Transport of Information Carriers in Semiconductors and Nanodevices pdf pdf

Transport of
Information-Carriers
in Semiconductors and
Nanodevices
Muhammad El-Saba
Ain-Shams University, Egypt

A volume in the Advances in Computer and
Electrical Engineering (ACEE) Book Series

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Library of Congress Cataloging-in-Publication Data
Names: El-Saba, Muhammad, 1960- author.
Title: Transport of information-carriers in semiconductors & nanodevices / by
Muhammad El-Saba.
Description: Hershey, PA : Engineering SCience Reference, 2017. | Includes
bibliographical references.
Identifiers: LCCN 2016057816| ISBN 9781522523123 (hardcover) | ISBN
9781522523130 (ebook)
Subjects: LCSH: Electron transport. | Photon transport theory. |
Semiconductors--Transport properties.
Classification: LCC QC611.6.E45 E423 2017 | DDC 621.3815/2--dc23 LC record available at https://lccn.loc.
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Table of Contents

Preface .................................................................................................................................................. vii
Chapter 1
Introduction to Information-Carriers and Transport Models .................................................................. 1
Chapter 2

Semiclassical Transport Theory of Charge Carriers, Part I: Microscopic Approaches ........................ 72
Chapter 3
Semiclassical Transport Theory of Charge Carriers, Part II: Macroscopic Approaches .................... 138
Chapter 4
Quantum Transport Theory of Charge Carriers.................................................................................. 188
Chapter 5
Carrier Transport in Low-Dimensional Semiconductors (LDSs) ....................................................... 274
Chapter 6
Carrier Transport in Nanotubes and Nanowires ................................................................................. 334
Chapter 7
Phonon Transport and Heat Flow ....................................................................................................... 379
Chapter 8
Photon Transport ................................................................................................................................. 450
Chapter 9
Electronic Spin Transport ................................................................................................................... 530

Chapter 10
Plasmons, Polarons, and Polaritons Transport .................................................................................... 587
Chapter 11
Carrier Transport in Organic Semiconductors and Insulators ............................................................ 617

Index ................................................................................................................................................... 676

vii

Preface

During the last decade, rapid development of electronics has produced new high-speed devices at
nanoscale dimensions. These nanodevices have tremendous applications in modern communication
systems and computers.
This book, Transport of Information-Carriers in Semiconductors and Nanodevices, is intended to be
the first in a series of 3 volumes titled Semiconductor Nanodevices: Physics, Modeling, and Simulation
Techniques.
The main purpose of this course is to develop an appreciation and a deep understanding for the conceptual foundations underlying the operation of emerging nanoelectronic devices.
I’ve decided to dedicate the first volume to talk about transport modelling, which can serve both
academicians and professionals. The next book will cover the Modeling and Simulation Techniques,
and will be rather dedicated for professionals and postgraduate students in device simulation. The third
book is about Physics and Operation of Modern Nanodevices. However, for the matter of completeness
in each book, I squeeze other volumes in a single chapter or as illustrative case studies.
In this book, we study the transport models of information carriers (e.g., electrons and photons) in

semiconductors and nanodevices. It contains a comprehensive discussion about carrier transport phenomena and includes some topics not previously assembled, altogether, in a single book.
I mean by information carriers, the particles or particle characteristics that carry and transport signals
in semiconductor materials and solid-state devices. For instance, the electronic charge in conventional
semiconductor devices, the electronic spin in spintronic devices and photons in optoelectronic devices.
In fact, the characteristic of any particle may be utilized for information transport. For example, a quantum bit (or qubit) of information can be manipulated and encoded in any of several degrees of freedom,
notably the photon polarization. In addition, other quasi particles, such as phonons (lattice vibration
waves) may be considered as information carriers, because they are capable of transporting thermal
energy from point to another in solid-state devices. In fact, some or all of these information carriers
may interact in the same device. Indeed, electrons and phonons interact in all semiconductors devices.
They also intervene, together with photons in photonic devices, like laser diodes. In the so-called spin
light emitting diode (spin LED), the electron spin plays a basic role with all the aforementioned types
of information carriers.
The main subject of this book is, therefore, focused around the transport equations, which govern
the transport of information carriers. These transport equations form the physical device models of all
semiconductor devices, including the emerging nanodevices. The TCAD (Technology Computer-Aided
Design) tools make use of these transport models to simulate the behavior of solid-state devices and
circuits, in terms of the device structure and external boundary conditions of bias voltage or current.

