ORIGINAL

NATIONAL UNIVERSITY OF SINGAPORE

EXAMINATION FOR

(Semester II: 2012/2013)

EE5431R - FUNDAMENTALS OF NANOELECTRONICS

Apr / May 2013 - Time Allowed: 2 Hours

INSTRUCTIONS TO CANDIDATES:

This paper contains FIVE (5) questions and comprises EIGHT (8) printed pages.

All questions are compulsory. Answer ALL questions.

This is a CLOSED BOOK examination with authorized material. The candidate isallowed to bring into the examination hall one A4 size sheet of formulae written on bothsides.

You may use the following data and formulae:

Magnitude of electron's charge le = 1.6 x10 -19 C

Permittivity of vacuum so = 8.85 x 10-12 F/m

Permeability of vacuum po 471- x 10 -7 T A--1 m

Boltzmann's constant k = 1.38 x 10-23 J IC/

Free electron mass me = 9.1 x 10-31 kg

Planck's constant h = 6.63 x 10-34 J s

e L2 (1 eV) = 1312.33; L = 10 nm

co1- 6(x' - x) f (x')dx' = f (x)

Continue on Page 2

EE5431R —Fundamentals of Nanoelectronics/ Page 2

Taking into account perturbation to the Hamiltonian by H', the resultant eigenenergiesand eigenfunctions are respectively given by

En = E'73, + (glfrig) 111,,H;ik

(E791 —

Hkr, 0 ,lEri ) =1E7D+ Ep lEk

k#n K

EP, and IE,9„) are respectively the eigenenergy and eigenfunction of the unperturbedHamiltonian and Hk = (E1,11H/lEn°).

Fermi's Golden rule27

wi,f = 1W(r)1 2 8(1-kof , — hco) + IH'(r)1 2 8(teluifi + tic())

kin

EE5431R —Fundamentals of Nanoelectronics / Page 3

Ql. The Spin Operator, S, has eigenvectors IR ) and IL) , where

SIR )= IR ) andSIL ) = )

The eigenvectors are orthonormal.

Using the completeness relation, express S in terms of the operators IR )(RI

and IL )(L I.

(2 marks)Show that S is Hermitian.

(1 mark)(iii) A new operator, P, is defined by P = IR )(LI + IL )(RI. Can P be an

observable? Explain your answer.(1 mark)

(iv) Let the eigenvectors of P be written as

1

I0) Val b2 (a1R ) + bIL ))

Show that 10 ) is normalized.(2 marks)

Find the values of a and b and hence the eigenvectors and theircorresponding eigenvalues.

(6 marks)Using the vector IR ) and IL ) as bases vectors for the 2-D space, determinethe matrix representing the operator .

(2 marks)Are S and P compatible operators? Explain your answer.

(2 marks)Given that

11 k ) = -(1x) + i lY

1IL)=-(Ix)-iIY))

where Ix ) and Iy ) are the x and y polarization states. Express 10 ) in termsof the I x ) and I y ) representations.

(2marks)(ix) Explain what the operator P does to ket vectors Ix ) and ly .

(2 marks)

EE5431R —Fundamentals of Nanoelectronics/ Page 4

Q2. Schrodinger's equation for a Dirac potential with energy —Vo (x) is given by

h 2 d20(x) vo8(x)0(x) = EiP(x)

2m dx2

The potential exists at -E < X < E with E -> 0 .

( i )

Show that the solutions for x < -E and x > E are given by

11)(x) =

Find )6 .

Hence, solve for the quantized energy E.

By normalizing the wavefunction ii (x), find the value of A.

(4 marks)

(4 marks)

(4 marks)Consider a perturbation of potential energy H' = —V1 8(x) to the energy- 170 6(x), find the correction to the total energy.

(4marks)

By comparing the solution obtained in Q2. Pt. (iv) by perturbation to the fullsolution obtained in Q2. Pt. (ii) with potential energy - (V0 + V1 )6(x), find

the value of(—vi

) so that the error obtained by perturbation method is lessvo

than 10%.(4 marks)

EE5431R —Fundamentals of Nanoelectronics / Page 5

Q3. Consider an infinite barrier quantum well with width L.

(i) Find the energy of the 1 st quantized level.

(2 marks)

(ii) Sketch the wavefunctions of the first and second energy levels.

(3 marks)

(iii) The hole and electron energy levels in a semiconductor quantum well

formed by two heterojunctions are shown in Fig. Q3.

Explain why the transitions e i hi and e 2 h 2 are allowed but not e 2 h1 or e1 h2 .

Consider Fermi's Golden Rule.

E0

e2

el

h1

h2

Ev

Fig. Q3.

(5 marks)

EE5431R —Fundamentals of Nanoelectronics/ Page 6

Q4. Fig. Q4 shows a two-dimensional system, in which each atom connects to six

nearest neighbor atoms with the distance of a nm. We wish to model this system

using a nearest neighbor Hamiltonian whose non-zero elements are

Hn,in = e if 2,/ =

Hn,n) = t if d -M

• •• •a • •• •• •• •• •

• • >x

• •

Fig. Q4

Find the primitive cell and its translation vectors for this 2D system.

(4 marks)

Based on (a), deduce the expressions of the reciprocal vectors.

(4 marks)

Sketch the 1 St Brillouin zone of the lattice.

(4 marks)

Deduce the expression of dispersion relation E(kx ,k , ) for this 2D system.

(9 marks)

(e) Deduce the expression of the effective mass of an electron at F point in thisband structure as the functions of t, a and h.

n

= a.

(6 marks)

EE5431R —Fundamentals of Nanoelectronics / Page 7

Q5. (a) Using Fig. Q5a. briefly explain the giant magnetoresistance (GMR) effect.

(4 marks)(b) Find the MR of the systems shown in Fig. Q5a as a function of spin polarization,

r — 1?P, where P =

(7 marks)

Parallel Configuration Anti-Parallel Configuration

I NM NM FM

r + 1?•

r i Channel rt NV

0A jA

Parallel R•

r Channc R V\ •v t z 'Vv'

Anti-parallel: r

Fig. Q5a

Question 5 is continued on Page 8

EE5431R —Fundamentals of Nanoelectronics/ Page 8

A one-dimensional conductor consisting of two different materials, A and B, isshown in Fig. Q5b. The dispersion relations, E(k), of these two materials can be

h 2=

k 2 2 2k,described as E A , and Ea =

h; E , where Ec > 0 and m is the

— 2m 2m

effective mass of material A and B. The wavefunction of the electron isexpressed as:

(1) (X) =

e ik lx + re

tei k

2x

x < 0,

x > 0,

Assume the total energy of the electron is E > E. Deduce the expressions of r

and I as the functions of k 1 and k2 .

(6 marks)

Material A Material B

x=o

Fig. Q5b

Based on the result in Q5. Pt. (c), deduce the expression of the near-equilibrium

conductance of this system as the functions of 1( 1 , k2 , q and h.

(6 marks)

End of Questions

END OF PAPER