Infochemistry E-Book

Konrad Szacilowski

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 790

Veröffentlichungsjahr: 2012

Beschreibung

Infochemistry: Information Processing at the Nanoscale, defines a new field of science, and describes the processes, systems and devices at the interface between chemistry and information sciences. The book is devoted to the application of molecular species and nanostructures to advanced information processing. It includes the design and synthesis of suitable materials and nanostructures, their characterization, and finally applications of molecular species and nanostructures for information storage and processing purposes.

Divided into twelve chapters; the first three chapters serve as an introduction to the basic concepts of digital information processing, its development, limitations and finally introduces some alternative concepts for prospective technologies. Chapters four and five discuss traditional low-dimensional metals and semiconductors and carbon nanostructures respectively, while further chapters discuss Photoelectrochemical photocurrent switching and related phenomena and self-organization and self-assembly. Chapters eight, nine and ten discuss information processing at the molecular level, and eleven describes information processing in natural systems. The book concludes with a discussion of the future prospects for the field.

Further topics:

• Traditional electronic device development is rapidly approaching a limit, so molecular scale information processing is critical in order to meet increasing demand for high computational power
• Characterizes chemical systems not according to their chemical nature, but according to their role as prospective information technology elements
• Covers the application of molecular species and nanostructures as molecular scale logic gates, switches, memories, and complex computing devices

This book will be of particular interest to researchers in nanoelectronics, organic electronics, optoelectronics, chemistry and materials science.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface: Why “Infochemistry”?

Acknowledgements

Chapter 1: Introduction to the Theory of Information

1.1 Introduction

1.2 Definition and Properties of Information

1.3 Principles of Boolean Algebra

1.4 Digital Information Processing and Logic Gates

1.5 Ternary and Higher Logic Calculi

1.6 Irreversible vs Reversible Logic

1.7 Quantum Logic

Refrences

Chapter 2: Physical and Technological Limits of Classical Electronics

2.1 Introduction

2.2 Fundamental Limitations of Information Processing

2.3 Technological Limits of Miniaturization

Refrences

Chapter 3: Changing the Paradigm: Towards Computation with Molecules

Refrences

Chapter 4: Low-Dimensional Metals and Semiconductors

4.1 Dimensionality and Morphology of Nanostructures

4.2 Electrical and Optical Properties of Nanoobjects and Nanostructures

4.3 Molecular Scale Engineering of Semiconducting Surfaces

Refrences

Chapter 5: Carbon Nanostructures

5.1 Nanoforms of Carbon

5.2 Electronic Structure and Properties of Graphene

5.3 Carbon Nanotubes

5.4 Conjugated and Polyaromatic Systems

5.5 Nanocarbon and Organic Semiconductor Devices

Refrences

Chapter 6: Photoelectrochemical Photocurrent Switching and Related Phenomena

6.1 Photocurrent Generation and Switching in Neat Semiconductors

6.2 Photocurrent Switching in MIM Organic Devices

6.3 Photocurrent Switching in Semiconducting Composites

6.4 Photocurrent Switching in Surface-Modified Semiconductors

Refrences

Chapter 7: Self-Organization and Self-Assembly in Supramolecular Systems

7.1 Supramolecular Assembly: Towards Molecular Devices

7.2 Self-Assembled Semiconducting Structures

7.3 Self-Assembly at Solid Interfaces

7.4 Controlling Self-Assembly of Nanoparticles

7.5 Self-Assembly and Molecular Electronics

Refrences

Chapter 8: Molecular-Scale Electronics

8.1 Electron Transfer and Molecular Junctions

8.2 Nanoscale Electromagnetism

8.3 Molecular Rectifiers

Refrences

Chapter 9: Molecular Logic Gates

9.1 Introduction

9.2 Chemically Driven Logic Gates

9.3 All-Optical Logic Gates

9.4 Electrochemical Logic Systems

Refrences

Chapter 10: Molecular Computing Systems

10.1 Introduction

10.2 Reconfigurable and Superimposed Molecular Logic Devices

10.3 Concatenated Chemical Logic Systems

10.4 Molecular-Scale Digital Communication

10.5 Molecular Arithmetics: Adders and Subtractors

10.6 Molecular-Scale Security Systems

10.7 Noise and Error Propagation in Concatenated Systems

Refrences

Chapter 11: Bioinspired and Biomimetic Logic Devices

11.1 Information Processing in Natural Systems

11.2 Protein-Based Digital Systems

11.3 Binary Logic Devices based on Nucleic Acids

11.4 Logic Devices Based on Whole Organisms

Refrences

Chapter 12: Concluding Remarks and Future Prospects

Refrences

index

This edition first published 2012

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Library of Congress Cataloging-in-Publication Data

Szacilowski, Konrad.

Infochemistry: information processing at the nanoscale / Konrad Szacilowski.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-71072-2 (hardback)

1. Molecular computers. I. Title. QA76.887.S93 2012 620′.5—dc23

2012007002

For Bela, Maria and Marek, with love

Preface: Why “Infochemistry”?

‘It is a very sad thing that nowadays there is so little useless information.’

Oscar Wilde

For many people, information processing using molecules seems a kind of science fiction. It is hard to imagine fancy netbooks, palmtops and other smart electronic gadgets replaced by jars of snot-like liquid or other gebuzina1. On the other hand each of us carry the most powerful information processing “device” that can be found anywhere: the brain. At this moment and in any foreseeable future, mimicking our brains with any artificial systems of any kind seems impossible. However, we should keep trying to force molecular systems to compute. While we will not be able to build powerful systems, all this effort can serendipitously yield some other valuable results and technologies. And even if not, the combination of chemistry and information theory paves an exciting path to follow. There is a quote attributed to Richard P. Feyman saying:“Physics is like sex. Sure, it may give some practical results, but that's not why we do it”. With infochemistry is it exactly the same!

When approximately half of the manuscript was ready I realized that it was going to be almost a “useless” book. For most chemists it may be hard to follow due to the large amount of electronics content, while for electronic engineers there is far too much chemistry in it. And both fields, along with solid-state physics, are treated rather superficially, but are spiced with a handful of heavy mathematics and a couple of buckets of weird organic structures. But then I found that rather optimistic sentence by Oscar Wilde which motivated me to complete this work.

