Lecture 2 OUTLINE Semiconductor Fundamentals (cont’d) – Energy band model – Band gap energy...

Lecture 2

OUTLINE• Semiconductor Fundamentals (cont’d)

– Energy band model– Band gap energy– Density of states– Doping

Reading: Pierret 2.2-2.3, 3.1.5; Hu 1.3-1.4,1.6, 2.4

Potential Energy Profiles

Discrete allowed energy levels

When two atoms are in close proximity, the upper energy levels are shifted to bonding and anti-bonding levels.

Lecture 2, Slide 2EE130/230A Fall 2013

V(r)1/r is mostly a coulombic potential btwn the positive nucleus & negative electrons.

N atoms

1 atom 2 atoms

many bonding/anti-bonding levels

Energy states in Si atom energy bands in Si crystal

Si: From Atom to Crystal

• The highest nearly-filled band is the valence band• The lowest nearly-empty band is the conduction band


R.F. Pierret, Semiconductor Fundamentals, Figure 2.5

Energy Band Diagram

• Simplified version of energy band model, showing only the bottom edge of the conduction band (Ec) and the top edge of the valence band (Ev)

• Ec and Ev are separated by the band gap energy EG

Ec

Ev

ele

ctro

n e

ne

rgy

distance


Electrons and Holes (Band Model)• Conduction electron = occupied state in the conduction band• Hole = empty state in the valence band• Electrons & holes tend to seek lowest-energy positions

Electrons tend to fall and holes tend to float up (like bubbles in water)

Lecture 2, Slide 5

Incr

easi

ng h

ole

ener

gy

Incr

easi

ng e

lect

ron

ener

gy

Ec

Ev

electron kinetic energy

hole kinetic energyreferencecP.E. EE

Ec represents the electron potential energy.

EE130/230A Fall 2013

Electrostatic Potential, Vand Electric Field, E

• The potential energy of a particle with charge -q is related to the electrostatic potential V(x):

)(1

creference EEq

V

qVP.E.

Lecture 2, Slide 6

dx

dE

qdx

dV c1

• Variation of Ec with position is called “band bending.”

0.7 eV

EE130/230A Fall 2013

• EG can be determined from the minimum energy of photons that are absorbed by the semiconductor

Measuring the Band Gap Energy

Band gap energies of selected semiconductors

Lecture 2, Slide 7

Semiconductor Ge Si GaAsBand gap energy (eV) 0.67 1.12 1.42

Ec

Ev

photonh > EG

EE130/230A Fall 2013

g(E)dE = number of states per cm3 in the energy range between E and E+dE

Near the band edges:

Density of States

28)(

2/3*,3 cDOSnc EEm

hEg

for E Ec

Lecture 2, Slide 8

Ec

Ev

dE

E

density of states, g(E)Ec

Ev

for E Ev 28

)(2/3*

,3EEm

hEg vDOSpv

Si Ge GaAs

mn,DOS*/mo 1.08 0.56 0.067

mp,DOS*/mo 0.81 0.29 0.47

Electron and hole density-of-states effective masses

EE130/230A Fall 2013

• When an electron is moving inside a solid material, the potential field will affect its movement.

• For low kinetic energy where p is the crystal momentum

i.e. a conduction electron behaves as a particle but with an effective mass m*

Schrödinger equation:

E : total energy : wave functionħ : reduced Planck constant

Effective Mass, m*

Vm

E 2

0

2

2

*2

2

m

pE


EG and Material Classification

• Neither filled bands nor empty bands allow current flow

• Insulators have large EG

• Semiconductors have small EG

• Metals have no band gap (conduction band is partially filled)

silicon

Lecture 2, Slide 10

EG = 1.12 eVEG = ~ 9 eV

silicon dioxideEc

Ev

Ec

Ev

metal

Ev

Ec

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• By substituting a Si atom with a special impurity atom (Column V or Column III element), a conduction electron or hole is created.

Doping

Donors: P, As, Sb

ND ≡ ionized donor concentration (cm-3)

Lecture 2, Slide 11

Acceptors: B, Al, Ga, In

NA ≡ ionized acceptor concentration (cm-3)

EE130/230A Fall 2013http://inventors.about.com/library/inventors/blsolar5.htm

Doping Silicon with a DonorExample: Add arsenic (As) atom to the Si crystal

The loosely bound 5th valence electron of the As atom “breaks free” and becomes a mobile electron for current conduction.


As

Si

Si SiSi

SiSi

SiSi

Doping Silicon with an Acceptor

The B atom accepts an electron from a neighboring Si atom, resulting in a missing bonding electron, or “hole”. The hole is free to roam around the Si lattice, carrying current as a positive charge.

Lecture 2, Slide 13

Example: Add boron (B) atom to the Si crystal

EE130/230A Fall 2013

B

Si

Si SiSi

SiSi

SiSi

Solid Solubility of Dopants in Si

ATOMS PER CUBIC CENTIMETER

F. A. Trumbore, Bell Systems Technical Journal, 1960


Doping (Band Model)

Ionization energy of selected donors and acceptors in silicon

Lecture 2, Slide 15

Donors AcceptorsDopant Sb P As B Al InIonization energy (meV)Ec-ED or EA-Ev

39 45 54 45 67 160

Ec

Ev

Donor ionization energy

ED

EA

Acceptor ionization energy

EE130/230A Fall 2013

Dopant Ionization



Charge-Carrier Concentrations

Charge neutrality condition: ND + p = NA + n

At thermal equilibrium, np = ni2 (“Law of Mass Action”)

Note: Carrier concentrations depend on net dopant concentration!


n-type Material (n > p)

ND > NA (more specifically, ND – NA >> ni):


p-type Material (p > n)

Lecture 2, Slide 19

NA > ND (more specifically, NA – ND >> ni):

EE130/230A Fall 2013

Carrier Concentration vs. Temperature



Terminology

donor: 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 semiconductorsuch that majority carrier concentration = net dopant concentration


Summary

• Allowed electron energy levels in an atom give rise to bands of allowed electron energy levels in a crystal.

– The valence band is the highest nearly-filled band.– The conduction band is the lowest nearly-empty band.

• The band gap energy is the energy required to free an electron from a covalent bond.

– EG for Si at 300 K = 1.12 eV

– Insulators have large EG; semiconductors have small EG


Summary (cont’d)

• Ec represents the electron potential energy

Variation in Ec(x) variation in electric potential V

Electric field

• E - Ec represents the electron kinetic energy

dx

dE

dx

dE vc


Summary (cont’d)

• Dopants in silicon:– Reside on lattice sites (substituting for Si)

– Have relatively low ionization energies (<50 meV) ionized at room temperature

– Group-V elements contribute conduction electrons, and are called donors

– Group-III elements contribute holes, and are called acceptors

Dopant concentrations typically range from 1015 cm-3 to 1020 cm-3


Lecture 2 OUTLINE Semiconductor Fundamentals (cont’d) – Energy band model – Band gap energy...

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