EE331-bandformation

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2.3 Energy band theory

Quantization Concept

plank constant

Core electrons

Valence electrons

2s2p

1sK

L

Quantization Concept

The shell model of the atom in which the electrons are confined to live within certain shells and in sub shells within shells.

The Shell Model

1s22s22p2 or [He]2s22p2

L shell with two sub shells

Nucleus

Band theory of solids

Two atoms brought together to form molecule

“splitting” of energy levels for outer electron shells

Energy Band Formation (I)

→ ← =

2

Splitting of energy states into allowed bands separated by a forbidden energy gap as the atomic spacing decreases.The electrical properties of a crystalline material correspond to specific allowed and forbidden energies associated with an atomic separation related to the lattice constant of the crystal.

Allowed energy levels of an electron acted on by the Coulomb potential of an atomic nucleus.

Energy Band Formation (I)

Broadening of allowed energy levels into allowed energy bandsseparated by forbidden-energy gaps as more atoms influence each electron in a solid.

Energy Band Formation

One-dimensional representation

Two-dimensional diagram in which energy is plotted versus distance.

Energy Band Formation (III)

Conceptual development of the energy band model.

Elec

tron

ene

rgy

Elec

tron

ener

gy

isolatedSi atoms

Si latticespacing

Decreasing atom spacing

s

p

sp n = 3

N isolated Si-atoms

6N p-states total2N s-states total

(4N electrons total)

Elec

tron

ene

rgy

Crystalline Si N -atoms

4N allowed-states (Conduction Band)

4N allowed-states (Valance Band)

No states

4N empty states

2N+2N filled states

Elec

tron

ene

rgy Mostly

empty

Mostlyfilled

Etop

EcEg

Ev

Ebottom

Energy Band Formation (II)

Strongly bonded materials: small interatomic distances. Thus, the strongly bonded materials can have larger energy bandgaps than do weakly bonded materials.

Energy Bandgapwhere ‘no’ states exist

As atoms are brought closer towardsone another and begin to bond together, their energy levels mustsplit into bands of discrete levelsso closely spaced in energy, theycan be considered a continuum ofallowed energy.

Pauli Exclusion Principle

Only 2 electrons, of spin ± 1/2, can occupy the same energy state at

the same point in space.

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The 2N electrons in the 3s sub-shell and the 2N electrons in the 3p sub-shell undergo sp3 hybridization.

Energy Band Formation (Si)

Energy levels in Si as a function of inter-atomic spacing

The core levels (n=1,2) in Si are completely filled with electrons.

conduction band(empty)

valence band(filled)


Energy levels in Si as a function of inter-atomic spacing

N electrons filling half of the 2Nallowed states, as can occur in a Metal.


Energy band diagrams.

A completely empty band separated by an energy gap Eg from a band whose 2N states are completely filled by 2N electrons, representative of an Insulator.

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2 s Band

Overlapping energy bands

Electrons2 s2 p

3 s3 p

1 s 1sSOLIDATOM

E = 0

Free electronElectron Energy, E

2 p Band3s Band

Vacuumlevel

In a metal the various energy bands overlap to give a single band of energies that is only partially full of electrons. There are states with energies up to the vacuum level where the electron is free.

Typical band structures of Metal

Metals, Semiconductors, and Insulators

Electron energy, E

ConductionBand(CB)Empty ofelectrons at 0 K.

ValenceBand(VB)Full of electrons at 0 K.

Ec

Ev

0

Ec+χ

Covalent bond Si ion core (+4e)

A simplified two dimensional view of a region of the Si crystal showing covalent bonds.

The energy band diagram of electrons in the Si crystal at absolute zero of temperature.

Typical band structures of Semiconductor


Band gap = Eg

Chap. 2 Carrier Modeling2.2 Semiconductor models

Carrier Modeling

Chap. 2 Carrier Modeling

Atomic Bonding in SolidsIonic bondingMetallic bondingCovalent bondingVan der Waals bonding Mixed bonding



Electron Motion in Energy Band

Energy Band Diagram

Direct and Indirect Energy bandgap

Electrons and Holes

Effective Mass

Impurity Doping (p-, n-type Semiconductors)

Electron motion in an allowed band is analogous to fluid motion in a glass tube with sealed ends; the fluid can move in a half-filled tube just as electrons can move in a metal.


E = 0 E ≠ 0

Current flowing


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E = 0 E ≠ 0

No fluid motion can occur in a completely filled tube with sealed ends.



