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FAKULTA ELEKTROTECHNIKY A KOMUNIKANCH TECHNOLOGIVYSOK UEN TECHNICK V BRN

Electronic Devices

Garant p edm tu: Doc. Ing. Jaroslav Bouek, CSc.

Auto i textu:Doc. Ing. Jaroslav Bouek, CSc.

RNDr. Michal Hork, CSc.

Brno 2.11. 2006


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2 FEKT Vysokho uen technickho v Brn

Contents1 ELECTRONIC DEVICES IN BACHELOR STUDY .................................................... 9

2 SEMICONDUCTOR FUNDAMENTALS .................................................................... 10

2.1 THE ELECTRONS IN FREE ATOMS .................................................................................... 102.2 THE ELECTRONS IN SOLIDS ............................................................................................ 102.3 BAND STRUCTURE OF SOLIDS ........................................................................................ 11

2.3.1 Band structures in different types of solids ............................................... 12 2.4 PREPARATION OF SEMICONDUCTOR MATERIALS ............................................................ 12

2.4.1 Intrinsic semiconductors ........................................................................... 13 2.4.2 Doped semiconductors .............................................................................. 14 2.4.3 Degenerately doped semiconductors......................................................... 15 2.4.4 Control question and example for capture 2.4 .......................................... 16

2.5 CARRIER DENSITY ......................................................................................................... 16

2.5.1 Density of states ........................................................................................ 16 2.5.2 The Fermi-Dirac distribution function ...................................................... 17 2.5.3 Filling of bands ......................................................................................... 18 2.5.4 Calculation of carrier densities ................................................................. 19 2.5.5 Calculation of Fermi energy level ............................................................. 21 2.5.6 Generation and recombination processes ................................................. 23

2.6 TRANSPORT OF CARRIERS .............................................................................................. 262.6.1 Carrier drift ............................................................................................... 27 2.6.2 Carrier diffusion ........................................................................................ 29 2.6.3 Control questions and examples ............................................................... 31

3 PN JUNCTION ................................................................................................................ 31 3.1 STRUCTURE AND PRINCIPLE OF OPERATION ................................................................... 31

3.1.1 Built in potential ........................................................................................ 33 3.2 ELECTROSTATIC ANALYSIS OF APN JUNCTION ............................................................. 34

3.2.1 The full-depletion approximation .............................................................. 34 3.2.2 Full depletion analysis .............................................................................. 35

3.3 FORWARD AND REVERSE BIAS ....................................................................................... 373.3.1 Forward bias ............................................................................................. 38 3.3.2 Reverse bias ............................................................................................... 39

3.4 IDEAL DIODE CURRENT .................................................................................................. 413.5 STATIC AND DYNAMIC RESISTANCE OFPN JUNCTION .................................................... 45

3.6 JUNCTION CAPACITANCE ............................................................................................... 453.7 DIFFUSION CAPACITANCE .............................................................................................. 473.8 DYNAMIC CHARACTERISTICS OFPN JUNCTION ............................................................. 48

3.8.1 Turn on process ......................................................................................... 48 3.8.2 Reverse recovery ....................................................................................... 49

3.9 REVERSE BIAS BREAKDOWN .......................................................................................... 503.9.1 Avalanche breakdown ............................................................................... 51 3.9.2 Zener breakdown ....................................................................................... 52 3.9.3 Second breakdown ..................................................................................... 54 3.9.4 Surface breakdown .................................................................................... 55

3.10 METAL-SEMICONDUCTORJUNCTION ............................................................................ 563.10.1 Ohmic contact for semiconductor devices................................................. 59 3.10.2 Control questions and examples for Chapter 3.10 .................................... 59


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Electronic Devices 3

4 SEMICONDUCTOR DIODES ...................................................................................... 61 4.1 POINT-CONTACT DIODES................................................................................................ 614.2 PLANAR DIODES............................................................................................................. 62

4.2.1 Examples for Chapter 4.2 .......................................................................... 64 4.3 PIN DIODES 654.4 ZENER DIODES ............................................................................................................... 65

4.4.1 Examples for Chapter 4.4 .......................................................................... 66 4.5 VARICAP OR VARACTOR DIODES .................................................................................... 66

4.5.1 Examples for capture 4.5 ........................................................................... 68 4.6 TUNNEL(ESAKI) DIODE ................................................................................................. 684.7 RECTIFIER DIODES ......................................................................................................... 69

4.7.1 Line-frequency diodes ................................................................................ 69 4.7.2 Fast recovery diodes .................................................................................. 70 4.7.3 Schottky diodes .......................................................................................... 71 4.7.4 Avalanche diodes ....................................................................................... 71

4.8 LEDS AND

PHOTOTIODES

.............................................................................................. 724.8.1 Light-emitting diodes and lasers ................................................................ 72 4.8.2 Photodiodes ............................................................................................... 72

4.9 I-V CHARACTERISTICS OF DIODE.................................................................................... 734.10 DIODE SUBSTITUTE CIRCUIT........................................................................................... 75

5 BIPOLAR JUNCTION TRANSISTOR ........................................................................ 75 5.1 STRUCTURE OF BIPOLAR TRANSISTOR ............................................................................ 755.2 OPERATION OF BIPOLAR TRANSISTOR ............................................................................ 78

5.2.1 Modes of operation .................................................................................... 78 5.2.2 Two types of bipolar junction transistor .................................................... 79

5.3 TECHNOLOGY OF BIPOLAR TRANSISTORS ....................................................................... 805.4 USE OFBJT 835.4.1 BJT as electronic amplifier ........................................................................ 83 5.4.2 BJT as electronic switch ............................................................................ 85 5.4.3 Current gain in BJT structure .................................................................... 85

5.5 CHARACTERISTIC CURVES OFBJT ................................................................................. 865.5.1 Early effect ................................................................................................. 88

5.6 OPERATION OFBJT AMPLIFIER ...................................................................................... 895.6.1 Operation point of BJT amplifier............................................................... 89 5.6.2 Voltage gain of the BJT amplifier .............................................................. 96 5.6.3 Power gain of BJT amplifiers .................................................................... 97 5.6.4 Control questions and examples for Chapter 5.6 ...................................... 99 5.7 OPERATION OFBJT AS CONSTANT CURRENT SOURCE .................................................. 100

5.8 BJT AS ELECTRONIC SWITCH ....................................................................................... 1015.8.1 Darlington transistor ............................................................................... 102 5.8.2 Design rules for BHT switch .................................................................... 103

5.9 VOLTAGE BREAKDOWN OFBJT STRUCTURE ............................................................... 1055.10 CONTROL QUESTIONS FORBIPOLARJUNCTIONTRANSISTOR ....................................... 107

6 FIELD-EFFECT TRANSISTORS .............................................................................. 108 6.1 JUNCTIONFIELDEFFECTTRANSISTOR(J-FET) ........................................................... 109

6.1.1 Output characteristics of JFET ................................................................ 109 6.1.2 The transfer characteristic ....................................................................... 112 6.2 INSULATEDGATE FIELDEFFECTTRANSISTOR(IGFET) .............................................. 116


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6.2.1 Operation of IGFET ................................................................................ 117 6.2.2 Enhancement and depletion mode ........................................................... 119 6.2.3 Characteristic curves for E-IGFET ......................................................... 120 6.2.4 Characteristic curves for D-IGFET ........................................................ 120 6.2.5 Available terminals by IGFET devices .................................................... 121

6.3 CONTROL QUESTIONS FORFIELD-EFFECTTRANSISTORS ............................................. 1227 BIPOLAR SWITCHING DEVICES............................................................................ 122

7.1 THYRISTOR 1237.1.1 Operation of thyristor .............................................................................. 123 7.1.2 Thyristor switching .................................................................................. 124 7.1.3 Applications ............................................................................................. 125 7.1.4 Triac ........................................................................................................ 125 7.1.5 Diac ......................................................................................................... 126

8 RESULTS AND EXAMPLES ...................................................................................... 128

8.1 EXAMPLES FORSEMICONDUCTOR DIODES ................................................................... 1288.1.1 Solution Chapter 4.2.1 ............................................................................ 128 8.1.2 Solution Chapter 4.4.1 ............................................................................ 129

8.2 EXAMPLES FORBIPOLARJUNCTIONTRANSISTOR ....................................................... 1318.2.1 Solution Chapter 5.6.4 ............................................................................ 131 8.2.2 Examples for BJT as electronic switch.................................................... 134 8.2.3 Example for Operation of BJT as constant current source ..................... 135

8.3 EXAMPLES FORFIELD-EFFECTTRANSISTORS .............................................................. 1358.3.1 Example for Operating point of JFET ..................................................... 135 8.3.2 Example for Operating point of MOSFET .............................................. 137 8.3.3 Example for JFET as analogue switch .................................................... 138

9 REFERENCES ............................................................................................................... 139


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Seznam obrzk F IG . 2.1: THEORETICAL BAND MODEL OFIV. ELEMENTS UNIT IN DEPEND ON THE GRID

