Organic Electronics 1 E-Book

Thien-Phap Nguyen

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 270

Veröffentlichungsjahr: 2021

Beschreibung

Due to their special properties, organic semiconductors enable the development of large-area, low-cost devices, paving the way for flexible and nomadic applications that advantageously replace those made with traditional semiconductors. This book describes the properties and deposition methods of organic semiconductors, transparent conductive materials or metals which are used in the fabrication of organic devices. The physical processes (optical, electrical and interface) that control the mechanisms in the formation and transport of the charge carriers of the materials are studied and explained in detail. Organic Electronics 1 introduces the fundamental and applied aspects of the field of organic electronics. It is intended for researchers and students in university programs or engineering schools specializing in electronics, energy and materials.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Introduction

1 Semiconductor Theory

1.1. Introduction

1.2. Review of the basic concepts of crystalline semiconductors

1.3. P–N junction

1.4. Impurities and defects

1.5. Metal/semiconductor contact

1.6. Semiconductors under non-equilibrium conditions

1.7. Space charge current

1.8. Hopping conduction

2 Materials

2.1. Introduction

2.2. Organic materials

2.3. Conjugated polymers

2.4. Energy bands

2.5. Small molecules

2.6. Design and engineering of organic materials

2.7. Hybrid materials or nanocomposites

2.8. Transparent and conductive materials

2.9. Materials for encapsulation

3 Optical Processes

3.1. Introduction

3.2. Interaction between light and molecules

3.3. Optical processes

3.4. Excitons

3.5. Experimental techniques

4 Electronic Processes

4.1. Introduction

4.2. Charge carrier injection process

4.3. Charge transport process

5 Interface Processes

5.1. Introduction

5.2. Formation of organic semiconductor/metal interfaces

5.3. Surface characterization techniques

5.4. Interface engineering

5.5. Conclusion

List of Acronyms

References

Index

End User License Agreement

List of Illustrations

Chapter 1

Figure 1.1.

Generation and recombination process

Figure 1.2. Energy band diagram of a P–N junction. For a color version of this f...

Figure 1.3. Structural defects in semiconductors: a) vacancy; b) interstitial at...

Figure 1.4. Trapping and recombination mechanisms: a) capture of an electron; b)...

Figure 1.5. The pulse applied to the junction and the variation of the junction’...

Figure 1.6. Changes in the SCR of the junction during the trapping process: 1) b...

Figure 1.7. The process of light emission in SCs, with and without defects. For ...

Figure 1.8. The parameters in the energy band diagram for metals and SCs. For a ...

Figure 1.9. The parameters in the energy band diagram for metals and SCs. For a ...

Figure 1.10.

Schottky barrier at a metal/SC interface with the Schottky effect

Figure 1.11. Electron transport in a contact between a metal and an N-type SC. F...

Figure 1.12. The change in the concentration of excess carriers as a function of...

Figure 1.13. Diffusion of excess carriers within the thickness of an SC lit on o...

Figure 1.14. Electronic conduction regime of the space-charge limited current. a...

Chapter 2

Figure 2.1. Basic structure of an SC-based light-emitting diode: (a) inorganic S...

Figure 2.2. Structures of polymers. For a color version of this figure, see www....

Figure 2.3. Trans

and

cis

configurations of polyacetylene

Figure 2.4. Hybridization of the carbon atom and hybridization mechanism of sp2....

Figure 2.5.

Atomic orbitals π and π

*

of the π bond

Figure 2.6. Structure of polyacetylene: a) chemical bonds between atoms; b) orbi...

Figure 2.7. Structure of benzene: a) chemical bonds between atoms; b) orbitals o...

Figure 2.8.

Molecular structures of poly(phenylene)

Figure 2.9. Different techniques for deposition of polymer films in a solution: ...

Figure 2.10.

Wave functions of atomic and molecular orbitals

Figure 2.11. Energy levels of the isolated atom, the molecule formed by two atom...

Figure 2.12. Representation of solitons: a) energy of a dimer consisting of two ...

Figure 2.13. Formation of bipolarons by solitons. For a color version of this fi...

Figure 2.14. Doping mechanisms for N-type and P-type organic SCs. For a color ve...

Figure 2.15.

Molecular structure of small molecules of benzene cycles

Figure 2.16. Examples of polymers and small molecules and their use in electroni...

