An analytical approach to quantum mechanical tunneling ...

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1-1-1991

An analytical approach to quantum mechanical tunneling time in An analytical approach to quantum mechanical tunneling time in

electronic devices electronic devices

Prabhaharan Thanikasalam University of Nevada, Las Vegas

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An analytical approach to quantum mechanical tunneling tim e in electronic devices

Thanikasalam, Prabhaharan, M.S.

University of Nevada, Las Vegas, 1992

U M I300 N. Zeeb Rd.Ann Arbor, MI 48106

AN ANALYTICAL APPROACH TO QUANTUM

MECHANICAL TUNNELING TIME IN ELECTRONIC

DEVICES

by

Prabhaharan Thanikasalam

A thesis subm itted in partia l fulfillment

of the requirem ents for the degree of

Master of Science in

Electrical and Computer Engineering

D epartm ent of Electrical and C om puter Engineering

University of Nevada, Las Vegas

A ugust, 1992

T he thesis of P rab h a h a ra n T han ikasa lam for th e degree of M aster of Science

in E lec trica l and C o m p u te r E ngineering is app roved .

■ g ■ ____

C h airp erso n . R. V en^arfasubram anian. P h .D

\ \I

\ L i u , F - U * ___________ ____________E x am in in g C o m m ittee M em ber. A bdol R ah im K hoie. P h .D

_______________________________E x am in in g C o m m iu e e M em ber, A shok Iyer, P h .D

G ra d u a te F acu lty R ep resen ta tiv e . Ajoy K u m a r D a tta , P h .D

________________G ra d u a te D ean , R onald WT S m ith , P h .D

U niversity of N evada, Las Vegas

A ugust. 1992

ii

<|A/VVV'-gr

A B S T R A C T

S ta rtin g from th e a n a ly tic a l so lu tio n to th e T im e-In d ep en d e n t Schrodinger E q u a­

tion , and ex p lo itin g th e ana logy betw een th e tran sm iss io n line eq u a tio n s an d th e

tim e -in d ep en d en t S chrod inger w ave eq u a tio n , an a n a ly tic a l expression for th e A v­

erage P a rtic le T raversa l (A P T ) tim e , t a p t , th ro u g h a re c tan g u la r p o ten tia l b a rrie r

region, u n d er no bias, is derived , in te rm s of th e b a rr ie r w id th , p o ten tia l, an d th e

inc iden t energy of th e e lec tro n . T h is app roach is ex te n d e d to derive an an a ly tica l

expression for th e A P T tim e th ro u g h a resonan t tu n n e lin g s tru c tu re , tw o sy m m etri­

cal rec tan g u la r p o te n tia l b a rrie rs sandw iching a p o te n tia l well, u n d er no bias. T h e

resu lts of th e single p o te n tia l b a rrie r trav e rsa l tim e are co m p ared w ith th a t of o th e r

approaches. T h e A P T tim e is inversely p ro p o rtio n a l to th e tran sm issio n coefficient,

and satisfies physically in tu itiv e energy lim its . For th e re so n an t tu n n e lin g s tru c tu re ,

th e A P T tim e is m in im u m a t re so n an t energies, an d th e t a p t is inversely p ro p o rtio n a l

to th e tran sm issio n coefficient. T h e m ax im u m frequency of o scilla tion is e s tim a ted

an d co m p ared for som e of th e ex p e rim en ta lly s tu d ied re so n an t tu n n e lin g s tru c tu re s

based on th e A P T tim e . T h e ag reem en t is excellen t.

A C K N O W L E D G M E N T S

I sincerely th a n k m y m a jo r professor Dr. V en k a tasu b ram an ian for his d irec tio n ,

m ystical ta len t of m ak in g th e m o st difficult an d su b tle p ro b lem s seem triv ia l, an d for

his incred ib le p a tien c e an d en d u ran ce in clarify ing m ost of m y ideas. I also th a n k D r.

A bdol R ah im K hoie for several discussions, th a t I h ad w ith h im , du rin g th e course

of m y study . I th a n k D r. M arc C ahay of U n iversity of C in c in n a ti for a n u m b er of

s tim u la tin g in d irec t d iscussions an d advice.

I th a n k D r. A shok Iyer for financially su p p o rtin g m e th ro u g h th e S um m er of 1991,

du rin g w hich I was ab le to co m p le te m ost p a r ts of m y thesis . S incere th an k s a re due

to D r. Ajoy K u m ar D a t ta for serv ing on m y th esis co m m itte e . I am g ra te fu l to Dr.

C hangfeng C hen an d D r. S tep h en Lepp of th e P hysics D e p a rtm e n t for th e d iscussions

I had w ith th e m d u rin g th e course of th is research w hich gave us th e confidence th a t

we were p roceed ing in th e rig h t d irection .

F inally , I w ish to exp ress m y g rea t ap p rec ia tio n for th e gestu res of m y friends an d

confidants for th e ir co n s tru c tiv e and so m etim es-to rm en tin g critic ism s w hich really

k ep t m e going, an d above all, p rev en ted m e from feeling bored an d lonely.

V

Contents

A b s tr a c t .......................................................................................................................................... iii

A cknow ledgm ents......................................................................................................................... iv

1 INTRODUCTION 1

1.1 Q uantum Mechanics and Schrodinger’s Wave E q u a tio n ...................................... 1

1.2 Tunneling P h e n o m e n a .................................................................................................... 2

1.3 Resonant Tunneling Phenom ena ............................................................................... 4

1.4 Overview of the T h e s is .................................................................................................... 6

2 LITERATURE OVERVIEW 7

2.1 Dwell T i m e ......................................................................................................................... 7

2.2 Phase-delay T im e .............................................................................................................. 8

2.3 B uttiker-Landauer T i m e ................................................................................................. 9

2.4 Collins-Barker M onte-Carlo Simulation T i m e ....................................................... 9

2.5 Average Particle Traversal (A P T ) T i m e ................................................................. 10

3 THEORETICAL FORMULATION 11

3.1 In tro d u c tio n ........................................................................................................................ 11

3.2 F o rm alism ............................................................................................................................ 11

3.3 Transmission Line Analogy ......................................................................................... 13

3.3.1 The Q uantum Mechanical Wave Im p e d a n c e ............................................ 14

3.3.2 Steady S tate Probability C urrent D e n s i ty ................................................ 15

3.3.3 Average Particle Traversal T i m e ................................................................. 16

vi

4 SINGLE BARRIER 17

4.1 Calculation of the Complex Coefficients of Wave Function Solutions . . . . 17

4.2 A PT Time Expression for the Single B a rr ie r .......................................................... 20

4.2.1 The A P T Tim e for E < VQ................................................................................. 21

4.2.2 The A P T Tim e for E > VQ................................................................................ 21

4.3 Various energy limits of t a p t ....................................................................................... 22

5 DOUBLE BARRIER 24

5.1 Calculation of the Complex Coefficients of Wave Function Solutions . . . . 24

5.2 The t a p t for a Resonant Tunneling S tru c tu re ...................................................... 26

5.2.1 The t a p t f°r the Left Barrier Region ........................................................ 27

5.2.2 The t a p t for the Potential Well R e g io n .................................................... 28

5.2.3 The t a p t for the Right Barrier R eg io n ........................................................ 29

6 RESULTS, COMPARISONS and DISCUSSIONS 31

6.1 Single B a r r ie r ..................................................................................................................... 31

6.1.1 Comparison of Dwell tim e, Phase-delay tim e, Buttiker-Landauer time,

t a p t , and the Classical time for, E < V a .................................................... 32

6.1.2 Comparison of Dwell tim e, Phase-delay tim e, B uttiker-Landauer time,

t a p t , and the Classical tim e for E > V0 .................................................... 32

6.1.3 Effect of the barrier w idth on the t a p t .................................................... 33

6.2 Double B a r r i e r .................................................................................................................. 34

6.2.1 Comparison of t a p t with Experim ental R e s u lts ...................................... 34

6.2.2 Effect of barrier w idth on the A PT t i m e .................................................... 35

6.2.3 Effect of well w idth on the A P T tim e ....................................................... 36

7 CONCLUSION 56

BIBLIO G RAPH Y................................................................................................. 57

8 APPENDIX A 63

8.1 Complex Coefficients of Wave Function S o lu t io n s ............................................... 63

vii

List of Figures

1.1 Q uantum mechanical tunneling: (a ) potential barrier of height V0 and thick­

ness W; (b ) wave function $ for an electron with energy E < V0, indicating

a non zero value of the wave function beyond the b a r r ie r ................................... 3

1.2 The I-V characteristics of a tunnel diode, a -b is the linear resistance region;

b -c is the negative differential resistance, (N D R), region; and c-d is the

exponential region. I p and I v are the peak and valley current. Vp and Vj are

the peak and forward voltage, respectively................................................................. 3

1.3 Q uantum mechanical tunneling in a resonant tunneling structu re with a bar­

rier w idth of b and well width of w ............................................................................... 4

3.1 Transmission line analogy: (a ) transm ission line circuit equivalent with a

load im pedance of Z l , (b ) quantum mechanical system with a potential step

barrier configuration.......................................................................................................... 12

4.1 Conduction band edge profile for a single rectangular potential barrier with

the corresponding wave function solution for different regions............................ 18

5.1 Conduction band edge profile for a sym m etrical double rectangular potential

barrier w ith the wave function solutions for different regions.............................. 25

