Solid State & Microelectronics Technology 9789815079876, 9789815079890, 9789815079883

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Solid State & Microelectronics Technology Authored by

Sunipa Roy Electronics & Communication Engineering, Guru Nanak Institute of Technology, Kolkata, West Bengal, India

Chandan Kumar Ghosh Department of Material Science and Technology, School of Materials Science & Nanotechnology, Jadavpur University, Kolkata, West Bengal India

Sayan Dey Electrical Engineering Department, Columbia University, New York, USA

&

Abhijit Kumar Pal Applied Electronics and Instrumentation, Future Institute of Engineering and Management, Kolkata, West Bengal, India

Solid State & Microelectronics Technology Authors: Sunipa Roy, Chandan Kumar Ghosh, Sayan Dey and Abhijit Kumar Pal ISBN (Online): 978-981-5079-87-6 ISBN (Print): 978-981-5079-88-3 ISBN (Paperback): 978-981-5079-89-0 © 2023, Bentham Books imprint. Published by Bentham Science Publishers Pte. Ltd. Singapore. All Rights Reserved. First published in 2023.

BSP-EB-PRO-9789815079876-TP-394-TC-12-PD-20230630

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CONTENTS PREFACE ................................................................................................................................................... i CHAPTER 1 FUNDAMENTALS OF SEMICONDUCTOR PHYSICS ................................................ INTRODUCTION ............................................................................................................................. Classification, Crystal Structure and Miller Indices of Semiconducting Materials .................................................................................................................................... Basic Quantum Mechanics for Semiconductors ........................................................................ Sommerfeld’s Free Electron Model ........................................................................................... Kronig – Penney Model: The Origin of the Band Gap .............................................................. Bloch’s Theory: Electrons in Three-Dimensional Periodic Potential and Band Structure..................................................................................................................................... Band Structure of Face-Centered Cubic Crystals....................................................................... BAND, NUMBER OF STATES IN A BAND, BAND FILLING ................................................... DIRECT AND INDIRECT BAND GAP SEMICONDUCTOR ..................................................... Intrinsic Semiconductor: Density of States, Fermi – Dirac Statistics and Fermi Energy ....................................................................................................................................... EXTRINSIC SEMICONDUCTOR .................................................................................................. CARRIER’S CHARACTERISTICS: CHARGE, EFFECTIVE MASS ....................................... CARRIER TRANSPORT PARAMETERS: DRIFT, MOBILITY, CARRIER LIFETIME, SCATTERING ............................................................................................................. CHARGE CARRIER UNDER ELECTRIC FIELD: BOLTZMANN TRANSPORT EQUATION .............................................................................................................. CHARGE CARRIER TRANSPORT DUE TO CONCENTRATION GRADIENT: DIFFUSION ....................................................................................................................................... HALL EFFECT: DETERMINATION OF TYPE CARRIER AND ITS DENSITY ........................................................................................................................................... CONCLUSION .................................................................................................................................. QUESTIONS ..................................................................................................................................... REFERENCES ................................................................................................................................. CHAPTER 2 FUNDAMENTALS OF p – n JUNCTION ........................................................................ INTRODUCTION ............................................................................................................................. Physics of p – n Diode: Depletion Region, Built – in –Potential ............................................... Current-voltage Characteristics of Biased p – n Junction .......................................................... Zero Biased Condition (𝐕𝐃 = 𝟎𝐕) .............................................................................................. Forward Biased Condition (𝐕𝐃 > 𝟎 𝐕) ....................................................................................... Reverse Biased Condition (𝐕𝐃 < 𝟎 𝐕)........................................................................................ Contact for p – n diode: choice of suitable material ................................................................... Resistance of p – n Diode: Static, Dynamic, and Average Ac Resistance ................................. PIECEWISE-LINEAR ANALYSIS OF DIODE CHARACTERISTIC ....................................... CAPACITANCE OF P – N JUNCTION UNDER THE REVERSE BIASED CONDITION: TRANSITION CAPACITANCE ............................................................................ CAPACITANCE OF P – N JUNCTION UNDER THE FORWARD BIASED CONDITION: DIFFUSION CAPACITANCE ............................................................................... BREAK DOWN OF REVERSE BIASED P – N JUNCTION: AVALANCHE AND ZENER MECHANISM ........................................................................................................... DIODE SWITCHING ....................................................................................................................... CONCLUSION ................................................................................................................................. QUESTIONS ..................................................................................................................................... REFERENCES ................................................................................................................................. CHAPTER 3 METAL SEMICONDUCTOR CONTACTS SCHOTTKY DIODES ............................. INTRODUCTION ............................................................................................................................. Metal-Semiconductor Junction .................................................................................................. METAL SEMICONDUCTOR JUNCTION .................................................................................... CONCLUSION ................................................................................................................................. WIDTH OF THE DEPLETION REGION IN A SCHOTTKY BARRIER

1 1 3 11 15 16 20 26 29 31 32 36 42 44 49 53 55 56 57 58 59 59 59 65 66 67 73 74 74 79 81 85 86 88 89 89 90 91 91 91 96 104

DEVICE ............................................................................................................................................ IDEALITY FACTOR ...................................................................................................................... METAL SEMICONDUCTOR/SCHOTTKY DIODE ................................................................... CONCLUSION ................................................................................................................................. REVIEW QUESTIONS ................................................................................................................... REFERENCES .................................................................................................................................

104 107 112 118 118 121

CHAPTER 4 JUNCTION FIELD EFFECT TRANSISTOR .................................................................. 122 INTRODUCTION ............................................................................................................................. 122 FET Fundamentals .................................................................................................................... 122 The Field Effect Transistor (FET) can be Broadly Classified into the Following Categories ................................................................................................................................. 124 Constructional Features of N- Channel JFET............................................................................ 125 Midpoint Bias............................................................................................................................ 140 Ideal Current-voltage Relationship ........................................................................................... 140 SHORT ANSWER TYPE QUESTIONS ................................................................................. 150 FILL IN THE BLANKS ........................................................................................................... 160 TRUE/FALSE ........................................................................................................................... 161 REFERENCES .......................................................................................................................... 161 CHAPTER 5 METAL OXIDE FIELD EFFECT TRANSISTOR (MOSFET) ...................................... 162 INTRODUCTION ............................................................................................................................. 162 MOS Diode ............................................................................................................................... 163 FABRICATION OF N-MOS ............................................................................................................ 163 Selection of Substrate ............................................................................................................... 164 Deposition of Photoresist .......................................................................................................... 164 Lithography .............................................................................................................................. 165 Window Formation ................................................................................................................... 165 Deposition of the Gate Oxide and Polysilicon .......................................................................... 165 Formation of n+ Region and Metal Contact .............................................................................. 166 Observation ............................................................................................................................... 168 POLYSILICON ................................................................................................................................ 169 THE IDEAL MOS CAPACITOR ................................................................................................... 174 FLAT BAND ..................................................................................................................................... 176 IDEAL MOS DIODE THRESHOLD VOLTAGE VT ................................................................... 194 LONG CHANNEL MOSFET .......................................................................................................... 203 QUESTIONS ..................................................................................................................................... 246 CONCLUSION ................................................................................................................................. 247 REFERENCES ................................................................................................................................. 247 CHAPTER 6 SEMICONDUCTOR DEVICES......................................................................................... 249 INTRODUCTION ............................................................................................................................. 249 PIN Diode .................................................................................................................................. 249 Microwave Switches.................................................................................................................. 251 Photodetector ............................................................................................................................. 252 Tunnel Diode (Esaki Diode) ...................................................................................................... 252 CONCLUSION  ................................................................................................................................. 278 REVIEW QUESTIONS ................................................................................................................... 278 REFERENCES ................................................................................................................................. 279 CHAPTER 7 SILICON ................................................................................................................................. 280 INTRODUCTION ............................................................................................................................. 280 Crystal Systems ......................................................................................................................... 281 Crystal Systems and Miller Indices ........................................................................................... 282 Types of Crystal Structures........................................................................................................ 282 Face-centred Cubic Crystal Structure ........................................................................................ 282 CONCLUSION ................................................................................................................................. 299 POINTS TO REMEMBER .............................................................................................................. 299 REVIEW QUESTIONS ................................................................................................................... 300 REFERENCES ................................................................................................................................. 301

CHAPTER 8 OXIDATION …………………………………………………………………………………………………… 302 INTRODUCTION ............................................................................................................................. 302 Thermal Oxidation ..................................................................................................................... 303 Growth Mechanism and Kinetics of Oxidation.......................................................................... 306 Thin Oxide Growth .................................................................................................................... 311 Properties of Oxides Oxide-induced Defects ............................................................................. 311 Characterization of Oxide Films ................................................................................................ 312 CONCLUSION ................................................................................................................................ 319 POINTS TO REMEMBER ............................................................................................................. 319 REVIEW QUESTIONS ................................................................................................................... 320 REFERENCES ................................................................................................................................. 321 CHAPTER 9 DIFFUSION…………..…………………………………………………………………………………………… 322 INTRODUCTION ............................................................................................................................. 322 Mechanisms of Diffusion ............................................................................................................... 324 Interstitial Diffusion ....................................................................................................................... 324 Vacancy Diffusion ......................................................................................................................... 325 Steady-State Diffusion and Diffusion Flux .................................................................................... 325 Diffusion in Semiconducting Materials ......................................................................................... 328 CONCLUSION ................................................................................................................................ 329 POINTS TO REMEMBER ............................................................................................................. 330 REVIEW QUESTIONS ................................................................................................................... 331 REFERENCES ................................................................................................................................. 331 CHAPTER 10 ION IMPLANTATION…………..………………………………………………………………………… 332 INTRODUCTION ............................................................................................................................. 332 Range Theory Ion Stopping ........................................................................................................... 332 Damage .......................................................................................................................................... 336 Channeling ..................................................................................................................................... 337 Recoils ........................................................................................................................................... 339 Instrumentation .............................................................................................................................. 339 CONCLUSION ................................................................................................................................ 344 POINTS TO REMEMBER ............................................................................................................. 345 REVIEW QUESTIONS ................................................................................................................... 346 REFERENCES ................................................................................................................................. 346 CHAPTER 11 MEMS IN IMPROVED EFFICIENCY…………..……………………………..…………………… 347 INTRODUCTION ............................................................................................................................. 347 MEMS Materials ............................................................................................................................ 348 Ceramic Material Fabrication Process ........................................................................................... 351 Fabrication Method – Micromachining.......................................................................................... 355 CONCLUSION ................................................................................................................................ 367 REVIEW QUESTIONS ................................................................................................................... 367 REFERENCES ................................................................................................................................. 368 CHAPTER 12 LITHOGRAPHY……………………………………………………………………………………………… 369 INTRODUCTION ............................................................................................................................. 369 Optical Lithography ....................................................................................................................... 370 Types of Photoresists: the Negative and Positive Photoresists....................................................... 373 Photolithography Process ............................................................................................................... 375 Some Advanced Lithographic Techniques .................................................................................... 380 CONCLUSION ................................................................................................................................ 388 REVIEW QUESTIONS ................................................................................................................... 389 REFERENCES ................................................................................................................................. 389 SUBJECT INDEX ....................................................................................................................................... 390

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PREFACE Solid-state and Microelectronics technology attempts to fill the gap between a general solid-state physics book and real-life application by providing detailed explanations of the electronic, vibrational, transport and optical properties of semiconductors. The approach is physical and instinctive rather than prescribed. The enhanced application of semiconductors to the different electronic industries helped to explore various aspects of microelectronics technology which was made possible with the availability of solid-state devices known for their versatile applications. This technological advancement also demanded the proper understanding of new device physics as the properties of materials alters significantly at this reduced dimension. These dimensions are comparable to the electron wavelength of motion, and hence the device characteristics are governed by the confinement of the electron wave function, commonly referred to as the quantum confinement effect. For the purpose of commercialization, integration of different electronic components on the same platform is very much essential to develop an integrated device using a standard IC fabrication process. Thus, low power, low cost, highly sensitive and miniaturized electronic system on chip using silicon is possible. This book consists of thirteen chapters. The most apparent package for PART I consists of chapter 1 to 6, and PART II consists of chapter 7 to 12. The goal was to give a brief idea about the microfabrication technology needed to fabricate solid-state devices through part II. Chapter 1 provides an introduction to Semiconductor Physics fundamentals. Basic quantum mechanics have been introduced in order to explain the fundamental properties of semiconducting materials. In this context, Sommerfeld’s free electron theory has been considered. Although this model corroborates with a few experimental observations, but can’t differentiate between semiconductors, insulators and metal. Then Kronig – Penney, who successfully explains the deviation on the basis of the band concept, has been considered. Followed by Kronig – Penney model, Bloch’s theory has been introduced, and it well explains the origin of conduction and valence bands. From this concept, different types of semiconducting materials, e.g., direct and indirect band gap semiconductors, n- and p-type semiconductors, etc., have emerged. Here,

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properties of charge carriers, such as their charge, effective mass, etc., have also been discussed. Knowing these parameters, conductivity expression and related scattering phenomena influencing conductivity have been briefly elaborated. Chapter 2 briefly discusses the fundamentals of the p – n junction, the electric field across the junction, equilibrium carrier concentration on each side, etc. Expression of built-in potential in terms of carrier density has been derived. Magnitudes of the current under zero, forward, and reverse biased conditions have been calculated. Unlike a resistor, p – n junction corresponds to static, dynamic and average ac resistance, and they have been briefly discussed here along with protocol to examine their values. These p – n junctions also exhibit capacitance, namely transition capacitance and diffusion capacitance. Herein, these parameters and their relevance in terms of current-voltage characteristics of p – n junction and applicational aspects, have been elaborated. Chapter 3 discusses metal-semiconductor contacts. Schottky and Ohmic contact with a detailed band diagram has been presented. I-V characteristics are also given to interpret the contact behaviour. Chapter 4 deals with JFET and its I-V characteristics expressed in a detailed manner with necessary equations. JFET parameters are another important factor to understand the characteristics of it which have been mentioned. Small signal model of JFET illustrated in a very lucid manner. Chapter 5 discusses the fabrication of MOSFET and its principle operations based on the concept of metal-oxide-semiconductor technology. Further the discussion is focused on the details mathematical modelling of MOS capacitors, device characteristics and the process of channel length modulation and its application. The conversation is continuing on the concept of CMOS technology and its combination with the transistor – the BiCMOS technology. Chapter 6 discusses advanced semiconductor devices where Semiconductor resistivity can be changed by the incorporation of an electric or magnetic field, by revelation to light or heat, or by mechanical distortion. The doping of silicon significantly modulates the characteristics of semiconductors to have different types of devices like LED, Tunnel diode, Solar cells and many more, which have been discussed in this section. Chapter 7 introduces silicon as an electronic material for microelectronic device fabrication. It discusses, in-depth, the various aspects of crystalline silicon, wafer

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manufacturing and their identification techniques. Finally, the various steps involved in a microfabrication process are also discussed. From this chapter, the readers are expected to learn how to process a silicon wafer successfully and proceed with customized device fabrication. Chapter 8 introduces the readers to the oxidation process focusing on silicon. It establishes the importance of oxidation in a silicon process and discusses in depth the thermal oxidation process and its growth mechanism. Towards the end, a detailed discussion of the oxide film characterization and its properties has been provided. After reading this chapter, the readers are expected to have a sound knowledge of the oxidation process as a whole and its significance in the silicon processing industry. Chapter 9 introduces the readers to the diffusion process which is used for doping intrinsic silicon. It discusses in depth Fick’s Law of diffusion and the different types of diffusion that may be observed. Towards the later part of the chapter, the process of diffusion in semiconductors is discussed in detail, explaining the diffusion-assisted doping process commonly employed for semiconductors, especially silicon. Chapter 10 discusses in depth the ion implantation process of silicon processing technology. The chapter gives an in-depth insight into various aspects of the ion implantation process and ion-implanted silicon systems commonly encountered in a silicon process. The readers, after reading this chapter, will have a sound understanding of the ion implantation process and its various aspects. Chapter 11 represents the concept of MEMS technology. Fabrication technique, including bulk and surface micromachining with CMOS compatibility issues, has been elaborated. The most important etch-stop techniques have also been elaborated in this chapter.

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Chapter 12 provides the idea about photo-resist, and its properties. It also narrates some advanced lithographic techniques, including the most common one, optical lithography. Sunipa Roy Electronics & Communication Engineering, Guru Nanak Institute of Technology, Kolkata, West Bengal, India Chandan Kumar Ghosh Department of Material Science and Technology, School of Materials Science & Nanotechnology, Jadavpur University, Kolkata, West Bengal India Sayan Dey Electrical Engineering Department, Columbia University, New York, USA & Abhijit Kumar Pal Applied Electronics and Instrumentation, Future Institute of Engineering and Management, Kolkata, West Bengal, India

Solid State & Microelectronics Technology, 2023, 1-58

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CHAPTER 1

Fundamentals of Semiconductor Physics Abstract: In recent times, crystalline semiconductors have played a major role in device fabrication for all purposes, where crystal structure plays the most crucial role. Herein, all kinds of fundamental unit cell structures have been briefly discussed, whereas Miller indices have been introduced to illustrate the orientation of the crystal structures. Basic quantum mechanics have been introduced in order to explain the fundamental properties of semiconducting materials. In this context, Sommerfeld’ free electron theory has been considered. Although this model corroborates with a few experimental observations, but can’t differentiate between semiconductors, insulators and metals. Then Kronig – Penney, which successfully explains the deviation on the basis of the band concept, has been considered. Following the Kronig – Penney model, Bloch’s theory has been introduced, and it well explains the origin of conduction and valence bands. From this concept, different types of semiconducting materials, e.g., direct and indirect band gap semiconductors, n- and p-type semiconductors, etc., have emerged. Here, properties of charge carriers, such as their charge, effective mass etc., have also been discussed. Knowing these parameters, conductivity expression and related scattering phenomena influencing conductivity have been briefly elaborated.

Keywords: Band, Crystal structure, Fermi – Dirac statistics, Scattering, Semiconductor. INTRODUCTION A semiconductor is a sub-group of materials with electrical conductivity between metal and insulators. The parameter which differentiates semiconductors from metal and insulator others is the band gap of reasonable value. Unlike metal and insulators, the electrical conductivity of semiconducting materials is very sensitive to temperature and easily tunable, and thus they have been attracted due to their opportunities in various devices. Though the study of semiconductors started in the year 1930, the real essence of semiconducting materials came when the transistor was invented in the year 1947. Later, with the invention of silicon bipolar transistors, semiconductor devices grow rapidly. After successful integration with computers in the late 1970 – 1980s, the demand for semiconductor materials with memory effect had increased dramatically. Late in the 1960s and 1970s, Leo Esaki and his peer group developed a new crystal growth method, molecular beam epitaxy, to develop semiconducting materials, which opened new concepts of quantum well, superlattice based semiconducting devices [1]. In the late 1980s, the concept of high-mobility transistors came up and triggered a wide variety of applications, including superlattice-based receivers for satellite broadcasting. Sunipa Roy, Chandan Ghosh, Sayan Dey and Abhijit Kumar Pal All rights reserved-© 2023 Bentham Science Publishers

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Besides Si-based large-scale integrated memory, metal-semiconductor field-effect transistors (MESFET) based GaAs had emerged as an important material for mobile phones in the 1990s. Till then, GaAs became a boom for next-generation technology in the 21st century. After the discovery of the quantum Hall effect, also known as the integer quantum Hall effect, two-dimensional electron gas systems of semiconducting materials opened new applicational opportunities. During the last decade, the fractional quantum Hall effect came into the picture with its several scopes. Recently, we have technologies to fabricate various micro- and nanostructures of semiconducting materials that give several new quantum phenomena like Aharonov-Bohm, ballistic transport, electronic interference, etc. In recent times, many technologies have emerged out of these phenomena. It is very well known to all of us that semiconductors play an important role in information technology and all kinds of electronic gadgets around us, such that we can’t talk about anything without semiconductor devices. Herein, it may be stated that the performances of semiconductor-based devices and related technologies significantly depend on several physical parameters of the semiconducting materials. Therefore, to explore different technological aspects and devices of these semiconducting materials, we have to understand their basic physics. Early initiatives were taken around the 1950s to understand the optical and electrical properties of semiconducting materials, and it was resolved during the 1960s that all the fundamental properties of the semiconducting materials were related to energy band structures which is nothing but energy (E) – wave vector (k) relation, called E(k) diagram, for the electrons within semiconducting materials. In fact, this band structure allows us to differentiate between an insulator, a semiconductor and a metal. In this context, it may be stated that the microscopic properties of electrons are conveniently described by band structure. During 1955s, cyclotron resonance experiments carried out on semiconducting materials, gave the first experimental information about the E(k) diagram of the valence band, i.e., the lowest occupied energy levels of semiconducting materials. This experiment also revealed the existence of degeneracy of valence bands and successively concepts of two types of holes, i.e., heavy-hole and light-hole bands. In this context, it may also be noted that there were several controversies during the early stage of the development of semiconductor-based devices. For example, when transistors were invented, it was unclear whether germanium was direct or indirect band gap semiconducting material. Later on, researchers adopted versatile tools for detailed, in-depth information about the band structure of semiconducting materials. In the present time, several optical and electronic spectroscopic techniques like UV-Vis spectroscopy, photoluminescence spectroscopy, cathodoluminescence spectro scopy, x-ray photoelectron spectroscopy etc., are used to obtain information about band structure experimentally. Herein, it may be stated that in addition to

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experimental processes, several empirical pseudopotential and quantum mechanical perturbation methods had been simultaneously developed for a theoretical understanding of the band structure of the semiconducting materials. Among them, k·p and Kronig – Penny technique, developed during late 196os, became the most popular. Presently, density functional theory is being widely used to calculate band structure and related optical, electrical and magnetic properties of semiconducting materials. Herein, it may be stated that the band structure of any semiconducting material is crucial as it determines every property, hence every applicational possibility of the semiconducting materials depends on the band structure. Therefore, we have to understand the fundamental of the band structure of semiconducting materials, which is mainly carried out by solving Schrödinger’s equation. However, solving Schrödinger’s equation to understand electron behavior in a semiconducting material with ~ 1023 atoms/cm3 is very complicated. To simplify this tedious job, symmetries of semiconductors that involve translational, rotational and reflection are being utilized. Here, Group theory is the tool that facilitates the task. As our primary objective of this chapter is to understand band structure and related electronic properties, we must have initial knowledge based on the symmetry of the semiconducting materials and group theory. So primarily, these will be discussed, and then band structure, and electronic behavior will be developed. Classification, Crystal Structure and Miller Indices of Semiconducting Materials Like, other materials, semiconductors consist of a large number of the same or different types of atoms. However, depending on atomic arrangements, they are broadly classified into three categories: amorphous, single crystalline and polycrystalline. In amorphous semiconductors, there is no long-range ordering of atoms within the materials, and one part has a completely different look from the other part (Fig. 1a). Atoms are arranged with lone ordering in a single crystalline material in all three directions (Fig. 1b). Unlike amorphous, all sections of a single crystalline material are identical. Polycrystalline materials lie in between amorphous and single crystalline materials. Briefly, it comprises crystalline subsections that are disjoined relative to one another (Fig. 1c). Herein, it may be stated all three kinds of the same materials have their own advantages suitable for particular applications [1]. For example, amorphous Si has high-concentration dangling bonds; those in the presence of an electric field produce electron-hole pairs, hence they are very useful for varies switching devices like liquid crystal displays (LCD). In addition, amorphous Si exhibits excellent uniformity during the thin film, which also makes them beneficial to enhance the brightness of LCD

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panels. In contrast, polycrystalline Si is beneficial for metal – oxide – semiconductor – field – effect – transistors (MOSFETs). Though amorphous semiconductors have several advantages for many applications, however, crystalline semiconductors are mostly used, and in this discussion, we would consider crystalline semiconductors only.

