Introduction to Microcontrollers & Embedded Systems

Electronic Engineering Materials & Nanotechnology

Chapter 5

ENERGY BAND THEORY & Classification of Solids Contents: 5-1. Introduction 5-1.1. Energy Levels & Energy Bands 5-1.2. Reference Energy Levels (Vacuum level & Fermi level) 5-2. Crystal Hamiltonian Operator 5-2.1. Adiabatic Approximation 5-2.2. Jellium Model 5-2.3. Hartree Equation (One-electron Model) 5-2.4. Hartree-Fock Equation 5-2.5. Self-consistent Field Method 5-2.6. Post Hartee-Fock Methods 5-3. Bloch Functions 5-4. Kronig-Penney Model 5-5. Effect of Crystal Size & Fine Structure on Energy Bands 5-6. Analogy with Circuit Theory 5-7. Energy Band Structure Calculation Methods 5-7.1. Energy Band Structure (Approximate Methods) .

.

5-7.2.

i. Gas of Free electrons (Fermi Gas) ii. Empty lattice model iii. Quasi-free electron (QFE) model iv. Pseudo-potential method v. Tightly-bound electron (TBE) model vi. The k.p method

Energy Band Structure (Ab initio Methods) i. Full-potential Linearized Augmented Plane Waves ii. Density Functional (DFT) method iii. Local Density Approximation (LDA) -305-

Prof. Dr. Muhammad EL-SABA


Chapter 5

Contents of Chapter 5 (cont.): 5-8. Electron Dynamics & Effective Mass 5-8.1. 1-dimensional Model 5-8.2. 3-dimensional Model (Tensor mass) 5-9. Actual Band Structure and Fermi Surface in 3-D crystals 5-10. Case Studies: Band Structure of Si, Strained Si, GaAs 5-10.1. Band Structure of Si 5-10.2. Band Structure of Strained Si i. Effect of Biaxial Stress ii. Effect of Uniaxial Stress

5-10.3. Band Structure of SiC 5-10.4. Band Structure of GaAs 5-11. Hole Concept 5-12. Density of States in Energy Bands 5-13. Classification of Solid Materials 5-14. Classification of Semiconductors 5-15. Direct- and Indirect-gap Semiconductors 5-16. Energy Band Alignment in Heterojunctions 5-17. Graded Bandgap Semiconductors 5-18. Superlattices & Bloch Oscillations 5-19. Low-Dimensional Semiconductors (LDS’s) 5-20. Energy Band Structure of Graphene 5-21. Energy Band Structure of Carbon Nanotubes (CNT’s) 5-22. Energy Band Structure in Silicon NanoWires 5-23. Organic Semiconductors 5-24. Energy-band Structure of Amorphous Semiconductors 5-25. Topological Insulators 5-26. Effect of Magnetic Fields on Energy Bands Structure 5-27. Effect of Magnetic Field on Energy Bands in LDS’s 5-28. Effect of Magnetic Field on Energy Bands in Nanotubes 5-29. Summary 5-30. Problems 5-31. Bibliography

-306Prof. Dr. Muhammad EL-SABA


Chapter 5

Energy Band Theory & Classification of Solids 5-1. Introduction The basis for discussing transport in semiconductors is the underlying electronic band structure of the material arising from the solution of the many body Schrödinger equation. As we have seen earlier, the solution of Schrödinger equation for a particular system results in a set of wave eigenfunctions  n and energy eigenvalues En. The relation between the electron energy and the electron wave-vector, E(k), which we call dispersion relation of the system, is obtained by substituting the solution of into the Schrödinger equation. In isolated atoms, the electrons are orbiting around their nuclei with certain energy levels. In this case, the E(k) relation consists of a set of discrete (quantized) points in the k space. When isolated atoms are brought together, to form a solid, various interactions (due to attraction and repulsion forces) occur between neighboring atoms. As a consequence of these interactions, the energy levels of individual atoms are transformed into common energy bands. The energy band structure of a solid describes ranges of energy that an electron is forbidden or allowed to have. The band structure of a material determines its electrical and optical properties, among other characteristics. 5-1.1. Energy Levels and Energy Bands As the electron clouds of individual atoms in a solid are overlapped, the height of potential wells of different atoms is lowered and electrons of higher energies are no longer belonging to a specific atom. In contrast, the electrons that are found in the deep energy levels are somewhat tightly bound to their nuclei. Figure 5-1 depicts the splitting of energy levels in an array of diamond atoms. As a consequence of the overlapping process, the electron energy levels are split into dense groups of levels, known as energy bands. In fact the exclusion principle dictates that no two electrons in a given interacting system may have the same quantum state. Therefore, atoms of identical quantum states modify their energy states such that the Pauli Exclusion Principle is satisfied. In fact, the original discrete energy levels of such electrons are split into dense groups of adjacent levels, which -307Prof. Dr. Muhammad EL-SABA


