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1

Chapter 2: Semiconductor Fundamentals

2.1 Introduction2.2 Crystals and Crystal Structures2.3 Energy Bands2.4 Density of States2.5 Carrier Distribution Functions2.6 Carrier Densities2.7 Carrier Transport2.8 Carrier Recombination and Generation2.9 Continuity Equation 2.10 The Drift-Diffusion Model2.11 Semiconductor Thermodynamics


2.1 Introduction

Carrier densityCrystal structuresEnergy bandsDensity of statesDistribution function

Carrier transportDriftDiffusionRecombination/Generation


2.2 Crystals and Crystal Structures

Bravais lattice2D3DMiller indices

Cubic crystalsDiamond latticePacking density

Types of solid materials

Amorphous materialNo order“Frozen” liquid state

Poly-crystalline materialShort range order onlySmall crystalline regions with random

orientationCrystalline material

Both short and long range order

2

10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.5

Bravais lattice construction

32rr

2ar

1ar2132 23 aar rrr

+=

α

Figure 2.2.1 The construction of lattice points using unit vectors


2D Bravais Lattices

2ar

1ar

o90 2ar

1aro90 2ar

1aro90

2ar

1ar

o120 2ar

1arα

(a)

(d) (e)

(b) (c)

Figure 2.2.2 The five Bravais lattices of two-dimensional crystals: (a) square, (b) rectangular, (c) centered rectangular, (d) hexagonal and (e) oblique


2D Bravais Lattices

Name Number of Bravais lattices Conditions

Square 1 a1 = a2 , α = 90°

Rectangular 2 a1 ≠ a2 , α = 90°

Hexagonal 1 a1 = a2 , α = 120°

Oblique 1 a1 ≠ a2 , α ≠ 120°, α ≠ 90°

Table 2.2.1 Bravais lattices of two-dimensional crystals


3D Bravais Lattices

Table 2.2.2 Bravais lattices of three-dimensional crystals (14 total)

Name Number of

Bravais lattices

Conditions Primitive Base-centered

Body-centered

Face-centered

Triclinic 1 a1 ≠ a2 ≠ a3 , α ≠ β ≠ γ

Monoclinic 2 a1 ≠ a2 ≠ a3 , α = β = 90° ≠ γ

Orthorhombic 4 a1 ≠ a2 ≠ a3 , α = β = γ = 90°

Tetragonal 2 a1 = a2 ≠ a3 α = β = γ = 90°

Cubic 3 a1 = a2 = a3 , α = β = γ = 90°

Trigonal 1 a1 = a2 = a3 , α = β = γ < 120° ≠ 90°

Hexagonal 1 a1 = a2 ≠ a3 , α = β = 90° , γ = 120°

3


3D Bravais LatticesTriclinic

Monoclinic

Base-centered

a1 ≠ a2 ≠ a3

α ≠ β ≠ γ

a1 ≠ a2 ≠ a3

α = β = 90ο ≠ γ


3D Bravais Lattices

Orthorhombic

Base-centered Face-centeredBody-centered

a1 ≠ a2 ≠ a3 α = β = γ = 90ο


3D Bravais Lattices

Tetragonal

Body-centered

a1 = a2 ≠ a3 α = β = γ = 90ο


Cubic Crystals

(a) (b) (c)

a a a

Figure 2.2.3 The simple cubic (a), the body-centered cubic (b)and the face centered cubic (c) lattice

SimpleCubic

Body-centeredCubic

Face-centeredCubic

a1 = a2 = a3 α = β = γ = 90ο

4


3D Bravais Lattices

Trigonal

a1 = a2 = a3

α = β = γ < 120ο ≠ 90ο

Hexagonal

a1 = a2 ≠ a3

α = β = 90o γ = 120ο


Miller Indices

x

z

y

p

r

q

x

z

y

p

r

q

Figure 2.2.4 Intersections of a plane and the x, y and z axes, as used to determine the Miller indices of the plane.

)rA

qA

pA(

Calculate inverses

Miller indices Example

)346(_

p = 2, q = -3, r = 4

Multiply with smallest common denominator, A = 12

1/2, -1/3, 1/4

Minus signs are replaced with a bar above the number


Diamond Lattice

Figure 2.2.5 The diamond lattice of silicon and germanium

a


The diamond lattice

a

)346(_

Model detail Front view

a /2

109.5o

5


Zincblende Lattice

aFigure 2.2.6 The zincblende crystal structure of GaAs and InP


Packing Density of Cubic Crystals

Radius Atoms/ unit cell

Packing density

# Neighbors

Simple cubic 2

a 1 % 52 6

=π 6

Body centered cubic 4

3 a 2 % 68 8

3=

π 8

Face centered cubic 4

2 a 4 % 74 6

2=

π 12

Diamond 83 a 8 % 34

163

=π 4

3

3

34

cellunit theofVolumeatoms of VolumeDensity Packing

a

rN

π==

1631)