SPACING 11F IG . 2.2: ELECTRON AND HOLE GENERATION FOR INTRINSIC SEMICONDUCTOR ................ 13

F IG . 2.3: SI CRYSTAL LATTICE WITH ONE ATOM OF DONOR AND BAND MODEL FORN TYPE 14F IG 2.4: SI CRYSTAL LATTICE WITH ONE ATOM OF ACCEPTOR AND BAND MODEL FORP

TYPE 15F IG 2.5: GENERAL ENERGY DEPEND OF GC(E) AND GV(E) ................................................ 17F IG . 2.6: ENERGY DEPEND OFFERMI-DIRAC FUNCTION A) T 0 K, B) GENERAL DEPEND

17F IG . 2.7: BAND FILLING OF BANDS FOR INTRINSIC SEMICONDUCTOR AND BOTH DOPED P-

TYPE AND N-TYPE .............................................................................................................. 18F IG . 2.8: THERMAL DEPENDENCE OF INTRINSIC CARRIERS DENSITY FORGAAS, SI ANDGE

21F IG . 2.9: A) THE LEVEL OF FERMI ENERGY IN SILICON IN DEPENDENCE ON

CONCENTRATION OF IMPURITIES. B) TEMPERATURE DEPENDENCE OFFERMI ENERGY ISTAKEN IN MARKED POINTS FOR BOTHN-TYPE ANDP-TYPE. ............................................... 22

F IG . 2.10: TEMPERATURE DEPENDENCE OF CARRIERS DENSITY FORN-TYPESEMICONDUCTOR IN LINEAR SCALE A1/T SCALE FOR TEMPERATURE. ............................... 22

F IG . 2.11: THE CARRIER MOTION IN THE SEMICONDUCTOR IN THE PRESENCE OF ANELECTRIC FIELD. ................................................................................................................ 27

F IG . 2.12: DRIFT CURRENT OF HOLES THROUGH P-TYPE SEMICONDUCTOR ......................... 27F IG . 2.13: ELECTRICAL FIELD DEPENDENCE OF MOBILITY FOR ELECTRONS AND HOLES IN

SILICON BY TEMPERATURE300 K. ..................................................................................... 28F IG . 2.14: THE MOBILITY OF ELECTRONS AND HOLES IN SILICON AT ROOM TEMPERATURE IN

DEPENDENCE ON DENSITY OF DONOR AND ACCEPTOR IMPURITIES(BY TEMPERATURE300 K). 29

F IG . 2.15: THE SCHEME OF THE DIFFUSION PROCESS AND CORRESPONDING CURRENTDENSITY FOR ELECTRONS AND HOLES. ............................................................................... 30

F IG . 3.1: ASYMMETRICPN JUNCTION............................................................................... 32F IG . 3.2: ABRUPTPN JUNCTION IN FORWARD AND REVERSE BIAS: A) CONFIGURATION; B)

SPACE CHARGE; C) ELECTRIC FIELD; D) JUNCTION POTENTIAL .......................................... 38F IG . 3.3: CARRIER TRANSPORT OVER THE ENERGETIC BARRIER IN CASE OF FORWARD(A)

AND REVERSE(B) BIAS. ...................................................................................................... 41F IG . 3.4: MINORITY CARRIER DENSITY IN CASE OF FORWARD BIAS(A) AND REVERSE BIAS

(B). 42F IG . 3.5: I-V CHARACTERISTIC OF IDEALPN JUNCTION IN LINEAR(A) AND

SEMILOGARITHMIC(B) SCALE ............................................................................................ 44F IG . 3.6: VOLTAGE DEPENDENCE OF JUNCTION CAPACITY(OR BARRIER CAPACITY). ........ 47F IG . 3.7: VOLTAGE OVERSHOOT AS CONSEQUENCE OF RAPID GROW OF CURRENT DURING

SWITCH ON. (A) DIODE FORWARD VOLTAGE / DOTTED LINE IS SUPPOSED POWER LOSS /. (B) ESTIMATION OF FORWARD RECOVERY TIME TRR ................................................................. 49

F IG . 3.8: DEFINITIONS OF THE RECOVERY CHARACTERISTICS ........................................... 50F IG . 3.9: PRINCIPLE OF AVALANCHE MULTIPLICATION ..................................................... 51F IG . 3.10: BREAKDOWN VOLTAGE VERSUS DOPING DENSITY OF AN ABRUPT ONE-SIDEDPN

JUNCTION(A) AND LINEARLY GRADED JUNCTION(B). ........................................................ 52


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F IG . 3.11: PRINCIPLE OFZENER BREAKDOWN VISUALIZED BY MEANS OF BAND DIAGRAM. A) PN JUNCTION IN THERMAL EQUILIBRIUM. B) THE SAME JUNCTION WITH REVERSE BIAS ANDVALUE OF REVERSE VOLTAGE CORRESPONDING TOZENER BREAKDOWN. ......................... 53

F IG . 3.12: THERMAL STABILITY OF THEPN JUNCTION ....................................................... 55F IG . 3.13: SCHOTTKY BARRIER. (A) BAND DIAGRAM OF METAL AND SEMICONDUCTOR. (B)

BAND DIAGRAM OF METAL-SEMICONDUCTOR JUNCTION IN THERMAL EQUILIBRIUM. ........ 56F IG . 3.14: METAL-SEMICONDUCTOR JUNCTION. BAND DIAGRAMS .................................... 58F IG . 3.15: BAND DIAGRAM OF METAL-SEMICONDUCTOR JUNCTION.. (A) BY FORWARD BIAS.

(B) BY REVERSE BIAS. (C) I-V CHARACTERISTIC. .............................................................. 59F IG . 3.16: CONTACT SYSTEM METAL-SEMICONDUCTOR. (A) RECTIFYING BY LOW DOPING

LEVEL. (B) OHMIC(WITH TUNNEL TRANSITION) BY HEAVILY DOPED SEMICONDUCTOR. .... 59F IG . 4.1: SEMICONDUCTOR(A) OF GERMANIUM POINT CONTACT DIODE. (B) GERMANIUM

POINT CONTACT DIODE WITH GOLD WIRE(C) PLANAR SILICON DIODE ............................... 62F IG . 4.2: TECHNOLOGIES FOR PLANAR DIODES. (A) ALLOYED JUNCTION. (B) DIFFUSION

MESA TECHNOLOGY. (C) DIFFUSION PLANAR TECHNOLOGY. (D) EPITAXIAL PLANARTECHNOLOGY. (E) SCHOTTKY DIODE MADE BYEPITAXIAL PLANAR TECHNOLOGY. ONPICTURES(A-D) THERE IS NOT DRAWN AN ALUMINUM CONTACT LAYER. IN PICTURE(E) THEALUMINUM LAYER IS A PART OF THE DIODE STRUCTURE. .................................................. 63

F IG . 4.3: DOPANTS DENSITY DEPENDENCE OF BREAKDOWN VOLTAGE BYPN JUNCTION .. 65F IG . 4.4: CONNECTION OF VARICAP DIODE TO RESONANTRC CIRCUIT. ........................... 67F IG . 4.5: CHARACTERISTIC CURVE OF A TUNNEL DIODE COMPARED TO THAT OF A

STANDARDPN JUNCTION. ................................................................................................. 69F IG . 4.6: PRINCIPAL SCHEMES OF LINE FREQUENCY RECTIFIERS. (A) HALF WAVE

RECTIFIER. (B) FULL WAVE RECTIFIER WITH CENTER TAPPED TRANSFORMER. (C) FULLWAVE RECTIFIER WITH A BRIDGE RECTIFIER. ..................................................................... 70

F IG . 4.7: I-V CHRACTERISTICS PHOTOVOLTAIC AND PHOTOCONDUCTIVE MODES ............ 73F IG . 4.8: I-V CHARACTERISTICS OF SEMICONDUCTOR DIODE WITH DIFFERENT SCALES. (A)

LOW VOLTAGE AND LOW CURRENT SCALE. (B) COARSE SCALE FOR VOLTAGE. (C) COARSESCALE FOR BOTH VOLTAGE AND CURRENT. (D) DIFFERENT SCALES FOR CURRENT ANDVOLTAGE. 73

F IG . 5.1: PNP BIPOLAR JUNCTION TRANSISTOR. (A) THE CURRENT THROUGH BASE REGION. (B) IN FORWARD ACTIVE BIAS MODE OPERATION THE COLLECTOR TAKES OVER A GREATPART OF THE EMITTER CURRENT. ....................................................................................... 76

F IG . 5.2: NORMAL MODE OPERATION OF THEBJT ............................................................ 77F IG . 5.3: COMMON EMITTER CONFIGURATION FORNPN AND PNP TYPES OFBJT IN

NORMAL(ACTIVE FORWARD) MODE OPERATION ............................................................... 80F IG . 5.4: DIFFUSION TRANSISTORS. (A) ALLOY JUNCTION TRANSISTOR. (B) DIFFUSED BASE

TRANSISTOR. (C) D

IFFUSED MESA TRANSISTOR. (D

) D

IFFUSED PLANAR TRANSISTOR. (E)