Figure 2.17.

Types of chain arrangement in poly(hexylthiophene)

Figure 2.18. Preparation of composites by dispersion method: a) materials; b) di...

Figure 2.19.

Structure of a polyoctahedral silsesquioxane (POSS)

Figure 2.20. Nanocomposites using a nanostructured substrate: (a) nanostructured...

Figure 2.21.

a) Nanowires; b) nanorods; c) nanorod arrays

Figure 2.22. Hydrothermal fabrication of ZnO nanorods: a) buffer layer-covered s...

Figure 2.23. Transparent and conductive electrodes: a) unordered array of metal ...

Figure 2.24. Encapsulation of organic electronic devices: a) by a glass slide; b...

Chapter 3

Figure 3.1. Potential energy of a diatomic molecule as a function of the inter-a...

Figure 3.2. Diagram of the energy and electronic transitions in the process of a...

Figure 3.3. Energy diagram and possible electronic transitions in a molecule acc...

Figure 3.4. Overlap of the emission spectrum of the donor molecule D and the abs...

Figure 3.5. Process of energy transfer between molecules: a) transfer by Förster...

Figure 3.6. Excitons: a) Frenkel; b) charge transfer; c) Wannier–Mott. For a col...

Figure 3.7. The steps of the exciton formation: a) initial electron–hole pair; b...

Figure 3.8. Schematic representation of rate constants of photophysical processe...

Figure 3.9. Transitions in the processes of: a) infrared absorption; B) Rayleigh...

Chapter 4

Figure 4.1. Schematic representation of the movement of charge carriers in an or...

Figure 4.2. Injection process of charge carriers in a simple diode structure. Fo...

Figure 4.3. Energy band diagram of electronic devices: a) electron-only; b) hole...

Figure 4.4. Energy band diagram of the electronic device with transport layers o...

Figure 4.5. a) Intra-chain and inter-chain movements of electrons; b) energy dia...

Figure 4.6. Density of states following the Gaussian disorder model (GDM). For a...

Figure 4.7. a) Schematic setup for measuring time of flight; b) variation of non...

Figure 4.8. a) CELIV measurement setup; b) applied voltage ramp and CELIV spectr...

Figure 4.9. The conduction regimes limited by space charges in an SC: a) contain...

Figure 4.10.

Types of trap distributions: Gaussian and exponential

Figure 4.11. Schematic representation of the effective capture cross-section: a)...

Figure 4.12.

Steps for measuring the thermally stimulated current

Figure 4.13. The principle of fractional TSC current measurement: temperature cy...

Figure 4.14. The principle of charge-based deep-level transient spectroscopy: a)...

Figure 4.15.

Variati on of capacitance C

(

ω

)

and its derivative

as a function o...

Figure 4.16.

Variation of

as a function of the direct voltage V

DC

. 1) trap-fre...

Chapter 5

Figure 5.1. Schematic representation of an N-type semiconductor/metal contact ac...

Figure 5.2. Diagram of the energy bands of the metal/SC contact with alignment o...

Figure 5.3. Formation of the interface dipole layer: a) metal with a low work fu...

Figure 5.4. Schematic representation of the ICT model with the transition from t...

Figure 5.5.

Energy levels: a) in the metal; b) at the metal/SC interface

Figure 5.6. Example of the structure of an organic solar cell: a) normal; b) inv...

Figure 5.7. a) Structure of a self-assembled monolayer; b) example of the use of...

List of Tables

Chapter 2

Table 2.1.

Comparison of the properties of organic and inorganic SCs

Chapter 4

Table 4.1.

Metal work function and potential barrier for an MEHPPV/metal contact

Guide

Cover

Table of Contents

Title Page

Copyright

Introduction

Begin Reading

List of Acronyms

References

Index

End User License Agreement

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Series Editor

Robert Baptist

Organic Electronics 1

Materials and Physical Processes

Thien-Phap Nguyen

First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2021

The rights of Thien-Phap Nguyen to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2020948483

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-321-9

Introduction

Organic electronics can be defined as a branch of general electronics that focuses on studying the properties and applications of organic semiconductors. These materials, referred to in common usage as plastics, are, in fact, a distinct class separate from that of ordinary plastic materials. They may be small molecules, or conjugated polymers, which have an electronic structure comparable to that of conventional semiconductors. Therefore, their physical properties are very similar to the properties of these latter materials. However, organic materials are generally amorphous and their electrical conductivity is much lower than that of traditional semiconductors. As a result, they have been neglected for a long time despite the discovery of some of their notable physical properties, such as the discovery of electroluminescence in anthracene in the 1960s.