6.1 The conduction band edge profile of a single rectangular potential barrier of

width 200.A and height 0 . 3 e V ....................................................................................... 37

viii

6.2 P lot of the (a ) Transmission Coefficient, and the (b ) A P T tim e, for a single

rectangular potential barrier for E < Va, with barrier w idth 200A and barrier

height Q.3eV ...................................................................................................................... 38

6.3 P lot of the (a ) Transmission Coefficient, and the (b ) A P T tim e, for a single

rectangular potential barrier for E > V0, with barrier w idth 200A and barrier

height 0.3eV ..................................................................................................................... 39

6.4 P lot of the Traversal Times : (a ) Dwell Time, (b ) Phase-Delay Tim e, (c)

B uttiker-Landauer Tim e, (d ) A P T tim e, and (e ) Classical Tim e for E < F0,

w ith barrier width 200A and barrier height 0 .3 eF .................................................. 40

6.5 P lot of the Traversal Tim es : (a ) Dwell Time, (b ) Phase-Delay Time, (c )

B uttiker-Landauer Time, (d ) A P T time, and (e ) Classical Tim e for E > V0,

with barrier width 200A and barrier height 0.3eV .................................................. 41

6 .6 P lot of the Traversal Tim es : (a ) Dwell Time, (b ) Phase-Delay Tim e, (c )

B uttiker-Landauer Tim e, (d ) A P T time, (e ) Classical Tim e, and the ( f )

Transmission Coefficient for E > V0, w ith barrier w idth 200A and barrier

height 0.3eV .......................................................................................................................... 42

6.7 3-Dimensional surface plot of the A PT tim e, for the case of E < V 0 w ith the

barrier height l.O eF and barrier width in the range 25A to 250A ..................... 43

6 .8 3-Dimensional surface plot of the A P T tim e, for the case of E > V0 with the

barrier height l .QeV and barrier w idth in the range 25A to 250A ..................... 44

6.9 The conduction band edge profile of a sym m etrical double rectangular po­

tential barrier s tructu re with a barrier height of 0.956eF, barrier width of

30A and a well width of 100A........................................................................................ 45

6.10 Plot of the (a ) Transmission Coefficient and the (b ) A P T tim e for a sym­

m etrical double rectangular potential barrier structure w ith a barrier height

0.956eV, barrier width 30A and a well w idth 100A, for E < V Q........................ 46

6.11 T he conduction band edge profile of a sym m etrical double rectangular po­

tential barrier structu re w ith a barrier height 0.23eV, barrier w idth 50A and

a well width 50A.................................................................................................................. 47

ix

6.12 Plot of the (a ) Transmission Coefficient and the (b ) A PT time for a sym ­

m etrical double rectangular potential barrier structu re with a barrier height

0.23eV, barrier width 50A and a well width 50A, for E < V 0............................. 48

6.13 The conduction band edge profile of a sym m etrical double rectangular po­

tential barrier s tructu re with a barrier height l.OeV, barrier width 25A and

a well width 45A................................................................................................................. 49

6.14 P lot of the (a ) Transmission Coefficient and the (b ) A PT tim e for E < V0 for

a sym m etrical double rectangular potential barrier structure with a barrier

height l.OeV, barrier w idth 25A and a well w idth 45A......................................... 50

6.15 The conduction band edge profile of a sym m etrical rectangular double po­

tential barrier structu re of barrier height O.ZeV, barrier width in the range

from 30A to 100A and well width 30A, for E < V0................................................. 51

6.16 3-Dimensional surface plot of the A PT tim e for E < V0. The barrier height

is 0 .3ey and the barrier w idth in the range from 30A to 100A and the well

width is 30A.......................................................................................................................... 52

6.17 The conduction band edge profile of a sym m etrical rectangular double po­

tential barrier s tructu re of barrier height l.OeV, barrier width 30A and the

well width in the range from 30A to llO A ................................................................. 53

6.18 3-Dimensional surface plot of the A PT tim e for E < V0. The barrier height

is l.OeV and barrier w idth 30A and the well w idth in the range from 30A to

llO A ........................................................................................................................................ 54

6.19 3-Dimensional surface plot of the (a ) A PT tim e and the (b ) transm ission

coefficient for E < V a. T he barrier height is l.OeV and barrier width is 30A

and well width in the range from 30A to llO A ........................................................ 55

1

C hapter 1

INTRODUCTION

1.1 Q uantum M echanics and Schrodinger’s W ave Equation

T he S c h ro d in g e r E q u a tio n is analogous to a classical energy conservation equation, and

describes the dynamics of a quantum particle. A quantum particle is one whose wave­

length is very small compared to the dimensions of the system. W hen the dimensions of

the dynam ic system of the particle are extrem ely small, the classical mechanics does not

explain m any of the experim ental observations, such as diffraction and tunneling. The

Schrodinger equation was developed to explain these physical phenomena. The time-

dependent Schrodinger equation in one-dimension for a quantum particle subjected to a

potential, V (x ) , is given by:

+ * (■ > *< *»= « =

where m m is the effective mass of the particle, h is the modified P lanck’s constant, and

E is the energy of the particle. T he tim e-independent Schrodinger equation which holds

good when the to ta l energy of the particle is independent of tim e, is given by:

d2iS{x) , , lT / ._ i i + _ r [ £ - n i )W x ) = 0

The boundary conditions are: ^ n(a: — ± 0 0 ) = 0 and \I/nU’) and are continuous

everywhere, —0 0 < x < 0 0 . The number of solutions to this equation are infinite. The eigen

values and functions are denoted with n as the index - E n and The physical meaning

of ^ n i x ) is th a t \ ^ n(x ) \2dx provides the probability of finding the quantum particle between

x and x + dx with unity probability of finding the particle in —00 < x < 0 0 .

1.2 T unneling P henom ena

In a single potential barrier comprising of G a A s / A l A s / G a A s , conduction band edge profile

shown in F igu refl.l], as the height of the potential barrier is finite, is now zero a t the

A l A s / G a A s barrier interface. $ n and 'n are continuous and non-zero a t each boundary of

the barrier, and and U>'n are non-zero within and beyond the potential barrier. Since

has a non-zero value to the right side of the barrier as shown in F igurefl.l], is non­

zero, implying th a t the probability of finding the particle with E < V 0 beyond the barrier

region is finite. According to classical mechanics, the probability of finding the particle with

E < V0 beyond the first A l G a A s / G a A s interface is zero, since such a real space transfer

of the particle through a potential barrier region is p rohibitted classically. The physical

mechanism by which the particle, with E < V0, penetrating a finite potential barrier is

called quantum m echanical tunneling through the barrier. The tunneling probability is

directly related to the energy of the particle, E , relative to V0 and the barrier w idth, d.

The first device proposed, based on the tunneling phenom ena, was the tunnel diode.

The Tunnel diode is often called the Esaki diode after L. Esaki [20, 23], who in 1973

received the Nobel prize for his work on this effect. The basic structure is a p+n + diode

with p and n regions are degenerately doped so th a t the depletion layer region is very thin.

Due to thin depletion layer, electrons in the conduction band can tunnel through the thin

depletion region to the valence band electrons. A tunnel diode exhibits the critical feature

of negative differential resistance (NDR), over a portion of i t ’s I — V characteristics, as

shown in Figure[1.2]. In NDR region, the I — V characteristics exhibits a negative slope,

i.e., the quantity is negative.

3oo

(a)

L + W

E xponential decrease inside barrier

i/' t6 0 beyond barrier<b)

Figure 1 .1 : Quantum mechanical tunneling: (a ) potential barrier of height V0 and thickness W ; (b ) wave function $ for an electron with energy E < F0, indicating a non zero value of the wave function beyond the barrier

Figure 1.2: The I-V characteristics of a tunnel diode, a -b is the linear resistance region; b -c is the negative differential resistance, (NDR), region; and c-d is the exponential region. I p and I v are the peak and valley current. Vp and V/ are the peak and forward voltage, respectively.

4

Resonant E n e rg y Levels

w

Figure 1.3: Q uantum mechanical tunneling in a resonant tunneling structure with a barrier width of b and well width of w.

1.3 R esonant T unneling P h en om en a

W ith the advent of Molecular Beam E pitaxy (M B E), it is possible to grow th in layers of

A lA s sandwiched between GaAs, thus creating a po tential barrier in the conduction band

edge profile, as shown in F igure[l.l]. The thickness of the A l A s layers can be as small as

lOA, which is 4 mono layers of AlAs . Double barrier structures as shown in Figure[1.3],

successfully grown by M BE, in which a G a A s layer (well) is sandwiched between two A l A s

barrier layers, quantized energy levels, d ictated by quantum mechanics, exist w ithin the

well region. W hen the incident energy of the particle in the free propagating region ou t side

the potential barrier equals one of the quantized energies, the transm ission probability is

unity, i.e., resonant tunneling results. For energy values o ther than the resonant ones, the

transm ission probability is less th an unity. This is referred to as non-resonant tunneling

phenomena.

R. Tsu and L. Esaki[20] proposed a superlattice structu re in 1969 for application in

negative differential resistance (NDR) devices. In 1972, L.Esaki et. al. reported for the

first time, the observation of NDR in a G a A s / A l G a A s superlattice[22]. Two years later

5

L.L. Chang et. a l ., observed NDR at tem peratu re below 77°K in a double barrier resonant

' tunneling diode, i.e. two periods of a G a A s / A l G a A s superlattice.

