CHAPTER - I

INTRODUCTION: William Bardforded in California and obtained

Ph.D. in 1936 from the Massachusetts Institute of Technology, USa.

Shickley’s research had been centred in many areas of semiconductor

Physics, e.g., energy bands in semiconductors i order and disorder an alloys i

theory of vaccum theory on ferromagnetic domains i and various topics in

tranistor Physics.

He was awarded the Nobel Prize an Physics in 1956 for his researches

on semiconductors and discovery of transistor effect along with John Bard

een (1908- 1991) and walter Houser Brattain (1902- 1987).

This work revolutionized the world of electronics wide energy Brand

Gap Electronics Devices definitions. Fan Ren (University of Florida, USA)

& Johnc Zolper (DARPA, USA) his book provides a summary of the current

state of the art in sic and GAN and identify future areas of development. The

remarkable improvements an material quality and device peroformance in

the last few years show the promise of these technologies for areas that si

cannot operate because of gt’s smaller band gap. We feel that this collection

of chapters provides and excellent introduction to the field and is an

1

outstanding reference for those peforming research on wide bandgap

semiconductors.

In this book, we bring together numerous experts in the field to review

progression sic and GAN electronic devices and noval detectors. Professor

Morkoc reviews the growth and characterization of nitrides followed by

chapters from professor Karmalkar and Professor Gaska on High Electron

Mobility Transisters Professor Pearton and Co-workers on ultro-high

breakdown voltage GAN based rectifier and the group of professor.

Abernathy on emerging MOS devices in the nitride system Dr. Baca from

Sandia National Laboratories and Dr. Chang from Agilent review the use of

mixed group V-Nitride as the base layer in novel Hetereojunction Biopolar

Transistors. There are 3 chapter on sic, including professor skowronski on

growth and characterization. Professor chow on power Schottky and pin

rectifiers and professor cooper on power MOSFETS. Professor Dupuis and

professor Campbell give an overview of short wavelength nitride based

detectors. Finally Jihyun kim and co-workers describe recent progress in

wide bandgap semiconductor spintronics where one can obtain room

temperature ferromagnetism and exploit the spin of the electron an addition

to its charge.

2

We thank W.de Heer, CT White, F.Liu, S.G. Louie M. Hybertsen, K.

Bolotin and P. Jarrillo- Herrero for helpful discussion. This work is

supported by ONR (No. N000150610138), FENA, DOE (No. DEFG02-

05ER46215), NSF CAREER (No. DMR- 0349232), NSEC(No. CHE-

0117752) and the New York Office of Science Technology and Academic

Research (NYSTAR). M. Han is supported by the National Science

Foundation.

We investigate electronic transport in lithographically patterned

graphically ribbon structures where the lateral confinement of charge carries

creats on energy gap near the charge neutrality point. Individual graphically

layer are contacted with metal electrodes & patterned into ribbons of varying

width and different crystallographic orientations. The temperature dependent

conductance measurement show layers energy gap opening narrower

ribbons. The size of these energy gaps are investigated by measuring the

conductance in the non-linear response regime at low temperatures. We find

that the energy gap scales inversely with the ribbon width, thus

demonstrating the ability to engineer the band gap of graphically

nanostructures by lithographic processes. In solid state physics & related

applied field, a band gap, also called an energy gap or band is an energy,

range in a solid where no electron states exist for insulator & semiconductor

3

the band gap generally refers to energy difference (in electron volts) between

top of valence band and the bottom of the conduction band; it is the amount

of energy required to free an outer shel electron from its orbit about the

nucleus to a free state.

In semiconductor Physics a material with a small, but not null or

negative, band gap (arbitrarily defined as <3ev, although some definitions

place the upper limit 4ev) is referred to as a semiconductor.

4

CHAPTER - II

THE ENERGY BAND GAPS SOME DEFINITIONS:

Band:

A range of some physical variable, as of radiation wavelength or

frequency. A range of very closely spaced electron energy levels in

solids, the distribution and nature of which determine the electrical

properties of a material.

Conduction:

The transmission or conveying of something through a medium or

passage, esp. of electric charge or heat through a conducting medium

without perceptible motion of the medium itself.

Dopant:

A small quantity of a substance, such as phosphorus, added to another

substance, such as a semiconductor, to alter the latter's properties.

Energy:

The work that a physical system is capable of doing in changing from

its actual state to a specified reference state, the total including, in

5

general, contributions of potential energy, kinetic energy, and rest

energy.

Gap:

A suspension of continuity; hiatus. A conspicuous difference;

disparity.

Model:

A tentative description of a system or theory that accounts for all of its

known properties.

Valence:

The capacity of an atom or group of atoms to combine in specific

proportions with other atoms or groups of atoms. A valence electron is

in the outer or next outer shell of an atom and can participate in

forming chemical bonds with other atoms.

THE BAND GAP ENERGY:

The band gap energy is the energy needed to break a bond in the crystal.

When a bond is broken, the electron has absorbed enough energy to leave

the valence band and "jump" to the conduction band. The width of the band

gap determines the type of material (conductor, semiconductor, insulator)

you are working with.

6

Band gap:

In condensed matter physics and related applied fields, a band gap, also

called an energy gap or bandgap, is an energy range in a solid where no

electron states exist. For insulator the band gap generally refers to the energy

difference (in electron volts) between the top of the valence band and the

bottom of the conduction band; it is the amount of energy required to free an

outer shell electron from its orbit about the nucleus to a free state.

7

CHAPTER - III

ENERGY BANDS:

Energy bands consisting of a large number of closely spaced energy levels

exist in crystalline materials. The bands can be thought of as the collection

of the individual energy levels of electrons surrounding each atom. The

wavefunctions of the individual electrons, however, overlap with those of

electrons confined to neighboring atoms. The Pauli exclusion principle does

not allow the electron energy levels to be the same so that one obtains a set

of closely spaced energy levels, forming an energy band. The energy band

model is crucial to any detailed treatment of semiconductor devices. It

provides the framework needed to understand the concept of an energy

bandgap and that of conduction in an almost filled band as described by the

empty states.

