2Electronic Devices_sample Chapter

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Eighth Edition

GATEELECTRONICS & COMMUNICATION

Electronics Devices

Vol 4 o f 10

R. K. KanodiaAshish Murolia

NODIA & COMPANY


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GATE Electronics & Communication Vol 4, 8eElectroncis DevicesRK Kanodia & Ashish Murolia

Copyright By NODIA & COMPANY

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To Our Parents


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Preface to the Series

For almost a decade, we have been receiving tremendous responses from GATE aspirants for our earlier books:

GATE Multiple Choice Questions, GATE Guide, and the GATE Cloud series. Our first book, GATE Multiple

Choice Questions (MCQ), was a compilation of objective questions and solutions for all subjects of GAT E

Electronics & Communication Engineering in one book. The idea behind the book was that Gate aspirants who

had just completed or about to finish their last semester to achieve his or her B.E/B.Tech need only to practiceanswering questions to crack GATE. T he solutions in the book were presented in such a manner that a student

needs to know fundamental concepts to understand them. We assumed that students have learned enough of

the fundamentals by his or her graduation. The book was a great success, but still there were a large ratio of

aspirants who needed more preparatory materials beyond just problems and solutions. This large ratio mainly

included average students.

Later, we perceived that many aspirants couldnt develop a good problem solving approach in their B.E/ B.Tech.

Some of them lacked the fundamentals of a subject and had difficulty understanding simple solutions. Now,

we have an idea to enhance our content and present two separate books for each subject: one for theory, which

contains brief theory, problem solving methods, fundamental concepts, and points-to-remember. The second book

is about problems, including a vast collection of problems with descriptive and step-by-step solutions that can

be understood by an average student. This was the origin of GATE Guide(the theory book) and GATE Cloud(the problem bank) series: two books for each subject. GATE Guideand GATE Cloudwere published in three

subjects only.

Thereafter we received an immense number of emails from our readers looking for a complete study package

for all subjects and a book that combines both GATE Guideand GATE Cloud. This encouraged us to present

GAT E Study Package (a set of 10 books: one for each subject) for GAT E Electronic and Communication

Engineering. Each book in this package is adequate for the purpose of qualifying GATE for an average student.

Each book contains brief theory, fundamental concepts, problem solving methodology, summary of formulae,

and a solved question bank. The question bank has three exercises for each chapter: 1) Theoretical MCQs, 2)

Numerical MCQs, and 3) Numerical Type Questions (based on the new GATE pattern). Solutions are presented

in a descriptive and step-by-step manner, which are easy to understand for all aspirants.

We believe that each book of GAT E Study Package helps a student learn fundamental concepts and develop

problem solving skills for a subject, which are key essentials to crack GAT E. Although we have put a vigorous

effort in preparing this book, some errors may have crept in. We shall appreciate and greatly acknowledge all

constructive comments, criticisms, and suggestions from the users of this book. You may write to us at rajkumar.

[email protected] and [email protected].

Acknowledgements

We would like to express our sincere thanks to all the co-authors, editors, and reviewers for their efforts in

making this project successful. We would also like to thank Team NODIA for providing professional support forthis project through all phases of its development. At last, we express our gratitude to God and our Family for

providing moral support and motivation.

We wish you good luck !

R. K . Kanodia

Ashish Murolia


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SYLLABUS

GATE Electronics & Communications

Energy bands in silicon, intrinsic and extrinsic silicon. Carrier transport in silicon: diffusion current, drift current,mobility, and resistivity. Generation and recombination of carriers. p-n junction diode, Zener diode, tunnel diode,BJ T, J FET, MOS capacitor, MOSFET, LED, p-I-n and avalanche photo diode, Basics of LASERs. Devicetechnology: integrated circuits fabrication process, oxidation, diffusion, ion implantation, photolithography, n-tub,p-tub and twin-tub CMOS process.

IES Electronics & Telecommunication

Electrons and holes in semiconductors, Carrier Statistics, Mechanism of current flow in a semiconductor, Halleffect; J unction theory; Different types of diodes and their characteristics; Bipolar J unction transistor; F ield effecttransistors; Power switching devices like SCRs, GTOs, power MOSFETS; Basics of ICs - bipolar, MOS and CMOS

types; basic of Opto Electronics.

