2.1 Atoms 1

21 Atoms 1

- The atomic model used as a basis for understanding the properties of matter has its origins in the particle scattering of Ernest Rutherford

- Niels Bohr developed a dynamic model for the simplest of atoms the hydrogen atom Using a blend of classical and quantum theory

- The total energy of the electron is made up of its kinetic energy and its electrostatic potential energy The value of is taken as zero when the electron is so far removed from the nucleus Ieat lsquoinfinityrsquo that interaction is negligible

Hence

rwmrvm

re

ee2

2

20

2

4

k pp

kp

(21)

(22)

Substituting from equation (21) gives

From the quantum condition for angular momentum

which substitutes with equation (21)leads to

or

22

0

2

21

4vm

re

0

2

8 (23)

nhwrme 2

42

22

30

2

22

4 rmhn

rmew

e

22

2

41

hnem

r o

e

(25)

(24)

21 Atoms 2

which substitutes into equation (23) to give

lsquode Broglie wavelengthrsquo

Electron states are described by the solutions of the following equation which was developed by Erwn Schrodinger and which bears his name

2220

2

4 132 nh

eme

(26)

mvh

(27)

0)(22

2 phm (28)

21 Atoms 3

This from of the Schrodinger equation is independent of time and so is applicable to steady state situation The symbol denotes the operator

To apply equation (28) to the hydrogen atom it is first transformed

into polar coordinates and then solved by the method of separation of the variables This involves writing the solution in

the form

2

zyx

)( r

)()()()( rRr (29)

21 Atoms 4

- Pauli Exclusion Principle states that there cannot be more than one electron in a given state defined by particular set of values for and For a given principal quantum number there are a total of available electronic

-Electrons occupy states such that following 1The value is maximum allowed by the Pauli Exclusion

PrincipleIe the number of unpaired spins is a maximum 2 The value is the maximum allowed consistent with rule 1 3 The value of when the shell is less than half-full and when it is mire than half-full When the shell is just

half-full the first rule requires so that

1 mln s n22n

S

LSL

SL 0L SJ

21 Atoms 5

-page 9 -page 10~11 with reference

22 The arrangement of ions in ceramics

Fig 21(a) in simple cubic packing the centres of the ions lie at the corners of cubes formed by eight ionsit is generally found that anion lattices will accommodate oversize cations more readly than undersize cations so that the tolerance to the relatively small ion is exceptional

Fig 21(b) and (b)The oxygen ions are more closely packed together in the close-packed hexagonal and cubic structures

Fig 21 Packing of ions

(a)simple cubic packing showing an interstice with eightfold coordination

(b)hexagonal close packing

(c)cubic close packing showing a face-centred cubic cell

4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

Fig 22 octahedra arrangements in

(a)perovskite-type structures

(b) And (c)hexagonal 3BaTiO2TiO

Fig 22(a) on this basis the rutile from of consists of columns of edge-sharing octahedra linked by shared coners of the units

Fig22(b)A hexagonal from of where the lattice is hexagonal close packedcontains layers of corner-sharing groups(Fig22(c))

2TiO

6TiO

3BaTiO 3BaO

6TiO

23 Spontaneous polarization Value of a crystal property depend in the direction of measurement Crystals having cubic symmetry are optically isotropic For these reasons a description of the physical behaviour of a material has to

be based ion a knowledge of crystal structure Purpose to distinguish polar crystals spontaneously polarized and possess a u

nique polar axis from the non-polar variety Of the 32 crystal classes 11(centrosymmetric) and non-polar variety Of the remaining 21 non- centrosymmetric classes 20(piezoelctric of these 10

are polar) An idea of the distinction between polar and non-polar structures can be gained

from Fig23

Fig23 (a) Non-polar array (b)(c) polar arrays The arrows indicate the direction of spontaneous polarization Ps

Piezoelectric crystals when stressed polarized or change in polarization When an electric field is applied become strained

The 10- polar crystal types(pyroelectric piezoelectric) because of their spontaneous polarization

Pyroelectric crystal change in temperature =gt change in spontaneous polarization additional features cannot be predicted from crystal structuresome material that the direction of the spontaneous polarization can be changed by an applied electric field or mechanical stresschange due to electric field = said to lsquoferroelectricrsquochange due to stress = said to lsquoferroelasticrsquo

Poling process conditions of temperature and time static electric field after poling =gt ferroelectric ceramic makess it capable of poezoelectric pyroelectric electro-optical behaviour

23 Spontaneous polarization

The changes in direction of the spontaneous polarization

-The tetragonal(4mm) structure allows six direction-The rhombohedral(3m) allows eight direction-both tetragonal and rhombohedral crystallites are present at a transition point the number of alternative crystallographic directions rises to 14 and the extra alignment attained becomes of practical significance

24 Transitions-Ionic size and the forces are both temperature dependent and may change sufficiently fo

r a particular structure to become unstable and to transform to a new one-If a system is described in terms of the Gibbs function G then because the molar entrop

ies and molar volumes of the two phase do mot change the change in G for the system can be

written dG =- SdT + Vdp-It follows that

-lsquofirst-orderrsquo transition if there is a discontinuity in the derivative of G(T) there is a change in

entropy at constant temperature which implies latent heat-lsquosecond-orderrsquo transition when the first derivative of the Gibbs function is continuous bu

t the second derivative is discontinuous

T

G G TG

TG

T T TFree-energy changes at transitions (a)fist-order transition (b) change in S at constant T and consequently latent heat(c)second-order transition (d)continuous change in entropy and so no latent heat (discontinuity in )