In this section, we present the free electron model and the Kronig-Penney

model. Then we discuss the energy bands of semiconductors and present a

simplified band diagram. We also introduce the concept of holes and the

effective mass.

8

FREE ELECTRON MODEL:

The free electron model of metals has been used to explain the photo-

electric effect. This model assumes that electrons are free to move within

the metal but are confined to the metal by potential barriers. The

minimum energy needed to extract an electron from the metal equals

qM, where M is the workfunction. This model is frequently used when

analyzing metals. However, this model does not work well for

semiconductors since the effect of the periodic potential due to the atoms

in the crystal has been ignored.

PERIODIC POTENTIALS:

The Kronig-Penney model: The analysis of periodic potentials is

required to find the energy levels in a semiconductor. This requires

the use of periodic wave functions, called Bloch functions which are

beyond the scope of this text. The result of this analysis is that the

energy levels are grouped in bands, separated by energy band gaps.

The behavior of electrons at the bottom of such a band is similar to

that of a free electron. However, the electrons are affected by the

presence of the periodic potential. The combined effect of the periodic

9

potential is included by adjusting the value of the electron mass. This

mass will be referred to as the effective mass.

ENERGY BANDS OF SEMICONDUCTORS:

Energy band diagrams of common semiconductors: The energy

band diagrams of semiconductors are rather complex. The detailed

energy band diagrams of germanium, silicon and gallium arsenide.

The energy is plotted as a function of the wavenumber, k, along the

main crystallographic directions in the crystal, since the band diagram

depends on the direction in the crystal. The energy band diagrams

contain multiple completely-filled and completely-empty bands. In

addition, there are multiple partially-filled band.

Simple energy band diagram of a semiconductor: The energy band

diagrams shown in the previous section are frequently simplified

when analyzing semiconductor devices. Since the electronic

properties of a semiconductor are dominated by the highest partially

empty band and the lowest partially filled band, it is often sufficient to

only consider those bands. This leads to a simplified energy band

diagram for semiconductors.

Temperature dependence of the energy bandgap: The energy

bandgap of semiconductors tends to decrease as the temperature is

10

increased. This behavior can be understood if one considers that the

interatomic spacing increases when the amplitude of the atomic

vibrations increases due to the increased thermal energy. This effect is

quantified by the linear expansion coefficient of a material. An

increased interatomic spacing decreases the average potential seen by

the electrons in the material, which in turn reduces the size of the

energy bandgap. A direct modulation of the interatomic distance -

such as by applying compressive (tensile) stress - also causes an

increase (decrease) of the bandgap.

The temperature dependence of the energy bandgap, Eg, has been

experimentally determined yielding the following expression for Eg as

a function of the temperature, T:

Doping dependence of the energy bandgap: High doping

densities cause the bandgap to shrink. This effect is explained by the

fact that the wavefunctions of the electrons bound to the impurity

atoms start to overlap as the density of the impurities increase. For

instance, at a doping density of 1018 cm-3, the average distance

between two impurities is only 10 nm. This overlap forces the

energies to form an energy band rather than a discreet level.

11

METALS, INSULATORS AND SEMICONDUCTORS:

Once we know the bandstructure of a given material we still need to find

out which energy levels are occupied and whether specific bands are

empty, partially filled or completely filled.

Empty bands do not contain electrons. Therefore, they are not expected

to contribute to the electrical conductivity of the material. Partially filled

bands do contain electrons as well as available energy levels at slightly

higher energies. These unoccupied energy levels enable carriers to gain

energy when moving in an applied electric field. Electrons in a partially

filled band therefore do contribute to the electrical conductivity of the

material.

Completely filled bands do contain plenty of electrons but do not

contribute to the conductivity of the material. This is because the

electrons cannot gain energy since all energy levels are already filled.

In order to find the filled and empty bands we must find out how many

electrons can be placed in each band and how many electrons are

available. Each band is formed due to the splitting of one or more atomic

12

energy levels. Therefore, the minimum number of states in a band equals

twice the number of atoms in the material. The reason for the factor of

two is that every energy level can contain two electrons with opposite

spin.

To further simplify the analysis, we assume that only the valence

electrons (the electrons in the outer shell) are of interest. The core

electrons are tightly bound to the atom and are not allowed to freely

move in the material.

ELECTRONS AND HOLES IN SEMICONDUCTORS:

As pointed out in metals, insulators and semiconductors,

semiconductors differ from metals and insulators by the fact that they

contain an "almost-empty" conduction band and an "almost-full" valence

band. This also means that we will have to deal with the transport of

carriers in both bands.

To facilitate the discussion of the transport in the "almost-full" valence

band of a semiconductor, we will introduce the concept of holes. It is

important to understand that one could deal with only electrons if one is

willing to keep track of all the electrons in the "almost-full" valence

13

band. After all, electrons are the only real particles available in a

semiconductor.

The concepts of holes is introduced in semiconductors since it is easier to

keep track of the missing electrons in an "almost-full" band, rather than

keeping track of the actual electrons in that band. We will now first

explain the concept of a hole and then point out how the hole concept

simplifies the analysis.

Holes are missing electrons. They behave as particles with the same

properties as the electrons would have when occupying the same states

except that they carry a positive charge.

THE EFFECTIVE MASS CONCEPT:

Electrons with an energy close to a band minimum behave as free

electrons, since the E-k relation can be approximated by a parabola. They

accelerate in an applied electric field just like a free electron in vacuum.

Their wavefunctions are periodic and extend over the size of the material.

The presence of the periodic potential, due to the atoms in the crystal

without the valence electrons, changes the properties of the electrons.

Therefore, the mass of the electron differs from the free electron mass,

m0. Because of the anisotropy of the effective mass and the presence of

14

multiple equivalent band minima, we define two types of effective mass:

1) the effective mass for density of states calculations and 2) the effective

mass for conductivity calculations.

DETAILED DESCRIPTION OF THE EFFECTIVE MASS

CONCEPT:

Introduction: The effective mass of a semiconductor is obtained by

fitting the actual E-k diagram around the conduction band minimum

or the valence band maximum by a paraboloid. While this concept is

simple enough, the issue turns out to be substantially more complex

due to the multitude and the occasional anisotropy of the minima and

maxima. In this section we first describe the different relevant band

minima and maxima, present the numeric values for germanium,

silicon and gallium arsenide and introduce the effective mass for

density of states calculations and the effective mass for conductivity

calculations.