This book treats the interface between chemistry and information sciences. There are other books which can be located in this field, including my favourites Ideas of Quantum Chemistry by Lucjan Piela and Information Theory of Molecular Systems by Roman F. Nalewajski. In my book I have tried to show how diverse properties of chemical systems can be used for the implementation of Boolean algebraic operations. The book can be divided into three main sections. The first section (Chapters 1–3) explores the basic principles of the theory of information, the physical and technological limits of semiconductor-based electronics and some alternative approaches to digital computing. The next section (Chapters 4–8) is intended to show how the properties of materials commonly used in classical electronics are modified at the nanoscale, what happens at the molecule/semiconductor interface and how these phenomena can be used for information processing purposes. Finally, the last section (Chapters 9–11) are (I hope) a comprehensive review of almost all molecular logic systems described in the chemical literature from 1993 (the seminal Nature paper by Amilra Prasanna de Silva) to November 2011, when my 2 was over.

Notes

1. (Slovak) unidentified disgusting semi-liquid substance of unpleasant smell, swill

2. An original title of a novel by Aleksey Nikolayevich Tolstoy. The title was translated as “The Road to Calvary”, but its literal meaning is rather “walking through torments”.

Acknowledgements

I would like to thank my wife Bela and kids Maria and Marek for their patience, help and support during the preparation of this manuscript. Without their love and understanding this book could not have been written.

I would also like to express my gratitude to my teachers and mentors for their efforts and devotion. First of all I should mention my grandfather Stefan Polus, who showed me the wonderful world of electronics, my PhD supervisor Professor Zofia Stasicka, who introduced me to the realm of inorganic photochemistry and my postdoctoral mentor, Professor John F. Endicott who taught me careful data analysis and skepticism. Large parts of this book were written at The Faculty of Non-Ferrous Metals, AGH University of Science and Technology. Therefore I address my thanks to the Dean of the Faculty, Professor Krzysztof Fitzner for his patience and support.

This book could not have been written without financial support. Most of the manuscript was prepared with support from AGH-UST within contract No. 11.11.180.509/11. Many results presented in this book were obtained within several research projects funded by The Polish Ministry of Science and Higher Education (grants Nos. 1609/B/H03/2009/36, 0117/B/H03/2010/38 and PB1283/T09/2005/29), The National Centre for Research and Development (grant No. NCBiR/ENIAC-2009-1/1/2010), The European Nanoelectronics Initiative Advisory Council JU ENIAC (Project MERCURE, contract No. 120122) and The European Regional Development Fund under the Innovative Economy Operational Programme (grant No. 01.01.02-00-015/09-00), both at The Faculty of Chemistry, Jagiellonian University and The Faculty of Non-Ferrous Metals, AGH University of Science and Technology.

Last but not least I would like to thank my copy-editor Jo Tyszka and the Wiley editorial and production team: Rebecca Stubbs, Emma Strickland, Sarah Tilley, Richard Davies, Tanushree Mathur and Abhishan Sharma.

Thank you!

Chapter 1

Introduction to the Theory of Information

‘Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.’

Albert Einstein

1.1 Introduction

Information processing is the most important and the most energy consuming human activity. Our brains contain approximately 3 × 109 neurons and each of them has approximately 104 connections with other neurons. This impressive network is dense, as each cubic millimetre of neural tissue contains up to 109 synaptic junctions. While the brain constitutes only 2% of body mass, is consumes 20% of the energetic demand at rest. We really must need our personal ‘CPUs’ as Mother Nature invests so much energy in nervous systems. The importance of information storage, transfer and processing has been greatly appreciated through the ages. Nowadays various techniques have revolutionized all aspects of information processing via digital electronic technologies.

It is impossible to find a direct relation between brains and computers, but both systems show some functional and structural analogies. Their building blocks are relatively simple and operate according to well-defined rules, the complex functions they can perform is a result of the structural complexity (i.e. is an emergent feature of the system) and communication between structural elements is digital. This is quite obvious for electronic computers, but spikes of action potential can also be regarded as digital signals, as it is not the amplitude of the signal, but the sequence of otherwise identical pulses that carries information.

1.2 Definition and Properties of Information

We all intuitively use and understand the notion of information, but it defies precise definition. The concept of information has many meanings, depending on the context. It is usually associated with language, data, knowledge or perception, but in thermodynamics it is a notion closely related to entropy. Its technical definition is usually understood to be an ordered sequence of symbols. Information can be also regarded as any kind of sensory input for humans, animals, plants and artificial devices. It should carry a pattern that influences the interaction of the system with other sensory inputs or other patterns. This definition separates information from consciousness, as interaction with patterns (or pattern circulation) can take place in unanimated systems as well.

While the psychological definition of information is ambiguous, the technological applications must be based on strict definitions and measures. Information can be regarded as a certain physical or structural feature of any system. It can be understood as a degree of order of any physical system. This (structural) form of information is usually regarded as a third (along with matter and energy) component of the Universe. Every object, phenomenon or process can be described in terms of matter (type and number of particles), energy (physical movements) and information (structure). In other words, information can be another manifestation of a primary element. In the same way that the special theory of relativity expresses the equivalence of mass and energy (1.1) [1],

(1.1)

the equivalence of energy and information can be shown within information theory (vide infra).

The most precise definition of information is given by the syntactic theory of Hartley and Shannon. According to this theory, information is a measure of the probability of a certain event. It is the amount of uncertainty removed on occurrence of an event or data transmission. The less probable the event, the higher its information value. According to Hartley the amount of information (Ii) given by an event xi can be formulated as (1.2) [2, 3]:

(1.2)

where pi denotes the probability of an event xi and r is the base of logarithm. Such an expressed amount of information is also a measure of the entropy associated with the event xi. The average amount of information carried by an event from a set of events (X) is the weighted average of entropies of all the events within this set (1.3):

(1.3)

This definition automatically defines the unit of information. Depending on the logarithm base the basic information units are bit (r = 2), nit (r = e) and dit (r = 10). With r = 2 the information content of the event is measured as the number of binary digits necessary to describe it, provided there is no redundancy in the message.