S entropy V volume P pressure

PTGS )(

Latent heat

25 Deffects in crystals1251 Non-stoichiometry-Manganese dioxide is a well-established compound but it always contains less than the stoichiometric amount of oxygen The positive charge deficiency can be balanced by vacant oxygen site-Iron monoxide always contains an excess of oxygen The charge excess can be balanced

-LiNbO3 The O ions are hexagonally close packed with a third of the octahedral sites occupied by Nb a third by Li and a third empty This can be deficient in lithium down to the level Li094Nb1012O3 There is no corresponding creation of oxygen vacanciesinstead the Nb content increases sufficiently to preserve neutrality-BaTiO3 there is marked difference in charge and size between the two cations corresponding to differences between their lattice sitesto complete solid solutions over the whole possible range where the ions are identical in charge and close in size and can only occupy one type of available lattice site such as Zr and Ti in Pb(ZrxTi1-x)O3

5+

4+

+

5+

252 Point defects

25 Deffects in crystals2

-Point defects They occur where atoms are missing(vacancies) or occupy the interstices between normal sites(interstitials)rsquoforeignrsquo atom are also point defects-Line defectsor dislocations They are spatially extensive and involve disturbance of the periodicity of the lattice

-Frenkel and Schottky Defects paired anions and cations bull Electronic neutrality must be maintained in crystal bull Defects must come in pairs to maintain Q=0bull Cation-vacancy + Cation-interstitial = Frenkel Defect (Q=0)bull In AX-type crystals bull Cation-vacancy + Anion-vacancy = Schottky Defect (Q=0)

-The equilibrium concentrations of defects in a simple binary oxide MO

-The notation of Kroger and Vink

-In summarya chemical equation involving defects must balance in 3 respects1the total charge must be zero2there must be equal numbers of each chemical species on both sides3the available lattice sites must be filledif necessary by the introduction of vacant site

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

nSampnF Schottky and Frenkel defect concentrations respectivelyΔHSampΔHFenthalpy change accompanying the formation of the associated defects (cation vacancy+ anion vacancy and ion vacancy + interstitial ion)N concentration of anions or cations N`concentration of available interstitial sites

interstitial vacancy

Net positive negative`

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

-The introduction of an acceptor Mn on a Ti site in BaTiO3 can be expressed as -which replaces the equilibrium equation for the pure crystal -Since BaO=BaBa+Oo equation(215) simplifies to -The equilibrium constant for equation(216) is

-KA is expressed as a function of temperature by

-The replacement of Ba in BaTiO3 by the donor La is represented by -The equilibrium constant KD is

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

ΔHA change in enthalpy of the reactionK`A temperature-insensitive constant

n electron concentration

BaOOMn 232 OOTiBa VOMnBa 5`22

BaOTiO 22 2 OTiBa OTiBa 622

32OMn OO VOMn 2

26 Electrical conductuion1261 Charge transport parameters-The current density j will be given by j=nQv

-If the drift velocity of the charge is proportional to the force acting on them v=uE j=nQuE

-For materials for which nQu is constant at constant temperature Ohmrsquolaw

u mobility

Ej σ(conductivity) = nQu =(1ρ)AlGR 1

A Q v

v

EFlow of charge in a prism

-The room temperature values of conductivity characteristic of the broad categories of material

-16-12-8-4048

300 1000 TK

MetalsSemi-conductorsand semi-insulators

insulators

Conductivities of the various classes of materialshading indicates the range of values at room temperature

Material class Example Conductivitylevel

dσdT Carrier type

Metals AgCu High Smallnegative Electronssemiconductors SiGe Intermediate Largepositive ElectronsSemi-insulators ZrO2 Intermediate Largepositive ions or electrons

Insulators AlO3 Very low Very large positive

Ions or electronsFrequently

lsquomixedrsquoConductivity characteristics of the various classes of material

)log( 1Sm

26 Electrical conductuion2

262 Electronic conduction(a) Band conduction

(a)Atomic levels having identical energies merging to a broad band of levels differing slightly in energy as free atoms condense to form a crystal (b) band structure At equilibrium interatomic spacing in a crystal

Bands arising from inner electron levels

Partly filled band

Empty conductionband

Forbidden zone

Valence band

EF

Schematic electron energy band structures for (a) a metallic crystal and (b) a semiconducting or insulating crystal

Energy

-The electron density in the conduction

-The Fermi-Dirac function F(E)

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

Z(E)dE total number of states in the energy range dE around E per unit volume of the solid

EF Fermi energy(229)

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

Band structure with electrons promoted from the valence to the conduction band

- E-EFgtgtkT at room temperature kT≒0025eV E-EF over the 02eVthe term +1 can be omitted from equation(229)- The excited electrons and holes occupy states near the bottom of the conduction band and the top of the valence band respectively The electrons and holes behave as free particles for state distribution function- The upper limit of the integration in equation (228) is taken as infinity since since the probability of occupancy of a state by an electron rapidly approaches zero as the energy increases through the band Under these assumption

-If we put ni=pi

-A more rigorous treatment shows that

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

NcampNveffective state densities for electrons in the conduction band and hole in the valence band

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

amp he mm the effective electron and hole masses

- under conditions in which EF≒ the center of the band gap

- From equation (225) the conductivity can be written

-theory and experiment show a temperature dependence for u lying typically in the range which is so weak compared with that for n (and p) that for most purpose it can be ignored

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

epuenu he ue amp uh electron and hole mobilities

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

(b) The effect of dopants- The configuration resembles a hydrogen atom for which the ground state(n=1) Energy - doping nnep the equilibrium relation e`+ h nil -rsquonilrsquo indicates a perfect crystal with all electrons in their lowest energy states from equation(239)