Fig. (1). Amorphous (a), single crystalline (b) and polycrystalline (c) semiconducting materials.

Though the use of Si as semiconducting material was started long ago, presently, several materials have been identified as semiconductors suitable for various applications. A part of the periodic table, represented in Table 1, reveals the elements with the semiconducting property. Briefly, Si and Ge from group IV, number of group III – V compounds, such as GaAs, GaN, AlAs, etc., and II-VI compounds, e.g., ZnO, ZnS, CdS, etc., several interesting trends can be noticed here. For example, all group IV semiconductors exhibit pure covalent character, and the band gap decreases with the increasing size of the atoms. Herein, C has the highest band gap ~ 6 eV, while Sn exhibits the lowest band gap, close to zero. For this reason, Sn often shows semi-metallic characters. Among group III – V semiconductors, the band gap has been found in the following order AlP > GaP > AlAs > GaAs, i.e., the band gap decreases if we go down the periodic table. A similar trend has been observed for group II-VI semiconductors. In addition, it has also been observed that the band gap increase with increasing ionicity of the materials. Table 1. Selected portion of the periodic table. Group Element

II -

III

IV

V

VI

B

C

N

O

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(Table 1) cont.....

-

Al

Si

P

S

Zn

Ga

Ge

As

Se

Cd

In

Sn

Sb

Te

Generally, single crystalline and polycrystalline semiconductors correspond to periodic atomic arrangement which can be described by a smallest unit that reproduces the whole system on repetition. This smallest portion, the building block of whole system, is defined as unit cell. The shape and size of the unit cell is described by three vectors a, b and c, called crystallographic axes of the unit cell. These crystallographic axes have well-defined lengths (a, b, c) and angle between them (α, β, γ), called lattice constants or lattice parameters of the unit cell. However, the choice of unit cell is not unique; it may have a different form. As an example, we consider one-dimensional periodic arrangement of atoms, illustrated in Fig. (2). In this case, unit cell may be considered as marked by red marks. In reality, all the semiconducting materials are extended in three dimensions; hence three-dimensional unit cell structure is constructed. Unit cell may not be primitive (smallest unit cell) always. In some cases, it is easier to deal with a bigger unit cell with orthogonal sides instead of the primitive unit cell with non-orthogonal sides.

Fig. (2). One-dimensional periodic arrangement of atoms, two different unit cell configurations are shown in red marks.

In general, periodic arrangements of atoms of a three-dimensional semiconducting material are expressed by two important parameters: basis and lattice points. Basis, defined as the minimum set of atoms to construct a unit cell, can be of anything ranging from atoms to large molecules like deoxyribonucleic acid / ribonucleic acid, while point lattice, a set of points (R), generated from three non-coplanar primitive translational vectors a, b and c according to 𝐑 = n1 · 𝐚 + n2 · 𝐛 + n3 · 𝐜, where, n1 , n2 and n3 are the integers, expresses translational symmetry and the whole system is being generated from basis set after translating them by multiples

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of primitive vector of their combination. Actually, lattice points are a set of imaginary lattice points having fixed relation with atoms of crystals in space and are regarded as the skeleton on which the actual crystal is built. Here, actual atoms composing the crystal and their periodic arrangements in space are ignored. This set of lattice points can be constructed as follows: we imagine a space divided with three sets of parallel and equally spaced planes. Such imagined planes would divide space into a set of cells with identical shapes, sizes and orientations. During intersection, these planes would produce a set of lines that would intersect further at some fixed points (shown in Fig. 3). These so-formed points have an identical environment and are termed lattice points.

Fig. (3). Schematic representation of lattice points.

By dividing space by these three sets of planes, it is possible to form unit cells of various shapes and sizes that, in consequence, have different axial lengths and angles, and each lattice point is located at the corner of the unit cell. It has been observed such permutation produces seven different types of unit cells corresponding to seven crystal systems in which the crystal structure of all semiconducting materials can be classified. Later, French crystallographer Bravais identified that a unit might contain more than one lattice point in the unit cell; they may be in the center or face of the unit cell. Based on this, Bravais categorized crystal structures into fourteen classes. After him, such lattice points are named Bravais lattice points. Those satisfy all the requirements of periodicities and related symmetry operations (translation, rotation and reflection) that the crystal structures undergo. Depending on the number of atoms, unit cells are classified into primitive and non-primitive unit cells, and they are indexed with different symbols. In Table 2, we have described all fourteen Bravais lattice points and consequently shown in Fig. (4).

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Fig. (4). Bravais lattice structure. Table 2. Description of crystal system and Bravais lattice.

System

Axial lengths and angles

Bravais lattice

Lattice Symbol

Simple

P

Body-centered

I

Face-centered

F

Simple

P

a=b≠c, α=β=γ 900

Body-centered

I

Three unequal axes at right angles

Simple

P

Body-centered

I

Base-centered

C

There are equal axes at right angles Cubic

a=b=c,α =β=γ=900 Three axes at right angles, two equal

Tetragonal

Orthorhombic

a≠b≠c, α=β=γ = 900

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(Table 2) cont.....

Face-centered

F

Simple

R

Simple

P

Simple

P

Body-centered

C

Simple

P

Three equal axes, equally inclined Rhombohedral

a = b =c , α = β = γ ≠ 900 Two equal coplanar axes at 1200,

Hexagonal

third axis at right angles a = b ≠ c , α = β = 90 0 , γ = 1200 Three unequal axes, One pair not at right angles

Monoclinic

a ≠ b ≠ c , α = γ = 90 0 ≠β

Three unequal axes, unequally inclined Triclinic

and none at right angles a≠b≠c, α≠β≠γ ≠ 900

The direction of any line within a lattice generally is described by first drawing a line parallel to it and passing through the origin and then specifying the coordinates of any point on the line through the origin. Let’s consider a line through the origin of the unit cell and any point with coordinates u v w (these may not be an integer). Then this set of coordinates u v w, written within square bracket [u v w], are indices to be used to denote the direction of the line. In this context, it may be stated that whatever may be the value of u v w, it is possible to obtain the smallest integers by

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either multiplication or division. As an example, [½ ½ 1], [1 1 2] and [2 2 4] denote the same direction, but [112] is the most preferred one.

Fig. (5). Miller indices of planes, marked with lines, are (a) (100), (b) (001), (c) (010) and (d) (111).

Not only lines but also planes, called crystallographic planes, are needed to be specified very often. This was first introduced by English crystallographer William Hallowes Miller in 1939. According to him, the plane is tilted with respect to crystallographic axes and as these axes represent the reference frame, therefore the orientation of the plane, most conveniently, is given the distances, measured from the origin, at which it intercepts the three axes. Briefly, by expressing these distances in terms of fractions of the axial length, we obtain the number independent of the particular axial lengths, involved in the given lattice. Here, the reciprocal of the intercepts, termed Miller indices, refer orientation of the plane through the lattice. Miller indices, generally denoted by h k l, are written as (h k l). Better still, we consider Miller indices of a plane to be represented by (h k l), then the intercepts of 1/h, 1/k, 1/l with the axes, and, if the axial lengths are a, b, c, then the actual intercepts on the plane are a/h, b/k, c/l. As an example, Miller indices of the planes (shown in Fig. (5)) corresponding to a simple cubic unit cell, marked with a line, are (100), (001), (010) and (111). Herein, it can be stated that the concept of Miller indices is a very useful concept to describe electronic properties, particularly electronic conductivity and optical absorption of semiconductor crystals with anisotropic lattice parameters. Conventionally, other representations stand here: [h k l], and {h k l} represent direction, a family of directions, and a family of planes. In this context, it may be stated that most of the technologically used semiconductors, such as Si, Ge, GaAs etc., correspond to diamond and zincblende in which each atom is surrounded by four equidistant nearest atoms. Lattices are considered two interpenetrating face-centered cubic lattices. Among several

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semiconductors, Si being widely used in integrated circuits (IC), and central processing unit (CPU) dominates in the commercial market. Within of Si crystal, each Si atom is surrounded by four Si atoms, whereas GaAs are widely utilized in laser diodes, IC due to their superior electron transport and high optical absorption coefficient consisting of one sublattice, Ga and other of As. It has been noticed that unit cells of GaAs may be generalized to other groups III – V-based semiconductors like GaN, AlN, AlAs etc. In recent times, a completely different set of semiconductors in the form ZnO, CdS, ZnS etc., also have wide applications. These are generalized as II-VI groups of semiconductors with wurtzite structure. Commonly, wurtzite structure gets formed from two interpenetrating hexagonal close-packed lattices with two-sublattice composed of cation (Zn, Cd etc.) and anion (O, S etc.) respectively [2]. Wurtzite structure corresponds to a tetrahedral arrangement of four equidistant nearest neighbors, similar to the zincblende structure. Example 1: Direct lattice vectors of a simple cubic unit cell are a = ax, b = ax and c = az. Example 2: Direct lattice vectors of a face-centered cubic unit cell are 𝐚 = 1 2

a (𝐲 + 𝐳), 𝐜 =

1 2

1 2

a (𝐱 + 𝐲), 𝐛 =

a (𝐳 + 𝐱).

Example 3: Direct lattice vectors of a body-centered cubic unit cell are 𝐚 = 𝐛=

1 2

a (−𝐱 + 𝐲 + 𝐳), 𝐜 =

1 2

1 2

a (𝐱 + 𝐲 − 𝐳),

a (𝐱 − 𝐲 + 𝐳).

Example 4: Miller indices of planes parallel to xy, yz and zx planes are (0 0 1), (1 0 0) and (0 1 0), respectively.

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Basic Quantum Mechanics for Semiconductors Understanding of the Atomic Structure In order to understand the fundamentals of semiconducting materials, let’s start with the basic construction of an atom. In 1911, Rutherford found that an atom consists of a nucleus of positive charge surrounding a positive core, with electrons having negative charge moving around. As a specific case, we consider hydrogen atoms constituting a positively charged nucleus, called a proton, and a sole electron. In this case, the charge of the nucleus and electron should be the same, but of opposite sign such that the hydrogen atom maintains charge neutrality. Due to the opposite charge, electron of the hydrogen atom is always attracted by the positive core. It can be shown that the resultant path generally appears as circle, similar to the planetary picture of the planet about the sun in classical mechanics. In this classical picture, an electron is experiencing a force, thus, it would always be accelerated. According to classical electromagnetic law, an accelerated electron would radiate energy; eventually, the electron would fall into the nucleus. Therefore, in spite of early prediction, the Rutherford model doesn’t provide atomic stability. Later, Bohr resolved the stability problem with the following three postulates: (i) Electrons only possess certain discrete energy levels. While they are at these levels, they don’t radiate and remain stationary. (ii) During the transition from one stationary to another stationary state, they emit radiation with energy equal to the energy difference between two states divided by Planck’s constant. (iii) Condition for the stationary state is given by the condition that angular momentum of the electron in stationary states should be an integral multiple of Planck’s constant divided by 2π, known as reduced Planck’s constant. Combining all these postulates, the energy of nth level, Wn , is found to be Wn = −

𝑍e4

1

8ħ2 ϵ20 n2

, where ‘Z’ and ‘e’ are the atomic numbers and electronic charge,

respectively. Atomic number Z generally refers to the number of electrons in an atom. And ħ and 𝜖0 are the reduced Planck’s constant (ħ = h/2π) and free space permittivity. For each value of n, a pictorial representation of energy levels, known as energy – a level diagram, is drawn. The lowest energy level is called ground level, while others are excited level. Herein, it may be stated that if the electron is gained more and more energy, it moves away from stationary states into excited states and after receiving sufficient energy, it gets detached from the field of the nucleus. The minimum energy required in this process, is called ionization potential; hydrogen atom ionization potential is found to be 13.06 eV. The most

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convenient way to supply ionization is a collision with other electrons or light. In this context, it may be stated that in spite of early success, the Bohr model can’t explain fine spectral properties. Later, Schrödinger’s equation was utilized here, and those properties have been successfully explained. The solution of Schrödinger’s equation includes three quantum numbers to specify the wave function of an electron space as follows: Principal quantum number: n = 1, 2, 3 … Azimuthal quantum number l = 0, 1, 2, ….. , (n – 1) and Magnetic quantum number: ml = 0, ±1, ±2, ….. , ± l After the discovery of spin of an electron, spin quantum number (±1/2) is used for a complete representation of the electron. In this picture, an electron with a given n is said to belong in the same electron shell, commonly represented by the letters K, L. M, N ….. for n = 1, 2, 3, 4 … levels. Then each shell is subdivided into several subshells, denoted by s, p, d, and f according to the value on l = 0, 1, 2, 3, ….respectively. Each sub-shell is distinguishable only in the presence of a magnetic field and corresponds to two energy levels in cases of parallel and anti-parallel orientation of the electrons in the presence of a magnetic field. As an example, we consider n = 1, which gives only l = 0 states only, and the state doesn’t split by a magnetic field. If we consider the spin of the electrons, then this state is occupied by two electrons with an opposite spin quantum number. They are called 1s state. Such a way of arranging electrons in various shells is known as electronic configuration. Example 5: Electronic configuration of C atom having Z = 6: 1s22s22p2 Example 6: Electronic configuration of Si atom having Z = 14: 1s22s22p63s23p2 Example 7: Electronic configuration of Na atom having Z = 11: 1s22s22p63s1 Example 8: Ground state energy of the hydrogen atom is -13.6 eV, and that of the first excited state is -3.39 eV. Understanding of Semiconductor As stated earlier, most semiconductors are crystalline in nature, consisting of repetitive patterns of atoms or molecules. The concept of discrete energy levels

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similar to free atoms is no longer valid here. This is because the potential characterizing the electron is the resultant contribution from every atom of the semiconducting system. Earlier classical kinetic theory of gas was applied to understand the electronic contribution to material’s properties; however, many discrepancies were observed between theoretical formalism and experimental results [3]. For example, experimental evidences show linear temperature (T) dependence of electronic contribution to specific heat in low-temperature region. However, kinetic theory illustrates that the electronic contribution to specific heat should be independent of temperature. Another example is the electronic contribution to the magnetic susceptibility of metallic systems (Pauli paramagnetism), which experimentally is found to be independent of temperature. But classical kinetic theory reveals that it has inverse dependence on T. These discrepancies were removed when quantum mechanical formulations were used to understand material’s properties, till then, it was used for most of the materials, including semiconductors. In this context, it may be stated that quantum mechanical to understand material’s characteristics starts from writing the Hamiltonian (H) of the semiconducting materials, followed by solving Schrödinger’s equation to obtain wave function that contains every information of the semiconductor, and we have to evaluate them. Hence to understand material’s characteristics, we have to solve Schrödinger’s equation. In a practical semiconducting system containing ~ 1023 atoms, there are several interactions such as electron-electron, electron – lattice etc. So considering all the interactions, Schrödinger’s equation for the perfect crystalline semiconducting system may be written in the following form: HΦ(𝐑, 𝐫) = EΦ(𝐑, 𝐫)

(1)

where, Φ(𝐑, 𝐫) is the wave function of the system, which describes every property/phenomenon of the system and Hamiltonian (H) is given by equation (2) ∑𝑖

𝑝𝑖2 2𝑚𝑖

+ ∑𝑗

𝑃𝑗2 2𝑀𝑖

+

1 2

∑𝑗,𝑗 ′

𝑍𝑗 𝑍𝑗′ 𝑒 2 4𝜋𝜀0 |𝑅𝑗 − 𝑅𝑗′ |

− ∑𝑗,𝑖

𝑍𝑗 𝑒 2 4𝜋𝜀0 |𝑟𝑖 − 𝑅𝑗 |

+

1 2

∑𝑖,𝑖 ′

𝑒2 4𝜋𝜀0 |𝑟𝑖 − 𝑟𝑖′ |

(2)

In this Hamiltonian, the first and second terms represent the kinetic energies of electrons and nuclei, respectively. Third, fourth and fifth terms denote the potential energy of the system due to nuclei – nuclei, electron – nuclei and electron-electron electrostatic interactions and summations include all electrons and nuclei. However, for the realistic, practical semiconducting system, it is very difficult to solve equation (1). Hence, several approximations have been proposed here to solve the equation.

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In this context, the first approximation proposed is the Born – Oppenheimer approximation, also known as the adiabatic approximation. Here, ions, due to heavier mass, exhibit very low response time, while electron’s response is very fast, i.e., ions can’t follow the motion of the electrons and only experience time-averaged adiabatic electronic potential. Then, according to the Born – Oppenheimer approximation, the total wave function of the system, 𝛷(𝑹, 𝒓), can be decomposed into total electronic wave function (ξ(𝐫)) and total ionic wave function (χ(𝐫)). The next approximation is to differentiate electrons into core electrons and valence electrons. Core electrons represent electrons of filled orbitals, i.e., 1s2, 2s2, 2p6 electrons corresponding to Si. They are strongly localized around nuclei and don’t have a direct contribution to any of the properties such as band gap, luminescence, electrical conductivity etc. In this context, it may be stated that these properties mostly originate from valence electrons. However, the participation of core electrons is considered as lumped with nuclei to form an ion core, and they contribute to the electrostatic potential energy of valence electrons. Another approximation that further simplifies the problem is the total electronic wave function (ξ(𝐫)) can be written as the product of the individual electron’s wave function (Ψ(𝐫)), hence to understand electron’s property, we have to solve Schrödinger’s equation for each electron. Thus solving Schrödinger’s equation for 1023 electrons has been simplified into a solution of Schrödinger’s equation for one electron. With this background, Schrödinger’s equation for each electron is written as: [

p2 2m

+ V(r)] Ψ(𝐫) = EΨ(𝐫)

(3)

where V(r) is the average electrostatic field, experienced by an electron within the semiconducting material. In this simplification, also known as mean-field approximation, 𝑉(𝑟) is the same for each valence electrons, and it invokes two different contributions: one from interaction with other electrons, Ve−e (r), and others from interaction with ions, Ve−ion (r), i.e., V(r) = Ve−e (r) + Ve−ion (r). Briefly, Ve−ion (r), )), also termed as electron – lattice interaction, is the effective potential that synergistically depends on a change of the nuclei and core electrons, while 𝑉𝑒−𝑒 (𝑟), known as electron-electron interaction, includes average electrostatic potential acting on an electron from all other valence electrons present in a semiconducting material. In this context, it may be stated that any regular crystalline semiconducting material consists of periodically arranged positively charged nuclei, hence Ve−ion (r) should have periodicity similar to lattice parameters.