In all scientific or technological disciplines, an idea, concept or creation can arise with a change or progression and profoundly alter the course of the evolution of the discipline – and sometimes even the course of the history of science. Such was the case in 1947 when the researchers John Bardeen, William Shockley and Walter Brattain invented the transistor, a device that would revolutionize traditional electronics. This invention is what first allowed for the design of circuits that provided logical functions, then for these circuits to be integrated into complex systems capable of handling sophisticated tasks, and finally for these miniaturized systems to be incorporated into the portable electronic devices that we now use every day. A similar progression occurred in organic electronics in 1987, when Tang and Van Slyke used the evaporation of small molecules in a vacuum to create diodes that emitted light in the form of a thin film that operated with a turn-on voltage lower than 10 V. Their work demonstrated that, in practice, organic devices are suitable for optoelectronic applications in the same way as their inorganic counterparts. A few years later, Burroughs et al. (1994) demonstrated that by using conjugated polymers deposited as thin films from a solution, they could also produce light-emitting diodes with a low operating voltage. This work paved the way for film depositing techniques using solutions and led to the production of devices through printing developed later in laboratories. For this domain, we will use the descriptors “flexible” or “printed” to indicate the discipline of organic electronics. Advances in research have been very rapid and have shown remarkable results, not only in the understanding of physical processes in materials and components but also for the creation of new electronic devices now available on the market. Today, organic electronics has become its own field, one that will prove to be important technologically and economically in the near future.

In France, education on organic electronics at the university level is developing, but remains limited in comparison with other European countries. It should be noted that these courses are generally offered in universities that carry out industrial research and development activities on organic electronic materials and devices. With regard to the books written on this discipline, there are very few titles available in French.

This book, divided into two volumes, was written with the goal of introducing organic electronics to students and researchers who are interested in this new discipline. It is organized into the following sections: Chapter 1 of Volume 1 provides an overview of the basic notions of the theory of traditional semiconductors, of which some of the material presented will be used later on. The materials used for the creation of devices are described in Chapter 2. The physical processes, which take place in the volume and interface of the layers of devices, are presented and explained in Chapters 3, 4 and 5. Volume 2 presents the primary applications of organic materials in optoelectronic devices. The first chapter of this volume centers on organic light-emitting diodes (OLEDs), the second on organic solar cells (OSCs or OPVs) and the third on organic field-effect transistors (OFETs). Chapter 4 deals with the practical and economic aspects of the industrialization of organic components. It also includes a discussion on the environmental aspects of the use of organic materials and devices.

As the title indicates, this book is not intended to provide a detailed and complete description of the materials, physical processes and applications involved in organic electronics. This would be far beyond the scope of what can be addressed within a single book. The choice of topics and of the extent to which the subjects are discussed were made by the author, based on his own experience from research and as an instructor. Interested readers will be able to find more detailed discussions on certain topics using the references provided. A list of acronyms used in the text is also included at the end of this book.

I would like to thank the people who devoted their time to proofreading and providing comments and suggestions, which allowed me to improve the writing of this book and the way in which certain areas are presented: Philippe Lerendu, Serge Lefrant, René Leparoux, Agnès Bournigal-Giret, Maxime Bayle and Jean-Luc Duvail. I would also like to thank ISTE for proposing this project, which I hope will help to arouse public interest in the new and promising possibilities offered by organic electronics in the scientific and technological communities in France and around the world.

1Semiconductor Theory

This chapter provides a brief overview of the basic concepts of semiconductor theory with regard to particular physical characteristics that allowed the creation of electronic components that would revolutionize various technologies starting in the middle of the 20th Century. We will now describe the operation of P–N junctions, which are the essential components for the creation of electronic devices. This basic knowledge will then allow us to address physical processes in organic materials and then to understand the operation and use of devices based on new organic semiconductors.