The average amount of information in the system is related to the system entropy by (1.4):

(1.4)

where kB is the Boltzmann constant. As derived by Landauer, the energetic equivalent binary transition of one bit at T = 298 K amounts to (1.5) [4]:

(1.5)

More precisely, this is the minimum amount of energy that must be dissipated on erasure of one bit of information. This calculation, initially based on the Second Law of Thermodynamics was later generalized on the basis of the Fokker–Planck equation concerning the simplest memory model and a Brownian motion particle in a potential well [5]. The same value has also been derived microscopically without direct reference to the Second Law of Thermodynamics for classical systems with continuous space and time and with discrete space and time, and for a quantum system [6]. Interestingly, exactly the same value is obtained as the energetic limit required for switching of a single binary switch [7]. Any physical system which can be regarded as a binary switch must exist in two stable, equienergetic states separated by an energy barrier (Figure 1.1a). The barrier must be high enough to prevent thermal equilibration of these two distinct states. At any temperature, mechanical vibration of atoms and the thermal electromagnetic field may induce spontaneous switching between the states.

Figure 1.1 Energy diagram of a binary switch (a) (Adapted from [5] Copyright (1995) American Physical Society) and schematic representation of a physical system corresponding to a binary switch immersed in a thermal bath with two components: radiation and atomic vibrations (b). Adapted from [7] Copyright (2006) Springer Science + Business Media.

The probability of this spontaneous process can be quantified by the error probability Πerr obtained from the Bolzmann distribution (1.6) [7]:

(1.6)

The minimum barrier height (below this value the two states are indistinguishable) can be calculated assuming the error probability is equal to 0.5. The value of the limiting energy thus obtained for a binary switching process is identical to the energetic equivalent of one bit of information (1.7).

(1.7)

At very low temperatures (T → 0 K), however, the Landauer principle is not valid because of quantum entanglement [8]. Recently measured energetic demand for single bit processing in conventional computers (Pentium II, 400 MHz) amounts to 8.5 × 10−11 J bit−1 [9]. The combination of Equations (1.1) and (1.5) yields a mass equivalent of information amounting to 3 × 10−38 kg bit−1.

The above definition of information assumes a finite number of distinct events (e.g. transmission of characters) and so may somehow represent a digitalized form of information. The digital representation of information, along with Boolean logic (vide infra) constitutes the theoretical basis for all contemporary computing systems, excluding the quantum approach.

The strict mathematical definition of information concerns only information in the sense of a stream of characters and other signals, and is not related to its meaning. There are, however three distinct levels at which information may have different meanings. According to Charles S. Pierce and Charles W. Morris we can define three levels of information: syntacticlevel, semantic level and pragmaticlevel. Information on the syntactic level is concerned with the formal relation between the elements of information, the rules of corresponding language, the capacity of communication channels and the design of coding systems for information transmission, processing and storage. The meaning of information and its practical meaning are neglected at this level. The semantic level relates information to its meaning, and semantic units (words and groups of words) are assigned more or less precisely to their meaning. For correct information processing at the syntactic level semantics are not necessary. On the pragmatic level the information is related to its practical value. It strongly depends on the context and may be of economical, political or psychological importance. Furthermore, at the pragmatic level the information value is time dependent and its practical value decreases with time, while correct prediction of future information may be of high value [10, 11].

1.3 Principles of Boolean Algebra

One of the definitions of the amount of information (Equation (1.2)) in the case of r = 2 implies that the total information contained in a system or event can be expressed using two symbols, for example binary digits. This situation is related to prepositional calculus, where any sentence has an attributed logic value: TRUE or FALSE. Therefore Boolean algebra based on a two-element set and simple operators can be used for any information processing. Unlike algebra, Boolean algebra does not deal with real numbers, but with the notions of truth and falsehood. These notions, however, are usually assigned symbols of 0 and 1, respectively. This is the most common symbolic representation, but others (e.g. ⊥, ; TRUE, FALSE) are also in use. Furthermore, the numerical operations of multiplication, addition and negation are replaced with the logic operations of conjunction (∧, AND, logic product), disjunction (∨, OR, logic sum) and complement (¬, NOT). Interestingly, the same structure would have algebra of the integers modulo 2; these two algebras are fully equivalent [12, 13]. The operation can be easily defined if Boolean algebra is understood as the algebra of sets, where 0 represents an empty set and 1 a complete set. Then, conjunction is equivalent to the intersection of sets and disjunction to the union of sets, while complement is equivalent to the complement of a set [10]. These operations can be simply illustrated using Venn diagrams (Figure 1.2).

Figure 1.2 Venn diagrams of set A (a), its complement (b), and the union (disjunction) (c) and intersection (conjunction) (d) of two sets, A and B.

Conjunction in Boolean algebra has exactly the same properties as multiplication in algebra. If any of the arguments of the operation is 0 (i.e. FALSE) the operation yields 0, while if both arguments are equal to 1, the result of conjunction is also 1. Disjunction, unlike addition, yields 1 if both arguments are unity, while in other cases its properties are similar to addition. The properties of the complement operation can be described as follows (1.8), (1.9):

(1.8)

(1.9)

Put simply, this operation exchanges the two Boolean values, that is ¬0 = 1 and ¬1 = 0. Therefore the double complement yields the initial logic value (1.10):

(1.10)

Boolean algebra is based on a set of axioms: associativity, commutativity, distributivity, absorption and idempotence. Furthermore, it assumes the existence of neutral elements and annihilator elements for binary operators.

The associativity rule states that the grouping of the variables in disjunction and conjunction operations does not change the result (1.11), (1.12).

(1.11)

(1.12)

Moreover, the operations of disjunction and conjunction are commutative, that is the result does not depend on the order of arguments (1.13), (1.14):

(1.13)

(1.14)

Both operations are distributive over the other one, that is (1.15), (1.16):

(1.15)

(1.16)

While the first distributivity law (1.15) is rather intuitive and true also in ordinary algebra, the other (1.16) is true only in Boolean algebra.

The absorption law is an identity linking a pair of binary operations (1.17):

(1.17)

Therefore Boolean algebra with two elements (0 and 1) and two commutative and associative operators (∨ and ∧), which are connected by the absorption law, is a lattice. In every lattice the following relation is always fulfilled (1.18):

(1.18)

Therefore the ordering relation ‘ ≤ ’ can be defined as follows (1.19):

(1.19)

The ∨ and ∧ operators can be defined as infimum and supremum of sets of arguments, respectively (1.20), (1.21):

(1.20)

(1.21)

While in binary logic this is quite intuitive, this analysis is necessary to understand basic binary operators in ternary and higher logic systems (vide infra). The binary Boolean operators are idempotent, that is when applied (many times) to one logic variable, its logic value is preserved (1.22), (1.23):

(1.22)

(1.23)

For each binary operator there exists a neutral element, which does not change the value of the logic variable. For disjunction this element is 0, while for conjunction it is 1 (1.24), (1.25):

(1.24)

(1.25)

Annihilators are the elements that destroy information contained in Boolean variables (1.26), (1.27):

(1.26)

(1.27)

Boolean algebra is dual, the exchange of 0 and 1 or ∧ and ∨ operators will also result in also a Boolean algebra, but with different properties. Concomitant interchange of values and operators, however, yields the same algebra. This so-called De Morgan duality can be formulated as follows (1.28), (1.29):

(1.28)

(1.29)

De Morgan duality has very important practical consequences. It allows construction of any Boolean function from only two operators: ¬, and either ∨ or ∧. This is especially important in electronics, because the realization of any combinatorial function requires only a limited number of building blocks (e.g. all binary functions can be achieved via combination of several NAND gates, vide infra).