Preface

The utilization of TCAD tools is essential because they accelerate the R&D cycle and nowadays, they
become essential more than ever. In fact, the device simulation has three main purposes; to understand
the underlying physics of a device, to depict the device characteristics and to predict the behavior of new
devices. Actually, the advent of new nanodevices has been an everyday occurrence. For example, some
versions of the 6th generation of Intel Core processors, is manufactured using a 14nm process. Projecting
the advance of semiconductor industry for the next few years, we expect to see nanodevices approaching
the size of a few atoms (1nm). The devices at such nanoscale display special quantum properties which
are completely different from the case of bulk systems. Therefore, the availability of powerful transport
models, which account for the underlying quantum effects, is very important for the simulation of such
nanodevices. Everybody working in the field of modeling and simulation of state-of-the-art devices feels
that current TCAD tools should be pushed beyond their present limits.
Almost all scientists in the field of semiconductors, agrees that a rigorous study of carrier transport
in nanodevices needs a many-body quantum description. Such a description requires the solution of a
huge number of equations describing each carrier of the system. Actually, the description of transport
in a real device should include the real number of carriers in both the device and its contacts to the
external world, and this is beyond the ability of typical computing platforms. Therefore, many levels of
approximation that sacrifice some vital information about the physics of transport process are necessary.
The figure below illustrates the hierarchy of main transport approaches, which are used in describing
carrier transport in semiconductors and nanodevices.
Many authors distinguish between three classes of transport models, namely;

•
•
•

Quantum models,
Kinetic models, and
Macroscopic (fluid dynamics) models.

The quantum approach lies at the top level of transport theories, for many-body problems. To treat
quantum problems, a mean-field (e.g., the Hartee-Fock potential) approximation is usually adopted to
Figure 1.

viii

Preface

transform the many-body system into one-electron problem. The Non-equilibrium Green function (NEGF)
method is very popular as a quantum approaches. Above this are quantum kinetic approaches such as the
Liouville-von Neumann equation of motion for the density matrix, or Wigner distribution that contain
quantum correlations but retain the form of semiclassical approaches. When we move from quantum to
classical description of carrier transport, information concerning the phase of the electron and its nonlocal behavior are lost, and electronic transport is treated in terms of a localized particle framework.
The semiclassical transport theory is based on the Boltzmann transport equation (BTE), which represents a kinetic equation describing the time evolution of the distribution function of particle. The BTE
has been the primary framework for describing transport in semiconductors and semiconductor devices
with micro-scale dimensions. There are then approximations to the BTE, given by moment expansions
of the BTE which lead to the hydrodynamic, the drift-diffusion, and relaxation time approximation approaches to transport. Finally, the so-called compact models come at an empirical level as circuit models
for circuit simulation.
This book consists of 11 chapters, which are organized as follows.
Chapter 1: Introduction to Information-Carriers and Transport Models
Chapter 2: Semiclassical Transport Theory of Charge Carriers (Part I: Microscopic Approaches)
Chapter 3: Semiclassical Transport Theory of Charge Carriers (Part II: Macroscopic Approaches)
Chapter 4: Quantum Transport Theory of Charge Carriers
Chapter 5: Carrier Transport in LDS and Nanostructures
Chapter 6: Carrier Transport in Nanotubes and Nanowires
Chapter 7: Phonon Transport and Heat Flow
Chapter 8: Photon Transport
Chapter 9: Spin Transport and Spintronic Devices
Chapter 10: Polarons, Plasamons, and Polaritons Transport
Chapter 11: Carrier Transport in Organic Semiconductors and Insulators

Figure 2.