Conduction band e - e - e- e- e-

+ + ++ +Valence band

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

Planar representation of a silicon crystal doped With P giving rise to a Psi defect

Effect of n-and p-type doping on the band structure of a semiconductor (eg silicon)

2202

4

32 hemeE

)`exp()(]`][[kTEgkTKnphe Eg band gap(at 0K)

K` independent of temperature

(239)

5+

Several cases of oxide systems in which the conductivity is controlled by the substitution of aliovalent cations (Chapter 4)

A limitation to present research is the non-availability of oxides that approach the parts in 109 purity of available silicon crystals

The study of semiconduction in oxides has necessarily been carried out at high temperatures( gt500C) because of the difficulties of making measurements when they have become highly resistive at room temperature

26 Electrical conductuion8-Semiconducting oxide -

1 Oxygen pressure (1) The features of oxide semiconductors is the effect on their behaviour of the

external oxygen pressure

Po2 Oxygen pressure Pa Atmosphere pressure σ Conductivity

Fig 212 Conductivity of undoped BaTiO3 as a function of pO2 and T

The general shape of the curves in Fig212 can be explained on the assumption that the observed conductivity is determined by electron and hole concentrations

The electron and hole mobilities depend only on temperature

1 Oxygen pressure (2) An estimate of K(T) can be arrived at as follow Combining equations (235) and (240) leads to (241) It follows that the value nm of n corresponding to a minimum σm in σ is given

by (242) which on substituting in equation (241) gives

(243) Combining equations (235) and (243) gives (244)

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

1 Oxygen pressure (3)

Equation (244) enables the relative contributions of electrons and holes to the conductivity to be estimated from the ratio of the conductivity to its minimum value without having to determine K(T)

If σ= σm α=1 then uhpm=uenm

When n is large

(246) When p is large (247) ue = 0808T-32exp(-ε ukT)m2V-1s-1 Eu= 202kJmol-1 (0021eV)

This give ue= 15X10-6m2V-1s-1 at 1000degC and 24X10-6m2V-1s-1 at 600degC uh is likely to be about 05ue

nue

ee

pue

hh

1 Oxygen pressure (4) The further analysis of the dependence of σ on po2 for BaTiO3 is mainly based o

n recent work by Smyth[3] Controlled factor for Conductivity po2 n p VOuml VTldquoildquo(VBldquoa)

The various po2 regions are now considered separately for the 1000degC isotherm of acceptor-doped or nominally pure BaTiO3

A - B in Fig213(a) the equilibrium reduction equation is OO hArr frac12 O2(g) + Vouml + 2ersquo (248)

which by the law of mass action leads to Kn = n2[Vouml]pO2

12 (249)

where Kn is the equilibrium constant

The oxygen vacancy concentration is regarded as insignificant compared with that arising through loss of oxygen

Therefore since (250)

61312)2( OpKnn

Fig 213 Schematic representation of the dependence of n p [VOuml] and [VTldquoildquo] on pO2 for (a) acceptor-doped and (b) donor-doped BaTiO3(After Smyth)

B ndash D in Fig213(a) the oxygen vacancy concentration now determined by the acceptor impurity concentration [Arsquo] is little affected by changes in pO2 and remains sensibly constant

(251) The p-type contribution to semiconductivity arises through the oxidation reaction i

nvolving take up of atmospheric oxygen by the oxygen vacancies according to Vouml + frac12 O2(g) hArr OO + 2hrsquo (252)

leading to p = [Vouml]12 Kp

12pO214 (253)

At pO2 ≒ 100Pa n = p σ = σm and the material behaves as an intrinsic D-E

Over this pO2 regime the discussion is more speculative since measurement against which the model can be checked have not been made

412

21

][

O

pV

Knn

1 Oxygen pressure (7) In the region D-E the dominating defect changes from VOuml to VTldquoildquo since the oxygen vac

ancies due to the acceptors are now filled The conductivity is largely governed by acceptor concentration and may be independent

of pO2 over a small pressure range

In the E-F region the equilibrium is O2(g) hArr VT

ldquoildquo + 2OO + 4hrsquo (254)

so that Krsquop = p4[VT

ldquoildquo]pO2

-1 (255)

which because p ≒ 4[VTldquoildquo] leads to

p = (4Krsquop)15pO215

Measurements in the region 10-17PaltpO2lt105Pa as shown in Fig212 show good agreement between the σ-pO2 slopes and the calculated

n-pO2 and p-p02 relations given above Increased acceptor doping moves the minimum in the σ-pO2 towards lower pressures

2 Donor-doped BaTiO3(1) The effect of pO2 on the conductivity of a

donor-doped system has been studied for lanthanum-substituted BaTiO3 as shown in

Fig214 for 1200degC The behaviour differs from that for acceptor -doped material (i) There is a shift of the curves towards higher oxygen pressures (ii) When at higher lanthanum the conductivity is independent of pO2

At low pressures the curves coincide with those of the lsquopurersquo ceramic At the lowest pO2 values (AB) loss of oxygen from the crystal is accompanied by

the formation of Vouml and electrons according to equations (248) and (250)

2 Donor-doped BaTiO3(2)

As pO2 is increased n falls to the level controlled by the donor concentration so that n≒[La

