∂ 2 E ∂px as the inverse effective mass (not necessarily parabolic), then 1 Gφ = q 2E i nL m* Finally, the recombination term is: 0 nφ − nφ I Rφ ≡ = n (viii) τI τφ 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 ∂I n I d 1 2 =− − qnL υ x + q 2 nL E x − n . (ix) * ∂t dx m τI ( ) Now let’s simplify this equation. Assume slow variations in time:: 1 d 2 In = q2 τ I nLE x + τ I qnL υ x * dx m and define the mobility as 1 µn ≡ q τ I . m* With these assumptions, the current equation becomes ( I n = nL qµ nE x + q τ I ( d 2 nL υ x dx ) ) 2 If the quantity υ x is independent of position, then I n = nL qµ nE x + qDn dnL , dx where ECE ­656 15 Fall 2013 Mark Lundstrom 2 υ x Dn ≡ τ 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 qµ nE x + qDn µn ≡ q τ I Dn ≡ τ I dnL dx 1 m* 2 υx Question: Does this work for a metallic carbon nanotube? 7) When deriving the momentum balance equation in 3D, a tensor, 1 υi p j Wij = ∑ f ( r , p, t ) Ωp 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 ­direction. Solution: The general balance equation is ∂nφ = −∇ • Fφ + Gφ − Rφ . ∂t (i) (ii) (iii) (iv) For Wxx , we use υ x px 1 * 2 = m υ x . 2 2 The associated quantity is Wxx φ ( p) = 1 ∑ φ ( p) f x, p, t = Wxx . Ωp The associated flux is 1 ⎛1 1 2...
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