1.1. Introduction

In electronics, the basic materials are semiconductors (SCs). They differ from metals in their dependence on the temperature of their electric characteristics. Essentially, when the temperature increases, the resistivity of metals increases, while the resistivity of SCs decreases. The theoretical contributions made by quantum mechanics allowed the electrical properties of SCs to be studied and explained based on the theory of energy bands. Unlike metals, in which the charge carriers are electrons, SCs can also carry positive charges, known as holes, that contribute to the electrical conductivity of these materials. The concentration of the carriers can also be changed by “doping”, that is, the incorporation of the selected impurities in a determined quantity. This process not only allows the conductivity of the SC to be changed, but also favors the transport of one type of charge carrier at the expense of the other. An SC is characterized as N-type if the majority of the carriers are electrons and as P-type if the majority of the carriers are holes.

The combination of an N-type and P-type SC is a P–N junction, which is the fundamental element of all traditional electronic components. The assembly of these elements, prepared with other materials, makes it possible to create components that perform special functions, and the design of the circuits with these components leads to multiple applications of various different types.

The miniaturizing of components has given rise to the field known as microelectronics, the technology for microscale component manufacturing while maintaining the same characteristics as conventional components. This new technology has significantly reduced the size of miniaturized electronic circuits, as well as certain devices that can be made portable. With the advent of integrated circuits, discrete components can now be installed on boards using a minimum amount of space and electrical connections. This evolution in the size of electronic components follows the projections of Moore’s law, which holds that the density of the components able to be installed will double about every two years. At the same time, this reduction in size leads to a reduction in the amount of material used. With the introduction of new digital technologies, and especially nanotechnology, technological advances in electronics seem to be limitless. However, this progress occurs more slowly as the size of the components decreases and, once they shrink to the size of a few tens of nanometers, the components can no longer be guaranteed in terms of their size and quality. It then becomes a question of modifying or changing the material used in the SC or the structure of the components, or better yet, the way they are built.

In the late 1980s, a scientific paper by an American team reported the results of measurements made on a light-emitting diode using, instead of a crystalline SC, an organic material: tris (8-hydroxyquinoline) aluminum (III) or Alq3. The article gave the first-ever description of the use of a non-crystalline SC in a device that was both practical and functional, as the diode was able to emit light under an applied voltage of about 10 volts. What was unique about this experiment was its use of a thin film structure for the active material, a first in the design of electronic components and devices. A few years later, researchers in the UK published their discovery of electroluminescence in conjugated polymers, which spurred on a considerable number of studies on organic SCs, both with regard to the study of materials and the study of the particular physical processes of the devices based on organic SCs. The enthusiasm of the scientific community for this new field of electronics allowed for significant advances in knowledge and applications to be made over a period of about ten years. Display screens entered the market in the early 2000s, followed shortly thereafter by organic photovoltaic panels. This interest arose due to the promising possibilities offered by organic electronic devices, which first appeared at a time when the problems in traditional electronics were beginning to become clear.

In this book, we will present the fundamental notions of organic electronics along with those of electronics based on crystalline SCs, as well as certain notable differences that correspond to the specific properties of organic materials. We will also focus on the most recent applications of this new field of electronics, which do not compete with traditional electronics, but instead complement and reinforce them to a certain extent.

We will present several basic concepts from the study of crystalline SCs (CSCs) in order to address the various different aspects of organic SCs (OSCs). It should be note that, for reasons of clarity, these abbreviations will be used to distinguish the two materials when there may be a possible confusion in the description. Otherwise, we will use the abbreviation SC to refer to the SC component. We will review the essential and useful points on the properties of CSCs, as well as P–N and metal/SC junctions. The principle of electronic devices based on OSCs or CSCs will be discussed in Chapters 1, 2 and 3 of Volume 2.

1.2. Review of the basic concepts of crystalline semiconductors

SCs used in traditional electronic devices are crystals, that is, materials made from atoms, ions or molecules arranged in an orderly and repetitive manner in a three-dimensional network. As a result, the charge carriers in motion within these materials are subjected to a periodic potential field, and their energy is restricted to energy domains, called allowed bands. This concept introduces the notions of the conduction band (CB), the valence band (VB) and the forbidden band or energy gap. The arrangement of allowed bands and the gap value allow us to distinguish conductive materials or metals, insulators and SCs.