In 1983, Sollner et. al .[48] reported a large NDR region in the I — V characteristics

of G a A s / A l G a A s double barrier resonant tunneling device, with a peak-to-valley ratio of

6:1 a t 25°K . Moreover, the current response m easured at a driving frequency of 2.5 THz

was rem arkably similar to the response expected from DC m easurem ents, indicating the

potential for high frequency applications[25]. A year later, Sollner et. al. reported the first

high frequency oscillations generated by a resonant tunneling device at frequencies up to 18

GHz[47]. In 1985, Shewchuk et. al. reported the first room tem perature observation of the

NDR in a G a A s / A l G a A s system[43, 44].

The dem onstration of resonant tunneling phenom ena has led to a num ber of proposals for

devices with a third term inal to control the NDR characteristics. Capasso et. al. proposed a

heterojunction bipolar transistor with a single quantum well in the base region [13, 14, 15].

T here are o ther resonant tunneling devices proposed by Luryi et. al .[34], Ray et. al.[42]

using M OCVD a t 300° K. Tsuchiya et. al .[49] reported room -tem perature observations of

negative differential resistance in 1985.

Bonnefoi et. al .[5, 6 ], proposed a device; Negative Resistance S tark Effect T ransistor

(N E R SET). This is a double barrier resonant tunneling device with an ex tra (base) contact.

Since this base is shielded by a thick 1000 — 1500A potential barrier, the base current is

negligible. N akata et. al. proposed a triode w ith a m etal-insulator superlattice in the base,

acting as an artificial semiconductor[35]. This device, called resonant electron transfer triode

(R E T T ), is excted to perform well in high-speed applications because of low resistivity of

m etal contacts. Due to the periodicity of the m etal-insulator superlattice, an artificial

conduction band in the base region is formed. W hen the device is biased such th a t em itter

Fermi level is aligned with the artificial conduction band in the base, electrons resonantly

tunnel from em itte r to collector.

M agnetic-field-induced resonant tunneling was discussed by Ram aglia et. al .[39]. Recent

paper by Glazer et. al. discusses the case of tunneling through highly transparen t double

barriers[24]. Ranfagni et. al. reported delay-tim e m easurem ents in narrow wave guides as

a test of tunneling through single barrier[40, 41]. A thorough recount of the history of

6

resonant tunneling can be found in Reference[2 1 ].

In the context of tunneling devices, the two physical quantities of interest to device

physicists and engineers are the tunneling current and the tunneling time. The subject of

this thesis is the la tte r quantity, the tunneling time. Specifically, based on average particle

velocity during tunneling, the tunneling time for a single and double barrier are studied.

Henceforth, this tunneling time is called the Average Particle Traversal tim e or A PT time.

1.4 O verview o f th e T hesis

A brief literature survey, discussing various theoretical approaches to tunneling times is

presented in C hapter 2. The analogy between the solutions to the one-dimensional time-

independent Schrodinger equation and transmission line equations is exploited and the

derivation of A P T tim e based on the analogy are discussed in C hapter 3. Analytical ex­

pressions for A PT tim e for the case of a single potential barrier and a symmetrical double

barrier under no bias are derived in Chapters 4 and 5, respectively. Results and discus­

sion are presented in C hap ter 6 . Conclusions along with the proposal for future work are

presented in C hapter 7.

C hapter 2

LITERATURE OVERVIEW

The prospect of high-speed devices based on resonant tunneling structures has brought new

urgency to understand every aspect of tunneling phenom ena for the form ulation of the the­

ory of dynamics of such systems. The question which is relevant to the dynamics of such a

system and th a t has resulted in a wealth of literature is “ H o w long d o e s it ta k e fo r a

p a r tic le to tu n n e l? ” .

The recent theoretical work on the tunneling times has centered around one-dimensional

models. W ithin this limited area of research, there are atleast six different approaches sug­

gested in the literature. These approaches are : the dwell time[45], the phase-delay time[26,

50], the B uttiker-Landauer traversal time[7, 8 , 9, 10], the complex traversal time[37, 38, 46],

the Collins-Barker M onte-Carlo sim ulation tim e[16 ,17,18], and the Average Particle Traver­

sal (A PT ) tim e. No two of these approaches agree[27, 28, 29, 32, 33]. . In this chapter, five

approaches are reviewed and contrasted.

2.1 D w ell T im e

The dwell time[45] is, in the context of the scattering of particles with fixed energy, the time

spent in any finite region of space, averaged over all the incoming particles. Thus, the dwell

tim e can serve as a reference point in any discussion on tunneling times. This describes the

average tim e a particle dwells within the barrier irrespective of it either reflects or transm its

at the end of i t ’s stay. The dwell tim e, T d w e u , is defined as:

NT d w e l i — j

where N is the number of particles within the barrier region and J is the incident flux of

the particles. The Tdweu for a single rectangular barrier can be shown to be equal to[7]:

_ m ' k , 2 a d ( a 2 — k 2) + k 2 sinh(2 ad ) dweil h a 1 4k2a 2 + k% sinh2 (ad )

where m* is the effective mass, h is the modified Planck’s constant, k =

a = y j 2m ancj ko = with E being the energy of the incident electrons. V0

and d are the barrier height and w idth, respectively. The dwell tim e in a resonant tunneling

s tructu re is discussed by Pandey et. al. [36].

2.2 P hase-delay T im e

T he other well established tunneling tim e concept is the phase-delay time[26, 50]. A time

delay for the scattering process can be calculated by following the peak of a wave packet

via. the m ethod of stationary phase[26]. Phase-delay time is the tim e interval between

the tim e the peak of the incident wave enters the barrier and the time the peak of the

transm itted wave appears beyond the barrier. The expression for the phase-delay tim e for

single rectangular barrier is given by[7]:

_ m* 2 a d k 2( a 2 — k 2) + k* sinh(2 ad ) phase h k a 4k 2a 2 + k* sinh2(ad )

A strong deform ation of the wave packet will result when the wave packet in teracts with

a thick barrier. This deformation m ay shift the peak of the wave from k in the incident

wave to k' in the transm itted wave, with k ^ k ' . Thus, the traversal time calculated by this

m ethod of following the peak of the wave packet becomes meaningless, as the same particle

is not used for the time delay measurem ent.

2.3 B uttiker-L andauer T im e

B uttiker and Landauer[7, 8 , 9, 10] considered tunneling through tim e-dependent rectangular

barrier with a small oscillating com ponent added to the sta tic barrier height. For a slowly

varying potential, the additional time dependence of the transm itted wave is caused by the

variation of the transm ission probability with the height of the barrier. If the potential

oscillates fast com pared to the traversal tim e (u> >> 1 / r ) , then the particles see a time-

independent barrier of average height V0. For slowly varying potential (u> << 1 / r ) , the

tunneling particles see an effective tim e-dependent sta tic barrier of height V(t) . Identifying

the transition frequency at which the static barrier becomes an oscillating barrier for the

particles, provides one with the inverse of a traversal tim e. T he expression for B uttiker-

Landauer traversal tim e for a single rectangular barrier is given as:

T B - L = (T~dwell + Tz ) ^

where rz is given by:

_ m ’k 2 (a 2 - k 2) sinh2 (ad ) + (k 2d a / 2 ) sinh(2 ad) z h a 2 4 k2a 2 + k% sinh2 (ad )

In this approach, the particles th a t are tunneling and those which are reflected are

differentiated.

2.4 C ollins-B arker M onte-C arlo S im ulation T im e

In this approach, a Gaussian wave packet with a particular standard deviation k, is made

to impinge on a potential barrier and the time delay associated between the entrance of

10

the peak of the incident wave and the appearance of the peak of the transm itted wave be­

yond the potential barrier, is com puted numerically [16, 17], using the conventional Monte-

Carlo approach. Based on excellent agreem ent of M onte-Carlo tim e and phase-delay time,

it was concluded th a t the phase-delay time result originally obtained by Wigner[50] and

Hartman[26] is the best expression to use for a wide param eter range of barriers, energies

and wave packets.

2.5 A verage P artic le Traversal (A P T ) T im e

In this approach, an average tunneling velocity at steady s ta te is defined as v av(x) =

with J being independent of x, and j ) | 2 is the probability density function a t any

point x along the barrier[31]. The analogy between the solution to the tim e-independent

Schrodinger equation and the steady s ta te transm ission line equation for a loss-less homoge­

neous transm ission line with a load, is exploited. A quantity analogous to the characteristic

impedance of the transm ission line called, Q uantum Mechanical Wave Im pedance (QM W I),

Z (x ) , is derived in term s of the complex coefficients of the solutions to the wave function

as a function of a:[3, 31]. T he v av{x) is then related to Z (x ) and the A P T time, t a p t , is

obtained as:

where d is the w idth of the barrier.

The integral expression given by Eq.[2.1] was used to obtain the tunneling time through

a delta function by Anwar et. al .[4] numerically. The tunneling through an em itter-base

t a p t ( 2 . 1 )

junction of a H eterojunction Bipolar T ransistor was investigated by Cahay et. al. using

numerical in tegration of the Eq.[2.1][ll, 12].

11

C hapter 3

THEORETICAL

FORMULATION

3.1 In troduction

In this chapter, a discussion of the analogy between the the solution to the tim e-independent

Schrodinger equation and transm ission line equations in one-dimension are presented. Ex­

ploiting this analogy, a quan tity called the Q uantum Mechanical Wave Im pedance (QM W I)

is derived in term s of the wave function solutions. Using QM W I, the average tunneling

velocity is derived and is used to derive the A PT t im e ,^ / ^ .