Most semiconductors can be described as having one band minimum

at k = 0 as well as several equivalent anisotropic band minima at k

0. In addition there are three band maxima of interest close to the

valence band edge.

15

Effective mass for conductivity calculations

The effective mass for conductivity calculation is the mass, which is

used in conduction related problems accounting for the detailed

structure of the semiconductor. These calculations include mobility

and diffusion constants calculations. Another example is the

calculation of the shallow impurity levels using a hydrogen-like

model.

As the conductivity of a material is inversionally proportional to the

effective masses, one finds that the conductivity due to multiple band

maxima or minima is proportional to the sum of the inverse of the

individual masses, multiplied by the density of carriers in each band,

as each maximum or minimum adds to the overall conductivity. For

anisotropic minima containing one longitudinal and two transverse

effective masses one has to sum over the effective masses in the

different minima along the equivalent directions.

16

CHAPTER – IV

ENERGY GAP :

1. In Semi conductor Physics:

Semiconductor Band Structure

A material with a small, but not null or negative, band gap (arbitrarily

defined as < 3 eV, although some definitions place the upper limit at 4 eV) is

referred to as a semiconductor. A material with a large band gap is called an

insulator.

17

In semiconductors and insulators, electrons are confined to a number of

bands of energy, and forbidden from other regions. The term "band gap"

refers to the energy difference between the top of the valence band and the

bottom of the conduction band; electrons are able to jump from one band to

another. In order for an electron to jump from a valence band to a

conduction band, it requires a specific minimum amount of energy for the

transition. The required energy differs with different materials. Electrons can

gain enough energy to jump to the conduction band by absorbing either a

phonon (heat) or a photon (light).

The conductivity of intrinsic semiconductors is strongly dependent on the

band gap. The only available carriers for conduction are the electrons which

have enough thermal energy to be excited across the band gap.

Band gap engineering is the process of controlling or altering the band gap

of a material by controlling the composition of certain semiconductor alloys,

such as GaAlAs, InGaAs, and InAlAs. It is also possible to construct layered

materials with alternating compositions by techniques like molecular beam

epitaxy. These methods are exploited in the design of heterojunction bipolar

transistors (HBTs), laser diodes and solar cells.

The distinction between semiconductors and insulators is a matter of

convention. One approach is to think of semiconductors as a type of

18

insulator with a narrow band gap. Insulators with a larger band gap, usually

greater than 3 eV, are not considered semiconductors and generally do not

exhibit semiconductive behaviour under practical conditions. Electron

mobility also plays a role in determining a material's informal classification.

The band gap energy of semiconductors tends to decrease with increasing

temperature. When temperature increases, the amplitude of atomic

vibrations increase, leading to larger interatomic spacing. The interaction

between the lattice phonons and the free electrons and holes will also affect

the band gap to a smaller extent. The relationship between band gap energy

and temperature can be described by Varshni's empirical expression,

, where Eg(0), α and β are material

constants.

In a regular semiconductor crystal, the band gap is fixed owing to

continuous energy states. In a quantum dot crystal, the band gap is size

dependent and can be altered to produce a range of energies between the

valence band and conduction band. It is also known as quantum confinement

effect.

Band gaps also depend on pressure. Band gaps can be either direct or

indirect, depending on the electronic band structure.

19

1.1 Mathematical interpretation

Classically, the ratio of probabilities that two states with an energy

difference ΔE will be occupied by an electron is given by the Boltzmann

factor:

where:

is the exponential function

is the energy difference

is Boltzmann's constant

is temperature

At the Fermi level (or chemical potential), the probability of a state being

occupied is ½. If the Fermi level is in the middle of a band gap of 1 eV, this

ratio is e -20 or about 2.0•109 at the room-temperature thermal energy of 25.9

meV.

1.2 Photovoltaic cells

The band gap determines what portion of the solar spectrum a photovoltaic

cell absorbs. A luminescent solar converter uses a luminescent medium to

20

down convert photons with energies above the band gap to photon energies

closer to the band gap of the semiconductor comprising the solar cell.

1.3 List of band gaps

Material Symbol Band gap (eV) @ 300K

Silicon Si 1.11

Selenium Se 1.74

Germanium Ge 0.67

Silicon carbide SiC 2.86

Aluminum phosphide AlP 2.45

Aluminium arsenide AlAs 2.16

Aluminium antimonide AlSb 1.6

Aluminium nitride AlN 6.3

Diamond C 5.5

Gallium(III) phosphide GaP 2.26

Gallium(III) arsenide GaAs 1.43

Gallium(III) nitride GaN 3.4

Gallium(II) sulfide GaS 2.5 (@ 295 K)

Gallium antimonide GaSb 0.7

Indium(III) nitride InN 0.7

Indium(III) phosphide InP 1.35

Indium(III) arsenide InAs 0.36

Zinc oxide ZnO 3.37

Zinc sulfide ZnS 3.6

Zinc selenide ZnSe 2.7

21

Zinc telluride ZnTe 2.25

Cadmium sulfide CdS 2.42

Cadmium selenide CdSe 1.73

Cadmium telluride CdTe 1.49

Lead(II) sulfide PbS 0.37

Lead(II) selenide PbSe 0.27

Lead(II) telluride PbTe 0.29

Copper(II) oxide CuO 1.2

2. In photonics and phononics:

In photonics band gaps or stop bands are ranges of photon frequencies

where, if tunneling effects are neglected, no photons can be transmitted

through a material. A material exhibiting this behaviour is known as a

photonic crystal.

Similar physics applies to phonons in a phononic crystal.

22

CHAPTER – V

ENERGY (BAND) GAP: What do we mean by "allowed" and "forbidden

energies" or equivalently what is an "energy (band) gap."  See the page entitled

Energy (Band) Gap.

Viewpoint #1: The gap energy is the ionization energy required to generate

two complementary charge carriers - electron and hole.