**********


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CONTENTS

CHAPTER 1 SEMICONDUCTORS IN EQULIBRIUM

1.1 INTRODUCTION 1

1.2 SEMICONDUCTOR MODELS 1

1.2.1 Bonding Model 1

1.2.2 Energy Band Model 2

1.3 CARRIERS 2

1.3.1 Carrier Properties 3

1.4 INTRINSIC SEMICONDUCTOR 3

1.5 DOPING 3

1.5.1 n-type Semiconductor 4

1.5.2 p-type Semiconductor 4

1.6 COMPENSATED SEMICONDUCTOR 5

1.7 FERMI FUNCTION 5

1.7.1 Energy Dependence of Fermi Function 5

1.8 EQUILIBRIUM CARRIER CONCENTRATIONS 7

1.8.1 Intrinsic Carrier Concentration 9

1.8.2 Extrinsic Carrier Concentration 9

1.9 ENERGY BAND DIAGRAM FOR INSULATOR, SEMICONDUCTOR, AND METAL 10

1.9.1 Insulator 10

1.9.2 Semiconductor 10

1.9.3 Metal 10

1.10 POSITION OF FERMI ENERGY LEVEL 10

1.10.1 Fermi Energy Level for n-type Semiconductor 11

1.10.2 Fermi Energy Level for p-type Semiconductor 12

1.10.3 Variation of Fermi Level with Temperature 12

1.11 CHARGE NEUTRALITY 13

1.11.1 Determination of Thermal Equilibrium Electron Concentration as a Function of ImpurityDoping Concentration 13

1.11.2 Determination of Thermal Equilibrium Hole Concentration as a Function of ImpurityDoping Concentration 13

1.12 DEGENERATE AND NON DEGENERATE SEMICONDUCTORS 14

1.12.1 Non-degenerate Semiconductor 14

1.12.2 Degenerate Semiconductor 14


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1.13 IMPORTANT PROPERTIES AND STANDARD CONSTANTS 15

EXERCISE 1.1 17

EXERCISE 1.2 24

EXERCISE 1.3 27

SOLUTIONS 1.1 31

SOLUTIONS 1.2 47

SOLUTIONS 1.3 60

CHAPTER 2 SEMICONDUCTORS IN NON EQUILIBRIUM

2.1 INTRODUCTION 65

2.2 CARRIER DRIFT 65

2.2.1 Motion of Carriers in a Crystal 65

2.2.2 Drift Current 66

2.3 CARRIER MOBILITY 67

2.3.1 Mobility due to Lattice Scattering 68

2.3.2 Mobility due to Ionized Impurity Scattering 68

2.3.3 Mobility Variation Due to Electric field 69

2.4 CONDUCTIVITY 69

2.5 RESISTIVITY 69

2.6 CARRIER DIFFUSION 70

2.6.1 Diffusion Current Density for Electron 70

2.6.2 Diffusion Current Density for Hole 70

2.6.3 Diffusion Length 70

2.7 TOTAL CURRENT DENSITY 71

2.8 THE EINSTEIN RELATION 71

2.9 BAND BENDING 72

2.10 QUASI-FERMI LEVELS 73

2.11 OPTICAL PROCESSES IN SEMICONDUCTORS 74

2.11.1 Absorption 74

2.11.2 Emission 74

2.12 AMBIPOLAR TRANSPORT 74

2.13 HALL EFFECT 75

2.13.1 Hall Field 75

2.13.2 Hall Voltage 76

2.13.3 Hall Coefficient 76

2.13.4 Applications of Hall effect 76


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EXERCISE 2.1 77

EXERCISE 2.2 86

EXERCISE 2.3 92

SOLUTIONS 2.1 98

SOLUTIONS 2.2 111

SOLUTIONS 2.3 125

CHAPTER 3 PN JUNCTION DIODE

3.1 INTRODUCTION 133

3.2 BASIC STRUCTURE OF THE pn-JUNCTION 133

3.2.1 Space Charge Region in pnjunction 134

3.3 ZERO APPLIED BIAS 134

3.3.1 Energy Band Diagram for Zero Biased pn junction 134

3.3.2 Built-in Potential Barrier for Zero Biased pn junction 135

3.3.3 Electric Field in Space Charge Region 135

3.3.4 Space Charge Width 136

3.4 REVERSE APPLIED BIAS 136

3.4.1 Energy Band Diagram for Reverse Biased pn J unction 136

3.4.2 Potential Barrier for Reverse Biased pn J unction 137

3.4.3 Space Charge Width 137

3.4.4 Electric Field 137

3.4.5 J unction Capacitance 137

3.5 FORWARD APPLIED BIAS 138

3.5.1 Energy Band Diagram for Forward Biased pn J unction 1383.5.2 Excess Carrier Concentration 138

3.5.3 Ideal pnJ unction Current 138

3.5.4 Ideal Current-Voltage Relationship 139

3.6 SMALL-SIGNAL MODEL OF THE pnJUNCTION 139

3.6.1 Diffusion Resistance 140

3.6.2 Small-Signal Admittance 140

3.7 COMPARISON BETWEEN PN JUNCTION CHARACTERISTICS FOR ZERO BIAS, REVERSE BIAS,

AND FORWARD BIAS 140

3.8 JUNCTION BREAKDOWN 141

3.8.1 Zener Breakdown 141

3.8.2 Avalanche Breakdown 141

3.9 TURN-ON TRANSIENT 141

3.10 SOME SPECIAL PN JUNCTION DIODE 142

3.10.1 Tunnel Diode 142

3.10.2 PIN Diode 144


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3.10.3 Varactor Diode 144

3.10.4 Schottky Diode 145

3.11 THYRISTORS 146

3.11.1 Silicon Controlled Rectifier (SCR) 146

3.12 TRIAC 150

3.13 DIAC 151

EXERCISE 3.1 153

EXERCISE 3.2 163

EXERCISE 3.3 169

SOLUTIONS 3.1 182

SOLUTIONS 3.2 202

SOLUTIONS 3.3 220

CHAPTER 4 BJT


4.2 BASIC STRUCTURE OF BJT 233

4.2.1 Typical Doping Concentrations for BJ T 234

4.2.2 Depletion Region 234

4.3 TRANSISTOR BIASING 235

4.3.1 Active Region 236

4.3.2 Saturation Region 236

4.3.3 Cut-off Region 236

4.3.4 Reverse Active Region or Inverse Region 237

4.4 OPERATION OF BJT IN ACTIVE MODE 237

4.4.1 Transistor Current Relation 238

4.5 MINORITY CARRIER DISTRIBUTION 240

4.5.1 Minority Carrier Distribution in Forward Active mode 241

4.5.2 Minority Carrier Distribution in Cut-Off Mode 241

4.5.3 Minority Carrier Distribution in Saturation Mode 242

4.5.4 Minority Carrier Distribution in Reverse Active Mode 242

4.6 CURRENT COMPONENTS IN BJT 243

4.6.1 DC Common-Base Current Gain 243

4.6.2 Small Signal Common Base Current Gain 243

4.6.3 Common Emitter Current Gain 245

4.7 EARLY VOLTAGE 245

4.8 BREAKDOWN VOLTAGE 246

4.8.1 Punch-Through Breakdown 246

4.8.2 Avalanche Breakdown 246


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4.9 IMPORTANT PROPERTIES AND STANDARD CONSTANTS 247

EXERCISE 4.1 249

EXERCISE 4.2 261

EXERCISE 4.3 266

SOLUTIONS 4.1 274

SOLUTIONS 4.2 294

SOLUTIONS 4.3 308

CHAPTER 5 MOSFET


5.2 TWO TERMINAL MOS STRUCTURE 317

5.3 ENERGY BAND DIAGRAM FOR MOS CAPACITOR 318

5.3.1 Energy Band Diagram for MOS Capacitors with the p-type Substrate 318

5.3.2 Energy Band Diagram for MOS Capacitors with the n-type Substrate 319

5.4 DEPLETION LAYER THICKNESS 320

5.4.1 Space Charge Width for p-type MOSFET 320

5.4.2 Space Charge Width for n-type MOSFET 320

5.5 WORK FUNCTION DIFFERENCES 321

5.5.1 Work Function Difference for p-type MOS Capacitors 321

5.5.2 Work Function Difference for n-type MOS Capacitors 321

5.6 FLAT BAND VOLTAGE 322

5.7 THRESHOLD VOLTAGE 323

5.7.1 Threshold Voltage for MOS Structure with p-type Substrate 323

5.7.2 Threshold Voltage for MOS Structure with n-type Substrate 323

5.8 DIFFERENTIAL CHARGE DISTRIBUTION FOR MOS CAPACITOR 324

5.8.1 Differential Charge Distribution in Accumulation Region 324

5.8.2 Differential Charge Distribution in Depletion Region 324

5.8.3 Differential Charge Distribution in Inversion Region 325

5.9 CAPACITANCE-VOLTAGE CHARACTERISTICS OF MOS CAPACITOR 325

5.9.1 Frequency Effects onC-VCharacteristics 326

5.10 MOSFET STRUCTURES 327

5.11 CURRENT-VOLTAGE RELATIONSHIP FOR MOSFET 3295.11.1 n-channel Enhancement Mode MOSFET forV VGS T 329

5.11.3 Ideal Current-Voltage Relationship for MOSFET 330

5.11.4 Transconductance 332

5.12 IMPORTANT TERMS 332

5.13 IMPORTANT CONSTANTS AND STANDARD NOTATIONS 334


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EXERCISE 5.1 335

EXERCISE 5.2 345

EXERCISE 5.3 351

SOLUTIONS 5.1 355

SOLUTIONS 5.2 372

SOLUTIONS 5.3 389

CHAPTER 6 JFET


6.2 BASIC CONCEPT OF JFET 393

6.2.1 n-channel J FET 393

6.2.2 p-channel J FET 394

6.3 BASIC JFET OPERATION 394

6.3.1 J FET Operation for ConstantVDSand VaryingVGS 394

6.3.2 J FET Operation for V 0GS= and VaryingVDS 395

6.4 DEVICE CHARACTERISTIC 397

6.4.1 n-channel J FET Characteristic 397

6.4.2 p-channel J FET Characteristic 398

6.5 IDEAL DC CURRENT-VOLTAGE RELATIONSHIP FOR DEPLETION MODE JFET 398

6.6 TRANSCONDUCTANCE OF JFET 399

6.7 CHANNEL LENGTH MODULATION 399

6.7.1 Depletion Legnth 399

6.7.2 Small Signal Output Impedance 399

6.8 EQUIVALENT CIRCUIT AND FREQUENCY LIMITATIONS 399

6.8.1 Small-Signal Equivalent Circuit 399

6.8.2 Frequency Limitation Factors and Cutoff Frequency 400

EXERCISE 6.1 401

EXERCISE 6.2 405

EXERCISE 6.3 407

SOLUTIONS 6.1 412

SOLUTIONS 6.2 422

SOLUTIONS 6.3 428

CHAPTER 7 INTEGRATED CIRCUIT


7.2 BASIC MONOLITHIC INTEGRATED CIRCUIT 433

7.3 FABRICATION OF A MONOLITHIC CIRCUIT 434


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7.4 EPITAXIAL GROWTH 436

7.5 OXIDATION 436

7.5.1 Dry oxidation 436

7.5.2 Wet oxidation 436

7.6 MASKING AND ETCHING 436

7.7 DIFFUSION OF IMPURITIES 437

7.7.1 Diffusion Law 438

7.7.2 Complementary Error Function 438

7.7.3 The Gaussian Distribution 438

7.8 ION IMPLANTATION 439

7.9 THIN FILM DEPOSITION 440

7.9.1 Evaporation 440

7.9.2 Sputtering 440

7.9.3 Chemical Vapour Deposition (CVD) 441

7.10 PN JUNCTION DIODE FABRICATION 441

7.11 TRANSISTOR CIRCUIT 442

7.11.1 Monolithic Integrated Circuit Transistor 442

7.11.2 Discrete Planar Epitaxial Transistor 442

EXERCISE 7.1 444

EXERCISE 7.2 452

EXERCISE 7.3 453

SOLUTIONS 7.1 458

SOLUTIONS 7.2 464

SOLUTIONS 7.3 465

***********


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Chap

Semiconductors in

Equilibrium

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GATE STUDY PACKAGE Electronics & Communication

Sample Chapter of Electronic Devices(Vol-4, GATE Study Package)

1.1 INTRODUCTION

Equilibrium or thermal equilibrium, implies that no external forces such

as voltages, electric fields, magnetic fields, or temperature gradients are

acting on the semiconductor. This chapter deals with the semiconductor in

equilibrium. Following topics are included in this chapter:

Semiconductor Models: bonding model, energy band model.

Electron and hole carriers; its properties: charge, effective mass. Intrinsic semiconductor

Extrinsic semiconductor: n-type semiconductor, p-type semiconductor,

compensated semiconductor

Effect of donor and acceptor impurities

Fermi function: energy dependence

Equilibrium carrier concentration: electron concentration in conduction

band, hole concentration in valence band.

Energy band diagram for insulator, semiconductor, and conductor

Fermi energy level: position of Fermi energy level, variation of Fermi

energy level with temperature

Charge neutrality

Degenerate and non-degenerate semiconductors

1.2 SEMICONDUCTOR MODELS

In this section, we introduce and describe two very important models that

are used extensively in the analysis of semiconductor devices.

1.2.1 Bonding Model

The isolated Si atom, or a Si atom not interacting with other atoms, was

found to contain four valence electrons. The implication here is that, ingoing from isolated atoms to the collective crystalline state, Si atoms come

to share one of their valence electrons with each of the nearest neighbours.

This results in covalent bonding, or equal sharing of valence electrons with

nearest neighbors. T he bonding model is shown in Figure 1.1.