3.2 Form alism

Let a flux of electrons, w ith energy E , be incident on the potential barrier, as shown in

Figure [3.1]. The dynam ic equation governing the electron system is the tim e-dependent

Schrodinger equation. T he Schrodinger equation at steady s ta te is given by:

d2$(a:) 2m *r „ T_.- 3^ + -i F [ r - r W J * W = o (3 .1)

12

V,In c id e n t E le c tro n s

R e fle c te d E le c tro n s

R e g io n 1 R e g io n 2

i = 0

(a ) (b )

Figure 3.1: Transmission line analogy: (a ) transm ission line circuit equivalent with a load impedance of Z l , (b ) quantum mechanical system with a potential step barrier configura­tion.

The solution to this time-independent Schrodinger equation in region 1 with conduction

band edge potential profile, as shown in Figure[3.1], can be w ritten as:

®(ar) = A +(eax - pe~ax) (3.2)

where a = 7 + j(3 = j \ J 2n}̂ xH E - V0)

p is the wave am plitude reflection coefficient, a is the propagation constant, and E is the

energy of the incident electrons. 7 and (3 are the real and im aginary parts of the propagation

constant a, respectively.

In particular, the wave equation for regions 1 and 2 can be w ritten as:

$ i ( s ) = A i ( e aiX - pe~aix) x < 0 (3.3)

* 2 (a ) = A+e° 21 x > 0 (3.4)

respectively,

where a { = j - W) = l i + j d ,

m* (z),V i,(i = 1 , 2 ) are the effective mass, and the potential, respectively, for the i th

region. Here, 7 ; and Qt are the real and im aginary parts of the propagation constant a,- for

the ith region.

There is no reflection of the wave in region x > 0, because the region is homogeneous

and of infinite extent. Applying the boundary conditions at x = 0, $ 1 (2 = 0) = = 0)

and ^ ( a : = 0 )/m ^ = $ '2(x = 0 ) ! m \ , an expression for p is obtained as:

D ifferentiating Eq.[3.2] with respect to x and m ultiplying both sides of the equation by

3.3 T ransm ission Line A nalogy

The expressions for voltage, V(a;), and current, I ( x ) , along the homogeneous lossless tran s­

mission line w ith generalized d istributed im pedance, are given by:

_ [q 2 / ^ 2 ~ oil/ml] [a 2/m"2 + a i l m ,{]

(3.5)

a factor an expression for $ (x ) is obtained as:

(3.6)

where Z0 = -M-

I ( x ) = I +(eax - T te~ax) (3.7)

V{ x) = I +Z 0(eax + Yte~ax) (3.8)

where

14

where I \ is the wave am plitude reflection coefficient.

Z l and Zo are the load and characteristic impedance of the transm ission line, respec­

tively. Com paring the expressions for \P(.r) and $ (x ) given by Eqs.[3.2] and [3.6] respec­

tively, with the expressions for / ( x) and V ( x ) for transm ission line given by Eqs.[3.7] and

[3.8], respectively, it is observed th a t they are analogous. Zo in Eq.[3.2] and [3.6] is the

quantum mechanical analog of the characteristic impedance of the transm ission line, Zo

given by, - \ / Z ] Y where Z and Y are the series impedance and the shunt adm ittance, per

unit length of the transm ission line, respectively.

3.3.1 The Quantum M echanical Wave Im pedance

At any plane x , the Q uantum Mechanical Wave Impedance (QM W I) [31] can be obtained

from Eq.[3.2] and [3.6] as:

« * > = i 5 5 <»•»>

Z ( x ) can be re-w ritten as:

2 h V' (x)j m* ( x ) $ (x )

= Z( x ) = R(x) + j X { x ) (3.10)

where R ( x ) and X ( x ) are the real and im aginary parts of Z( x ) a t any point x, looking in

the positive x direction.

M ultiplying both sides of Eq.[3.10] by jm2/[— , Eq.[3.10] modifies to:

= j'k(x ) + V(x) (3.11)

where k ( x ) = m and rj(x) = m are the propagation and attenuation constants

of the wave function, respectively.

Integrating Eq.[3.11] from x = 0 to any x, \P(a:) can be w ritten as:

« (* ) = 9 0efo v^)dxej f gz K(x)dx (3.12)

15

where 4/(a;) is the wave function at any point x. and is the incident wave function at

the x = 0 boundary.

3.3.2 Steady State Probability Current D ensity

The wave function, ^(a:), can be used to express the probability current density, J{x) as

follows. Using Eq.[3.12], the steady s ta te probability current density, J ( x ) , a t any point x

can be w ritten in term s of R( x ) and rj(x) as:

where ^ " ( x ) is the complex conjugate of the wave function, ®(x). At steady s ta te , the

current continuity equation necessitates th a t the probability current density everywhere

along the barrier be equal in the absence of any generation or recom bination mechanisms.

Using Eqs.[3.12] and [3.13], J{x) can be w ritten as:

The current density, J ( x ) , can be defined in term s of an average steady sta te velocity,

v av(x), and probability density as:

$ ' ( * ) $ ( * ) ] = | | ® 0 | 2R ( x ) e 2f o r,{x)dx (3.13)

J = ± R ( x ) M x ) \ 2 (3.14)

(3.15)

This equation is similar to the drift current density equation in term s of the drift velocity,

and charge density. Considering the probability current density again:

(3.16)

J ( x ) can be modified to:

J{x ) = A fle[S (*)**(*)] = ±Re[ V{ x )F{ x ) ] (3.17)

Thus, J ( x ) is analogous to the average power in the transm ission line. V ( x ) is the

1 6

voltage, I ' ( x ) is the complex conjugate of the current a t any x along the transm ission line.

3.3.3 Average Particle Traversal T im e

Comparing Eqs.[3.14] and [3.15], vav(x) can be w ritten as :

v av{x) = ^-R(z) (3.18)

In o ther words, the average velocity of the particle at any x is one half the real part

of the QM W I. The t a p t required for a particle to move an elemental distance, d x , a t any

point x is given by[3]:

dxdTa p t = ---- -r-; (3.19)

V a v ( x )

Using Eqs.[3.16] and [3.19], an integral expression for the time required to traverse a

distance L can be obtained as[3]:

t a p t = f d r = 2 f (3.20)Jo Jo R{x)

where R( x ) is the real part of the Q uantum Mechanical Wave Im pedance, Z(x) . It is

noted th a t using Eq.[3.10], Z( x ) can be obtained from the wave function solution to the

Schrodinger equation, given by Eq.[3.3]. Identifying the real p a rt, R(x) , and using Eq.[3.20],

t a p t cai* be obtained for any structure.

17

C hapter 4

SINGLE BARRIER

In this chapter, the A PT tim e, t a p t , through a single rectangular potential barrier is

considered. Based on the theoretical form ulation, discussed in C hapter 3, an analytical

expression for the t a p t is derived. In order to calculate the t a p t for the barrier region,

the real p a rt of the QM W I should be known. T he real p art of the quantum mechanical

wave im pedance can be obtained from the wave solution to the Schrodinger equation in

th a t particu lar region. By knowing the real part of the QM W I, and using Eq.[3.20], an

analytical expression for the t a p t is derived in term s of the incident energy, barrier height

and width.

4.1 C alcu lation o f th e C om plex C oefficients o f W ave Func­

tio n Solutions

The solutions to the tim e-independent Schrodinger equation for a rectangular potential

barrier s tructu re shown in Figure[4.1] is given by:

* i ( a r ) = e ikx + A e ~ i k x . . . . . . . . . . . . . . . . . . X < 1 T

18

fcl II

B e ux -f- C'c~a~

eikx _|_ A e-ikx D e ikx

£ — 0 R egion 1 ' {Region 2 R egion 3

x = 1 H II O H li. d ' 2

Figure 4.1: Conduction band edge profile for a single rectangular potential barrier w ith the corresponding wave function solution ‘for different regions.

$ 2(z ) = B e ax + C e~a x .................. -£■ < x < ^ (4.2)

V z {x) = D eikx ..................x > ^ (4.3)

where A, B , C, and D are the complex coefficients, a is the attenuation constant given by

\ J 2m k is the propagation constant given by \ J 2rr/*iE , V0 and d are the height and

width of the potential barrier, respectively, and E is the incident energy of the particle.

Boundary Conditions

The boundary conditions a t i = y and x = | are th a t the wave function, $ ( 2 ), and the

derivative of the wave function, $ ’(x) be continuous, which are given by:

« ! (* = - d / 2 ) = $ 2(a; = - d / 2 ) .................* = (4.4)

19

J$ 2 ( 2 = d/2) = ^ 3(x = d / 2 ) x = - (4.5)

^ ( a : = - d / 2 ) - 'S>2(x = ~ d / 2 ) .................x = - - (4.6)

= d/2) = $ '3(x = d / 2 ) * = ^ (4.7)

Substitu ting the respective wave function solutions from Eqs.[4.2] and [4.3] in Eqs.[4.4] -

[4.7], a t the interface x — | , an analytical expression for B and C can be obtained as:

D e ' W ( a + ik)2 ae<*d!2 ̂ *

c = (4 .9)2ae~ad/ 2

An expression for the complex coefficient A can be obtained by using the continuity condi­

tion on ̂ (a:) and ^ ( z ) a t the interface x = — | as follows:

e - i k d /2 + A e i k d /2 _ B e - a d /2 + C e c d /2 ^ 4 1 0 )

ike~ ikdf2 - i k A e ikd/2 = a B e ~ adl 2 - a C e ad/2 (4.11)

Solving Eqs.[4.10] and [4.11] simultaneously and using Eqs.[4.8] and [4.9], the complex

coefficient A can be obtained in term s of the the complex coefficient D as:

