CH02 Carrier modeling [호환 모드]

Chapter 2. Carrier Modeling

Chapter 2.

Carrier Modeling

Sung June [email protected]

http://helios.snu.ac.krp

1


Subjectj

1. 양자역학은 반도체 이론에서 어떻게 쓰이는가?

2. 반도체는 어떤 결합을 하고 있으며 에너지 대역은 어떻게 형성되는가?

3. 정공(hole)은 어떻게 생성되고 그의 전자와의 관계는? 또 이들의 유효질량이란 무엇인가?

4. 진성 반도체에서 보다 많은 전하 캐리어가 필요할 때는 어떻게 하는가??

5. 전자의 에너지에 대한 분포는 어떤 모양인가?

6. 열적 평형이란 무엇이고 이때 전자 및 정공의 농도는 어떻게 계산되는가?가?

2


Contents

The Quantization Concept

Semiconductor Models

Carrier PropertiesCarrier Properties

State and Carrier Distributions

Equilibrium Carrier Concentrations

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Carriers: electron, holeEquilibrium: no external voltages magnetic fields stresses orEquilibrium: no external voltages, magnetic fields, stresses, or

other perturbing forces acting on the semiconductor. All observables are invariant with time.

• The Quantization ConceptIn 1913 Niels Bohr hypothesized that “quantization” of theIn 1913 Niels Bohr hypothesized that quantization of the

electron’s angular momentum was coupled directly to energy quantization

Lh

,3,2,1,6.13)4(2 22

0

40 =−=−= neV

nnqmEn πε

n is the energy quantum number or orbit identifierThe electron volt (eV) is a unit of energy equal to 1.6×10-19 joules

h)4(2 0 nnπε

4

( ) gy q j


The hydrogen atom – idealized representation showing the first three allowed electron orbits and the associated energy quantizationand the associated energy quantization

5


Semiconductor ModelsSemiconductor Models

• Bonding ModelSi i d i h di d l i hibi b di hSi atoms incorporated in the diamond lattice exhibit a bonding that

involves an attraction between each atom and its four nearest neighborsg

Any given atom not only contributes four shared electrons but must also accept four shared electrons from adjacent atoms

6


Visualization of a missing atom or point defect

Breaking of an atom-to-atom bond and freeing of an electron

• Energy Band ModelStarting with N isolated Si atoms and conceptually bringing theStarting with N-isolated Si atoms, and conceptually bringing the

atoms closer and closer together, Pauli exclusion principle makes a progressive spread in the allowed energies

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E b d th d i i i i t l l dEnergy bands: the spread in energies gives rise to closely spaced sets of allowed states conduction band, valence band, band gap (or forbidden gap)( g p)

In filling the allowed energy band states, electrons tend to gravitate to the lowest possible energies

8

Isolated Si atoms Si lattice spacing


Th l b d i l t l t l fill d ith l t d thThe valence band is almost completely filled with electrons and the conduction band is all but devoid of electrons

The valence band is completely filled and the conduction band p ycompletely empty at temperatures approaching T=0 K

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• CarriersWhen a Si-Si bond is broken and the associated electron is free to

wander the released electron is a carrierwander, the released electron is a carrierIn terms of the band model, excitation of valence band electrons

into the conduction band creates carriers; electrons in the conduction band are carriers

Completely filled valence band : no currentThe breaking of a Si-Si bond creates a missing bond or voidg gMissing bond in the bonding scheme, the empty state in the

valence band, is the second type of carrier– the holeBoth electrons and holes participate in the operation of mostBoth electrons and holes participate in the operation of most

semiconductor devices

10


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• Band Gap and Material ClassificationThe major difference between materials lies not in the nature of the

energy bands but rather in the magnitude of the energy gapenergy bands, but rather in the magnitude of the energy gapInsulators: wide band gap. The thermal energy available at room

temperature excites very few electrons from the valence band into the fconduction band; thus very few carriers exist inside the material

Metals: very small or no band gap exists at all due to an overlap of the valence and conduction bands. An abundance of carriers excellent conductors

Semiconductors; intermediate caseAt 300K E =1 42 eV in GaAs E =1 12 eV in Si E =0 66 eV in GeAt 300K, EG=1.42 eV in GaAs, EG=1.12 eV in Si, EG=0.66 eV in Ge.Thermal energy, by exciting electrons from the valence band into

the conduction band, creates a moderate number of carriers

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Carrier PropertiesCarrier Properties