ix

Preface

I start with the classical approaches and end with the quantum description for composite quasi particles, such as polarons, plasmons, and polaritons.
Each chapter starts with a recap of concerned concepts and provides the state of the art advances in
the field as well as some case studies and overview of the literature. Some physical and mathematical
notes are inserted (without interrupting the main context) to clarify the jargons, that are unavoidably
utilized in such a specialized book.
In Chapter 1, I review the fundamental properties of semiconductors, and explain the transport phenomena within the framework of the classical Drudé model. The Drudé classical model is frequently
introduced to describe the electrical conductivity in solids. This model is still very relevant because free
particle picture can still be used as far as we can assume parabolic energy bands with a suitable effective
mass, near equilibrium. In fact, the Drudé model succeeded to explain (to some extent) the electrical
conductivity, the thermal conductivity, the Hall Effect, as well as the dielectric function and the optical
response of solids. Everything we explain in this chapter about semiconductor properties and carrier
transport is correct to the zero order approximation. In order to get into the details of carrier transport in
semiconductor devices, we proceed in the following chapters, and search for a master transport equation,
in two vertices, namely: the semiclassical and quantum transport theories.
In Chapter 2, I cover the essential aspects of charge carrier transport through solid materials, within
the semiclassical transport theory. We start with a review of the semiclassical approaches that leads to
the concepts of drift velocity, drift mobility, electrical conductivity and thermal conductivity of charge
carriers in metals and semiconductors. The semiclassical transport theory is based on the Boltzmann
transport equation (BTE). The Boltzmann transport equation can be derived from the Lowville equation,
which describes the evolution of the distribution function changes in phase space and time. I discuss the
various approximations and phenomenological approaches which make the equation useful and solvable for semiconductor devices. For instance, I present the spherical harmonic expansion (SHE) and the
Monte Carlo (MC) stochastic Methods as well as the microscopic relaxation-time approximation (RTA),
which leads to the conventional drift-diffusion model (DDM).
In Chapter 3, I discuss the hydrodynamic model (HDM) for semiconductor devices, which plays an
important role in simulating the behavior of the charge carrier in nano devices. This model consists of a
set of nonlinear conservation laws for the particle density, current density, and energy density. The hydrodynamic model for semiconductors is an inexpensive alternative tool for two- and three-dimensional
device simulation. The set of hydrodynamic equations (HDEs), which is derived from the first few moments of the semiclassical BTE, is indeed more accurate than the conventional DDM and less complex
than the direct solution of the BTE (by e.g., the SHE and Monte Carlo Methods).
In Chapter 4, I present the quantum transport approaches, which are necessary to simulate nanodevices including tunneling and other quantization effects. The quantum transport theory originates from
several directions, including the quantum Liouville (von Neumann) equation, the Feynman path integral
as well as the Wigner-Boltzmann transport equation (WBTE). The quantum Liouville equation describes
the temporal evolution of the density operator. The density matrix operator is the favorite mathematical instrument in quantum statistical physics. The so-called Pauli Master equation (PME) is derived
from the quantum Liouville equation. The PME is frequently used to describe irreversible processes in
quantum systems. The kinetic equation for the Wigner distribution function including scattering effects