Ba] as shown in the following equation La2O3 hArr 2 La

Ba + 2OO + frac12 O2(g) + 2ersquo (257) When n is constant over B-C corresponding to the plateau in the curves of Fig214 there are changes in the energetically favoured Schottky disorder so th

at [Vouml]proppO2-frac12 according to equation (249) and [VT

ldquoildquo]proppO2

frac12 At C the condition

4[VTldquoildquo] = [La

Ba] (258) is established from the equilibrium 2La2O3 + 4TiO2 hArr 4 La

Ba + 3TiTi + VTldquoildquo + 12OO + lsquoTiO2rsquo (259)

where lsquoTiO2rsquo indicates incorporation in a separate phase Both [VT

ldquoildquo] and [Vouml] remain sensibly constant over the range C-E so that according

to equation (255) p = KrdquoppO2

frac14 and n= KrdquoppO2-frac14 (260)

At still higher values of pO2 (E-F) the dependence of p on pO2 is same with equation (256)

3 Properties of doped BaTiO3(1) The n- and p-type substituents at low concentrations have important effects o

n the room temperature behaviour of BaTiO3

Acceptor-doped material Iow oxygen pressure without losing its high resistivity at room temperature Piezoelectric properties under high compressive stress Oxygen vacancies are also associated with the fall in resistance that occurs at temperatures above 85 under high DC fields

Donor-doped material The basis of positive temperature coefficient (PCT) resistors The insulating dielectrics formed with high donor concentrations have a low oxygen vacancy content and are therefore less prone to ageing and degradation

The effects of aliovalent substituents in PbTiO3 and Pb(Ti Zr)O3 are broadly speaking similar to those in BaTiO3

4 Band model There is less confidence then elemental semiconductors in band models for th

e oxide semiconductors because sufficiently precise physical and chemical characterization of the materials is often extremely difficult

In addition measurements are necessarily made at high temperatures where knowledge of stoichiometry impurity levels dislocation content defect association and other characteristics is poor

Fig215 shows a tentative band model for doped barium titanate

Fig215 Tentative band model for doped BaTiO3(energes in electronvolt)

(d) Polaron conduction The band model is not always appropriate for some oxides and the electron

or hole is regarded as lsquohoppingrsquo from site to site

lsquoHoppingrsquo conduction occurs when ions of the same type but with oxidation states differing by unity occur on equivalent lattice sites and is therefore likely to be observed in transition metal oxides

The addition of Li2O to NiO leads to an increase in conductivity as illustrated in fig216

1

-1

-2

0 5 10 15

[Li+]at (261) Fig 2 16 Resistivity of NiO as a function of lithium content

kTEu Aexp

263 Ionic conduction(1) Crystals Ionic conduction depends on the presence of vacant sites into which ions can

move In the absence of a field thermal vibrations proportional to kT cause ions and vacancies to exchange sites

The Nernst ndash Einstein equation links this process of self-diffusion with the ion drift σi caused by an electric field

(262) where Di is the self- or trace-diffusion coefficient for an ion species i Qi is the

charge if carries and Ni is its concentration Features that contribute to ionic mobility are small charge small size and latti

ce geometry A highly charged ion will polarize and be polarized by the ions of opposite cha

rge as it moves past them and this will increase the height of the energy barrier that inhibits a change of site

kTQN

Diii

i

263 Ionic conduction(2) Some structures may provide channels which give ions space for movement

The presence of vacant sites assists conduction since it offers the possibility of ions moving from neighbouring sites into a vacancy which in consequence moves in the opposite direction to the ions (Fig217)

The cations usually have to pass through the relatively small gap between three O2- ions to reach any neighbouring cation vacancy

Fig217 Energy barriers to ionic transport in a crystal

(a) in the absence of a field and (b) with applied field E

263 Ionic conduction(3) The crystal is highly ionic in character the barrier is electrostatic in origin and ion

in its normal lattice position is in an electrostatic potential energy lsquowellrsquo (Fig217) When an electric field EE is imposed barrier heights are no longer equal and the j

ump probability is higher for the jump across the lower barrier (in the illustrated case to the right) of height Ej ndash ΔEj where

ΔEj = e E a2 (263) Since we know bias in jump probability in one direction it is not difficult to arrive

at the following expression for the current density

(264)

in which nvN is the fraction of Na+ sites that are vacant and A is a constant describing the vibrational state of the crystal Since it is assumed that the vacancy is part of the Schottky defect then nv = ns and hence using equation (212) we obtain

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

263 Ionic conduction(4) Because the temperature dependence of σ is dominated by the exponential t

erm the expression for conductivity is frequently written

(267) in which Ei = Ej + ΔHs2 is an activation energy and σ0 is regarded as tempera

ture independent

Glasses 1 Conductivity σ depends upon temperature 2 σ decreases as the size of the mobile ion increase 3 σ decreases as the concentration of blocking ions increases

Mixed-phase materials In practice ceramics are usually multiphase comprising crystalline phase gla

sses and porosity The overall behaviour depends on the distribution as well as properties of the

se constituents

kTEiexp0

265 Schottky barriers to conduction

Fermi-Dirac fuction

(269)

where EF is the Fermi energy

The energy oslashm required to remove an electron with the Fermi energy to a point outside the metal with zero kinetic energy is called the lsquowork functionrsquo of the metal

Excited semiconductor the effective work function oslashs of the semi-conductor is the energy difference between the Fermi energy and the vacuum level

1

1exp)(

kTEEEF F

Schottky barriers

In the vicinity of the junction typically within 10-6 - 10-8m depending on the con

centration of n dopant the donors are ionized

Electrons moving up to the junction from the semiconductor then encounter an energy barrier ndash a Schottky barrier ndash of height |e|Ub

Fig219 Metal - n - type semiconductor junction(oslashmgtoslashs) (a)before contact and (b) after contact

Metal-semiconductor At metal-semiconductor current-voltage characteristic is

following

I

Reverse bias Forward biasU

Fig220 Current-voltage characteristic for a metal-semiconductor rectifying junction