EPITAXIAL PLANAR TRANSISTOR. (F) EPITAXIAL PLANAR TRANSISTOR IN INTEGRATEDCIRCUIT 81

F IG . 5.5: DIFFUSED PLANAR TRANSISTOR TECHNOLOGY PROCESS FORNPN TYPETRANSISTOR. (A) OXIDATION. (B) SELECTIVE ETCHING. (C) DIFFUSION OF BASE. (D) SHAPING THE MASK FOR EMITTER DIFFUSION. (E) DIFFUSION OF EMITTER. (F) SHAPING THEMASK FOR THE CONTACTS. (G) DEPOSITION OF METAL CONTACTS .................................... 83

F IG . 5.6: BJT AS AMPLIFER. CONFIGURATIONS: (A) COMMON EMITTER(CE) (B) COMMONBASE(CB) (C) COMMON COLLECTOR(CC) ....................................................................... 84

F IG . 5.7: BT CONNECTION................................................................................................ 84F IG . 5.8: BIPOLAR JUNCTION TRANSISTOR: CHARACTERISTICS IN COMMON EMITTER

CONFIGURATION................................................................................................................ 86F IG . 5.9: EARLY EFFECT. DETERMINATION OFEARLY VOLTAGE. ..................................... 88


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F IG . 5.10: OPERATING POINT OFBJT: THE SIMPLEST FIXED-BIAS CIRCUIT ........................ 90F IG . 5.11: BJT OPERATING POINT: COLLECTOR FEEDBACK(OR VOLTAGE OR PARALLEL

FEEDBACK) 90F IG . 5.12: BJT OPERATING POINT: VOLTAGE DIVIDER(RB1,RB2) ..................................... 91F IG . 5.13: BIPOLAR TRANSISTOR AS AN AMPLIFIER(CE CONFIGURATION). ........................ 91F IG . 5.14: DETERMINATION OF EMITTER-COLLECTOR VOLTAGE BY MEANS OF LOAD LINE. (A) AMPLIFIER(B) SWITCH. ............................................................................................... 97F IG . 5.15: BJT AS CONSTANT CURRENT SOURCE. (A) (B) WITH NEGATIVE FEEDBACK ONRE

100F IG . 5.16: BJT AS CONSTANT CURRENT SOURCE: CURRENT MIRROR. ............................... 100F IG . 5.17: BIPOLAR TRANSISTOR AS A SWITCH(RESISTIVE LOAD) .................................... 102F IG . 5.18: PRINCIPLE SCHEME OFDARLINGTON TRANSISTOR ........................................... 103F IG . 5.19: BJT SWITCH WITHSCHOTTKY DESATURATION DIODE. ..................................... 105F IG . 5.20: DIFFERENT CIRCUITS TO INCREASE THE BREAKDOWN VOLTAGE OFBJT. ......... 106F IG . 5.21: DIFFERENT CIRCUITS TO INCREASE THE BREAKDOWN VOLTAGE OFBJT .......... 106F IG . 5.22: SAFEOPERATINGAREA IN OUTPUTCE CHARACTERISTICS .............................. 107F IG . 6.1: THEJUNCTIONFIELD-EFFECT TRANSISTOR(JFET) ......................................... 109F IG . 6.2: JFET: CHANNEL SHAPES IN ACTIVE MODE ....................................................... 110F IG . 6.3: JFET: CHANNEL SHAPES IN ACTIVE MODE ....................................................... 111F IG . 6.4: OUTPUT CHARACTERISTICS OFN-CHANNELJFET ........................................... 112F IG . 6.5: OUTPUT CHARACTERISTICS OFN-CHANNELJFET ........................................... 113F IG . 6.6: IGFET/MOSFET TRANSISTORS, STRUCTURES AND TYPES .............................. 116F IG . 6.7: ENHANCED-MOSFET WITH N-CHANNEL: CHARGE ACCUMULATION BENEATH

THE GATE 117F IG . 6.8: ENHANCED-MOSFET WITH N-CHANNEL: THE GATE ACTION.......................... 118F IG . 6.9: ENHANCED-MOSFET WITH N-CHANNEL: SATURATION MODE ....................... 119F IG . 6.10: E-MOSFET WITH N-CHANNEL: OUTPUT AND TRANSFER CHARACTERISTICS ... 120F IG . 6.11: D-MOSFET WITH N-CHANNEL: OUTPUT AND TRANSFER CHARACTERISTICS ... 121F IG . 6.12: MOSFET WITH THE BULK(SUBSTRATE) ELECTRODE ...................................... 121F IG . 7.1: BIPOLAR SWITCHING DEVICES. (A) THYRISTOR. (B) TRIAC. (C) DIAC ............... 122F IG . 7.2: THYRISTOR. (A) THE STRUCTURE. (B) THE SUBSTITUTE SCHEME ..................... 123F IG . 7.3: THYRISTOR. SWITCHING CHARACTERISTIC. ..................................................... 124F IG . 7.4: THYRISTOR. PHASE ANGLE TRIGGERED CONTROL ............................................ 125F IG . 7.5: DIAC. (A) STRUCTURE. (B) V-I CHARACTERISTIC. ............................................ 127


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Seznam tabulekT AB . 1: THE WORK FUNCTION OF SELECTED METALS AND BARRIER HEIGHT IN CASE OF

DIFFERENT SEMI-CONDUCTIVE MATERIALS ....................................................................... 56T AB . 2: BIPOLAR JUNCTION TRANSISTOR: MODES OF OPERATION .................................. 79


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1 Electronic devices in bachelor study

From the invention of the transistor at Bell Labs in 1947 the semiconductor technologyhave seen tremendous development. On that way high-density microprocessors developed bythe mid of 1980s were only a further milestone aimed in fact to up to day technology whichstruggle with physical limits in many technological steps. Surely, silicon and its compoundsusing in semiconductor technology are to time the best explored material the mankind know.

This course will focus on practical knowledge of semiconductor technology and onoperation and use of semiconductor devices. The first part is addressed to semiconductorphysics and to basic knowledge concerning the semiconductors. In the second part somephenomena of semiconductor physics are discussed namely PN junctions, diode and transistorfundamentals, bipolar and unipolar devices and its operation in different operationalconditions. Basic physical models of the operation of semiconductor devices such as diodes,

MOS transistors, and bipolar junction transistors will be presented including the analysis anddesign of important circuits which utilize these devices.The subject Electronic Devices is established in second semester of bachelor study.

Because of that there is a certain overlap with other subjects dealing with theory of circuits,electronics, physics and material science. On the other hand the subject Electronic Devices inmany aspects brings the knowledge acquired in these subjects together.

In following chapter atomic and electronic structure of materials is dealt shortly todefine energetic band structure of materials. With help of energy bands diagrams the chargedensity in semiconductor is computed and charge transport in semiconductors is explained.

Third chapter aims to explain the operation of semiconductor junctions. Charge build upin the depletion zone in PN junction is dealt thoroughly and junction parameters as built involtage, junction width, barrier and diffusion capacitances and dynamic resistance arederived. Breakdown mechanisms and dynamic properties of PN junction are discussed tounderstand the operation of diodes in different circuits.

In following chapter manufacture processes for different diode types are discussedshortly. Examples of different diode circuits are given.

In fifth chapter the bipolar junction transistor is dealt. Its operation is explained and thebasic knowledge about transistor circuits as amplifiers, electronic switchers and constantcurrent sources is given. Computation of transistor circuits is explained using examples oftypical transistor circuits.

Similarly operation of both unipolar transistor types, JFET and IGFET is explained innext chapter. The depletion and enhanced IGFET types are discussed and computation oftransistor circuits is explained using both graphic and numerical methods.

In seventh chapter is explained operation of bipolar switches with emphasis on thyristorand triac devices. The turn on mechanisms of thyristor are dealt thoroughly. Power controlusing thyristor and triac circuits is discussed and dynamic properties of different types ofthyristors are shortly dealt.


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2 Semiconductor fundamentals

Aim of this chapter is to explain basic definitions from material and semiconductorscience. Atomic and electronic structure of materials is dealt shortly to define energetic bandstructure of materials. Coming out from energy band structure charge density insemiconductor is computed and charge transport in semiconductors is explained.

To understand the fundamental concepts of semiconductors modern physics must beapplied to solid materials. We need to know how many fixed and mobile charges are presentin the material and we need to understand the transport of the mobile carriers through thesemiconductor

In this chapter we start from the atomic structure of semiconductors and explain theconcepts of energy bands, energy band gaps and the density of states in an energy band. Wealso show how the current in an almost filled band can more easily be analyzed using theconcept of holes. Next, we discuss the probability that energy levels within an energy band

are occupied. We will use this probability density to find the density of electrons and holes ina bandTwo carrier transport mechanisms will be considered. The drift of carriers in an electric

field and the diffusion of carriers due to a carrier density gradient will be discussed.Recombination mechanisms and the continuity equations are then combined into the diffusionequation. Finally, we present the drift-diffusion model, which combines all the essentialelements discussed in this chapter

2.1 The electrons in free atoms

The electrons in free atoms can be found in only certain discrete energy states. Thesesharp energy states are associated with the orbits or shells of electrons in an atom. The Bohrmodel successfully predicted the energies for the hydrogen atom, Nevertheless there are somesignificant failures. The precise details of spectra and charge distribution must be left toquantum mechanical calculations, as with the Schrodinger equation. Despite this imperfectionthe Bohr model gives us a basic conceptual model of electrons orbits and energies.