x

Preface

is called the Wigner-Boltzmann transport function (WBTE). After solving the WBTE, and calculating
the Wigner distribution function (WDF), we can calculate the spatial density of carriers and current, as
well as the average value of any microscopic physical parameter.
Based on the WBTE, the quantum corrected Boltzmann equation, the quantum hydrodynamic
model (QHDM), and the density gradient (DG) approximation can be obtained. Also, the WDF may be
defined as the energy integral of the Green’s function. The Green’s function approach can be used to
give the response of a system to a constant perturbation in the Schrödinger equation. The so-called nonequilibrium Green’s function (NEGF) formalism is a very powerful technique to evaluate the transport
properties of quantum systems in both thermodynamic equilibrium and non-equilibrium conditions. At
the end of this chapter, I present the multi-band transport models and the major band structure calculation methods. This includes the ab initio models, such as the density functional theory (DFT), and the
approximate methods, such as, the tight binding (TB) model, pseudopotential methods, as well as the
GW approximation.
In Chapter 5, I demonstrate the carrier transport phenomena in low-dimensional semiconductors
(LDS), where, free electrons are only permitted to move in one or two dimensions. I describe some
LDS structures, such as quantum wells, quantum wires and quantum dots and the transport models of
charge carriers across them. I discuss the conductance of LDS systems, using the Landauer formalism
(for 2-terminal devices) or the generalized Landauer-Büttiker formalism (for multi-terminal devices). I
also describe some quantum effects that take place in such nanostructures, such as quantum Coulomb
blockade, Aharonov–Bohm, Shubnikov-De Haas oscillations and Kondo effects.
In Chapter 6, I handle transport across Carbon nanotubes (CNT’s), which are one of the most interesting materials in nanotechnolog. Nanotubes and nanowires with dimensions on the nanometer length
scale cannot be treated as classical conductors because their diameters are as small as the mean free
path length (between collisions), but their length is large for the full quantum treatment. Therefore, such
mesoscopic structures need a special framework of transport models, which we discuss in this Chapter.
In Chapter 7, I investigate phonon transport and thermal conductivity in semiconductor structures
and nanodevices. Microscopic approaches such as the Peierls-Boltzmann transport equation (phonon
BTE) and phonon Monte-Carlo simulation can capture quasi-ballistic phonon transport. These models
are valid only when heat transport is diffusive and the characteristic length scales are much larger than
the phonon mean free path. When phase coherence effects cannot be ignored, these semiclassical approaches fail and result in erroneous results. Therefore, I handle the topic of ballistic (non-diffusive)
phonon transport for nanoscale structures and nanodevices.
Chapter 8 is dedicated for photonic transport models. Accurate modeling of photonic devices is essential for the development of optical components in fields like communications, sensing, biomedical
instrumentation, consumer electronics and defense. The specific challenge of optoelectronic device
simulation lies in the combination of photonics and electronics, including the sophisticated interaction
of photons and electrons. Mathematical models for photon transport include the Monte Carlo simulation
method, numerical solution of the semiclassical and quantum transport equations, as well as phenomenological models. Macroscopic photonic transport analysis requires the consideration of seven independent
variables: three spatial directions, two angular directions, frequency and time.

xi

Preface

The so-called radiative transport equation (RTE) is an integro-differential equation that describe the
diffusion and scattering of photons inside matter. The diffusion approximation alleviates the solution of
this equation. Also, the optical Bloch equations, which are sometimes called the Maxwell-Bloch equations (MBEs), describe the dynamics of two-state quantum systems interacting with the electromagnetic
modes of an optical system. Within a semiclassical approach, where the light field is treated as a classical electromagnetic field and the material excitations are described quantum mechanically, all quantum
effects can be treated microscopically on the basis of the semiconductor Bloch equations (SBEs). The
quantum approaches are based on some sort of dynamic wave equations (Schrodinger-like or Heisenberglike) in the microscopic level or the SBEs in the macroscopic level. The so-called dynamics-controlled
truncation (DCT) formalism is another successful microscopic approach that describes coherent correlations in optically excited semiconductors. On the other hand, the most successful approach to study
incoherent effects and correlations in highly excited semiconductors is the nonequilibrium Green’s
functions (NEGF) approach. We discuss these models with illustrative examples, to show the features
and weakness of each model.
In Chapter 9, I present a full quantum and a semiclassical description of spin transport, which explains
how the motion of carriers gives rise to a spin current. The so-called two-component spin-drift-diffusion
model (SDDM) is a simple semiclassical and straight-forward method for spin transport modeling. The
semiclassical models can be useful for investigation of a broad class of transport problems in semiconductors, but they do not include effects of a spin phase memory. The quantum approach of spin density
matrix with spin polarization vector of a spin state is more appropriate for this case. The classical Bloch
equations for spin transport are the analogue of the classical BTE for particle transport. They can be extended to time-dependent non-equilibrium quantum transport equations, using a suitable non-equilibrium
quantum distribution function, like the spin magnetization quantum distribution function (SMQDF). The
so-called spinor-BTE resembles the Boltzmann kinetic equation with spin-orbit coupling in a magnetic
field together with spin-dependent scattering terms. By taking the macroscopic moments of the spinorBTE, we can get a density-matrix based version of the SDDM. The last section of this chapter covers
the latest proven spintronic devices, such as spin-FET, MRAM and spin LED.
In Chapter 10, I present the semiclassical and full quantum models of composite quasi particles, such
as polarons, plasmons and polaritons. These information carriers play a significant role in the emerging
nanoelectronic and nanophotonic devices and systems.
In Chapter 11, I focus the attention on the carrier transport in organic semiconductors and insulator
materials. Organic semiconductors are hydrocarbon molecular crystals or polymers. In order to understand charge transport in organic solids, I review the transport and tunneling mechanisms in disordered
materials. Thereupon, I discuss the recent transport models, such as the semiclassical and quantum
formalisms of Marcus theory.
As the reader can see from the above description of this book, I tried to give a balanced amount of
theory for almost all known transport models of charge carriers, phonons, photons and spin in semiconductors and nanodevices. However, I did my best to avoid drowning the reader into the minor mathematical
details. I consider this a critical point, because even the specialized reader may get bored from the arcane