A = D ( a 2 + k 2) sinh(ad) i2ka

Using Eqs.[4.8],[4.9], and [4.12] in Eq.[4.10], an analytical expression for the complex con­

stan t D can be w ritten as:

2 0

D — '2ika% ,kd(k 2 — a 2) sinh(ad) + i 'lka cosh(ad)

4.2 A P T T im e E xpression for th e S ingle Barrier

The solution to the Schrodinger equation in the potential barrier region is given by:

tf2(*) = B e ax + C e~ ax (4.14)

R eiterating the definition of Q uantum Mechanical Wave Impedance (QM W I) given by

Eq.[3.10]:

Z ( x ) = (4.15)j m ' $ 2 (3;)

Considering the fractional p a rt of Z( x ) and using Eq.[4.14] and [4.15], an analytical expres­

sion for can be obtained as:

^ ( a ) a[Be°* - Ce~°*]®2(a) [Beax + C e~ ax] K }

Substitu ting for B and C in term s of D from Eqs.[4.8] and [4.9] an expression for the QM W I

can be w ritten in term s of the attenuation constant a and the propagation constant k as:

$ 2(2 )̂ _ <*[(<* + ik )ea(x - (a — ik)e a x̂ d/ 2)] ®2(a ) [(a + ik)ea(x~dl 2> + ( a - i k ) e - Q(x~d/ 2)}

Eq.[4.17] can be modified to:

$ 2 (2 ) _ a fa s in l^ a a /) + ^ c o s ^ a a / ) ]$ 2 (1 ) [a cosh(aa:') + ifcsinh(ax ')]

where x = x — |

Eq.[4.18] can also be w ritten as:

’5P2(a:) _ a[(a2 + k 2) s i n ^ a s 1) cosh(aa;/) + ika] $ 2 (2 ) [a 2 cosh2(aa:') + k 2 sinh2 (aa;')]

(4.17)

(4.18)

(4.19)

2 1

Multiplying by the real part of Z(x) , R(x) , can be obtained as:

Re[Z(x )] = ■ (4.20)

The t a p t through the barrier can be obtained by substitu ting Eq.[4.20] into Eq.[3.20] as

follows:

„ r m * irJ - ^ 2 h a > k ^

The above expression for t a p t is analytically integrable for all values of incident energy of

the particle.

4 .2 .1 T h e A P T T i m e fo r E < V0.

When the energy of the incident electron, E , is less than the barrier height, V0, the a tten ­

uation constant, a, is a real quantity, and the electron wave function is decaying in nature.

Then, Eq.[4.21] can be integrated to obtain t a p t given by:

4 .2 .2 T h e A P T T im e fo r E > V0.

W hen the energy of the incident electron, E , is more than the barrier height, V0, a is an

im aginary quantity, and the electron wave function is propagating in nature. Then Eq.[4.21]

can be analytically integrated to obtain the following expression for t a p t -

t APT = ( 7 T 3 r ) P 2 + Q2) sinh(2ad) + 2a d (a 2 - fc2)]4 ha3k(4.22)

t $ F = ( ^ k ) [ 2 k B d { k 2 + ^ ~ k B ) s™(2kBd)] (4.23)

where kp is the propagation constant given by y 2m* ^ V°K

4.3 Various energy lim its o f t a p t

The limiting values for the t a p t for three cases of the incident energy of the particle, viz.

E —> 0, E —*• V0, and E —- oo can be com puted analytically. The derivation of these limits

are discussed in this section.

E — 0.

When the incident energy of the particle approaches zero, E — 0, from Eq.[4.22] the

propagation constant k tends to zero, hence t \ p t given by Eq.[4.22], tends to the following

limit:

tA p t — oo (4.24)

E —* oo.

W hen the incident energy of the particle approaches infinity, E oo, the propagation

constant in the barrier, kp — oo. The corresponding limit of t a p t is:

*

TAPT (4.25)B

m 'dt a p t t — Tciass{cai (4.26)

The ta p t tends to the classical tim e which is defined as the tim e it takes for a particle of

same energy and effective mass to traverse a distance equal to the barrier w idth, d, in the

absence of the barrier.

E - ^ V 0.

W hen the incident energy, E, of the particle tends to the barrier height, V0, the limiting

values for the t a p t can be obtained from either Eqs.[4.22] or [4.23] as follows[19]:

From Eq.[4.‘29], the t a p t is finite when E —*■ V0 as k is finite. The limiting values of Tdweii,

Tphase - delay, TB-L aad Tciassicai were obtained for three cases of limiting energies, E 0,

E — oo and E — V0 and listed in Table I for comparison of these values. A detailed

comparison is made in C hapter 6 .

Table I. The limits for the traversal times; the dwell time, Tdweii, the phase-delay time,

Tphase—d e l a y > the Buttiker-Landauer tim e. r p - L , th e classical traversal tim e, Tcia s s ica) , and

the A PT time, t a p t ■, for various incident energy limits.

01

E —► oo E ^ V 0

Tdweii 0 m* d T\k

m* k 0 / 4d3 +6d/fcp . h V 12+ 3 k ld * >

Tphase —delay oo m* d hk

m " k„ ( 4d3 + 6 d / k 3 , h I 12+ 3 k ld * >

t b - l/ m " k* \ ( a 2 — k 2) s in h 2(ad)+ (ofdfc^ /2 )s inhf2Qd) ' h a 2 / 4 k 2a 2+ k* s in h 3(a d )

m ' dhk y J d S + r ?

Tciassicai OO m* d hk

m* d hk

t a p t OO m* d hk

m * d r i , ( ^ d ) 2 iT F L1 + 3~J

24

C hapter 5

DOUBLE BARRIER

In this chapter, an analytical expression for the t^ p t through a sym m etrical double rectangu lar

po ten tia l barrier s truc tu re is obtained by using an approach sim ilar to th a t used for a single barrier

case in C hap ter 4.

T he conduction band edge profile of a sym m etrical double rectangular po tential barrier is shown

in Figure [5.1]. Firstly, the solution to the Schrodinger equation is obtained analytically. T hen the

solution is used to ob ta in the real p art of the Q M W I, which in tu rn is used to ob ta in v av(x). vav(x)

is used to ob ta in an analytical expression for the t ^p t ■

5.1 C alcu lation o f th e C om plex C oefficients o f W ave Func­

tio n Solutions

T he wave function solution to Schrodinger equation in the five regions shown in Figure[5.1] is given

by:

$ 1(a:) = ei i l + A e -a 'x a: < 0 (5.1)

•25

y o

B e ax -1- C e~ax F e ax + G c~ax

eikx £ e-ikxl l l l l l l l l l l l l l ! D eikx + E e~ ikx

l l l l l l l l l i l l lH e 'kx

Region 10

Region 2 Region 3 Region 4 Region 5

x = 0 x = d x = d + di x = 2 d d\

Figure 5.1: Conduction band edge profile for a sym m etrical double rectangular potential barrier with the wave function solutions for different regions.

<$2(x) = B ea x + C e - a x 0 < x < d (5.2)

* 3(a:) = Deikx + E e~ ikx ................. d < x < ( d + d i ) (5.3)

$ 4(z) = Feax + Ge~a x (d + di) < x < (dj + 2d) (5.4)

^ 5(®) = H e ik x ................. x > (di + 2d) (5.5)

where d and d\ are the barrier and well w idths, respectively, and V0 is the height o f the barrier.

A pplying the boundary conditions viz. the wave function, ^ (a :), and the derivative of the wave

function, (x), are continuous a t the interfaces x = 0, x = d, x = d + d\ , and a t x = 2d + d \ t

analytical expressions for the com plex coefficients A, B, C, D, E, F, and G, can be ob tained in

2 6

term s of the com plex constan t H, which is the wave function transm ission coefficient. The analytical

solutions to the com plex coefficients are:

piZkd i f Lt /y | 4 L*A = — ----- -r(------- -)[[(Jb2 — a 2) + (a — i&)2e,2fcd|] sinh(a(i) + i2fcarcosh(ard)] — ( ------ — )(5.6)

i2koteaa a — ik a — ik

B = ^ ’̂ / ^ [[(fc2 — q2 ) + e,2kdl(a — ffc)2] s in h (ad ) + t2fcacosh(o-d)] (5.7)

TTpi2kd(n _C = — i4 /.Q2 e- ad— [[(k2 — « 2) + e’2 d l(ar + ifc)2]s in h (a d ) + i2fcacosh(ad)] (5.8)

HeikdD = — [(fc2 — a 2) s inh (ad ) + i2ka cosh(ad)] (5.9)

i i j k ( 3 d + 2 d i )

E = f t k a (<*2 + k ) si nil (ad ) (5.10)

H e i H 2 d + d l ) ( a + i k )

2 aea(2d+d,)

H e i k ( 2 d + d l ) ( a _ i k )

2 a e - “ (2d+d*) ̂ ^

5.2 T he t a p t for a R esonant T unneling S tructure

In order to derive an ana ly tica l expression for the t a p t < the Q uan tum M echanical Wave Im pedance

(QM W I) should be com puted. T he QM W I can be ob tained from the wave function solution involving

the com plex constan ts B , C , D , E, F, and G. T he analytical expressions for the real and im aginary

parts of the com plex coefficients are given in A ppendix A. T he real p a rts are subscripted ‘1’ and the

27

im aginary parts are subscripted ‘2 ’. In order to calculate the to ta l t a p t through the structu re , the

t a p t 's for the two barriers and the well region are calculated individually and added up as follows:

_ t o t a l —lb . —w e l l , —rb / c i o \t APT — t APT + r APT + A P T (5.13)

where t^ p t , t^ p p , and t ’ ap t are the traversal tim es in the left barrier region, well region, and right

barrier region, respectively. T he t a p t 's , t^ p t , t%p p and r r̂ PT are calculated using the integral

expression for t a p t given by Eq.[3.20].