83(

3481

34

33

33 πππ ==

aa

arN

621)

42(

3441

34

33

33 πππ ==

aa

arN

831)

43(

3421

34

33

33 πππ ==

aa

arN

61)

2(

3411

34

33

33 πππ ==

aa

arN


2.3 Energy Bands

Free electron modelKronig-Penney modelReal energy band diagramsSimplified energy band diagramTemperature and doping dependenceMetals, insulators and semiconductorsEffective mass


Free electron model of a metal

MqΦ

VACUUME

E

0 L

x

States filledwith electrons

Figure 2.3.1 The free electron model of a metal.

6


Energy levels versus lattice constant

E

Ec

a

a0

2p orbital

2s orbitalEv

closely-packed states

Figure 2.3.2 Energy levels in a carbon crystal versus lattice constant, a.


Band diagrams of common semiconductors

Figure 2.3.7 Energy band diagram of (a) germanium, (b) silicon and (c) gallium arsenide

a) b) c)


Simplified band diagram

E Evacuum

Ec

Ev

Conduction

qχ

Eg

band

{{{

Energybandgap

Valenceband

Figure 2.3.8 A simplified energy band diagram used to describe semiconductors.

Vacuum level

Conduction band edge

Valence band edge

ElectronAffinity


Temperature dependence of the energy bandgap

00.20.40.60.8

11.21.41.6

0 200 400 600 800 1000 1200

Temperature (K)

Ene

rgy

Ban

dgap

(eV

)

GaAsSi

Ge

βα

+−=

TTETE gg

2

)0()(

Germanium Silicon GaAs

E g(0) (eV) 0.7437 1.166 1.519α (meV/K) 0.477 0.473 0.541β (K) 235 636 204

Figure 2.3.9 Temperature dependence of the energy bandgap of germanium (Ge), silicon (Si) and gallium arsenide (GaAs)

7


Doping dependence of the energy bandgap

-250

-200

-150

-100

-50

0

1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20

Doping Density (cm-3)

Ban

dgap

cha

nge

(meV

)

1016 1017 1018 1019 1020

Ge

GaAs

SikTNqqNE

ssg εεπ

22

163)( −=Δ

Figure 2.3.10 Doping dependence of the energy bandgap of silicon (Si), gallium arsenide (GaAS) and germanium (Ge)


Band DiagramsE

Metal Semiconductor Insulatora) b) c) d)

Eg

Conductionband

Valenceband

Figure 2.3.11 Possible energy band diagrams of a crystal.

Half filledband

Overlappingbands

Filled bandSmall energy gap

Filled bandLarge energy gap


Electron and Hole transport

E

Allmostempty band

{{{

Energybandgap

Allmostfilled band

E

Figure 2.3.12 Energy band diagram in the presence of a uniform electric field.

EqdxdE

=

∑=

statesfilled

1ivb (-q)v

VJ

))()((1

statesempty

states all

∑ −−∑ −= iivb vqvqV

J

∑ +=

statesempty

)(1ivb vq

VJ

Current in an almost filled band


Effective Mass Germanium Silicon GaAs

Smallest energy bandgap at 300 K Eg (eV) 0.66 1.12 1.424

Electron effective mass for density of states calculations

0

*,

mm dose

0.55 1.08 0.067

Hole effective mass for density of states calculations

0

*,

mm dosh

0.37 0.811 0.45

Electron effective mass for conductivity calculations

0

*,m

m conde0.12 0.26 0.067

Hole effective mass for conductivity calculations

0

*,m

m condh0.21 0.386 0.34

Table 2.3.2 Effective mass of carriers in germanium, silicon and gallium arsenide (GaAs)

8


2.4 Density of States

Derivation of 3D density of states


kx

kz

ky

Number of states in k-spaceL

nkxπ

= , n = 1, 2, 3, …

Figure 2.4.1 Calculation of the number of states with wavenumber less than k.