CHAPTER 1SEMICONDUCTORS IN EQULIBRIUM


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Semiconductors in

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Figure 1.1: Bonding Model of Silicon

1.2.2 Energy Band Model

Starting with N-isolated Si atoms and conceptually bringing the atoms closer

and closer together, one finds the inter atomic forces lead to a progressively

spread in the allowed energies.

1. The spread in energies give rise to closely spaced sets of allowed states

known as energy bands.

2. The distribution of allowed states consists of two bands separated by an

intervening energy gap.

3. The upper band of allowed states is called the conduction band; the lower

band of allowed state is called the valence band; and the interveningenergy gap is called the forbidden gap or band gap.

4. The valence band is completely filled and the conduction band is

completely empty at temperature approaching KT 0= .

Figure 1.2: Simplified Version of the Energy Band Model

1.3 CARRIERS

We are now in a position to introduce and to visualize the current carrying

entities within semiconductor. In a semiconductor, the two carriers are:

1. Electrons: In bonding model of semiconductor when Si-Si band is brokenand the associated electron is free to wander about the lattice, the

released electron is a carrier. Equivalently in term of the energy band


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model, excitation of valence band electrons into the conduction band

creates carriers; that is electron in the conduction band are carrier.

2. Holes:The breaking of a Si-Si bonds also creates a missing bond or void

in the bonding structure and in term of the energy band model where

the removal of an electron from the valence band create an empty state.

The missing band in the bonding scheme, the empty state in the valence

band, is the second type of carrier found in semiconductors called the

hole.

1.3.1 Carrier Properties

Having formally introduced the electron and hole in this section, we study

about the nature of these carriers.

Charge

Both electrons and holes are charge entities. Electrons are negatively charged,

holes are positively charged, and the magnitude of the carrier charge is

q . C16 10 19#= -

Effective Mass

When an external field is applied to a crystal, the free electron or hole inthe crystal responds, as if its mass is different from the true mass. T his mass

is called the effecti ve massof the electron or the hole. Following are some

important points about effective mass:

POINTS TO REMEMBER

1. By considering this effective mass, it will be possible to remove the

quantum features of the problem.

2. The effective mass allows us to use Newtons law of motion to

determine the effect of external forces on the electrons and holes

within the crystal.

1.4 INTRINSIC SEMICONDUCTOR

A semiconductor is said to be intrinsic if it contains no impurities and

no crystalline defects. In an intrinsic semiconductor, the equilibrium

concentration n0 of electrons in the conduction band is the same as the

equilibrium concentration of holes p0in the valence band. i.e.

n0 p ni0= =

where n0=number of electron/ cm3

p0=number of holes/ cm3

1.5 DOPING

Doping, in semiconductor terminology is the addition of controlled amounts

of specific impurity atoms with the express purpose of increasing either

the electron or the hole concentration. Depending on the characteristic of

dopants, semiconductors are classified as n-type and p-type semiconductors.


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1.5.1 n-type Semiconductor

To increase the electron concentration, one can add either phosphorus,

arsenic or antimony atoms to the Si crystal, these are the donor (electron-

increasing) dopants or n-type impurit y. Then-type semiconductor is created

by introducing the donor impurities in an intrinsic semiconductor.

EFFECT OF DONOR IMPURITIES

1. If a pentavalent impurity (phosphorus, arsenic or antimony) is added

to an intrinsic semiconductor, the four covalent bonds are still present.

However, it creates an additional fifth electron due to the impurity

atom. This remaining electron is relatively free to move within thematerial.

2. When donor impurities are added to a semiconductor, allowable energy

levels are introduced a very small distance below the conduction band

. T he energy band diagram of n-type semiconductor is shown in figure

below.

3. If intrinsic semiconductor material is doped with n-type impurities,

not only does the number of electron increase, but the number of

hole decreases below that which would be available in the intrinsic

semiconductor. The reason for the decrease in the number of holes

is that the larger number of electron present increases the rate of

recombination of electron with holes.

1.5.2 p-type Semiconductor

To increase the hole concentration, one can add either boron, gallium or

indium atoms to the Si crystal, these are the acceptor (hole-increasing)

dopants or p-type impurity. The p-type semiconductor is created by

introducing the acceptor impurities in an intrinsic semiconductor.

EFFECT OF ACCEPTOR IMPURITIES

1. If a trivalent impurity (boron, gallium or indium) is added to anintrinsic semiconductor, only three of the covalent bonds can be filled;

and the vacancy that exists in the fourth bond constitutes a hole.

2. When acceptor or p-type, impurities are added to the intrinsic

semiconductor, they produce an allowable discrete energy level which

is just above the valence band. T he energy band diagram of p-type

semiconductor is shown in figure below.


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MAJORITY AND MINORITY CARRIERS

1. Inn-type semiconductor, the electrons are called the majority carriers,

and the holes are called the minority carriers.

2. In p-type material, the holes are the majority carriers and the electrons

are the minority carriers.

1.6 COMPENSATED SEMICONDUCTOR

A compensated semiconductor is one that contains both donor and

acceptor impurity atoms in the same region. We classify the compensated

semiconductors as

1. An n-type compensated semiconductor occurs when N N>d a.

2. A p-type compensated semiconductor occurs when N N>a d.

3. If N Na d= , we have a completely compensated semiconductor that has

the characteristics of an intrinsic material.

1.7 FERMI FUNCTION

T he Fermi function f E^ hspeci fi es how many of the existi ng stat es at t heenergy Ewil l be fi l led wit h an electron .

or

f E^ hspecif ies, under equi li br ium condit ions, the probabi li ty t hat an avail ablestate at an energy Ewi l l be occupied by an electron.

Mathematically, the Fermi function is simply a probability distribution

function, defined as

f E^ he1

1/E E kT F

=+

-^ hwhere E=Any energy level

EF=Fermi energy or Fermi level

k=Boltzmann constant ( . /eV kk 8 617 10 5#= - )

T=Temperature in kelvin (K)

1.7.1 Energy Dependence of Fermi Function

We analyse the energy dependence of Fermi function for the following two

cases:


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CASE I: KT 0"Let us begin by investigating the Fermi functions energy dependence for

T 0" .

For all energies E E< F,

kT

E EF- " 3-

and f E E< F

^ h

e1

1 1"+

=3-

For all energies E E> F,

kT

E EF- " 3+

and f E E> F^ h / e1 1 0" + =3This result is plotted in figure below.

Figure 1.3: Energy Dependence of Fermi Function for KT 0"

CASE II: KT 0>Examining the Fermi function, we make the following observations.

1. If E EF= , then

f EF^ h /1 2=2. If E kTE 3F$ + , then

/e kTE EF-^ h 1>> ; f E^ h e

/E E kT F

, - -

^ h Consequently, above kTE 3F + , the Fermi function or filled-state

probability decays exponentially to zero with increasing energy moreover,

most-states at energies kT3 or more above EFwill be empty.

3. If E kTE 3F# - , then

e /E E kT F-^ h 1


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Equilibrium Distribution of Carrier

In the following diagram, the equilibrium distribution of carrier is illustrated.

Figure 1.5: Equilibrium Distribution of Carrier

1.8 EQUILIBRIUM CARRIER CONCENTRATIONS

The distribution (with respect to energy) of electrons in the conduction

band is given by the density of allowed quantum states times the probability

that a state is occupied by an electron, i.e.

n E^ h g E f Ec= ^ ^h h ...(1.1)where f E^ h=Fermi-Dirac probability functionand g Ec h=Density of quantum states in the conduction band

Similarly, the distribution (with respect to energy) of holes in the valence


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band is the density of allowed quantum states in the valence band multiplied

by the probability that a state is not occupied by an electron, i.e.

p E^ h g E f E 1v= -^ ^h h6 @ ...(1.2)Integrating the above two equations [(1.1) and (1.2)], we obtain the

electron and hole concentrations respectively.

Electron Concentration in Conduction Band

The total electron concentration per unit volume in the conduction band is

found by integrating the distribution function given in equation (1.1), i.e.

n0 g E f E dEc= ^ ^h h#where f E^ h

e11

/E E kT F=

+ -^ h

e /E E kT F- - -^ hSo, the thermal-equilibrium density of electron in the conduction band

is obtained as

n0h

mE E e dE

4 2 */

/nc

E E kT

E3

3 2

F

c

p= -

3- -^ ^h h#

hm kT

e22 *

//n E E kT

2

3 2Fcp

= - -

^^ h h; Eor n0 N e

/ /c

E E kT Fc=

-^ h

where the parameter Ncis called the effective density of states function inthe conduction band, given as

Nch

m kT2 2* /n2

3 2p

= ; ENOTE :

Since g E dEc_ i represents the number of conduction band states/ cm3lying in the EtoE dE+ energy range, and f E_ ispecifies the probability that an available state as an energyEwill be occupied by an electron. So, g E f E dEc_ _i i gives the number of conduction bandelectrons/ cm3lying in the EtoE dE+ energy range. Thus, g E f E dEc_ _i i integrated over

all conduction band energies must yield the total number of electrons in the conductionband. A similar statement can be made relative to the hole concentration.

