solid state electronic materialselectronic structure and band energy

to describe electrons and their electrical properties in a solid

qualitative band model quantitative bond model

Kimia Bahan Semikonduktor – 2010 – Dr. Indriana Kartini

Band Theory of Solids

Energy Levels

Valence band electrons are the furthest from the nucleus and have higher energy levels than electrons in lower orbits.

The region beyond the valence band is called the conduction band.

Electrons in the conduction band are easily made to be free electrons.

Isolated Semiconductor Atoms Silicon and Germanium are electrically neutral;

that is, each has the same number of orbiting electrons as protons.

Both silicon and germanium have four valence band electrons, and so they are referred to as tetravalent atoms. This is an important characteristic of semiconductor atoms.

Semiconductor Crystals Tetravalent atoms such as silicon, gallium arsenide, and

germanium bond together to form a crystal or crystal lattice.

Because of the crystalline structure of semiconductor materials, valence electrons are shared between atoms.

This sharing of valence electrons is called covalent bonding. Covalent bonding makes it more difficult for materials to move their electrons into the conduction band.

2 major binding forces:

Binding forces coming from electron-pair bonds (covalent bonding) For elemental semiconductors: C(diamond), Si

and Ge typically around 4 eV in semiconductor device

Ionic bonding/heteropolar bonding For ionic solids such as the nitride, oxide and

halide insulators, and compound semiconductors

• the motion of electrons (1023) in the solids determines the electrical characteristics of the solid state electronic devices and integrated circuit

• in vacuum, the motion of a few separately objects Newton Law; F = ma – classical law of mechanics

• for solids there is particle density – classical law must be extended

in a solid high packing density

• in a volume of about 1 cm3, there are 1023 electrons and ions packed

• in a vacuum tube, there are only 109-1010 electrons• consequences in solids:

– very small interparticle distances ((1023)-1/3=2.108 cm)– high interparticle forces (interacting particles)– high interparticle collision (about 1013 per second)

• high particle density in solid system condensed matter

current or wave generated in solids resulted from averaged motion of electrons statistical mechanics

Kristal (lattice of ions)

e- scatter in the periodic lattices

interacting particlesberlaku persamaan Schrodinger:H = E solved approximately

Band Diagram – electron standing wavesallowed energies bands

forbidden energies band-gaps

Kristal fotonik (matriks dan bola mempunyai sifat dielektrik yang berbeda)

photons scatter in the periodic lattices

non-interacting particlesberlaku persamaan Maxwell:

solved exactly

Band Diagram – standing wavesallowed frequencies bands

forbidden frequencies band-gaps

1 e- atom quantized energy

• uncertainties with small distances• large number of particles

Extrapolation on 1 crystal

allowed bands and forbidden bands

Wave mechanics applied (Schrodinger eqn.) and statistic mechanics

Electronic energy levels are arranged in allowed and forbidden bands

multielectron system (~ 1023/cm3)

discrete energy

results of statistical mechanic analysis at thermodynamic equilibrium give the Fermi-Dirac quantum distribution of the electron kinetic energy in a solid (condensed matter) and Boltzmann classical distribution of electrons and particles in a gas (dilute matter)

Math solution to quantum mechanic eqns model 1 electron

energy level of 1 electron

Applied :• Planck eqn. (EMR energy and

quantized particle wave) E = h• de Broglie eqn. (EMR

momentum and particle wave ~ 1/)

p = h/

ELECTRONIC SOLIDS

1 ELECTRON

band energyenergy level of 1 electron

Bands formation

As the two atoms interact overlap the two e- interact

interaction/perturbation in the discrete quantized energy level

splitting into two discrete energy levels

r0 represents the equilibrium interatomic distance in the crystal

• at r0 : allowed band consists of some discrete energy level

• Eg.: System co. 1019 atoms 1e, the width of allowed band energy at r0 = 1 eV

• if assumed that each e- occupies different energy level and discrete energy level equidistance allowed bands will be separated by 10-19 eV

allowed band

• The difference of 10-19 eV too small allowed bands to be quasi-continue energy distribution

Bands of atom 3e-

As 2 atoms get closer, electron interaction was started from valence electron, n=3

At r0 : 3 allowed bands separated by forbidden were formed

pita

ene

rgi t

erbo

lehk

an pita energi terlarang

Splitting energi pada atom 14Si 4 elektron valensi 3s2 3p2

3s2 : n=3 l=0

3p2 : n=3 l=1

At reduced distance : 3s and 3p interacted dan overlap 4 quantum state of upper bands (CB) and 4 quantum state of lower bands (VB) 4 valence e- of Si will occupy lower band