Chapter 5

we called energy bands. These dense groups of levels (or energy bands) are belonging to the crystal rather than to individual atoms. Usually, these energy bands are separated by successive forbidden regions, that we call energy gaps. Therefore, the starting point to understand the energy band structure of solids is the potential energy diagram of an electron in an isolated neutral atom and the corresponding energy levels.

Fig. 5-1. Energy levels splitting due to the interaction of adjacent atoms and overlapping of their probability density functions p(r). Energy

Energy

2p2 2s2

Ec

1s2

1s2

Ev R

Nucleus (a)Isolated diamond atom.

Ro

Ro

(b) Array of diamond atoms.

Fig. 5-2. Energy levels in an isolated diamond atom and in an array of diamond atoms. -308Prof. Dr. Muhammad EL-SABA


Chapter 5

Figure 5-3 shows the development of the electron energy bands in diamond crystals from the discrete energy levels of an isolated atom1. As shown in figure, when the separation between atoms is decreased, the energy levels of single atoms (1s, 2s, etc.) are split into bands of a huge number of adjacent energy levels.

Fig. 5-3. Schematic of energy level splitting and formation of energy bands in a diamond crystal of N atoms as a function of interatomic distance R. The distance Ro is the interatomic separation at the state of equilibrium between atomic forces.

Figure 5-4, depicts the energy band diagram at the state of equilibrium, where the interatomic distance R = Ro. The highest filled energy band, in the energy band diagram, is usually called: valence band. Also, the energy band just above the valence band is called the conduction band. The valence and conduction band are separated by a region called the energy gap. The height of energy gap is given by:

Eg = Ec - Ev

(5-1)

where Ec is the lowest (bottom) level in the conduction band and Ev is the highest (top) level in the valence band as shown in figure 5-4. Thus, in crystalline solids, the E-k relation, at a certain point in the physical crystal lattice, is usually characterized by a sequence of alternating allowed energy 1.

Note that this splitting schematic is not general. It is adapted from the conventional band theory



Chapter 5

bands and energy gaps. For three-dimensional crystals, the k-space is threedimensional. The E-k relation of electrons inside a solid crystal, in the various directions of the k-space, is usually called the energy band structure of that solid. E Energy gap

Conduction Band Ec

Eg

Ev

Band of allowed energy levels

Valence Band V.B.

Fig. 5-4. Schematic of the energy band diagram (versus crystal spatial position).

By studying the energy band structure of electrons inside crystalline solids, one can understand their electrical properties. For instant, the number of electrons in the conduction band, the electrons average velocity and the electron current in a certain crystalline solid can be all calculated if one knows the E-k relationship (or the energy band structure) of that solid. Therefore, our fundamental goal is to determine the electrical properties of semiconductor materials, which are necessary to develop the currentvoltage characteristics of semiconductor devices. 5-1.2. Reference Energy Levels (Vacuum level and Fermi level) Before proceeding in the mathematical details of energy bands in solids, it is wise to settle down some reference energy levels from which we measure the other energy levels and the whole band structure in solids. We start by defining the vacuum energy level as the potential energy of an isolated electron in vacuum at rest. In crystals, electrons are arranged in energy bands, which are separated by energy gaps. We have seen so far in Chapter 3, the arrangement of electron energies follows a probability distribution function. At equilibrium the distribution function of electron energies is called the Fermi-Dirac distribution. According to the Fermi-Dirac distribution, fn(E), all energy levels under a certain reference energy level (called the Fermi level and denoted by EF) -310Prof. Dr. Muhammad EL-SABA


