C1 Homogeneous Semiconductor - Revision [compatibility Mode] 4n3v6a

CHAPTER 2 (ANDERSON) Homogeneous Semiconductor

SEMICONDUCTOR FUNDAMENTALS COVALENT BONDINGS Neighboring atoms share their outermost electrons with each other. Eg. : Silicon atom share the electrons in the third shell with another 4 Si atoms in order to stabilize.

SEMICONDUCTOR FUNDAMENTALS ENERGY BANDS

Discrete states of the isolated atoms broadened into energy bands. Eg. : In Si case, the third shell splits into two bands in crystalline Si. The lower band is called the valence band, and the upper band is called the conduction band, which is split off from the valence band by an energy of 1.12 eV (band gap)

SEMICONDUCTOR FUNDAMENTALS ENERGY BANDS Electron affinity

Ionization energy Conduction band edge (lowest energy)

Energy gap

Valence band edge (highest energy) • Electron affinity : Energy difference between the vacuum level and the vacant state of the lowest energy. • Ionization energy : minimum energy required to excite an electron from the top of valence band to the vacuum level. • Energy gap / band gap : minimum energy required to excite an electron from valence band to conduction band.

SEMICONDUCTOR FUNDAMENTALS ENERGY BANDS

Electrons need to be at the conduction band in order to conduct current. Conductivity of these materials depends on the width of their energy gap.

BAND STRUCTURE OF SEMICONDUCTOR BANDGAP & MATERIAL CLASSIFICATION Metals – Insulators – Semiconductors: classified based on the band gap Metals : very narrow/ overlapping EV and EC Insulators: very wide band gap. Semiconductors : EG in between metal and insulator

Example of semiconductor at room temperature: GaAs EG = 1.42 eV Si EG = 1.12 eV Ge EG = 0.66 eV

BAND STRUCTURE OF SEMICONDUCTOR ENERGY BAND MODEL

• Energy band diagram shows the bottom edge of conduction band (CB), Ec and the top edge of valence band (VB) EV. • Ec and Ev are separated by the band gap energy Eg. BANDGAP • EG = Energy required to cross the band gap in the energy band model = Energy required to break a bond in the bonding model • EG = EC -EV

BAND STRUCTURE OF SEMICONDUCTOR ELECTRONS IN CONDUCTION BAND (CB) , HOLES IN VALENCE BAND (VB) • Electrons must be in the CB to be carriers. Electrons in VB are not carriers. • Empty states or holes in VB are carriers. Empty states in CB are not holes, thus, not carriers.

Electrons in CB

Holes in VB

INTRINSIC SEMICONDUCTOR • “Intrinsic” – extremely pure semiconductor material. The electrons and holes of intrinsic semiconductors are from the semiconductor material itself. • : n0 = number of electrons/cm3 p0 = number of holes/cm3 • Intrinsic material in equilibrium: n0 = p0 = ni Electron and hole created simultaneously. • Intrinsic material electron concentration, ni, (/cm3) represents the number of electrons (/cm3) that are carriers. ni are the number of electrons /cm3 that are excited by temperature and have entered the conduction band from the valence band. • ni typical values at room temperature: 2 x 106 / cm3 for GaAs 1 x 1010/ cm3 for Si 2 x 1013/ cm3 for Ge

INTRINSIC SEMICONDUCTOR • At 0 K, no excitation • Bonding model shows all bonds not broken •Energy band model shows valence band completely filled Energy Band Model

Bonding Model

INTRINSIC SEMICONDUCTOR ELECTRONS AS CARRIERS

When there is excitation According to bonding model: Si-si bond is broken. Electron is freed and free to wander within crystal. Electron is now a carrier. According to energy band model: Electron in valence band acquires enough energy to overcome the band gap and jumps into the conduction band. Electrons in the conduction band are carriers. When there is a free electron, According to bonding model: nearby shared electrons fill in the broken bond, leaving a broken bond. The moving broken bond hole. According to energy band model: when electrons leave the valence band, an empty state is left and are filled in by nearby valence electron. Empty states in the valence band are holes.

INTRINSIC SEMICONDUCTOR ELECTRON AND HOLE DISTRIBUTION WITH ENERGY

Ef = Ei

n=p

Intrinsic Material •Distribution is zero at band edges •Reaches peak value close to band edges then decays to zero into the band •Near midgap, electrons and holes approximately equal

INTRINSIC SEMICONDUCTOR The intrinsic concentration is too small a number to cause satisfactory current flow. Consider silicon: Just to illustrate • ni 1 x 1010 means 1 x 1010 broken bonds • Volume density 5 x 1022 atoms/ cm3 • Each atom has 4 bonds • 1 cm3 has 2 x 1023 bonds, of which 1 x 1010 are broken • Less than 1 broken bond for each 1013 bond To achieve a desired carrier concentration, impurities are put into the materials making it extrinsic.

