Lecture03 EnergyBand Model

EEE 41 Lecture 3 The Energy Band Model

UP EEEI 1 EEE 41 Lecture 3 (Alarcon 2014)

Today •  Energy-‐band model •  Band-‐gap energy •  Density of states •  Doping

•  Read: Chapter 2

UP EEEI EEE 41 Lecture 3 (Alarcon 2014) 2

Silicon: From Atom to Crystal


•  Energy states in a Si atom à energy bands in a Si crystal

•  The highest nearly-‐filled band is the valence band •  The highest nearly-‐empty band is the conducKon band a = 2.35 Å = 0.235 nm

[conceptualphysics.in]

Energy Band Diagram


Ec

Ev electron

ene

rgy

distance

Simplified version of the energy band model, with •  BoVom edge of the conducKon band (Ec) •  Top edge of the valence band (Ev) •  Ev and Ec are separated by the band gap energy EG

Recap •  In a pure Si crystal •  ConducKon electrons and holes are formed in pairs •  Holes can be considered as posiKvely charged mobile parKcles which exist inside a semiconductor

•  Both holes and electrons can conduct current •  SpliCng of allowed atomic energy levels occur in a crystal •  SeparaKon between energy levels is small, thus we can consider them as bands with conKnuous energy levels •  Highest nearly-‐filled band is the valence band •  Lowest nearly-‐empty band is the conducKon band

•  Energy band diagram •  Shows Ev and Ec, and are separated by the band gap energy EG


Band Gap and Material ClassificaKon


[auris-‐new.univ-‐lemans.fr]

Ec

Ev

EG

•  Filled bands and empty bands do not allow current flow •  Insulators have large EG •  Semiconductors have small EG •  Metals have no band gap

•  ConducKon band is parKally filled

Measuring Band Gap Energy •  EG can be determined from the minimum energy (hν) of photons that are absorbed by the semiconductor


Band gap energies of selected semiconductors Semiconductor Ge Si GaAs

Band Gap (eV) 0.67 1.12 1.42

Ec

Ev

EG photon (hν > EG)

electron

hole

Density of States


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

gc (E) =mn

* 2mn* E −Ec( )

π 23

gv (E) =mp

* 2mp* Ev −E( )

π 23

E ≥ Ec

E ≤ Ev

Near the band edges:

mn* =1.18m0

mp* = 0.81m0

m0 = 9.1×10−31kg

EffecKve masses: 0 5 10 15

x 1050States per cm3 per eV

Density of States

Ev

Ec

1eV below Ev

1eV above Ec

Eg = 1.12eV

gc(E)gv(E)

Doping •  By subsMtuMng a Si atom with a special impurity atom, a conducMon electron or hole is created


[eere.energy.gov]

Example: •  Phosphorous is a Group V

element •  5 valence electrons •  Electron “donor”

•  Other donors: As, Sb

Doping Si with Donors •  Example: Adding Phosphorous (P) atoms to the Si crystal


[diranieh.com]

•  The loosely bound 5th valence electron of the Phosphorous atom “breaks free”

•  Becomes a mobile electron for current conducKon

•  The Phosphorous ion (P+) is immobile

Doping Si with Acceptors •  Example: Adding Indium (In) atoms to the Si crystal


[diranieh.com]

•  The Indium atom (Group III) accepts an electron from a neighboring Si atom

•  Results in a missing bonding electron or hole

•  The hole is free to roam around the lahce carrying current as a posiKve charge

•  The Indium ion (In-‐) is immobile

Donor / Acceptor Levels (Band Model)


Ec

Ev

ED

EA

Donor ionizaKon energy

Acceptor ionizaKon energy

IonizaKon energy of selected donors and acceptors in Silicon:

Donor Acceptor

Dopant Sb P As B Al In

IonizaKon Energy (meV) 39 45 54 45 67 160

Charge Carrier ConcentraKons •  ND: ionized donor concentraMon (cm-‐3) •  NA: ionized acceptor concentraMon (cm-‐3)

•  Charge neutrality condiMon:

•  At thermal equilibrium:


ND + p = NA + n

np = ni2 (“Law of Mass AcKon”)

n = ND − NA

2+

ND − NA

2"

#$

%

&'2

+ ni2

p = NA − ND

2+

NA − ND

2"

#$

%

&'2

+ ni2

Note: •  Carrier concentraKons

depend on net dopant concentraKon |NA – ND|

N-‐Type Material •  Assume ND >> NA and (ND – NA) >> ni


P-‐Type Material •  Assume NA >> ND and (NA – ND) >> ni


Terminology •  Donor: impurity atom that increases n •  Acceptor: impurity atom the 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 T = 300K is 1.12eV •  Insulators have large EG, semiconductors have small EG

•  Dopants in Si •  Reside on lahce sites (subsKtuKng for Si atoms) •  Group V elements contribute conducKon electrons (donors) •  Group III elements contribute holes (acceptors) •  Very low ionizaKon temperatures (ionized at room temperature) •  Typical dopant concentraKons: 1014 cm-‐3 to 1020 cm-‐3 •  Si density: 5 x 1022 cm-‐3


Next MeeKng •  Thermal Equilibrium •  Carrier DistribuMons and ConcentraMons •  The Fermi Level


Lecture03 EnergyBand Model

Documents

band gap conduckon band

energy band model

lled band

band edges

conduckon band ec

eeei eee

valence band ev ev

energy states