2.2 The electrons in solids

According the Bohr model electrons of a single free-standing atom occupy atomicorbitals, which form a discrete set of energy levels. If several atoms are brought together intoa molecule, their atomic orbitals split. Reason for this behavior is that the electron energylevels can not to be the same - Pauli exclusion principle does not allow it. Consequently weobtain a set of energy levels ordered so closely that an energy band is formed instead oforiginal discrete energy levels. This produces a number of molecular orbitals proportional tothe number of atoms. Energy bands are thus the collection of the individual energy levels ofelectrons surrounding each atom. When a large number of atoms (of order 1020 or more) arebrought together to form a solid, the number of orbitals becomes exceedingly large, and thedifference in energy between them becomes very small.


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Fig. 2.1: Theoretical band model of IV. elements unit in depend on the grid spacing

Any solid has a large number of bands, but, as we will see further, only few lie atenergies which are significant for electronic properties of solids. As seen from Fig. 2.1 bandswidths and positions depends upon the properties of the atomic orbitals of atoms from whichthe solid is composed and upon the distances and angles in crystalline structure of solid.Moreover, allowed bands may overlap, producing a single large band.

2.3 Band structure of solids

The energy band model is very important feature for semiconductor devices science. Todescribe semiconductors behavior a simplified energy band diagram is used usually. There arethe valence band and conduction band separated by an energy gap between them.

For band diagram following parameters are important:EV Energy of the valence band edge indicated by a horizontal lineEV Energy of the conduction band edge indicated by a horizontal lineEG Energy gap between valence and conduction band EG andEvac Vacuum levelThe uppermost occupied band in an insulator or semiconductor is called the valence

band by analogy to the valence electrons of individual atoms. The lowermost unoccupied

band is called the conduction band because only when electrons are excited to the conductionband the current can flow in these materials. Energy interval between conduction and valencebands is called forbidden band gap. Forbidden band gap strongly influence the electrical andoptical properties of the material. Electrons can transfer from one band to the other by meansof carrier generation and recombination processes.

Electrons at energy bands under the valence band are tightly bound to the atom and arenot allowed to freely move in the material. It means that only the valence electrons (theelectrons in the outer shell) are of interest for electronic devices. Because there are twoelectrons with opposite spin on each energy level and the band is formed by splitting of one ormore atomic energy levels the minimum number of states in a band equals twice the number

of atoms in the material. For material properties such electrical and thermal conductivity isimportant how the energy levels in bands are occupied.


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The difference between insulators and semiconductors is that the forbidden band gapbetween the valence band and conduction band is larger in an insulator. Because one of themain mechanisms for electrons to be excited to the conduction band is due to thermal energy,the conductivity of semiconductors and insulators is strongly dependent on the temperature ofthe material. Owing to the large bandgap the probability of electron excitation is much less in

case of insulator so that fewer carriers of electric charge are found there and the electricalconductivity is less. Semiconductors on the other hand contain (relatively) small number ofthermally excited charge carriers. Consequently they have an "almost-empty" conductionband and an "almost-full" valence band. With this feature they differ from both metals andinsulators.

Metals contain a band that is partly empty and partly filled regardless of temperature.Therefore they have very high conductivity.

2.3.1 Band structures in different types of solids

Although electronic band structures are usually associated with crystalline materials,quasi-crystalline and amorphous solids may also exhibit band structures. However, theperiodic nature and symmetrical properties of crystalline materials makes it much easier toexamine the band structures of these materials theoretically. As a result, virtually all of theexisting theoretical work on the electronic band structure of solids has focused on crystallinematerials. In semiconductor industry practically all production is made using mono-crystallineor crystalline materials.

a) An important feature of an energy band diagram, which is not included on thesimplified diagram, is whether the conduction band minimum and the valence band maximumoccur at the same value for the wavenumber. If so, is called direct. If not, the energy bandgapis called indirect. This distinction is of interest for optoelectronic devices as LED and laserdiode. Direct bandgap materials provide much more efficient absorption and emission oflight. Mostly used semiconductor silicon and also germanium have indirect energy bandgap.Gallium arsenide and other semiconductors of AIIIBV group have a direct bandgap.

b) The energy bandgap of semiconductors tends to decrease as the temperature isincreased. By elevated temperature the inter-atomic distances increase as consequence ofincrease of the atomic vibrations. This effect cause linear expansion of a material. Byincreased inter atomic distances decreases the average potential seen by the electrons in thematerial, which in turn reduces the size of the energy bandgap. (The temperature dependenceof the energy bandgap, Eg, has been experimentally determined.)

c) A direct modulation of the inter-atomic distance caused by compressive or tensilestress can also bring an increase or decrease of the energy bandgap.

d) The energy bandgap of semiconductors tends to decrease by high doping densities.The average distance between two impurities is very small in this case, impurities thereforeinteract each to other forming an energy band rather than a discreet level. The effect starts bydoping density higher then 1018 cm-3.

2.4 Preparation of semiconductor materials

The level of chemical purity needed is extremely high because the presence ofimpurities even in very small proportions can have large effects on the properties of thematerial. A high degree of crystalline perfection is also required, since faults in crystalstructure (such as dislocations, twins, and stacking faults) change the properties of the


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material. Crystalline faults are a major cause of defective semiconductor devices. The largerthe crystal, the more difficult it is to achieve the necessary perfection. Current massproduction processes use crystal ingots between four and twelve inches (300 mm) in diameterwhich are grown as cylinders and sliced into wafers.

Because of the required level of chemical purity, and the perfection of the crystalstructure which are needed to make semiconductor devices, special methods have beendeveloped to produce the initial semiconductor material. A technique for achieving highpurity semiconductors includes growing the crystal using a special process called Czochralskiprocess. An additional step that can be used to further increase purity is zone refining. In zonerefining part of a solid crystal is melted. The impurities tend to concentrate in the meltedregion, while the desired material recrystallizes leaving the solid material more pure and withfewer crystalline faults.

2.4.1 Intrinsic semiconductors

Intrinsic semiconductor (or i-type semiconductor), is a pure semiconductor without anysignificant structure defects and dopant species. The presence and type of charge carriers isthen determined by the material itself. In thermal equilibrium the amount of free electronsdepends on the bandgap and on the temperature.

in pn == ( 2.1 )

When electron is excited to move freely in the semiconductor lattice than by atom offree electron origin there remain a vacant place in the chemical bond between semiconductoratoms. Such vacant place can be filled by an electron from neighboring atom, the vacant placewith positive charge displaces, new vacant place can be filed again.... etc., finally resulting inhopping of valence electrons from one bonds to the other and in moving of the positive

charge. This moving positive charge is called hole to differ from electron motion inconductivity band.

Fig. 2.2: Electron and hole generation for intrinsic semiconductor

Holes behave as particles with the same properties as the electrons would have exceptthat they carry a positive charge. However it is important to understand that the chargemovement is done by displacement of electrons which are the only real charged particlesavailable in a semiconductor. Because the mechanism of charge movement is more complexin this case, holes have allways lower mobility then electrons. The concepts of holes wasintroduced in semiconductors science because it is easier to keep track of holes, rather thankeeping track of the actual electrons in almost full valence band.


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In case of intrinsic semiconductors the amount of electrons and holes is roughly equaland depends on the bandgap and on the temperature. Intrinsic semiconductors conductivity istherefore given by present crystal defects and by thermal excitation.

2.4.2 Doped semiconductors

An N-type semiconductor is obtained by adding an impurity of valence-five elementsto the semiconductor in order to increase the number of free electrons. Atoms of the dopingmaterial give away weakly-bound outer electrons. This type of doping agent is labeled asdonor material since it gives away (donates) one of its electrons.

Fig. 2.3: Si crystal lattice with one atom of donor and band model for N type

To understand how n-type doping is accomplished, consider the case of silicon (Si). Siatoms have four valence electrons, each of which is covalently bonded with one of four

adjacent Si atoms. If an atom with five valence electrons (P, As or Sb) is incorporated into thecrystal lattice in place of a Si atom, then that atom will have four covalent bonds and oneunbonded electron. This extra electron is only weakly bound to the atom and can easily beexcited into the conduction band.

At normal temperatures, virtually all such electrons are excited into the conductionband. Since excitation of these electrons does not result in the formation of a hole, the numberof electrons in such a material far exceeds the number of holes. In this case the electrons arelabeled as the majority carriers and the holes as the minority carriers. Note that each movableelectron within the semiconductor is never far from an immobile positive dopant ion, and theN-doped material normally has a net electric charge of zero.

A P-type semiconductor is obtained in the same process as N-type only the dopingspecies is a trivalent atom (B, In, Al). The result is that one electron is missing from one ofthe four covalent bonds normal for the silicon lattice. As a consequence, when trivalent atomis added, it takes away (accepts) weakly-bound outer electrons from neighboringsemiconductor atoms to complete the fourth bond. Such dopants are therefore called acceptors . When the dopant atom accepts an electron, one bond electron from theneighboring atom is lost. This process result in the formation of another bond with missingelectron. This vacant bond is called hole and because of one missing electron it has positivecharge.