EXTRINSIC SEMICONDUCTOR • Doping: process of adding controlled amount of specific impurity atoms to increase n or p • Dopants to increase n are called DONORS. • Dopants to increase p are called ACCEPTORS. • For group IV s/c material, donors from group V, acceptors from group III

EXTRINSIC SEMICONDUCTOR n-type s/c (DONORS impurities)

Bonding Model : •Column V – 5 valence electrons. 4 fits, 1 weakly bound. •At room temp, weakly bound electron is readily freed as a carrier •Electron concentration increases. •Hole concentration does not increase. •No increase, no bonds broken. •Donor ion is “ionized”. +vely charged.

EXTRINSIC SEMICONDUCTOR n-type s/c (DONORS impurities)

Energy Band Model : •Column V element adds ED close to conduction band • At T=0 K, ED filled with electrons from donor • At room temp, electrons at ED have sufficient energy to (all) enter the conduction band. • Electrons in conduction band are carriers hence n increases • No increase in p. Electrons from ED do not leave empty states in the valence band.

EXTRINSIC SEMICONDUCTOR p-type (ACCEPTORS impurities)

Bonding Model : •Column III – 3 electrons. 3 fits in. One more. Missing bond. •At room temp, nearby valence electrons fill in the bond, leaving a broken bond. •Hole concentration increases. •Electron concentration does not increase. No increase. Broken bond does not free an •electron. •Acceptor ion “ionized”. Becomes –vely charged. Note that the electrons and holes are carriers and are free to move within the crystal. However, the donor ions and acceptor ions are fixed and are not free to move – the ions are not carriers.

EXTRINSIC SEMICONDUCTOR p-type (ACCEPTORS impurities)

Energy Band Model : •Column III element adds EA close to valence band •At T=0 K, EA empty •At room temp, electrons in valence band have sufficient energy to fill up EA but this electrons are not carriers. •However, the electrons filling up EA leaves behind empty states in the valence band thus creating holes i.e. p increase. •No increase in n. Electrons in EA are not carriers. Only electrons inconduction band are considered carriers.

EXTRINSIC SEMICONDUCTOR HOW DO ELECTRONS AND HOLES POPULATE THE SEMICONDUCTOR Allowed states refer to where the electrons are allowed to be along the energy levels

Density of states S(E) show how the allowed states are distributed in the valence band and in the conduction band. S(E) represents number of allowed states per unit volume per unit energy around energy E ie. the density of states at energy E In the bandgap, the density of states are zero. Presence of states does not mean presence of electrons Fermi -Dirac probability function f(E) is the probability that an electron occupies a given state at energy E in an allowed band.

How many states exist at energy E

S(E)

f(E)

Probability the state at energy E will be filled

EXTRINSIC SEMICONDUCTOR DENSITY STATE OF FUNCTIONS The equations representing the two graphs are:

From the graph, In conduction band, S(E) is zero at EC In valence band, S(E) is zero at EV Deeper into the band, S(E) increases with square root of energy Now that we know how and where the allowed states are situated, we need to know probability of the existence of electrons and holes at the conduction band, valence band and bandgap. Only then can we know how the electrons and holes are distributed.

EXTRINSIC SEMICONDUCTOR FERMI-DIRAC STATISTIC Now that we know the number of available states at each energy, how do the electrons occupy these states? We need to know how the electrons are “distributed in energy”. Again, Quantum Mechanics tell us that the electrons follow the “Fermi-Distribution Function”

EXTRINSIC SEMICONDUCTOR FERMI-DIRAC STATISTIC E5 EF E4 E3 E2 E1

As T=0 K :

As T=0 K :

Consider when E<Ef [(E-Ef)/kT]→exp(-∞)=0 resulting f(E)=1

• A quantum state being occupied in energy level E1-E4 is unity • E5 is zero • Ef must be above E4 and less than E5

Consider when E>Ef [(E-Ef)/kT]→exp(∞)= ∞ resulting f(E)=0

Fermi level EF is not an allowed and existing energy level. It is conceptual, and it is created to allow mathematical calculation of the electron and hole distribution.

