Question

Question 3: Assume a transistor's channel made of a 2-D carbon-nanosheet, with a length of 100 nm, and a width of 5 nm. If given the following energy/momentum relationship: |P E= + EC E > EC where P is the momentum, Ec is the energy level at the edge of the conduction band. 1) Plot the Energy momentum relationship. [5 marks] 2) Derive an expression for N(E). [10 marks] 3) Deduce an expression for the density of states D(E). [10 marks] 4) Plot the density of states D(E) as a function of the energy E. (5 marks] 5) What is the value of the material bandgap? Does the bandgap change with the change of the channel dimensions? Explain your answer. (15 marks]

Answer 1

© Assuming transistors channel made OF 2-D Carbon Nanosheet. Given length= loonm and width=5nm. momentum relationship is E-IP+ Ec ; EXE 4 08, (E-Ec) = 1P1 Given energy ox, PCE-Ec) = 11 IPI=P when pro =-p when PLO So, Case: P = (E-E) when P70 case-2:-P = 0 (E-E) When PLO 08, P- (EC-E) not possible because here E(TE. 0 Let us first plot the energy momentum relationship. i-e p = 0 (E-Ec) when E= Ec then p=0 Suppose E= 2ev and Ec= lev then p=0 E=lev and Ec = Oev then p=0 E=3er and Ec= zev then p=0 E=3ev and Ec= lev then p=20 then P=30 P1 E = 3 ev and Ec= Oev 78 54 407 34 24 $ 3, 30) - 0.0! 0 T - 2 3 4 5 6 7 (E-E) ev

Ans. Let us derive an expression for N(E). The total number of electrons in the energy reange E to EtdE ork to ktak or P to ptdp can be written in 2D TK² TK² L² 40 K²L² (24) 4T? We know the relation p=tk or, p² = t ² k² 08, K²= P² ħ2 2ME 7² 8² 22X (E-Ec)² SO, N(E) = Dale P?L? 40h2 In previous question 0 we have a relation i-e P= P(E-E) Now N(E)= $(E-E) L 0?L 47 h² h² Let us deduce an expression for the density of States. DCE). We know that DCE)= dN(E) h2 08, O(E) = 26 () (E-E) Let us now plot the D(E) as function of energy E Ans d [822K (E-Ec) dE dE DCE) Ans- constant E