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Sommerfeld’s Free Electron Model The first solution of Schrödinger’s equation was carried out by Sommerfeld with the assumption that their electron experiences no electrostatic interaction i.e. Ve−ion (r) = 0 and Ve−e (r) = 0. Therefore according to this approximation, also known as free electron approximation, V(r) = 0 in equation (3) for electrons within the material, while outside the material V(r) = ∝. The reason behind to choose V(r) = ∝ is that electron doesn’t come out of the material; this is only possible if outside potential energy be infinite. For further simplicity, we consider one-dimensional semiconducting materials ranging from x = 0 to x = L i.e., ‘L’ represents length of the system. Here V = 0 in between 0 ≤ x ≤ L, otherwise V = ∝. Thus, Schrödinger’s equation for the electronic wave function (Ψ(x)) within the semiconducting material may be written as −

ħ2 d2 Ψ(x) 2m

dx2

= EΨ(x)

(4)

where, ħ, 𝑚 and E are Planck’s constant, mass and energy of the electron, respectively. A general solution of equation (4) can be written as Ψ(x) = Asin(kx) + Bcos(kx) 2mE

where k = √

ħ2

(5)

denotes wave vector of the electron and constants A, and B are

determined from boundary conditions. At boundaries, V = ∝, hence the probability of finding an electron, |Ψ(x)|2 , at boundaries should be zero, i.e., Ψ(x) = 0 at x = 0 and x = L. Using boundary condition at x = 0, we have B = 0, while from Ψ(x) = nπ 0 at x = L, we may derive k = , where n = 1, 2, 3 …. Later condition yields E = ħ2

L

nπ ( )2 2m L

i.e., energies of electrons are no longer continuous, rather, they are quantized, i.e., electrons can stay inside the materials with definite energy levels. This contradicts the classical theory, which doesn’t have such restrictions on energy. From another condition, i.e., the probability of finding an electron within 2

the material should be unity, we can find 𝐴 = √ ; thus complete wave function 𝐿

representing an electron within a semiconducting material can be written as 2

𝑛𝜋

𝐿

𝐿

Ψ(x) = √ sin(

x)

(6)

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As of fundamental quantum mechanics, Ψ(x) describes every property of the electron, and to some extent, the Sommerfeld model was very successful to corroborate experimental results those was not possible according to classical theory. As an example, classical kinetic theory involves all electrons contributing to specific heat and according to equipartition theory each degree of freedom 1 corresponds heat capacity of k B T, where k B is the Boltzmann constant and T 2 represents absolute temperature. If the system contains ‘n’ number of electrons with 3 three degrees of freedom, then total heat content (U) of the system U = nk B T. 2 𝜕𝑈 3 Therefore, specific heat capacity (𝐶𝑣 ) is given by 𝐶𝑣 = = nk B which 𝜕𝑇 2 contradicts the linear temperature dependency of 𝐶𝑣 in low-temperature regions. But Sommerfeld theory gives electronic energy levels, occupied by electrons. As electrons follow Pauli Exclusion Principle, so each level would be occupied by one electron (presently, we are not considering spin of the electron) and the highest occupied energy level, generally termed as Fermi energy, would be dependent on the number of electrons within the system. In this context, it may be stated that the electrons at the bottom of the energy levels don’t acquire sufficient thermal energy to transition to an energy level above Fermi's energy. So all the electrons would not contribute to 𝐶𝑣 , only electrons near Fermi energy would contribute, and the number of electrons involved in this process should be ~ k B T. Therefore, if we calculate, 𝐶𝑣 would be proportional to T in well-agreement with experimental result. Though the model correctly describes several features of the electronic phenomenon, however few other experimental results can be explained on the basis of Sommerfeld theory. As an example, the origin of band gap, i.e., why some materials exhibit bad gap while others don’t have band gap can’t be explained by the Sommerfeld theory. Hence, a more appropriate quantum mechanical theory was proposed considering electron – lattice interaction [4]. Kronig – Penney Model: The Origin of the Band Gap The first essence of the band gap was realized when Kronig and Penney solved equation (3) considering electron – lattice interaction, thus V(r) ≠ 0. For simplicity, if we consider a one-dimensional crystalline system consisting of periodically arranged atoms, then V(x) could be assumed, as represented in Fig. (6). As shown, periodic V(x) consists of V(x) = 0, separated by V(x) = V0 while a + b represents periodicity, i.e., the lattice constant of these one-dimensional crystals.

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Fig. (6). Simple one-dimensional potential V(x) used in the Kronig–Penney model

Herein, Schrödinger’s equations of one electron wave function Ψ(x) within the region 0 ≤ x ≤ a can be written as (3): −

ħ2 d2 Ψ(x) 2m

dx2

= EΨ(x)

(7a)

And within the region – b ≤ x ≤ a −

ħ2 d2 Ψ(x) 2m

dx2

+ V(x)Ψ(x) = EΨ(x)

(7b)

Now the general solution of equation 7(a), which is represented by ΨI (x) is ΨI (x) = AeiKx + Be−iKx where K =

√2𝑚𝐸 . ħ

And solution of equation 7(b), illustrated by ΨII (x), is ΨII (x) = CeiQx + De−iQx

Where, Q = 𝑑Ψ(x) 𝑑𝑥

(8a)

√2𝑚(V0 − 𝐸) ħ

(8b)

. Using boundary conditions, i.e., continuity of Ψ(x) and

at boundaries x = 0, we have A+B=C+D

(9a)

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And

iK(A – B) = Q (C – D) (9b) Here, one approximation was proposed by famous scientist Bloch, and after his name, the theory that has been developed here is known as Bloch’s theory. According to Bloch's theory, electronic wave function Ψ(x) of electrons moving in periodic potential should be of periodic nature, and it would be written as Ψ(x) = uk (x)eikx

(10)

Therefore, from equation (10), we may write Ψ(x + a + b) = Ψ(x)eik(a+b) and putting x = – b, we have: Ψ(a) = Ψ(−b)eik(a+b)

(11)

Using boundary conditions at x = a, we have AeiKa + Be−iKa = (𝐶𝑒 −𝑄𝑏 + 𝐷𝑒 𝑄𝑏 )𝑒 𝑖𝑘(𝑎+𝑏)

(12a)

And iK(AeiKa − Be−iKa ) = Q(Ce−Qb − DeQb )eik(a+b)

(12b)

In this context, it may be stated that for four unknown coefficients A, B, C and D, we have four equations 9(a), 9(b), (12a) and (12b), thus solutions for them only exist if determinants of the coefficients are zero, hence we have Q2 + K2 2QK

sinh(Qb) sin(Ka) + cosh(Qb) cos(Ka) = cosk(a + b)

(13)

In the limit b → 0, V0 → ∞ with assumption bV0 to be constant, equation (13) may be simplified into cos(ka) = P Q2 ba

sinKa Ka

+ cos(Ka)

(14)

Where, P = . In deriving this equation, we have used the approximations Q ≫ 2 K and Qb ≪ 1. In this context, we have plotted the right hand with Ka as an independent variable, shown in Fig. (7).

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Fig. (7). Graph of right hand side of equation 14 as a function of Ka.

Now it can be stated that left-hand side of equation (14) must be within ±1, thus right-hand side of equation (14) must be within this limit, which is only possible for a certain range of K values. There are some K points for which right-hand side of the equation falls outside the region of ±1. Herein, it can be stated that K giving right-hand side of equation 14 outside ±1 provides forbidden states, hence they correspond to forbidden states. Now, if we re-scale the independent axis using the √2𝑚𝐸

, then we can state that we have allowed energy values, separated relation K = ħ by forbidden energy states. Such energy gap is referred to as a band gap. Therefore, it can ben stated that Ve−ion (x) is the origin of band gap. It is important to note that 𝑛𝜋 𝑘= for integer values of n at the edge for each band for which left-hand side of 𝑎 equation (14) is equal to ±1. These fixed values of k appear at the boundaries of what is defined as Brillouin zones. In the limit P → 0, which generally arrears in ħ2 k2

case of nearly free electrons, defined by V0 → 0, E – k relation turns into E = 2m i.e. E – k relation of free electrons. In Fig. (8), we have represented between E – k diagrams of free electrons and the Kronig – Penney model. The figure shows that the solutions of equations (7) and (8) within the shaded regions may be considered portions of the parabola that have broken by energy gaps and are distorted in shape, and in the case of nearly free electrons, it closely resembles the parabola. In addition, it may be further concluded that the energy gap increases with V0 in the crystalline solids, which generally appear for atoms with fewer electrons. Less number of electrons corresponds to less screening of nuclei of the atoms. It has been observed that V0 increases for semiconductors having ionic character. Though Kronig – Penney model describes electrons in periodic potential and explains the origin of band of semiconducting materials, real essence comes only when we consider the solution of Schrödinger’s equation of three-dimensional semiconductors, which will be considered in the next section.

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Fig. (8). Comparison between parabolic E – k for free electrons and Kronig – Penney model.

Bloch’s Theory: Electrons in Three-Dimensional Periodic Potential and Band Structure As stated in the previous section, a more sophisticated, realistic theory is urgently required for better understanding semiconducting materials. In this context, Bloch proposed a new quantum mechanical theory in the 1928s for electrons within semiconducting materials. Herein, he considered the interaction between electron and ion to solve Schrödinger’s equation for three-dimensional materials. However, it must be mentioned here that Ve−ion (r) should be periodic as ions are periodically arranged in a well-crystalline three-dimensional semiconducting material. As well as, it should follow the required symmetries of the crystal structure. Initially, it was thought of as a tedious job, but later symmetry of the crystal structure was realized to be beneficial to solve Schrödinger’s equation and to calculate band structure which is nothing but E(k) diagram of an electron. Briefly, symmetry refers to geometrical transformations of crystal structure that leave it invariant. The most important symmetry is translational symmetry, where a crystal structure remains unchanged after translation. In addition, inversion symmetries also play a deterministic role in determining band structure, i.e., in getting a feasible solution from the Schrödinger equation for semiconducting materials having inversion symmetry. Most of the semiconducting materials exhibit rotational symmetry, hence their band structures get simplified due to rotational symmetry. Let’s see how these symmetries are beneficial for calculating the wave function of the electrons in semiconducting materials. In this context, we first consider translational symmetry and discuss its importance in the electronic wave

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function. Actually, this was first introduced by Bloch when he tried to solve Schrödinger’s equation considering Ve−ion (r) which exhibit perfect periodicity similar to lattice parameters. Bloch assumed that when an electron is moving in such a periodic potential, it’s wave function would no longer be a plane wave, rather, it should follow translational symmetry. In fact, translational symmetry plays a significant role here to determine electronic wave function in this situation as follows: For simplicity, we consider a one-dimensional perfect periodic system with periodicity ‘a’ i.e. Ve−ion (x + a) = Ve−ion (x). In order to understand its effect on wave function, we define translation operator TR which will provide translational invariance of any function f(x) as given by the following equation: TR f(x) = f(x + R)

(15)

With this background, Bloch proposed that instead of plane wave giving uniform probability everywhere, electron moving in periodic Ve−ion (x) should have |Ψ(x)|2 similar to Ve−ion (x). In consequence, electron wave function should contain a term giving this periodic probability. Here, Bloch suggested the wave function of the electrons moving in Ve−ion (x) should have form Ψ(x) = exp(k . x)u𝐤 (x), where uk(x) should have periodicity similar to Ve−ion (x) i.e. u𝐤 (x) = u𝐤 (x + a). Thus, |Ψ(x)|2 is governed by its amplitude u𝐤 (x). Here, suffix ‘k’ represents the wave vector of electron. From the definition of translation operator, we have TR Ψ(x) = Ψ(x + a) = exp(ik . a)Ψ(x)

(16)

Therefore equation (16) illustrates that Ψ(x) is the eigenfucntion of TR with eigenvalue exp(ik . a). As the Hamiltonian, given in equation (3), remains invariant under translation by ‘a’, thus it commutes with TR . Then according to the quantum mechanical rule, the eigenfucntion of TR would also be the eigenfucntion of Hamiltonian of equation (3). Consequently, ‘k’ is also a vector quantity for threedimensional semiconductors and can be written in terms of reciprocal lattice 2π vectors 𝐚∗ , 𝐛∗ , 𝐜 ∗ along three directions as 𝐤 = (nx · 𝐚∗ + ny · 𝐛∗ + nz · 𝐜 ∗ ), N where 𝑛𝑥 , 𝑛𝑦 , and 𝑛𝑧 = 0, ±1, ±2 ….. and N represents the total number of electrons. Here, reciprocal lattice vectors are related to direct space lattice vectors on semiconducting materials of any crystal structure by the following relations: 𝐛×𝐜 𝐜×𝐚 𝐚 ×𝐛 𝐚∗ = , 𝐛∗ = , 𝐜∗ = (17) 𝐚·(𝐛 × 𝐜)

𝐚·(𝐛 × 𝐜)

and they have to satisfy the following relations:

𝐚·(𝐛 × 𝐜)

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𝐚∗ · 𝐚 = 𝐛∗ · 𝐛 = 𝐜 ∗ · 𝐜 = 1 (18a) ∗ ∗ ∗ ∗ 𝐚 ·𝐛=𝐚 ·𝐜=⋯=𝐜 ·𝐚= 𝐜 ·𝐛=0 (18b) Three-dimensional reciprocal wave vector (G), written in terms of 𝐚∗ , 𝐛∗ and 𝐜 ∗ as 𝐆 = 2π(𝑛𝑥 · 𝒂∗ + 𝑛𝑦 · 𝒃∗ + 𝑛𝑧 · 𝒄∗ ), is used to express any periodic function, particularly three-dimensional periodic potential and electronic wave function, according to Fourier transformation theory. This transformation is nothing but to elaborate the whole calculation in a simplified manner. As per Fourier transformation, we may write Ve−ion (𝐫) and u𝐤 (𝐫) as given in equations (19a) and (19b): Ve−ion (𝐫) = ∑𝐆 V𝐆 exp(−i𝐆 · 𝐫) (19a) and 𝒖𝐤 (𝐫) = ∑𝐆 u𝐤 𝐆 exp(−i𝐆 · 𝐫) (19b) where, V𝐆 and u𝐤 𝐆 , the Fourier coefficients, are obtained from inverse Fourier transformation according to equations (20a) and (20b): 1

V𝑮 = ∫ exp(i𝐆 · 𝐫)Ve−ion (𝐫)d𝐫 𝛺

(20a)

and 𝑢𝑘 𝑮 =

1 𝛺

∫ exp(i𝐆 · 𝐫)𝑢𝑘 (𝐫) d𝐫

(20b)

where ‘Ω’ represents the volume of the unit cell. In case of G = 0 in equation (20a), we have zeroth – order Fourier coefficient(V𝐆 = 0 ) representing the average potential energy of an electron within Ω, thus putting G = 0 in equation (20a), we have the expression for V𝐆 = 0 1

V𝐆 = 0 = ∫ Ve−ion (𝐫)d𝐫 𝛺

(21)

In the simplified Sommerfeld model, V𝐆 = 0 was assumed to be zero to solve Schrödinger’s equation. But, a more realistic solution of Schrödinger’s equation arises for Ve−ion (𝐫) ≠ 0, which can be understood as follows: with Ve−ion (𝐫) ≠ 0, we have Schrödinger’s equation, given in equation (22) [−

ħ2 2m

∇2 + Ve−ion (𝐫)] Ψ(𝐫) = E(𝐤)Ψ(𝐫)

(22)

Considering Bloch’s hypothesis and equation (20b), we may write Ψ(𝐫) in the following form:

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Ψ(𝐫) =

1 √𝛺

=

exp(i𝐤. 𝐫) ∑ u𝐤 𝐆 exp(−i𝐆 · 𝐫) 𝐆

1 √𝛺

∑𝐆 u𝐤 G exp[i(𝐤 − 𝐆). 𝐫]

(23)

1

Here, we have multiplied the wave function by for normalization, similar to √𝛺 equation (6). From equations (22) and (23), we have −

ħ2 1 2m √Ω

∑𝐆 u𝐤 𝐆 ∇2 exp[i(𝐤 − 𝐆). 𝐫] + 𝐆). 𝐫] = E(𝐤)

1

1 √𝛺

√𝛺

∑𝐆,𝐆 ′ V𝐆′ 𝑢𝐤 𝐆 exp(−i𝐆′ · 𝐫) exp[i(𝐤 −

∑𝐆 𝑢𝐤𝐆 exp[i(𝐤 − 𝐆). 𝐫

(24)

Rearranging equation (24), we may write 1 √𝛺

∑𝐆 [

ħ2 2𝑚

(𝐤 − 𝐆)2 − 𝐸(𝐤) + ∑𝐆′ V𝐆′ exp(−i𝐆′ . 𝐫)] × u𝐤𝐆 exp[i(𝐤 − 𝐆). 𝐫 = 0

(25)

Now multiplying both sides of equation (25) with exp[−i(𝐤 − 𝐆′′ ). 𝐫 and integrating over Ω, we have [

ħ2 2m

(𝐤 − 𝐆′′ )2 − E(𝐤)] u𝐤

𝐆′′

+ ∑𝐆 ′ V𝐆′ −𝐆′′ u𝒌

𝑮′

=0

(26)

In this integration, we have used the following orthogonality relation, given by 1 ∫ Ω Ω

exp[i(𝐆 − 𝐆′ ). 𝐫] d𝐫 = δ𝐆𝐆′

(27)

Within free electron approximation, V𝐆′ −𝐆′′ = 0; thus putting V𝐆′ −𝐆′′ = 0 in equation ħ2

(26), we have E(k) = k 2 and Ψ(𝐫) = 2m with Bloch’s theory.

1 √Ω

𝑢𝑘 𝑮 = 𝟎 exp(i𝐤 · 𝐫) in well agreement

In the nearly free electron approximation, potential energy is assumed to be very close to zero, but exactly not equal to zero and Fourier coefficients V(G) are assumed to have value close to zero, except V𝐆 = 0 . Therefore, substituting E(k) = ħ2

2m

k 2 and considering non-zero 𝑢𝐤𝐆=𝟎 in equation (26), we may write [

ħ2 2𝑚

(𝐤 − 𝐆′′ )2 −

ħ2 2𝑚

𝐤 2 ] 𝑢𝐤

𝐆′′

≠𝟎

+ V(𝐆′′ )𝑢𝐤𝐆=𝟎 = 0

(28)

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Thus from equation (28), we may derive the following expression for 𝑢𝐤 𝑢𝐤

𝐆′′ ≠ 𝟎

=

V(𝐆′′ )𝑢𝐤𝐆=𝟎

ħ2 2 [𝐤 −(𝐤 − 𝐆 ′′ )2 2𝑚

𝐆′′

: (29)

]

In this nearly free electron approximation, we still assume that the electronic wave 1 function is given by Ψ(𝐫) = 𝑢𝐤𝐆=𝟎 exp(i𝐤 · 𝐫), thus 𝑢𝐤 ′′ has non-negligible √Ω

𝐆

≠𝟎

values. In this context, it may be stated that 𝑢𝐤 ′′ would have non-zero value if 𝐆 ≠𝟎 (𝐤 − 𝐆′′ )2 ≈ 𝐤 2 and it is mostly represented by the following condition: (𝐤 − 𝐆′′ )2 = k 2

(30)

The condition, referred in equation (30), is well known as the Bragg reflection condition, which is mainly used to determine Brillouin zones, unit cell in the reciprocal space of the semiconducting crystals. First Brillouin, constructed with the smallest G, represents the smallest unit cell in the reciprocal lattice space. For electron’s wave vector satisfying condition (30), we have to consider 𝑢𝐤 ′′ and 𝐆 ≠𝟎 𝑢𝐤𝐆=𝟎 neglecting others, and then we have the following two relations: [

ħ2

2𝑚

𝐤 2 − E(𝐤)] 𝑢𝐤𝐆=𝟎 + V(−𝐆)𝑢𝐤

𝐆′′ ≠ 0

=0

(31a)

and [

ħ2 2𝑚

(𝐤 − 𝐆′′ )2 − E(𝐤)] 𝑢𝐤

𝐆′′

+ V(𝐆)𝑢𝐤𝐆=𝟎 = 0

(31b)

Both these two equations give non-zero value of 𝑢𝐤𝐆=𝟎 and 𝑢𝐤 ′′ only when the 𝐆 determinant of their coefficient as given by equation (32) vanishes. ħ2

[

2𝑚

𝐤 2 − E(𝐤) V(𝐆)

V(−𝐆) ħ2 2𝑚

]=0

(𝐤 − 𝐆′′ )2 − E(𝐤)

(32)

From equation (32), we have E(𝐤) =

1 ħ2

[

2 2𝑚

{𝐤 2 + (𝐤 − 𝐆′′ )2 } ± √(

ħ2 2 2 ) {𝐤 2𝑚

− (𝐤 − 𝐆 ′′ )2 }2 + 4|𝑉(𝐆 ′′ )|2 ] (33)

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Here, we have used the condition that V(−𝐆) = V(𝐆). At Brillouin zone boundary, 2 given by (𝐤 − 𝐆′′ )2 = 𝐤 2 i.e. 2𝐤. 𝐆′′ = 𝐆′′ , we have expression for E(𝐤) as given by equation (33): E(𝐤) =

ħ2 2𝑚

𝐤 2 ± |V(𝐆′′ )|

(34)

Therefore equation indicates that there exists an energy gap of 2|V(𝐆′′ )| at the Brillouin zone boundary. In order realize the origin of band gap, we consider simplest situation of one-dimensional crystalline semiconducting material. Here, G is given by G= 2πn/a (n = 0, ±1, ±2, ±3, ..) where ‘a’ represents lattice parameter and we have the following expression for E(k) 1 ħ2

E(k) = √(

[

2 2𝑚

ħ2 2 2 ) {k 2𝑚

{k 2 + (k − 𝟐πn/a)2 } ±

− (k − 𝟐πn/a)2 }2 + 4|𝑉(G ′′ )|2 ]

Equation (35) indicates that E(k) ≈

ħ2 2𝑚

(35)

k 2 gets validated for every value of ‘k’ πn

except at the zone boundary, which is given by the condition k 2 = (k − )2 or k a = 2πn/a. Importantly, equation (35) corresponds ± sign. Briefly, considering the sign of the square root, in the region k < (k – G)2, we have to consider minus sign, but the region k > (k – G)2 gives a positive sign in equation (35). Thus, E(k) relation may be summarized as E(k) =

ħ2 2𝑚

k 2 − |𝑉(G′′ )| when k ≤ nπ/a

(36a)

and E(k) =

ħ2 2𝑚

k 2 − |𝑉(G′′ )| when k ≥ nπ/a

(36b)

A plot of energies in equation (36) as a function of k, as represented in Fig. (9a), is known as the electronic band structures of the semiconducting materials. A band structure plot where ‘k’ gets varied over all possible values is defined as the extended zone scheme. However, it may be stated that the choice of ‘k’ in indexing a wave function is not unique. In fact, k and k + 2nπ/a represent the same wave function which originates from the translational symmetry of the crystalline semiconducting materials. Here, another technique to plot energies as a function of ‘k’ has been developed. In this new technique, k is replaced by k’ = k – 2nπ/a to limit k’ to the range [-π/a. π/a]. This interval of k-space is defined as the first Brillouin zone and the band structure plot within the interval of [-π/a. π/a], as represented in Fig. (9b), is known as the reduced zone scheme. It should be

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emphasized here that the magnitude of the energy gap depends on |𝑉(G′′ )|. Different materials exhibit different |𝑉(G′′ )|, hence they correspond to different band gaps. After Bloch’s theory, different techniques, such as k.p, perturbations etc., have been developed to calculate the band structure of semiconducting materials (Fig. 9c). Presently, density functional theory is being widely adopted for band structure calculation.