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Fig 2.4: Si crystal lattice with one atom of acceptor and band model for P type

Each hole is associated with a nearby negative-charged dopant ion, and thesemiconductor remains electrically neutral as a whole. When a sufficiently large number ofacceptor atoms are added, the holes greatly outnumber the thermally-excited electrons. Theholes are then the majority carriers, while electrons are the minority carriers in P-typematerials.

In general, an increase in doping concentration cause an increase in conductivity due tothe higher concentration of carriers available for conduction. By doped semiconductors thecarrier concentration is determined by the concentration of dopant introduced to an intrinsicsemiconductor. In addition the dopant indirectly affects also other electrical properties.

Doping a semiconductor crystal introduces allowed energy states within the band gapbut very close to the energy band that corresponds with the dopant type. Donor impuritiescreate states near the conduction band while acceptors create states near the valence band (seethe Fig. 2.3 andFig 2.4 ). The gap between these energy states and the nearest energy band isusually referred to asdopant-site bonding energy or EB and is relatively small. For example,the EB for boron in silicon bulk is 45 meV, compared with silicon's band gap of about 1.12eV. Because EB is so small, it takes little energy to ionize the dopant atoms and create freecarriers in the conduction or valence bands. Usually the thermal energy available at roomtemperature is sufficient to ionize most of the dopant atoms.

2.4.3 Degenerately doped semiconductors

Degenerately (very highly) doped semiconductors have conductivity levels comparableto metals and are often used in modern integrated circuits as a replacement for metal. Oftensuperscript plus and minus symbols are used to denote relative doping concentration insemiconductors. For example, n+ denotes an n-type semiconductor with a high (oftendegenerate) doping concentration. Similarly, p would indicate a very lightly doped p-typematerial. It is useful to note that even degenerate levels of doping imply low concentrations ofimpurities with respect to the base semiconductor. In crystalline intrinsic silicon, there areapproximately 51022 atoms/cm. Doping concentration for silicon semiconductors rangefrom 1013 cm-3 up to 1018 cm-3. Doping concentration above about 1018 cm-3 is considereddegenerate at room temperature.


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2.4.4 Control question and example for capture 2.4

1. a) What are doped semiconductors? b) Why we use them?

2. What are intrinsic semiconductors?

3. a) What are doped semiconductors? b) Explain how they are produced

4. Explain the origin of the electron conductivity of solids

5. Explain the origin of hole conductivity of solids

2.5 Carrier density

To calculate the density of carriers in a semiconductor we need to know the number ofavailable states at each energy and we also need to know the probability that at a given energythe state at is occupied by an electron. The number of electrons at each energy level will bethen obtained by multiplying the number of states with the probability that a state is occupiedby an electron.

It is obvious that the number of energy levels is very large and depends on the volumeof the semiconductor. Therefore is very convenient to calculate the number of states per unitenergy and per unit volume. This parameter is called density of states.

2.5.1 Density of states

For given energy level density of states in a semiconductor equals the number of states

per unit volume.Density of energy states in a band is not uniform. It approaches zero at the band

boundaries, and is generally greatest near the middle of a band.For energy close to the lower edge of the conduction band the density of states is

( )C C

C nnC E E E E konst

E E mm E g =

= ,.

2)( 32

**

h ( 2.2 )

And likewise for energy close to upper edge of the valence band:

( )V V

V p p

V E E E E konst E E mm

E g =

= ,.2

)( 32**

h ( 2.3 )

Here gV(E) gC(E) are density of states in conduction and valence bands respectively, m*and m* are effective mass for electrons and holes and h is Dirac constant.


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Fig 2.5: General energy depend of gC(E) and gV(E)

Shape of both functions is seen on the Fig 5. It is obvious that for the narrow energyrange from E to E + dE the number of allowed energy levels per unit volume of the materialis g(E) dE.

2.5.2 The Fermi-Dirac distribution function

Although the number of states in all energy bands is effectively infinite, in an unchargedmaterial the number of electrons is equal only to the number of protons in the atoms of thematerial. Therefore not all of the states which can be occupied by electrons are filled at anytime. The likelihood of any particular state being filled at any temperature is given by theFermi-Dirac distribution function:

=

kT F E E

E f

exp+1

1)( ( 2.4 )

where:k is Boltzmanns constantT is the temperature,EF is the Fermi energy (or 'Fermi level').The Fermi-Dirac distribution function is visualized onFig 2.6 . Regardless of the

temperature, f(EF) = 1/2. At T=0, the distribution is a simple step function. At nonzerotemperatures, the step "smooths out", so that an appreciable number of states below the Fermilevel are empty, and some states above the Fermi level are filled. In condition of thermalequilibrium the Fermi function therefore gives the probability that an energy level at energy Eis occupied by an electron. The system is thus characterized by its temperature, T, and itsFermi energy, EF.

Fig. 2.6: Energy depend of Fermi-Dirac function a) T 0 K, b) general depend


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Note:

a) Electrons are Fermions, the half-integer spin particles, which obey the Pauliexclusion principle.

The Pauli exclusion principle postulates that only one Fermion can occupy a single

quantum state. The states with the lowest energy are filled first, followed by the next higherones. At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximumenergy, which we call the Fermi level. No states above the Fermi level are filled.

b) For energies more than a few times kT below the Fermi energy the Fermi functionhas a value of one.

c) If the energy equals the Fermi energy Fermi function equals 1/2. Finally, it decreasesexponentially for energies which are a few times kT larger than the Fermi energy.

d) Near the absolute zero at T = 0 K the Fermi function is a step function.e) At finite temperatures the transition is more gradual and by high temperatures the

slope of the curve is less and less giving thus high probability of high energy states.2.5.3 Filling of bands

E g E f E n V )()( = ( 2.5 )

Where gC(E) is the density of states in the valence band.Probability of having a hole occupied state equals the probability that a particular state

is not filled by electron because holes correspond to empty states in the valence band. Using[1- f(E)] for hole instead f(E) the hole density per given energy, p(E), equals:

)()](1[)( E g E f E pV

= ( 2.6 )

Where gV(E) is the density of states in the valence band.Graphic visualization of band filling of bands for intrinsic semiconductor and both

doped p-type and n-type respectively is visualized onFig 2.7 .

Fig. 2.7: Band filling of bands for intrinsic semiconductor and both doped p-type and n-type


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2.5.4 Calculation of carrier densities

The density of carriers can be obtained by integrating the density of carriers per unitenergy over all possible energy states within a given band. The integral is taken from thebottom of the conduction band Ec, to the top of the conduction band for electrons in theconduction band and, similar way, from the upper energy of the valence band E

V to the

bottom of valence band for holes in valence band.

( ) ( )=roof E

C E C dE E g E f n ( 2.7 )

Where gC(E) is the density of states in the conduction band and f(E) is the Fermifunction.

For holes, using [1-/f(E)] for probability of occupancy, we get:

( )[ ] ( ) =V E

bottom E

V dE E g E f p 1 ( 2.8 )

After substitution for density of states and FD distribution function we obtain:

( )[ ]

=roof E

C E F

C 2

n

kT E - E +1

dE E - E mmn n

/ exp2

3h ( 2.9 )

( )[ ]( )[ ]

=V E

bottom E F

V F 2

p p

kT E - E +1

dE E - E kT E - E mm p

/ exp / exp2

3h ( 2.10 )

And after integration the result is:] / )(exp[ kT E E N n F C C = ( 2.11 )

] / )(exp[ kT E E N p C F V = ( 2.12 )

where NC is the effective density of states in the conduction band and NV is theeffective density of states in the valence band.

The effective density of states NC , NV are as follows:

2 / 3

2 / 3

222 T konst

h

kT m N nC =

=

( 2.13 )

2 / 3

2 / 3

222 T konst

h

kT m N nV =

=

( 2.14 )

Note that the effective density of states is temperature dependent.

Equations (( 2.11 )( 2.12 )) are not much convenient for practical computation. In caseof intrinsic semiconductor we can substitute EF = Ei and n = p = ni . We obtain:


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] / )(exp[ kT E E N n iC C i = ( 2.15 )

] / )(exp[ kT E E N n C iV i = ( 2.16 )

and:) / exp( / exp( kT E nkT E N iiC C = ( 2.17 )

) / exp() / exp( kT E nkT E N iiV V = ( 2.18 )

Using the equations ( 2.17 ) and ( 2.11 ) densities of carriers as a function of theintrinsic density and the intrinsic Fermi energy Ei can be derived:

] / )exp[( kT E E nn iF i = ( 2.19 )

] / )exp[( kT E E n p F ii = ( 2.20 )

Equations ( 2.19 ) and ( 2.20 ) can be used to find the electron and hole density in asemiconductor (in thermal equilibrium.) Value of intrinsic concentration is usually, asimportant semiconductor parameter, known. Of course to computation we need the level ofFermi energy.

By multiplying the expressions for the electron and hole densities in a non-degeneratesemiconductor, as in equations ( 2.19 ) and ( 2.20 ) one obtains:

2in pn = ( 2.21 )

For any non-degenerate semiconductor the product of the electron and hole densityequals the square of the intrinsic carrier density. This important property is referred to as themass action law.