xii

Preface

mathematical proofs. In fact, the basic interest of the readers who have an engineering background, is
to discover and then to know when and how to apply the different transport models. For these readers I
wrote this book. However, when, some mathematical details are important I mention them, in brief, as
a note, so that the reader can bypass them without interrupting the main subject.
Although this book is primarily theoretical in approach, I frequently refer to experimental results,
which show the variation of transport parameters as well as their measurement methods. I also supplement each chapter, with one or two case studies of real devices that aid understanding of the treated
theory in this chapter. The book has many illustrations and diagrams to clarify the presented transport
models, and comprehensively referenced for further study.
This one-stop book (for almost all semiconductor transport models) is dedicated for engineers and
researchers in solid-state physics and nanodevices, as well as students in nanoelectronics and nanotechnology.
Muhammad El-Saba
Ain-Shams University, Egypt

xiii

1

Chapter 1

Introduction to InformationCarriers and Transport Models

1. OVERVIEW AND CHAPTER OBJECTIVES
During the last decade, the rapid development of electronics technology has produced several new
devices at nanoscale dimensions (nanodevices). Nanodevices, are the tiny devices whose dimensions
are in the order of nanometers (or less than 100nm however). The information carriers in these devices
are the particles or quasi particles that can carry and transport information objects or signals. The most
famous example of an information carrier is the electron charge in conventional semiconductor devices.
Also, photons in photonic and optoelectronic devices and the electron spin in spintronic devices can
be considered as information carriers. In addition, other quasi particles, such as phonons (quasi particles associated with lattice vibration waves) may be considered as information carriers, because they
are capable of transporting energy from point to another in solid-state devices. The recent research in
nanodevices is focused around the control of such information carriers and to exploit their features to
build new devices with superior characteristics in terms if speed and integration density. Naturally, great
efforts have been dedicated to understanding the transport mechanisms of such information carriers in
semiconductors and nanostructures.
The transport theory of information carriers forms the basis of any physical device model. The
transport models are used in Technology/Computer-Aided Design (TCAD) tools to simulate the device
behavior, in terms of its structure and geometry as well as external boundary conditions of voltage and
current. In fact, the transport of information carriers is a non-equilibrium phenomenon, where the role
of external forces plays a crucial role. External forces which drive the device out of equilibrium may be
electromagnetic in origin, such as the electric fields associated with an applied bias, or the excitations
of electrons by optical sources. Alternately, thermal gradients and electrochemical potentials may also
provoke the transport of charge carriers and therefore create external currents and voltages drops, across
the device. The Figure1 depicts the role of carrier transport models in TCAD simulation tools and how
they are used to calculate the current-voltage (I-V) and capacitance-voltage (C-V) characteristics of a
DOI: 10.4018/978-1-5225-2312-3.ch001

Copyright © 2017, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.

Introduction to Information-Carriers and Transport Models

Figure 1. Electronic design automation (EDA) lifecycle and carrier transport modeling in TCAD Tools;
in the EDA section, DFT=Design for Testability, LVS=Layout versus Schematic, DRC=Design Rule
Checking and GDSII is a design file format.