3334)(

812 kLN ×××××= π

π


Density of states derivation33

34)(

812 kLN ×××××= π

π

dEdkk

dEdk

dkdN

dEdN )(== 23L ππ

*

22

2=)(

mkkE h , providing

km

dEdk

2

*

=h

and h

Emk*2

=

0for ,281)(2/3*

33≥== EEm

hdEdN

LEg π

ccc EEEEmh

Eg ≥−= for ,28)( 2/3*3

π

cc EEEg <= for ,0)(

Conduction Band

Derivation (3 Dimensions)


2.5 Carrier Distribution Functions

Fermi-Dirac distributionExample

Impurity distribution functionBose-Einstein distribution functionMaxwell-Boltzmann distribution function

9


Statistical Thermodynamics• Describes a large collection of particles

– Parameters• Temperature, volume, number/type of particles

– Statistical thermodynamics• All configurations are equally likely

• Needed concepts– Thermal equilibrium– Basic laws– Thermodynamic identity (energy conservation)– Fermi energy, EF– Thermal voltage, Vt


10/1010/10

1/102/104/106/108/109/10

0/100/100/10

ExampleN = 10E = 23T > 0K

N = 10E = 20T = 0K

1234567

0

89

10

1234567

0

89

10


1/102/104/106/108/109/10

10/101234567

10/100

0/108

0 %

100 %80 %60 %40 %40 %20 %

1 2 3 4 5 6 70 8 9 10

0/1090/1010

Example (cont.)


The Fermi-Dirac distribution function

0%

20%

40%

60%

80%

100%

120%

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

E-E F (eV)Pr

obab

ility

600 K

150 K300 K

kTEEE

Ff

/)()(

e11−+

=

f(EF) = 1/2

Figure 2.5.1 The Fermi function at three different temperatures

10


Example (cont.)

Energy(eV)

0

5

10

15

#1 #24#2 #3 #4 #23#22#21

Figure 2.5.2 Eight of the 24 possible configurations in which 20 electrons can be placed having a total energy of 106 eV

N = 20E = 106 eV


Example (cont.)All 24 possible configurations for N = 20 and E = 106 eV


Example (cont.)

0%

20%

40%

60%

80%

100%

120%

0 5 10 15 20 25

Energy (eV)

Prob

abili

ty (%

)

Figure 2.5.3 Probability versus energy averaged over the 24 possible configurations (circles) fitted with a Fermi-Dirac function (solid line) using kT = 1.447 eV and EF = 9.998 eV


Other distribution functionsImpurity distributions Bose-Einstein distribution

For indistinguishable integer spin particles

Maxwell-Boltzmann distributionfor distinguishable particles

Donors

Acceptors with degenerate valence band

kTEEEMB

Ff

/)()(

e1

−=

1e1

/)()(

−=

− kTEEEBE

Ff

kTEEEacceptor

Faaf

/)()(

e411

−+=

kTEEEdonorFd

df /)(21

)(e1

1−+

=

11


Other distribution functionsMaxwellBolzmann

FermiDirac

BoseEinstein

MaxwellBolzmann

FermiDirac

BoseEinstein

0%

25%

50%

75%

100%

125%

150%

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Energy (eV)

Occ

upan

cy p

roba

bilit

y

0.01

0.1

1

10

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Energy (eV)O

ccup

ancy

pro

babi

lity

Figure 2.5.4 Probability of occupancy versus energy of the Fermi-Dirac, the Bose-Einstein and the Maxwell-Boltzmann distribution

kTE /e−


2.6 Carrier Densities

Carrier density integralNon-degenerate semiconductorsDegenerate semiconductors

Joyce-Dixon approximationIntrinsic semiconductors

Mass action lawIntrinsic material as a reference

Doped semiconductorsDonors and acceptorsGeneral solutionTemperature dependence


Electron and hole densities per unit energy

)()()( EfEgEn c=

)](1[)()( EfEgEp v −=

ccec EEEEmh

Eg ≥−= for ,28)( 2/3*3

π

vvhv EEEEmh

Eg ≤−= for ,28)( 2/3*3

π

electrons

holes


Carrier density integral

E F

0.0E+00

1.0E+21

2.0E+21

3.0E+21

4.0E+21

5.0E+21

6.0E+21

-0.2 -0.1 0 0.1 0.2 0.3 0.4

Energy (eV)D

ensi

ty (c

m-3

eV-1

)

0%

20%

40%

60%

80%

100%

120%

Prob

abili

tyf(E)n(E)

g c (E)

6 x 1021

5 x 1021

4 x 1021

3 x 1021

2 x 1021

1 x 1021

0 x 1021

Figure 2.6.1 The carrier density integral. Shown are the density of states, gc(E), the density per unit energy, n(E), and the probability of occupancy, f(E).