Hole Concentration in Valence Band

The thermal equilibrium concentration of holes in the valence band is foundby

p0 g E f E dE 1v F= -^ ^h h6 @#where f E1 E- ^ h

e11

/E E kT F=

+ -^ h

e kTE EF

- -

-b lSo, the thermal equilibrium concentration of holes in the valence band is

p0h

mE E e dE

4 2 * /pv

E

kT

E E

3

3 2Fv p

= -3-

- -^ bh l#

-

h

m kTe2

2 */

pkT

E E

2

3 2F vp

=

-e bo lor p0 N e

/v

E E kT F v=

- -^ hwhere Nv is called the effective density of states function in the valence

band, given as


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Nvh

m kT2

2 */

p

2

3 2p

= e o

1.8.1 Intrinsic Carrier Concentration

For an intrinsic semiconductor, the concentration of electrons in the

conduction band is equal to the concentration of holes in the valence

band. These parameters are usually referred to as the intrinsic electron

concentration and intrinsic hole concentration, i.e.

ni pi=Fermi energy level for the intrinsic semiconductor is called the intrinsic

Fermi energy, or E EF Fi= . So, we have

n0 n N e/

i cE E kT Fic

= = - -^ h ...(1.3)

and p0 p n N e /

i i vE E kT Fi v

= = = - -^ h ...(1.4)

Hence, multiplying equations (1.3) and (1.4), we get

ni2 N N e e / /c v

E E kT E E kT Fi Fi v c=

- - - -^ ^h hor ni

2 N N e N N e / /c vE E kT

c vE kTv gc

= =- - -^ h ...(1.5)

or ni N N e/ kT

c vE 2G

= -

Temperature Dependence of Intrinsic ConcentrationSubstituting the values of Ncand Nvin equation (1.5), we have

ni2 .

m

m mT e233 10

* * //n p E kT43

02

3 23 G

#= -e o ...(1.6)

Since, EG E TG0 b= -where EG0is the magnitude of energy gap at 0 K. So, by substituting this

relation into equation (1.6), we get

ni2 A T e /E kT0

3 G0=

-

where A 0 .m

m me233 10

* * //n p k43

2

3 2

#=b-^ eh o

and b has the dimension of electron volt per degree kelvin.

1.8.2 Extrinsic Carrier Concentration

Let us derive a general form of equations for the thermal-equilibrium

concentration of electrons and holes in extrinsic semiconductors. For an

extrinsic semiconductor, we have

n0 N e/

cE E kT Fc

= - -^ h

If we add and subtract an intrinsic Fermi energy ( )EFi in the exponent

of above equation, we can write

n0 N ec kTE E E E Fi F F i c

=

- - + -^ ^c h h mor n0 N e e

/ /c

E E kT E E kT Fi F Fi c=

- - -^ ^h h ...(1.7)Since, the intrinsic carrier concentration is given by [from equation (1.3)] ni N e

/c

E E kT Fic=

- -^ hHence, by substituting it in equation (1.7), we get the thermal equilibrium

electron concentration as

n0 n e/

iE E kT F Fi

= -^ h

Similarly, we obtain the thermal equilibrium hole concentration as

p0 n e/

iE E kT F Fi

= - -^ h


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1.9 ENERGY BAND DIAGRAM FOR INSULATOR, SEMICONDUCTOR, AND

METAL

Figure 1.6 shows the energy band diagram of an insulator, a semiconductor,

and a metal. T he characteristic of these materials are described in the

following texts.

Figure 1.6: Energy Band Diagram of (a) an Insulator (b) a Semiconductor (c) a Metal

1.9.1 Insulator

A very poor conductor of electricity is called an insulator. For insulator, the

width of the forbidden energy region is high (6 eV).

1.9.2 Semiconductor

A substance whose conductivity lies between the metal and insulator is called

semiconductor. The width of the forbidden energy region for semiconductor

is relatively small eV1.

^ h. The energy bandgap for silicon and germanium

semiconductor are tabulated below.Table 1.1: Energy Bandgap of Semiconductors

Semiconductors Energy bandgap at temperature (T) Energy bandgap at room

temperature ( )KT 300=

Silicon . . TE T 121 360 10G4

#= - -^ h EG . eV11=

Germanium . . TE T 0785 223 10G4

#= - -^ h Eg . eV072=

1.9.3 Metal

An excellent conductor is a metal. The band structure of a metal contains

overlapping valence and conduction bands, as shown in Figure 1.6 (c).

1.10 POSITION OF FERMI ENERGY LEVEL

We can now determine the position of the Fermi energy level as a function

of the doping concentration. The position of the Fermi energy level within

the bandgap can be determined by using the equations already developed

for the thermal equilibrium electron and hole concentration.


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1.10.1 Fermi Energy Level for n-type Semiconductor

For an n-type semiconductor, we define the carrier concentration as

n0 N e/

cE E kT Fc

= - -^ h

or E Ec F- lnkT nNc

0= b l ...(1.8)

If the donor concentration, Nd n>> i

then, n0 Nd-

Therefore, equation (1.8) becomes

E Ec F- lnkTNN

d

c= b l

Figure 1.7 (a) shows the position of Fermi energy level for n-type

semiconductor. Now, we may develop a slightly different expression for the

position of the Fermi level. Since,

n0 n e/

iE E kT F Fi

= -^ h

So, E EF Fi- lnkT nn

i

0= a k

Figure 1.7: Illustration of Fermi Energy Level for (a) n-type Semiconductor, (b) p-typeSemiconductor

Following are some important points about position of Fermi level in an

n-type semiconductor.

CHARACTERISTICS OF FERMI LEVEL IN n-TYPE SEMICONDUCTOR

1. The distance between the bottom of the conduction band and the

Fermi energy is a logarithmic function of the donor concentration.

2. As the donor concentration increases, the Fermi level moves closer to

the conduction band.

3. If the Fermi level moves closer to the conduction band, then the

electron concentration in the conduction band is increasing.

4. If we have a compensated semiconductor, then the Nd term in

equation is simply replaced by N Nd a- , or the net effective donor

concentration.

5. The difference between the Fermi level and the intrinsic Fermi level is

a function of the donor concentration.

6. If the net effect donor concentration is zero, i.e. N N 0d a- = then

n ni0 = and E EF Fi= .

7. A completely compensated semiconductor has the characteristics of

an intrinsic material in term of carrier concentration and Fermi level

position.

8. For an n-type semiconductor, n n> i0 , E F>F Fi, and therefore theFermi level is above EFi.


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1.10.2 Fermi Energy Level for p-type Semiconductor

For a p-type semiconductor, the carrier concentration is

p0 N e/

vE E kT F v

= - -^ h

or E EF v- lnkT pNv

0= b l ...(1.9)

If the acceptor concentration, Na n>> i

then, p0 Na,

Therefore, equation (1.9) becomes

E EF v- lnkTNN

a

v= b l

Figure 1.7 (b) shows the position of Fermi energy level for p-type

semiconductor. Now, we may develop a slightly different expression for the

position of the Fermi level. Since,

p0 n e/

iE E kT F Fi

= - -^ h

So, E EFi F- lnkT np

i

0= b l

Following are some important points about position of Fermi level in a

p-type semiconductor.

CHARACTERISTICS OF FERMI LEVEL IN p-TYPE SEMICONDUCTOR

1. The distance between the Fermi level and the top of the valence band

energy for a p-type semiconductor is a logarithmic function of the

acceptor concentration.

2. As acceptor concentration increases, the Fermi level moves closer to

the valence band.

3. If we have a compensated p-type semiconductor, then the Naterm

in equation is replaced by N Na d- or the net effective acceptor

concentration.

4. We can also derive an expression for the relationship between the Fermi

level and the intrinsic Fermi level in term of the hole concentration.

p0 n e/

iE E kT F Fi

= - -^ h

E EFi F- lnkT np

i

0= b l

5. For a p-type semiconductor, p n> i0 , E E>Fi F, and therefore the

Fermi level is below EFi.

1.10.3 Variation of Fermi Level with Temperature

The Fermi energy level EFfor a semiconductor varies with the temperature

in following manner:1. The intrinsic concentration ni is a strong function of temperature, so

that EFis a function of temperature also.

2. As the temperature increases, niincreases, and EFmoves closer to the

intrinsic Fermi level.

3. At high temperature, the semiconductor material begins to lose its

extrinsic characteristics and begins to behave more like an intrinsic

semiconductor.


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4. At the very low temperature, freeze-out occurs; the fermi level goes

above Edfor the n-type material and below Eafor the p-type material.

5. At absolute zero degrees, all energy states below EFare full and all the

energy states above EFare empty.

1.11 CHARGE NEUTRALITY

In thermal equilibrium, the semiconductor crystal is electrically neutral.

The electrons are distributed among the various energy states, creating

negative and positive charges but the net charge density is zero. This charge

neutrality condition is used to determine the thermal equilibrium electron

and hole concentration as a function of the impurity doping concentration.