No flow can occur in either the completely filled or completely empty tube.

Fluid can move in both tubes if some of it is transferred from the filled tubeto the empty one, leaving unfilled volume in the lower tube.

Fluid analogy for a Semiconductor


Carrier Modeling

Chap. 2 Carrier Modeling

Atomic Bonding in SolidsIonic bondingMetallic bondingCovalent bondingVan der Waals bonding Mixed bonding




Energy Band Diagram

Direct and Indirect Energy bandgap

Electrons and Holes

Effective Mass

Impurity Doping (p-, n-type Semiconductors)

Carrier Flow for Metal


Carrier Flow for Semiconductors.mov

Carrier Flow for Semiconductor

Carrier Flow for Metals.mov


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Insulator Semiconductor Metal

Typical band structures at 0 K.


Ease of achieving thermal population of conduction band determines whether a material is an insulator, metal, or semiconductor.

Material Classification based on Size of Bandgap

Insulator Semiconductor

Metal


10610310010-310-610-910-1210-1510-18 109

Semiconductors Conductors

1012

AgGraphite NiCrTeIntrinsic Si

Degeneratelydoped Si

Insulators

Diamond

SiO2

Superconductors

PETPVDF

AmorphousAs2Se3

Mica

Alumina

Borosilicate Pure SnO2

Inorganic Glasses

Alloys

Intrinsic GaAs

Soda silica glass

Manyceramics

MetalsPolypropylene


Range of conductivities exhibited by various materials.

Conductivity (Ωm)-1


Problem 2.18 (text book)

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r

PE(r)

x

V(x)

x = Lx = 0 a 2a 3a

0aa

Surface SurfaceCrystal

PE of the electron around an isolated atom

When N atoms are arranged to form the crystal then there is an overlap of individual electron PEfunctions.

PE of the electron, V(x), inside the crystal is periodic with a period a.

The electron potential energy [PE, V(x)], inside the crystal is periodic with the same periodicity as that of the crystal, a. Far away outside the crystal, by choice, V = 0 (the electron is free and PE = 0).

Energy Band Diagram

E-k diagram, Bloch function.

Moving through Lattice.mov

Energy Band Diagram

E-k diagram, Bloch function.

[ ] 0)(222

2

=Ψ⋅−+Ψ xVEm

dxd e

h

Schrödinger equation

...3,2,1)()( =+= mmaxVxV

Periodic Potential

xkikk exUx )()( =Ψ

Periodic Wave functionBloch Wavefunction

There are many Bloch wavefunction solutions to the one-dimensional crystal each identified with a particular k value, say kn which act as a kind of quantum number.

Each ψk (x) solution corresponds to a particular kn and represents a state with an energy Ek.

Ek

kš /a–š /a

Ec

Ev

ConductionBand (CB)

Ec

Ev

CB

The E-k D iagram The Energy BandD iagram

Em pty ψ k

O ccupied ψ kh+

e-

Eg

e-

h+

hυ

V B

hυ

V alenceBand (V B)

The E-k curve consists of many discrete points with each point corresponding to a possible state, wavefunction Ψk (x), that is allowed to exist in the crystal.

The points are so close that we normally draw the E-k relationship as a continuous curve. In the energy range Ev to Ec there are no points [Ψk (x), solutions].

Energy Band Diagram

E-k diagram of a direct bandgap semiconductor

Energy Band Diagram

The bottom axis describe different directions of the crystal.

Si Ge GaAs

The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal.


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E

CB

k–k

Direct Bandgap

GaAs

E

CB

VB

Indirect Bandgap, Eg

k–k

kcb

Si

E

k–k

Phonon

Si with a recombination center

Eg

Ec

Ev

Ec

Ev

kvb VB

CB

ErEc

Ev

Photon

VB

In GaAs the minimum of the CB is directly above the maximum of the VB. direct bandgapsemiconductor.

In Si, the minimum of the CB is displaced from the maximum of the VB.indirect bandgap semiconductor

Recombination of an electron and a hole in Si involves a recombination center.

Energy Band Diagram

E-k diagram


Direct and Indirect Energy Band Diagram

(a) Direct transition with accompanying photon emission.(b) Indirect transition via defect level.


Energy Band

A simplified energy band diagram with the highest almost-filled band and the lowest almost-empty band.

valence band edge

conduction band edge

vacuum level

χ : electron affinity


Only the work function is given for the metal.Semiconductor is described by the work function qΦs, the electron affinityqχs, and the band gap (Ec – Ev).

