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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

2 atoms

1 atom

Discrete allowed energy levels

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

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

N atoms

many bonding/anti-bonding levels

EE130/230A Fall 2013

Lecture 2, Slide 2

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 states in Si atom energy bands in Si crystal

EE130/230A Fall 2013

Lecture 2, Slide 3

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

electron energy

Ev

distance

EE130/230A Fall 2013

Lecture 2, Slide 4

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)

electron kinetic energy

Ec represents the electron potential energy.

Ec

Increasing electron energy

Increasing hole energy

Ev

hole kinetic energy

EE130/230A Fall 2013

Lecture 2, Slide 5

Electrostatic Potential, Vand Electric Field, E

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

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

0.7 eV

EE130/230A Fall 2013

Lecture 2, Slide 6

Measuring the Band Gap Energy

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

Ec

photon

hn > EG

Ev

Band gap energies of selected semiconductors

EE130/230A Fall 2013

Lecture 2, Slide 7

Density of States

E

dE

Ec

Ec

density of states, g(E)

Ev

Ev

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

Near the band edges:

Electron and hole

density-of-states effective masses

for E Ec

for EEv

EE130/230A Fall 2013

Lecture 2, Slide 8

Effective Mass, m*

- 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

Y: wave function

ħ : reduced Planck constant

EE130/230A Fall 2013

Lecture 2, Slide 9

EG and Material Classification

silicon dioxide

silicon

metal

- 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)

Ec

Ec

Ev

EG = 1.12 eV

EG = ~ 9 eV

Ev

Ec

Ev

EE130/230A Fall 2013

Lecture 2, Slide 10

Acceptors: B, Al, Ga, In

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

ND ≡ ionized donor concentration (cm-3)

NA ≡ ionized acceptor concentration (cm-3)

http://inventors.about.com/library/inventors/blsolar5.htm

EE130/230A Fall 2013

Lecture 2, Slide 11

Doping Silicon with a Donor

Example: Add arsenic (As) atom to the Si crystal

Si

Si

Si

Si

As

Si

Si

Si

Si

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

EE130/230A Fall 2013

Lecture 2, Slide 12

Doping Silicon with an Acceptor

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

Si

Si

Si

Si

B

Si

Si

Si

Si

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.

EE130/230A Fall 2013

Lecture 2, Slide 13

Solid Solubility of Dopants in Si

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

Atoms per cubic centimeter

EE130/230A Fall 2013

Lecture 2, Slide 14

Doping (Band Model)

Donor ionization energy

Ec

ED

EA

Ev

Acceptor ionization energy

Ionization energy of selected donors and acceptors in silicon

EE130/230A Fall 2013

Lecture 2, Slide 15

Dopant Ionization

R.F. Pierret, Semiconductor Fundamentals, Figure 2.13

EE130/230A Fall 2013

Lecture 2, Slide 16

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!

EE130/230A Fall 2013

Lecture 2, Slide 17

n-type Material (n > p)

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

EE130/230A Fall 2013

Lecture 2, Slide 18

p-type Material (p > n)

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

EE130/230A Fall 2013

Lecture 2, Slide 19

Carrier Concentration vs. Temperature

R.F. Pierret, Semiconductor Fundamentals, Figure 2.22

EE130/230A Fall 2013

Lecture 2, Slide 20

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 semiconductor

such that majority carrier concentration = net dopant concentration

EE130/230A Fall 2013

Lecture 2, Slide 21

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

EE130/230A Fall 2013

Lecture 2, Slide 22

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

EE130/230A Fall 2013

Lecture 2, Slide 23

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

EE130/230A Fall 2013

Lecture 2, Slide 24

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