EXTRINSIC SEMICONDUCTOR FERMI-DIRAC STATISTIC E5 E4 E3 E2 E1

When T > 0K •Electron gain thermal energy so that some electrons can jump to higher energy level. •Two electron from E4 have gained enough energy to jump to E5 and one from E3 to E4 •As temperature change the distribution of electrons versus energy changes. •E=Ef resulting f(E)= 1/2

EXTRINSIC SEMICONDUCTOR ELECTRON AND HOLE DISTRIBUTION WITH ENERGY

Extrinsic: n-type material •Distribution is zero at band edges •Reaches peak value close to band edges then decays to zero into the band •Above midgap, electrons more than holes

Extrinsic: p-type material •Distribution is zero at band edges •Reaches peak value close to band edges then decays to zero into the band •Below midgap, holes more than electrons

EXTRINSIC SEMICONDUCTOR ELECTRON AND HOLE DISTRIBUTION WITH ENERGY The electron and hole distribution n(E)= S(E)f(E) p(E)=S(E)[1-f(E))=S(E)fp(E) The total number of electron in CB / hole in VB, at equilibrium is :

S

S

EXTRINSIC SEMICONDUCTOR ELECTRON AND HOLE DISTRIBUTION WITH ENERGY The total number of electron in CB / hole in VB, in of Nc nd Nv

•| no= equilibrium electron concentration/ cm3 •| NC= “Effective” density of CB states •| NV= “Effective” density of VB states •| po= equilibrium hole concentration/ cm3 •| EC-EF= distance of Fermi level from edge of CB •| EV-EF= distance of Fermi level from edge of VB

o

o

: The closer Ef moves up to Ec, the larger n is; The closer Ef moves down to Ev, the larger p is.

CARRIER CONCENTRATION For INTRINSIC semiconductor :

no = po = ni and Ei = Ef then :

n0 =

p0 =

NON-For NON DEGENERATE semiconductor :

Ei =

( Ec + Ev ) kT N v + ln 2 2 Nc

Ei =

( Ec + Ev ) 3  m * dsh  + kT ln  2 4  m * dse 

CARRIER CONCENTRATION

DEGENERATE & NON-DEGENERATE nopo=ni2

Nondegenerate semiconductor

Known as Law of Mass Action •A semiconductor is said to be nondegenerate if the probability of any state in the conduction band being occupied by an electron is small, or the probability of any state in the valence band being occupied by an hole is small. •Then implies that the Fermi level is at least 2.3kt below the conduction band edge and at least 2.3kT above the valence band edge. *2.3kT or 3 kT

CARRIER CONCENTRATION NON--DEGENERATE semiconductor : For NON For n-type semiconductor, as long as ND >> ni, we can make the approximation that no ≅ND when ND>>ni Similar to p-type material, as long as NA >> ni, we can make the approximation that

 no  E f − Ei = kT ln   ni  No increase , so does Ef - Ei

po ≅NA when NA>>ni Wheter the material is doped n-type or p-type, however, as long as the doping level is not degenerate, it is still true that nopo=ni2

CARRIER CONCENTRATION Carrier concentrations at high temperature (total ionization) degenerate semiconductor

non-

Space charge neutrality : p0+ND=no+NA

po-no+ND-NA=0

For n-type:

For p-type:

majority carrier :

 N D − N A  N D − N A  2 no = +   + ni  2 2    2

1 2

majority carrier : 2  N A − N D  N A − N D  2 po = +   + ni  2 2   

Minority carrier for both n and p-type using : nopo= ni2

1 2

CARRIER CONCENTRATION

CARRIER CONCENTRATION Carrier concentrations at low temperature semiconductor Space charge neutrality : p0+ND +pA =no+NA + nD po+ pA – no - nD+ND-NA=0 pA= hole still attached to the acceptor nD= un-ionized donor

non-degenerate

DEGENERATE & NON-DEGENERATE N: D:

Non-degenerate s/c dopant atoms are far apart enough to assume there is no interaction between dopant atoms. The states for the higher donor levels can overlap if the doping concentration is high enough (dopant atoms close enough together).

N: D:

Condition for non-degenerate:EC-EF>2.3kT and EF-EV>2.3kT. When EC-EF and EF-EV is less than 2.3kT, the semiconductor is “heavily” doped @ degenerate.

N:

Maximum non-degenrate doping for silicon: ND ~1.6x1018 / cm3 NA ~ 9.1x1017 / cm3

D:

Degenerate s/c are heavily doped, often noted by n+ or p+. The heavier a s/c is doped, the closer EF gets to the band edges. If the Fermi level is closer than 2.3kT to band edges than almost all the formulas that has been introduced for carrier concentration is not applicable.

CARRIER CONCENTRATION For COMPENSATED semiconductor : Semiconductor that contain both donors and acceptors are called “compensated semiconductor”. Uncompensated semiconductor refers to a material with single doping, or single dominant doping. For uncompensated semiconductor, •| if doped with ND, n0=ND •| if doped with NA, p0=NA For compensated semiconductor, •| If ND>NA, n-type compensated •| If NA>ND, p-type compensated

no=ND-NA

ND>NA

n type

Majority

po=NA-ND

NA>ND

p type

carrier

nopo=ni2

Minority carrier

It is assumed that doping concentration is much greater than the intrinsic carrier concentration. NDNA or NA-ND.>>ni Example: In device fabrication of BJT ~ n-type is next doped with p then next with n.