• ChargeElectron (-q), hole (+q), q=1.6×10-19 C( q), ( q), q

• Effective MassAn electron of rest mass m0 is moving in aAn electron of rest mass m0 is moving in a

vacuum between two parallel plates under the influence of E,

0dq mdt

= − =F E v

Conduction band electrons moving between the two parallel end faces of a semiconductor crystal under the influence of an appliedfaces of a semiconductor crystal under the influence of an applied electric field

Electrons will collide with atoms a periodic deceleration

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I dditi t th li d l t i fi ld l t i t lIn addition to the applied electric field, electrons in a crystal are also subject to complex crystalline fields

The motion of carriers in a crystal can be described by Quantum y yMechanics.

If the dimensions of the crystal are large compared to atomic dimensions the complex quantum mechanical formulation for thedimensions, the complex quantum mechanical formulation for the carrier motion between collisions simplifies to yield an equation of motion identical to Newton’s 2nd equation, except that m0 is replaced by an effective carrier mass

ndqE mdt

∗= − =F vElectron effective massdt

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F h l ith d * *For holes with –q q and The carrier acceleration can vary with the direction of travel in a

crystal multiple components

nm pm

y p pDepending on how a macroscopic observable is related to the

carrier motion, there are, for example, cyclotron resonance effective masses conductivity effective masses density of state effective massmasses, conductivity effective masses, density of state effective mass, among others.

Material /m m∗ /m m∗

Density of State Effective Masses at 300 K

Material 0/nm m 0/pm m

Si 1.18 0.81Ge 0.55 0.36

GaAs 0.066 0.52

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• Carrier Numbers in Intrinsic MaterialIntrinsic semiconductor = pure (undoped) semiconductorAn intrinsic semiconductor under equilibrium conditionsAn intrinsic semiconductor under equilibrium conditions

in p n= =

etemperaturroomatSiincmni310101 −×=

The electron and hole concentrations in an intrinsic semiconductor are equal:are equal:

Si atom density: 5×1022 cm-3, total bonds: 2×1023 cm-3 and ni~ 1010

cm-3 1/1013 broken in Si @ 300 K

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• Manipulation of Carrier Numbers - DopingThe addition of specific impurity atoms with the purpose of

increasing the carrier concentrationincreasing the carrier concentration

Common Si dopants.

Donors (Electron-increasing dopants) Acceptors (Hole-increasing dopants)

P B Column V Column IIIAs Ga

Sb In Al

Column Velements

Column IIIelements

Column V element with five valence electrons is substituted for a Si atom four of the five valence electrons fits snugly into the bondingatom , four of the five valence electrons fits snugly into the bonding structure

The fifth donor electron, however, does not fit into the bonding

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structure and is weakly bound


At 300 K th d l t i dil f d t d hAt 300 K, the donor electron is readily freed to wander hence becomes a carrier (!! no hole generation)

The positively charged donor ion left behind (cannot move)p y g ( )

The Column III acceptors have three valence electrons and cannot pcomplete one of the bonds.

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Th C l III t dil t l t f bThe Column III atom readily accepts an electron from a nearby bond, thereby completing its own bonding scheme and in the process creating a hole that can wanderg

The negatively charged acceptor ion cannot move, and no electrons are released in the hole-creation process

Weakly bound? : a binding energy ~0 1 eV or lessWeakly bound? : a binding energy ~0.1 eV or less

The positively charged donor-core-plus-fifth electron may be likened yto a hydrogen atom

11 8Kε = =4

B H 12 2

1 0.1 eV2(4 )

n nm q mE EK m Kπ ε

∗ ∗

| = − = − nh

11.8r SKε = =

≅ ≅

20

S 0 0 S2(4 )K m Kπ ε h


Donors lEBl Accepters lEBlDonors lEBl Accepters lEBlSb 0.039eV B 0.045eVP 0.045eV Al 0.067eVAs 0.054eV Ga 0.072eV

In 0.16eV

Table 2.3

Dopant-Site Binding Energies

The bound electron occupies an allowed electronic levels at an

E E |E |

Energies

energy ED=Ec-|EB|

(1/ 20) (Si)c D B GE E E E− = ≅

All donor sites filled with bound electrons at T 0 KAs the temperature is increased, with more and more of the weakly

bound electrons being donated to the conduction bandAt 300 K, the ionization is all but totalThe situation for acceptors is completely analogous