Fig. (9). Energy band structure of one-dimensional crystal obtained from equation (36) in extended zone scheme (a) and reduced zone scheme (b). Energy band gap at the zone boundary is schematically represented in (c).

Example: 9 Reciprocal lattice vectors of a simple cubic unit cell with direct lattice vectors a = ax, b = ax and c = az, according to equation (17), are 𝐚∗ =

2π a

𝐱 , 𝐛∗ =

2π a

𝐲 and 𝐜 ∗ =

2π a

𝐳

Band Structure of Face-Centered Cubic Crystals As an example, we consider the band structure evolution of Diamond (C), Si, and Ge having direct lattice in the form of face-centered cubic. In this case, primitive lattice vectors consist of a a

𝐚 = ( , , 0) 2 2

𝑎 𝑎

𝐛 = (0, , ) 2 2

(37a) (37b)

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And

𝑎

𝑎

𝐜 = ( , 0, ) (37c) 2 2 where ‘a’ represents the length of the side of the smallest cube of this face-centered cubic lattice and, according to equation (17), respective reciprocal lattices given by 1 𝐚∗ = ( )(1, 1, − 1) (38a) ∗

𝑎 1

𝐛 = ( )(−1, 1, 1) 𝑎

(38b)

and 1

𝐜 ∗ = ( ) (1, −1, 1) 𝑎

(38c)

Fig. (10). First Brillouin zone of face centered cubic lattice.

Thus, reciprocal lattice vectors are identified to show body-centered cubic symmetry. Herein, Brillouin zone (shown in Fig. 10), constructed according to equation (30), exhibits several high symmetry points, often denoted by some letters. Conventionally, high symmetry points and lines inside the first Brillouin zone are denoted by Greek letters, while points on the surface are represented by Roman letters. As an example, high symmetry directions such as [100], [110] and [111]

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𝛬

𝛴

within the first Brillouin zone are represented by Γ ↔ X, Γ ↔ L and Γ ↔ K ħ2

respectively. Within empty lattice approximation, we have E(𝐤) = k 2 which 2𝑚 gets modified as given in equation (39) in a reduced zone scheme ħ2

E(𝐤) =

2𝑚

(𝐤 + 𝐆)2

(39)

Here, several reciprocal lattice vectors from the lowest order as given by the following expressions: 2𝜋 𝐆0 = (0, 0, 0) (40a) 𝑎

𝐆3 =

2𝜋 𝑎

𝐆4 =

(±1, ±1, ±1)

2𝜋 𝑎

(±2, 0, 0)

(40b) (40c)

Using equation (39) and (40), E(k) along [100] directions shown in Fig. (11a) can be written: 𝐆0 : E(k) = k 2x 𝐆3 : E(k) = (𝑘𝑥 ± 1)2 + (±1)2 + (±1)2

(41a) (41b)

(k + 1)2 + 2 ∶ 4 − fold degenerte ={ x (k x − 1)2 + 2 ∶ 4 − fold degenerte 𝐆4 : E(k) = (k x ± 2)2 (41c) k 2x + 4: 4 − fold degenerate = {(k x − 2)2 ∶ non − degenerate (k x + 2)2 : non − degenerate As stated earlier, empty lattice approximation doesn’t generate band gap, thus we can’t see any gap in the Brillouin zone boundary. But the real band structure of Si crystals as a prototype of face-centered cubic lattice is represented in Fig. (11b). Here, band structure has been calculated using k.p method.

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Fig. (11a). Empty lattice-bands of face-centered cubic lattice structure. , and illustrate reciprocal lattice vectors of G3, G4 and G8, respectively, while the number within the first bracket represents the degeneracy of the wave functions. (b) Actual band structure of Si.

BAND, NUMBER OF STATES IN A BAND, BAND FILLING From the previous section, we came to know that electronic states are no longer continuous in terms of energy within semiconductors, rather, some closely spaced energy states are allowed, while others are not forbidden. The allowed closely spaced energy states form a band, i.e., it can be stated that we have several bands, separated by forbidden states. Depending on the number of electrons within a semiconducting material, bands are occupied with electrons (schematically shown in Fig. (12a – c)). In this context, the highest energetic band, occupied with electrons, is defined as the valence band, while the lowest unoccupied energetic band is referred to as the conduction band, and the energy difference between these two bands is termed as energy band gap, which is the most useful parameter for the practical utility of semiconducting materials. Here, the next question that would come is how many states exist within each band. Let’s see: From the boundary condition as prescribed in Sommerfeld’s model, we have an nπ expression of k, given by k = , where n = 1, 2, 3 … and according to Bloch’s L π

nπ

𝑎

L

theory, k is also written as k = . Using these relations, we can write

=

π a

and

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L

consequently we have n = . Thus it can be stated that the number of electron in a each band is given by the ratio of total length of the semiconductor and lattice parameter [4]. In more generalized three-dimensional semiconductors, this number would be defined as ratio of volume of whole semiconductor and unit cell volume. 2L Importantly, if we consider spin of electrons, then actual number would be n = . a

In this context, it may be stated that all states are not occupied by electrons, i.e., number of electrons in a band is not equal to a number of states. However, it may be stated that in the low-temperature region, electrons would occupy lowest energy levels, while higher energy bands would remain empty. For example, we consider Si semiconducting material containing 14 electrons per atom. In this particular system with an even number of electrons per atom, all low-lying energy bands are occupied and the highest occupied band is completely filled, but the next energy band remains empty. Similar conclusions can also be drawn for Ge, diamond, and group II – VI and III – V semiconductors. However, there are always some exceptions. Let’s consider Al Ga and In from group III – V semiconductors. For these elements, there are odd numbers of electrons per atom. Hence, the highest occupied energy bands would half-filled. In this context, it may be stated that in the case of completely filled bands, for each electron moving in + x direction, there is another electron moving in the opposite direction with equal momentum. Thus, the resultant momentum would be zero; hence there would be zero resultant currents for completely filled bands, even in the presence of an external electric field. But the current wouldn’t be zero in the case of half or partially-filled bands. In this context, it may be stated there is a chance that electrons get thermally excited from the valence band into conduction, leaving valence and conduction bands partially filled. Hence, valence, as well as conduction, would carry current under this situation. However, this probability is appreciable for semiconductors with a very low magnitude of the energy band gap. But, most of the semiconductors exhibit band gap ~ eV, while thermal energy lies in the range of meV, thermal excitation is not sufficient for the electronic transition from the valence to the conduction band. Hence, most of the semiconductors are insulators in the intrinsic case.

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Fig. (12). The degree of filling of the energy bands in (a) semiconductors, (b) insulators and (c) metals at temperatures approaching 0 K. Available electron states in the hatched regions are filled with electrons, and the energy states at higher energies are empty.

Formation of energy bands and differentiation among metal, semiconductor and insulator can also be understood alternatively as follows: when atoms form a crystal, their outer-shell orbitals get changed significantly as all atoms share it. The new energy levels result in a band of closely spaced energy states instead of widely separated discrete energy levels of isolated atoms. In semiconductors like Si and Ge, outer-shell consists of 2s and 2p electrons. If we consider a semiconducting system of N atoms, then 2N electrons would completely fill 2N possible s-levels, all with the same energy levels. Since p orbital has six possible states, 2N would fill one-third of the 6N possible p states of the same energy levels. Thus Si and Ge are electrically conductive. DIRECT AND INDIRECT BAND GAP SEMICONDUCTOR As has been observed from Fig. (13), the conduction band minima (CBM) and valence band maxima (VBM) of Si crystal are not in the same k-point. In fact, this is not a very unusual phenomenon. Here, depending on the position of CBM and VBM, we can differentiate all semiconducting materials into two categories: direct and indirect band gap semiconductors. In direct band gap semiconductors like GaAs, GaN, CdTe etc., VBM and CBM lie at the same k-point, whereas they are in different k-points in the case of indirect band gap semiconductors such as Si, GaP etc. In this context, it may be stated that the distinct difference between direct and indirect band gap semiconductors has a significant impact on several optoelectronic devices such as photovoltaic, and light emitting diodes where light absorption takes

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place. Light absorption and generation processes differ significantly in these two types of semiconductors. Very briefly, it can be stated that during the light absorption phenomenon, light quanta, called photon, excites one electron into the conduction band, leaving hole in valence band. Off course, energy of photon should be greater than band gap of that particular material. In the case of indirect band gap semiconductors, during excitation process, there would be a shift of momentum of the electron. As photon momentum is very less, hence conservation of momentum gets satisfied by the quanta of lattice vibration, called phonon. In this context, it may be stated that there exist phonons in a crystal with some definite momentum and energy, i.e., phonon with arbitrary momentum and energy doesn’t exist. As an example, in Si crystals, 2𝜋 VBM lies at k = 0 point, whereas CBM is situated at k = point. Therefore, light 𝑎 2𝜋

absorption induces change in electron’s momentum ~ ħ . Though phonons with 𝑎 several moments are present in semiconductors, there may be a situation where a phonon with momentum, required to satisfy the momentum conservation rule, is absent. In this situation, the electron excitation process is inhibited, and the photon continues to propagate through the semiconductor crystal with being absorbed. It has been generally observed that indirect band gap semiconductors possess a very low absorption coefficient of light, thus, they have limited usage in opto-electronic devices. In light-emitting devices, the reverse phenomenon is involved. Briefly, an excited electron in the conduction band recombines with a hole in the valence band after emitting extra energy as a photon. Herein, the energy of the photon, i.e., color of radiation depends on the band gap of the semiconductors. In the case of indirect band gap semiconductors, this recombination involves phonons to validate momentum conservation rule and facilitates non-radiative transition increasing temperature of the semiconductors during operation. In this context, it may be stated that direct band gap semiconductors have several advantages over direct band gap semiconductors in device applications. Intrinsic Semiconductor: Density of States, Fermi – Dirac Statistics and Fermi Energy In a pure semiconductor, also called an intrinsic semiconductor, most of the optoelectronic properties are determined by electrons in bands. It has been identified that available energy states within the energy bands are not energetically equispaced, rather, they have some distribution. Hence to know the occupancy of

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bands by electrons and related phenomenon, we have to determine the occupancy of individual available energy states within each band. In this context, the concept of density of states, N(E), defined as the number of states with energy interval between E and E + dE in each band, has been adopted and it is being defined as follows: in k-space, density of allowed points is uniform, assuming that the surface of constant energy are spherical, then volume of k-space with energy in between E 8𝜋3

and E + dE is 4πk 2 dk. Volume of a k-point is , where V represents the volume 𝑉 of the crystal and each k-point is occupied with two electrons (considering spin degeneracy), thus N(E) can be written as N(E)dE =

8πk2 8π3

Substituting E(k) using the relation E(k) =

dk = ħ2 2meff

k2 π2

dk

(42)

k 2 , we have

2meff 3/2 1/2 ) E dE ħ

N(E)dE = (

(43)

Where, E is referenced from the bottom conduction band for electrons and from top of valence band for holes. However, a few points to be noted here is that N(E) increases with the square root of energy. Here we have used parabolic E(k) dependence which is the consequence of a nearly free electron. The slope of the parabola depends on the effective mass of electron in the conduction band and on the effective mass of the hole for valence band. Thus it is different for different semiconductors, hence it would be considered an intrinsic property of a particular semiconductor material. The next point, which is important in this context, is the occupancy of energy states. It has been identified that electrons do not equally occupy all N(E); rather, they correspond to different occupancy by electrons which may be described by Fermi – Dirac distribution f(E) under thermal equilibrium at temperature T f(E) =

1+exp(

1 E− EF ⁄k T) B

(44)

Where EF denotes Fermi energy. The concept of thermal equilibrium can be understood as follows: we consider crystal lattice with lattice vibrations, the quantum of energy of which is called phonon, which transfers energy to electrons

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in the crystals. These electrons initially occupy excited states and then immediately transfer their energy back to the lattice, and a thermal equilibrium gets established. Like all other electronic phenomena, f(E) and EF also play a significant role in semiconducting materials. In general, f(E) = ½ at E = EF irrespective of temperature T, however, transitions from occupied states to unoccupied states increase with increasing T due to increased thermal activation of electrons from lower energy states to higher energy states. However, the concept is tedious in semiconducting materials. We have plotted f(E) for semiconductors with an energy axis in the vertical direction. Here, Ec and Ev denote energies of conduction band minima and valence band maxima, respectively. In the case of semiconducting materials, EF lies in between conduction band minima and valence band maxima, hence presently, f(E) indicates the probability of electron occupancy in the bands. In any band, the probability of electron deficiency, i.e., hole occupancy of an energy state, is given by 1 – f(E), while the probability of finding an electron at an energy state is f(E).

Fig. (13). Band diagram is plotted along with Fermi – Dirac distribution for semiconductors.

Using equations (40) and (41), we can calculate total number of electrons, N, in conduction band as follows: N=

1 2𝜋2 ℎ3

𝐸 1/2 𝑑𝐸 c 1+exp(E− EF⁄ kB T)

∞

(2meff.e )3/2 ∫E

(45)

where, meff.e is the effective mass of an electron in the conduction band. The solution of equation (45) is very tedious; hence we have to adopt an analytical method with assumption. In most semiconducting systems, different in electronic energy levels E − EF ~ eV, while thermal energy k B T ~ meV. Hence, for all

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practical purposes, we may assume (E − EF ) ≫ k B T, thus [1 + E − EF E − EF exp ( ⁄k T)] ≈ ( ⁄k T). Thus equation (45) may be simplified into B B N=

∞ 1 E −E 3/2 (2m ) ∫ E1/2 exp ( F ⁄k T) dE eff,e B 2π2 h3 Ec

EF − Ec

= Nc exp( 2(2πm

⁄k T) B

(46)

k T)3/2

eff,e B Where Nc = denotes energy density at the band edge, more h3 precisely within few 𝑘𝐵 𝑇 above the conduction band edge.

In a similar fashion, we may also calculate a number of holes (P) in the valence band. As a consequence of equation (45) and (46), we may write an expression for total number of holes (P) in valence band as equation (47) P=

Ev 1 3/2 (2m ) ∫ N(E)(1 − f(E))dE eff.h 2π2 h3 −∞

Ev − EF

= Nv exp(

⁄k T) B

(47) 2(2πm

k T)3/2

eff,h B Where, meff.h represents mass of hole in valence band and Nv = h3 denotes density of states at the valance band edge. From equations (46) and (47), we may derive the expression for EF as follows:

EF = =

Ec + Ev k B T Nv + ln( ) 2 2 Nc Ec + E v 2

+

kB T 2

meff.h 3/2 ) meff.e

ln(

(48)

It may be stated from equation (48) that EF like the metallic system doesn’t refer highest occupied energy states, rather, it is a parameter to represent type conductivity. Briefly, equation (48) states that EF lies in between conduction and valence band edges. The exact position is determined by effective masses of charge carriers, i.e., it should vary for different semiconducting materials. For simplicity, if we consider the effective masses of the charge carriers to be the same, then EF should be exactly in the middle between the conduction and balance band edges.

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However, equations (46) and (47) can be used to determine the intrinsic carrier concentration of undoped semiconductors, called intrinsic carrier concentration (Ni ), where electron concentration (N) should be the same as hole concentration (P). Using these two equations, we have an expression for intrinsic carrier concentration Ni2 = N. P as given below: Ni2 = Nc exp (

EF − Ec

⁄k T) . Nv exp ( B

Ev − EF

⁄k T) = Nc . Nv exp ( B

−Eg

⁄k T) (49) B

where, Eg denotes the band gap of the semiconducting material. As a practical example, if we consider Eg ~ 1 eV, then Ni is calculated to be approximately 1011 cm-3 indicating that thermal excitation is not sufficient for carrier generation, thus, for all practical purposes use of intrinsic semiconductors is very much limited. As an example, Ni is measured to be 2 × 1013 , 2 × 1010 and 2 × 106 /cm3 for Ge, SI and GaAs at room temperature. Hence, doping is highly essential to increase carrier concentration to make semiconductors practically usable. EXTRINSIC SEMICONDUCTOR Intrinsic semiconductors don’t have significant free charge carrier concentration for electrical conduction; hence they possess very low electrical conductivity. However, carrier concentration can be moderately tailored by doping, and such tuneability of charge carriers in semiconducting material is the heart of all semiconductor devices [1].

B

Fig. (14). n- and p-type doping in Si crystal.

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The principle effect of doping, as illustrated in Fig. (14), can be explained as follows: we start with Si as an example. In a perfect Si crystal, each Si atom is surrounded by four Si atoms with which the central Si atom shares its four electrons to form covalent bonds. Thus it can be stated all the electrons are engaged in making a bond. If a phosphorous atom with five electrons substitutes one Si atom, then its four electrons form a bond, while the fifth electron is free and is responsible for increasing the conductivity of the Si crystal. Here, electrons are the majority charge carrier, referred to as n-type conductivity. On the other hand, if we consider the substitution of Si by aluminum with three electrons in its outer cell, then an electron is transferred from Si atom to boron, leaving a deficiency of electrons, called hole, on the Si atom. In this case, the hole enhances the conductivity of Si crystal; specifically, it is defined as p-type conductivity. Here, it may be stated that it is possible to change the type of conductivity within a semiconductor, unlike metals having electrons only as a charge carrier. It has been observed that these donors and acceptor levels lie near conduction and valence band edge, and they are called shallow levels. When semiconductors are doped with donor or acceptor atoms, then their corresponding energy levels are introduced within in forbidden in between conduction and valence band. In general, donor and acceptor energy states are denoted by ED and EA respectively shown in Fig. (15). In this context, the energy differences Ec − ED and EA − Ev representing donor and acceptor binding energies illustrates the required energy to excite electron and hole into conduction and valence bands. It has been observed doping with any atoms is not suitable to obtain n- or p-type conductivities. Only donor atoms giving energy just below the conduction band and acceptor atoms generating energy levels just above the valence band are considered useful dopants. This is because these states Ec − ED and EA − Ev ≪ k B T i.e. k B T is sufficient for thermal excitations of electrons and holes in the conduction and valence band, respectively. A donor level filled with electron is referred to as neutral and has positive charge when empty. Similarly, an acceptor level is neutral when empty and carries negative when occupied by electrons.

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Fig. (15). Donor and acceptor levels in n-type and p-type semiconductors.