1.4 Digital Information Processing and Logic Gates

1.4.1 Simple Logic Gates

The simple rules discussed in preceding sections allow any binary logic operations to be performed, and all the complex logic functions can be produced using the basic set of functions: OR, AND and NOT. It is not important whether the information is encoded as electric signals (classical electronics), light pulses (photonics) or mechanical movements. The only important issues are the distinguishability of signals assigned to logic values, and the principles of Boolean algebra. Any physical system whose state can be described as a Boolean function of input signals (also Boolean in nature) is a logic gate. Therefore it is not important if the signals are of electrical, mechanical, optical or chemical nature [14]. Information can be represented by transport of electric charge (classical electronics), ionic charge (electrochemical devices), mass, electromagnetic energy and so on. Furthermore, the actual state of any physical system can be also regarded as a representation of information, for example electrical charge, spin orientation, magnetic flux quantum, phase of an electromagnetic wave, chemical structure or mechanical geometry [15]. Usually the term ‘logic gate’, however, is associated with electronic devices capable of performing Boolean operations on binary variables.

There are two types of one-input electronic logic gates: YES (also called a buffer) and NOT. The YES gate transfers the unchanged signal from the input to the output, while the NOT gate computes its complement (Table 1.1).

Table 1.1 Truth tables, symbols and Venn diagrams for YES and NOT binary logic gates.

There are 16 possible combinations of binary two-input logic gates, but only eight of them have any practical meaning. These include OR, AND and XOR, as well as their combinations with NOT: NOR, NAND, XNOR, INH and IMP (Table 1.2).

Table 1.2 Truth tables, symbols and Venn diagrams for two-input binary logic gates.

The OR gate is one of the basic gates from which all other functions can be constructed. The OR gate produces high output when any of the inputs is in the high state and the output is low when all the inputs are in the low state. Therefore the gate detects any high state at any of the inputs. It computes the logic sum of input variables, that is it performs the disjunction operation.

The AND gate is another of the principal logic gates, it has two or more inputs and one output. The AND gate produces high output (logical 1) only when all the inputs are in the high state. If any of the inputs is in the low state the output is also low (Figure 1.6b). The main role of the gate is to determine if the input signals are simultaneously true. Other words, it performs the conjunction operation or computes the logic product of input variables.

A more complex logic function is performed by exclusive-OR (XOR) gate. This is not a fundamental gate, but it is actually formed by a combination of the gates described above (usually four NAND gates). However, due to its fundamental importance in numerous applications, this gate is treated as a basic logic element and it has been assigned a unique symbol (). The XOR gate yields a high output when the two input values are different, but yields a low output when the input signals are identical. The main application of the XOR gate is in a binary half-adder, a simple electronic circuit enabling transition from Boolean logic to arithmetic.

The whole family of logic gates is formed by the concatenation of OR, AND and XOR gates with the NOT gate, which can be connected to the input or output of any of the above gates. The various connection modes and resulting gates are presented in Table 1.2. Gates resulting from concatenation with NOT are obviously not basic gates, but due to their importance they are usually treated as fundamental logic gates together with NOT, OR and AND logic gates.

Along with fundamental and NOT-concatenated devices (Table 1.2) there are several other devices which are not fundamental (or cannot even be described in terms of Boolean logic), but are important for construction of both electronic and non-classical logic devices.

The FAN-OUT operation drives a signal transmitted through one line onto several lines, thus directing the same information into several outputs (Figure 1.3a). A SWAP gate (Figure 1.3b) is a two-input two-output device; it interchanges values transmitted through two parallel lines. This device is especially interesting from the point of view of reversible computation, as it is an element of the Fredkin gate (vide infra). While the FAN-OUT and SWAP operations are seen to be extremely simple devices in electronic implementations (forked connectors and crossed insulated connectors, respectively), in molecular systems it is not that straightforward. FAN-OUT, for example, requires replication of the signalling molecule [16]. A useful device, which is regarded as universal in quantum computing with cellular automata, is the MAJORITY gate (Figure 1.3c). This is a multiple-input single output device. It performs the MAJORITY operation on input bits, that is yields 1 if more than 50% of inputs are in the high state, otherwise the output is zero. A three-input majority gate can be regarded as a universal gate as it can be easily transformed into OR and AND gates (vide infra).

Figure 1.3 Schematics of FAN-OUT (a), SWAP (b) and MAJORITY (c) gates.

1.4.2 Concatenated Logic Circuits

Single logic gates, even with multiple inputs, allow only basic logic operations on single bits of information. More complex operations, or on larger sets of bits require more complex logic systems. These systems, usually called combinatorial circuits, are the result of connecting several gates. The gates must, however, be connected in a way that eliminates all possible feedback loops, as the state of the circuit should depend only on the input data, not on the device's history. The most important circuits are the binary half-adder and half-subtractor, and the full adder (Figure 1.4a) [17]. These circuits enable arithmetic operations on bits of information in a binary fashion, which is one of the pillars on which all information technology has been built.

Figure 1.4 Logic circuits of half-adder (a), half-subtractor (b) and binary comparator (c).

The half-adder is a device composed of two gates: AND and XOR. It has two inputs (two bits to be added) and two outputs (sum and carry). The half-subtractor is a related circuit (the only difference lies in one NOT gate at input) which performs the reverse operation: it subtracts the value of one bit from the other yielding one bit of difference and one bit of borrow (Figure 1.4b).

An interesting device, closely related to the half-adder and the half-subtractor is the binary comparator. It takes two bit inputs (x and y) and yields two bit outputs, which are determined by the relationship between the input quantities. If x = y one output is set to high (identity bit) and the other to low (majority bit). If x > y the identity bit is zero, while the majority bit equals 1. In the case of x < y both output bits are 0 (Table 1.3, Figure 1.4c).