23

Viewpoint #3: The gap energy is a manifestation of the discrete character of

basic atomic energy states.

As distance between atoms gets smaller --->

24

Distance between atoms

25

Three useful references on this viewpoint from the Georgia State

University's HyperPhysics project:

Energy Bands:Insulators and Semiconductors I

Energy Bands: Insulators and Semiconductors II

Interatomic spacing and semiconductors

26

CHAPTER – VI

ELECTRONIC BAND STRUCTURE :

In solid-state physics, the electronic band structure (or simply band

structure) of a solid describes ranges of energy that an electron is

"forbidden" or "allowed" to have. It is due to the diffraction of the quantum

mechanical electron waves in the periodic crystal lattice. The band structure

of a material determines several characteristics, in particular its electronic

and optical properties.

1. Why bands occur in materials

The electrons of a single isolated atom occupy atomic orbitals, which form a

discrete set of energy levels. If several atoms are brought together into a

molecule, their atomic orbitals split, as in a coupled oscillation. This

produces a number of molecular orbitals proportional to the number of

atoms. When a large number of atoms (of order 1020 or more) are brought

together to form a solid, the number of orbitals becomes exceedingly large,

and the difference in energy between them becomes very small, so the levels

may be considered to form continuous bands of energy rather than the

discrete energy levels of the atoms in isolation. However, some intervals of

27

energy contain no orbitals, no matter how many atoms are aggregated,

forming band gaps.

Within an energy band, energy levels are so numerous as to be a near

continuum. First, the separation between energy levels in a solid is

comparable with the energy that electrons constantly exchange with phonons

(atomic vibrations). Second, it is comparable with the energy uncertainty

due to the Heisenberg uncertainty principle, for reasonably long intervals of

time. As a result, the separation between energy levels is of no consequence.

Several approaches to finding band structure are discussed below:-

2. Basic concepts

Figure 1: Simplified diagram of the electronic band structure of metals, semiconductors, and insulators.

28

Figure 2: First Brillouin zone of FCC lattice showing symmetry labels

Figure 3: More complex representation of band structure in silicon showing wavevector dependence. Going from left to right along the

horizontal, the wavevector is tracing out a particular one-dimensional path through the three-dimensional "first Brillouin zone" (see Figure

2). On the vertical axis is the energies of bands at that wavevector.

29

Any solid has a large number of bands. In theory, it can be said to have

infinitely many bands (just as an atom has infinitely many energy levels).

However, all but a few lie at energies so high that any electron that reaches

those energies escapes from the solid. These bands are usually disregarded.

Bands have different widths, based upon the properties of the atomic orbitals

from which they arise. Also, allowed bands may overlap, producing (for

practical purposes) a single large band.

Figure 1 shows a simplified picture of the bands in a solid that allows the

three major types of materials to be identified: metals, semiconductors and

insulators.

Metals contain a band that is partly empty and partly filled regardless of

temperature. Therefore they have very high conductivity.

The lowermost, almost fully occupied band in an insulator or semiconductor,

is called the valence band by analogy with the valence electrons of

individual atoms. The uppermost, almost unoccupied band is called the

conduction band because only when electrons are excited to the conduction

band can current flow in these materials. The difference between insulators

and semiconductors is only that the forbidden band gap between the valence

band and conduction band is larger in an insulator, so that fewer electrons

30

are found there and the electrical conductivity is lower. Because one of the

main mechanisms for electrons to be excited to the conduction band is due

to thermal energy, the conductivity of semiconductors is strongly dependent

on the temperature of the material.

This band gap is one of the most useful aspects of the band structure, as it

strongly influences the electrical and optical properties of the material.

Electrons can transfer from one band to the other by means of carrier

generation and recombination processes. The band gap and defect states

created in the band gap by doping can be used to create semiconductor

devices such as solar cells, diodes, transistors, laser diodes, and others.

2.1 Symmetry

A more complete view of the band structure takes into account the periodic

nature of a crystal lattice using the symmetry operations that form a space

group. The Schrödinger equation is solved for the crystal, which has Bloch

waves as solutions:

,

where k is called the wavevector, and is related to the direction of motion of

the electron in the crystal, and n is the band index, which simply numbers

the energy bands. The wavevector k takes on values within the Brillouin

31

zone (BZ) corresponding to the crystal lattice, and particular

directions/points in the BZ are assigned conventional names like Γ, Δ, Λ, Σ,

etc. These directions are shown for the face-centered cubic lattice geometry

in Figure 2.

The available energies for the electron also depend upon k, as shown in

Figure 3 for silicon in the more complex energy band diagram at the right. In

this diagram the topmost energy of the valence band is labeled Ev and the

bottom energy in the conduction band is labeled Ec. The top of the valence

band is not directly below the bottom of the conduction band (Ev is for an

electron traveling in direction Γ, Ec in direction X), so silicon is called an

indirect gap material. For an electron to be excited from the valence band to

the conduction band, it needs something to give it energy Ec − Ev and a

change in direction/momentum. In other semiconductors (for example

GaAs) both are at Γ, and these materials are called direct gap materials (no

momentum change required). Direct gap materials benefit the operation of

semiconductor laser diodes.

Anderson's rule is used to align band diagrams between two different

semiconductors in contact.

32

2.2 Band structures in different types of solids

Although electronic band structures are usually associated with crystalline

materials, quasi-crystalline and amorphous solids may also exhibit band

structures. However, the periodic nature and symmetrical properties of

crystalline materials makes it much easier to examine the band structures of

these materials theoretically. In addition, the well-defined symmetry axes of

crystalline materials makes it possible to determine the dispersion

relationship between the momentum (a 3-dimension vector quantity) and

energy of a material. As a result, virtually all of the existing theoretical work

on the electronic band structure of solids has focused on crystalline

materials.

2.3 Density of states

While the density of energy states in a band could be very large for some

materials, it may not be uniform. It approaches zero at the band boundaries,

and is generally highest near the middle of a band. The density of states for

the free electron model in three dimensions is given by,

33

2.4 Filling of bands

Although the number of states in all of the bands is effectively infinite, in an

uncharged material the number of electrons is equal only to the number of

protons in the atoms of the material. Therefore not all of the states are

occupied by electrons ("filled") at any time. The likelihood of any particular

state being filled at any temperature is given by Fermi-Dirac statistics. The

probability is given by the following expression:

where:

kB is Boltzmann's constant,

T is the temperature,

μ is the chemical potential (in semiconductor physics, this quantity is

more often called the "Fermi level" and denoted EF).

