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# The sheet carrier density is 2 2 kf n kf ns gv

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Unformatted text preview: T = 0 K, all of the states with k < k F are occupied and all for k > k F are empty. 2 The area of occupied k- space is π k F In 2D, each state occupies an area in k- space of ( 2π ) 2 A So the number of states occupied is: ECE- 656 2 Fall 2013 Mark Lundstrom 8/24/2013 ECE 656 Homework 2: (Week 2) (continued) 2 π kF N= × 2 × gV 2 ( 2π ) A 2) where the factor of 2 is for spin degeneracy and gV is the valley degeneracy. The sheet carrier density is 2 2 kF N kF nS = = × gV = (because valley degeneracy is 2 for both cases consider here.) A 2π π Assume a finite temperature and work out the sheet carrier densities, nS , for: 2a) Electrons in the conduction band of a 2D parabolic band semiconductor 2b) Electrons in the conduction band (E > 0) of graphene. Solution: 2a) parabolic energy bands ∞ ∞ ⎛ m* ⎞ 1 nS = ∫ D2 D ( E ) f0 ( E ) dE = ∫ ⎜ gV 2⎟ ( E − EF ) k B T dE π ⎠ 1+ e EC EC ⎝ ∞ ⎛ m* ⎞ 1 nS = ⎜ gV 2⎟ ∫ ( E − EF ) ⎝ π ⎠ EC 1 + e kB T ∞ ⎛ m* ⎞ 1 dE = ⎜ gV 2⎟ ∫ ( E − EC + EC − EF ) ⎝ π ⎠ EC 1 + e kB T dE define: ηF = E F − EC k BT η= E − EC k BT dη = dE k BT dE = k BTdη wit...
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