1.11.1 Determination of Thermal Equilibrium Electron Concentration as a

Function of Impurity Doping Concentration

We assume complete ionization. T he charge neutrality condition is expressed

by equating the density of negative charges to the density of positive charges,

i.e.

n Na0 + p Nd0= +

or n N p N a d0 0+ - - 0= ...(1.10)

From mass action law, we have

p0 nni

0

2

=

Substituting it in equation (1.10), we get

n N Nnn

a di

00

2

+ - - 0=

or n N N n n d a i02

02

- - -^ h 0=Thus, by solving the above quadratic equation, we obtain the electron

concentration as

n0N N N N n

24d a d a i

2 2

=- + - +^ ^h h

CASE I: N N 0a d= =Substituting N N 0a d= = in above expression, we get

n0 ni!=Since, the electron concentration must be a positive quantity, so

n0 ni=

CASE II: N 0a=For N 0a= , the electron concentration becomes

n0 N N n2 4d d i

2 2

= + +

1.11.2 Determination of Thermal Equilibrium Hole Concentration as a Function

of Impurity Doping Concentration

Again, from mass action law, we have

n0 pni

0

2

=

Substituting it in the equation (1.10), we obtain the hole concentration as


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p N N p n a d i02

02

- - -^ h 0=or p0

N N N N n

24a d a d i

2 2

=- + - +^ ^h h

CASE I: N N 0a d= =Substituting N N 0a d= = in above expression, we get

p0 ni!=Since, the hole concentration must be a positive quantity, so

p0 ni=

CASE II: N 0d =For N 0d = , the hole concentration becomes

p0N N n

24a a i

2 2

= + +

1.12 DEGENERATE AND NON DEGENERATE SEMICONDUCTORS

We may define the degenerate and non-degenerate semiconductors in

following ways:

1.12.1 Non-degenerate Semiconductor

When the concentration of dopant atoms added is small compared to the

density of host or semiconductor atoms, the impurities introduce discrete,

non-interacting donor energy states in then-type semiconductor and discrete

non-interacting acceptor states in the p-type semiconductor. These types of

the semiconductors are referred to as non-degenerate semiconductor.

1.12.2 Degenerate Semiconductor

When the donor concentration is increased, the band of donor states widens

and may overlap the bottom of the conduction band. This overlap occurs

when the donor concentration becomes comparable with the effective densityof states. The two types of degenerate semiconductors are defined as

Degeneraten-Type Semiconductor

When the concentration of electrons in the conduction band exceeds the

density of states Nc, the Fermi energy lies within the conduction band.

This type of semiconductor is called a degenerate n-type semiconductor.

Figure 1.8 (a) shows the energy band diagram for a degenerate n-type

semiconductor.

Figure 1.8:Energy Band Diagram for (a) Degeneraten-type Semiconductor, (b) Degenerate

p-type Semiconductor


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Degenerate p-Type Semiconductor

In a similar way, as the acceptor doping concentration increases in a p-type

semiconductor, the discrete acceptor energy states will split into a band of

energies and may overlap the top of the valence band. T he Fermi energy

will lie in the valence band when the concentration of holes exceeds the

density of states Nv. T his type of semiconductor is called a degenerate p-type semiconductor. Figure 1.8 (b) shows the energy band diagram for a

degenerate p-type semiconductor.

1.13 IMPORTANT PROPERTIES AND STANDARD CONSTANTS

Following are some important properties and standard values used in

determination of semiconductor parameters in the exercises of the chapter.

Table 1.2: Some Standard Constants

Avogadros number .N 602 10A23

#= + atoms per gram molecular

weight

Boltzmanns Constant . / . /J K eV Kk 138 10 862 1023 5# #= =- -

Electronic Charge (Magnitude) . Ce 160 10 19#= -

Free Electron Rest Mass . kgm 911 10031

#= -

Permeability of Free Space /H m4 1007

#m p= -

Permittivity of Free Space . / . /F cm F me 885 10 885 10014 12

# #= =- -

Plancks Constant . .J s eV sh 6625 10 4135 1034 15# #- -= =- -

Thermal Voltage KT 300=^ h . voltVekT 00259t = = , . eVkT 00259=

Table 1.3 : P roperties of Silicon, Gallium Arsenide, and Germanium KT 300=_ i

Property Si GaAs Ge

Atoms cm 3-^ h .50 1022# .442 1022# .442 1022#Dielectric Constant 11.7 13.1 16.0

Bandgap Energy (eV) 1.12 1.42 0.66

Effective Density of States in

Conduction Band, cmNc3-^ h .28 10

19# .47 10

17# .104 10

19#

Effective Density of States in

Valence Band, cmNv3-^ h .104 10

19# .70 10

18# .60 10

18#

Intrinsic Carrier Concentrationcm 3-^ h .15 10

10# .18 106# .24 1013#

Electron Mobility, nm /cm V s2

-^ h 1350 8500 3900Hole Mobility, pm /cm V s

2-^ h 480 400 1900

Effective Mass (Density of States) 1.08 0.067 0.55


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Table 1.4: Carrier Modeling Equation Summary

Density of States and Fermi Function

g Eh

m m E E 2* *c

n n c

2 3p=

-^ ^h h , E Ec$g E

h

m m E E 2* *v

p p v

2 3p=

-

^ ^

h h

, E Ev#

f Ee1

1/E E kT F

=+

-^ ^h h

Carrier Concentration Relationships

Nh

m kT22

* /

cn

2

3 2

p= ; E

Nh

m kT2

2

* /

vp

2

3 2

p= = G

n N e/

cE E kT F c

= -^ h

p N e/

vE E kT v F

= -^ h

n n e/

iE E kT F i

= -^ h

p n e/

iE E kT i F

= -^ h

ni, np-Product, and Charge Neutrality

n N N e /

i c vE kT2G

= - np ni

2= p n N N 0d a- + - =

n, p, and Fermi Level Computational Relationships

n N N N N

n2 2

/d a d a

i

22

1 2

= -

+ -

+b l; E lnE E E kTm

m

2 43

*

*

ic v

n

p=

++ e o

For N N>>d a, N n>>d i;

n Nd-

p n Ni d2

-

/ /ln lnE E kT n n kT p n F i i i- = = -^ ^h h

For N N>>a d, N n>>a i;

p Na-

/n n Ni a2

-

/lnE E kT N n F i d i - = ^ h,N N N n >> >>d a d i

/lnE E kT N n i F a i - = ^ h,N N N n >> >>a d a i

***********


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EXERCISE 1.1

MCQ 1.1.1 In intrinsic semiconductor at 300 K, the magnitude of free electronconcentration in silicon is about

(A) per cm15 104 3# (B) per cm5 1012 3

#

(C) . per cm145 1010 3# (D) . per cm145 106 3

#

MCQ 1.1.2 Two initially identical samples A and B of pure germanium are doped with

donors to concentrations of 1 1020# and 3 1020

# respectively. I f the hole

concentration in A is 9 1012# , then the hole concentration in B at the same

temperature will be

(A) m3 1012 3# - (B) m7 1012 3#

-

(C) m11 1012 3# - (D) m27 1012 3# -

MCQ 1.1.3 Given the effective masses of holes and electrons in silicon respectively as

m*p . m056 o= , .m m108*n o=

What will be the position of the intrinsic Fermi energy level with respect to

the center of the bandgap for the semiconductor at 300KT = ?

(A) 0.029 eV0 below the centre

(B) 0.0128eVabove the centre

(C) 0.0128eVbelow the centre

(D) 0.029 eV0 above the centre

MCQ 1.1.4 What will be the position of Fermi energy level, EFiwith respect to the

center of the bandgap in silicon for 200KT = ?

(A) 0.0 eV085 below the centre

(B) 0.0128eVabove the centre

(C) 0.0128eVbelow the centre

(D) 0.0 eV085 above the centre

Common Data For Q. 5 and 6The electron concentration in silicon at 300KT = is 5 10 cmn0

4 3#=

- .

MCQ 1.1.5 What will be the hole concentration (in cm 3- ) in silicon ?

(A) 109 15# (B) 3 10 9#

(C) .4 5 10 15# (D) 3 10 5#

SAN. SHARMA428/21

B.L.THERAJA37/23


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MCQ 1.1.6 The material is

(A) p-type (B) n-type

(C) intrinsic (D) cant be determined

MCQ 1.1.7 If the Fermi energy in silicon is 0.22eVabove the valence band energy, what

will be the values of n0and p0for silicon at 300KT = ?

n0(in cm 3- ) p0(in cm 3- )

(A) 2.27 104# 2.13 1015

#

(B) 2.13 10 cm15 3# - 2.27 104#

(C) . 10 cm104 4 3# - 2.8 1015#

(D) 2.8 1015# 1.04 104

#

Common Data For Q. 8 and 9

Consider 0.25eVE Ec F- = in gallium arsenide (GaAs) at 400KT = .