It has been observed that if the charge carrier has to propagate, there remains potential energy for charged donor or acceptor site, which in a generalized way, may be written as V= −

e2 4πϵr ϵ0 r

× (screening factor) + V2

(50)

𝑟

The screening factor has a typical form of 𝑒 − 𝛿 , where the characteristic length 𝛿 depends on the presence of other charges in its surroundings. The potential energy V2 takes into account the contribution of a particular character of an impurity atom. In case of low impurity concentration, 𝛿 ≈ ∞ and V2 = 0. Therefore, above mentioned potential energy gets simplified into V= −

e2

(51)

4πϵr ϵ0 r

Herein, potential energy simply looks like the potential energy of an electron moving around a positively charged nucleus. Hence, simple quantum mechanical calculation illustrates that the binding energies of donor/acceptor states and Bohr radius (𝑎0 ) are given by Ec − ED / EA − Ev = −

meff e4 32𝜋2 ε2r ϵ20 ħ2

=−

13.6 ϵ2r

eV

(52)

And 𝑎0 =

4𝜋ħ2 𝜖𝑟 𝜖0 meff e2

= 0.529𝜖𝑟 Å

(53)

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Where, meff represents the effective mass of electron (hole) in the conduction (valence) band in the case of n-type (p-type) semiconductors, while 𝜖𝑟 and 𝜖0 denote the relative permittivity and static dielectric constant, respectively. As an example, for an n-type dopant in Si, the binding energy of the donor is found to be approximately 0.05 eV, while 𝑎0 is observed to be equal to several lattice constants. Thus, electron in donor level moves around the doped site. In this context, it may be stated that the increase of Bohr radius well corroborates a posteriori of choice of the potential energy, given in equation (51), which utilized dielectric constant and thus assumes that the charge carrier “sees” the crystal medium and corresponds large orbit around defect site of opposite charge. There is another possibility where donor or acceptor states don’t contribute electrons in the conduction band or hole in the valence band. In this particular case, charge carriers are bound closer to the nucleus of the defect site. This may happen when the nucleus carries multiple charges or carries possess a large effective mass. Importantly, the above picture on the basis of the dielectric constant is not justified then, and the impurity energy levels can no longer be expressed in terms of equation (51). Instead, the theory for deep impurity states involves potential close to the nucleus, the core electrons that matter more binding electrons strongly. It has been identified that Ec − ED or EA − Ev exhibits a very high value of binding energy ~ eV and are generally called deep-level impurity levels. Due to high binding energies, it becomes very difficult to excite them thermally, but they often act as carrier trapping centers or carrier recombination centers. Though these states are not beneficial for electrical conductivity, they play a significant role in lightemitting devices. These n- and p-type doping change concentrations of charge carrier and it is being considered that EF levels adjust it-self to maintain charge neutrality. In case of ptype doping EF would move towards valence band edge, while in case of n-type doped semiconductor, EF would shift towards conduction band edge (shown in Fig. (16)).

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Fig. (16). Position of Fermi level in case of (a) n-type and (b) p-type semiconductors.

Most importantly, in doped semiconductors, electron concentration (N0 ) is no longer equal to hole concentration (P0 ), but equation (49) is still applicable as it is independent of position of EF . Thus we may use equation (49) for doped semiconductors as N0 . P0 = Ni . Pi = Ni2 = Nc . Nv exp (

−Eg

⁄ ) kBT

(54)

We now consider an intermediate temperature region where all donors are thermally excited in an n-type semiconductor, so we may write N0 ≈ ND , where ND denote the concentration of donors. Thus from equation (54), we may deduce P0 =

N2i

ND

. N0 ≫ Ni in an n-type semiconductors implies N0 ≫ P0 ,

therefore electrons is called the majority carrier, while hole is the minority carrier. Herein, we can use f(E) to calculate the concentration of ionized donor states as follows: let ND+ be the concentration of donors contributing electron in the conduction band. From the charge neutrality condition, we have N + ND+ = P, where N and P represent intrinsic electron and hole concentration, respectively. According to Fermi – Dirac statistics, ND+ = ND [1 − 𝑓(𝐸)] and it can be explicitly written as:

(55)

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ND+ = ND [1 − 𝑓(𝐸)] = ND [1 −

1 ED − EF 1+exp( ⁄k T) B

]

(56)

Although exact value of ND+ can be calculated from equation (56) with the help of equation (49), (52) and (55), but it can be generally stated that all donor levels would be ionized if EF lies well below of E𝐷 . However, with increasing ND+ , EF would be shifted towards the conduction band edge and ultimately EF would be merged with ED and then no ND would be further ionized. In an extreme case of a highly doped semiconductor, EF gets moved into the conduction band, and then the semiconductor becomes degenerate. Under this condition, the above-mentioned relations are no longer valid, and the semiconductor gives metallic behavior. Similar to n-type semiconductors, we can deduce the following expressionN0 = P2i

N𝐴

, where N𝐴 is the concentration of acceptor levels. Herein it can also summarize

that hole is the majority carrier, whereas the electron acts as the minority carrier. Similar to equation (56), we may find out expressions for charged acceptor levels (NA+ ) as given below: NA+ = NA [1 −

1 E F − EA 1+exp( ⁄k T) B

]

(57)

It has been observed that extrinsic semiconductors behave differently in different temperature regions with respect to intrinsic semiconductors (shown in Fig. (17)). As an example, most dopants aren’t ionized in the low-temperature region, however, a slight change in temperature would result in a significant change in electrical conductivity. In the intermediate temperature region, most of the dopants are ionized, leaving few un-ionized, therefore, electrical conductivity would increase if we increase the temperature, but the variation would not be similar to the low-temperature region. In high-temperature regions, all the dopants are ionized; hence any change in electrical conductivity would be tuned by the excitation of an electron from the valence band into the conduction band. In this particular temperature range, the electrical conductivity of extrinsic semiconductors would be similar to intrinsic semiconductors.

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Fig. (17). Variation of carrier concentration with respect to temperature for intrinsic and extrinsic semiconductors.

Example 9: Ionization energy of donors in Si (𝜖𝑟 = ) is 0.096 eV as calculated from equation (52). Example 10: Bohr radius of an electron in n-type semiconductor is calculated to be 6.295 Å from equation (53). CARRIER’S CHARACTERISTICS: CHARGE, EFFECTIVE MASS In semiconducting materials, the existence of two types of charge carrier has been found. These are electrons having a negative charge and holes with a positive charge, while the magnitude of the charge (1.6 × 10-19 Coulomb in MKS unit) is the same for both. Like charge, mass of the charge carrier is another important property of electron and hole. But unlike charges, mass of holes/electrons isn’t the same for all semiconducting materials. In fact, it significantly depends on a particular semiconducting material, though electron’s rest mass in vacuum conditions is measured to be 9.1 × 10-31 kg. Indeed, instead of mass of electron/hole within a

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semiconducting material, effective mass more appropriately defines the essence and can be realized as follows [3]: we consider two parallel plates in a vacuum condition and an electron with mass 𝑚0 is moving due to an electric field ζ. According to Newton’s law, we may write an equation of motion for the electron as – e ζ = 𝑚0 f = 𝑚 0

dv

(58)

dt

Where ‘v’ and ‘f’ are the velocity and acceleration of the electron and ‘t’ represents time. Instead of vacuum, if we just incorporate a semiconducting slab between two plates and is asked whether (58) is valid or not. The answer would be no as because electrons moving inside the semiconducting slab would collide with periodically arranged positively charged nuclei causing deceleration. In addition, electrons would not experience electric field ζ as in the case of vacuum; more precisely, they would be screened within the slab. Truly, the concept of effective mass has been realized after the development of quantum mechanics through band structure formulation as follows: the modification starts with consideration of effective mass (meff ) instead of 𝑚0 as described in equation (59): – e ζ = meff f = meff

dv

(59)

dt

In quantum mechanics, ‘v’, in general, is expressed in terms of group velocity (vg ) of the wave corresponding to the electron, given by vg =

𝑑𝜔

(60)

𝑑𝑘

where ′𝜔′ representing an angular frequency of the electron is related to its energy (E) by the relation E = ħω. Now substituting 𝜔 in equation (60), we have vg =

1 𝑑𝐸

(61)

ħ 𝑑𝑘

Fundamentally, acceleration (f) of electron is the time derivative of vg i.e. f=

dvg dt

=

1 d

dE

( )=

ħ dt dk

1 d

dE dk

( )

ħ dk dk dt

=

1 d2 E 1 dp ħ

dk2

ħ dt

=

1 d2 E dp ħ2

dk2

dt

=

1 d2 E ħ2 dk2

meff f (62)

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In equation (60), we have used the well-known de Broglie’s relation p = ħk, and it may be re-written as 2 meff = ħ ⁄𝑑2 𝐸

(63)

d𝑘 2

It can be inferred from equation (63) that meff of the electron significantly depends on the slope of the E(k) plot. For different materials, the slope is different, hence meff correspondence different values for different materials and also for different bands. Herein, it may be stated effective mass of the carrier in the conduction band, i.e., electron (meff,e ) is completely different from the effective mass of the carrier in valence band, i.e., hole (meff,h ). It has been observed that meff,h is always greater than meff,e in semiconductors. In this context, it may also be stated that the slope of E(k) diagram is different at different symmetry points in an anisotropic semiconducting material, hence meff of the electron would also be different in different directions. In recent times, with the advancement of science and technology, many sophisticated techniques have been available to grow semiconducting material with a controlled growth orientation. So researchers are now concentrating to modulate growth orientation in order to tune meff for better device performance. CARRIER TRANSPORT PARAMETERS: DRIFT, MOBILITY, CARRIER LIFETIME, SCATTERING Drift, the motion of a charged particle under the influence of an electric field, can be understood as follows: in the presence of ζ across a semiconductor crystal, electrons and holes are accelerated in the opposite and same direction of ζ, respectively. However, during motion, electrons as well as holes, will be scattered by impurities, lattice vibrations etc., and their motion gets decelerated, i.e., the motion would be in a disjoined fashion involving periods of accelerating and decelerating collisions. An actual microscopic model for such motion is very tedious, but there exists a reasonably good macroscopic theory to describe the overall motion of the charge carriers. Averaging over all charge carriers at any given time, it is stated that the resultant motion could be written in terms of a constant drift velocity, generally denoted by vd . It is the time-averaged constant velocity with which charge carriers move under ζ. It has been noticed that drift is superimposed with a thermal motion of the charge carriers, which is approximately 1/1000th of the velocity of the light. As thermal motion is random, their average should be zero. Thus, no current is detected in the absence of ζ. Semiconductors are

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most appropriately characterized by current density (J), defined as current per unit area, rather than current. Commonly, the current density is a vector quantity and is expressed in terms of vd by the following relation: J = e(nvd,n + pvd,p )

(64)

Where, vd,n and vd,p are the drift velocity of electrons and holes, respectively. As vd arises due to ζ, hence their dependency is important here. In fact, it has been found experimentally that vd increases linearly with ζ for low values of ζ and finally it becomes saturated, independent of ζ. The following relation between vd and ζ has been derived μ0 ζ vd = (65) 1 μ ζ

[1+ (v 0 )δ ]

𝛿

sat

Where, μ0 , δ and vsat are the proportionality constant, power of the exponent and saturation velocity, respectively. The value of δ is found to be 2 for electrons and 1 μ ζ for holes. It can be inferred from equation (50) that we have ( 0 )δ ≫ 1 in the high vsat

μ0 ζ δ ) vsat

field limit (ζ → ∞), thus vd ≈ vsat . And in the low field region, (

≪ 1,

therefore equation (65) gets simplified into vd = μ0 ζ

(66)

Here, the proportionality constant μ0 is defined as mobility (μ) of the charge carriers and current density can be written with the help of equation (64) and (66) as J = e(nμe + pμh )ζ

(67a)

Where, μe and μh represent mobility for electron and hole, respectively and mostly, it is observed that μe is greater than μh . ′μ′ lying in the range of 1 and 1000 cm2V1 -1 s significantly dependents on the motion-impeding scattering of the charge carriers with impurity ions, lattice vibrations etc. In this context, conductivity, denoted by σ, is often defined as the current density per unit electric field strength and its inverse is called resistivity (ρ). Thus from equation (67a), we have an expression for σ as follows: σ = e(nμe + pμh )

(67b)

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Conceptually, μ refers to the freedom of the movement of charge carriers, and it has been noticed that μ decreases with increasing motion-impeding scattering. In semiconducting materials, μ gets influenced by several scattering processes, such as scattering with thermally agitated lattice atoms, ionized impurities etc. It should be emphasized that the scattering of charge carriers with agitated lattice atoms includes displacements of atoms from their equilibrium sites, while the influence of the internal field of atomic lattices is taken care of in the effective mass of charge carriers. Irrespective of semiconductors, μ is found to be independent of doping in the low doping concentration range; while above a certain doping concentration, μ gets reduced. On the other hand, scattering with agitated lattice doesn’t depend on impurity concentration. Actually, these two scattering mechanisms synergistically influence the resistance of any semiconductors, where total resistance (R total ) of the semiconductor can be expressed as R total = R lattice scattering + R ionized impurity scattering

(68)

where, R lattice scattering and R ionized impurity scattering represent contributions of lattice scattering and ionized impurity, respectively. In addition, μ also depends on temperature. Experimental studies show that overall μ decrease with increasing temperature following power law dependence as T −2.23 . But component-wise, thermal agitation of the atoms that decreases with decreasing T reduces R lattice scattering . On the other hand, scattering from ionized impurity increases with decreasing T and consequently R ionized impurity scattering gets increased. Therefore, it may be stated that these two scattering mechanisms are differentiable from temperature dependence. In this context, another term which is being often considered for a better understanding of the influence of several scattering mechanisms on the transport phenomenon of any semiconductors is the average lifetime (< 𝜏 >) of charge carrier, and it is related to μ by the following relation μ=e

meff

(69)

Basically, < 𝜏 > signifies the average lifetime of a charge carrier between two successive scatterings. From equation (69), we may conclude that an increase in scattering reduces < 𝜏 >, hence mobility gets decreased. As per the equation, μ is inversely related to the effective mass of the charge carriers suggesting that μe should be higher than μe in any semiconductors. In different semiconducting

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materials, a semiconductor with a lower effective mass shows higher mobility. As an example, effective masses of electrons are smaller in GaAs in comparison with Si, hence GaAs give higher μ in GaAs. In practical situations, charge carriers often encounter several scattering processes, thus μ can be written as 𝑒

μ=

meff

〈

1

〉

(70)

1

∑𝑗

where, the summation includes scattering processes. Keeping similarities with equation (68), we can broadly classify contributions from several scattering processes into < 𝜏 > as impurity contribution and lattice vibration contribution. In n- or p-type doped semiconductors, impurities (donors and acceptors) exist in ionized conditions at room temperature. These ionized impurities act as motionimpeding scattering centers for charge carriers and significantly influence the < 𝜏 > of charge carriers. In this context, Brooks – Herring have considered screened potential of these impurity ions with charge ‘z’ and find out < 𝜏 >, denoted by < τ >ionized impurity,BH , as 1 ionized impurity,BH 8ϵm

=

z2 e4 nI 16πϵ2 √2m

eff

E −3/2 [log(1 + ϑ) −

ϑ

]

1+ϑ

(71)

k T

eff B where, ϑ = E. In this expression ϵ represents the dielectric constant of the ħ2 e2 semiconductor, which includes electronic contribution and ionic polarization due to lattice vibrations, nI is the impurity concentration and E denotes the energy of the charge carrier. Few years later, a modified expression was derived by Conwell and Weisskopf. They have neglected the screening effect of the impurity atom and derived the following expression for < 𝜏 >, usually denoted by < τ >ionized impurity,CW , using Rutherford’s scattering cross-section relation

1 ionized impurity,CW

Where, Em =

ze2 4πϵrm

=

z2 e4 nI 16πϵ2 √2meff

2E 2

E −3/2 ln [1 + (

E𝑚

) ]

(72)

. In equation (72), rm denotes the cut off distance of the

scattering potential, which is related with nI by the formula nI = (2rm )−3 . In the low-temperature region, thermal energy is not sufficient to excite donors and acceptors, i.e., they act as neutral impurities. Instead of Coulomb interaction, charge

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re-distribution acts as motion-impeding scattering centers and < τ > for this interaction, denoted by < τ >neutral impurity , has the following form: 1 neutral impurity

=

20ħ meff

nnI anI

(73)

Where anI represents the Bohr radius of charge carrier moving around the impurity site. Contribution from electron–lattice, interaction to < τ > can be understood as follows: atom of a regular crystalline semiconductor vibrates around its equilibrium positions at T ≠ 0 K. When the atom is displaced from the equilibrium position, a restoring force, proportional to displacement, acts on it according to Hooke’s law. During the random motion of the atoms, vibration immediately gets damped. However, in the case of regular motion, vibration continues with small vibrational energy. In a semiconductor with more than one atom in its unit cell, if all the atoms vibrate in the same direction giving zero resultant displacement, then the vibrational mode is known as acoustic mode and quantum energy of acoustic vibrational mode is called an acoustic phonon. When a charge carrier gets scattered off due to its interaction with an acoustic phonon, the charge carrier may lose its energy, hence acoustic phonon may act as motion-impeders. Herein, an expression of < τ > due to this particular scattering, denoted by < τ >scoustic phonon , has been derived by Bardeen and Shockley as follows: 1 acoustic phonon

=

(2meff )3/2 D2ac kB T 2πħ4 ρv2s

E1/2

(74)

Where, Dac denotes the deformation potential, whereas 𝑣𝑠 and ρ are the velocity of sound and density of semiconducting material, respectively. In contrast to acoustic vibration, we consider displacement atoms in the opposite directions to each other, which results in zero motion of the center of mass. If the atoms are of ionic nature, then this vibrational mode, called an optical mode of vibration, induces an electric field. Quantum of optical vibrational energy is known as optical phonon and the corresponding < τ >, denoted by < τ >polar optical phonon , is given by the following expression: 1 polar optical phonon

with x0 =

ħ𝜔𝐿𝑂 𝑘𝐵 𝑇

≈

𝐞 meff

3√πmeff E ex0 +1 8√2kB T

ex0 −1

(75)

, where ħ𝜔𝐿𝑂 represents the optical phonon energy. Here,

derivation has been carried with the assumption that 𝐸 ≫ ħ𝜔𝐿𝑂 . In this context, it

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Solid State & Microelectronics Technology 49

may be stated that all these scattering exhibits a crucial role during charge carrier transport which may be better realized from microscopic theory as discussed below. Example 11: Given that electron and hole mobilities of intrinsic Ge are 3800 and 1800 cm2V-1s-1. If intrinsic carrier concentration be 2.5 × 1013 cm-3, conductivity (σ) of intrinsic Ge is 0.0224 Ω-1cm-1 as per equation (67b).

Example 12: If donor type impurity is being incorporated into the above-mentioned Ge semiconductor at a ratio of 1:108, then hole density (p) from the relation p = calculated to be 1.42 × 1014 cm-3.

n2i NA

is

Example 13: In the above example, conductivity (σ) can be calculated from equation (67b) as 0.268 Ω-1cm-1. CHARGE CARRIER UNDER TRANSPORT EQUATION

ELECTRIC

FIELD:

BOLTZMANN

We have noticed from equation (59) that the velocity of a charge carrier, generally expressed by vg , depends on slope of the E(k) at particular k point, hence velocity of all the charge carriers (electron in conduction band or hole in valence band) are not equal, therefore it may be stated that velocity should be a function of E. For simplicity, if we consider the movement of electrons in the conduction band only, then the number of electrons with velocity 𝐯(E) would be N(E)f(E) i.e., differential current density may be written as 𝑑𝐉 = N(E) f(E) 𝐯(E) and total current density as given in equation (62) may be written as d𝐉 = −𝑒N(E) f(E) 𝐯(E)dE

(76)

For simplicity, if we consider flow of electrons in x-direction only, then equation is given by ∞

Jx = −𝑒 ∫0 N(E) f(E) 𝑣𝑥 (E)dE

(77)

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In the absence of ζ, 𝒗𝒙 along +x and –x directions are equally probable, hence their average would be zero i.e., not current flows in the absence of ζ. In the presence of ζ, the acceleration of the electrons may be written as 𝑑𝒗𝒙 (E) eζ = − (78) 𝑑𝑡

meff eζ

i.e. 𝒗𝒙 (E) = −

meff

t

(79)

In this context, it may be stated that if we leave the system for a long time (t →∞), then equation (79) indicates that the velocity of the electron would approach to infinity, but this doesn’t happen in a practical situation that can be understood as follows: herein, we define a probability function, f(E), of charge carrier which is basically same as Fermi – Dirac distribution function to denote state of the charge carriers. In the presence of ζ, electrons initially would be accelerated and would gain energy, thus their distribution function would be changed. If f ′ (E) represents the distribution function of new state, then the rate of change of f(E) cane be written as dfelectric field (E) (f ′ (E) − f(E))⁄ = lim (80) ∆𝑡 dt ∆𝑡→0 In deriving equation (80), no scattering mechanism is being considered; hence it is often referred to as change due to the drift velocity of the charge carriers. Being accelerated and gaining velocity, carriers immediately would be scattered off through different scattering mechanisms, as discussed in the previous section. And as the electric field ζ is switched off, charge carriers would immediately lose their energy through scattering and would return the initial stated, represented by f(E). If < τ > represents average time, taken by the charge carriers to be returned from the state f ′ (E) into initial state f(E), then the rate of change of f(E) can be written as dfscattering (E) dt

=

f′ (E) − f(E)

(81)

Herein, equilibrium would be established in the presence of ζ when there would be no change in f(E). Therefore, considering two contributions felectric field (E) and fscattering (E) into probability f(E) as f(E) = felectric field (E) + fcollision (E), we have in equilibrium dfelectric field (E) dt

+

dfscattering (E) dt

=0

(82)

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From equations (81) and (82), we have dfscatteing (E)

=

dt

f′ (E) − f(E)

dfelectric field (E)

=−

(83)

dt

After rearranging equation (83), we have f ′ (E) = f(E) −

dfelectric field (E) dt

(84)

Change in variable in equation (84) gives f ′ (E) = f(E) −

dfelectric field (E) 𝑑𝑣𝑥

< τ >= f(E) +

𝑑𝑡

dvx

dfelectric field (E) eζ dvx

meff

< τ > (85)

𝑑𝑣

In the last step, we have substituted 𝑥 from equation (70). Therefore, substituting 𝑑𝑡 f ′ (E) from equation (85) into equation (77), we may obtain expression for Jx as follows: ∞

Jx = −e ∫0 N(E) [f(E) + ∞

dfelectric field (E) eζ meff

dvx ∞

= −e ∫0 N(E) [f(E)] 𝑣𝑥 (E)dE − e ∫0 N(E) [

< τ >] 𝑣𝑥 (E)dE

dfelectric field (E)

eζ

dvx

meff

< τ >] 𝑣𝑥 (E)dE

(86)

Briefly, first term of equation (86) gives Jx at ζ = 0 and it would be zero. Second term gives Jx , hence we may write ∞

dfelectric field (E) eζ

Jx = −e ∫0 N(E) [

meff

dvx

dfelectric field (E) dE

∞

= −e ∫0 N(E) [

dE

< τ >] 𝑣𝑥 (E)dE eζ

dvx meff

< τ >] 𝑣𝑥 (E)dE

(87)

If a charge carrier moves in three-dimensional isotropic semiconducting crystal, its 1 3 energy E is given by E = meff (vx2 + vy2 + vz2 ) ≈ meff vx2 , since for this 2

case

vx2

=

vy2

meff vx (E)2 =

= 2E 3

vz2 .