N-type semiconductor sandwiched A sandwich comprising a semiconductor between two metallic

electrodes presents the same effective barrier irrespective of the sense of an applied voltage

The situation is illustrated in Fig221 Current ndash voltage characteristic in Fig222

Fig221 n-type semiconductor sandwiched between two metal electrodes

Fig222 Current-voltage characteristic for back to back Schottky barriers

Junction n-type and p-type

Junction is that between n and p types of the same semiconductor

The situation before and after contact is illustrated in Fig223

Fig223 Junction between an n-type and a p-type semiconductor (a) before contact (b) after contact

About n-p junction reverse bias Forward bias in Fig224

Fig224 n ndash p junction (a) reverse bias (b) forward bias

27 Charge displacement processes271 dielectric in static electric fields(a)Macroscopic parameters

1Atomic polarizationAtomic polarization occurs in all materialsit is a smalldisplacement of the electrons in an atom relative tothe nucleus2ionic polarization ionic polarization involving the relative displacementof cation and anion sublattices 3dipolar polarization dipolar materials such as water can become polarizedbecause the applied electric field orients the molecules4space charge polarizationspace charge polarization involves a limited transport ofcharge barriers until they are stopped at a potential barrierpossibly a grain boundary or phase boundary

Fig 225 Various polarization processes

271 dielectric in static electric fields

The dipole moment P of the dipole

polarized material can be regarded as made up of elementary dipolar prismsthe dipole moment per unit volume of material is termed the polarization P and can vary from region to region From Fig 226 the magnitudes of the vectors are given by

or

in general where n is the unit vector normal to the surface enclosing the polarized material and directed outwards from the material

xQp

VxAp pp ppVp

pnp

Fig 226 Elementary prism of polarized material

From Gaussrsquos theorem the electric field E between and normal to two parallel plates carrying surface charge density and separated by a vacuum is

In (b) the polarization charge density appearing thus the effective charge density giving rise to E is reduced to so that

Total charge density is equivalent to the magnitude of the dielectric

displacement vector D

0E

0 pTE

pT T

T

pED 0

Fig 227 The role of the dielectric in a capacitor

If the dielectric lsquolinearrsquo so that polarization is proportional to the electric field within the material which commonly case

It follows from that

And since

is the total charge on the capacitor plate therefore the capacitance is

Since vacuum has zero susceptibility the capacitance is

The permittivity of the dielectric is defined by

and is the relative permittivity(or dielectric constant)of the dielectric

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

(b)From induced elementary dipoles to macroscopic properties

applied external fielddepolarizing field internal macroscopic field ( )

aEdpEmE dpa EE

bull In reality the atomic nature of matter dictates that the local field which is known as the Lorenz field

bull is the contribution from the charges at the surface of the spherical cavity

bull is due to the dipoles within the boundary can be shown to be

bullHowever certain crystals of high symmetry and glasses it can be and for these cases

bullIn the more general case it is assumed that

bull is the lsquointernal field constantrsquo

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

Fig 228 The lsquolocalrsquo field in a dielectric

The dipole moment P induced in the entity can now be written

If it is assumed that all entities are of same type and have a density N then

or

In the particular case for which rearrangement of equation leads of the Clausius-Mosotti relationship

Ferroelectrics possess very high permittivity values which vary considerably with both applied field strength and temperature The permittivity reaches a peak at the Curie-Weiss law

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

272 Dielectrics in alternating electric fields

(a)Power dissipation in a dielectric )sin(0 wtU

Fig 229 Sinusoidal voltage applied to a perfect capacitor

If voltage described then the current is and leads U by Since the instantnaneous power drawn from the voltage source is the time average power dissipated is where

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

Fig 230 phasor diagram for a perfect capacitor

The applied voltage at a given point in time is represented by a horizontal line and the instantnaeous current by a vertical line since it leads the voltage by

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Time average power loss is

Intergrating equation gives

Since and

For disspated power density in the dielectric

is termed the lsquoloss factorrsquo of the dielectric and is the lsquodielectric conductivityrsquo

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

Fig 231 phasor diagram for a real capacitor

(b) The complex permittivity

can represent a complex sinusoidal voltage The time differential of U is given

The instantaneous charge on a lsquolosslessrsquo vacuum capacitor is

And Complex relative permittivity It follows

that

As indicated in Fig 232 It can be seen from the figure that

The current in phase with U can be written

So that the current density is given by

Average dissipated power density is given by

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

Fig 232 Capacitative and lsquolossrsquo components of total current I

(c) Frequency and temperature dependence of dielectric properties

(1) Resonance effects If the damping (resistive) force is assumed to be

proportional to the velocity of the moving charged particle the equation becomes

Solving and ignoring the transient term yields

Since -ex(t) is the induced dipole moment per atom the complex polarization is

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

Fig 233 Variation in and with frequency close to a resonance frequency

r

0w

(2) Relaxation effects

ds

d

PP

Fig 234 schematic one-dimensional representation of the electrostatic potential in a glass

Polarization processes occurs in ceramics for which the damped forced harmonic motion approach is inappropriate

for example because of the random structure of glass the potential energy of a cation moving through a glass can be shown schematicalliy as in Fig 234

Fig 235 Development of polarization by slow diffusional processes

Figure 235 illustrates how on the application of a field and following the initial instantaneous atomic and ionic polarization

The slow diffusional polarization approaches its final static value

dPdsP

)(tPP dds

)(1 tPPP ddsd

It assumed that time t the polarization develops at a rate proportional to

In which is a proportionality constant Integrating with initial condition when t=0 gives