Table 1.3 Truth table for binary half-adder, half-subtractor and comparator.

The appropriate connection of two binary half-adders or binary half-subtractors results in two more complex circuits, the binary adder and the binary subtractor, respectively. A full adder consists of two half-adders and an OR gate (Figure 1.5a). The circuit performs full addition of three bits yielding two-bit results. Similarly, a full subtractor is built from two half-subtractors and an OR gate (Figure 1.5b). This device can subtract three one-bit numbers yielding a two-bit binary result. The schematics of the binary full adder and full subtractor are shown in Figure 1.5 and the corresponding logic values in Table 1.4.

Figure 1.5 Electronic diagrams for binary full adder (a) and full subtractor (b). HA stands for half-adder and HS for half-subtractor, respectively. In the case of subtractors, a stands for subtrahend and b for minuend.

Table 1.4 Truth table for full adder and full subtractor.

The simple concatenated logic circuits show the infinite possibilities of combinations of simple building blocks (logic gates) into large functional circuits.

Figure 1.6 Schematic of a sequential information processing device (an automaton). The simplest memory forf the device can be realized by a feedback loop (dashed arrow), feeding some of the outputs of the device to the input.

1.4.3 Sequential Logic Circuits

A circuit comprised of connected logic gates, devoid of feedback loops (memory), is a combinatorial logic circuit, a device whose output signal is a unique Boolean function of input variables. A combinatorial logic circuit with added memory forms a sequential logic circuit, often referred to as an automaton (Figure 1.6). Memory function can be simply obtained by the formation of a feedback loop between the outputs and inputs of individual gates within the circuit. The output state of an automaton depends on the input variables and the inner state (memory) of the device.

The memory function can be simply realized as a feedback loop connecting one of the outputs of the logic device to one of the inputs of the same device. The simplest (but not very useful) memory cell can be made on the basis of an OR gate via feeding back the output to the input (Figure 1.7).

Figure 1.7 Looped OR gate as a model of the simplest memory cell (a) and RS-type flip-flop built from two NOR gates (b).

Initially the device yields an output of 0. However, when the input is set to high, the output also switches to the high state. As the output is directed back to the input, the device will remember the state until power-off. Loops involving XOR and NAND gates tend to generate oscillations. These oscillations render the circuits unusable. This problem, however, can be simply solved in feedback circuits consisting in two gates (NOR, NAND, etc.) and two feedback loops (Figure 1.7b). This circuit is the simplest example of a latch (flip-flop), a device, the state of which is a Boolean function of both the input data and the state of switch. This device can serve as a simple memory cell and after some modification can be used as a component of more complex circuits: shift registers, counters, and so on.

The two inputs of the latch, named R and S (after set and reset) change the state of the outputs in a complex way (Table 1.5), provided they are never equal to 1 at the same time (i.e. R = S = 1) as this particular combination of inputs results in oscillations of the latch.

Table 1.5 Truth table for the R-S latch.

In the case of most input combinations, the output state of the device is not changed, but the (1,0) state induces 0 → 1 switching, while the (0,1) state results in 1 → 0 switching.

1.5 Ternary and Higher Logic Calculi

Binary logic can be generalized for any system with a finite number of orthogonal logic states. In ternary logic any sentence may have three different values: FALSE, TRUE or UNKNOWN. Analagous to binary logic, numerical values can be associated with these values, as shown in Table 1.6.

Table 1.6 Numerical representations of ternary logic values in unbalanced and balanced system.

Logic valueNumerical representationUnbalancedBalancedFALSE0−1UNKNOWN½, #0TRUE11

Logic operations are defined in an analogous way to the case of binary logic:

The unary ternary operator NOT is defined as (1.30):

(1.30)

while the binary ternary operators are defined as follows:

(1.31)

(1.32)

(1.33)

The logic values for unary and binary ternary operators are shown in Tables 1.7 and 1.8.

Table 1.7 Truth table for the unary ternary NOT.

ANOT Alogic valuesFALSETRUEUNKNOWNUNKNOWNTRUEFALSEnumerical representation01½½10

Table 1.8 Truth table for the binary ternary OR, AND and XOR operators.

In any multivalued logic the unary and binary operators can be defined as follows:

(1.34)

(1.35)

(1.36)

(1.37)

where T0 represents the numerical value associated with the TRUE value. These definitions hold for any ordered unbalanced numerical representation of a multinary logic system.

The main advantage of ternary logic consists in lower demand for memory and computing power, however the electronic implementation of ternary logic gates is not as straightforward as in the case of binary logic gates. In the second half of the twentieth century Russian Setun () and Setun-70 (-70) computers, based on ternary logic, were developed at the Moscow State University [18].

Along with ternary logic, so called three-valued logic has been developed. Three-valued electronic logic combines two-state Boolean logic with a third state, where the output of the gate is disconnected from the circuit. This state, usually called HiZ or Z (as this is a high impedance state) is used to prevent shortcuts in electronic circuits. The most common device is a three-state buffer (Figure 1.8, Table 1.9).

Figure 1.8 The electronic symbol for a three-state buffer and its functional equivalent.

Table 1.9 Truth table for the three-state buffer.

InputsOutputab00HiZ10HiZ010111

1.6 Irreversible vs Reversible Logic

The energetic equivalent of information (vide supra) is dissipated to the environment when information is destroyed. This is one of the fundamental limits of information processing technologies (see Chapter 3). In order to avoid this limit, computation should be performed in such a way that no bits of information are destroyed. This approach is usually called reversible computing, but another term, non-destructive computing, is also in use. It concerns all the computational techniques that are reversible in the time domain, so the input and the output data are interchangeable. First of all, this approach implies that the number of inputs of the device equal the number of outputs. Other words, the output of the reversible logic device must contain original information supplied to the input. In this sense amongst classical Boolean logic gates only YES and NOT can be regarded as reversible (cf. Table 1.1).

All of these gates can be described in terms of permutation of states, therefore they can be easily described by unitary matrices (Pauli matrices) [19, 21]. The construction of such matrices is shown in Figure 1.10.

The unitary matrices represent mapping of input states into output states, as reversible logic functions can be regarded as bijective mapping of n-dimensional space of data into itself. This ensures that no information is lost during processing, as the mapping is unique and reversible.