The Fermi level naturally is the level at which the electrons and protons are

balanced.

At T=0, the distribution is a simple step function:

34

At nonzero temperatures, the step "smooths out", so that an appreciable

number of states below the Fermi level are empty, and some states above the

Fermi level are filled.

3. Band structure of crystals

3.1 Brillouin zone

Because electron momentum is the reciprocal of space, the dispersion

relation between the energy and momentum of electrons can best be

described in reciprocal space. It turns out that for crystalline structures,

the dispersion relation of the electrons is periodic, and that the Brillouin

zone is the smallest repeating space within this periodic structure. For an

infinitely large crystal, if the dispersion relation for an electron is defined

throughout the Brillouin zone, then it is defined throughout the entire

reciprocal space.

4. Theory of band structures in crystals

The ansatz is the special case of electron waves in a periodic crystal lattice

using Bloch waves as treated generally in the dynamical theory of

diffraction. Every crystal is a periodic structure which can be characterized

by a Bravais lattice, and for each Bravais lattice we can determine the

reciprocal lattice, which encapsulates the periodicity in a set of three

reciprocal lattice vectors ( , , ). Now, any periodic potential

35

which shares the same periodicity as the direct lattice can be expanded out as

a Fourier series whose only non-vanishing components are those associated

with the reciprocal lattice vectors. So the expansion can be written as:

where for any set of integers (m1,m2,m3).

From this theory, an attempt can be made to predict the band structure of a

particular material, however most ab initio methods for electronic structure

calculations fail to predict the observed band gap.

4.1 Nearly-free electron approximation

In the nearly-free electron approximation in solid state physics interactions

between electrons are completely ignored. This approximation allows use of

Bloch's Theorem which states that electrons in a periodic potential have

wavefunctions and energies which are periodic in wavevector up to a

constant phase shift between neighboring reciprocal lattice vectors. The

consequences of periodicity are described mathematically by the Bloch

wavefunction:

where the function is periodic over the crystal lattice, that is,

36

.

Here index n refers to the n-th energy band, wavevector k is related to the

direction of motion of the electron, r is position in the crystal, and R is

location of an atomic site.

4.2 Tight-binding model

The opposite extreme to the nearly-free electron approximation assumes the

electrons in the crystal behave much like an assembly of constituent atoms.

This tight-binding model assumes the solution to the time-independent

single electron Schrödinger equation Ψ is well approximated by a linear

combination of atomic orbitals .

,

where the coefficients are selected to give the best approximate solution

of this form. Index n refers to an atomic energy level and R refers to an

atomic site. A more accurate approach using this idea employs Wannier

functions, defined by:

;

in which is the periodic part of the Bloch wave and the integral is over

the Brillouin zone. Here index n refers to the n-th energy band in the crystal.

37

The Wannier functions are localized near atomic sites, like atomic orbitals,

but being defined in terms of Bloch functions they are accurately related to

solutions based upon the crystal potential. Wannier functions on different

atomic sites R are orthogonal. The Wannier functions can be used to form

the Schrödinger solution for the n-th energy band as:

4.3 KKR model

The simplest form of this approximation centers non-overlapping spheres

(referred to as muffin tins) on the atomic positions. Within these regions, the

potential experienced by an electron is approximated to be spherically

symmetric about the given nucleus. In the remaining interstitial region, the

potential is approximated as a constant. Continuity of the potential between

the atom-centered spheres and interstitial region is enforced.

A variational implementation was suggested by Korringa and by Kohn and

Rostocker, and is often referred to as the KKR model.

4.4 Order-N spectral methods

To quote RP Martin: "The concept of localization can be imbedded directly

into the methods of electronic structure to create algorithms that take

advantage of locality … For large systems, this fact can be used to make

38

"order-N" or O(N) methods where the computational time scales linearly in

the size of the system"

4.5 Density-functional theory

In recent physics literature, a large majority of the electronic structures and

band plots are calculated using density-functional theory (DFT), which is

not a model but rather a theory, i.e., a microscopic first-principles theory of

condensed matter physics that tries to cope with the electron-electron many-

body problem via the introduction of an exchange-correlation term in the

functional of the electronic density. DFT-calculated bands are in many cases

found to be in agreement with experimentally measured bands, for example

by angle-resolved photoemission spectroscopy (ARPES). In particular, the

band shape is typically well reproduced by DFT. But there are also

systematic errors in DFT bands when compared to experiment results. In

particular, DFT seems to systematically underestimate by about 30-40% the

band gap in insulators and semiconductors.

It must be said that DFT is, in principle an exact theory to reproduce and

predict ground state properties (e.g., the total energy, the atomic structure,

etc.). However, DFT is not a theory to address excited state properties, such

as the band plot of a solid that represents the excitation energies of electrons

injected or removed from the system. What in literature is quoted as a DFT

39

band plot is a representation of the DFT Kohn-Sham energies, i.e., the

energies of a fictive non-interacting system, the Kohn-Sham system, which

has no physical interpretation at all. The Kohn-Sham electronic structure

must not be confused with the real, quasiparticle electronic structure of a

system, and there is no Koopman's theorem holding for Kohn-Sham

energies, as there is for Hartree-Fock energies, which can be truly

considered as an approximation for quasiparticle energies. Hence, in

principle, DFT is not a band theory, i.e., not a theory suitable for calculating

bands and band-plots.

4.6 Green's function methods and the ab initio GW approximation

To calculate the bands including electron-electron interaction many-body

effects, one can resort to so-called Green's function methods. Indeed,

knowledge of the Green's function of a system provides both ground (the

total energy) and also excited state observables of the system. The poles of

the Green's function are the quasiparticle energies, the bands of a solid. The

Green's function can be calculated by solving the Dyson equation once the

self-energy of the system is known. For real systems like solids, the self-

energy is a very complex quantity and usually approximations are needed to

solve the problem. One such approximation is the GW approximation, so

called from the mathematical form the self-energy takes as the product Σ =

40

GW of the Green's function G and the dynamically screened interaction W.