Chapter 5

are occupied at absolute zero temperature. Therefore, fn(E) =1 when E EF at T=0. That’s electrons fill the energy bands of solids, at very low temperatures, up to the Fermi energy level EF, and the bands are empty for energies that exceed EF. Figure 5-5 depicts the location of Fermi energy level, EF, with respect to the reference vacuum energy level, Eo, in different solids. The difference between the vacuum level and the Fermi level in metals is usually termed the work function ( M = Eo - EF). Also, the difference between the vacuum level and the conduction band edge in semiconductors and insulators is usually termed the electron affinity ( = Eo - EF). Eo Vacuum level

Eo

 m

 s

EF Metal

Ec EF

Vacuum

Semiconductor or Insulator

Vacuum

Fig. 5-5. Vacuum energy level at metal and semiconductor/insulator–vacuum interface. Table 5-1: Work function of some metals

Table 5-2: Electron affinity of some semiconductors



Chapter 5

5-2. Crystal Hamiltonian Operator We know that the one particle Schrödinger equation governs the time evolution of a complex-valued wavefunction  on the configuration space ( ). The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function V on :

 E  =Η 

(5-2)

or

For many particles, the equation is the same except that the Hamiltonian  and are now on configuration space, ,where N is the Η as well as number of particles. The large number of particles involved in a typical solid makes the direct solution of the Schrödinger equation completely impossible. Therefore, we are naturally led to the development of a one-electron approximation. It is however, necessary to relate the one-electron theory to the actual manyelectron situation, and to estimate the corrections to the results of the oneelectron approximation. In this section we demonstrate how such a oneelectron approximation is developed from the first principles. We start by defining the Hamiltonian for the entire problem of solid state physics. It consists of the kinetic energy of all particles in the solid and of their interaction energies. The solid is comprised of two groups of electrons: valence electrons, which contribute to chemical bonding and core electrons which, are tightly bound in the closed shells of the lattice ions. We usually consider the valence electrons and the lattice ions as independent constituents of the solid. This is the first approximation, we usually make in solid state physics. For simplicity, we shall call the valence electrons, the electrons. The Hamiltonian then consists of the kinetic energy of these valence electrons and all ions as well as their interaction (potential) energies.

 Η = Hel + Hion + He-i + Hext

(5-3a)

where we omitted the operators cap sign for the sake of simplicity. For a non-perturbed system we have Hext = 0. For electrons we have: -312Prof. Dr. Muhammad EL-SABA


Chapter 5

Hel = He (K.E.)+ He-e (electron-electron interaction)

pk2 1   + 4  o k 2m

e2 rk - rk `

 k, k `

(5-3b)

where rk and rk` are the coordinates of the kth and k`th electrons, respectively. Similarly, for ions, we have:

Hion = Hi (K.E.)+ Hi-i(ion-ion interaction) Pi 2  + 1  Vi Ri  R j 2 i,j 2M i i





(5-3c)

where Ri and Rj are the coordinates of the ith and jth ions, respectively. Concerning the electron-ion interaction energy, it may be written as follows:

H e-i =  Ve-i rk  Ri 

(5-3d)

k,i

The ion-ion interaction term Hi-i may be subdivided into two terms, one accounts to the interaction when the ions are in their equilibrium positions; the other is a correction to account for the vibrations of the lattice ions (phonons).

H i -i = H o i -i + H ph , H e-i = H o e-i + H e- ph

(5-3e)

Unfortunately it is not always possible to rigorously solve the quantum mechanical problem. Approximations have to be made. In solid state theory, two simplifications are commonly made. In solving a given problem individual terms of the Hamiltonian may be neglected or only partially considered or handled subsequently as perturbations. This simplified problem is then further simplified by using the symmetry of the lattice. The particular approximation needed here depends on the questions we are seeking to answer and on the nature of the solid material we want to describe. -313Prof. Dr. Muhammad EL-SABA


Chapter 5

5-2.1. Adiabatic Approximation We can consider that the solid state problem can be divided into two components: 

Movement of electrons (electron gas) in a stationary lattice, and



Movement of ions in a uniform space charge of electrons.