In a conductor, the density or concentration of electrons is of the same order of magnitude as the density of atoms, while in an insulator, the ratio of the densities of free electrons to atoms is less than 10-20. An SC can be considered as a bad conductor at room temperature if its electrical conductivity varies between 10-10 and 103 S/cm, whereas the conductivity of metals varies between 104 and 106 S/cm. This poor conductivity results from the low density of electrons in the CB and the holes in the VB, because the energy required for creating an electron–hole pair (or exciton) using thermal generation is relatively large for most SCs. To explain the physical properties and processes of SCs, we will use silicon (Si) as an example, noting that its properties and processes are applicable to other SCs.

1.2.1. Intrinsic semiconductors

In a lattice of pure silicon, the bonds between atoms are covalent. Each Si atom is surrounded by four neighboring atoms that exchange valence electrons with it, so that all atoms have eight valence electrons. This configuration ensures the stability of the silicon crystal lattice. A number of electrons that have acquired sufficient energy (provided by the energy supply from an external source that can be heat, electricity or light) can break their bonds and become free. An electron released in this way passes from the VB to the CB and creates a hole in the VB through the generation process.

Figure 1.1.Generation and recombination process

Let ni and pi be the concentrations of electrons and holes of the intrinsic SCs. We can write ni = pi because there are as many electrons in the CB as there are holes in the VB that are created through the generation process. The intrinsic concentrations verify the mass action law as follows:

[1.1]

The electrons and holes thus created in the SC can then disappear by recombination. This process allows an electron in the CB to take the place of a hole in the VB by losing an amount of energy theoretically equal to the energy of the SC gap. This energy can be recovered either thermally (phonon emission) or optically (photon emission). When the SC is in thermal equilibrium, the concentrations of CB electrons and VB holes are stable because the rates or speeds of generation and recombination of electron–hole pairs are constant. This concentration is a function of temperature T, governed by the relation:

[1.2]

where A is a constant, EG is the SC gap and k is Boltzmann’s constant.

1.2.2. Extrinsic semiconductors

By adding impurities to the intrinsic SC, the concentrations of electrons and holes, and therefore their electrical conductivity, can be controlled. In comparison with metals, SCs contain fewer charge carriers, but the fact that their concentrations can be adjusted to desired values is technologically essential for electronic device applications.

For silicon, a “doping“ using the elements from group V of the periodic table helps to promote conductivity by electrons, while reducing the number of holes. Each atom of impurity, called a donor (D), offers the possibility of providing a free electron to the silicon CB when it is ionized. The doped SC is referred to as N-type. Let ND be the concentration of donors and the concentration of ionized donors. The concentration of free electrons due to doping in the CB of the SC is equal to . Similarly, the use of an element from group III of the periodic table promotes conductivity through the holes. After ionization, a doping atom, known as an acceptor (A), provides a hole for the VB and the SC is called P-type. Let NA be the concentration of acceptors and the concentration of ionized acceptors. The concentration of free holes due to doping in the VB of the doped SC is equal to .

The ionization of impurities is a thermally activated process, and in principle, the concentration of charge carriers in the bands depends on the temperature of the SC. However, the ionization energy of typical impurities is low (of the order of several tens of milli-electron volts, or meV) so that they are virtually all ionized at room temperature (T=300 K). At this temperature, the concentration of electrons due to doping of an N-type SC is therefore n = ND and the concentration of holes due to doping of a P-type SC is therefore p = NA.

In a doped SC, the creation of electron–hole pairs by generation and recombination processes occurs at the same time as the ionization of impurities. The total concentration of electrons in the CB is the sum of two concentrations: one due to the ionization of donors and the other due to generation/recombination that we denote as n0. Similarly, the total concentration of holes in the VB is the sum of two concentrations, and p0.

The charge neutrality condition is written as:

[1.3]

At room temperature, the ionization of impurities is practically achieved, and the concentrations of electrons and holes due to doping are and . At thermal equilibrium, the product of the concentrations of electrons and holes confirms the mass action law [1.1], but since the concentration of carriers created by ionization is dominant, the concentrations n0 and p0 are no longer equal. For an N-type SC, equations [1.1] and [1.3] result in:

[1.4]

As ND ≫ ni and ND ≫ NA, we therefore obtain:

[1.5]

For a P-type SC, we obtain:

[1.6]

1.2.3. Fermi level

The Fermi level (EF) is defined as the highest energy level occupied by electrons at temperature T0 = 0 K. It allows us to determine how the electrons in an SC are distributed at a given temperature in the band diagram. For instance, at room temperature, the Fermi level of an intrinsic SC is located approximately in the middle of the energy gap. For an N-type SC, it is close to the bottom level EC of the CB, and for a P-type SC, it is close to the top level EV of the VB.