This approach is more pertinent when addressing the calculation of band

plots (and also quantities beyond, such as the spectral function) and can also

be formulated in a completely ab initio way. The GW approximation seems

to provide band gaps of insulators and semiconductors in agreement with

experiment, and hence to correct the systematic DFT underestimation.

4.7 Mott insulators

Although the nearly-free electron approximation is able to describe many

properties of electron band structures, one consequence of this theory is that

it predicts the same number of electrons in each unit cell. If the number of

electrons is odd, we would then expect that there is an unpaired electron in

each unit cell, and thus that the valence band is not fully occupied, making

the material a conductor. However, materials such as CoO that have an odd

number of electrons per unit cell are insulators, in direct conflict with this

result. This kind of material is known as a Mott insulator, and requires

inclusion of detailed electron-electron interactions (treated only as an

averaged effect on the crystal potential in band theory) to explain the

discrepancy. The Hubbard model is an approximate theory that can include

these interactions.

41

4.8 Others

Calculating band structures is an important topic in theoretical solid state

physics. In addition to the models mentioned above, other models include

the following:

The Kronig-Penney Model, a one-dimensional rectangular well model

useful for illustration of band formation. While simple, it predicts

many important phenomena, but is not quantitative.

Bands may also be viewed as the large-scale limit of molecular orbital

theory. A solid creates a large number of closely spaced molecular

orbitals, which appear as a band.

Hubbard model

The band structure has been generalised to wavevectors that are complex

numbers, resulting in what is called a complex band structure, which is of

interest at surfaces and interfaces.

Each model describes some types of solids very well, and others poorly. The

nearly-free electron model works well for metals, but poorly for non-metals.

The tight binding model is extremely accurate for ionic insulators, such as

metal halide salts (e.g. NaCl).

42

CHAPTER – VII

THE ENERGY BAND MODEL:

Material Classification: Crystalline materials can be classified according to

their bandgap.

An insulator is a poor conductor since it requires a lot of energy, 5-8 eV, to

excite the electrons enough to get to the conduction band. We can say that

the width of the band gap is very large, since it requires that much energy to

traverse the band gap, and draw the band diagram respectively.

A metal is an excellent conductor because, at room temperature, it has

electrons in its conduction band constantly, with little or no energy being

applied to it. This may be because of its narrow or nonexistent band gap, the

conduction band may be overlapping the valence band so they share the

electrons. The band diagram would be drawn with Ec and Ev very close

together, if not overlapping.

The reason semiconductors are so popular is because they are a medium

between a metal and an insulator. The band gap is wide enough to where

current is not going through it at all times, but narrow enough to where it

does not take a lot of energy to have electrons in the conduction band

creating a current.

43

CHAPTER – VII

ENERGY BAND-GAP ENGINEERING OF GRAPHENE

NANORIBBONS:

The recent discovery of graphene, a single atomic sheet of graphite,

has ignited intense research activities to elucidate the electronic

properties of this novel twodimensional (2D) electronic system.

Charge transport in graphene is substantially different from that of

conventional 2D electronic systems as a consequence of the linear

energy dispersion relation near the charge neutrality point

(Dirac point) in the electronic band structure. This unique band

structure is fundamentally responsible for the distinct electronic

properties of carbon nanotubes (CNTs) .

When graphene is patterned into a narrow ribbon, and the

carriers are confined to a quasi-one-dimensional (1D) system, we

expect the opening of an energy gap. Similar to CNTs, this energy

gap depends on the width and crystallographic orientation of the

graphene nanoribbon (GNR). However, despite numerous recent

theoretical studies, the energy gap in GNRs has yet to be

investigated experimentally.

44

In this Letter, we present electronic transport measurements of

lithographically patterned GNR structures where the lateral

confinement of charge carriers creates an energy gap. More than

two dozen GNRs of different widths and crystallographic orientations

were measured. We find that the energy gap depends strongly on

the width of the channel for GNRs in the same crystallographic

direction, but no systematic crystallographic dependence is

observed. The GNR devices discussed here are fabricated from

single sheets of graphene which have been mechanically extracted

from bulk graphite crystals onto a SiO2=Si substrate. Graphene

sheets with lateral

sizes of _20 _m are contacted with Cr=Au (3=50 nm) metal

electrodes. Negative tone e-beam resist, hydrogen silsesquioxane

(HSQ), is then spun onto the samples and patterned to form an etch

mask defining nanoribbons with widths ranging from 10–100 nm

and lengths of 1–2 _m. An oxygen plasma is introduced to etch

away the unprotected graphene, leaving the GNR protected beneath

the HSQ mask [Fig. (a)].

In this Letter, we study two different types of device sets:

device sets P1-P4 each contain many ribbons of varying width

running parallel [Fig. (b)], and a device sets D1 and D2 have ribbons

of uniform width and varying relative orientation [Fig. (c)]. In either

45

case, each device within a given set is etched from the same sheet

of graphene, so that the relative orientation of the GNRs within a

given set is known.

We remark that each GNR connects two blocks of wider (_0:5 _m)

graphene, which are in turn contacted by metal electrodes. Thus,

unlike CNTs, Schottky barrier formation by the metal electrodes is

absent in our GNR devices. Furthermore, multiple contacts on the

wider block of grapheme allow for four-terminal measurements in

order to eliminate the residual contact resistance (_1 k_). A heavily

doped silicon substrate below the 300 nm thick SiO2 dielectric layer

serves as a gate electrode to tune the carrier density in the GNR.

The width (W) and the length of each GNR were measured using a

scanning electron microscope (SEM) after the transport

measurements were performed.

46

Figure: (color online). (a) Atomic force microscope image of GNRs in set P3 covered by a protective HSQ

etch mask. (b) SEM image of device set P1 with parallel GNRs of varying width. (c) SEM image of

device set D2 containing GNRs in different relative crystallographic directions with uniform width (e)– (f ) Conductance of GNRs in device set P1 as a function of gate voltage measured at different temperatures. The

width of each GNR is designated in each panel.