Therefore

substituting

dE dvx

2

= 3meff vx

and

in equation (87), we have

Jx = −

e2 ζ

∞ ∫ N(E) meff 0

[

dfelectric field (E) dE dE

dvx

< τ >] 𝑣𝑥 (E)dE

writing

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=−

2 e2 ζ

∞

3 meff

∫0 N(E) [

Roy et al.

dfelectric field (E) dE

According to Fermi – Dirac statistics, f(E) = degenerate

semiconductors, (𝐸 − 𝐸𝐹 ) ≫ 𝑘𝐵 𝑇,

approximated as f(E) ≈ may be written as

2 e2 ζ

2 e2 ζ

1

3 meff 𝑘𝐵 𝑇

𝑒

(𝐸𝐹 ) ⁄𝑘 𝑇 𝐵

1 (𝐸− 𝐸𝐹 ) ⁄𝑘 𝑇 𝐵 1+ 𝑒

(88) and for all non-

therefore

f(E)

may

be

Using the expression of f(E), equation (88)

∞ N(E) − ∫ 0 3 meff

Jx =

=

(𝐸𝐹 −𝐸) ⁄𝑘 𝑇 𝐵 . 𝑒

< τ >] E dE

[

d(𝑒

(𝐸𝐹 −𝐸) ⁄𝑘 𝑇 𝐵 )

dE

∞

∫0 N(E) [𝑒

< τ >] E dE

(−𝐸) ⁄𝑘 𝑇 𝐵

< τ >] E dE

(89)

Using N(E) from equation (43) and considering < τ > ≈ E −3/2 for electron– ionized impurity scattering as given in equations (71) and (72), equation (89) may be simplified into Jx =

2 e2 ζ

1

3 meff 𝑘𝐵 𝑇

=

2 e2 ζ

𝑒

(𝐸𝐹 ) ⁄𝑘 𝑇 2meff 3/2 ∞ 𝐵 ( ) ∫0 ħ 1

3 meff 𝑘𝐵 𝑇

𝑒

E1/2 𝑒

𝐸𝐹 ⁄𝑘 𝑇 2meff 3/2 ∞ 𝐵 ( ) ∫0 ħ

(−𝐸) ⁄𝑘 𝑇 𝐵

E1/2 𝑒

E −3/2 E dE

−𝐸⁄ 𝑘𝐵 𝑇

E −3/2 E dE

(90)

To evaluate the above integration, we consider 𝑞 = 𝐸⁄𝑘 𝑇 and after substituting it 𝐵 in equation (90), we have an expression for Jx as given below Jx ≈

𝐸𝐹 ⁄𝑘 𝑇 2meff 3/2 ∞ −𝑥 𝐵 ( ) ∫0 𝑒 dx 3 meff ħ 2 e2 ζ 𝐸𝐹⁄𝑘 𝑇 2meff 3/2 𝐵 ( ≈ 𝑒 ) ħ 3 meff 2 e2 ζ

𝑒

(91a)

And consequently, the expression for σ is found to be as follows: 𝜎≈

2 e2 3 meff

𝑒

𝐸𝐹 ⁄𝑘 𝑇 2meff 3/2 𝐵 ( ) ħ

(91b)

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Thus, we have derived equation (91b) to represent σ when the scattering of charge carriers from ionized impurities is solely being considered. However, it may be generalized for other scattering processes. CHARGE CARRIER TRANSPORT GRADIENT: DIFFUSION

DUE

TO

CONCENTRATION

Diffusion is a process by which free charge carriers tend to spread out or redistribute themselves from one part of the semiconductor to another part as a result of their thermal motion, migrating on a macroscopic scale from a localized region of high carrier concentration to a region of low carrier concentration. This particular transport mechanism is often encountered in semiconductor diodes [3]. Similar to drift at ζ = 0, diffusion involves random motion of charge carrier, however if the process is allowed for a longer time, then there would be the net average flow of charge carrier due to concentration difference. But major difference is that no external electric field is involved here. Similar to other diffusion cases, Fick’s law, equation (92) can also be used for better understanding the related phenomenon: φ(x) = −D

dc(x)

(92)

dx

where, φ(x) denotes flux of charge carriers (number of carriers crossing per unit dc(x) area per unit time), while D and represent diffusion coefficient (square meter dx per second) and concentration gradient, respectively. Herein, the negative sign of dc(x) equation (92) indicates that the diffusion occurs in the direction of decreasing . dx As the charge carriers have charge, thus diffusion would result in diffusion currents as given by the equation (92): J(x)diffusion = −|e|

dn(x) dx

(93)

Where J(x)diffusion representS diffusion current. Unlike drift, electron and hole contribute diffusion currents in opposite directions. An interesting situation would arise when both drift and diffusion currents coexist in a semiconducting material. This would be realized practically when an electric field gets developed across a semiconductor having a concentration difference of charge carrier due to doping gradient. Briefly, an electric field, called built – in – electric field, develops due to the concentration gradient of thermal excited charge carriers, thus current is resulted from the synergistic effect of built – in – electric field and carrier concentration gradient and resultant current densities are given by

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J(x)diffusion+drift = −|e|D

dc(x) dx

+ |e|c(x)μζ(x)

(94)

Since, both diffusion coefficient and mobility are not statistical parameters to present the transport phenomenon of semiconductors; hence they are mutually dependent on each other. Therefore, under thermodynamic equilibrium conditions, we may set net current across the semiconductor zero, thus from equation (94), we can write |𝑒|D

dc(x) dx

= −|𝑒|c(x)μζ(x)

(95)

In general, equation (43) and (44) are used to evaluate carrier concentration within any semiconductor materials, however they provide uniform concentration throughout the semiconductor crystal, indicating that Fermi energy should remain constant under equilibrium conditions. Thus, present c(x) suggests that Ec and Ev are no longer constant as stated in equations (43) and (44), rather they should be a function of x i.e. E𝑐 (x) and Ev (x). As a gradient exists for each charge carrier, therefore both E𝑐 (x) and Ev (x) would bend according to electron and hole concentration, respectively, referring to the phenomenon of band bending, which is very common in the case of p – n junction devices. This phenomenon of bending Ec (x) and Ev (x) is defined as band bending. Herein, it may be stated that the accumulation of charge generates a built – in – electric – field which has a strong correlation with the energy band diagram. If the interaction between charge carriers dV(x) and lattice is ignored, then the built – in – electric field ζ(x) = − , where V(x) dx is the potential energy of the free charge carriers that, in consequence, may be dV(x) 1 dE expressed as = , where E is the energy of the charge carriers. Therefore, dx

|e| dx 1 dE

substituting ζ(x) = −

|e| dx

, followed by D μ

=

dc(x) dx

=

c(x) dE kB T dx

in equation (95), we have

kB T |e|

(96)

Equation (96), known as Einstein relation, yields that ′μ′ and ‘D’ are related by a kB T universal constant at a fixed temperature. |e| , also referred to as thermal voltage at temperature T, plays a crucial role for charge carriers in macrodevices.

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HALL EFFECT: DETERMINATION OF TYPE CARRIER AND ITS DENSITY Commonly, if a piece of semiconductor or metal is placed in a magnetic field and a current flows through it, then a voltage develops in a perpendicular direction to the flowing current and magnetic field. This effect is known as Hall Effect after the name of Edwin Hall, who discovered the phenomenon in the year 1879, and voltage is called Hall voltage, denoted by VHall . The main advantage of Hall Effect is that it gives information about the nature of the charge carrier, whether a semiconductor is a n- or p-type, along with information about ‘σ’ of the material.

Fig. (18). Hall measurement setup.

The origin of the Hall Effect, rather a Hall voltage, can be understood as follows: ‘I’ and ‘B’ be the current passing through a semiconductor bar along +ve x direction and magnetic field in the + ve z direction respectively, then the charge carriers would experience a force in the –ve y direction. The force is called the Lorentz force with its magnitude is given by |e|𝐯𝐝 ×B. Now, depending on the type of semiconductor, |e| would be positive or negative. Briefly, if we consider n-type semiconductor where electrons are the majority carriers, then electrons would experience a downward force and would be accumulated on –ve y-surface, making this surface negatively charged with respect to +ve y-surface. Therefore, it can be stated that the accumulation of electrons gives rise to this Hall voltage [4]. Herein, if the polarity of +ve y-surface experimentally measured to be positive, then it can be inferred that the semiconductor is of n-type. In this context, it may be stated that the accumulation of free charge carriers at the surfaces can’t continue indefinitely. As accumulated charge sets an electric field (ζHall ) which would oppose further accumulation of charge, hence equilibrium would be reached where Lorentz force

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on the carriers gets balanced by ζHall . Therefore, we can write at an equilibrium condition |e|vd Bz = |e|ζHall Herein, Hall coefficient (R H ) is defined as R H =

(97) ζHall Bz Jx

and for n-type

semiconductors where Jx = n (−vd ) (−|𝑒|), R H is given by RH =

1 n |e|

(98)

In this context, it may be stated that electron density ‘n’ can be determined from equation (98) as well as we can say ζHall is directed towards –ve y-direction. But in the case of p-type semiconductor direction of ζHall would be the opposite. Therefore, from the polarities of the surfaces, we can conclude about the type of charge carriers. In real semiconducting materials, often bound carriers are present, however, their concentration is mostly very lower in comparison with free carriers. Thus, Hall Effect is mostly contributed by free-charge carriers. CONCLUSION In this chapter, we have discussed different crystal structures of semiconductors and quantum mechanical theory to describe their electronic structure. The free electron model provides a better understanding of material’s properties in comparison to earlier classical theory, but it can’t explain the origin of the band gap of semiconductors. Then Kronig – Penney model, which successfully describes the origin, was introduced. In subsequent sections, the fundamental of band structure calculation, the concept of indirect and direct band gap and the density of states have been elaborated. After a detailed discussion about the fundamentals of semiconductor’s property, carrier transport mechanisms in the presence of an external electric field as well as charge carrier concentration gradient, have been discussed. In this regard, several terminologies and scattering phenomena, related to charge carrier transport, have been depicted.

Fundamentals of Semiconductor

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QUESTIONS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Draw the following planes and directions in a tetragonal unit cell: (0 0 1), (0 1 1), (1 1 3) and [1 1 0], [2 0 1] along with cell axes. Write down the electronic configuration for Sn and Ge atoms. Calculate ionization energy (energy required to remove an electron) of hydrogen atom. Find out ground state energy of He atom. An electron is confined within a finite well potential. Solve the Schrödinger’s equation to find wave function of the electron. Show that paramagnetic susceptibility of metals is independent of temperature. What is the difference between metal and semiconductor? Why all the semiconductors have zero conductivity at 0K? Why semiconductors are more advantageous than metals in device fabrication? What is the difference between intrinsic and extrinsic semiconductors? Explain the significance of Pauli Exclusion Principle. A crystalline Si semiconductor is doped with 1014 boron atoms per cm3. Calculate carrier concentration of Si at 300K. What do you mean by mobility? State its dimension. State the relation between intrinsic hole and electron concentration. How would you prepare p- and n-type semiconductor? How does conductivity vary with temperature? What is diffusion current? Give it dimension. Define average life time of a carrier. What are the parameters to influence life time of a carrier? How life time of a carrier is related to conductivity of a material? What is the advantage of Hall Effect experiment? Given that hole concentration of extrinsic Ge is 2.5 × 1015 cm-3. Calculate Hall coefficient of it.

58

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Calculate Hall voltage of an n-type Si semiconductor. Given that majority carrier concentration 1017 cm-3, magnetic field 0.1 Wbm-2, d = 4 mm and electric field 5 Vcm-1.

REFERENCES [1] [2] [3] [4]

Basic Semiconductor Physics, Chihiro Hamaguchi, Springer-Verlag Berlin Heidelberg 2010. Advanced theory of semiconductor devices, Karl Hess, A John Wiley & Sons, INC., Publication, 2000. Electronic Devices and Circuit theory, R. Boylestad and L. Nashelsky, Prentice Hall, New Jersey, 1995. Electronic Principles, A. Malvino, D.J. Bates McGraw-Hill Education, 2016.

Solid State & Microelectronics Technology, 2023, 59-90

59

CHAPTER 2

Fundamentals of p – n Junction Abstract: In this chapter, fundamentals of the p – n junction, electric field across the junction, equilibrium carrier concentration on each side, etc., have been briefly discussed. Expression of built-in potential in terms of carrier density has been derived. Magnitudes of the current under zero, forward, and reverse biased conditions have been calculated. Unlike a resistor, p – n junction corresponds to static, dynamic and average ac resistance, and they have been briefly discussed here along with protocol to examine their values. These p – n junctions also exhibit capacitance, namely transition capacitance and diffusion capacitance. Herein, these parameters and their relevance in current-voltage characteristics of p – n junction and applicational aspects have been elaborated.

Keywords: p – n junction, Biasing, capacitance, Built-in-potential, Graded region. INTRODUCTION Although semiconductor devices having one type of charge carrier, like photodetector, etc., show potentiality in different applications, however, most semiconductor devices use both types of charge carriers; thus, they are constructed of an n- and p-type regions, joined together, and the interfacial boundary is known as p – n junction. Therefore, the p – n junction is the heart of all the p – n diode devices; hence, for a proper understanding of the p – n devices, it is highly essential to have detailed knowledge of the p – n junction. Physics of p – n Diode: Depletion Region, Built – in –Potential In the previous chapter, we discussed the p- and n-type semiconducting materials. Diode gets formed when these two types of materials are joined together. The interface between n-and p-type materials is called the p–n junction (Shown in Fig. (1)). In this context, it may be stated that this p – n junction leads to various inventions, including diodes, transistors, integrated circuits, etc. Thus, a good understanding of the p – n junction is essential for all semiconductor devices.

Sunipa Roy, Chandan Ghosh, Sayan Dey and Abhijit Kumar Pal All rights reserved-© 2023 Bentham Science Publishers

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Fig. (1). Schematic view of p – n junction.

As stated in the previous chapter, trivalent atoms are generally used to dope in Si for p-type conductivity, whereas pentavalent atoms are for n-type conductivity. Conventionally, as illustrated in Fig. (1), donor ions in the n-type region are represented by a plus sign as they exhibit a positive charge after donating electrons. In contrast, electrons orbiting the donor are characterized by a negative sign. Similarly, acceptors within the p-type region are characterized by a minus sign as they become negatively charged after accepting electrons, and orbiting holes surrounding the acceptor ions are characterized by a positive sign. As both n- and p-type regions are electrically neutral; hence they should be equal in number within each region. In reality, a minor concentration of electrons is present in the p-type region, while a minor concentration of holes exists in the n-type region. Thus, in general, it is stated that electrons are the majority carrier and holes are the minority carrier within the n-type semiconductor. At the same time, reverse distribution happens for the p-type region. Due to the concentration gradient, electrons get diffused into the p-type region from the n-type region, while holes diffuse opposite to reach equilibrium. Electrons immediately recombine because there are so many holes in the p-type region. As a result, immobile acceptor ions within the p-type region carrying negative charge become uncovered. Similarly, hole migration into the n-type region uncovers the positive charge of immobile donor ions (as shown in Fig. (2)). These unneutralized Ions in the neighborhood of the p – n junction is uncovered charges [1].

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Fig. (2). Schematic diagram of a p – n junction. The yellow portion denotes the depletion region.

Since this region is depleted of free charges, it is often referred to as the depletion region, the space-charge region, or the transition region. It has been noted that the depletion region corresponds very narrow width on both sides of the junction, and beyond this region rest of both n- and p-type regions remain as usual in a neutral state. This region is the heart of all p – n diode or transistor devices. It has been observed that the depletion region's width and density (ρ) mainly depend on the distribution of immobile ions, with zero charges at the junction. Two different distributions, step-graded and linear-graded doping, are being formulated here (discussed later). If the concentration of acceptor ions (NA ) is different from that of donors (ND ), then space charge is asymmetric about the junction; however, in this case, the charge neutrality condition, as given in equation (1), predicts that the width of space charge is on both sides of the junction. ND xn = N𝐴 xp

(1)

Where xn and 𝑥𝑝 represent the width of the space charge in the n-type and p-type regions, respectively; this particular relation has assumed zero free carriers within the depletion region. As mentioned earlier, for any p – n diode, the charge distribution of the depletion region plays the most crucial role in the current-voltage characteristics. Herein, this distribution constitutes a layer of an electric dipole, giving rise to electric lines of flux from immobile positively charged donor ions of the n-type region into immobile negatively charged acceptor ions of the p-type region (shown in Fig. 2). Thus, the built-in – electric field (𝜉built−in−potential ) plays the most important role in every charge transport characteristic through the p – n diode. However, in this

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context, it may be stated diffusion of majority charge carriers can't continue indefinitely; here, an equilibrium would be reached that may be understood as follows: initially, very few majority carriers diffuse the interfacial junction, i.e., 𝜉built−in−potential is low enough to limit the diffusion of the majority of carriers across the junction. However, as time passes, the diffusion of more and more majority carriers strengthens 𝑡ℎ𝑒 𝜉built−in−potenti al. Ultimately, a critical 𝜉built−in−potential would be achieved where the majority of carriers' diffusion is restrained completely. At the same time, 𝜉built−in−potential causes the drift of the minority carriers in a direction opposite to the majority carriers. Stated otherwise, diffusion of majority carriers from either side is equal and opposite to drift of minority carriers in steady state conditions, giving zero resultant currents. The p – n diode acts as an open circuit device. The equilibrium can be interpreted in terms of band edges and Fermi levels (EF ) as follows: for simplicity, we consider Si as the prototype simplest semiconductor with n- and p-type region and respective positions of EF as shown in Fig. (3a), called flatband diagram. In this case, the conduction band edge is separated with respect to the vacuum level by energy |e|χ, called the work function of Si, where χ represents the electron affinity of Si. As stated in the previous chapter, EF lies close to the conduction band edge in the ntype region, while it appears just above the valence band edge in the case of the ptype region. Initially, the flatband diagram doesn’t represent the equilibrium situation as majority carriers since the energy of electrons in the n-type region and holes in the p-type region get lowered after crossing the junction. The diffusion continues until equilibrium is reached when the EF of both sides becomes the same, as shown in Fig. (3b). The missing conduction and valence band edges have been completed in Fig. (3c), assuming their monotonic variation near the junction point and zero slopes at the two ends of the central region. This phenomenon is known as band bending. Though the exact form of band bending is completely unknown, the variation represented in Fig. (3c) is the most accepted one [2]. In this context, it may be stated that a voltage drop, commonly known as built-in potential Vbuilt−in−potential ), occurs across the depletion region under equilibrium conditions, and as a thumb rule, it is defined as the difference between band edges of the two regions, as shown in Fig. (3d).

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Fig. (3a). Left hand and right side represent p-type and n-type semiconductors, respectively (before joining). Fig. 3(b): Alignment of Fermi energy after joining. Fig. 3(c): Alignment of conduction and valence band edges at the junction. Fig. 3(d): Build-in potential at the junction

Though the thumb rule, as stated in Fig. (3d), works well for most of the semiconductors, an actual expression for Vbuilt−in−potential and associated electric field 𝜉built−in−potential can be derived in the following way: through the junction, carriers flow due to 𝜉built−in−potential and diffusion, and an equilibrium condition, no current flows through the junction; hence we can write current density J(x)diffusion+drift as J(x)diffusion+drift = −|e|Dn

dn(x) dx

+ |e|𝑛(x)μn 𝜉built−in−potential (x) = 0

(2)

Where, Dn and μn represent diffusion coefficient and mobility of electrons, respectively and 𝑛(x) is the carrier concentration. From equation (2), we have an expression for 𝜉built−in−potential (x) as follows:

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𝜉built−in−potential (x) =

Roy et al. Dn

dn(x)

𝑛(𝑥)μn

dx

(3)

Therefore, Vbuilt−in−potential (x) with the help of equation (3), can be written as Vbuilt−in−potential (x) = − ∫ 𝜉built−in−potential (x)dx

= −∫

Dn dn(x)

(4a)

μn n(x)

From Einstein’s boundary condition, we have

Dn μn

=−

𝑘𝐵 𝑇 |𝑒|

, thus substituting this in

equation (4), we have Vbuilt−in−potential (x) =

𝑘𝐵 𝑇 |𝑒|

∫

dn(x)

(4b)

n(x)

If ( )XP and xn be the width of depletion regions in p- and n-type regions, respectively, with electron densities n(xn ) = ND and 𝑛(−𝑥𝑝 ) =

𝑛𝑖2

𝑁𝐴

respectively,

then from equation (4b), we have the expression for Vbuilt−in−potential ( ) as Vbuilt−in−potential = In equation (5),

𝑘𝐵 𝑇 |𝑒|

𝑘𝐵 𝑇 |𝑒|

ln

NA ND n2i

(5)

is often called the thermal equivalence of voltage and is denoted

by Vth . At room temperature (T = 300K), VTh = 26 mV at absolute temperature. In this expression, ni is the intrinsic carrier concentration of the semiconductor, while NA and ND represent acceptor and donor concentrations. According to equation (5), the variation of Vbuilt−in−potential across the junction looks as presented in Fig. (4a), where associated 𝜉built−in−potential ( ) is shown in Fig. (4b).