However it is valid only for non-degenerate semiconductors in thermal equilibrium.From this relation it is possible to find the hole density if the electron density is known or viceversa.

The value of intrinsic density can be estimated from product of equations ( 2.11 ) and (2.12 ) :

) / exp(] / )(exp[2 kT E N N kT E E N N n GV C V C V C i == ( 2.22 )

And after square root:

( ) )2 / exp(2 / exp 2 / 3 kT E T konst kT E N N n GGV C == ( 2.23 )

Intrinsic density depends on temperature and semiconductor bandgap EG. For silicon byroom temperature intrinsic density is about ni = 1,45.1010 cm-3.


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Fig. 2.8: Thermal dependence of intrinsic carriers density for GaAs, Si and Ge

2.5.5 Calculation of Fermi energy level

In case ofintrinsic semiconductor we can find the intrinsic Fermi energy using theabove equations ( 2.11 ) and ( 2.19 ) for the intrinsic electron and hole density:

] / )exp[(] / )exp[( kT E E N kT E E N F V V C F C = ( 2.24 )

From this equation we can obtain EF easily. After substitution for NV and NC we get:2 / 3

*

*

ln22

ln22

+

+=

+

+=

n

pV C

C

V V C F m

mkT E E

N

N kT E E E ( 2.25 )

The intrinsic Fermi energy is half way between the conduction and valence band edge(midgap energy). The intrinsic Fermi energy can also be expressed as a function of theeffective masses of the electrons and holes in the semiconductor.

By doped (extrinsic) semiconductor we can suppose that carriers density equals thedensity of impurities. So for electrons (n = ND) we get:

] / )exp[( kT E E n N iF i D = ( 2.26 )

The position of Fermi energy EF in respect to intrinsic energy Ei is:) / ln( i DiF n N kT E E = ( 2.27 )

And similarly for holes) / ln( i AF i n N kT E E = ( 2.28 )


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Position of Fermi energy in semiconductor bandgap depends on concentration (density)of impurities.

The dependence of Fermi level for silicon is visualized onFig. 2.9 . By lowconcentration the change is very rapid (the same as logarithm does). By high density ofimpurities the dependence in semilogarithmic plot is almost linear. When the distance ofFermi energy from the edge of valence or conductivity band is less then about 3KT, thesemiconductor is regarded as degenerated.

Fig. 2.9: a) The level of Fermi energy in Silicon in dependence on concentration ofimpurities. b) Temperature dependence of Fermi energy is taken in marked points for both N-type and P-type.

T1 and T2 correspond with carriers activation from impurities and from semiconductorlattice respectively (see alsoFig. 2.10 ) . The level of intrinsic energy Ei grows withtemperature (see equ. ( 2.25 )). The bangap energy diminish with temperature with coefficientapproximately 10-4 eVK-1.

Fig. 2.10: Temperature dependence of carriers density for N-type semiconductor in linearscale a 1/T scale for temperature.


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2.5.6 Generation and recombination processes

Carrier generation and recombination result from interaction between electrons andother carriers, or with the lattice of the material, or with optical photons. As the electronmoves from one energy band to another, the energy which is gained or lost in this processtake different forms. The form of energy transformed during electron transition distinguishesvarious types of generation and recombination.

2.5.6.1 Generation of free carriersIn thermal equilibrium electrons can gain energy for transition from valence to

conductivity band only from thermal vibrations of the semiconductor lattice. The number ofgenerated carriers is therefore very low. In addition to thermal generation there are othermechanism described as follows.

Carriers can be generated byilluminating the semiconductor with light. The energy ofthe incoming photons is used to bring an electron from a lower energy level to a higher energylevel. If the photon energy is large enough, an electron raised from the valence band into anempty conduction band state, and one electron-hole pair is generated. The photon energyneeds to be larger than the bandgap energy to satisfy this condition. The excess energy, E ph

E G, is added to the electron and the hole in the form of kinetic energy. As the energy of thephoton is given off to the electron, the photon does not exist any longer.

Because of light absorption the light intensity in a semiconductor decreases withdistance from surface. The calculation of the generation rate of carriers therefore requires firsta calculation of the optical power within the structure from which the generation rate can thenbe obtained.

Note: After impact to the semiconductor photons are mostly absorbed generating a pair of free

carriers. Absorption is the active process in photodiodes, solar cells, and other semiconductorphotodetectors. Nevertheless a photon can also stimulate a recombination event, resulting in agenerated photon with similar properties to the original photon responsible for the event.Original photon is not absorbed by such interaction. This process results in stimulatedemission which is basic mechanism for laser action in laser diodes.

Carrier generation or ionization due to ahigh-energy beam consisting ofcharged particles is similar except that the available energy can be much larger than the bandgapenergy so that multiple electron-hole pairs can be formed. The high-energy particle graduallyloses its energy and eventually stops. This generation mechanism is used in semiconductor-based nuclear particle counters. As the number of ionized electron-hole pairs varies with theenergy of the particle, one can also use such detector to measure the particle energy.

Finally, there is a generation process calledimpact ionization , the generationmechanism that is the counterpart of Auger recombination. Impact ionization is caused by anelectron (hole) with an energy, which is much larger (smaller) than the conduction (valence)band edge. The excess energy is given off to generate an electron-hole pair through a band-to-band transition. This generation process causes avalanche multiplication in semiconductordiodes under high reverse bias: As one carrier accelerates in the electric field it gains energy.The kinetic energy is given off to an electron in the valence band, thereby creating anelectron-hole pair. The resulting two electrons can create two more electrons which generate

four more causing an avalanche multiplication effect. Electrons as well as holes contribute toavalanche multiplication.


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2.5.6.2 Recombination of free carriersRecombination of electrons and holes is a process by which both carriers annihilate

each other: As a result of this process electrons ends to occupy the empty state associated witha hole. This process proceeds through one or multiple steps. The energy difference betweenthe initial and final state of the electron is released several ways:

a) Radiative recombination - electron energy is emitted in the form of a photon.b) Non-radiative recombination - electron energy is given to one or more phonons.c) Auger recombination - electron energy is transformed to kinetic energy of another

electron.Band-to-band recombination occurs when an electron moves from its conduction

band state into the empty valence band state associated with the hole. The recombination ratedepends on the density of available electrons and holes. Both carrier types need to beavailable in the recombination process. Therefore, the rate is expected to be proportional tothe product ofn and p. Also, in thermal equilibrium, the recombination rate must equal thegeneration rate since there is no net recombination or generation.

Note: Band-to-band transition mechanism is typically also for a radiative transition in direct

bandgap semiconductors. During radiative recombination a photon is emitted with thewavelength corresponding to the energy released. This effect is called spontaneous emissionan it is the basis of LEDs operation. Because the photon carries relatively little momentum,radiative recombination is significant only in direct bandgap materials.

Trap-assisted recombination occurs when an electron falls into a "trap", an energy

level within the bandgap caused by the presence of a foreign atom or a structural defect. Oncethe trap is filled it cannot accept another electron. The electron occupying the trap, in a secondstep, moves into an empty valence band state, thereby completing the recombination process.One can envision this process as a two-step transition of an electron from the conduction bandto the valence band. The impurity state can absorb differences in momentum between thecarriers, and so this process is the dominant generation and recombination process in siliconand other indirect bandgap materials. The energy is exchanged in the form of lattice vibration,or phonon, exchanging thermal energy with the material. This process is referred as Shockley-Read-Hall (SRH) recombination.

Auger recombination involves three particles: an electron and a hole, which recombineand third particle which overtake the resulting energy excess without moving to anotherenergy band. After the interaction, the third carrier normally loses its excess energy to thermalvibrations. The involvement of a third particle affects the recombination rate so that we needto treat Auger recombination differently from band-to-band recombination. The density of theelectrons or holes, which can receive the released energy from the electron-hole annihilationis very important. Practically Auger recombination takes place only by very high densities ofcarriers. Moreover, the Auger generation process is not easily produced, because the thirdparticle would have to begin the process in the unstable high-energy state.

Recombination at surfaces and interfaces can have a significant impact on thebehavior of semiconductor devices. This is because surfaces and interfaces typically contain alarge number of recombination centers because of the abrupt termination of thesemiconductor crystal, which leaves a large number of electrically active states. In addition,


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the surfaces and interfaces are more likely to contain impurities since they are exposed duringthe device fabrication process.

2.5.6.3 Thermal equilibrium

Each of previoulsy described recombination mechanisms can be reversed leading tocarrier generation rather than recombination. A single expression will be used to describerecombination as well as generation for each of the above mechanisms.

In addition, there are generation mechanisms,which do not have an associatedrecombination mechanism , such as generation of carriers by light absorption or by a high-energy electron or particle beam. These processes are referred to as ionization processes.Impact ionization, which is the generation mechanism associated with Auger recombination,also belongs to this category.

To denote the carriers density in thermal equilibrium we will use 0 index for bothholes and electrons. Then we will have for thermal equilibrium condition:

200 inn p = ( 2.29 )

nnn += 0 ( 2.30 )

p p p += 0 ( 2.31 )

When the difference is positive we talk about an injection of carriers. When thedifference is negative the process is called extraction.

