Semiconductors

Chapter 8 of Solymar

Introduction

• Classification of solids by conductivity– Conductors: metals

– Insulators: dielectrics

– Semiconductors

• Semiconductors led to the information age through transistors and integrated circuits

• Electrical properties of semiconductors can be controlled by the impurities we put in– Purification a prerequisite

– Purity in ppb range

– Doping techniques required

Intrinsic Semiconductors

• Definition– Intrinsic: pure, undoped– Extrinsic: doped, with impurity

• Silicon as an example– Silicon has 4 valence electrons

• They form 4 covalent bonds with 4 neighboring silicon atoms• The 4 bonds are equally spaced, leading to a tetrahedral structure

– In band theory• All 4 valence electrons are in the valence band at 0 K, so the valence

band is full and the conduction band is empty• A 1.1 eV energy gap between valence band and conduction band• To excite electrons from valence band to conduction band, extra

energy is needed: thermal energy or optical energy

Questions

• What is the number of electrons in the conduction band at temperature T?

• What is the number of holes in the valence band at temperature T?

• How to do it?

Density of states Z(E)Occupancy probability F(E)

Number of electrons at energy E Z(E)F(E)

Total number of electrons ∫Z(E)F(E)dE

Density of States

• Assume electrons in the conduction band behave like free electrons, but with an effective mass m*

– Density of states– Reference at top of valence band (Fig. 8.1)

• For holes• Total number of electrons

– The upper and lower limits

21ge )EE(C)E(Z

)kkk(m2

E 2z

2y

2x

2

323*ee h/)m2(4C

21h )E(C)E(Z 323*

hh h/)m2(4C

?

?e dE)E(F)E(ZN

From bottom of the conduction bandto top of the conduction band

Simplifications

• F(E) decays exponentially when E >> EF

– In conduction band, E >> EF

• We don’t know the top of the conduction band– Since F(E) is so small deep in conduction band, we can replace top

of conduction band with

• Ce ( E – Eg )1/2 only valid for the bottom of the conduction band– Since F(E) is so small deep in conduction band, we can use the

equation without too much error

• F(E) is difficult to integrate– Since E >> EF in conduction band, E – EF >> kT

V.B.

C.B.

EF

]kT/)EE(exp[~1]kT/)EEexp[(

1)E(F F

F

Derivation

• Z(E), F(E), and Z(E)F(E) as functions of E (Fig. 8.2)

• With all the simplifications

– With a new variable

– where– This is the number of electrons in conduction band

kT

EE g

gE F

21gee dE]kT/)EE(exp[)EE(CN

]kT/)EE(exp[N

d)exp(]kT/)EE(exp[)kT(CN

FgC

0

21Fg

23ee

232*eC )h/kTm2(2N

Hole Concentration

• Integration from - to 0

– Where

• For an intrinsic semiconductor

)kT/Eexp(NN FVh

232*hV )h/kTm2(2N

he NN

Intrinsic Semiconductor

• In an intrinsic semiconductor, Ne = Nh

– From which

• kT small &

– EF is roughly in the middle of the band gap in an intrinsic semiconductor

)kT/Eexp(N]kT/)EE(exp[N FVFgC

2

E~)

mm

ln(kT43

2

EE g

*e

*hg

F

1~mm

*e

*h

)kT2/Eexp(NN ghe

Extrinsic Semiconductors

• What happens if a small amount of impurity atoms is added to an intrinsic semiconductor

• If a small amount of Group V atoms is introduced into Si– Antimony (Sb), arsenic (As), or phosphorous (P)

– Each Group V atom replaces a Si atom

– 4 of the valence electrons of Group V atoms are used for covalent bonding (Fig. 8.3)

– The 5th lone electron becomes loosely bound to the impurity atom, since it has 9 electrons in outer shell

– The lone electron can easily escape the impurity atom and becomes a free electron

Donor Atom

• The energy required to free the lone electron from the impurity atom into the conduction band is small– The band gap represents the energy required to free a

valence electron from a Si atom into conduction band

– Since Group V atoms donate electrons, they are called donors, and their energy level ED is called the donor level

V.B.

C.B.