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Fig. (4). Variation of (a) built-in-potential and (b) built-in-electric field across the junction.

Example 1 At room temperature (T = 300K), built-in-potential in case of typical Si semiconductor with 𝑛𝑖 = 1010, NA = ND = 1015 is calculated to be ~ 0.6V using equation (5). Current-voltage Characteristics of Biased p – n Junction p – n junction is basically a two-terminal device that facilitates current flow depending on the magnitude of the external bias voltage (VD ). Herein, it may be stated that VD opens three possibilities: zero, forward, and reverse biased conditions, each giving different current-voltage characteristics. In this section, all these conditions will be briefly discussed.

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Zero Biased Condition (𝐕𝐃 = 𝟎𝐕) Under VD = 0V condition, the majority carriers, say, for example, electrons in the case of an n-type semiconductor, are attracted by the positive ions in the depletion region of this region and are repelled by negative ions of the depletion ions of the p-type region. Though large numbers of electrons are present in the n-type region, however, a very small fraction of them has sufficient kinetic energy to cross the depletion region of the p-type region. It is also true for a hole in the p-type region also. Thus it can be generalized that most of the majority carriers can't be found beyond the depletion region of other sides; only a few can be found. Now, if the terminals of the p – n diode are joined externally, then a very low magnitude of current would flow through the diode, then it is referred to as 'forward current.' On the other hand, minority carriers, say, for example, holes in the n-type region, if found within the depletion region of the n-type side, would be immediately drifted by 𝜉built−in−potential to cross the junction as well as the depletion region of the ptype side; hence they would be found beyond the depletion region of the p-side. If the terminals are connected externally, then a current would flow due to minority carriers, commonly known as 'reverse current.' As these two currents flow in opposite directions, hence the resultant current through the diode would be zero under zero bias conditions. This junction condition is often referred to as Dynamic equilibrium, and zero current flow at dynamic equilibrium condition can be understood as follows [3]: In general, the p – n diode appears to be short-circuited at VD = 0. Under this condition, no current flows through the diode, and the electrostatic potential across the junction remains unchanged and is similar to the open-circuit condition. The phenomenon can be better understood thermodynamically as follows: as VD = 0, therefore required energy to heat metallic wire would have to be supplied by p – n junction potential, i.e., Vbuilt−in−potential , and consequently, the junction has to be cooled off. According to thermodynamics, simultaneous heating and cooling of the same system are impossible, indicating zero current flow through the circuit. Stated otherwise, some of the voltage drops within the closed loop should be zero, i.e., Vbuilt−in−potential is compensated by metal–to–semiconductor contact potential in the case of Ohmic contact. Hence it is not possible to measure the difference in contact potential directly using a voltmeter. However, it has been observed that minority carriers are constantly being generated by thermal energy; hence any change in temperature results change in Dynamic equilibrium, and then resultant current flows due to minority carriers are referred to as leakage current.

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Forward Biased Condition (𝐕𝐃 > 𝟎 𝐕)

Fig. (5). p – n junction under forward biased condition.

Under forward-biased conditions (VD > 0 V), the n-type region is connected to a negative potential, while the p-type side is to the positive potential of an external power source, as shown in Fig. (5). In this condition, the net potential at the p – n junction gets lowered, which can be understood as follows: here EF of both sides no longer line up but are shifted with respect to an unbiased condition, as represented in Figs (6a and b). Briefly, this shift is upward for the n-type region with respect to the p-type region, and band edges get modified accordingly. Such shift lowers the barrier height of the junction potential, which in consequence, facilitates the injection of electrons from the n-side to the p-side and leads to a flow of current in the circuit. Such current is referred to as current due to the injection of carriers in the p – n junction.

Fig. (6). Band diagram of p – n junction under (a) equilibrium and (b) forward bias. While Fermi levels line up in equilibrium in the presence of an external potential, the levels shift by an amount proportional to the applied potential.

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Fig. (7). Schematic diagram for the distribution of charge under forward-biased conditions.

Under the forward biased condition, variation of charge concentration along the length of the junction, schematically shown in Fig. (7), can be understood as follows: in equilibrium condition, carrier concentrations (majority and minority) of n- and p-type region under unbiased conditions are generally represented by the following expressions: n2i

in n-type region: nmajority = ND and pminority =

(6a)

ND

and in p-type region: pmajority = NA and nminority =

n2i

(6b)

NA

After injection, the concentration of majority carriers, i.e., electrons in the n-type region and hole in the p-type region, increases significantly as the potential barrier gets decreased due to forward bias, majority carriers across the junction and become minority after entering into the other side. In general, these minority concentrations can be expressed as follows [4]: in n-type region: pminority = pmajority in p−type region exp(−

e(Vbuilt−in−potential − VD ) kB T

)

(7a)

and in p-type region: nminority = 𝑛majority in n−type region exp(−

e(Vbuilt−in−potential − VD ) kB T

) (7b)

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These excess minority carriers diffuse into more distance before recombination and contribute to the forward current. In general, the distance traveled by these minority carriers before recombination is called minority carrier diffusion length (L), expressed by L = √D τ, where D and 𝜏 are the diffusion coefficient and lifetime of the charge carriers. Interestingly, if the length of the n- or p-type region is larger than the respective carrier diffusion path length, then minority carrier concentration falls exponentially as follows: in n-type region: pminority = pminority (x = 0)exp(−

x Lℎ

)

(8a)

and x

and in p-type region: 𝑛minority = 𝑛minority (x = 0)exp(− ) L𝑒

(8b)

where, pminority (x = 0) and 𝑛minority (x = 0) represent the minority concentration at the junction and the distance x is being measured from the junction in both regions. Le and Lℎ are the diffusion length of electrons and holes, respectively. From equations (6a) and (8a), we may write excess hole concentration in n-type region as ∆pminority (𝑥) = ∆pminority (x = 0)exp(−

x Lh

)

(9a)

Similarly, excess electron concentration in p-type region can be written as x

∆nminority (𝑥) = ∆nminority (x = 0)exp(− )

(9b)

Le

In this context, it may be stated that base minority concentrations, given by equations (6a) and (6b), is very low; hence the excess minority concentrations, expressed by equation (9a) and (9b), are mostly contributed from flow majority carriers from other sides. Therefore, diffusion current due to these minority carriers, for example, hole in the n-type region (JD,hole ), can be expressed by the following expression according to Fick's law of diffusion. JDiffusion,hole = −eDh

dpn (x) dx

= −eDh

d∆∆pminority (x) dx

=

eDh 𝐿ℎ

∆𝑝minority (x = 0)exp(−

x Lℎ

)

(10a)

Similarly, we can derive expression of diffusion current due to electron in p-type region as

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JDiffusion,electron = −eDe

dn𝑝 (x) dx

= −eD𝑒

Roy et al.

d∆nminority (x) dx

=

eDe 𝐿𝑒

x

∆nminority (x = 0)exp(− ) (10b) L𝑒

It can be inferred from equations (10a) and (10b) that the sum of electron ad hole current (shown in Fig. 6) is constant and independent of position. However, the exact expression of the sum of the hole and electron currents can be evaluated as follows [4]: At junction (x = 0), from equation (10a) and (10b), we have JDiffusion,hole =

eDh 𝐿ℎ

∆pminority (x = 0) =

eDh 𝐿ℎ

[pminority (x = 0) − pminority ]

(11a)

and JDiffusion,electron =

eDe 𝐿𝑒

∆nminority (x = 0) =

eDe 𝐿𝑒

[nminority (x = 0) − nminority ] (11b)

Using expressions (7a) and (7b), we have eVD

[pminority (x = 0) − pminority ] = pminority [exp (

kB T

) − 1]

(12a)

and eVD

[nminority (x = 0) − nminority ] = 𝑛minority [exp (

kB T

) − 1]

(12b)

After substituting expressions of pminority and 𝑛minority as given in equations (6a) and (6b), equations (11a) and (11b) may be written as JDiffusion,hole =

eDh n2i 𝐿ℎ ND

[exp (

eVD kB T

) − 1]

(13a)

) − 1]

(13b)

and JDiffusion,electron =

eDA n2i Le NA

eVD

[exp (

kB T

Summing equations (13a) and (13b), we have the expression for the resultant current through the junction JDiffusion = JDiffusion,hole + JDiffusion,electron = [

Now equation (14) may be written as

eDh n2 i 𝐿ℎ ND

+

eDA n2 i ] Le NA

[exp (

eVD kB T

) − 1] (14)

Introduction

Solid State & Microelectronics Technology 71 eVD

JDiffusion = JS [exp ( Dh

Where, JS is given by JS = 𝑒n2i [

Lh ND

+

eDA Le NA

kB T

) − 1]

(15)

]. Equation (15), known as the

Schockley equation after the name William Schockley who invented this equation in the year 1949, gives the current through p –n diode under forward biased conditions. In this context, it must be mentioned that equation (15) has been derived assuming that widths of the n- and p-type region are typically larger than carrier diffusion length; this type of diode is known as a long diode. On the other hand, there may also be an exit diode with n- and p-type regions smaller than diffusion lengths; then, in deriving expression for JS , we have to substitute Lh and Le by dimensions of the respective regions. It has often been observed that minority carriers recombine in the depletion region during diffusion, but this effect is not included in equation (15). However, this has a significant effect on current-voltage characteristics and is highly dependent on the applied voltage. In fact, it increases exponentially with VD . Herein, considering the recombination of charge carriers, equation (15) is modified in the following form: JDiffusion = JS [exp (

eVD

) − 1]

ηkB T

(16)

Where η, known as the ideality factor, is found to be 1 in the case of diffusioncontrolled current, while η = 2 when the minority recombination process predominates current through p – n diode. In reality, with an increase of external bias voltage, the width of the depletion region decreases continuously, resulting in numerous numbers of electrons flowing beyond a critical value of VD , called threshold or cutin or offset or breakpoint voltage, denoted by VDγ . Beyond this threshold, the current through p – n diode (JD ) increases exponentially, which is well-expressed by equation (16) for all semiconductor-based p – n diodes. In addition, constant η is found to be 1 for Ge and 2 for Si, indicating that JDiffusion is controlled by diffusion for Ge junction, while recombination dominates within Si. Herein, it may be stated that η plays an important role in diode characteristics. As an example, it may be stated from equation (16) that JD rises more rapidly with VD for Si in comparison with Ge. In the case of VD > 0 V, the first term dominants over the second term; thus, JD increases exponentially, corroborating experimental results of all semiconductorbased p – n diodes. Under VD = 0 V condition, we have JD = 0 from equation (16), in well-agreement with the previous discussion. However, in the current-voltage

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characteristics of the diode, VDγ depending on particular semiconductors, plays a crucial role in diode applications. As an example, VDγ is found to be 0.2 and 0.6 V for Ge and Si, respectively. It is the general tendency that semiconductors with higher JS exhibits lower VDγ . Herein, for large values of VD , we have ηeVD /k B T"; thus, equation (16) may be simplified into the following form after taking the logarithmic of both sides, log(JD ) = log(JS ) +

ηeVD kB T

(17)

Hence, from the slope of the log(JD ) versus VD plot, we can determine η, and from the intercept of the straight line on the VD axis, we can evaluate JS . In fact, equation (17) is more useful for determining diode parameters experimentally. If we look into the current-voltage characteristics of the diode, we can notice that in the high current region, an increment of VD doesn't yield a significant enhancement of JD similar to that of the low current region, and it is an indication of the resistive behavior of the p – n diode. Briefly, it may be stated that in the low current region, the voltage drop across the semiconductor is negligible; hence VD mostly reduces Vbuilt−in−potential giving very sensitive JD on VD . As implicated in equation (17), temperature, an important parameter influencing diode characteristics, often plays a crucial role in the case operation of a p – n diode at a higher temperature. As JD increases with temperature, thus maintain JD constant at elevated temperature, VD dV has to be lowered. In fact, /dt, an important parameter in this context, is found to be approximately 2.5 mV/ºC for both Si and Ge to maintain constant JD . In dV general, | D| decreases with increasing T for most of the semiconductor crystals. dT

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2.2.3 Reverse biased condition (𝐕𝐃 < 𝟎 𝐕)

n-Re g i on

p-Re g i on

Fig. 8: p – n junction under reverse biased condition. Under reverse-biased conditions (VD < 0 V), n- and p-type regions are connected to the positive and negative potential of an external power supply, as shown in Fig. (8). In this situation, VD is in the same direction as that of Vbuilt−in−potential , and it causes the majority of carriers to move away from the junction spreading the width of the negative and positive charge region of n- and p-type sides, respectively. Briefly, a large number of majority carriers, i.e., electrons in the n-type region and holes in the p-type region, are withdrawn by the external power supply; thus, more and more numbers of immobile ions become uncovered, widening the depletion region. In the energy band diagram, the corresponding shift of EF is shown in Fig. (6), which is completely opposite to that under forward-biased conditions. Due to the high potential barrier, negligible diffusion current due to the majority carrier flows through the diode. However, a small current, called reverse saturation current (JS ), which is independent of the reverse bias voltage, is observed here. The origin of such current is generally attributed to thermally generated minority carriers within the depletion region, followed by drift due to VD . This current under reverse biased condition, also referred to as drift current, is expressed by the following relation [2]: Dh

JS = en2i [

Lh ND

+

eDA Le NA

]

(18)

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As the generation of minority carriers significantly depends on temperature, hence reverse bias current is found to be increasing with the increase of the operating temperature, i.e., the resistance of a semiconductor crystal is said to be decreasing with an increase in temperature under reverse biased conditions. Experimentally, JS is found to be increased by 8% and 11% for Si and Ge, respectively, due to per degree rise of temperature. Though in some p – n diodes, leakage current contributes 𝑡𝑜 JS , it doesn’t depend on temperature. Hence from the temperature dependence experimental data, we can differentiate these components. Commonly, for n-type semiconductors, JS is simplified into the following relation: JS = en2i [

Dh Lh ND

]

(19)

Current flow under reverse-biased conditions may also be understood from the energy band diagram as follows: when a reverse bias voltage VD is applied, the potential energy barrier gets increased by the amount |e|VD for the flow of majority carriers, i.e., electrons in the n-type region and hole in the p-type region. However, the potential energy acts downhill for minority carriers facilitating their flow. 2.3 Contact for p – n diode: choice of suitable material Though we have discussed currents under different biased conditions, the most crucial part in this point is the electrical contact with the external battery by which this biasing has to be carried out. Normally, metal wire is used to make the electrical contacts, but it has been noticed that a contact potential develops at these junctions. Such contact shows non-linear current-voltage characteristics with rectifying nature. Hence the proper choice of metallic wire giving non-rectifying characteristics is very important in order to obtain correct electrical features from p – n diodes. Therefore, we have to choose metallic wire such the contact potential across should be constant, independent of the direction and magnitude of the current. This type of contact is called Ohmic contact. In the case of Ohmic contact, VD solely changes the height of the potential barrier at the junction as there is no voltage drop at the metal-semiconductor junction as well as the semiconductor bulk. Commonly, it is very difficult to find out suitable metals for Ohmic contact with semiconductor crystals; very few exist in nature for Ge and Si diodes. Resistance of p – n Diode: Static, Dynamic, And Average Ac Resistance Due to nonlinearity in I – V characteristics of p – n diode, the resistance of a p – n diode varies from one point of operation to another point depending on the type of

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applied voltage or signal. Commonly three different types of resistance are illustrated here: (i) static resistance, (ii) dynamic resistance, and (iii) average resistance. In the presence of dc applied voltage across the p – n diode, the operating point remains constant on the I – V plot. In this particular case, like other Ohmic devices, static resistance, defined as the ratio of voltage (VD ) to current (ID ), is being calculated from the reciprocal of the slope of the line joining the operating point and origin as shown in Fig. (9).

Fig. (9). Determination of static resistance at a particular operating point.

Commonly, static resistance, also known as dc resistance, at or below the knee is much higher than the resistance corresponding to the vertical rise section of the characteristic current-voltage curve. Static resistance within the reverse-biased region is always found to be very high in comparison with that of the forwardbiased region. As static resistance varies widely with an operating point, thus it is not considered to be a useful, practical parameter to present diode characteristics, particularly when an ac bias voltage is applied across the diode. In this case, dynamic resistance (rdynamic ) or incremental resistance or ac resistance, defined as the reciprocal of the slope of the current-voltage characteristic curve, is expressed by rdynamic =

dVD dID

(20)

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As the current-voltage characteristic curve is not linear, thus rdynamic is not constant; rather, it highly depends on a particular VD . rdynamic is of particular importance for applied ac voltage and can be understood as follows:

Fig. (10). ac voltage across the p – n junction at Q-point.

It is obvious from the previous discussion that the static resistance of a p – n diode is independent of the shape of the characteristic in the neighborhood region of the point of operation. Instead of dc voltage, if a sinusoidal voltage is applied across the diode, then the point of operation would move instantaneously up and down, as shown in Fig. (10). In this context, a point, known as a 'quiescent-point' and is denoted by Q-point, is considered to represent the point of operation with no varying signal and the straight line drawn tangent to the curve at Q-point (shown in Fig. (11)) as defined in equation (20) denotes rdynamic . Importantly, the change in voltage and current would be as small as possible and symmetric about Q-point.

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Solid State & Microelectronics Technology 77

Fig. (11). Tangent line at the Q-point.

It is the general feature of the curve that the steeper the slope, the less change in VD for a given change in ID , indicating less value of rdynamic , i.e., rdynamic would be quite low in the vertical region, while it would be much higher in the low current region. In this context, dynamic conductance (𝑔dynamic ) which is reciprocal of rdynamic (g dynamic =

1 rdynamic

), can be written in the following form from equation

(16) as g dynamic =

dID dVD

=

ηe kB T

ηeVD

IS [exp(

kB T

)] =

ηe kB T

Equation (21) depicts that under high reverse bias voltage (|

ηeVD

would be very low. During the high forward biased condition high, and equation (21) may be approximated as rdynamic ≈

kB T ηeID

[ID + IS ]

kB T

|

(21)

"1),the dynamic

s,the dynamic

is very

(22)

Therefore, it can be concluded from equation (22) that rdynamic varies inversely with ID . It can be further simplified that rdynamic ≈

26 mV ID

(23)

for η = 1 as in the case of Ge. In this context, it may be stated that the Ohmic body resistance of the semiconductor crystal is the same order of magnitude or sometimes

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even higher than dynamic . Equation (23) implies that rdynamic can be simply evaluated by substituting the quiescent value of current without knowing any other characteristics of the diode. Importantly, it should be mentioned that for ID below the knee of the curve, equation (23) is inappropriate. However, rdynamic , defined in equation (22), doesn’t include the resistance of the semiconducting material itself, often called body resistance, and the contact resistance between the semiconductor and external metallic connector. These two additional resistances are added in series with rdynamic , and it is redefined by the following relation [4]: rdynamic ≈

26 mV ID

+ rS

(24)

where rS represents total resistance from the above-mentioned two components. Commonly, it is found that rS ~ 0.1 – 0.15 Ω for high-power diodes and ~ 2 – 5 Ω for general-purpose low-power diodes. But in most of the diodes, rdynamic " rS , hence for all practical purposes, rS contribution is neglected. Very often, it is found that the input signal is very high, and then the resistance associated with this high signal is the average dynamic resistance (𝑟average,dynamic ) or average ac resistance. In contrast to the previous two resistances, 𝑟average dynamic s are determined by a straight line drawn between two intersections, joined by the maximum and minimum input voltage, as shown in Fig. (12).

Fig. (12). Straight line between limits of operation.

Example 2 Changes in ID and VD are found at 10 mA and 0.02 V; then, dynamic resistance is 2 Ω according to equation (20).

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Solid State & Microelectronics Technology 79

Example 3 ID and VD are found to be 2 mA and 0.5 V; then, static resistance is found to be 250 Ω. PIECEWISE-LINEAR ANALYSIS OF DIODE CHARACTERISTIC Though static, dynamic resistances are used to specify p – n junction diode, however more accurate information is obtained from piecewise linear diode characteristic, which has been developed on the basis of large-signal approximation. This technique of analyzing diode characteristics, followed by obtaining an equivalent circuit, is to approximate the characteristic feature of the p – n diode by a straight-line, as illustrated in Fig. (13), and the corresponding resultant equivalent circuit is defined as a piecewise-linear equivalent circuit.

Fig. (13). Current-voltage characteristic of a p – n junction diode.