This is the basis of the so-called Born-Oppenheimer (adiabatic) approximation, which is often used to provide a justification for system decoupling. The physical background of this approximation is discussed in chapter 2 of this book. 5-2.2. Jellium Model According to the adiabatic approximation, Hel can be written as follows:

H el =  k

2 2 1 k + 2m 8 o

e2



rk  rk `

k, k `

+ H ve

(5-4)

where H+ve includes the effect of the ions positive space charge and the interaction of the electrons with this space charge. The electron gas is considered here to be embedded in a constant positive charge background. This model is described as Jellium. Many of the properties of metals can be described by mean of this approximation. On the other hand, the movement of ions can be treated in a similar way. So, Hi can be written as follows:

H ion=  i

2 1  i2 + 2M i 2

 V R i

i

 R j  + H ve

(5-5)

i,j

where H-ve includes the effect of negative space charge of electron (considered fixed) and the interaction of this charge with ions. It should be noted that the two terms have H-ve and H+ve exactly cancel each other. As far as the motion of electrons is concerned, we can therefore adopt a Schrödinger equation for the electrons:

H el 

H e-i   = Ee  -314-


(5-6)


Chapter 5

Here He-i is considered with fixed coordinates of ions. The remaining coupling between electrons and ions vibration is the interaction He-ph. This coupling can be dealt with in most cases by the perturbation theory. This theory will be described in a following chapter. 5-2.3. Hartree Equation (One-Electron Approximation) In the absence of electron-electron interactions, the electron gas problem (Jellium model) would decouple into one-electron problem, which describes the motion of an electron in a given potential field. In this case, the Hamiltonian term which accounts for the electron-electron interaction is neglected, so that:

 Η

2 2 -  k +  V rk =  H k k 2m k k

(5-7)

Then, the following Schrödinger equation:

H

k

 = E (5-8)

k

can be reduced, by substituting:

H

k

 = E

k

(5-9)

and

 r1 ..rN  =  i r1  2 r2 ... N rN 

(5-10)

into the following one-electron equations:

Hk  k rk  = Ek  k rk 

(5-11)

However, the many-body problem (the electron gas which is embedded in a homogenous, positively charged medium) could be reduced to one-body problem (one-electron problem) which includes an energy term to account for the electron interaction. Such a reduction is achieved by the Hartree (or Hartree-Fock) approximation. In this approximation, the Schrödinger equation for a single particle writes: -315Prof. Dr. Muhammad EL-SABA


Chapter 5

 2 2     V ( r ) H  2m  k r   Ek k r   

(5-12)

where VH is the Hartree potential, which is given by:

VH (r )  V (r ) 

2

e 4o

 j k

 j r 

2

r  r

dV

(5-13)

Here r and r` are the positions of the ith and kth electrons, respectively. The above equation is called the Hartree equation. It describes the motion of the kth electron at the position r in the potential field V(r) of the lattice ions, and in the Coulomb potential of an average distribution of all other electron: (j=1, 2, .., N, with j≠k) It should be noted that the Hartree equation does not account for the Pauli Exclusion Principle. The spin of electrons is not considered in this equation. If two electrons are interchanged, the sign of the total wave N function will remain unchanged (     K ). This representation is suitable k 1

for bosons but not for Fermions (e.g. electrons) where the total wave function  must change its sign when two particles (electrons) are interchanged. 5-2.4. Hartree-Fock Equation In order to account for the Pauli-exclusion principle we replace the usual expression of the total wave function 

 =  1 r1  ....  N rN 

(5-14)

by the following Slater determinant, which satisfies the Pauli-exclusion principle:

 1 q1  . .  N qi  =

1 N!

.

.

. .  i q N  . .  N q N  -316-


(5-15)


Chapter 5

Here, the coordinates qk include the spatial coordinates rk as well as the spin coordinates of the electrons. Also the factor in front of the determinant is added for normalization purposes. If two electrons are interchanged, two columns of the determinate are interchanged, and  changes sign. Also if two electrons have the same coordinates, two columns will be identical, and willvanish. Now the interaction term accounts for the Pauli exclusion term and the Schrödinger equation reads:  2 2 e   V r    4 o  2m



 r     HF r , r  r-r 

 d V   k r  = Ek k r  

(5-16)

where

 r     kH   - e  k r  k

2

(5-17)

k

and

 r  r r  r   e   r  r  * j

  HF

k

* k

* k

k

j

j, k

(5-18)

k

The above form of Schrödinger’s equation is called the Hartree-Fock equation. Also, the following potential is called Hartree-Fock potential.