NOT and SWAP gates have already been discussed in the preceding section. More complex is the C-NOT (controlled NOT) gate, also known at the Feynman gate. This two-input two-output device (1.38) transfers one of the input bits directly to the output, but the second bit is replaced with its complement if the first input is in the high state. This output is thus a XOR function of the inputs.

(1.38)

The C-NOT gate, however, is not universal, as any combination of C-NOT gates cannot perform all the basic Boolean operations (cf. Table 1.2). Introduction of one more control line to the C-NOT gate results in CC-NOT (controlled-controlled NOT, Figure 1.9d). This gate, called also the Toffoli gate, is universal, as it can perform all simple Boolean functions. Its unitary matrix is shown as Equation (1.39).

(1.39)

A device which combines the ideas of SWAP and C-NOT is a so called Fredkin gate (Equation (1.40), Figure 1.9d). It can be shown that this gate is universal as well.

(1.40)

The universality of Toffoli and Fredkin gates is, however, not a unique feature. Let us look at three-input-three-output binary devices. Altogether there are 224 = 16 777 216 different truth tables for 3 × 3 logic devices. Reversible logic gates must, however, map directly each input state to a different output state (cf. Figure 1.10). This makes 8! = 40 320 different reversible logic gates in a 3 × 3 device. Only 269 of them are not fundamental, that is their combinations cannot generate all Boolean functions.

Figure 1.9 Selected reversible logic gates presented in Feynman quantum gate notation [19, 20]: NOT (a), SWAP (b), C-NOT(c) CC-NOT, that is Toffoli gate (d) and Fredkin gate (e).

Figure 1.10 Construction of unitary permutation matrices representing reversible logic operations in the case of NOT (a) and SWAP (b) logic gates.

1.7 Quantum Logic

The elemental unit of information in quatum systems is the qubit (quantum bit). Contrary to the bit, its value is not confined to one of the two allowed states of ‘0’ and ‘1’, but the state of any qubit may be any linear combination of and eigenstates (1.41):

(1.41)

where c0 and c1 are complex coefficients normalized to unity. Even though a qubit has discrete orthogonal eigenstates of and , it can be regarded as an analogue variable in the sense that it has a continuous range of available superpositions (1.41). This state of a qubit collapses to or if the information is read from the qubit (i.e. when any measurement of the qubit is performed). In other words, upon measurement a qubit loses its quantum character and reduces to a bit [22]. The graphical representation of a qubit as a point on a Bloch sphere is shown in Figure 1.11.

Figure 1.11 Bloch sphere representing a single qubit. The orthogonal and eigenstates are located at the opposite poles of the sphere, while any superpositions are located on the surface of the sphere. Adapted from Ref. [24].

Quantum systems containing more than one qubit exist in a number of orthogonal states corresponding to the products of eigenstates. A two-qubit system, for example, may have four eigenstates: ,, and . Unlike classical systems, interference between individual qubits will result in quantum states of the form (1.42):

(1.42)

Furthermore, two interfering qubits can exist in an entangled state, when the result of measurement on one qubit determines the result of the measurement on the other, like in the Einstein–Podolski–Rosen state (1.43):

(1.43)

All classical logic gates can also be implemented in quantum systems. There are, however, logic operations that can be only performed on qubits [23]. The operation of these functions can be understood with the help of the Bloch representation of a qubit (Figure 1.11) and can be regarded as various symmetry operations on the qubit vector.

The first and the simplest gate of this type is . Its unitary matrix has the form (1.44):

(1.44)

This operation, which has no classical equivalent, has the following property (1.45):

(1.45)

that is, it flips the vector in the Bloch space by (cf. Figure 1.11), while the NOT gate results in π radian flipping along the x axis. There are two other possible rotations, along y and z axes, respectively, with the corresponding matrices (1.46):

(1.46)

respectively. Another important gate is the Hadamard gate with a unitary matrix of the form (1.47):

(1.47)

The operation of the gate is simple: from or states the gate yields the superposition of and with equal propabilities, that is (1.48)–(1.49):

(1.48)

(1.49)

A universal quantum logic gate that corresponds to the Toffoli gate is the Deutsch quantum gate(1.50) [25]:

(1.50)

Reversible (or non-destructive) information processing with application of the above and other devices is not restricted by the Shannon–Landauer–von Neuman energetic limit (cf. Equation (1.5) and Section 3.1) and therefore the thermodynamic limits of binary information processing do not apply to quantum computation [26].

References

1. Einstein, A. (1905) Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Ann. Phys., 323, 639–641.

2. Shannon, C.E. (1948) A mathematical theory of communication. Bell. Syst. Tech. J., 27, 379–423.

3. Hartley, R.V.L. (1928) Transmission of information. Bell. Syst. Tech. J., 7, 535–563.

4. Landauer, R. (1961) Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 5, 183–191.

5. Shizume, K. (1995) Heat generation by information erasure. Phys. Rev. E., 52, 3495–3499.

6. Piechocinska, B. (2000) Information erasure. Phys. Rev. A., 61, 062314.

7. Cavin, R.K. III, Zhirnov, V.V., Herr, D.J.C. et al. (2006) Research directions and challenges in nanoelectronics. J. Nanopart. Res., 8, 841–858.

8. Allahverdyan, A.E. and Nieuwenhuizen, T.M. (2001) Breakdown of the Landauer bound for information erasure in the quantum regime. Phys. Rev. E., 64, 056117.

9. Landar', A.I. and Ablamskii, V.A. (2000) Energy equivalent of information. Cyber Syst. Anal., 36, 791–792.

10. Waser, R. (2003) Properties of information, in Nanoelectronics and Information Technology (ed. R. Waser), Wiley-VCH Verlag GmbH, Weinheim, pp. 11–23.

11. Szaciłowski, K. (2008) Digital information processing in molecular systems. Chem. Rev., 108, 3481–3548.

12. Zhegalkin, I.I. (1927) O tekhnike vychisleniya predlozhenii v simvolicheskoi logike (in Russian, On the technique of calculating propositions in symbolic logic). Mat. Sbor., 43, 9–28.

13. Stone, M.H. (1936) The theory of representation for Boolean algebras. Trans. Amer. Math Soc., 40, 37–111.

14. Hillis, W.D. (1999) The Pattern on the Stone. The Simple Ideas that Make Computers Work, Perseus Publishing, Boulder.