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The situation for acceptors is completely analogous


Visualization of (a) donor and (b) acceptor action using the energy band model

22


xΔ

EC

ED

0.045 eV(P)

D

Donor site

(Si) 1.12 eV

T = 0 KT = 300 K

EV

T = 0 KT = 300 K

23x


• Carrier-Related TerminologyDopants: specific impurity atoms that are added to semiconductors

in controlled amounts for the purpose of increasing either the electronin controlled amounts for the purpose of increasing either the electron or the hole concentration

Intrinsic semiconductor: undoped semiconductorExtrinsic semiconductor: doped semiconductorDonor: impurity atom that increases the electron concentrationAcceptor: impurity atom that increases the hole concentrationp p yn-type material: a donor-doped materialp-type material: a acceptor-doped materialMajority carrier: the most abundant carrier in a givenMajority carrier: the most abundant carrier in a given

semiconductor sample; electron in an n-type material and hole in a p-type material

Minority carrier: the least abundant carrier in a given semiconductor sample; holes in an n-type material and electrons in a p-type material

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p type material


State and Carrier DistributionsState and Carrier Distributions

• Density of StatesyHow many states are to be found at any given energy in the energy

bands?The state distribution is an essential component in determiningThe state distribution is an essential component in determining

carrier distributions and concentrationsThe results of an analysis based on quantum mechanical

consideration: density of states at an energy E

n n cc c2 3

2 ( )( ) ,

m m E Eg E E E

∗ ∗ −= ≥c c2 3

p p v

( ) ,

2 ( )( ) ,

g

m m E Eg E E E

π∗ ∗ −

= ≤

h

25

v v2 3( ) ,g E E Eπ

≤h


Difference between and stem from differences in the masses.

)(Egc )(Egv

c( )g E dE represents the number of conduction band states/cm3 lying in the energy range between E and E+dE

v( )g E dE represents the number of valence band states/cm3 lying in the energy range between E and E+dE

≤

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• The Fermi Functiontells one how many states exist at a given energy E

Fermi function f(E) specifies the probability that an available state)(Eg

Fermi function f(E) specifies the probability that an available state will be occupied by an electron

F( ) /

1( )1 E E kTf E

e −=+

EF=Fermi energy k=Boltzmann constant (k=8.617×10-5 eV/K)T t tF1 e+ T = temperature

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(i) f(EF)=1/2(ii) If (iii) If

( ) / ( ) /3 , 1 and ( )F FE E kT E E kTFE E kT e f E e− − −≥ +

( ) / ( ) /3 , 1 and ( ) 1 : 1 ( ) emptyF FE E kT E E kTFE E kT e f E e f E− −≤ − − −

>>

<<≅≅( )

(iv) At T=300 K, kT=0.0259 eV and 3kT=0.0777 eV << EG

, ( ) ( ) p yF f f

• Equilibrium Distribution of Carriers• Equilibrium Distribution of Carriers

)](1)[(&)()( EfEgEfEg vc −All carrier distributions are zero at the band edges, reach a peak

value very close to Ec or Ev

Wh E i iti d i th h lf f th b d ( hi h )When EF is positioned in the upper half of the band gap (or higher), the electron distribution greatly outweighs the hole distribution

Equal number of carriers when EF is at the middle

28


29


Abb i t d f hi Th t t b f i l d tAbbreviated fashion; The greatest number of circles or dots are drawn close to Ec and Ev, reflecting the peak in the carrier concentrations near the band edges

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Equilibrium Carrier ConcentrationsEquilibrium Carrier Concentrations

• Formulas for n and ph b f d i b d l / 3 l i( ) ( )E f E dE : the number of conduction band electrons/cm3 lying

in the E to E+dE range( ) ( )cg E f E dE

[ ]

top

c

v

c ( ) ( )E

E

E

n g E f E dE= ∫

∫ [ ]v

bottonv ( ) 1 ( )