V `(r )  V r  

e 4 o



 r`    HF r , r  r  r`

d V

(5-19)

The Hartree-Fock equation may be then written in the following simple form:

 2 2       V` r  2m   k r  = Ek r    V`(r) is sometimes called the pseudo-potential.


(5-20)


Chapter 5

5-2.5. Self-Consistent Field Method The calculation of energy bands is based, in principle, on the Hartree-Fock equations. The set of Hartree-Fock equations forms a quite complicated nonlinear integro-differential system. In fact the solution of such equations must be carried out using iterative numerical methods. The self-consistent field (SCF) method results in the wave eigenfunctions and energy eigenvalues satisfying the Hartree (or the Hartree-Fock) equation. The solution of this equation is undertaken as follows: 1Choose a set of eigen-functions  (k0 ) as a zero-order approximation (ground-state functions) for  k . 2Calculate an initial estimate of the Hartree (or the Hartree-Fock) potential, VH(0) , using equation (5-19) as follows:

V

(0) H

r   V r  +

e2 4  o

 j k

 j o  r  r - r`

dV`

(5-21)

3Substitute the initial estimate of Hartree potential VH(0) into the Hartree equation (5-13) and solve to obtain a set of new functions  k1 ,  2 2  1 (0) 1     V r H    k r  = Ek  k r   2m 

(5-22)

4Calculate a new estimate of Hartree potential, VH(1) , using the new set of functions  k1 , etc. 5-

Proceed until you reach the allowed error in  k1 or Ek .

6-

The so-obtained wave functions are called the self-consistent wave functions.

If the Hartree-Fock potential V`(r) is taken as the total potential energy of an electron in the crystal (due to other electrons and ions), then we can write:

 2 2     V ' ( r )  . (r )  E. (r ) 2 m   -318Prof. Dr. Muhammad EL-SABA

(5-23)


Chapter 5

This equation is very interesting because it yields valid results without having to determine the self-consistent functions, if we start with an acceptable pseudo potential V(r). It can be shown that V`(r) in the crystal lattice is a periodic function of lattice coordinates. This feature is exploited in the so-called “Bloch functions” that we shall discuss in a next section. Other derived and simplified methods, like the density functional theory (DFT) or the tightly-bound electron (TBE) method, have been developed in such the way that they either neglect or approximate some of the integrals, which are involved in the HF and SCF methods. The approximate methods will be discussed in section 5-7 of this chapter. 5-2.6. Post-Hartee-Fock Methods We have seen in the above discussion that the introduction of the selfconsistent potential (Hartree or Hartree-Fock potential), enables the problem of a system of interacting particles to be reduced to a one-particle problem. The Hartree–Fock based models have been successfully applied to atoms and molecules. Therefore, the Hartee-Fock (HF) is considered as one of the ab initio computational methods in solid-state physics. The main two problems of the technique are the neglect of electrons and ions correlation effects and the high computational demands. In fact the Hartree–Fock methods can only be applied to systems with small numbers of atoms. The so-called post-Hartree–Fock methods are a category of computational methods, which are developed to improve the Hartree–Fock and selfconsistent field methods. They add electron correlation terms to account for the repulsion between electrons rather than being averaged in the Hartree– Fock method. For the great majority of systems, in particular for excited states and processes, the correlation term is the most important. The postHartree–Fock methods give more accurate results than the traditional HF method, with the price of numerical computational costs. The accuracy and reliability of the above methods is hard to assess. In large scale atomic calculations where one deals with hundreds or perhaps thousands of atoms, and where translational symmetry may be broken, new approaches are required. In Chapter 9, we present some formulations of quantum mechanics that significantly reduces the number of variables, but still enables us to calculate electronic properties of electronic systems and devices with high degrees of accuracy. -319Prof. Dr. Muhammad EL-SABA


Chapter 5

5-3. Bloch Functions When we try to find out the wave functions for electrons in a crystalline solid, the task may seem impossible because of the huge number of atoms and electrons. However, the problem is greatly simplified if we consider that the electrons of the solid belong to the crystal rather than to individual atoms. If the total potential energy function V(r) of the lattice is periodic, as shown in figure 5-6, then the solution is greatly simplified by using the socalled Bloch functions. Bloch showed that if the lattice potential V(r) is periodic such that:

V  r  = V  r + R

(5-24)

where R= n1 a1 + n2 a2 + n3 a3 and a1, a2 & a3 are the basic lattice vectors, then the solution of the one-electron Schrödinger equation can be written in the following form :

 k r  = e j k.r uk r 

(5-25)

where uk (r) is a periodic function and has the same periodicity as the crystal lattice, i.e.,

uk r  = uk r  R

(5-26)

In fact the functions uk are three-fold periodic and should satisfy:

V r  =

 u r  exp  j G hkl

hkl

. r

(5-27)

h,k,l

where Ghkl is the reciprocal lattice vector, which is given by the relation: Ghkl = hb1 + kb2 + lb3 and b1, b2 & b3 are the reciprocal lattice unit vectors. As we have seen so far in Chapter 1, the product Ghkl.Rijk (or simply G.R) is equal to 2N, where N is an integer. The functions uk (r) are sometimes called the atomic part of the Bloch functions because they contain the essential features of the unit cell and reflect the atomic properties. In contrast, the plane-wave part of  (r), i.e. exp(jk.r), reflects the large scale variations of  (r), and becomes significant for free electrons. -320Prof. Dr. Muhammad EL-SABA


Chapter 5

Fig. 5-6. Schematic of the crystal periodic potential.

Note 5-1. Bloch Functions versus Wannier Functions The Bloch wave functions ukn(r) are the eigenfunctions of the Schrödinger equation with periodic potential. They are characterized by the wavevector k and the band index n. The Wannier function Wn(r-R) are equivalent set of orthogonal functions, which are defined as follows:

Wn r-R =

1 V

 j k.R e u nk r .dk  BZ

(5-28a)

The integration is taken over the first Brillouin zone (BZ) of volume V. There exist , for each band n, a set of Wannier functions, all identical except each translational lattice vector R. Alternatively, the set of orthogonal Wannier functions may be defined as follows:

WR r  =

1 N

 j k.R e u k r   k

(5-28b)

where uk (r) has the same periodicity as the crystal lattice and N is the number of primitive cells in the crystal. The Wannier functions are not eigenfunctions of the Hamiltonian. However they may serve as a convenient basis for the expansion of electron states in certain regimes.



Chapter 5

5-4. Kronig-Penney Model Even when using the Bloch functions approach, the mathematics would be almost impossible if one were to use realistic periodic potentials. In the Kronig-Penny model (1931), the crystal potential V(r) is modeled by a periodic chain of square wells. Fig. 5-7 depicts the Kronig-Penny model in one dimension. Here we have:

V(x) = Vo for = 0 for

-b < x < 0 0 5eV

Ev

Ev V.B.

V.B.

Semiconductors

Insulators

(b) Non-Conductors. Fig. 5-49. Classification of solids to metals, semiconductors and insulators.



Chapter 5

The energy gap in semiconductors is usually in the order of 1 eV, as shown in Table 5-7. Hence, the thermal excitations can raise a few electrons from valence band to conduction band in semiconductors. Therefore, the properties of a semiconductor, is determined from the shape of the energy band diagram near bottom of conduction band and top of valence band. On the other hand, insulators are characterized by a huge energy gap in the order of 5eV. Table 5-7. Energy gap and resistivity of some materials.

Material SiO2 Si 3N4 GaN Poly acetylene

Ga As Si Ge In As Al Cu

Eg (eV)

Category Insulator Insulator Compound Semiconductor Polymer Semiconductor Compound III-V Semiconductor Element IV Semiconductor Element IV Semiconductor Compound III-V Semiconductor Metal Metal


 (.cm)