Eg represents the width of forbidden band = bandgap energy

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Bonding In Metals: Lithium

according to Molecular

Orbital Theory

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Sodium According to Band Theory

Conduction band:

empty 3s antibonding

Valence band:

full 3s bonding

No gap

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Magnesium

3s bonding and antibonding should be full

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Magnesium

Conduction band:

empty

Valence band:

full

No gap: conductor

Conductor

Classification of solids into three types, according to their band structure

insulators: gap = forbidden region between highest filled band (valence band) and lowest empty or partly filled band (conduction band) is very wide, about 3 to 6 eV;

semiconductors: gap is small - about 0.1 to 1 eV;

conductors: valence band only partially filled, or (if it is filled), the next allowed empty band overlaps with it

Band structure and conductivity

Band gaps of some common semiconductors relative to the optical spectrum

0 1 2 3 4

InSb Ge Si

GaAs

CdSeGaP

CdS SiC ZnS

Eg (eV)

7 3 25 1 0,5 0,35 (m)

Infrared UltravioletVisible

TiO2

Energy band gap

• determines among other things the wavelengths of light that can be absorbed or emitted by the semiconductors– Eg GaAs = 1.43 eV corresponds to light wavelengths

in the near infrared (0.87 m)

– Eg GaP = 2.3 eV green portion of the spectrum

• The wide variety of semiconductors band gap tunable wavelength electronic devices– broad range of the IR and visible lights LEDs and

lasers

Electron Distribution

• Considering the distribution of electrons at two temperatures:– Absolute zero - atoms at their lowest energy level.– Room temperature - valence electrons have absorbed enough

energy to move into the conduction band.• Atoms with broken covalent bonds (missing an electron) have a hole

present where the electron was. For every electron in the conduction band, there is a hole in the valence band. They are called electron-hole pairs (EPHs).

• As more energy is applied to a semiconductor, more electrons will move into the conduction band and current will flow more easily through the material.

• Therefore, the resistance of intrinsic semiconductor materials decreases with increasing temperature.

• This is a negative temperature coefficient.

At 0°K, each electron is in its lowest possible energy state, and each covalent bounding position is filled.

If a small electric field is applied, the electrons will not move → silicon is an insulator

If the temperature increases, the valence electrons will gain some thermal energy, and breaks free from the covalent bond → It leaves a positively charged hole.

In order to break from the covalent bond, a valence electron must gain a minimun energy Eg: Bandgap energy

• For elemental/intrinsic semiconductor of Si and Ge: the filled valence band of 4 + 4 = 8 electrons

• For non-intrinsic semiconductor: the filled valence band of 8 electrons constructed by combination of elements of group II-VI and III-V

• the E for the bandgap will differ from the elemental semiconductors

• the bandgap will increase as the tendency for the e- to become more localised in atom increases (a function of constituent electronegativities)

Compound Semiconductor: combination of elements

Impurities

• strongly affects the electronic and optical properties of semiconductor materials– used to vary conductivities from apoor

conductor into a good conductor of electric current

• may be added in precisely controlled amounts doping

Evaluation of both properties needs prior understanding of the atomic arrangement of atoms

in the materials – various solids

Kimia Bahan Semikonduktor - Indriana Page 28

Empirical relationship between energy gap and electronegativities of the elements

Metallic conductance (Sn)

Elemental semiconductors(Si, Ge, etc)

Insulators:-Elemental (diamond, C)

-Compound (NaCl)

Compound semiconductors(GaAs, CdS, etc.)

Kimia Bahan Semikonduktor - Indriana Page 29

Impurity and Defect Semiconductor:

Creating band gap through electronegativity effect

P-typen-type

Kimia Bahan Semikonduktor - Indriana Page 30

Semiconductor Doping• Impurities are added to intrinsic semiconductor materials to improve

the electrical properties of the material.

• This process is referred to as doping and the resulting material is called extrinsic semiconductor.

• There are two major classifications of doping materials.– Trivalent - aluminum, gallium, boron– Pentavalent - antimony, arsenic, phosphorous

Kimia Bahan Semikonduktor - Indriana Page 31

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Figure 13.29: Effect of doping silicon.

(a) donation of electrons from donor level to conduction band; (b) acceptance of valence band electrons by an acceptor level, and the resulting creation of holes; (c) donor and acceptor atoms in the covalent bonding model of a Si crystal.

Energy band model and chemical bond model of dopants in semiconductors