Where is a relaxation time If assumed that the polarizing field is it can be shown that

In which is the value of the permittivity measured at low frequencies or with a static field applied It can be integrated to give

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

Fig 236 variation in permittivity with for a dielectric showing lsquoDebyersquo relaxation

If the transient is neglected it leads to

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

Fig 236 show graphically Debye equations

The relaxation frequency is because the polarization occurs by the same temperature-activated diffusional processes which give rise to DC conductivity depends on temperature through an exponential factor

1rw

)exp(0 kTA

Fig 237 permittivity dispersion and dielectric loss for a glass

r

)( wr

Fig 238 variation of and with frequency Space charge and dipolar polarizations are relaxation processes and are strongly temperature dependent ionic and electronic polarizations are resonance processes and sensibly temperature independent Over critical frequency ranges energy dissipation is a maximum as shown by peaks in

r

bull various polarization processes which lead to dielectric dispersion and attendant energy dissipation are summarized in fig 238

bullIn conclusion it is opportune to mention the relationship between the refractive index n and the relative permittivity

2nr

-Fig242 Variation in the potential energy of Ti4+ aling the c axis

-Tetragonal BaTiO3 the energy if the Ti4+ion in terms of its position along the c axis takes the form of two wells-Applied field in the opposite direction to the polarization may enable a Ti4+ion to pass over the energy barrier between the two states and so reverse the direction of the polarity at that point -When this happens the energy barriers for

neighboring ions are reduces and the entire region affected by the field will eventually switch into the new direction

Fig243 (a) surface charge associated with spontaneous polarizaion

(b) formation of 180deg domains to minimize electrostatic energy

273 Barium titanate- the prototype ferroelectric ceramic

Fig243(a)(b) These regions are called domains and the whole configuration shown comprises 180deg domains

bull Surface mosaic carrying apparent charges of opposite sign resulting in a reduction in Ed and in energy

bull The presence of mechanical stress in a crystal results in the development of 90deg domains configured so as to minimize the strain

bull The configurations ca be modified by imposing either an electric or a mechanical stress

bull A polycrystalline ceramic that has not been subjected to a static field behaves as a non-polar material even though the crystals comprising it are polar

bull One of the most valuable features of ferroelectric behaviour is that ferroelectric ceramics can be transformed into polar materials by applying a static field

bull Poling and depoling processes are illustrated schematically in Fig244

-Fig244 Poling in a two-dimensional ceramic (a) unoriented material (b)oriented by 180deg domain changes (c) oriented by 180deg and 90deg domain changes (d) disoriented by stress

-The random directions of the crystallographic axes of the crystallites of a ceramic limit the extent to which spontaneous polarization can be developed

-The fractions of the single-crystal polarization value polar axes alignments 083(tetragonal) 091(orthorhombic) 087(rhombohedral) perovskites structure

-The orientations occurring in a simple domain structure are shown schematically in Fig245(b)

Fig245 (b) schematic diagram of 180deg and 90deg domains in barium titanate

bullThe thickness of the layer separating the domains is of the order of 10nm but varies with temperature and crystal puritybullThe wall energy is of the order 10mJm2

Fig246 (a) Hysteresis loops for a single-domain single crystal of BaTiO3

bullThe almost vertical portions of the loop are due to the reversal of the spontaneous polarization as reverse 180 degdomains nucleate and grow

bullThe horizontal portions represent saturated states crystal is single domain with a permittivity of 160 obtainable in the polar direction

bullThe coercive field at room temperature - 50Hz 01MVm saturation polarization(027Cm )1

Fig246 (b) Hysteresis loops for BaTiO3 ceramic

-The coercive field is higher and the remanent polarization is lower than for a single crystal

-Both 180 degand 90 degchanges take place during a cycle and are impeded by the defects and internal strains within the crystallites

-The hysteresis loss single crystal 01MJm

-Rapid rise in temperature at 1000Hz power 100MWm

1

3

bullThe dissipation factor in ceramic at high fields very high

bullBut at the 100Vmm level tanδ less than 01 for undoped material

bullA unusually characteristic of ferroelectric materials properties change with Time

1

ta

tp

)(log

010 t

tap t0 arbitrary zero for the timeUsually a ltlt 00510log eaa

bullThe property negative for permittivity positive for Youngrsquos modulus

bullAfter sufficient lapse of time negligible =gt mechanical electrical stresses exceeding the Curie point

bullVery advantage of ceramic ferroelectrics properties can be modified by adjusting the composition and ceramic microstructure

bullAdditions and the substitution of alternative cations effects =gt

1 Shift the Curie point and other transition temperatures 2 Restrict domain wall motion 3 Introduce second phases or compositional heterogeneity 4 Control crystallite size 5 Control the oxygen content and the valency of the Ti ion

bullThe effects are important for the following reasons =gt

1 Changing the Curie point enables the peak permittivity to be put in a temperature range in which it can be exploited Ba2+ of BaTiO3 substitution of Sr2+ = lowers Tc substitution of Pb+ increase Tc

Fig247 The effect in the Curie point of the substitutiom of either strontium ot lead for barium in BaTiO3

Fig248 The effect of grain size on the permittivityof a BaTiO3 ceramic

2 Dissipation factor due to domain wall motion Fe3+ Ni2+ Co3+ can occupy Ti4+ sites reduce

3 CaZrO3 additions broadening of the permittivity-temperature peak Materials contain regions of variable composition that contribute a range if Curie points so that the high permittivity is spread over a wider temperature range

4 Cations that have a higher valency than those they replace (gt05 cat) Ti4+ =gtNb5+ Ba2+=gtLa3+ inhibit crystal growth Rising the permittivity level below the Curie pointltFig248gt Crystal size is also controlled by sintering conditions