A simple model for the recombination-generation mechanisms states that therecombination-generation rate is proportional to the excess carrier density. It acknowledgesthe fact that no net recombination takes place if the carrier density equals thethermalequilibrium value .

Example 2.1:

For silicon with donor density ND = 10 14 cm -3 we will have injection of electrons andholes n = p = 10 9 cm -3. For this semiconductor material n0 ND = 10 14 cm -3 a p0 ni2 / ND 10 6 cm -3. Then n = n0 + n n0 10 14 cm -3

Solution

p = p0 + p p =10 9 cm -3

When we comparen.p product in equilibrium and under carriers injection we see thatthe injection of majority carriers cause practically no change while injection of minoritycarriers brings the semiconductor far from thermal equilibrium.

n0 p0 = ni2 = 10 14 10 6 cm -3 = 10 20 cm -3

np =10 14 10 9 cm -3 = 10 23 cm -3


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Conclusion: Minority carriers influence thermal equilibrium.

Based on considerations above it is obvious that the resulting expression for therecombination or generation of electrons in a p-type semiconductor can be written as:

n

p

RG

p n

t

n

=

( 2.32 )

p

n

RG

n nt

n

=

( 2.33 )

where the parameter can be interpreted as the average time after which an excessminority carrier recombines.

For each of the different recombination mechanisms the recombination rate can besimplified to this form when applied to minority carriers in a "quasi-neutral" semiconductor.The above expressions are therefore only valid under these conditions.

Because of recombination processes the lifetime of minority carriers is much shorterthan lifetime of majority carriers. In medium quality silicon the lifetime of minority carriers isabout 1s. In case of structurally and chemically pure materials the lifetime can be about 1ms, but when some structural defects or impurities (traps) are present the lifetime can be asshort as 1 ns.

Note: 1. The recombination rates of the majority carriers equals that of the minority carriers

since in steady state recombination involves an equal number of holes and electrons.Therefore, the recombination rate of themajority carriers depends on the excess-minority -carrier-density as the minority carriers limit the recombination rate.

2. Recombination in a depletion region and in situations where the hole and electrondensity are close to each other cannot be described with the simple model and the moreelaborate expressions for the individual recombination mechanisms must be used.

2.6 Transport of carriers

Electric current is caused by motion of free carriers. In principle there are two transportmechanisms of carriers in semiconductor:

1. The motion of carriers can be caused by an electric field. The transport mechanism iscalled ascarrier drift .

2. When the motion is caused by difference in carriers concentration we describe themechanism ascarrier diffusion . This carrier transport mechanism is due to random motion ofthe carriers in consequence of the thermal energy in semiconductor lattice.

When neglecting generation and recombination phenomena the total current in asemiconductor equals the sum of the drift and the diffusion current for both holes andelectrons.


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2.6.1 Carrier drift

In the absence of electric field, the carriers exhibit random motion depending on theirthermal energy. In an electric field the random motion still exists but in addition, there is amotion caused by electrostatic force. The carriers first accelerate until they are moderate dueto collisions and lattice vibrations. Consequently they reach a constant average velocity,v.Holes move in the direction of the applied field because of their positive charge. Electronsmove in the opposite direction.

The carrier motion in the semiconductor in the presence of an electric field is visualizedin Fig. 2.11 .

Because of thermal motion the carriers constantly change direction and velocity due toscattering and do not exactly follow a path along the electric field lines.

Fig. 2.11: The carrier motion in the semiconductor in the presence of an electric field.a) macroscale; b) microscale

A typical drift velocity in semiconductors is not much higher than about 104 cm/s. Thisis much less than the typical thermal velocity at room temperature which reach up to 107 cm/s.

Fig. 2.12: Drift current of holes through p-type semiconductor

Investigating motion of carriers caused by an electric field we can consider the averagevelocity,vd of the carriers. Then, for example, considering the area A with uniform currentflow (seeFig. 2.12 ) for holes the drift current is given as:

Aqpv I d drift p =, ( 2.34 )

Where p is the hole density in the semiconductor.And the hole current density:

d drift p qpv J =, ( 2.35 )


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By low electric field intensity, the velocityvd is proportional to the applied electricfield. Velocity to field ratio is called mobility, . The drift velocity for holes can be nowdescribed as:

E v pd = ( 2.36 )Where p is hole mobility.Then the drift current density for electrons and holes respectively is:

pE q J pdrift p =, ( 2.37 )

nE q J ndrift n =, ( 2.38 )

Here p is electron mobility.The linear relationship between the applied field and the average carrier velocity is valid

only for relatively low electric field - see theFig. 2.13 . In high electric field the carriers aredecelerate due to collisions and lattice vibrations.

Fig. 2.13: Electrical field dependence of mobility for electrons and holes in silicon bytemperature 300 K.

Because the motion of the hole is in fact the hopping of electrons between valencebonds the mechanism of hole motion is more complicated and because of that the holes havealways lesser mobility then electrons. In case of silicon the difference is considerable. Forelectrons the mobility isn = 1300 cm2V-1s-1 while for holes the mobility isp=490 cm2V-1s-1.

The scattering mechanisms are as follows:1. At high electric fields the velocity of carriers saturates because of scattering with

semiconductor lattice.2. Additional scattering occurs at the surface of a semiconductor due to surface or

interface scattering mechanisms.3. Efficient scattering centers are impurities in the semiconductor lattice. Ionized donors

and acceptors are here typical example. Larger impurity concentrations result in a lower

mobility. The mobility of electrons and holes in silicon at room temperature in dependence ondensity of donor and acceptor impurities is shown inFig. 2.14 .


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Fig. 2.14: The mobility of electrons and holes in silicon at room temperature in dependenceon density of donor and acceptor impurities (by temperature 300 K).

Electrical conductivity of semiconductors is given by sum of both electron and hole

current: E pnq J J J pndrift ndrift pdrift )(,, +=+= ( 2.39 )

Here the electrical conductivity is:)( pnq pn += ( 2.40 )

In case of doped (extrinsic) semiconductors the density of majority carriers is given bydopant density while density of minority carriers several order lesser. It means that theminority carriers have practically no influence on electrical conductivity of extrinsicsemiconductor. Consequently, the electrical conductivity for N-type and P-type

semiconductor respectively is: Dn N q = ( 2.41 )

A p N q = ( 2.42 )

2.6.2 Carrier diffusion

Carrier diffusion is based on random motion of the carriers which is due to the thermalenergy in the semiconuctor lattice. When the distribution of carrier density is not uniform this

random motion cause the net flow of carriers from regions where the density is high toregions where the density is low.Diffusion of carriers is controlled by the gradient of the carrier density. For example

such gradient can be consequence of different doping density or in the case of a thermalgradient in the semiconductor device.

In material with a uniform distribution of carrier density this process does not yield anet flow as the carriers replace each other during its random motion.

Diffusion process and corresponding density of electrical current is schematicallyvisualized on theFig. 2.15 .


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Fig. 2.15: The scheme of the diffusion process and corresponding current density forelectrons and holes.

For the particle density,c, the diffusion process is expressed by the first Fick law:

c Dgrad z

c

y

c

x

c

D J =

=

,, ( 2.43 )Where D [m2s-1] is diffusion coefficient.Considering the gradient of carrier density the diffusion current density for electrons

and holes respectively is:

dxdn

qD J ndif n =, ( 2.44 )

dxdp

qD J pdif p =, ( 2.45 )Where D n is diffusion coefficient for electrons and D p is diffusion coefficient for holes.The total current density is obtained by adding the diffusion current and the drift current

for both electrons and holes:

)()(dxdp

Ddxdn

Dqn pqE J pnn p X X ++= ( 2.46 )

Despite that diffusion is caused by thermal energy and the carriers drift is caused by anelectric field

diffusion and drift mechanism are related. Apparent reason there is that the sameparticles and the same scattering mechanisms are involved. Between diffusion coefficient andmobility is a well known relationship, often referred to as the Einstein relation:

T p

p

n

n U q

kT D D ===

( 2.47 )

Here Kt/q is e thermal voltageU T . For temperature of 300 K the value ofU T isapproximately 26 mV.

Formula ( 2.45) is useful for some considerations in theory of semiconductors; seefurther.


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2.6.3 Control questions and examples

1. Two pieces of the same semiconductor (silicon) one of them is p-type and one ofthem is n-type, contain impurities with constant (homogeneous) concentrations

ND = NA. Which of the pieces has greater resistance? Explain.

2. Explain how the Fermi level position depends on the type of semiconductor andon the impurity concentration.

3. Explain how the Fermi level position depends on temperature.

4. a) Draw a typical graph of the temperature dependence of carrier concentration(both electrons and holes) in n type semiconductor with ND >> ni. Chooseappropriate temperature interval to demonstrate the effect of all possibleactivation energies.

5. Explain the effect of impurity atoms and lattice imperfections at semiconductors.

3 PN Junction

Aim of this chapter is to explain operation of semiconductor junctions. Charge build upin the depletion zone in PN junction is explained thoroughly. Junction parameters as built involtage, junction width, barrier and diffusion capacitances and dynamic resistance arederived. Breakdown mechanisms and dynamic properties of PN junction are dealt.