Metals vs. Semiconductors

Pertinent energy levels

Metal Semiconductor

work functionwork function electron affinity


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e–hole

CB

VB

Ec

Ev

0

Ec+χ

Eg

Free e–hυ > Eg

Hole h+

Electron energy, E

hυ

Electrons and Holes

A photon with an energy greater then Eg can excitation an electron from the VB to the CB.

Each line between Si-Si atoms is a valence electron in a bond.When a photon breaks a Si-Si bond, a free electron and a hole in the Si-Sibond is created.

Generation of Electrons and Holes

Chap. 2 Carrier Modeling2.3 Carrier properties

Electrons: Electrons in the conduction band that are free to move throughout the crystal.

Holes: Missing electrons normally found in the valence band(or empty states in the valence band that would normally be filled).

Electrons and Holes

These “particles” carry electricity. Thus, we call these “carriers”

Effective Mass (I)

An electron moving in respond to an applied electric field.

E

E

within a Vacuum within a Semiconductor crystal

dtdmEqF v

0=−= dtdmEqF n

v∗=−=

It allow us to conceive of electron and holes as quasi-classical particles and to employ classical particle relationships in semiconductor crystals or in most device analysis.

Carrier Movement Within the Crystal

Density of States Effective Masses at 300 K

Ge and GaAs have “lighter electrons” than Si which results in faster devices

( m0 = Electron rest mass, 9.11x10-31 kg )

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Effective Mass (II)

Electrons are not free but interact with periodic potential of the lattice.Wave-particle motion is not as same as in free space.

Curvature of the band determine m*.m* is positive in CB min., negative in VB max.

Moving through Lattice.mov

ωh=Ekp h=Plank-Einstein-De Broglie

Relation

Energy Band Diagram

The bottom axis describe different directions of the crystal.

Si Ge GaAs

The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal.


Exercise Indicate where the effective mass of the electron is greatest and

least on the band diagram. The motion of electrons in a crystal can be visualized and described in a quasi-classical manner.

In most instancesThe electron can be thought of as a particle.The electronic motion can be modeled using Newtonian mechanics.

The effect of crystalline forces and quantum mechanical properties are incorporated into the effective mass factor.

m* > 0 : near the bottoms of all bandsm* < 0 : near the tops of all bands

Carriers in a crystal with energies near the top or bottom of an energy band typically exhibit a constant (energy-independent) effective mass.

` : near band edge

Effective Mass Approximation

constant2

2

=

dk

Ed

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Covalent Bonding Covalent Bonding


Band Occupation at Low Temperature (0 K)



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Without “help” the total number of “carriers” (electrons and holes) is limited to 2ni.

For most materials, this is not that much, and leads to very high resistance and few useful applications.

We need to add carriers by modifying the crystal.

This process is known as “doping the crystal”.

Impurity Doping

The need for more control over carrier concentration

Regarding Doping, ...


Concept of a Donor “Adding extra” Electrons


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Binding energies in Si: 0.03 ~ 0.06 eV

Binding energies in Ge: ~ 0.01 eV

Binding Energies of Impurity

Hydrogen Like Impurity Potential (Binding Energies)

Effective mass should be used to account the influence of the periodic potential of crystal.

Relative dielectric constant of the semiconductor should be used (instead of the free space permittivity).

: Electrons in donor atoms

: Holes in acceptor atoms


Donor Acceptor

Impurity Doping



Band diagram equivalent view


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e–As+

x

As+ As+ As+ As+

Ec

Ed

CB

Ev

~0.05 eV

As atom sites every 106 Si atoms

Distance intocrystal

Electron Energy

The four valence electrons of As allow it to bond just like Si but the 5th electron is left orbiting the As site. The energy required to release to free fifth- electron into the CB is very small.

Energy band diagram for an n-type Si dopedwith 1 ppm As. There are donor energy levels

below Ecaround As+ sites.


n-type Impurity Doping of Si

just


Energy band diagram of an n-type semiconductor connected to a voltage supply of V volts.

The whole energy diagram tiltsbecause the electron now has an electrostatic potential energy as well.

Current flowingV

n-Type Semiconductor

Ec

EF − eV

A

B

V(x), PE (x)

x

PE (x) = – eV

EElectron Energy

Ec − eV

Ev− eV

V(x)

EF

Ev


Energy Band Diagram in an Applied Field


Concept of a Acceptor “Adding extra” Holes

All regions of

materialare neutrally

charged

One less bondmeans

the acceptor is electrically satisfied.