5 Higher-valency substituents = lead to low resistivity lower-valency substituents (Ti3+=gtMn3+) = act as acceptors enable high-redielectrics to be sinterd in atmospheres with low oxygen contents

274 Mixtures of dielectricsbulllsquoConnectivityrsquo classifying different types of mixture

bullAny phase in a mixture self-connected in zero one two three dimensions

bullConnectivity of 0 dispersed particles Connectivity of 3 medium surrounding particles

bullA disc containing a rod-shaped phase extending between its major surfaces

connectivity 1 with respect to the rods 3 with respect to the intervening phase

bullMixture stack of plates of two different phases connectivity of 2-2

bullIn all 10 different connectivities for mixtures of two phases(0-0 1-0 2-0 3-0 1-1 2-1 3-1 2-2 3-2 3-3) for mixtures of three phases(20) for mixtures of four phases(35)

bull3-0 system Maxwell deduced that permittivity( ) of random dispersion of spheres in matrix of relative is

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

For equation reduces to

1012 fV(2121)

274 Mixtures of dielectrics

Fig249 log resistivity versus volume fraction of conductive particles in an insulating matrix

-Convert the values of permittivity found for porous bodies to the value expected for fully dense bodies-Eq(2121) is less than about 01V (agreement)-Resistance-volume concentration relations for dispersions of conductive particle in insulating media(Fig249)-The resistivity remains high until a critical concentration in the neighbourhood of 005-02 is reached when it drops by several orders of magnitude-Transition from a dispersion of separated particles to one of connected aggregates

fV

Fig250 Equivalent structures for dielectrics with (a) 1-3 and (b) 2-2 connectivity

-Fig250 A capacitor containing a two-phase 1-3 dielctric consisting of rods of extending from one elctrode to the other in a medium of is equivalent in behaviour

-The structure consists of two capacitors in parallel so that

12

hA

hAm 2211

12)1( ffm VV

-Fig250(b) 2-2 connectivity dielectric with the main planes of the phases parallel to the electrodes is equivalent to the structure-Effectively two capacitors in series

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

Where n=+-1 or for a multiplicity of phases of partial volumes Vf1 Vf2hellipVfi

-The approximation is only valid for small values of x and n but nevertheless

xnxn ln1

i

ifim V ln

-Differentiation of equation of above eq With respect to temperature gives

i

fim

m TV

T

1

-Which gives the temperature coefficient of permittivity for a mixture of phases and although not in exact agreement with observation is a useful approximation

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Substituting from equation (21) gives

From the quantum condition for angular momentum

which substitutes with equation (21)leads to

or

22

0

2

21

4vm

re

0

2

8 (23)

nhwrme 2

42

22

30

2

22

4 rmhn

rmew

e

22

2

41

hnem

r o

e

(25)

(24)

21 Atoms 2

2220

2

4 132 nh

eme

(26)

mvh

(27)

0)(22

2 phm (28)

21 Atoms 3

the form

2

zyx

)( r

)()()()( rRr (29)

21 Atoms 4

1 mln s n22n

S

LSL

SL 0L SJ

21 Atoms 5

4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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2220

2

4 132 nh

eme

(26)

mvh

(27)

0)(22

2 phm (28)

21 Atoms 3

the form

2

zyx

)( r

)()()()( rRr (29)

21 Atoms 4

1 mln s n22n

S

LSL

SL 0L SJ

21 Atoms 5

4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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the form

2

zyx

)( r

)()()()( rRr (29)

21 Atoms 4

1 mln s n22n

S

LSL

SL 0L SJ

21 Atoms 5

4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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1 mln s n22n

S

LSL

SL 0L SJ

21 Atoms 5

4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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4Zr

3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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3

2

6

BaTiOTiOMO

6MO

2TiO

6TiO

3BaTiO 3BaO

6TiO

from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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from Fig23

T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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T

G G TG

TG

PTGS )(

Latent heat

5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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5+

4+

+

5+

252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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252 Point defects

s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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s