The PN junction is a basic part of all semiconductor electronic devices. One junction

device diode - can be used as a rectifier, as a voltage dependent capacitor and as currentdependent resistor. In addition, they can be used as optoelectronic receivers, solar cells, lightemitting diodes or laser diodes. PN junction can be also used as an isolation structure. BipolarJunction Transistors (BJTs) which are used as amplifier and switch have two junctions.Devices with more than two PN junctions are used in power electronics. Most known amongthem are tyristor and triac. PN Junction can be also found in Field-Effects-Transistors whichare not only in integrated circuits in each PC but also in LCD displays, mobile phones etc.

3.1 Structure and principle of operation

A PN junction consists of two semiconductor regions with opposite doping type - P-type with an acceptor density NA, and N-type with a donor density ND. To simplify theinvestigation we will assume that both regions are doped uniformly and that the transitionbetween regions P and N is abrupt. Such structure is caled an abrupt PN junction (seeFig.3.1a). Usually one side is higher-doped than the other. We will find later that in such a casethe device characteristics are determines mostly by the low-doped region.

In thermal equilibrium no external voltage is applied between the n-type and p-typematerial. The processes involved can be shortly summarized as follows:

1. Because of very high gradient of carrier density (seeFig. 3.1 b). the diffusion ofcarriers must take place. Both electrons and holes which are close to the metallurgical junction diffuse across the junction into the opposite part of the junction where they change to


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minority carriers. This way a thermal equilibrium is disturbed (by injection of minoritycarriers) and as consequence very strong recombination takes place.

Fig. 3.1: Asymmetric PN junction


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2. Carriers which diffuse to the opposite part of the junction leave behind the ionizeddopants namely positive donor ions in N-type and negative acceptor ions in P-type.

Consequently a narrow region around the metallurgical junction is depleted of mobilecarriers. Thus in thermal equilibrium there is positive charge of ionised donors in P-typeregion and negative charge of ionized acceptors in N-type region (seeFig. 3.1 c). Both chargeshave the same absolute value so the junction region is electrically neutral.

3. The charge due to the ionized donors and acceptors causes an electric field (seeFig.3.1d). which in turn causes a drift of carriers in the opposite direction.

4. The diffusion of carriers continues until the drift current balances the diffusioncurrent, thereby reaching thermal equilibrium.

5. Near the metallurgical junction is a narrow region where no carriers are present (as aconsequence of carriers diffusion) but only ionized donors and acceptors. We call this regionthe depletion region. (InFig. 3.1 ) the depletion region begins at x = -xp and reach up to x =xn.)

6. Because of built in charge there is a built-in potential across the depletion region (seeFig. 3.1 e). Sometime this potential difference is called Diffusion Potential.

7. In P-type semiconductor the Fermi level is near the valence band while in N-typesemiconductor the Fermi level is near the conductivity band. On the other hand there is animportant law in thermodynamic which predict that in thermal equilibrium the Fermi energymust have the same level in whole material volume. To fulfil both requirements the banddiagram must be bent near the metallurgical junction (see theFig. 3.1 f). The energydifference is given by built in (diffusion) potential E = q.U D.

3.1.1 Built in potential

In thermal equilibrium the diffusion of carriers is balanced by the drift current becauseof electric field in the depletion region:

0=== pn J J J ( 3.1 )

For electron and hole current density we obtain:

0,, =+=+= dxdn

qDnE q J J J nndif ndrift nn ( 3.2 )

0,, ==+= dxdpqD pE q J J J p pdif pdrift p p ( 3.3 )

Further we use the Einstein relation:

E =

=

=

qDq

dndx

Dn

dndx

kT q n

dndx

n

n

n

n 1 1 ( 3.4 )

And now we can compute the built in potential as:

( )

( )

[ ] ( )( )

U dxkT q n

dndx

dxkT q

dnn

kT q

n Dn

n

n

n= =

= =

+

+

+

+ E 1 ln ( 3.5 )

The limits for integration are:


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A

i p N

nnn

2

)( == ( 3.6 )

Dn N nn ==+)( ( 3.7 )Finally we obtain the built in potential (or diffusion potential) as:

( )U kT q

n nkT q

nn D n p

n

p

= =

ln ln ln ( 3.8 )

And using the equations 3.6 and 3.7 we obtain:

U kT q

N N

n D D A

i

=

ln 2 ( 3.9 )

Example 3.1:

By room temperature ( T = 300 K) is thermal voltage UT = kT/q 0,026 V . Consider aPN junction in silicon ( ni = 10 16 m-3 = 10 10 cm -3). P region is acceptor doped to thedensity NA = 10 21 m-3 = 10 15 cm -3 while N region is donor doped to the density ND = 10 21 m-3 = 10 15 cm -3 .

Using the equ. ( 3.9 ) we can compute built in potential:

= 20

1515

101010ln026,0 DU ( 3.10 )

Owing to logarithmic dependence on density of impurities for most silicon PN junctionsthe built in potentialU D does not much differ from about 0,6 V.

3.2 Electrostatic analysis of a PN junction

The electrostatic analysis of a PN junction provides knowledge about the charge densityand the electric field in the depletion region. It is also required to obtain the capacitance-voltage characteristics of the PN junction.

3.2.1 The full-depletion approximation

To simplify the solution we will make the assumption that the depletion region is fullydepleted and that the adjacent neutral regions contain no charge.

1. Full depletion approximation (which is used in all following considerations) assumesthat the depletion region around the metallurgical junction has well-defined edges. It also

assumes that the transition between the depleted and the quasi-neutral region is abrupt.


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2.We define the quasi-neutral region as the region adjacent to the depletion regionwhere the electric field is small and the free carrier density is close to the net doping density.

3. Coordinates x for depletion region boundary which are derived from the depletionlayer width in the p-type region and from the depletion region width in the n-type regionrespectively arex

pandx

n . In metallurgical junction x = 0. We obtain following relations:

1. for - x p x 0 N A >> n p or p p = -qN A 2. for 0 x xn N D >> nn or pn = qN D 3. for x < -x p a x > xn = 0

Now we can start the electrostatic analysis using an abrupt charge density profile.

3.2.2 Full depletion analysis

To the general analysis we need Poisson's equation:

=

dxdE

where is the charge density. Charge density is a function of the electron density, the holedensity and the donor and acceptor densities. To solve the equation, we have to express theCharge density by means of electron and hole density, n and p . Considering full depletionapproximation we obtain:

A N dxdE

= for - x p x 0 ( 3.11 )

D N dxdE

= for 0 x xn ( 3.12 )

0= E for x -x p a x xn ( 3.13 )

The electric field is obtained by integrating equation ( 3.11 ),( 3.12 ),( 3.13 ):

( ) 1C xqN

dxqN

x E A +

=

=

A for - x p x 0 ( 3.14 )

( ) E x qN dx qN x C D D= = + 2

for 0 x xn ( 3.15 )

The boundary conditions are that the electric field is zero at both edges of the depletionregion: E (-x p) = E ( xn) = 0 (see equ.( 3.13 )).

After substitution we get:

)(1

A p qN xC

= ( 3.16 )


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)(

2 Dn qN xC = ( 3.17 )

And finally for electric field in depletion region:

( ) ( ) x xqN

x p A

+= E for - x p x 0 ( 3.18 )

( ) ( ) x xqN x n D = E for 0 x xn ( 3.19 )

For x = 0 the electric field must be continuous. From this condition follows:

n D

p A x

qN x

qN

= ( 3.20 )

Equation ( 3.20 ) expresses the condition of charge neutrality. The net charge on bothsides of PN junction must be the same except the polarity. Equation ( 3.20 ) can be rewritten:

A

D

n

p

N N

x

x= ( 3.21 )

From this relation follows that the depletion layer extends to the less doped region. Lessdoped region is often labelled as high resistance region or also as base.

To compute the function for potential dependence we will use the relation between thepotential and the electrical field:

E dxdV

= ( 3.22 )

Using equations ( 3.14 ) we obtain:

( ) ( ) ( ) 32)(2 C x xqN

xd x x N q

dx x xV p A

x

x p

A x

x p

++=+==

E for - x p x 0 ( 3.23 )

( ) ( ) +=== x

xn

Dn

D x

xnn

C x xqN

dx x xqN

xd x xV 42)(

2)(

E for 0 x xn ( 3.24 )

The boundary conditions here are again that the electric field is zero at both edges of PN junction and that the potential difference between both edges isU D. When we set V(- x p) = 0then V( xn) = U D. Consequently we obtain the relations for potential in P and N regionrespectively.

( )22

)( x x N q xV p A += for - x p x 0 ( 3.25 )

Dn D U x x

qN xV += 2)(

2)(

for 0 x xn ( 3.26 )


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Now we can compute the width of depletion region. For x=0 the potential must becontinuous:

( ) Dn D p A U x xqN

x N q

+= 22 )(22

( 3.27 )

From equation ( 3.21 ) we can derive:

n A

D p x N

N x =

After substitution x p in equation ( 3.27 ) we obtain quadratic equation for unknown xn :

Dn D

A

n Dn

A

D A U xqN

N

xqN x

N N N q

+==

2222

)(22

.2

( 3.28

1 microelectr devices

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