ED

The Hydrogen Model

• The energy of an electron in a hydrogen atom is

• A donor atom is similar to a hydrogen atom– It loses and gains one electron– When it loses the lone electron, it has 1 unit of positive charge– When we apply the hydrogen model

– Donor level is at

– Table 8.1 Energy levels of donors and acceptors

22o

4

h8me

E

mm* os

eV05.0~h8em

EE 22s

4*

Dg

Group III Impurity

• If a small amount of Group V atoms is introduced into Si– Boron (B), aluminum (Al), indium (In)

– Each Group III atom replaces a Si atom

– All 3 valence electrons of Group III atoms are used for covalent bonding (Fig. 8.4)

– There is a missing electron, since it has 7 electrons in outer shell

– It is called a hole

– An electron in a neighboring bond may jump over, so the hole moves to the next bond

– The hole is weakly bound to the impurity atom

Acceptor Atom

• In the band theory, this hole level is slightly higher than the energy level at the Si atom, but– It is still low enough for other electrons to easily gain

excess energy to jump into it

– Since Group III atoms accept electrons, they are called acceptors, and their energy level EA is called the acceptor level

V.B.

C.B.

EA

Terminology

• A semiconductor can have either donors and acceptors– When electrons are the main charge carriers, it is called

n-type (negative type)

– When holes are the main charge carriers, it is called p-type (positive type)

Compensation

• By accident or purposely, a semiconductor can have both donors and acceptors– Electrons from donors will first populate the acceptors

and then go into the conduction band

– If it has more donor atoms than acceptor atoms, there will be electrons going into the conduction band: it’s n-type

– If it has more acceptor atoms than donor atoms, it’s p-type

– Donor atoms fill up acceptor atoms

– It’s called compensation V.B.

C.B.

ED

EA

• • • • • • • •• • • •••

• •

Charge Neutrality

• How to determine EF?• Charge neutrality

– The semiconductor is electrically neutral, so the total amount of positive charge is equal to the total amount of negative charge

– NA-: number of ionized acceptors

– ND+: number of ionized donors

• How to find NA- and ND

+

– NA-: number of occupied acceptor states

– ND+: number of unoccupied donor states

• Ne and Nh are given by

• Replace them in charge neutrality equation and solve

DhAe NNNN

)E(FNE AAA

)]E(F1[NE DDD

]kT/)EE(exp[NN FgCe )kT/Eexp(NN FVh

Simplification

• Usually in an n-type semiconductor, Ne >> Nh, ND+ >> NA

-

– EF as a function of ND and T– Not valid for intrinsic semiconductor, Ne ~ Nh

– Not valid for lightly doped semiconductor, ND ~ NA

– Not valid when E – EF ~ kT– If (EF – ED)/kT is a large negative number

EF increases with ln(ND)

De NN1

DFDFgC ]}kT/)EEexp[(1{N]kT/)EE(exp[N

DF N)kT/Eexp(ttancons

Example

• For Si, Eg = 1.15 eV, Eg – ED = 0.049 eV, ND = 1022 m-3

– Introducing

– Equation

becomes– Solve for EF

EF = 0.97 eV– Considerably above the middle of Eg, so E – EF >> kT

)kTE

exp(x F

)Bx1/(NAx D DF N)kT/Eexp(ttancons

Variation of EF w/ Temperature

• Eg and ED are both temperature dependent, but let’s for now ignore their effects

• If ND = 1021 m-3, density of lattice atoms 1028 m-3, Eg = 1 eV, Eg – ED = 0.05 eV– At low temperatures, all the conduction electrons come from

donors and little contribution from lattice atoms (Fig. 8.5)

Carrier concentration

The semiconductor seems to have a band gap of Eg – ED

– At high temperatures, all donors are ionized and many electrons jump from valence band to conduction band, Ne >> ND, and the semiconductor behaves like an intrinsic one

Carrier concentration

2/)EE(E DgF ]kT2/)EE(exp[ Dg

2/EE gF )kT2/Eexp( g

p-Type Semiconductor

• Fig. 8.6• Question

– Why the band gap decreases when temperature increases?