Ep g E f E dE= −∫

For n-type material

2 E E E dEm m∗ ∗ top

Fc

cn n( ) /2 3

21

E

E E kTE

E E dEm mn

eπ −

− =

+∫h

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Defining

3/ 2

nC 22 , the "effective" density of conduction band states

2m kTNπ

∗⎡ ⎤= ⎢ ⎥

⎣ ⎦h3/ 2

pV 2

2

2 , the "effective" density of valence band states2m kT

N

π

π

∗

⎣ ⎦

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

h

h2π⎢ ⎥⎣ ⎦h

At 300 K

Degenerate

2/30

*,

319, )/)(10510.2( mmcmN pnVC

−×=

Ec EF

EF

Degeneratesemiconductor

Nondegeneratesemiconductor

If 3 3v F cE kT E E kT+ ≤ ≤ −

F c( ) /C

E E kTn N e −=

Ev

EF

EFDegeneratesemiconductor

semiconductorv F

C( ) /

VE E kT

n N e

p N e −=

32


*Alternative Expressions for n and p

For intrinsic semiconductor, Ei=EF

( ) /E E kT

Alternative Expressions for n and p

kTEE iN /)( −i c

v i

( ) /i C

( ) /i V

E E kT

E E kT

n N e

n N e

−

−

=

=kTEE

iV

kTEEiC

vi

ic

enN

enN/)(

/)(

−=

=

F i( ) /i

( ) /

E E kT

E E kT

n n e −=i F( ) /

iE E kTp n e −=

• ni and the np Product

c v G( ) / /2i C V C V

E E kT E kTn N N e N N e− − −= =

G / 2i C V

E kTn N N e−= 2inp n=

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Intrinsic carrier concentrations in Ge, Si, and GaAs as a function of temperature

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• Charge Neutrality RelationshipFor the uniformly doped material to be everywhere charge-neutral

clearly requiresclearly requires

D A3

charge 0cm

qp qn qN qN+ −= − + − =cm

D A 0p n N N+ −− + − =

3

3

cmacceptors/ionizedofnumber

donors/cm ionized ofnumber

=

=−

+

A

D

N

N

3

3

total number of donors/cmtotal number of acceptors/cm

D

A

NN

=

=

pA

D A 0p n N N− + − = assumes total ionizationf d

pA

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D Apof dopant atoms


• Carrier Concentration CalculationsAssumptions: nondegeneracy and total ionization of dopant atoms

Two equations and two unknowns

22inpn

=

2i

D A 0n n N Nn

− + − =

2 2D A i( ) 0n n N N n− − − =

1/ 21/ 222D A D Ai2 2

N N N Nn n⎡ ⎤− −⎛ ⎞= + +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

1/ 2222i A D A Di2 2

n N N N Np nn

⎡ ⎤− −⎛ ⎞= = + +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

⎢ ⎥⎣ ⎦

p

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2 2n ⎝ ⎠⎢ ⎥⎣ ⎦


I t i i i d t (N 0 N 0)Intrinsic semiconductor (NA=0, ND=0) n=p=ni

Doped semiconductor where either ND-NA≈ND >> ni or ND-NA≈NA >> ni (usual)

D A D i,N N N n >> >>DNn ≅(nondegenerate, total ionization)

Di Nnp /2≅

A D A i,(nondegenerate, total ionization)N N N n

>> >>

Ai

A

Nnn

Np

/2≅

≅

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• Determination of EFone-to-one correspondence between EF and the n & pExact positioning of EiExact positioning of Ei

n p= p

i c v i( ) / ( ) /C V

E E kT E E kTN e N e− −=

c v Vi

C

ln2 2

E E NkTEN

⎛ ⎞+= + ⎜ ⎟

⎝ ⎠C2 2 N⎝ ⎠

3/ 2mN ∗⎛ ⎞ pc v 3 l

mE EE kT∗⎛ ⎞+

+ ⎜ ⎟pV

C n

mNN m∗

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

pc vi

n

ln2 4

E kTm∗= + ⎜ ⎟⎜ ⎟

⎝ ⎠

eVmmkT 00730)/ln()4/3( ** −=

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eVmmkT np 0073.0)/ln()4/3( =


D d i d tDoped semiconductors

kTEEi

iFenn /)( −=

F i i iln( / ) ln( / )E E kT n n kT p n− = = −

i

torssemiconductype-pfortorssemiconduc type-nfor

A

D

NpNn

≅≅

F i D iln( / )E E kT N n− =

torssemiconductypepfor ANp ≅

F i D i

i F A i

ln( / )ln( / )

E E kT N nE E kT N n− =

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The maximum nondegenerated doping concentrations318 /1061N317

318

/101.9

/106.1

cmN

cmN

A

D

×≅

×≅

40

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CH02 Carrier modeling [호환 모드]