One less bondmeans

the neighboring Silicon is left with

an empty state.


Hole Movement

Empty state is located next to the AcceptorAnother valence electron can fill the empty state located next tothe Acceptor leaving behind a positively charged “hole”.

The positively charged “hole” can move throughout the crystal.(Really it is the valance electrons jumping from atom to atom that creates the hole motion)



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Regionaround the“hole” hasone lesselectronand thus ispositivelycharged.

Hole Movement


Regionaround the“acceptor”hasone extraelectronand thus isnegativelycharged.



Band diagram equivalent view


B–h+

x

B–

Ev

Ea

B atom sites every 106 Si atoms

Distanceinto crystal

~0.05 eV

B– B– B–

h+

VB

Ec

Electron energy

p-type Impurity Doping of Si


Boron doped Si crystal. B has only three valence electrons. When it substitute for a Si atom one of its bond has an electronmissing and therefore a hole.

Energy band diagram for a p-type Si crystal doped with 1 ppm B. There are acceptor energy levels just above Ev around B- site. These acceptor levels accept electrons from the VB and therefore create holes in the VB.


Ec

Ev

EFi

CB

EFp

EFn

Ec

Ev

Ec

Ev

VB

Intrinsic semiconductors

In all cases, np=ni2

Note that donor and acceptor energy levels are not shown.

Intrinsic, n-Type, p-Type Semiconductors

Energy band diagrams

n-type semiconductors

p-type semiconductors


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CB

g(E)

E

EFp

Ev

Ec

EFn

Ev

Ec

CB

VB

Degenerate n-type semiconductor. Large number of donors form a band that overlaps the CB.

Heavily Doped n-Type, p-Type Semiconductors

Degenerate p-type semiconductor

Impurities forming a band


Impurity Doping


Donor / Acceptor Levels (Band Model)Ec

Ev

Donor Level

Acceptor Level

ED

EA

Donor ionization energy

Acceptor ionization energy

Ionization energy of selected donors and acceptors in siliconAcceptors

Dopant Sb P As B Al InIonization energy, E c -E d or E a -E v (meV) 39 45 54 45 67 160

Donors

Impurity Doping

Valence Band

Valence Band


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Impurity Doping

Position of energy levels within the bandgap of Si for common dopants.

Terminologydonor: impurity atom that increases n

acceptor: impurity atom that increases p

n-type material: contains more electrons than holes

p-type material: contains more holes than electrons

majority carrier: the most abundant carrier

minority carrier: the least abundant carrier

intrinsic semiconductor: n = p = ni

extrinsic semiconductor: doped semiconductor

Summary• The band gap energy is the energy required to free an electron from a covalent bond.– EG for Si at 300K = 1.12eV– Insulators have large EG; semiconductors have small EG

• Dopants in Si:– Reside on lattice sites (substituting for Si)– Group-V elements contribute conduction electrons, and are call

ed donors– Group-III elements contribute holes, and are called acceptors– Very low ionization energies (<50 meV)

ionized at room temperature

Dopant concentrations typically range from 1014 cm-3 to 1020 cm-3

Bandgap Energy: Energy required to remove a valence electron and allow it to freely conduct.Intrinsic Semiconductor: A “native semiconductor” with no dopants. Electrons in the conduction

band equal holes in the valence band. The concentration of electrons (=holes) is the intrinsic concentration, ni.

Extrinsic Semiconductor: A doped semiconductor. Many electrical properties controlled by the dopants, not the intrinsic semiconductor.

Donor: An impurity added to a semiconductor that adds an additional electron not found in the native semiconductor.

Acceptor: An impurity added to a semiconductor that adds an additional hole not found in the native semiconductor.

Dopant: Either an acceptor or donor.N-type material: When electron concentrations (n=number of electrons/cm3) exceed the hole

concentration (normally through doping with donors).P-type material: When hole concentrations (p=number of holes/cm3) exceed the electron

concentration (normally through doping with acceptors).Majority carrier: The carrier that exists in higher population (i.e. n if n>p, p if p>n)Minority carrier: The carrier that exists in lower population (i.e. n if n<p, p if p<n)Other important terms: Insulator, semiconductor, metal, amorphous, polycrystalline, crystalline

(or single crystal), lattice, unit cell, primitive unit cell, zincblende, lattice constant, elemental semiconductor, compound semiconductor, binary, ternary, quaternary, atomic density, Miller indices

Summary of Important terms and symbols

EE331-bandformation

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