5.2.1 The tapt for the Left Barrier Region

In order to calculate the t a p t tim e for the barrier region, the real p a rt of the Q M W I should be

com puted. Once the the real p a r t of the QM W I, Re[Z(x)], is known, it can be substitu ted into

Eq.[3.20] to ob ta in an integral expression for the t a p t for the barrier region.

T he solution to the tim e-independent Schrodinger equation for the left barrier region is given

by:

tf(x ) = { B x + i B 2)eax + (C i + iC2)e~ax (5.14)

where B y , C y are the real parts and B 2 , C 2 are the im aginary parts o f the com plex constants B and

C , respectively. The ^ ( x ) can be ob ta ined from E q .[5.14] as:

tf '(x ) = a [ { B y + iB 2)eax - (Ci + iC2)e~ax} (5.15)

Re[Z{x) \ can be obtained from Eq.[5.14], Eq.[5.15] and E q .[3.19] as:

fi>„r71 m _ 2 (g;C i - ByCo)__________ .[ )[( £ 2 + B !)e2“* + (Cf + C |) e - 2ai + 2{ByCy + B 2C 2) ̂ ( J

ta p t can be ob ta ined from Eq.[5.16] and Eq.[3.20] as:

•28

lt „ f d, m ’d x , t (B\ + + (C? + C | ) e - 2QI + 2(BxCx + B2C2),Ta p t = 2 J 0 {- 2 M )[---------------------------2(BiC\ — B~C2)--------------------------- ] (5 -1 0

Integrating Eq.[5.17]:

= W ^ - ^ c 3)][(^ )Ieaad- 1]- [̂ ] [ e" aa,,- i ] + M (fllC l+ g> C a)] (5-18)

5.2.2 The t A p t for the Potential W ell Region

T he solution to the tim e-independent Schrodinger equation for the well region is given by:

* (x ) = (Dx + iD2)eikx + ( E x + i E 2)e~ikx (5.19)

where Dx, Ex are the real parts and D 2, E 2 are the im aginary parts of the com plex constan ts D

and E, respectively. T he expressions for all the coefficients are given in appendix A. T he V ( x ) is

obtained from E q .[5.19] as:

« '( x ) = iJk[(Z>i + iD2)eikx - (Ex + iE 2)e~ikx] (5.20)

T he tAp t in the po ten tia l well region can be w ritten as:

r d + d x J

Ta p p =2 1 R ^ ) ] (5-21)

Using Eq.[5.19] and Eq.[5.20] in Eq.[3.9], and w riting the exponentials in term s of trignom etric

functions, m ultip ly ing by in the expression for Re[Z(x)\ in term s of the com plex coefficients, an

expression for the t a p t for the well region can be ob ta ined as:

29

f d + d j

T%px = j [p + q cos(2fcx) + rsm(2kx)]dx (5.22)

where p, q, and r are given by:

D\ + D\ + E\ + El P ~ Dl + D l - E l - k l (5'23)

+ D2E2 , e nA^q ~ D l + D l - E \ ~ E l ( ^

_ D xE 2 - D i E i

D\ + D'l — E'l — E l (5 5)

In tegrating Eq.[5.22], an analy tical expression for the tap t in the well region, is ob ta ined as

follows:

= (S i )[pdl + ( ^ ) [ sin[2fc(rf+ * ) ] - 8in(2W )] - (^ ) [ c o s [2 * (d + * ) ] - cos(2W)]] (5.26)

5.2.3 The t a p t for the Right Barrier Region

T he solution to the tim e-independent Schrodinger equation in the baarrier region is given by:

9( x ) = (Ft + iFn)eax + (G j + iGo)e~ax (5.27)

where Ft, G 1 are the real p a rts and F2, G 2 are the im aginary p a rts o f the com plex constants F and

G, respectively, ^ ’(x) can be obtained from Eq.[5.27] as:

# '( * ) = a [(Ft + iF2)eax - (G j + iG 2) e " “*] (5.28)

30

Re[Z(x)] can be ob ta ined from Eq.[5.27] and Eq.[5.28] as follows:

r !7( n HF3G 1 - F 1G2)________________ ,1 m* ) l ( F ? + W ) e 2ax + (G? + G l ) e ~ 2ax + 2 ( F i G i + F 2G 2) J

and the r ^ pT can be obtained as follows:

.rb n f 2d+d' , m ’ d x , r(F? + F?)e2ax + {G\ + Gl)e~2ax + 2 (F : G i + F 2G 2) 1TAPT J d+dl ( 2ha )[ 2(E2G 1 - F 1G 2) J

Integrating Eq.[5.30]:

[ ]4 f i a ( F 2G i - F i G 2)

^ F 1 + Fn j ^ 2a (r f i+2 r i ) _ e 2 a ( d ! + d ) j _ ^ 1 + G o j

^ - 2 a ( d 1+2d) _ e -2ofd+ d,)] + 2d(ir1G 1 + F 2G 2)]

(5.29)

(5.30)

(5.31)

Chapter 6

RESULTS, COMPARISONS and

DISCUSSIONS

In th is chapter, the tapt ob ta ined from the analytical expressions are com pared w ith previous

work reported in the litera tu re for the single and double barrier cases. The tapt for single barrier

is com pared w ith the dwell time[7, 45], the phase-delay time[7, 16, 26, 50], the B uttiker-L andauer

traversal time[7, 8, 9], and the classical traversal tim e. T he double barrier tapt is com pared w ith the

experim entally ob tained m axim um frequency of oscillation of som e of the structu res experim entally

grown and tested by Sollner et. al. [25, 47, 48].

6.1 Single Barrier

A plot of t a p t and transm ission coefficient vs. the norm alized incident energy for a rectangular

po ten tia l barrier is shown in Figure[6.2] for a barrier o f height 0.3eV and w idth of 200A as shown in

Figure[6.1], for the case of incident energy of the particle less th an the barrier height, i.e., E < V0.

T he transm ission coefficient increases w ith incident energy, E, as expected. T he t a p t decreases with

increasing energy. It is noted th a t the t a p t reaches infinity in the lim it o f zero energy like a classical

particle. A p lo t of t a p t and transm ission coefficient vs. the normalized incident energy is shown in

Figure[6.3] for the s truc tu re shown in Figure[6.1], where the incident energy of the particle is more

than the barrier height, i.e., E > V0. In the lim it of incident energy of the particle tending to the

barrier height, i.e. E —. V0, the t a p t is large bu t finite as given by E q.[4.29]. T he t a p t decreases

w ith the increasing incident energy. It is also noted th a t the t a p t oscillates slightly for large barrier

thicknesses. In the lim it of E —- oo, t a p t reaches the classical lim it as given by Eq.[4.26],

6.1.1 Com parison of Dwell tim e, Phase-delay tim e, Buttiker-Landauer

tim e, t a p t , and the Classical tim e for, E < V0

A p lo t of the dwell tim e, the phase-delay tim e, the B uttiker-L andauer tim e, the tapt and the

classical traversal tim e vs. norm alized incident energy is shown in Figure[6.4], for a rectangular

po ten tia l barrier w ith a barrier height of 0.3eV and a w idth of 200A as shown in Figure[6.1], when

E < V 0.

t a p t is always g rea ter than the classical tim e, t a p t >s infinity in the lim it o f no energy, E = 0,

im plying th a t the particle takes infinite tim e to traverse the distance when the particles possess no

energy a t all. For all incident energy values below the barrier height, Voi the dwell tim e and the

phase-delay tim e are less than the classical traversal tim e as shown in Figure[6.4]. W hereas, the

B uttiker-L andauer tim e is below the classical tim e for a range of incident energy, and above the

classical tim e for the rest of the incident energy interval.

6.1.2 Com parison of Dwell tim e, Phase-delay tim e, Buttiker-Landauer

tim e, t a p t , and the Classical tim e for E > V0

A plot of the dwell tim e, the phase-delay tim e, the B uttiker-L andauer tim e, the tap t and the

classical traversal tim e vs. norm alized incident energy are shown in Figure[6.5], for a rectangular

po ten tia l barrier w ith a barrier height o f O.SeV and a w idth of 200A as shown in Figure[6.1], when

E > V0. In th is case, all the traversal tim es are above the classical traversal tim e. A p lo t o f the

33

dwell tim e, the phase-delay tim e, the B uttiker-L andauer tim e, the t a p t , and the classical tim e w ith

the transm ission coefficient is shown in Figure[6.6], The dwell tim e, the phase-delay tim e and the

B uttiker-L andauer tim e a tta in a m axim um value when the transm ission coefficient is m axim um , and

reaches a m in im um when the transm ission is m inim um . W hereas, the t a p t a tta in s a m inim um when

the transm ission coefficient is m axim um , and reaches a m axim um when transm ission is m inim um . In

other words, A ccording to t a p t , the particle travels fastest a t resonant energies, whereas according

to the approaches, the particle travels fastest a t non-resonant energies. All the traversal tim es

approach the classical tim e lim it a t very high incident energies.

6.1.3 Effect of th e barrier width on the t a p t

The dependence of t a p t on the barrier w idth w ith E < V0 is shown in Figure[6.7], for a barrier

height of l .OeV and for a barrier w idth in the range of 25A to 250A.