However, drawing a straight line should not be exact, particularly in the knee region. But the resulting segments are close enough, which results in an excellent first approximation to the actual behavior of the diode. For the slopping section, is the average, the dynamic is particularly used to signify the resistance of the diode in the 'on' state, as represented in Fig. (13). Briefly, the ideal diode allows current flow in one direction only while limiting current flow in the reverse direction, i.e.,

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the diode remains in an open circuit state under reverse-biased conditions. As the diode doesn't reach the conduction state until the applied voltage is higher than thresh hold voltage; thus the external voltage source, preferably a battery, opposing the conduction direction appears in the equivalent circuit, as shown in Fig. (14).

Fig. (14). Components of the piecewise-linear equivalent circuit.

After reaching the conduction state, diode resistance is specified by average,dynamic . However, for all practical uses, the average dynamic is very low such that it may be ignored in comparison with other elements of the network; therefore, the removal of raverage dynamic from the circuit depicts that the equivalent circuit would appear as presented in Fig. (15).

Fig. (15). Equivalent circuit when average dynamic resistance is zero.

The next level of approximation includes ignorance of the threshold voltage that shifts the vertical current line toward zero voltage. Under this ideal diode situation (shown in Fig. (16a), the equivalent circuit is represented in Fig. (16b). In the semiconductor industry, the phrase "diode equivalent circuit" is very popular where the diode model is being used to represent existing devices.

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Solid State & Microelectronics Technology 81

Fig. 16(a). Current-voltage characteristic of an ideal diode, (b) equivalent circuit of an ideal diode.

CAPACITANCE OF P – N JUNCTION UNDER THE REVERSE BIASED CONDITION: TRANSITION CAPACITANCE Often p – n diodes are used in the presence of an ac signal. As there are numerous immobile charges, the diode often exhibits a capacitive effect that acts in parallel to the equivalent diode circuit, as shown in Fig. (17).

Fig. (17). Equivalent circuit considering capacitance of a reverse biased p – n junction.

In most of the diode cases, the capacitive shunt effects of the p – n diode are neglected in the low-frequency region due to the high reactant value, and it is mostly represented by open-circuit conditions. However, in the low-frequency region, reactants can’t be neglected as a new low-reactance shorting path gets opened in this region. Remembering from the fundamental, a capacitor gets formed when some insulating medium separates two metallic plates, and the formation capacitance can be corroborated in the following way [3]: As per our previous discussion, reverse bias takes away the majority of carriers from the junction boundary (can be considered as metallic plate), thereby uncovering more immobile charges (insulating medium), and the thickness of this space-charge layer increases with increasing reverse-biased voltage. Therefore, an

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increase of uncovered charge with voltage may be considered a capacitive effect, and the respective capacitance, generally termed as transition-region, space-charge, barrier, or depletion-region capacitance (CT ), may be expressed by the following differential equation: CT = |

dQ dVD

|

(25)

where, dQ and dVD represent differential change in charge and reverse biased voltage, respectively. As ID is defined as the differential change in charge per unit dQ time (ID = ), therefore substituting this in equation (25), we have dt

ID = C T

dVD dt

(26)

Hence, equation (26) depicts that knowledge of CT is highly essential to determine the current through the p – n diode. In this context, it may be stated that similar to resistance, capacitance is not constant; rather, it varies with applied reverse bias voltage; the larger the reverse voltage, the larger the space-charge width, and hence smaller the value of CT as represented in Fig. (18).

Fig. (18). Variation of capacitance with reverse bias voltage.

But, in the case of the forward-biased condition, an opposite trend is observed. Herein, CT is defined more accurately in differential form instead of simply the ratio of charge (Q) to voltage (VD ).

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Solid State & Microelectronics Technology 83

Fig. (19). Equivalent circuit of varactor diode.

This voltage-dependent CT has gotten attention in some applications, including LC resonant circuits, self-balancing bridge circuits, and a few special types of amplifiers, called parametric amplifiers, frequency multipliers, bandpass filters, and harmonic generators. Under the reverse biased condition, the equivalent circuit of these types of p – n junction diodes, also called varactors, varicaps, or voltacaps, as shown in Fig. (19), consists of resistance, denoted as reverse biased resistance (R R ) in parallel with CT and the whole is in series with the body resistance of the semiconductor crystal (R G ). Typically, R G ~ 5 to 15 Ω and CT ~ 10 to 50 pF at a reverse bias voltage of approximately 2 – 5 V. Under reverse-biased conditions, R R is mostly greater than a few MΩ, hence for all practical purposes, it is being neglected. The maximum operating frequency (fmax ) of the varactor diode 1 depending on R G and CT is given by fmax = with quality factor (Q) at 2𝜋RG CT

fmax

frequency ‘f’ is Q = . For practical uses in the high-frequency region, in f general, CT is kept low as much as possible due to the following reason: a varactor diode is being used to prevent transmission of a signal herein if CT has a large value, instead of being restrained, current flows through the circuit. What is important to deal with the varactor diode is its magnitude which may be calculated as follows: from equations (2) and (4), we may write for potential (V) in the transition region of the n-type side 𝑑2V 𝑑2𝑥

= −|𝑒|

ND 𝜀

(27)

Though equation (27) is primarily used to derive the expression of charge and capacitance, depending on the situation, two different transition junctions are classified here, and accordingly, the solution would be different. One is the step graded junction, and the other is the linear graded junction.

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Transition Capacitance for Step-graded Junction In step-graded junction, the density of immobile ions changes abruptly in either region, i.e., acceptor ions in the p-type region and donor ions in the n-type region. In this case, NA is kept unequal with ND intentionally. Assuming NA "ND and the distribution of immobilized charge from x = 0 to x = Wn , i.e., Wn represents the width of the spatial distribution of donor ions in the n-type region, then from Poisson's equation, we can correlate voltage with ND within the transition region. After integrating the equation, followed by the boundary condition that the electric field at junction boundary (x = Wn ) is zero, we have dV dx

= −|𝑒|

ND 𝜀

(𝑥 − Wn )

(28)

In the next step, using the fact that potential is zero x = 0 and after integrating equation (15), we may write. V = −|𝑒|

ND 2𝜀

(x 2 − 2Wn 𝑥)

(29)

From equation (29), we have an expression of potential at x = Wn , termed junction potential (Vjunction potential ) given by Vjunction potential = |𝑒|

ND 2𝜀

Wn 2

(30)

In this context, it may be stated that Vjunction potential can be explicitly written as Vjunction potential = V0 − VD , where V0 represents contact potential, and VD is the reverse biased voltage that has been applied externally. From equation (30), we may infer that Wn can be expressed in terms of VD as follows: 2𝜀

Wn = [|𝑒|N (V0 − VD )]1/2 D

(31)

εA

Using the well-known formula of capacitance CT = , where A and d are the area d and thickness of the dielectric slab, then from equation (31), we have the following form of CT as given by CT =

εA Wn

|𝑒|ND 𝜀

= 𝐴√

2(V0 − VD )

(32)

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Solid State & Microelectronics Technology 85

Thus, equation (32) depicts that CT increases with increasing VD . However, if ND is not neglected here, then the expression of Wn would be slightly modified as 2𝜀Vjunction potential

Wn 2 = [

|𝑒|

][

1 ND

+

1 NA

], and accordingly, equation (32) would be

modified. Experimentally, such type of step graded junction is achieved by incorporating impurities atoms into the host matrix, followed by heating at a very high temperature for a short period of time. Often, this type of junction is called an alloy or fusion junction. Linear Graded Junction In a linear graded junction, the net charge varies with distance in the transition region [4]. However, assuming ND = NA here, we may deduce the expression of a width corresponding to the transition region (W) using Poisson’s relation as W = √

6ε (V0 − VD ) |𝑒|ND

, and CT as |𝑒|ND 𝜀

CT = 𝐴√

6(V0 − VD )

(33)

In comparison with a single-step junction, a linear graded junction is often encountered between the collector and base of an integrated transistor and has more practical utility, particularly in the field of a bandpass filter, amplifier, adjusting bridge circuit, voltage-controlled oscillator, etc. CAPACITANCE OF P – N JUNCTION UNDER THE FORWARD BIASED CONDITION: DIFFUSION CAPACITANCE Under the forward-biased condition, injected charge carriers are accumulated near the junction just outside of the transition region. It has been noted that the charge, stored out the transition region, gets increased with increasing forward biased voltage giving the capacitive property, commonly defined as diffusion or storage capacitance (CD ). In general, it has been observed that CD has quite a higher magnitude than CT , while under reverse-biased conditions, the opposite situation happens. As in other cases, CD is defined as the ratio of incremental change in charge to increase of forward bias voltage VD and the expression of CD may be deduced as follows: let us assume that at a particular time, forward-biased voltage be VD and the voltage has been changed very slowly in time τF in such a way that current (ID ) remains the same as DC current flows at that particular voltage. During

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this change, the amount of charge Q D transit through the device at this particular moment, then we may write. Q D = ID (VD )τF

(34)

Hence, using equation (12), we may derive the expression of CD as follows: CD =

𝑑QD 𝑑VD

=

ID (VD ) 𝑑VD

τF = g τF

(35)

Where g is the dynamic conductance under forward biased conditions, it is obvious from equation (35) that CD is proportional to ID . In this derivation, if we consider the contribution of holes and electrons in p- and n-type regions, then we have a synergistic effect from both ends, and diffusion capacitance would be added up. To get an idea about the magnitude of CD in practical situations, we consider germanium crystal in which η = 1. If we further consider ID = 26 mA and g = 1 mho, then from equation (35), we have CD = τF . In practical situations, τF , approximated as the average lifetime of charge carriers, is generally of the order of 20 - 40 μs; thus, the CD is observed to be varied between 20 to 40 μF; thus, it is millions of times higher than CT . Importantly, the time constant, expressed by CD /g, is not very high for this capacitance because of high dynamic conductance. The origin of CD can be better understood as follows: we consider the accumulation of injected charge across the junction at a different time due to very slow varying voltage. Let ‘p’ and ‘n’ represent the steady state hole and electron concentration at an applied voltage V. If V gets changed into V + dV in time dt, then the concentration of charge carrier near junction would change significantly, but away from the junction, no significant change would be observed. BREAK DOWN OF REVERSE BIASED P – N JUNCTION: AVALANCHE AND ZENER MECHANISM It has been noted that reverse-biased voltage beyond a critical limit leads sharp rise of the characteristic diode current, as shown in Fig. (20). The critical voltage limit is called Zener voltage (VZ ), and the diode is known as the Zener diode. Here, the operating voltage and resistance are selected in such a way that the diode operation lies within the breakdown region. The characteristic current-voltage curve of the Zener diode depicts the small change in current due to large voltage variation. On the other way, it may be stated that diode current would accommodate itself for any change in supply voltage to regulate constant voltage. Hence, the Zener diode is

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being widely used as a voltage-reference or constant voltage device. In addition, this type of diode has adequate power dissipation capability.

Fig. (20). Zener region of p – n diode.

Commonly, two mechanisms have been proposed for a sharp increase of the current within the breakdown region. One of them is avalanche multiplication. According to this mechanism, with increasing reverse-biased voltage, the velocity of the minority carriers increases significantly. And eventually, their velocity and associated kinetic energy are sufficient enough at or beyond VZ to produce additional charge carriers via collisions with other stable atoms. Briefly, minority carriers, due to their extra energy, disrupt stable atoms making them ionized, which in consequence, generates new electron-hole pairs. These newly generated charge carriers again gain sufficient energy to produce charge carriers further. In this context, it may be stated that each carrier generates additional carriers. This cumulative carrier generation process is called avalanche, while the corresponding current is referred to as avalanche current. Therefore, it may be stated that by virtue of the avalanche mechanism, large currents are being generated [3]. Another mechanism that has been considered here is the Zener breakdown mechanism. Within this mechanism, if the charge carriers haven't gained sufficient energy may still rupture covalent bonds between electron and atom. Here, the electric field at the junction exerts a strong force on the electrons to make them free. In contrast to the avalanche mechanism, presently electric field at the junction causes breakdown, and it is called Zener breakdown. In general, this particular

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mechanism is found within a highly doped p – n diode, while the avalanche mechanism dominates within a lightly or moderately doped p – n diode. It has been found that in the case of Zener breakdown, VZ gets shifted towards the vertical axis with increasing dopant concentration, i.e., break-down voltage lies very close to zero for this particular set of the diode, commonly known as Zener diodes. Temperature is the other parameter that significantly influences the breakdown of the p – n diode. In this context, the temperature coefficient is defined to signify the percentage change in reference voltage per degree change in operating temperature. It is obvious that the coefficient may be either negative or positive, depending on the breakdown mechanism. It has been noted that for a diode operating at a voltage above 6 V, avalanche dominates, and in this case, the coefficient is positive. On the other hand, at a voltage below 6 V, Zener breakdown mostly governs the breakdown process, and within this voltage range of operation, the coefficient is negative. In the real field, this data is supplied by the manufacturer. DIODE SWITCHING The switching time, i.e., the transient time taken by the diode before achieving a steady state in response to the reversal of bias voltage, is a very crucial parameter for the diode operation [3]. In order to better understand the phenomenon, we particularly consider a sudden change in external bias voltage from forward biased condition into reverse biased condition. In the forward-biased condition, a large number of electrons migrate from the n-type region into the p-type region, while holes follow the opposite pathway. During migration, electrons reside in the p-type material and holes within the n-type material as minority carriers. While reversing the applied voltage in this situation, we expect the diode to be changed instantaneously from the conductive state into the non-conductive state. However, due to the presence of a large number of minority carriers on each side, diode current stays at a measurable level for a short period of time, generally defined as the storage time (t s ). Briefly, t s represents the time taken by the minority carriers to return to their majority state. In this context, it may be stated that the diode remains in the short-circuit condition within this time interval, and the level of current would depend on external resistances connected to the diode. After time t s , the current would reduce in level to that of a steady non-conductive state. This second part of the transient time is called the transition interval and is denoted by t t . In general, the recovery time of diode (t rr ) is defined as t rr = t s + t t shown in Fig. (21). In this context, it may be stated that t rr is a very important parameter for diode operation, particularly in switching applications. Most of the diodes have t rr within

Introduction

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the range of nanosecond to a microsecond. However, there are few diodes with t rr ~ 10-12 suitable for the high-field application.

Fig. (21). Reverse recovery time of diode.

CONCLUSION In this chapter, fundamentals of p – n junction diode, and current-voltage characteristics under zero, forward and reverse biased conditions have been discussed. Diode parameters like static, dynamic resistances etc., along with their physical significances, have been defined. Herein, diode capacitances under different biased conditions have been stated. Different mechanisms behind diode breakdown have been briefly elaborated. QUESTIONS 1. What is the difference between the characteristics of a simple switch and an ideal diode? 2. How much work is required to move a charge of 6 C across a potential difference of 5 V? 3. Describe in your own words the current through the p – n junction under forward and reverse biased voltage. 4. Calculate current density through a forward biased (0.75 V) Si diode (Js = 25 nA / mm2), operated at room temperature (T = 300K). 5. If the operating temperature of the Si diode is doubled, then calculate the percentage change in reverse biased voltage.

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6. What are the advantages of reverse biased p – n junction? 7.Why is Zener breakdown important in device fabrication?

REFERENCES [1] [2] [3] [4]

Basic Semiconductor Physics, Chihiro Hamaguchi, Springer-Verlag Berlin Heidelberg 2010. Advanced theory of semiconductor devices, Karl Hess, A John Wiley & Sons, INC., Publication, 2000. Electronic Devices and Circuit Theory, R. Boylestad and L. Nashelsky, Prentice Hall, New Jersey, 1995. Electronic Principles, A. Malvino, D.J. Bates McGraw-Hill Education, 2016.

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CHAPTER 3

Metal Semiconductor Contacts Schottky Diodes Abstract: Metal-Semiconductor-Junction is also called heterojunction since the material on each side of the junction is not identical. The normal pn junction diode concept can also be applied here. There are two probable types of metalsemiconductor junctions: Schottky junction and ohmic junction. When the workfunction of metal is greater than the work-function of a semiconductor, then this is called Schottky junction. Ohmic junctions are the junctions in which the work function of the metal is less than the work function of a semiconductor. Their band engineering discussed in detail the essentials of junction physics.

Keywords: Band engineering, Heterojunction, Schottky contact, Ohmic contact. INTRODUCTION Semiconductor materials are homogeneous throughout the structure, called a semiconductor homojunction. Similar energy bandgaps are the prime factor that is to be noted in homojunction. On the contrary, two different semiconductor materials are there to form a junction; then, it is called semiconductor heterojunction. In heterojunction, a combination of narrow bandgap material with wide bandgap material is found. Fig. (1) shows three probable combinations. When the forbidden band gap of wide bandgap material completely overlaps with the narrow bandgap materials, it is called straddling. A portion of the wide bandgap and narrow band gap overlap in staggered. In a broken gap, there is a bandgap itself that exists between the energy band gaps of two dissimilar semiconductors. Metal-Semiconductor Junction A good example of heterojunction is the Metal-semiconductor junction. This junction can be formed by depositing a metal (Al, Au, Ag, etc.) over a semiconductor. Deposition techniques may vary, like e-beam/thermal evaporation, sputtering, CVD, etc. Based on the work function of the metal, the type of junction varies, that is, the ohmic or Schottky junction. As the name suggests, if the contact Sunipa Roy, Chandan Ghosh, Sayan Dey and Abhijit Kumar Pal All rights reserved-© 2023 Bentham Science Publishers

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is ohmic, the current-voltage relationship is linear, and also, it is mandatory that the semiconductor is heavily doped because the metal work function is lesser than the semiconductor work function.

Fig. (1). Different types of heterojunction (a) straddling (b) staggered (c) broken gap.

So, Schottky diodes are formed by depositing metal over a lightly doped semiconductor. Schottky junction switches faster than pn junction as in Schottky junction, majority carriers only dominate, and there is the depletion region on one side of the junction only. The band chemistry [1-3] of a metal-semiconductor junction can be best understood by the energy band diagram to comprehend the electronic transport throughout the junction. Prior to that, the following factors are to be noted: Work Function (Φ) Work function is the most important parameter in understanding band chemistry. It is the gap between the vacuum level and the Fermi energy level of any energy band diagram and is symbolized by φ. It is the least energy required to diffuse one electron from a particular node in a solid to a node just outside its surface. For metal, it is denoted by φm or qφm, and for semiconductors, it is denoted by φs or qφs, where q indicates the charge of the electron. Work function and ionization energy

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are the same in metal. It is not constant for a particular metal as the location of the Fermi level may vary when doping concentration changes. Typical values of φm are given in the table below. Surface contaminations are mainly responsible for the varied work function. The photoemission spectroscopy (PES) technique is used to calculate the work function in a semiconductor. Electron Affinity (χ) Electron affinity of a semiconductor material can be defined as the energy needed out of an electron from the bottom of the conduction band to the vacuum. The vacuum level indicates the energy level of an electron when it is absolutely free. Electron affinity is denoted by χ or qχ, where q indicates the charge of the electron. The unit is an electron volt (eV). It is a native property of a material and doesn’t depend on doping, as the vacuum level, and the lower of the conduction band can’t be changed. Table 1 gives the value of the work function of some metals. Band Diagram Φms = Φm– Φs= metal semiconductor work function difference or just the difference between two Fermi levels and can't be zero as the average energy of an electron present in the metal is not the same as the average energy of an electron present in the semiconductor. Table 1. Examples of some metals. Work Functions of Some Metals Element

The Work Function, φm(volt)

Ag, silver

4.26

Al, aluminum

4.28

Au, gold

5.1

Cr, chromium Mo, molybdenum

4.5 4.6

Ni, nickel

5.15

Pd, palladium

5.12

Pt, platinum

5.65

Ti, titanium

4.33

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(Table 1) cont.....

W, tungsten

4.55

Electron Affinity of Some Semiconductors Element

The Work Function, φm(volt)

Ge, germanium Si, silicon GaAs, gallium arsenide

4.13 4.01 4.07

Alas, aluminum arsenide

3.5

When Φm>Φs and the two materials are brought into contact, electrons in the semiconductor must pass through a potential barrier of e(Φm– Φs) to flow from semiconductor to metal in order to establish equilibrium, which means to bring Fermi energy (EF) in the same level (Fig. 2a).

(a)

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(b) Fig. (2). (a) Energy band diagram of metal-n semiconductor junction for Φm>Φs (b) Energy band diagram of metal-p semiconductor junction for ΦmΦs, electrons are extracted from the semiconductor into the metal. If the semiconductor is of n-type, then electrons get depleted. A depletion region thus appears near the junction, and we obtain a behaviour similar to a p-n junction when an external bias is applied. This is shown in Fig. (4a). This situation is often called a rectifying contact or Schottky contact. If the semiconductor is p-type (Fig. 4b) and the work function of the metal is less than the work function (Φm0) condition, the operation of the metal-semiconductor junction is illustrated in Fig. (5). The Fermi energy of metal stays lower than the Fermi energy level of the semiconductor EF, which results in a reduction of the potential barrier ΦB across the semiconductor. Thus, it will be much easier for electrons to pass over the barrier and electrons diffuse from the semiconductor to the metal. Therefore, as the VF increases, the current will increase quickly, since more electrons will be able to overcome the surface barrier. There will be more electrons diffuse from the semiconductor to the metal, and a positive current will flow across the metal-semiconductor junction.

Fig. (5). Energy band diagram and carrier activity at forward biasing (VF>0).

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Reverse Biasing When a reverse bias is applied, i.e., a negative electrode connected to the metal (VF