Scattering

• The conductivity due to one type of charge carriers is

– m needs to be effective mass me*– Ne can be calculated is the mean free time– What determine ?

e

2

Nme

Scattering Mechanisms

• Lattice atoms– The higher the temperature, the larger the amplitude of atomic

vibration, the higher the scattering probability, the smaller the mean free time

– Thermal mean free time

• Ionized impurities– Scattering occurs when the electrostatic energy is comparable to

the thermal energy

– Scattering cross section– Impurity mean free time

21thDth Tv)vv(l 21

th )m/kT3(v 2121

thermal T~lT

2/kT3r4/Ze s2 kT6/Zer 2

s

22

2ss )

kT6Ze

(1

rS

23211s

21impurity T~TS~lT

Overall Mean Free Time

• Overall mean free time

– High t is desirable for high mobility, which means pure material and low temperature

• Effect of doping (Fig. 8.7)– Mobility decreases with doping– Smaller effective mass m* means higher mobility– The conductivity with both electrons and holes

impuritythermal

111

*h

hh2

*e

ee2

mNe

m

Ne

Mass Action Law

• The product of electron concentration and hole concentration

– It is a constant for a given semiconductor at a given temperature– For an intrinsic semiconductor, Ne = Nh = Ni

– For extrinsic semiconductors, when Ne increases, Nh must decrease– When Nh increases, Ne must decrease– When Ne increases, the recombination rate for holes increases, so

Nh decreases– Law of mass action

)kT/Eexp()mm()hkT2

(4NN g23*

h*e

32he

2ihe NNN )kT2/Eexp(NNN gVCi

III-V & II-VI Compounds

• So far we focus on Siand Ge– They are both Group IVA

• Compound semiconductor– Compound of Group III

& V– Compound of Group II

& VI– The bonding in II-VI is more ionic than in III-V, which is more

ionic than IV– Most important III-V compound semiconductor is GaAs– Eg for GaAs 1.40 eV, meaning more difficult to break a bond in

GaAs than in Si (1.12 eV)– Mobility in GaAs (Fig. 8.8)

)x()Lx(

Trend in III-V Semiconductors

• Iif P replaces As in GaAs, the band gap increases• If Sb replaces As in GaAs, the band gap decreases• If Al replaces Ga, band gap increases• If In replaces Ga, band gap decreases• The rule is

– The lower in the Periodic Table, the smaller the band gap– The lower in the Periodic Table, the larger the atom and the

weaker the binding force between the nucleus and the electrons, and the easier to excite an electron into the conduction band

Compound Semiconductors

• A compound of Group III and Group V is a binary semiconductor

• There are ternary and quaternary compound semiconductors

– Ternary: AlxGa1-xAs & GaAsyP1-y

– Quaternary: AlxGa1-xAsyP1-y

• Doping in GaAs by Si– If Si replaces Ga atoms, it’s n-type– If Si replaces As atoms, it’s p-type– Residual doping in GaAs

1016

1017

1018

100 101 102

p-typeModelModeln-type

Car

rier

Co

ncen

trat

ion

(cm

-3)

AsH3/Ga(CH

3)3 Ratio

GaAs(001)

Residual carbon forced into Ga or As sites by AsH3/TMG ratio

3AsHTMG P

)T(B

P

)T(Apn

II-VI Compound Semiconductors

• The bonding is more ionic, so the band gap is even bigger (Table 8.2)

– ZnSe, 2.6 eV– ZnS, 3.6 eV– ZnTe, 2.35 eV

Non-Equilibrium Processes

• So far all the discussions assume that the semiconductor is in thermal equilibrium

• A simple way to disturb the equilibrium is light (Fig. 8.9)– Three processes to produce excess carriers for enhanced

conduction - photoconduction– When the light is switched off, the number of carriers will fall

gradually to the equilibrium value– The time excess carriers reduce to e-1 of the original value is called

the lifetime of carriers

– If the semiconductor is locally illuminated, the illuminated region has more carriers than the surrounding regions and a diffusion current results

)/texp(NN oee

eNeDJ

Real Semiconductors

• Our band theory was based on simple rectangular crystals– Real crystals are more complicated– Si and Ge have diamond structure– E ~ k curve for Si looks quite different from the simple theory

(Fig. 8.10)– Minimum is not at kx = 0 as predicted in our model– Indirect band gap: needs a phonon to assist in photon absorption

and emission– E ~ ky curve looks different from E ~ kx curve– There are three valence bands with maximum at kx = 0: heavy,

light, and split-off– There are three types of holes: heavy, light, split-off– There are three effective masses for holes– In most device applications, it’s the average effective mass which

matters

Amorphous Semiconductors

• In amorphous semiconductors, there are crystallites (small crystals)– Crystallites lead to an energy gap– Small arrays of atoms give rise to spread band edges– Covalent bonds break off when orientation changes, so there are

dangling bonds and distorted bonds– They act as traps for both electrons and holes, and mobility is

reduced– Doping is difficult, since carriers from dopant atoms compensate

the dangling bonds– By introducing hydrogen into amorphous Si, hydrogen atoms can

saturate the dangling bonds– Doping amorphous Si thus becomes possible– Amorphous Si is for solar cells, xerox drums, and optoelectronics