certain device and interact with other electronic design automation (EDA) tools. Also, Figure 2 depicts
the different levels of transport models, in device simulation. As shown, the TCAD tools are based on
semiclassical and quantum transport models. These models range from ab-initio physical models, which
describe the transport of information carriers from first principles down to compact models that describe
the outer behavior (usually the I-V and C-V) of devices and circuits. The success of nanotechnology
to produce well-functioning nanodevices and systems is mortgaged by the availability of suitable and
efficient transport models that meet the challenges at the nanoscale.
As shown in Figure 2, the transport models cover a wide scale, from classical to quantum transport,
according to their accuracy and the required computational costs. Actually, a single description in the
hierarchy of transport models may not be suitable to provide the correct behavior of all devices.
Depending on the device length scale, the carrier transport may be semiclassical or purely quantum.
Nowadays, the most famous semiclassical approaches for the simulation of charge-carrier transport
in semiconductor devices are the drift-diffusion model (DDM), the hydrodynamic model (HDM), the
Spherical harmonic expansion (SHE) as well as the Monte Carlo method (MCM). DDM and HDM
descriptions of particle transport are macroscopic in nature and enable a quick computation of device
characteristics (in terms of macroscopic quantities like the carrier density). Depending on the particular
application, the macroscopic transport models are applicable to devices with characteristic lengths in

2

Introduction to Information-Carriers and Transport Models

Figure 2. Complexity (accuracy) of transport models versus computational time (cost)

the range of micrometers or some hundred nanometers, where microscopic-size and quantum effects
are not dominant. For even smaller devices, it is necessary to resort to microscopic approaches, which
are based on the semiclassical Boltzmann transport equation (BTE) or its quantum counterparts, e.g.,
the quantum Liouville equation (QBTE) or the Wigner BTE (WBTE).
The solution of the BTE by MCM or SHE approaches may yield accurate results for the transport
characteristics in many small devices. However, the semiclassical approaches (both microscopic and
macroscopic) fail as soon as quantum mechanical effects dominate and a description of the information
carriers as localized particles becomes invalid. Indeed, the description of carrier transport in modern
nanodevices requires sophisticated many–body quantum approaches. Clearly, the full quantum description
including the actual number of carriers in a device is beyond the ability of any computational platform
nowadays1. Therefore, approximations are necessary to simulate and predict the behavior of such devices.
In order to construct a successful approximation (model), we need to understand the phenomena behind
the real problem, and under which physical limits, the approximation can be assumed.
Hence, successive levels of approximation, that sacrifice some information about the exact nature of
transport, are sometimes utilized in any nanodevices. As shown in Figure 2, the quantum models range
from ab-initio models, such as density-functional theory (DFT), and the tight-binding (TB) models that
predict the band structure, to the quantum Liouville equation (QBTE) and its variant master equations
as well as the non-equilibrium green functions (NEGF) to predict the device characteristics.
When the appropriate transport model is selected and utilized by a suitable device simulator, we can
get the device input/output characteristics and understand the device behavior. Finally, the so-called
compact models are non-linear circuit models that capture the device behavior, and are suitable for
circuit simulation.
Although we assume a basic knowledge of solid-state physics in this Book, we start with the theoretical fundamentals of semiconductors. This Chapter is a general review of the fundamentals physics of
charge carrier transport in semiconductors, with emphasis on the classic transport models.
Upon completion of this chapter, students will

3

Introduction to Information-Carriers and Transport Models

•
•
•
•
•
•

Understand the concept of transport modeling and information carriers in semiconductors and
nanodevices.
Be familiar with the different models of information carrier transport.
Review the fundamentals of semiconductor physics, such as energy band structure, density of
states, drift and diffusion of charge carriers and carrier scattering mechanisms.
Explain the advantages and disadvantages of the classical transport theory of charge carriers in
metals and semiconductors.
Describe the electrical, thermal, magnetic and optical properties of metals and semiconductors, on
the basis of the simple Drudé model.
Decide what evidence can be used to support or refuse a carrier transport model.

2. CLASSIFICATION OF INFORMATION CARRIERS
The term Information Carriers has its origin in computer science and information technology and has
been applied in many different ways. In computer science, an information carrier is a means to keep
(store) information. However, I mean by information carriers in electronic devices, the particles or particle characteristics that can carry, transport or store signals within a device. For instance, the electron
charge in conventional semiconductor devices and the spin of electrons in spintronic devices as well as
photons in photonic devices are all examples of information carriers. In addition, other quasi particles,
such as phonons (quasi particles associated with lattice vibration waves) may be considered as information carriers, because they are capable of transporting energy from point to another in solid-state devices.
Other examples of information carriers are shown in Figure 3.
A charge carrier is a moving particle, which carries an electric charge. Examples are moving electrons,
ions and holes. In a conducting medium, an electric field can exert work (force) on the free particles,
causing a net motion of their charge through the medium; this is what is referred to as electric current.
In metals, the charge carriers are electrons. Free (or more precisely quasi free) electrons in good conductors are able to move about freely within the material. Free electrons can also be generated in vacuum
and act as charge carriers. As well as charge, an electron has another intrinsic property, called spin. A
spinning charge carrier produces a magnetic field similar to that of a tiny bar magnet.