0K

300K

300K

9 5 3.36 2.5 1.52 1.17 0.74 0.42 -

9 5 3.3 2.5 1.43 1.12 0.66 0.36 -

1014-1016 1014 1011 108 2x105 44 0.3 2.64x10-6 1.67x10-6


Chapter 5

5-14. Classification of Semiconductors Semiconductor materials can be classified according to their constituent atoms to: Elemental semiconductors (e.g., Si, Ge, and diamond), Compound semiconductors (e.g., GaAs, InSb, AlGaAs, AlGaAsSb). Simple or elemental semiconductors are all from the IV group in the periodic table (e.g., Si and Ge) and crystallize in a diamond structure. Compound semiconductors are composed of atoms of two or more elements, most frequently from the III and V groups (III-V compounds, like GaAs) or the II and VI groups (II-VI compounds, like CdS, ZnS and ZnTe). Also some compound semiconductors are composed of two or more elements from the IV group (IV-IV compounds, like SiC). When compound semiconductors are formed from two elements, such as GaAs, they are called binary compounds. Many of the III-V compound semiconductors crystallize in the zincblende structure, while others crystallize in the Wurtzite or rock salt structures. Note that Zincblende is a cubic structure while Wurtzite is non-cubic. Some IV-IV compound semiconductors, like silicon carbide SiC crystallize in close-packed structures having a special one-dimensional polymorphism, called polytypism. Polytype crystals are identical in the two dimensions of the close-packed planes and different in the perpendicular stacking direction. The SiC crystals have two famous structures, namely  -SiC and -SiC. The -SiC is a poly-type of SiC, which has a repetitive sequence of covalent bonded tetrahedra of SiC4 and Si4C. Also, -SiC has a zinc-blende structure. The -SiC is sometimes called 3C-SiC. This is the only SiC cubic structure8. All other non-cubic structures (e.g., rhombohedral or hexagonal structures, like 6H-SiC) are referred to as  -SiC. The Il-VI semiconductors encompass wide-band-gap materials (e.g. ZnS and ZnSe) as well as narrow- and zero-gap semiconductors (e.g. HgTe). Some II-VI compound semiconductors, like CdS, crystallize in the HCP (hexagonal closed packing) Wurtzite structure, while others, like ZnTe, crystallize as zincblende structure. Like IV-IV group and Ill-V compounds, the Il-VI compounds are tetrahedral sp 3 bonded, but in contrast to most of the former, the outermost cation states do influence the bonding 8

We consider that zincblende structures as two inter-penetrating face-centered cubes (FCC), separated by ¼ the cube length.



Chapter 5

Fig.5-50. Main semiconductor elements and compounds in the periodic table and there typical energy gaps



Chapter 5

When compound semiconductors are formed from more than one group III elements (randomly distributed on III sites) or more than one group V elements (randomly distributed on V sites), these compounds are called crystalline solid solutions. Crystalline solid solutions that have 3 elements are called ternary compounds, and have the notation AxB1-xC (e.g., AlxGa1-xAs). Note that x and 1-x are the mole fractions of the first two elements, which belong to the same group, in the crystal solution. Also, crystalline solid solutions that have 4 elements are called quaternary compounds, and have the notation AxB1-xCyD1-y (e.g., GaxIn1-xAsyP1-y). Here, the subscript x is the mole fraction of Ga in the group III elements and the subscript y is the mole fraction of As in the group V elements. For more details about the variation of Eg with mole fraction x, refer to table 5-8. Semiconductors can be also classified according to their bandgap width as follows: Narrow-gap semiconductors (like PbS, PbTe, and SnTe), Wide-gap semiconductors (like ZnS, GaN and SiC). Narrow gap semiconductors are frequently used in microwave and infrared detectors. Interestingly, a certain number of semiconductors, like HgTe, are referred to as null-gap semiconductors. Such materials are classified as semiconductors because they do not have partially filled bands at 0K. As for wide-gap semiconductors, they are currently used in power devices, optoelectronics and space applications. Interest in Group-III nitrides (like GaN) is motivated by the possible use of these materials in optoelectronic devices which operate in the blue to UV region of the electromagnetic spectrum. As mentioned above, the Group-III nitrides are normally grown in the wurzite structure, but can also be produced in the zincblende structure, which is more amenable to p-type doping and device fabrication.