Mobility Measurement

• Mobility definition– Measurements of voltage and distance give the electric field– Velocity measured by the time needed to move from point A to

point C– Haynes and Shockley method (Fig. 8.13)– With S open, a current flows across R. With S closed, there is a

sudden increase of current through R– Holes are injected into the n-type semiconductor. They travel

under the influence of B1– When they arrive at C, the current through R rises again– The time needed is recorded

E/vD

Four-Point Probe

• A less direct method (Fig. 8.14)– Ne measured by Hall measurement– Conductivity by 4-point probe– A current I is passed between contacts 1 and 4– The voltage drop V is measured between contacts 2 and 3– With equal probe spacing d

– At low fields, is a constant– At high fields, is a function of the electrical field

eN/

Vd2/I

dE/dvDaldifferenti

Hall Measurement

• It’s discussed in Chapter 1 of Solymar

– It measures charge density– Four contacts are made on the sample– Two opposite contacts pass a certain current– The other two contacts measure the Hall voltage

eN/1R eH

Energy Gap Measurement

• Conductivity as a function of temperature

– For an intrinsic semiconductor

– Plot ln vs 1/T, the slope is Eg/2kT

e)NN( hhee

)kT2/Eexp(Tttancons

)kT2/Eexp(NNNNN

g23

gVCihe

)kT2/Eexp(

)kT2/Eexp(eT)(ttancons

go

g23

he

Extrinsic Semiconductor

• At high temperatures, it is intrinsic in behavior• At low temperatures, it is pseudo-intrinsic

– Apparent band gap Eg = Eg – ED for n-type

• At intermediate temperatures, all impurity atoms are ionized– Conductivity is less dependent on temperature

• In a ln vs. 1/T plot, there are three regions with slopes of -Eg/2, ~0, and -(Eg – ED)/2 (Fig. 8.17)

nT

Optical Transmission

• Monochromatic light onto a semiconductor to measure transmission (Fig. 8.18)– When wavelength is small, the incident photons have enough

energy to excite electrons from valence band to conduction band, so transmission is low

– When wavelength is large, no absorption occurs and transmission is high

– The boundary is at = hc/Eg, at which there is a rise in transmission

– For GaAs– Direct band gap: maximum of V.B. and minimum of C.B. at the

same k (Fig. 8.19)– Indirect band gap: maximum of V.B. and minimum of C.B. not at

the same k

eV41.1nm880/hc/hcEg

Phonon

• Si has a gradual transmission edge (Fig. 8.18)– With an indirect band gap, momentum conservation comes in– Phonon is the quantum mechanical equivalent (particles) of

momentum (lattice vibration)– Phonon with frequency has the energy of ħ– The momentum operator is -iħ– The momentum of an electron is ħk– For a free electron– For an electron in a crystal, E ~ k relation more complicated

m2/kE 22

• For both phonons and photons– Their momenta are ħk– v is the velocity of the wave– For an optical wave with = 1 m, f = 31014 Hz, c = 3108 m/s

k = 61014 m-1

– For a sound wave (lattice vibration), zone boundary is at /a. If a = 3 nm

k = 1010 m-1

– This is much larger than photon momentum. Photon absorption is practically vertical in E – k diagram

– In direct gap semiconductors, photon absorption is a two-particle process: electron & photon

– In indirect-gap semiconductors, it is a three-particle process: electron, photon, & phonon (less likely to happen)

v/f/f2/2k

Phonon and Photon

• Minority carrier lifetime– If in Si, ND = 1022 m-3, Ni = 1016 m-3

Nh ~ 1010 m-3

– If 1015 m-3 electron-hole pairs are created by light

Nh = 1010 + 1015 = 1015 m-3, Ne = 1022 + 1015 = 1022

– Minority carriers are the ones which experience significant changes

– Photoconductivity– Measurement of minority carrier lifetime (Fig. 8.20)

– When light is switched off, the current I decays exponentially

p is the lifetime of holes

Minority Carrier Lifetime

hhhhee Nee)NN(

hNI

)/texp(II po

HW Assignment

• 8.2, 8.6, 8.7, 8.9, 8.15, 8.17