The t a p t decreases as the incident energy of the particle increases and the t a p t is inversely related

to the transm ission coefficient. T he t a p t is finite when the incident energy of the particle is equal

to the barrier height V0.

T he dependence of t a p t on barrier w idth, w ith incident energy of the particle m ore th an the

barrier height is shown in Figure[6.8], for a barrier height o f l.OeV' and for a barrier w idth in the

range from 25A to 250A. Sm all oscillations in the t a p t can be seen as the w idth of the po ten tia l

barrier increases.

The t a p t decreases as the incident energy of the particle increases. It is to be noted th a t as the

barrier thickness increases, sm all oscillations in the t a p t can be observed. These oscillations are

inversely p roportional to th e transm ission coefficient, i.e. t a p t valleys a t resonant energies whereas

the transm ission coefficient peaks.

34

6.2 D ouble Barrier

T he plot of the transm ission coefficient and t a p t is shown in Figure[6.10] for the structu re shown

in Figure[6.9], Five quasi-bound energy s ta tes can be observed below the barrier height. The

transm ission coefficient is unity a t these resonant energies levels. It is to be noted th a t t a p t is

m inim um a t the resonant energies. T hus the transm ission coefficient is inversely related to t a p t -

6.2.1 Comparison of tap t w ith Experim ental R esults

Structure 1: Barrier width = 50A, Well width = 50A, and Barrier height =

0 .23eV \

T he plot of t a p t and the transm ission coefficient vs. the incident energy of the particle is shown

in Figure[6.12], T he s truc tu re considered was a sym m etrical rectangular double potential barrier

w ith a barrier height of 0.23eV, barrier w idth of 50A and a well w idth of 50A as shown in Figure

[6.11]. T his sam e structu re is chosen as it is well characterized experim entally in term s of the

I — V characteristics and high frequency studies[47, 48]. It was reported by Sollner et. al.[47, 48]

th a t the m axim um frequency of operation of the resonant tunneling device is l .2THz. It was also

observed th a t there is one quasi-bound resonant energy s ta te a t 0.079leV which is less than the

barrier height. This value of 0.0791eVr agrees w ith th a t obtained from our analy tical solutions. A t

this resonant energy value, the t a p t is a m inim um and also th e transm ission coefficient a tta in s

unity. T he estim ated m axim um frequency of operation, from analy tical expression for t a p t < is

0.8T Hz , i.e., frequency corresponding to the resonant energy, which is in good agreem ent w ith the

experim ental value of 1.2T Hz . T h is m axim um frequency of opera tion was estim ated taking into

consideration only the t a p t • T he capacitance charging tim es a t the depletion layers are not taken

into consideration.

Structure 2: Barrier width = 25A, Well width = 45A, and Barrier height =

l.OeV.

T he plot o f tap t and the transm ission coefficient vs. the incident energy of the particle is shown in

Figure[6.14], T he structu re considered was asym m etrica l rectangular double po ten tia l barrier w ith a

barrier height o f l.OeV, barrier w idth of 25A and a well w idth of 45A as shown in Figure [6.13]. This

stru c tu re was fabricated and experim entally studied for high frequency oscillations by Sollner el.

al.[25, 47]. It was reported by Sollner et. al., th a t the m axim um frequency of operation of the above

resonant tunneling device is '2ATHz. It was also reported th a t there are two quasi-bound resonant

energy sta tes, one a t 0 .154eF and the o ther a t 0 .581eF below the barrier height. These values

agree well w ith the values obtained from other num erical solutions. T he transm ission coefficient is

m axim um and a tta in s the value o f un ity a t these resonant energy levels and tap t a t these resonant

energy levels is a m inim um . T he estim ated m axim um frequency of operation is I A T H z which is in

good agreem ent w ith the experim ental results. T he m axim um frequency of operation was estim ated

by tak ing in to consideration the t a pt only.

6.2.2 Effect of barrier w idth on the A PT tim e

A 3-dim ensional surface p lot of the t a pt is shown in Figure[6.16], for a range of the barrier w idths

from 30A to 100A, well w idth of 30A, and for a barrier height of 0.3eK, as shown in Figure[6.15]. The

ta p t approaches infinity when E —<■ 0. T he t ap t is inversely related to the transm ission coefficient.

T he form ation of troughs in the t ap t a t resonances indicate th a t a t these resonant energy levels,

the t a pt is m inim um . As the barrier thickness is increased, the form ation of the resonant energy

levels is m ore pronounced and tap t for a very thick barrier, a t the resonance, is m ore th an th a t for

a th in barrier.

6.2.3 Effect of well w idth on the A PT tim e

A 3-dim ensional surface plot of the t a p t is shown in Figure[6.18], for a range of well w idths from 30A

to 110A and a barrier w idth of 30A, and f t r a barrier height of l.OeV as shown in Figure[6.17]. A 3-

dim ensional surface plot of the t a p t along w ith the transm ission coefficient is shown in Figure[6.19]

for the sam e s truc tu re shown in Figure[6.17]. More quasi-bound s ta te s w ith E < V0 appear as the

well w idth increases, as shown in Figures[6.18] and [6.19]. T he form ation of troughs in the t a p t a t

resonances indicate th a t a t these resonant energy levels, the t a p t is m inim um , t a p t is m inim um a t

resonant energies. W hen the w idth of the po ten tia l well is sm all, the num ber of resonant energies is

sm all w ithin the po tential well. As the w idth of the po ten tia l well increases, the num ber of resonant

energy levels increase and the energy spacing between any two adjacen t resonant energy levels w ithin

the p o ten tia l well, A E, decreases.

37

0.3eV

200A

Figure 6.1: The conduction band edge profile of a single rectangular potential barrier of width 200A and height 0.3eV

Tra

vers

al T

ime

(Sec

onds

)38

le-02

le-03

le-04

le-05

e-06

le-07

le-08

le-09

le-10

le-11

le-12

le-13

0.00 0.20 0.40 1.000.60 0.80Normalized Energy (below the barrier)

Figure 6.2: P lot of the (a) Transmission Coefficient, and the (b ) A P T tim e, for a singlerectangular potential barrier for E < V0, with barrier width 200A and barrier height 0.3el^

Trav

ersa

l Ti

me

(Sec

onds

)39

180.00 1.00

0.95170.000.90160.000.85

150.000.80

140.000.75

130.00 0.70120.00 0.65110.00 0.60

100.00

0.5090.000.4580.000.40

70.000.35

60.000.30

50.00 0.2540.00 0.2030.00 0:15

20.00 0.10

0.0510.00

1.00 1.20 1.40 1.60 1.80 2.00

Normalized Energy (above the barrier)

Figure 6.3: P lot of the (a) Transmission Coefficient, and the (b ) A PT time, for a singlerectangular potential barrier for E > V0, with barrier width 200A and barrier height 0.3eV

Tran

smiss

ion

Coe

ffic

ient

Trav

ersa

l Ti

me

(Sec

onds

)40

le-03

le-04

le-05

le-06

le-07

le-08

le-09

le-10

le-11

le-12

le-13

le-14

le-15

le-16

0.00 0.20 0.40 0.60 0.80 1.00

Normalized Energy (below the barrier)

Figure 6.4: P lot of the Traversal Times : (a ) Dwell Time, (b ) Phase-Delay Tim e, (c)B uttiker-L andauer Tim e, (d ) A PT time, and (e ) Classical Time for E < V0, with barrierwidth 200A and barrier height Q.SeV.

41

2.5

<nTJOu11 le-13 in

2H(Stn!hD%

&2.5

le-14

1.00 1.20 1.40 1.60 1.80 2.00

Normalized Energy (above the barrier)

Figure 6.5: P lot of the Traversal Times : (a ) Dwell Tim e, (b ) Phase-Delay Tim e, (c )B uttiker-Landauer Tim e, (d ) A PT time, and (e ) Classical T im e for E > Va, w ith barrierwidth '200A and barrier height 0.3eV.

Trav

ersa

l Ti

me

(Sec

onds

)

42

x 1 0 ' 1 5

180.00

- 0.95170.00

0.90160.00

0.85150.000.80

140.000.75

130.000.70

120.000.65

110.00 0.60

100.00 0.55

0.5090.00

0.4580.000.4070.000.35

60.000.30

50.000.25

40.00 0.2030.00 0.15

20.00 0.10

H 0.0510.00

1.00 1.20 1.40 1.60 2.001.80Normalized Energy (above the barrier)

Figure 6.6: P lot of the Traversal Tim es : (a) Dwell Tim e, (b ) Phase-Delay Time, (c)B uttiker-Landauer Tim e, (d ) A P T tim e, (e ) Classical T im e, and th e ( f ) TransmissionCoefficient for E > V0, with barrier w idth 200A and barrier height 0.3eV.

Tran

smiss

ion

Coe

ffic

ient

Log

e (T

rave

rsal

Tim

e)

(Sec

onds

)

43

Figure 6.7: 3-Dimensional surface plot of the A PT tim e, for the case of E < V0 w ith thebarrier height l .OeV and barrier w idth in the range 25A to 250A.

44

; ■ x - 2 i

Figure 6.8: 3-Dimensional surface plot of the A P T time, for the case of E > V0 w ith thebarrier height l.OeV and barrier width in the range 25A to 250A.

45

m b '>**<'< 4

^ 30A 100A ^ 3 0 1 |*

Figure 6.9: The conduction band edge profile of a sym m etrical double rectangular potential barrier s tructu re with a barrier height of 0.956eV, barrier w idth of 30A and a well width of100A.