47

Figure: (color online). Conductance vs width of parallel GNRs (set P4) measured at Vg _ VDirac __50 V at three representative temperatures. The square and

triangle symbols correspond to T _ 300, and 1.6 K, respectively. Dashed lines represent the linear fits at each temperature. The insets show the conductivity (upper) and the inactive GNR width (lower) obtained

from the slope and x intercept of the linear fit at

48

varying temperatures. Dashed curves are shown in the insets as a guide to the eye.

Since the HSQ protective layer was not removed from the GNR for

this imaging, this measurement provides an upper bound to the true

width of the GNR.

The conductance G of the GNRs was measured using a

standard lock-in technique with a small applied ac voltage ( < 100

_V@8 Hz). Figure (d)–1(f) shows the measured G of three

representative GNR devices of varying width (W _ 24 _ 4, 49 _ 5, and

71 _ 6 nm) and uniform length (L _ 2 _m) as a function of gate

voltage Vg at different temperatures. All curves exhibit a region of

depressed G with respect to Vg. In ‘‘bulk‘‘ (i.e., unpatterned)

graphene, this dip in G is well understood and corresponds to the

minimum conductivity _4e2=h at the charge neutrality point, Vg _

VDirac, where e and h are the electric charge and Planck constant,

respectively. At room temperature, our GNRs exhibit qualitatively

similar G_Vg_ behaviors, showing a minimum conductance Gmin on

the order of 4e2=h_W=L_. Unlike the bulk case, GNRs with width

W<100 nm show a decrease in Gmin of more than an order of

magnitude at low temperatures. The narrowest GNRs show the

greatest suppression of Gmin. For example, for the GNR with W _ 24

_ 4 nm [Fig. 1(d)], a large ‘‘gap’’ region appears for 25<Vg < 45 V,

49

where Gmin is below our detection limits (<10_8 __1). This strong

temperature dependence of G_Vg_ in GNRs is in sharp contrast to

that of the ‘‘bulk’’ graphene samples where Gmin changes less than

30% in the temperature range 30 mK–300 K. The suppression of G

near the charge neutrality point suggests the opening of an energy

gap. We observe [Fig. 1(d)–1(f)] stronger temperature dependence

of G for a broader range of Vg values in narrower GNRs, suggesting

larger energy gaps in narrower GNRs.

Outside of the ‘‘gap’’ region near the Dirac point, the

conductance scales with the width of the GNR. Figure: shows the

conductance of a set of parallel GNRs, with widths ranging from 14–

63 nm, measured at two temperatures, T _ 1:6 and 300 K. The gate

voltage is fixed at Vg _ VDirac _ 50 V, which corresponds to a hole

carrier density of n _ 3:6 _ 1012 cm_2. The conductance is well

described by the linear fit G _ __W _ W0_=L (dashed line). Here _ and

W _ W0 can be interpreted as the GNR sheet conductivity and the

active GNR width participating in charge transport, respectively. The

sheet conductivity is _1:7 mS and decreases with decreasing

temperature, reaching _75% of the room temperature value at T _

1:6 K . The inactive GNR width W0 increases from 10 nm at room

temperature to 14 nm at 1.6 K. A reduced active channel width was

initially reported in GNRs fabricated on epitaxial multilayer

50

graphene films, where much larger inactive edges (W0 _ 50 nm)

were estimated than for our GNR samples. We suggest two possible

explanations for the finite W0 measured in our experiment: (i)

contribution from localized edge states near the GNR edges due to

structural disorder caused by the etching process and (ii) inaccurate

width determination due to over-etching underneath the HSQ etch

mask. To investigate this, we removed the HSQ etch mask from

several GNRs and found that the actual GNR is often _10 nm

narrower than the HSQ protective mask. This suggests that the

inactive region due to the localized edge states is small (<2 nm) at

room temperature and spreads to as much as _5 nm at low

temperatures.

We now discuss the quantitative scaling of the energy gap as

a function of GNR width. By examining the differential conductance

in the nonlinear response regime as a function of both the gate and

bias voltage, we can directly measure the size of the energy gap.

Figure (a) shows a schematic energy band diagram for a GNR with

source and drain electrodes. As the bias voltage, Vb, increases, the

source and drain levels approach the conduction and valence band

edges, respectively. When conduction (valence) band edge falls into

the bias window between the source and drain electrodes, electrons

(holes) are injected from source (drain) and the current I rises

51

sharply. The gate voltage adjusts the position of the gap relative to

the source-drain levels. Figure (b), (d) shows the conductance

versus Vg and Vb for three representative GNR devices of different

width measured at T _ 1:6 K. The color indicates conductivity on a

logarithmic scale, with the large dark area in each graph

representing the turnedoff region in the Vg-Vb plane where the both

band edges are outside of the bias windows. The diamond shape of

this region indicates that both Vb and Vg adjust the position of the

band edges relative to the source and drain energy levels,

analogous to nonlinear transport in quantum dots . As designated

by the arrows, the GNR band gap Egap can be directly obtained from

the value of Vb at the vertex of the diamond.

In order to obtain the quantitative scaling of Egap with respect

to W, we now plot E_1 gap against W in Fig.(e) for a set of 13 parallel

GNRs. The dashed line indicates a linear fit to the data,

corresponding to Egap _ _=_W _ W__, where we obtained _ _ 0:2

eVnm and W_ _ 16 nm from the fit. Recent density functional theory

studies predict that the energy gap of a GNR scales inversely with

the channel width, with a corresponding _value ranging between

0:2–1:5 eVnm, which is consistent with this observation. We also

note that W_ W0, in good agreement with the independent

estimation of the GNR edge effects above.

52

A similar scaling behavior holds even across GNR device sets

running in different crystallographic directions. Figure shows the

overall scaling of Egap as a function of W for six different device sets.