15. Waser, R. (2003) Logic gates, in Nanoelectronics and Information Technology (ed. R. Waser), Wiley-VCH Verlag GmbH, Weinheim, pp. 321–358.

16. Zaune, K.P. (2005) Molecular information technology. Crit. Rev. Solid State Mat. Sci., 30, 33–69.

17. Gibilisco, S. (ed.) (2001) The Illustrated Dictionary of Electronics, McGraw-Hill, New York.

18. Brousentsov, N.P., Maslov, S.P., Ramil Alvarez, J. and Zhogolev, E.A. (2010) Development of ternary computers at Moscow State University. [cited 2010 2010-03-26]; Available from: http://www.computer-museum.ru/english/setun.htm.

19. Feynman, R. (1996) Feynman Lectures on Computation, Addison-Wesley, Reading, Mass.

20. Barenco, A., Bennett, C.H., Cleve, R. et al. (1995) Elementary gates for quantum computation. Phys. Rev. A., 52, 3457–3467.

21. Kitaev, A.Y. (1997) Quantum computations: algorithms and error correction. Russ. Math. Surv., 52, 1191–1249.

22. Vedral, V. and Plenio, M.B. (1998) Basics of quantum computation. Progr. Quant. Electron., 22, 1–39.

23. Galindo, A. and Martín-Delgado, M.A. (2002) Information and computation: Classical and quantum aspects. Rev. Mod. Phys., 74, 347–423.

24. Vandersypen, L.M.K. and Chuang, I.L. (2004) NMR techniques for quantum control and computation. Rev. Mod. Phys., 76, 1037–1069.

25. Deutsch, D. (1985) Quantum computational networks. Proc. R. Soc. Lond. A., 425, 73–90.

26. Zhirnov, V.V., Cavin, R.K. III, Hutchby, J.A. and Bourianoff, G.I. (2003) Limits to binary logic switch scaling. A gedanken model. Proc IEEE., 91, 1934–1939.

Chapter 2

Physical and Technological Limits of Classical Electronics

‘Science may set limits to knowledge, but should not set limits to imagination’

Bertrand Russell

2.1 Introduction

Progress in the performance of electronic devices was for a long time well described by the empirical Moore's law [1]. It states that the economically justified number of transistors in a microchip grows exponentially in time. In the first formulation of 1965 it was postulated, that this figure doubles every 12 months, while now (2010) it has been scaled to approximately 24 months. Moore's law cannot be, however, valid forever. The development of information processing technologies (and hence the ultimate performance of the resulting devices) is limited by several factors. Some of the constrains are simply a result of fundamental principles, including the granular structure of matter, Einstein's special theory of relativity, Heisenberg's uncertainty principle and others. These limits are absolute in the sense that they impose constraints on any physical system and cannot be eliminated by technological progress. The only way to ignore these constraints may consist in changing the information processing paradigms and utilizing the apparent hindrances. For example, quantum phenomena (which are deleterious for classical electronic devices) can be utilized in quantum computing.

Another kind of limitation results from applied technologies and economics. Device cooling, doping inhomogeneity, crosstalk, latency and electron tunnelling are the best examples of such limitation. In contrast to the fundamental limits of computation, the technological limits can be overcome with appropriate technological development.

Along with progress in miniaturization of transistors and other active elements in integrated circuits (the more ‘Moore’ approach) progress in the combination of digital devices with analogue, MEMS, radio frequency, high voltage and sensory devices is also observed (‘more than Moore’ approach) [2, 3]. These new devices offer much larger functional versatility and diversification of microchips, including communication, sensing and prospective integration with biological materials and structures (biochips). These additional features incorporated into digital microchips do not scale in the same way as digital electronics, but offer new capabilities within single device.

The following sections discuss the most important limitations imposed on information processing devices by the fundamental laws of physics and technological imperfections.

2.2 Fundamental Limitations of Information Processing

The most obvious limitation in information processing results from Einsteins special theory of relativity [4]. As the speed of light in a vacuum (c = 299 792 458 m s−1) is the highest available speed, information cannot travel faster than light. This limits both the rate of information transfer, and the ultimate size of the device working with a predefined switching time [5]. For example, a device with a size of 1.2 μm cannot transmit information faster than 200 ps. In the case of electronic devices, the signal is even slower by a factor of at least two. Superconducting devices based on the Josephson junction can work at frequencies reaching 1000 GHz and these high frequencies limit the size of the device to approximately 0.3 mm [6]. This automatically imposes limits on the size of elementary components of the chip as larger size elements (built using classical silicon technologies) would result in a severe race condition (i.e. delays on different signal pathways resulting in desynchronization of the device). This limitation, however, concerns only semiconductor devices of classical architecture.

The energetic limit of computation, referred to as the Shannon–Landauer–von Neuman limit(2.1) quantifies the energy that must be dissipated in the thermal bath on destruction of one bit of information.

(2.1)

The temperature T is not the temperature of the device itself, but the temperature of the thermal bath which absorbs heat from the device. Therefore, any computing process which dissipates energy cannot operate with heat dissipation lower than 3 × 10−23 J per bit of information. The coolest thermal reservoir of unlimited heat capacity is the interstellar microwave background with a temperature of 3 K. More practical devices utilize thermal baths at room temperature, which results in minimal energy dissipation of 2.88 × 10−21 J· bit−1 (0.018 eV·bit−1).

Another limit imposed on any physical system results from quantum mechanics. The time necessary to switch any physical system to an orthogonal state is limited by Heisenbergs uncertainty principle (2.2) [7]:

(2.2)

where ΔE is the uncertainty of energy of the system and Δτ is the lifetime of a particular state [8]. Other words, any system with energy ΔE cannot evolve in a time shorter than Δτ. Therefore any binary logic process within a system of average energy ΔE cannot be performed in a time shorter than Δτ, and the ultimate performance of a computer is limited by its total energy [9]. Taking into account all these assumptions, Seth Lloyd presented an ingenious analysis of performance of a 1 kg laptop. The total energy of the computer can be derived from Einsteins mass–energy equivalence (2.3) [10]:

(2.3)

A computer of mass 1 kg therefore carries a total energy of 8.99 × 1016 J. Combination of (2.2) and (2.3) yields the minimum time for a binary switching process of 1.843 × 10−51 s. This figure is many orders of magnitude smaller than the Planck time(2.4):

(2.4)

that is the shortest physically possible time period, and also much shorter than a chronon (the quantum of time limiting the timescale of evolution of a physical system) (2.5) [11]:

(2.5)

On the other hand, information processing does not have to be either entirely serial or parallel, and the limit can be understood as information processing with a rate of 5.43 × 1050 binary operations per second. In this limit more serial or more parallel processing just means a different energy allocation between the various nodes (logic gates) of the hypothetical device, but has no real influence on its global computing performance [9].