MCQ 1.1.8 What will be the electron and hole concentrations in the material at

400KT = ?

n0(in cm 3- ) p0(in cm 3- )

(A) 2.27 104# 2.13 1015

#

(B) 2.13 10 cm15 3# - 2.27 104#

(C) . 10 cm519 14 3# - 2. 8 100 4#

(D) 2. 8 100 4# . 1051914

#

MCQ 1.1.9 If the value of n0, obtained in above question, remains constant then, what

will be the hole concentration at 300KT = ?(A) 10 cm7 3 3#

- -

(B) 9.67 10 cm3 3# - -

(C) 96.7 10 cm3 3# - -

(D) . 10 cm208 4 3# -

MCQ 1.1.10 If a germanium semiconductor is doped with the donor and acceptor

concentrations respectively as

Nd 5 10 cm 15 3#= - , N 0a= .

What will be the thermal equilibrium concentrations, n0

and p0at 300KT =

in the material ?

n0(in cm 3- ) p0(in cm 3- )

(A) 2. 8 100 4# 2.13 1015

#

(B) .1 10 cm1 5 11 3# - . 1050 15#

(C) 2.13 1015# 2. 8 1004

#

(D) . 1050 15# .1 10 cm1 511 3

# -


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MCQ 1.1.11 The doping concentrations in silicon semiconductor areNd 10 cm Na 15 3= = - .

What will be the concentrations of n0and p0in the material at 300KT = ?

n0(in cm 3- ) p0(in cm 3- )

(A) 1015 1015

(B) . 10 cm225 10 3# - 1015

(C) . 1015 10# . 101510

#

(D) 1015

. 10 cm22510 3

#

-

MCQ 1.1.12 Assume that gallium arsenide has dopant concentrations of 1 10 cmNd13 3

#= -

and 2.5 10 cmNa13 3

#= - at 300KT = . T he material is

(A) p-type with . , 0.216cm cmp n15 10013 3

03

#= =- -

(B) p-type with 0.216 , 1.5 10cm cmp n03

013 3

#= =- -

(C) n-type with 0.216 , 1.5 10cm cmp n03

013 3

#= =- -

(D) n-type with . , 0.216cm cmp n15 10013 3

03

#= =- -

MCQ 1.1.13 A sample of silicon at 450KT = is doped with boron at a concentration of

1.5 10 cm15 3# - and with arsenic at a concentration of 8 10 cm14 3#

- . The

material is

(A) p-type with 4.23 10 , 7 10cm cmp n011 3

014 3

# #= =- -

(B) p-type with 10 , 4.23 10cm cmp n7014 3

011 3

# #= =- -

(C) n-type with 4.23 10 , 7 10cm cmp n011 3

014 3

# #= =- -

(D) n-type with 10 , 4.23 10cm cmp n7014 3

011 3

# #= =- -

MCQ 1.1.14 A particular semiconductor material is doped at 2 10 cmNd13 3

#= - , N 0a=

, and the intrinsic carrier concentration is 2 10 cmni 13 3#= - . The thermal

equilibrium majority and minority carrier concentrations will be, respectively

(Assume complete ionization)

(A) 1. 10 , 0.216cm cmp n23013 3

03

#= =- -

(B) 0.216 , 3.24 10cm cmp n03

013 3

#= =- -

(C) . , 3.24 10cm cmp n123 10013 3

013 3

# #= =- -

(D) 3.24 10 , .2cm cmp n 1 3 10013 3

013 3

##= =- -

MCQ 1.1.15 Consider germanium with an acceptor concentration of 10 cmNa15 3

= - and

a donor concentration of N 0d = at 200KT = . T he Fermi energy of thematerial will be

(A) 0. 85 eV1 5 below the intrinsic Fermi level

(B) 0.0128eVabove the intrinsic Fermi level

(C) 0.0128eVbelow the intrinsic Fermi level

(D) 0. 85 eV1 5 above the intrinsic Fermi level


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MCQ 1.1.16 Consider germanium at 300KT = with doping concentrations of

10 cmNd14 3

= - and N 0a= . What will be the position of Fermi energy level

with respect to the intrinsic Fermi level for these doping concentrations ?


(B) 0.0 8 eV3 2 above the intrinsic Fermi level

(C) 0.0 8 eV3 2 below the intrinsic Fermi level


MCQ 1.1.17 If silicon is doped with phosphorus atoms at a concentration of 10 cm15 3-

then, what will be the position of the Fermi level with respect to the intrinsic

Fermi level in silicon at 300KT = ?


(B) 0.2877eVabove the intrinsic Fermi level

(C) 0.2877eVbelow the intrinsic Fermi level


MCQ 1.1.18 Gallium arsenide at 300KT = contains acceptor impurity atoms at a density

of 10 cm15 3- . Additional impurity atoms are to be added so that the Fermi

level is 0.45eV below the intrinsic level. The concentration and type of

additional impurity atoms will be respectively

(A) 9.368 10 cmNa14 3

#= - , acceptor

(B) 6.32 10 cmNa13 3

#= - , acceptor

(C) 9.368 10 cmNd14 3

#= - , donor

(D) 6.32 10 cmNd13 3

#= - , donor


For a particular semiconductor, the effective mass of electron is .m m14*n =(where mis electron mass at rest).

MCQ 1.1.19 What is the effective density of states in the conduction band at KT 300c= .

(A) . m415 1025 3# -

(B) . m208 1025 3# -

(C) . m427 1026 3# -

(D) . cm415 1020 3# -

MCQ 1.1.20 If . eVE E 025C F- = at KT 300c= , then what is the concentration of

electrons in the semiconductor ?

(A) . m133 1021 3# -

(B) . m267 1021 3# -

(C) . cm267 1015 3# -

(D) . m267 1022 3# -

MILMAN90/4.3


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MCQ 1.1.21 The effective masses of electron and hole in germanium are .m m055*n = and

.m m037*p= (where mis the electron rest mass) what will be the position

of the intrinsic Fermi energy lavel with respect to the centre of the bandgap

for the Germanium semiconductor at KT 300= ?

(A) 0.0154 eV above the centre

(B) 0.0154 eV below the centre

(C) 0.0077 eV above the centre

(D) 0.0077 eV below the centre

Common Data For Q. 22 to 24

For a particular material, . cmN 15 10C18 3

#= - , . cmN 13 10V

19 3#=

- and

bandgap . eVE 143G = at KT 300c= .

MCQ 1.1.22 What is the position of the Fermi level with respect to the top of the valence

band EV?

(A) 0.028 eV above the valence band edge EV

(B) 0.743 eV below the valence band edge EV

(C) 0.028 eV below the valence band edge EV

(D) 0.743 eV above the valence band edge EV

MCQ 1.1.23 What is the position of the Fermi level with respect to the conduction band

edge EC?

(A) 0.687 eV above EC

(B) 0.687 eV below EC

(C) 0.743 eV below EC

(D) 0.743 eV above EC

MCQ 1.1.24 What are the effective massesm*nand m*pof electron and hole respectively ?

m*n m*p

(A) . kg135 10 31# - . kg646 10 31#

-

(B) . kg123 10 21# - . kg646 10 31#

-

(C) . kg135 10 31# - . kg588 10 31#

-

(D) . kg123 10 31# - . kg588 10 31#

-

MCQ 1.1.25 The probability that an energy state is filled at E KTC + , is equal to the

probability that a state is empty at E KTC + . Where is the Fermi level EF^ hlocated ?

(A) E E KT 2F C= +

(B) E E KT 2F C= -

(C) E E KT F C= +

(D) E E KT 2F C= -

MILMAN93/4.4

MILMAN95/4.5

MILMAN95/4.5

MILMAN95/4.5C

PEIRRET71/2.6C


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MCQ 1.1.26 A silicon wafer is uniformly doped p-type with acceptor impurity

.per cmN 10A15 3

= At KT 0c, what is the equilibrium electron concentration ?

(A) / cm1015 3 (B) / cm105 3

(C) / cm1010 3 (D) 0,

MCQ 1.1.27 In a non-degenerate germanium sample maintained under equilibrium

conditions near room temperature, it is known that intrinsic concentration

/ ,cmn 10i13 3

= n p2= and N 0A = . What are the values of n (electron

concentration) and ND(Donor concentration) ?

n ND

(A) . per cm707 1012 3# .1 414 1013

# per cm3

(B) . per cm1 414 1013 3# . per cm0 707 1013 3

#

(C) . per cm2 828 1013 3# . per cm0 707 1013 3

#

(D) .1 414 1013# per cm3 . per cm1 414 1013 3#

MCQ 1.1.28 Which of the following sketches best describes the DNversusNDdependence

of electrons in silicon at room temperature ?


At room temperature KT 300=^ h, the probability that an energy state inthe conduction band edge EC^ hof silicon is 10 4- .

MCQ 1.1.29 The type of semiconductor is

(A) n-type (B) p-type

(C) intrinsic (D) cant be determine

MCQ 1.1.30 Assume the effective density of states function . cmN 286 10,C V19 3

#= - .

What is the value of doping concentration ?

(A) . cmN N 126 10D A10 3

#- = -

(B) . cmN N 126 10A D10 3

#- = -

(C) . cmN N 28 10D A15 3

#- = -

(D) . cmN N 28 10A D15 3

#- = -

PIERRET73/2.16A

PIERRET73/2.16C

PIERRET183/7

ANDERSON109/2.34


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An unknown semiconductor has bandgap . eVE 11g= and N Nc v= . It is

doped with cm1015 3- donors, where the donor level is 0.2 eV below EC.