Figure 3. Examples of information-carriers in electronic, spintronic, optoelectronic, thermoelectric and
quantum devices

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Introduction to Information-Carriers and Transport Models

In melted ionic solids or electrolytes, such as salt water, the charge carriers are ions, atoms or molecules
that have gained or lost electrons so they are electrically charged. Atoms that have gained electrons and
become negatively charged are called anions, while atoms that have lost electrons become positively
charged and called cations.
In semiconductors, electrons and holes (moving vacancies in the valence band) are the charge carriers.
In fact, holes are considered as mobile positive carriers in semiconductors. In semiconductor devices,
most of the electrical, thermal and electrical properties of interest have their origins from electrons (in
the conduction band) and holes (in the valence band).
Of course electrons and holes carry electrical charges as well energy. Other important energy carriers are phonons (lattice vibrations). Actually, the thermal energy transport in crystals occurs primarily
due to the vibration of atoms about their equilibrium positions. In semiconductors, the heat conduction
process takes place, primarily, through lattice vibrations (phonons).
The property of coherence was originally connected with light propagation in optics but now it is
defined in all types of waves. In quantum mechanics coherence is due to the nature of the wave functions,
which are associated with moving particles. Coherence means that the phase difference between wave
functions is kept constant for coherent particles. The delay over which the phase or amplitude wanders
by a significant amount is defined as the coherence time (usually termed τc), as shown in Figure 5. The
coherence length λc is defined as the distance the wave travels in time τc. The spatial coherence of a
wave is defined as the cross-correlation between two points in the wave for all times. The most popular
experimental technique which provides direct information about charge carrier coherence in semiconductors is four-wave-mixing (FWM) spectroscopy.

Figure 4. Information-carriers in electronics and spintronics

Figure 5. Illustration of the concept of phase coherence of the wave functions

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Introduction to Information-Carriers and Transport Models

3. CLASSIFICATION OF TRANSPORT MODELS
The nature of transport in a semiconductor device depends on the characteristic length of the device
active region. The carrier motion can be described with classical laws, when the length of the device
active region is much larger than the corresponding carrier wavelength. When the device dimensions
(or one of them) are comparable to the carrier wavelength, the carriers can no longer be treated as classical point-like particles, and the effects originating from the quantum- nature of propagation begin to
determine transport.
The appearance of quantum effects can be determined by comparing the device size2, L, to the electron mean-free path (λn), or the dephasing length (λϕ) or the de Broglie wavelength (λdB =h/p, where h is
Planck’s constant and p is the electron momentum). The dephasing length (or phase coherence length),
λϕ, is a physical quantity which describes the quantum interference and may be defined as follows:
Dephasing length → λϕ = √(Dn τϕ)

(1)

where Dn is the electron diffusion constant and τϕ is the dephasing (or phase-breaking) time. One way
to obtain the dephasing time (τϕ,) is to measure the magneto-resistance of the material (Pierret, 2003).
The quantum interference and strong coherence phenomena can be observed in nanostructures, when
λdB ≈ L, λn λT, λdB ≈ λn < λϕ ,

(3)

where λT is called the temperature length (or thermal correlation length) which is defined as follows:
Temperature length →λT =< ℏ Dn/kBTL

(4)

where ℏ=h/2π and kB is Boltzmann’s constant.
On the other hand, the semiclassical approach can be still used in small devices as long as:
λdB Δk (satisfying the Heisenberg uncertainty principle
Δx.Δk ≈1). This is the level of the Boltzmann transport equation (BTE), which is a kinetic equation
describing the time evolution of the distribution function of particles. The BTE has been the primary
framework for describing transport in semiconductors devices down to submicron scale. There are then
approximations to the BTE, given by hydrodynamic moments of the BTE which lead to the hydrodynamic
model (HDM), the drift-diffusion model (DDM), and relaxation time approximation approaches (RTA).
Figure 8 illustrates the details of transport models of different sophistication levels to describe the
transport of charge carriers in semiconductor devices.