46

le+OO -

le-02 -

le-03 -

^ le-04 - /cn

T3fi le-05 - '0uCL)

CO le-06 -

S le-07

60 le-08 -

S le-09 -

le-10 I-

le-12 -

le-13 j-

le-14

0.00 0.20 0.40 0.60 0.80 1.00

Energy (eV)

Figure 6.10: P lot of the (a ) Transmission Coefficient and the (b ) A P T tim e for a sym m et­rical double rectangular potential barrier s tructu re with a barrier height 0.956eV, barrierwidth 30A and a well w idth 100A, for E < V 0.

47

0.23eV

5 0 l I - 50A 50A

Figure 6.11: T he conduction band edge profile of a sym m etrical double rectangular potentialbarrier s tructu re with a barrier height 0.23eV, barrier w idth 50A and a well w idth 50A.

48

le+OO

le-01

le-02

e-03

le-04

le-05

le-06

le-07

C le-08

le-09

le-10

e-12

le-13

0.00 50.00 100.00 150.00 200.00 - 3

x 10Energy (eV)

Figure 6.12: P lo t of the (a ) Transmission Coefficient and the (b ) A PT tim e for a sym m et­rical double rectangular potential barrier s tructu re with a barrier height 0.23eV, barrierwidth 50A and a well w idth 50A, for E < V0.

49

BH BM i

25A L 45A J 25A

Figure 6.13: The conduction band edge profile of a sym m etrical double rectangular potentialbarrier structu re with a barrier height l.OeV, barrier w idth 25A and a well width 45A.

50

le+OO

! e-03

le-04

0)g le-07

bo le-08 .21 le-09C

H le-10

le-13

le-140.00 0.20 0.40 0.60 0.80 1.00

Energy (eV)

Figure 6.14: P lot of the (a ) Transmission Coefficient and the (b ) A P T tim e for E < V 0 fora sym m etrical double rectangular potential barrier structu re with a barrier height l.OeV',barrier width ‘25A and a well width 45A.

51

iflltllli

i$:Mw 0.3eV

miiiiilili

MRiHi■>| 3 0 A - 1 0 0 A |< — 30^1 - h* | 3 0 A - 1 0 0 l |< -

Figure 6.15: The conduction band edge profile of a sym m etrical rectangular double potentialbarrier structu re of barrier height 0 .3eF , barrier width in the range from 30A to lOOA andwell w idth 30A, for E < V0.

Loge

(Tun

nelin

g Ti

me)

(S

econ

ds)

52

Figure 6.16: 3-Dimensional surface plot of the A P T tim e for E < V0. The barrier height is0.3eV and the barrier width in the range from 30A to 100A and the well width is 30A.

53

■»| 30A |«30A - 1 1 0 i * | 30A

Figure 6.17: The conduction band edge profile of a sym m etrical rectangular double potential barrier structu re of barrier height l.OeV, barrier width 30A and the well w idth in the range from 30A to 110A.

Loge

(Tun

nelin

g T

ime)

(S

econ

ds)

54

Figure 6.18: 3-Dimensional surface plot of the A PT tim e for E < V0. The barrier height isl.OeV and barrier w idth 30A and the well width in the range from 30A to llOA.

55

Figure 6.19: 3-Dimensional surface plot of the (a ) A PT time and the (b ) transm ission coefficient for E < V 0. The barrier height is l.OeV and barrier width is 30A and well width in the range from 30A to 110A.

56

C hapter 7

CONCLUSION

S tarting from the analytical solution to the tim e-independent Schrodinger equation in one-dim ension,

and exploiting the analogy between the transm ission line equation and the solution to the Schrodinger

equation , an analytical expression for the Average Particle Traversal (A PT ) tim e, t a p t , was derived

in term s of the real p a rt o f the Q uan tum Mechanical Wave Im pedance (QM W I). This approach was

used to derive an analytical expression for the t a p t through a single rectangular po ten tia l barrier

under zero bias, and the results were com pared w ith the dwell tim e, the phase-delay tim e, the

B uttiker-L andauer tim e, the Collins-Barker num erical traversal tim e and the classical traversal tim e.

T he A P T tim e is inversely proportional to the transm ission coefficient, t a p t approaches infinity as

the incident energy tends to zero and as the energy goes to infinity, t a p t tends to the classical tim e.

T he A P T tim e is always more th an the classical traversal tim e. T he sam e approach was extended

to ob ta in an analytical expression for the t a p t through a sym m etric double rectangular po ten tia l

barrier s truc tu re under zero bias. T he A P T tim e is inversely proportional to the transm ission

coefficient and the t a p t a tta in s a m inim um a t resonant energies when the transm ission coefficient is

unity. T he m axim um frequency of oscillation of some of experim entally studied resonant tunneling

structures were com pared w ith those obtained using A P T tim e. T he agreem ent is good.

T he capacitance charging tim e a t the depletion regions were not taken into consideration. T he

57

effective masses were assum ed to be constant th roughout the struc tu re . T he tapt in the free

propagation regions, ahead and beyond the barriers, were not taken in to consideration. T he effect

of the effective m ass can be incorporated in this approach and an analy tical expression for the

tapt for a. barrier s truc tu re w ith different outer edges can possibly be derived. An analytical

expression can possibly be derived for the tapt through unsym m etrical double potential barriers.

Coupled-quantum -w eils effect can also be incorporated. As the pre-free-propagating region may

play an im portan t role, the traversal tim e through the pre-free-propagating region also should be

considered, especially when com parison w ith experim ents are m ade.

58

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63

C hapter 8

A PPE N D IX A

8.1 C om plex C oefficients o f W ave Function Solutions

T he com plex constan ts involved in the solutions to the Schrodinger Equation are given in th is section.

T he suffix 1 and 2 indicate the real and im aginary p arts o f the constants, respectively.

A = i2kaead^a ~ f k ^ k2 ~ + " *i') 2c ,2 id‘] sinh (ad ) + i '2 iacosh (od )] - (8-1)

f i pik2d(n I :L)B = — t'4 fca 2ead— ~ ®2) ^ e'2k<i' ( a ~ *’&)2]s in h (ad ) + i'2^acosh(ad)] (8.2)

f r ei2kd(n _ :l \C = — i4 f ^ 2 e-ad— tt(^2 — a 2 ) e'2kd' (Q + l^ )2] s in h (ad ) + i2ka cosh(ad)] (8-3)

H eikdD = — [(fc2 — a 2)s in h (a d ) + i2ka cosh(ad)] (8.4)

64

f j e i k ( 3 d + 2 d i )

( a 2 + k 2) s inh(ad) (8.5)

H e ikV d+d' ) ( a + i k )2 a e a(2d+di) ( 8 .6 )

f f e i k ( 2 d + d l ) ( a _ j f . )

2&e~ai2d+dl )(8.7)

B i = a ( k 2 — a 2) ( l — e ~ 2ad) s i n2(k d \ )

+ 2a 2fc(l — e ~ 2ad) s i n ( kd i ) cos ( kd i )

+ k ( k 2 — a 2) ( l — e - 2“ d)sin(&di)cos(&di)

+ a fc 2( l — e_2“ d) ( l — 2 sin 2(k d i ) )

- a k 2( l + e ~ 2ad)

B 2 = k ( k 2 - a 2) ( l - e ~ 2ad) s i n2( k d ^

+ 2a £ 2( l — e ~ 2ad) s i n ( k d i ) co s( kd \)

—a ( k 2 - a 2) ( l - e ~ 2ad) s i n ( k d i ) co s( kd i )

—a 2fc(l — e ~ 2ad) ( l — 2 sin2( t d i ) )

+ a 2fc(l + e ~ 2ad)

(8 .8 )

(8.9)

65

C\ — a ( k 2 — a 2)(e2ad — 1) cos2(kdi)

—2a2k(e2ad — l)sin(fct/i) cos(fcdi)

+fc(fc2 — a 2)(e2ad — l)sin(fc<ii) cos(fcdi)

+ a k 2{e2ad - 1 ) ( - 1 + 2 c o s 2 ( M i ) )

+ a k 2(e2ad + 1)

Ci = a ( k 2 — a 2)(e2ad — l)cos(&<fi)sin(fc(fi)

+ 2 a k 2(e2ad — l)s in (£ c fi)co s(M i)

—k(k2 — a 2)(e2ad — 1) cos2(kd\)

+ a 2k(e2ad — 1)(—1 + 2 cos2(kdi))

+ a 2k(e2ad+ 1)

2ka cos(kd) cosh(ad) + (k2 — a 2) sin(kd) sinh(arf) D l = -----------------------------

2 k a cosh(atf) sin(fcd) — (k2 — a 2) cos (kd) s in h (ad )D2 = ---------------------------- ---- -----------------------------

( a 2 + k 2) sin[£(3<i + 2.d\ )] s in h (o d ) El = 2^ --------------------

(8 . 10)

(8 . 11)

( 8 . 12)

(8.13)

(8.14)

66

—( a 2 + &2)cos[&(3<f + 2<fi)]sinh(ad)

a cos[k(2d -f <fi)] — fcsin[fc(2d + rfi)]2 a e o(2d+di)

k cos[k(2d -f d \ )] + asin[£(2f/ + rfi)] 2 a e “ (2d+d' )

a cos[£(2c/ + d i )] + k sin[£(2d + d \ )] 2 a e - a l2d+d l1

a sin[&(2d + d i )] — k cos[fc(2<f + d \ )] 2txe~ “ (zd+di)

(8.15)

(8.16)

(8.17)

(8.18)

(8.19)