12/10/09 1 ECE 495N, Fall 09 Fundamentals of Nanoelectronics Final examination: Wednesday 12/16/09, 7-9 pm in CIVL 1144. Cumulative, closed book. Equations listed below will be provided. Pauli spin matrices: (2x2) Identity matrix: x = 0 1 1 0, y = 0 i +i 0, z = 1 0 0 1 I = 1 0 0 1 Eigenvectors of. ˆ n x sin cos + y sin sin + z cos corresponding to eigenvalues +1 and -1 can be written as c s* s, c* respectively where c cos 2 ei/2, ssin 2 e+i/2 Basic equations of coherent transport µ 1 H µ 2 + 1. 1 =i[ 1 1 ] +, 2 =i[ 2 2 ] 2.G(E)=[EIH 1 2 ] 1, 3.A(E)=i[GG + ]=G 1 G + +G 2 G + 4.[G n (E)]= [G 1 G + ] f 1 +[G 2 G + ] f 2 1 2 Density of states /2 Electron density / 2 5.I i (E)= q h ((Trace[ i A])f i Trace[ i Gn ]) Current/energy 6. I(E)= q h Trace[ 1 G 2 G+ ](f 1 (E) f 2 (E)) 2-terminal current
12/10/09 2 Simpler version introduced earlier: 4a. n(e) = D(E) 1f 1 (E)+ 2 f 2 (E) 1 + 2 Electron density 5a. I i (E) = q i ( D(E)f i (E)n(E)) Current/energy 6a. I(E)= q 1 2 1 + 2 D(E)(f 1 (E) f 2 (E)) 2-terminal current In general H has to be replaced with H+U, and D(E) with D(E-U) where U has to be calculated self-consistently from an appropriate Poisson -like equation, but you can ignore this aspect in the following problems. Ballistic transport: 1 = 2 =v x /L. In diffusive regime di + /dx =di /dx =I/mfp, leading to a reduction in the current I =I + I by a factor mfp / (mfp+l). + C Q = q 2 de f D(E) Quantum Capacitance E G B = q2 h f(e)= + de f M(E) Ballistic Conductance E 1 e (Eµ)/kT +1 Fermi function P =(1/Z)exp((E µn )/kt) Law of Equilibrium h( k )= [ H nm ] m exp(i k.( d m d n )) Bandstructure D(E)= (E ( k )) Density of states k M(E)= (E ( k ))v z ( k )/L Density of modes k v z = k z Group velocity
12/10/09 3 There will be six problems on the test (6 x 5 = 30), evenly distributed over the semester. Non-MATLAB HW questions and past practice exams are a good guide to the kinds of questions on the test. In addition a few example questions are listed below to help you prepare. Solutions to these will also be posted. Problem 1 (Simple model for conduction): Suppose a material has a density of states (per unit energy), D(E)= AE and a density of modes (dimensionless), M(E)=BE, where A, B, and are constants. Assuming ballistic transport, obtain an expression for the conductance as a function of the electron density at T=0K. Hint: First, obtain separate expressions for the electron density and the conductance as a function of the equilibrium electrochemical potential µ. Problem 2 (Multi-electron picture): One-electron picture Multi-electron picture (state 11 excluded because it is energetically inaccessible) 01 10 Source I V I Drain 00 (a) A box has two degenerate energy levels both having energy. Use the multielectron picture to derive the correct expression for the maximum current if the electron-electron interaction energy is so high that no more than one electron can be inside the box at the same time. Your answer should be in terms of, assuming a bias polarity such that 1 < 2, as shown. (b) Assuming. if we reverse the polarity of the applied bias so that 1 > 2. will the current change? By what factor? Problem 3 (Law of Equilibrium): A channel has four energy levels all with the same energy, but the interaction energy is so high that no more than one of these levels can be occupied at the same time. Starting from the law of equilibrium, what is the average number of electrons in the channel if it is in equilibrium with chemical potential and temperature T? Your answer should be in terms of, and T.
12/10/09 4 Problem 4 (Bandstructure): Consider an infinitely long linear 1-D lattice (lattice constant: a) with one s-orbital per atom (assumed orthogonal) and having a site energy of E 0, so that the Hamiltonian looks like (t is real) Impose periodic boundary conditions and assume a solution of the form to find the dispersion relation E(k). Problem 5 (Density of states): A three-dimensional conductor (S: cross-sectional area, L: Length) has an E(k) relationship E =Bk 2. Find the density of states (per unit energy), D(E) and the density of modes (dimensionless), M(E) using periodic boundary conditions and assuming dimensions to be large enough that the sum can be replaced with an integral. Problem 6 (Subbands): A two-dimensional sheet having an E(k) relationship E =Bk 2 is rolled up along the x-axis to have a circumference c. Find the density of states as a function of energy. Assume periodic boundary conditions for y direction and replace sum with integral, but assume c small enough that the corresponding sum cannot be replaced with an integral. Problem 7 (Matrix model for conduction): (a) Consider an infinite wire modeled as a discrete lattice with points spaced by a -2-1 0 1 2 having a Hamiltonian with and (all other elements are zero), such that the dispersion relation is given by. What is the local density of states at the point 0, D(0,E)? (b) Suppose we cut the wire in part (a) into two separate semi-infinite wires as shown so that (other elements of the H-matrix remain unchanged). cut -2-1 0 1 2 What is the local density of states at the point 0, D(0,E)?
12/10/09 5 Problem 8 (spin): Consider a device with two spin-degenerate levels described by [H] = 0 0 and with one contact magnetized along +z with [ 1 ] = 0 0 0. Contact 2 is identical to contact 1, except that it is magnetized along +x instead of +z. What is the corresponding [ 2 ], in the same basis as [H] and [ 1 ]? Useful Relations: sin =2sin(/2)cos(/2), 1+cos =2cos 2 (/2), 1cos =2sin 2 (/2)