Heisenbergs uncertainty principle also limits the dimensions of basic computational device (switch) operating at the ESLN limit and using electrons as information carriers (cf. Equation (2.1)) [12]. In semiconducting structures only the kinetic energy of electrons is taken into account, while energy associated with the rest mass is neglected. Any semiconductor structure must be large enough to meet the following condition (2.6):

(2.6)

where Δp is the x-component of the momentum of the electron within the semiconducting structure (2.7):

(2.7)

Simple calculations yield the size of the smallest possible switch to be about 1.5 nm, which corresponds to a density of nmax = 4.7 × 1013 devices per square centimetre. Within the same limit the shortest switching time would be (according to Equation (2.2)) on the order of 4 × 10−14 s, which corresponds to a clock frequency of 250 THz. Operation under these conditions, however, would be associated with a power dissipation of (2.8):

(2.8)

This figure (especially in comparison with Sun radiation intensity of 6.3 kW cm−2 [13]) seems to be quite unrealistic. A further increase of element density would be possible at higher operating temperatures, but it would result in even larger power dissipation [12].

It is not only the ultimate size and the performance of information processing devices that are limited by fundamental laws. The same laws limit the information density, which in turn influences the performance of information storage devices. The amount of information that can be stored in any physical system is closely related to its entropy, which in turn reflects the number of available states of a system with average energy E. The number of available memory bits of any system can be calculated as follows (2.9) [9]:

(2.9)

where S is the total entropy of the system. The entropy in turn can be calculated using a canonical ensemble(2.10)–(2.12):

(2.10)

(2.11)

(2.12)

In a system of average energy E, a state of energy Ei can exist with a probability of pi, where Z is the partition function. For the ultimate computational performance calculations a system was chosen which preserves only total energy, angular momentum and electric charge. The baryon number may not be conserved in this model, as it assumes mass–energy equivalence. At a given temperature T the entropy is dominated by the contribution of particles with masses smaller than kBT/2c2 and the energy contributed by the jth such particle is given by (2.13):

(2.13)

where V is the systems volume and rj is equal to the number of particles/antiparticles in the species (that is, 1 for photons, 2 for electrons/positrons) multiplied by the number of polarizations (2 for photons, 2 for electrons/positrons) multiplied by a factor that reflects particle statistics (1 for bosons, 7/8 for fermions) [9]. The entropic contribution of this particle can in turn be expressed as follows (2.14):

(2.14)

The assumption that the entropy is dominated by black-body radiation photons results in a temperature of 5.87 × 108K, which according to (2.9) and (2.14) results in available memory space of I = 2.13 × 1031 bits [9]. This value is only a rough estimate, as it neglects the presence of any heavy particles. It is interesting to note, however, that the average relativistic mass of a photon in such a system is kBT/2c2 = 4.51 × 10−32 kg which is on the order of magnitude of the rest mass of an electron (me = 9.1 × 10−31 kg), so this ultimate device operates at conditions close to spontaneous creation of electrons and positrons.

The values presented above may not ever be achieved by any computing system, and definitely cannot be achieved by any semiconductor-based device. Operation of these ultimate devices (based mostly on hot radiation and quark–gluon plasma) would be as difficult as controlling a thermonuclear explosion. On the other hand, this analysis supports the previous discussion (see Chapter 1) on the physical nature of information.

Any practical device cannot use the whole rest mass as the energy for computation. Present electronic devices use only the kinetic energy of electrons and numerous degrees of freedom are used to encode every single bit of information. Therefore they are far away from the physical limits of computation discussed above. On the other hand, NMR quantum computing uses single nuclear spins to encode bits of information, and with a density of 1025 atoms per one kilogram of ordinary matter these devices are close to the physical limits for information densities [9, 14].

2.3 Technological Limits of Miniaturization

Nowadays silicon is the main material used in microelectronics. Devices made of silicon have certain limitations resulting from its physical properties (e.g. thermal conductance, charge carrier mobility, electrical permittivity, etc.) and others resulting from the production technologies.

The speed at which an electromagnetic wave propagates in a semiconductor limits the rate of electric signal transmission. It strongly depends on the relative electrical permittivity of the material by (2.15):

(2.15)

First of all, a classical transistor (or any other semiconductor-based device) must be large enough in order to benefit from the continuum of electronic states constituting a band (i.e. valence band and conduction band). Elements that are too small would suffer from a quantum size effect, which alters the electrical and optical properties of semiconductors (see Section 4.3). The most important feature which is strongly size-dependent is the bandgap energy, Eg. For any semiconductor it changes with the particle size as (2.16) [15]:

(2.16)

where Eg and are the bandgap energies of nanocrystals and microcrystals, respectively, R is the radius of a nanocrystal particle, me and mh are the effective masses of an electron and a hole, respectively, ε is the dielectric constant of the material and is the Rydberg energy, defined as (2.17):

(2.17)

Quantum size effects do not disqualify these systems, but make the design much more complex, as the principle of operation of such devices is size-dependent. Furthermore, in such small features control over the dopant concentration and distribution is extremely difficult [16, 17]. With a doping density of 1016 cm−3, a 1 × 1 × 1 μm silicon cube contains only 104 doping atoms. A cube of the same material with a 50 nm edge contains on average only 1.25 dopant atoms. Therefore a much higher doping level is required to assure more reproducible properties of small silicon devices. On the other hand, doping levels on the order of 1018 cm−3 (125 doping atoms per cube) yield degenerate semiconductors, which exhibit metallic properties [18]. The fluctuations of dopant concentrations within a cube of Δx edge are given by (2.18):

(2.18)

where σ is the standard deviation of the number of doping atoms, μ is the mean value and is the distance between individual dopant atoms. With decreasing cube size the standard deviation of the dopant concentration increases without bounds. In an extreme case some regions may be undoped, while the neighbouring regions may be heavily doped, which would result in dramatic changes in the electric field within a device. This is especially important in devices with thin depletion layer at p–n junctions. Uniform distribution of doping atoms in large semiconductor structures results in a depletion layer with only small thickness fluctuations (Figure 2.1a