)2

exp(`)(

)2

exp(

21

kTHNNn

kTHNn

FF

SS

AgiAg VAgAg hVV

eVVeVV

MM

OO

```

(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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(215)

][][]`[

32

2

OMnVTiMnK O

A

)exp(`kTHKK A

AA

`2)(22 221

32 egOOLaOLa OBa

][][

)exp(`32

21222

OLapnLaBa

kTHKK OD

DD

3+ 4+

(216)

3+

32OMn OO VOMn 2

u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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u mobility

A Q v

v

-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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-16-12-8-4048

300 1000 TK

insulators

dσdT Carrier type

)log( 1Sm

Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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Partly filled band

Forbidden zone

Valence band

EF

Energy

topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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topE

Eci dEEFEZn )()(

11)exp()(

kTEEEF F

(228)

+ + + + + + + + + + + + + + Valence band

Conduction band

- - - - - - - - - - - - - -

EFEg

Ec

Ev

Energy

-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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-If we put ni=pi

)exp()exp(kTEvENvp

kTEEcNcn F

iF

i

2EvEcEF

)ln(4

32

h

eF

mmkTEvEcE

he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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he mm

)2

exp(10)exp(10 2525

kTEg

kTEEcpini F

)2

exp()(10)( 25

kTEguueuuen hehei

5251 ~ TT

)2

exp(kTEgB

Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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Donor states

Acceptor states

Eq=001eV

Eg=11eV

Eg=001eV

2202

4

32 hemeE

(239)

5+

where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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where α=uhpuen

nTKunu

eh

e)(

)(2 TKuune

mh

)(42

TKuue

hem

21

e

When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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When n is large

nue

ee

pue

hh

12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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12 (249)

61312)2( OpKnn

12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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12pO214 (253)

412

21

][

O

pV

Knn

ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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ldquoildquo]pO2

-1 (255)

ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

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ldquoildquo]proppO2

1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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1

-1

-2

0 5 10 15

kTEu Aexp

kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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kTQN

Diii

i

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
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Slide 73

(264)

(266)

kTEE

TA

Nnj jv exp

2

1exp HsEkTT

Aj

ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
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Slide 8
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Slide 10
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Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
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Slide 56
Slide 57
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Slide 59
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ture independent

se constituents

kTEiexp0

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
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Slide 27
Slide 28
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Slide 32
Slide 33
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Slide 35
Slide 36
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Slide 41
Slide 42
Slide 43
Slide 44
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Slide 46
Slide 47
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Slide 51
Slide 52
Slide 53
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Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

Fermi-Dirac fuction

(269)

1

1exp)(

kTEEEF F

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
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Slide 12
Slide 13
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Slide 41
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Slide 51
Slide 52
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Slide 73

Schottky barriers

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
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Slide 16
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Slide 19
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Slide 23
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Slide 27
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Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
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Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
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Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

following

I

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
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Slide 30
Slide 31
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Slide 33
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Slide 35
Slide 36
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Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
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Slide 46
Slide 47
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Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
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Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
Slide 16
Slide 17
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Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
Slide 16
Slide 17
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Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
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Slide 17
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Slide 19
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Slide 22
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Slide 24
Slide 25
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Slide 27
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Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

or

xQp

VxAp pp ppVp

pnp

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
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Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

0E

0 pTE

pT T

T

pED 0

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
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Slide 52
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Slide 73

And since

EP e 0

EEED ee 000 )1( TD

hU

AQ

eT

0)1(

TQ

hA

UQC eT

0)1(

0C

hAC 00

)1(0 e re

10

r

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
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Slide 12
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Slide 23
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Slide 31
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Slide 33
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Slide 35
Slide 36
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Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
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Slide 46
Slide 47
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Slide 49
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Slide 51
Slide 52
Slide 53
Slide 54
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Slide 72
Slide 73

aEdpEmE dpa EE

dpmL EEEE pE

dE pE 03 p

0dE

00 33 PEEPEE dpamL

PEE mL

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
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Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
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Slide 72
Slide 73

or

LEP

)( PENNPP m

NN

Ep

em

1

0

031

0321

N

r

cr T

A

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
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Slide 23
Slide 24
Slide 25
Slide 26
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Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
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Slide 72
Slide 73

)cos(0 wtU

T

c dtwtwtIUT

P0 0 0)cos()sin(1T

cUdtIT

P0

1

90

PUIc

or

90

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
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Slide 12
Slide 13
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Slide 52
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Slide 71
Slide 72
Slide 73

TT

dtwtIwtUT

UIdtT

P0 000

)cos()sin(11

sin21

00IUP

Since and

cos0 cII CwUIc 0

tan21tan

21 2

00 wCUIUP c

tan21

02

0 rwEVP

tanrrw 0 tan

tan0 rAC w

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
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Slide 71
Slide 72
Slide 73

that

jwUjwtjwUU

)exp(0

)exp(0 jwtUU

0C

0UCQ UjwCCUQI 00

0

rj

UCwUCjwUCjwIr 0

0

tan

r

1I

UhAwI

r0

1

EwEwAI

r 0

1

tan21

21

02

0

02

0 rrAC wEwEEVP

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

)(

122222

00

2

wwww

mNe

r

0w

ds

d

PP

dPdsP

)(tPP dds

)(1 tPPP ddsd

)(tPd

1 0dP

)exp(1tPP dsd

)()1 0 tPEP drrsd

Ejw

tCP rrsd 0

1)exp(

)exp(0 jwtEE

or

and

jw

rrsr

11

)exp( tC

22

11

wrrs

r

22

1)(

ww

rrsr

1rw

)exp(0 kTA

r

)( wr

r

2nr

1

3

1

ta

tp

)(log

010 t

m 12

)(2

)(31

2121

212

f

fm V

V

)231

(2f

m

V

1012 fV(2121)

fV

12

hA

hAm 2211

12)1( ffm VV

22

2

11

1

Ah

m

1

11

21 )1( ffm VV

nf

nm VV 12)1(

i

nifi

nm V

xnxn ln1

i

ifim V ln

i

fim

m TV

T

1

Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Slide 13
Slide 14
Slide 15
Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
Slide 22
Slide 23
Slide 24
Slide 25
Slide 26
Slide 27
Slide 28
Slide 29
Slide 30
Slide 31
Slide 32
Slide 33
Slide 34
Slide 35
Slide 36
Slide 37
Slide 38
Slide 39
Slide 40
Slide 41
Slide 42
Slide 43
Slide 44
Slide 45
Slide 46
Slide 47
Slide 48
Slide 49
Slide 50
Slide 51
Slide 52
Slide 53
Slide 54
Slide 55
Slide 56
Slide 57
Slide 58
Slide 59
Slide 60
Slide 61
Slide 62
Slide 63
Slide 64
Slide 65
Slide 66
Slide 67
Slide 68
Slide 69
Slide 70
Slide 71
Slide 72
Slide 73

and so that

it follow

)exp(02

0 jwtQExmwxmxm

x

)()exp()(

220

0

wjwwmjwteEtx

P)()( txeNP

)(

122

00

2

wjwwmNe

e

wjwwmNe

r

)(1122

00

2

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Date post:	11-Feb-2016
Category:	Documents
Upload:	sinjin
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