Figure 7. Schematic of the extension of a particle wave packet and its Fourier transform in the k-space

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Introduction to Information-Carriers and Transport Models

Figure 8. Detailed illustration of information-carrier transport models

4. CHARGE-CARRIER TRANSPORT MODELS IN SEMICONDUCTORS
We know that any thermodynamic system is in thermal equilibrium forever unless it is acted upon by
external forces, i.e. when no exchange of energy is done with the exterior. We may consider a semiconductor in state of thermal equilibrium, as long as it is not acted upon by any external force field (e.g.,
electric field, magnetic field, electromagnetic field or light). However, the individual atoms and electrons
in a solid still exchange energy between themselves, even when no external force is applied. Therefore,
the equilibrium state is called “dynamic thermal equilibrium”.

4.1 Semiconductor Conductivity Model
A semiconductor is neither a true conductor nor an insulator, but half way between. The discovery of
semiconductor properties, dated back to Michael Faraday (1839) who noticed that the conductivity of
some materials decreases as temperature increases, inverse to the behavior of known metals. A variety of
substances, such as germanium (Ge), silicon (Si) and gallium arsenide (GaAs), exhibit semiconducting
properties. In this section we first review the model of conduction in semiconductors using the silicon
as an example. In fact silicon was established as a good semiconductor material about 80 years ago (the
1930s). At this time, Alan Wilson applied Felix Bloch energy band theory to study the energy band structure of silicon. Actually, the Si atom has 14 electrons distributed over energy levels of different orbitals
(1s2, 2s2, 2p6, 3s2, 3p2). The incomplete outer shell of silicon atom contains 4 electrons (3s2, 3p2). The
silicon lattice has a diamond lattice and its atoms have tetrahedral covalent bonds as shown in Figure 9.
In pure silicon lattice all electrons are bound, in the valence band, and there are no free charge carriers (no free electrons!) at zero absolute temperature (0K). Therefore, behaves like an insulator and
the application of an electric field does not result in electric current. In order to produce an electrical

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Introduction to Information-Carriers and Transport Models

Figure 9. Diamond lattice and covalent bonds in pure elemental semiconductors, like silicon

current in a semi-conductor, some valence electrons must be freed from their bonds. This can be done
by supplying the crystal by external energy, usually in the form of heat or light. The minimum energy
that is required to free an electron in a pure semiconductor is equal to the height of its energy gap Eg. In
Si, the energy gap is about 1.2eV at 300K. Each free electron in a pure semiconductor leaves a broken
bond (or a hole) as shown in Figure 10. Such a free electron roams everywhere in the crystal with equal
probability in all directions. A free electron can also recombine with a vacant bond (a hole) to produce
a bond, while transmitting its excess energy in the form of light quanta (photons) or lattice vibrations
(phonons)

e + hole

Recombination
 
→
bond
←

Generation

(6)

If an electric field ζ is applied to a crystal, the free electrons will be acted upon by a force F = -e.ζ
and they begin to drift against the field direction. If the concentration of free electrons in the conduction
band is n electrons per unit volume (electrons/cm3) and their average drift velocity is vn, then the electron
current density Jn (A/cm2) is given by:
Jn = - e n vn = σn .ζ

(7a)

where σn = - e n (vn / ζ) is called the electrical conductivity of electrons. Unlike metals the conductivity
of semiconductors depends actually on many ambient parameters such as temperature, illumination, etc.
Figure 10. Generation of an electron-hole pair by breaking a covalent bond

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Introduction to Information-Carriers and Transport Models

Regarding the valence band, it is more convenient to consider the motion of holes instead of the motion
of valence electrons, as shown in Figure 11. This is because the number of holes is usually much less
than the number of valence electrons4. If there are p holes (vacant bonds in the crystal lattice) per unit
volume in the valence band, then the current produced by the motion of valence electrons to fill in these
holes, against field direction, is equal to the current produced by the motion of holes, along the field
direction. Therefore, the hole current density Jp is given by:
Jp = e p vp = σp ζ

(7b)

where σp = e p (vp / ζ) is called the electrical conductivity of holes