Given that EFis 0.25 eV below ECat KT 300= .

MCQ 1.1.31 What is the concentration of electrons ?

(A) . cm8 733 1014 3# -

(B) . cm127 1014 3# -

(C) cm1015 3-

(D) cm105 3-

MCQ 1.1.32 What is the value of effective density of electron NC^ h?(A) . cm197 1018 3#

- (B) . cm1 359 1019 3# -

(C) . cm482 1018 3# - (D) . cm285 1019 3#

-

MCQ 1.1.33 What is the value of intrinsic carrier concentration ?

(A) . cm310 109 3# -

(B) . cm662 1019 3# -

(C) . cm759 104 3# -

(D) . cm8 142 109 3# -


A piece of intrinsic silicon at room temperature is kept at thermal equilibrium.

The position of energy level, Exis set exactly 0.6 eV above the intrinsic level

and band gap of intrinsic silicon . eVE 11g =^ h .

MCQ 1.1.34 What is the type of semiconductors if the probability of capture of an energy

state by an electron at Exis 50%.

(A) p-type, non degenerate

(B) n-type, non degenerate

(C) p-type, degenerate

(D) n-type, degenerate

MCQ 1.1.35 What is the doping concentration

(A) . cm1 725 1016 3# -

(B) . cm1 725 1020 3# -

(C) . cm2 879 1019 3# -

(D) . cm2 879 1016 3# -

***********

STREETMAN V106/3.20

BHATTACHA

RYA 119/3.14


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EXERCISE 1.2

QUES 1.2.1 The intrinsic carrier concentration, niat 200KT = for silicon is_ _ _ _ _cm104 3#

- .

QUES 1.2.2 The intrinsic carrier concentration, nifor germanium at 400KT = is _ _ _ _

cm104 3# - .

QUES 1.2.3 The intrinsic carrier concentration, ni at 600KT = for GaAs is_ _ _ _

cm1012 3# - .

QUES 1.2.4 The intrinsic carrier concentration in silicon is to be no greater than

1 10 cmni12 3

#= - . What will be the maximum temperature (in K) allowed

for the silicon ?

QUES 1.2.5 Two semiconductor materials have exactly the same properties except that

material A has a bandgap energy of 1.0eVand material Bhas a bandgap

energy of 1.2eV. The ratio of intrinsic concentration of material A to thatof material Bfor 300KT = will be _ _ _ _ _

QUES 1.2.6 The hole concentration in silicon at 300KT = is10 cm15 3- . T he concentration

of electrons in the material will be_ _ _ _ _ cm104 3# - .

QUES 1.2.7 Given the acceptor and donor concentrations in a germanium semiconductor

respectively as Na 10 cm 13 3= - , N 0d = . The thermal equilibrium hole

concentration in the material at 300KT = will be_ _ _ _ _ cm1013 3# - .

QUES 1.2.8 A silicon semiconductor is doped with the donor and acceptor concentrations

respectively as Nd 2 10 cm15 3#= - and N 0a= . The thermal equilibriumhole concentration in the material at 300KT = will be_ _ _ _ _ cm105 3#

- .

QUES 1.2.9 A silicon semiconductor has the dopant concentrationsNd 0= , 10 cmNa14 3

= - .

The thermal equilibrium electron concentration in the material at 00KT 4=will be_ _ _ _ _ cm1010 3#

- .


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QUES 1.2.10 In a sample of GaAs at 200KT = , we have experimentally determined that

n p50 0= and N 0a= . What will be the concentration of electrons (in cm3-

) in the material ?

QUES 1.2.11 Silicon at 300KT = is uniformly doped with phosphorus atoms at a

concentration 2 10 cm15 3# - and boron atoms at a concentration of

3 10 cm16 3

# -

. The thermal equilibrium concentration (in cm 3-

) of minoritycarriers will be_ _ _ _ _ cm103 3#

- .

QUES 1.2.12 In silicon at 300KT = , we have experimentally found that 4.5 10 cmn04 3

#= -

and 5 10 cmNd15 3

#= - . T he acceptor impurity concentration in the material

will be _ _ _ _ _ cm1016 3# - .

QUES 1.2.13 A GaAs device is doped with a donor concentration of 3 10 cm15 3# - . For the

device to operate properly, the intrinsic carrier concentration must remain

less than 5 percent of that electron concentration. What is the maximumtemperature (in K) that the device may operate ?

QUES 1.2.14 Silicon at 300KT = contains acceptor atoms at a concentration of

5 10 cmNa15 3

#= - . Donor atoms are added forming an n-type compensated

semiconductor such that the Fermi level is 0.215eVbelow the conduction

band edge. The concentration of donor atoms added is _ _ _ _ cm1016 3# - ?


Silicon at 300KT = is doped with acceptor atoms at a concentration of

7 10 cmNa 15 3#= - .

QUES 1.2.15 The difference between Fermi energy and valence band energy, E EF v-

equals to_ _ _ _ _ eV.

QUES 1.2.16 The concentration of additional acceptor atoms that must be added to

move the Fermi level a distance kTcloser to the valence-band edge will be

_ _ _ _ _ cm1016 3# - .

QUES 1.2.17 If silicon is doped with boron atoms at a concentration of 10 cm15 3- then, the

change in Fermi level, E EF Fi- will be_ _ _ _ _ eV.

QUES 1.2.18 Consider intrinsic germanium at room temperature (300 K). By what percent

does the conductivity increase per degree rise in temperature ?MILINANS87/4.1


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QUES 1.2.19 In an n-type silicon, the donor concentration is 1 atom per 2 108# silicon

atoms. Assume that the effective mass of the electron equals the true mass

and the density of atoms in the silicon is /atoms cm5 1022 3# . At what

temperature in Kc^ h will the Fermi level coincide with the edge of theconduction band ?

QUES 1.2.20 What is the ratio of the probability that a state is filled at the conductionband edge EC^ hto the probability that a state is empty at the valence bandedge EV^ hif the Fermi level is positioned at midgap ?

QUES 1.2.21 If Fermi energy level EFis positioned at EC(edge of the conduction band),

then the probability of finding electrons in states at the E KTC + will be_ _ _

QUES 1.2.22 For a non-degenerate semiconductor, the peak in the electron distribution

versus energy inside the conduction band occurs at /E KT2C + . What is

the ratio of the electron population in a non-degenerate semiconductor atE E KT 5C= + to the electron population at the peak energy ?

QUES 1.2.23 For a silicon sample maintained at KT 300= , the Fermi level is located 0.259eV above the intrinsic level and intrinsic concentration per cmn 10i

10 3= .

The hole concentration is_ _ _ _ _ per cm105 3# .

QUES 1.2.24 The probability of occupancy of a state at the bottom of the conduction

band in intrinsic silicon at room temperature is _ _ _ _ _ 10 10# - (assume at

room temperature . VKT

0026=

).

QUES 1.2.25 Two semiconductors A and B have the same density of states effective

masses. Semiconductor A has a bandgap energy of . eV10 and semiconductor

B has a bandgap energy of eV2 . The ratio of intrinsic concentration of

semiconductor A to semiconductor Bfor KT 300= will be_ _ _ _ _ 108# .

QUES 1.2.26 For a piece of GaAs (Gallium Arsenide) having a band gap . eVE 143g=. The minimum frequency of an incident photon that can interact with a

valence electron and elevate the electron to the conduction band is_ _ _ _ _

Hz1014# .

QUES 1.2.27 A piece of intrinsic silicon at room temperature is kept at thermal equilibrium.

The position of some random level Ex is to be fixed at . eV09 above the

valence band edge. T he doping concentration such that the probability of

capture of an energy state by an electron at Exis 50% is _ _ _ cm1016 3

# - ?

***********

MILMAN102/4.8

PEIRRET44/2.2

PEIRRET71/2.6

PIERRET71/2.8

PIERRET73/2.16D

ANDERSON76/2.2

ANDERSON109/2.36

BHATTACHARYA 115/3.11

BHATTACHARYA 117/3.13


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EXERCISE 1.3

MCQ 1.3.1 The mobility of charge-carriers has the unit(A) / secm volt 2- (B) secmvolt-

(C) / secm volt3 - (D) / secm volt2 -

MCQ 1.3.2 The total energy of a revolving electron in an atom can

(A) have any value above zero

(B) never be positive

(C) never be negative

(D) not be calculated

MCQ 1.3.3 Electronic distribution of an Si atom is

(A) 2, 10, 2

(B) 2, 8, 4

(C) 2, 7, 5

(D) 2, 4, 8

MCQ 1.3.4 Major part of the current in an intrinsic semi-conductor is due to

(A) conduction-band electrons

(B) valence-band electrons

(C) holes in the valence band

(D) thermally-generated electron

MCQ 1.3.5 Conduction electrons have more mobility than holes because they

(A) are lighter

(B) experience collisions less frequently

(C) have negative charge

(D) need less energy to move them

MCQ 1.3.6 Current flow in a semiconductor depends on the phenomenon of

(A) drift

(B) diffusion

(C) recombination

(D) all of the above

SAN. SHARMA26/4

B.L.THERAJA36/1

B.L.THERAJA36/4

B.L.THERAJA36/8

B.L.THERAJA36/9

B.L.THERAJA36/11


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MCQ 1.3.7 The process of adding impurities to a pure semiconductor is called