Four device sets (P1–P4) have parallel GNRs with W ranging from

15–90 nm, and two device sets (D1, D2) have GNRs with similar W

but different crystallographic directions. The energy gap behavior of

all devices is well described by the scaling Egap _ _=_W _ W__ as

discussed above, indicated by the dashed line. Remarkably, energy

gaps as high as _200 meV are achieved by engineering GNRs as

narrow as W _ 15 nm. Based on the empirical scaling determined

here, a narrower GNR may show an even larger band gap, making

the use of GNRs for semiconducting device components in ambient

conditions a possibility.

Finally, we remark on the crystallographic directional

dependence of Egap. The inset to Fig. 4 shows Egap versus the

relative orientation angle _ for two sets of GNRs. In principle, we

expect Egap___ for each set to be periodic in _, provided all GNRs in

the set have similar edge structures. However, experimental

observation shows randomly scattered values around the average Egap

corresponding to W with no sign of crystallographic directional dependence.

This suggests that the detailed edge structure plays a more important role

than the overall crystallographic direction in determining the properties of

53

the GNRs. Indeed, theory for ideal GNRs predicts that Egap depends

sensitively on the boundary conditions at the edges. The lack of directional

dependence indicates that at this point our device fabrication process does

not give us the atomically precise control of the GNR edges necessary to

reveal this effect. The interplay between the precise width, edge orientation,

edge structure, and chemical termination of the edges in GNRs remains a

rich area for future research.

Figure: (color online). (a) Schematic energy band diagram of a GNR with bias voltage Vb applied. The current Iis controlled by both source-drain bias Vb and gate voltage Vg. (b)–(d) The differential conductance (dI=dVb) of three representative GNRs from set P4 with W _ 22, 36, and

48 nm as a function of Vb and Vg measured at T _1:6 K. The light

54

(dark) color indicates high (low) conductance as designated by the color map. The horizontal arrows represent Vb _ Egap=e. (e) E_1 gap vs W obtained from similar analysis as (b)– (d), with a linear fit of the data.

55

In conclusion, we demonstrate that the energy gap in patterned graphene

nanoribbons can be tuned during fabrication with the appropriate choice of

ribbon width. An understanding of ribbon dimension and orientation as

control parameters for the electrical properties of grapheme structures can be

seen as a first step toward the development of graphene-based electronic

devices.

Figure: (color online). Egap vs W for the 6 device sets considered in this study: four (P1–P4) of the parallel type and two (D1,D2) with varying

orientation. The inset shows Egap vs relative angle _ for the device sets D1 and D2. Dashed lines in the inset show the value of Egap as

predicted by the empirical scaling of Egap vs W.

56

RESULTS

Figure : Semiconductor Band Structure

57

Figure: As distance between atoms gets smaller --->

58

Figure: Distance between atoms

59

Figure : Simplified diagram of the electronic band structure of metals, semiconductors, and insulators.

60

Figure: First Brillouin zone of FCC lattice showing symmetry labels

Figure: More complex representation of band structure in silicon showing wavevector dependence. Going from left to right along the

horizontal, the wavevector is tracing out a particular one-dimensional path through the three-dimensional "first Brillouin zone" (see Figure

2). On the vertical axis is the energies of bands at that wavevector.

61

Figure: (color online). (a) Atomic force microscope image of GNRs in set P3 covered by a protective HSQ

etch mask. (b) SEM image of device set P1 with parallel GNRs of varying width. (c) SEM image of

device set D2 containing GNRs in different relative crystallographic directions with uniform width (e)– (f ) Conductance of GNRs in device set P1 as a function of

62

gate voltage measured at different temperatures. The width of each GNR is designated in each panel.

63

Figure: (color online). Conductance vs width of parallel GNRs (set P4) measured at Vg _ VDirac __50 V at three representative temperatures. The square and

triangle symbols correspond to T _ 300, and 1.6 K, respectively. Dashed lines represent the linear fits at each temperature. The insets show the conductivity (upper) and the inactive GNR width (lower) obtained

from the slope and x intercept of the linear fit at varying temperatures. Dashed curves are shown in

the insets as a guide to the eye.

64

Figure: (color online). (a) Schematic energy band diagram of a GNR with bias voltage Vb applied. The current Iis controlled by both source-drain bias Vb and gate voltage Vg. (b)–(d) The differential conductance (dI=dVb) of three representative GNRs from set P4 with W _ 22, 36, and

48 nm as a function of Vb and Vg measured at T _1:6 K. The light (dark) color indicates high (low) conductance as designated by the color

map. The horizontal arrows represent Vb _ Egap=e. (e) E_1 gap vs W obtained from similar analysis as (b)– (d), with a linear fit of the data.

65

Figure: (color online). Egap vs W for the 6 device sets considered in this study: four (P1–P4) of the parallel type and two (D1,D2) with varying

orientation. The inset shows Egap vs relative angle _ for the device sets D1 and D2. Dashed lines in the inset show the value of Egap as

predicted by the empirical scaling of Egap vs W.

66

REFERENCES

Hilmi, Unlu. A Thermodynamic Model for Determining Pressure and

Temperature Effects on the Bandgap Energies and other Properties of

some Semiconductors. Solid State Electronics, Vol 35:9, 1343-1352.

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a b c d e f g h i j k l m n o p q r s t Streetman, Ben G.; Sanjay Banerjee (2000).

Solid State electronic Devices (5th edition ed.). New Jersey: Prentice

Hall. pp. 524. ISBN 0-13-025538-6.

J. Wu, W. Walukiewicz, H. Lu, W. Schaff, et. al.; Unusual Properties of

the Fundamental Bandgap of InN; Appl. Phys. Lett., 80, 3967 (2002).

Madelung, Otfried (1996). Semiconductors - Basic Data (2nd rev. ed.

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60883-4.

P. Ordejon Order-N tight-binding methods for electronic-structure and

molecular dynamics\

Because of difficulties in removing the HSQ etch mask and

imaging the underlying ribbon without damage, we were

67

able to measure the actual width of only a few GNRs

studied in this experiment.

See, e.g. L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen,

S. Tarucha, R. M. Westervelt, and N. S. Wingreen, in

Mesoscopic Electron Transport (Plenum, New York, 1997).

Although the relative crystallographic directions between

GNRs in a same graphene flake can be controlled by the

proper lithography processes, the absolute

crystallographic direction relative to the graphene lattice

is unknown.

68