1.2.4. Charge transport in semiconductors

In an SC, charge carriers are transported by two mechanisms that allow the flow of the electric current. The currents corresponding to these mechanisms are the drift current and the diffusion current.

1.2.4.1. Drift or electric current

When no electric field is applied, free charge carriers move through the SC through thermal agitation and collide with other carriers. The average speed of all carriers is zero and no current flows in the SC. When an electric field is applied, a carrier of charge q and mass m* is subjected to an electric force . By taking t = 0 at the moment of a collision and applying Newton’s second law of motion to the charge carrier, the equation for its instantaneous velocity is obtained:

We can then deduce the expression of the average velocity or drift velocity of the free carriers:

[1.7]

where τC is the time between two collisions or the lifetime of the carriers.

Expression [1.7] indicates that the velocity of the carriers is proportional to the applied electric field. It can also be written as:

[1.8]

where μ is the “mobility“ of the charge carrier. It expresses the ease of movement of a charge carrier in the material and equals its velocity under a unit electric field. The current density J or the amount of charges that flows per second through a unit area of the SC with a concentration of carriers ρ is:

[1.9]

Expressions [1.7], [1.8] and [1.9] apply to both types of carriers. For electrons of a concentration n, the current density is:

and for holes of a concentration p, the current density is:

The total density of the drift current is:

[1.10]

By introducing electrical conductivity using Ohm’s law, we can write:

Thus, we obtain:

[1.11]

1.2.4.2. Diffusion current

When charge carriers are not uniformly distributed in the SC, the carriers flow from high concentration regions to low concentration regions.

The diffusion of the carriers tends to lead to a uniform spatial concentration of charges in the material. The resulting current is called the diffusion current and the density of this current is governed by Fick’s law.

For electrons moving along the Ox axis:

[1.12]

For holes moving along the Ox axis:

[1.13]

where Dn and Dp are, respectively, the diffusion coefficients of electrons and holes. These coefficients are related to the mobility of carriers through the Einstein relation, which is written as:

[1.14]

1.3. P–N junction

A P–N junction is obtained by allowing an N-type SC with a donor concentration of ND to contact a P-type SC with an acceptor concentration of NA. Since the concentrations of electrons and holes are very different in both types of SC, the majority charge carriers migrate from one SC to another by means of diffusion.

At thermal equilibrium, an internal electric field is established at the junction and disrupts the diffusion. A space charge region (SCR) of a width λ, limited by the abscissa (− xp) and (xn), is created at the junction in which no free charge is present.

1.3.1. Space charge region

From the point of view of energy bands, since the Fermi energy level EFp of the P-type SC is lower than the Fermi level EFn of the N-type SC, the electrons of the N-type SC will move from the N region to the P region until the Fermi level is aligned; thus, the probabilities of energy level occupancy by the electrons are identical in the two SCs.

The same applies for holes. Therefore, the Fermi levels EFn and EFp are aligned when an equilibrium is reached in the junction.

Figure 1.2.Energy band diagram of a P–N junction. For a color version of this figure, see www.iste.co.uk/nguyen/electronics1.zip

The internal electric field Ei(x) can be determined by Gauss’s law and the potential V(x) in the junction by the relation:

[1.15]

When the junction is at thermodynamic equilibrium, the total current density JT = JC + JD is zero within the SCR. Using the Einstein relation [1.14] for the electrons and holes, and the expression of the law of mass action [1.4], it can be shown that the difference in energy between the two regions P and N is equal to:

with:

[1.16]

where Vbi is the built-in potential of the junction. It represents the potential barrier that the electrons of the CB of the N-type SC (or the holes of the VB of the P-type SC) must surmount to reach the corresponding allowed band of the other SC.

When the junction is biased by a voltage V0, the potential barrier is changed. Under forward bias, it decreases from V0 and becomes Vbi − V0