Chapter 5

5-15. Direct and Indirect Gap Semiconductors One of the most important observations on the E-k diagram of certain semiconductors is that the valence band maximum may be found at a different value of k than the principal conduction band. For instance, the silicon material has its top valence bands at k =0, while the bottom of principle conduction band lies at k =0.85 (2 /a) in directions [100], as shown in figure 5-51. Such semiconductors are called indirect-gap semiconductors. The electron transitions from the minimum point in the conduction band to the maximum point of the valence band in indirect-gap semiconductors require some change in k. Therefore, the conservation of crystal momentum (or wave vector) requires a phonon emission (or absorption) during the band-to-band transitions in indirect-gap semiconductors. This is only possible if there are occupied electron and hole states directly on top of each other. This is why the Si, which has an indirect gap, is not used as a light source in optoelectronic devices. However, there exists other type of semiconductors, like GaAs, whose energy gap is direct. That is to say, the top of valence band and the bottom of conduction band are situated at the same k-value. So, electrons, which make transitions from the conduction band to the valence band, do so without change in their k-value. This type of transitions is usually associated by an emission of photons.

Fig. 5-51. Electron transitions in direct (GaAs) and indirect (Si) semiconductors.



5-16. Energy Band Alignment in Heterostructures

Chapter 5 (A)

Heterojunction devices are special category of semiconductor devices, which consist of several layers of different semiconductors, with different energy gaps. The energy bands alignment at the interface of heterojunctions may be explained by the band affinity model (Anderson’s rule). This is based on the line-up of the electron affinity χ of materials in contact. The electron affinity is the energy required to take an electron from the bottom of the conduction band to the vacuum level where it can escape from the crystal. According to Anderson’s rule, the vacuum level of the two materials of a heterojunction should line up, such that:

Ec = χ = χ1 - χ2

&

Ev = Eg2 - Eg1 - Ec

(5-111)

For instance, if we have a GaAs - Al0.3Ga0.7As heterojunction, where the GaAs has χ = 4.07 eV and the Al0.3Ga0.7As has χ = 3.74 eV, then we predict that Ec= χ = 0.33eV at their interface. Also, the energy gap changes by Eg =1.79-1.42 = 0.37eV, and hence Ev = 0.04eV, as shown in figure 552. More examples are shown in figure 5-53.

Fig. 5-52. Alignment of energy bands at a heterstructure interface, according to the band affinity model (Anderson’s rule).

Note 5-7 Band Alignment Theories Actually, there exist 3 theories, for the heterojunction energy bands alignment. The Anderson rule (the band affinity model), is widelyaccepted in the literature. It dictates the align-up of the vacuum level of all materials in contact. Other theories presume the align-up of the conductionband edge, or the intrinsic level, which is mid-way between the conduction band and the valence band edges, of the two semiconductors in contact. -387Prof. Dr. Muhammad EL-SABA


Chapter 5

Fig. 5-53. Alignment of energy bands at the interface of AlN-GaN-InN and GaAs – AlxGa1-xAs hetero-junction, where Eg = 1.422 + 1.247x. For x = 0.3, Ecc = 0.33 eV, Eg = 0.37 eV and Evv = 0.04 eV at 300K. Table 5-8. Variation of important parameters of AlxGa.As1-x as function of the mole fraction x, for x < 0.45 and T = 300 K

Parameter Energy gap [eV]  Valley L Valley X Valley

Formula Eg

= = 1.422 + 1.247x = 1.707 + 0.642x = 1899 + 0.125x

Electron effective mass  Valley L Valley (total) X Valley (total)

mn*/mo = = 0.067 + 0.083x = 0.55 + 0.12x = 0.85 - 0.07x

Hole effective mass Dielectric constant

mp*/mo = 0.48 + 0.31 x 

r

= 13.1 - 3x



Chapter 5

Fig. 5-54. Bandgap offset of the hetero-junctions of some semiconductors.



Chapter 5

5-17. Graded Bandgap Semiconductor Structures

(A)

Some devices need alloy systems with very low strain-induced defects. This can be achieved in some compound semiconductors, like AlxGa1-xAs, GaxIn1-x, AsxP1-x and GaxIn1-xAsyP1-y, by grading the mole fraction indexes x and y. The resultant material is a graded band gap semiconductor. It is interesting to show that some compound semiconductors can be changed from direct to indirect gap according to the mole fraction of their constituent elements. For instance, figure 5-55 depicts the band structure of of GaAs1-xPx at different values of x. As shown in figure, the energy gap height Eg of such a compound semiconductor varies piecewise linearly with the mole fraction x such that:

Eg(x) = Eg + 1.25 x

for 0