(A) mixing (B) doping

(C) diffusing (D) refining

MCQ 1.3.8 Hall effect is observed in a specimen when it (metal or a semiconductor) is

carrying current and is placed in a magnetic field. The resultant electric field

inside the specimen will be in

(A) a direction normal to both current and magnetic field

(B) the direction of current

(C) a direction antiparallel to the magnetic field

(D) an arbitary direction depending upon the conductivity of the specimen

MCQ 1.3.9 An n-type semiconductor as a whole is

(A) positively charged

(B) negatively charged

(C) positively or negatively charged depending upon doping(D) electrically neutral

MCQ 1.3.10 In p-type semiconductor, there are

(A) no majority carriers

(B) electrons as majority carriers

(C) immobile negative ions

(D) immobile positive ions

MCQ 1.3.11 Fermi level represents the energy level with probability of its occupation of

(A) 0 (B) 50%

(C) 75% (D) 100%

MCQ 1.3.12 The resistivity of a semiconductor depends on the

(A) shape of the semiconductor

(B) atomic nature of the semiconductor

(C) width of the semiconductor

(D) length of the semiconductor

MCQ 1.3.13 An electron in conduction band has

(A) no charge

(B) higher energy than electron in the valance band

(C) lower energy than the electron in the valance band

(D) All of these

B.L.THERAJA36/12

B.L.THERAJA

37/24

SANJEE GUPTA2.24/6

SANJEE GUPTA2.24/10

SANJEE GUPTA3.29/1

S SALIVAHANA2.76/16

B.P.SINGH49/4


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MCQ 1.3.14 In an intrinsic silicon the band gap is

(A) 1.12 eV (B) 0.7 eV

(C) 2 eV (D) 0.2 eV

MCQ 1.3.15 Mobility of holes in intrinsic Si is

(A) . /m Vs0048 2

(B) . /m Vs0135 2

(C) /m Vs1350 2

(D) /m Vs480 2

MCQ 1.3.16 Fermi level in the intrinsic Si or Ge is(A) in the middle of the band gap

(B) near the valance band

(C) near the conduction band

(D) none of these

MCQ 1.3.17 The diffusion constant and the mobility of electron are related as

(A) / /D KT q n nm =

(B) / /D q KT n nm =

(C) /D KTq n nm =

(D) /D qKT n nm =

MCQ 1.3.18 Electron population in silicon is not

(A) zero in the forbidden band(B) zero in the conduction band at 0 K

(C) zero at the conduction band edge EC

(D) zero in the conduction band at room temperature

MCQ 1.3.19 Acceptor impurity atom in germanium results in

(A) increased for bidden energy gap

(B) reduced for bidden energy gap

(C) new narrow energy band slightly above the valence level

(D) new discrete energy level slightly above the valence level

MCQ 1.3.20 If NDand NA are the donor and acceptor concentrations respectively and

N N>D A , then net impurity concentration is

(A) N ND A- (B) N ND A+

(C) N NA D- (D) .N ND A

B.P.SINGH51/38

B.P.SINGH

52/45

B.P.SINGH52/57

B.P.SINGH55/100

B.P.SINGH55/105

G.K. MITHAL84/2.10

G.K. MITHAL85/2.24


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MCQ 1.3.21 If donor concentration NDequals acceptor concentration NA , the resulting

semiconductor is

(A) n-type (B) p-type

(C) both pand ntype (D) intrinsic

MCQ 1.3.22 In an intrinsic semiconductor, the concentration of free electrons in the

conduction band is given by

(A) N e /cE E kT c F-^ h (B) N e /c E E kT F c-^ h

(C) N e /cE E kT F v-^ h (D) N e /c E E kT v F-^ h

MCQ 1.3.23 In an intrinsic semiconductor, the concentration of holes in the valence band

is given by

(A) N e /vE E kT v F-^ h (B) N e /v E E kT F v-^ h

(C) N e /vE E kT V F-^ h (D) N e /v E E kT F v-^ h

MCQ 1.3.24 In an intrinsic semiconductor, the concentration of charge carriers equals

(A) A T e /E kT02 G0 (B) A T e /E kT0

3 G0

(C) A T e /E kT0 G0 (D) A T e/ /E kT

03 2 G0

MCQ 1.3.25 In an intrinsic semiconductor, forbidden energy gap EGequals

(A) E T/G01 2b- (B) E TG0 b+

(C) E TG0 b- (D) E T/

G01 2b+

where b is a positive number

***********

B.P.SINGH86/2.27

G.K. MITHAL131/3.5

G.K. MITHAL131/3.6

G.K. MITHAL131/3.7

G.K. MITHAL131/3.8


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SOLUTIONS 1.1

SOL 1.1.1 Correct option is (C).At 300 K, intrinsic carrier concentration is

ni . / cm15 1010 3

#=

Hence, free electron concentration is

ne . / cm15 1010 3

#=

SOL 1.1.2 Correct option is (A).

Since, sample A and Bare identical. So, by using mass action law, we have

n pA A n pB B=

The donor impurity in sample A is

NA m1 1020 3

#= -

Donor impurity in sample Bis

NB m3 1020 3

#= -

The hole concentration in sample A is

pA 9 1012

#=

So, the hole concentration in sample Bis

pB3 10

1 10 9 1020

20 12

#

# # #=^ ^h h

m3 1012 3#= -

SOL 1.1.3 Correct option is (C).

The concentration of electrons and holes are defined as

n0 expNkT

E Ec

c F= -

-^ h; E p0 expN

kT

E Ev

F v= -

-^ h; EAt Fermi level position, the electron and hole concentrations are equal. So,

we have

expNkT

E Ec

c F-

-^ h; E expNkT

E Ev

F v= -

-^ h; EIf we take natural log of both sides, then

EF lnE E kT NN

21

21

c vc

v= + +^ bh l ...(1)

Also, we know that

E E21

c v+^ h Emidgap=Substituting it in equation (1), we get

E EmidgapF - lnkTNN

21

c

v= b l ...(2)


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Again, we have

Nch

m kT2 2* /n2

3 2p

= c mand Nv

h

m kT2

2 */

p

2

3 2p

= e oTherefore, we obtain

NN

c

v

m

m

*

* /

n

p

3 2

= e oThus, by substituting the above expression in equation (2), we get E EmidgapF - lnkT

m

m

43

*

*

n

p= e o

..lnkT

mm

43

108056

o

o= c m

. .e43 00259 0656# #= -^ h

. eV00128=-

The negative sign indicate the Fermi Energy level is . eV00128 below the

centre of the band gap.


The concentration of electrons and holes are defined as

n0 expNkT

E Ec

c F= -

-^ h; E p0 expN

kT

E Ev

F v= -

-^ h; EAt Fermi level position, the electron and hole concentration are equal, i.e.

expNkT

E Ec

c F-

-^ h; E expNkT

E Ev

F v= -

-^ h; EIf we take natural log of both sides, then

EF lnE E kT NN

21

21

midgap

c vc

v= + +^ bh l

1 2 344 44

or E EmidgapF - lnkTNN

21

c

v= b l ...(1)

At T 300c= for silicon, we have

Nc .28 1019

#=

and Nv .104 1019

#=

Therefore, we obtain

NN

c

v..28104

=

Since the ratio does not depend on temperature, so at T 200=

, we get

NN

c

v..28104

=

Substituting it in equation (1), we have

E EmidgapF - . ..lne2

1 00259300200

28104

#= b bl l . eV00085=-

Thus, the intrinsic Fermi level is . eV00085 below the centre of the bandgap.


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SOL 1.1.5 Correct option is (C).

At KT 300= the electron concentration is

n0 cm5 104 3

#= -

At KT 300= intrinsic carrier concentration for silicon is

ni .15 1010

#=

By using mass action law, we obtain the hole concentration as

n p0 0 ni2

=

or p0 nni

0

2

=

.

5 10

15 104

10 2

#

#=^ h

. cm45 1015 3#= -


We have the electron and hole concentrations for the silicon as

n0 cm5 104 3

#= -

and p0 . cm45 10

15 3#=

-

So, we conclude that

p0 n> 0i.e. the concentration of hole is greater the concentration of electron. It

means hole are in majority in this material, hence it is p-type material.


Given that Fermi energy in silicon is 0.22 eV above the valence band energy,

i.e.

E EF v- . eV022=

So, we obtain the hole concentration as

p0 expNkT

E Ev

F v

= --^ h; E

...exp

ee104 10

0025902219

#= -: D . cm213 1015 3#=

-

Now, the energy bandgap for silicon is 1.12 eV, i.e.

Eg . eVE E 112c v= - =

Therefore, we obtain

E E E E c F F v - + -^ h . eV112=or E Ec F- . . . e

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