∫=∫=band conduction theof topband conduction theof top

)()()(cc E

cE

dEEfEgdEEnn

12


-2-1.5

-1-0.5

00.5

11.5

22.5

3

0.0E+00 5.0E+25 1.0E+26 1.5E+26 2.0E+26 2.5E+26

Density (cm-3eV-1)

Ene

rgy

(eV

)

0 0.2 0.4 0.6 0.8 1Probability of occupancy

f(E)E F

n (E)

p (E)

g c (E)

g v (E)

5 x 10250 10 x 1025 15 x 1025 20 x 1025 25 x 1025

Calculation of the electron and hole density

Figure 2.6.2 The density of states and carrier densities in the conduction and valence band

∫=∞

cco

EdEEfEgn )()(

∫ −=∞−

vE

vo dEEfEgp )](1)[(


Solution to the Fermi integral

Non-degenerate semiconductors: Ev +3kT < EF < Ec + 3kT

kTEE

co

cF

eNn−

= 2/32

*]

2[2

h

kTmN e

cπ

=

kTEE

vo

Fv

eNp−

= 2/32

*]

2[2

h

kTmN h

vπ

=

electrons

holes

with

with

∫+

−=∞

−c

FEkT

EEceo dE

e

EEmh

n

1

128 2/3*3

π

∫

+

−=∞−

−

v

F

E

kTEEvho dE

e

EEmh

p

1

128 2/3*3

π


Other solution to the Fermi integral

Joyce-Dixon approximation (Degenerate semiconductors)

... ))(93

163(

81ln 2 +−−+≅

−

c

o

c

o

c

ocF

Nn

Nn

Nn

kTEE

... ))(93

163(

81ln 2 +−−+≅

−

v

o

v

o

v

oFv

Np

Np

Np

kTEE

electrons

holes

Zero Kelvin solution

electrons

holes

cFcFe

o EEEEmn ≥−= for ,)(232 2/3

3

2/3*

2 hπ

FvFvh

o EEEEmp ≥−= ,)(232 2/3

2

2/3*

2 forhπ


Fermi integral comparison

1.E+19

1.E+20

1.E+21

-0.1 0 0.1 0.2

Fermi Energy (eV)

Car

rier

Den

sity

(cm

-3)

1021

1020

1019

Joyce-Dixonapproximation

Non-degeneratesemiconductor

Zero Kelvin solution

Numericsolution

Figure 2.6.3 Carrier (electron) density versus Fermi energy, for Εc = 0 eV.

NC

13


Intrinsic semiconductors

kTEE

co

cF

eNn−

= kTEE

vo

Fv

eNp−

=

kTEEcEEoi ci

iFeNnn /)(

)(−

= ==kTEE

vEEoiiv

iFeNpn /)(

)(−

= ==

kTEvci

geNNn 2/−=

)ln(21

2 c

vvci N

NkT

EEE +

+= )ln(

43

2 *

*

e

hvci

m

mkT

EEE +

+=

2/32

*]

2[2

h

kTmN e

cπ

= 2/32

*]

2[2

h

kTmN h

vπ

=

Carrier density for non-degenerate semiconductors

Intrinsic Fermi energy

Intrinsic carrier density, ni


Mass action law

2/)(i

kTEEvcoo neNNpn cv ==⋅ −

kTEE

co

cF

eNn−

= kTEE

vo

Fv

eNp−

=kTE

vcigeNNn 2/−=

Product of electrons and holes

Mass action law: Product of electrons and holes is constant

2ioo npn =⋅

Electron, hole and intrinsic carrier density


Intrinsic carrier density versus temperature

1.E-10

1.E-05

1.E+00

1.E+05

1.E+10

1.E+15

1.E+20

0 2 4 6 8 10 12

1000/T (1000/K)

Intr

insic

den

sity

(cm

-3)

1020

1010

10-10

germanium

GaAs

silicon100

100

Temperature (K)

2003001,000 kTEvci

geNNn 2/−=

Figure 2.6.4 Intrinsic carrier density versus temperature in gallium arsenide (GaAs), silicon and germanium.

2/32

*]

2[2

h

kTmN e

cπ

=

2/32

*]

2[2

h

kTmN h

vπ

=

Dotted lined without temperature dependence of the energy bandgap

βα

+−=

TTETE gg

2

)0()(


Intrinsic material as a reference

kTEEio iFenn /)( −= kTEE

io Fienp /)( −=

i

oiF n

nkTEE ln+=i

oiF n

pkTEE ln−=

kTEE

co

cF

eNn−

= kTEE

vo

Fv

eNp−

=

kTEEcEEoi ci

iFeNnn /)(

)(−

= == kTEEvEEoi

iviF

eNpn /)()(

−= ==

Taking the ratio of n0 and ni, the ratio of po and ni:

Solving for EF:

14


2D representation of the diamond lattice


Donors and acceptors in silicon:

-+

Phosphorous and ArsenicGroup V

Boron and AluminumGroup III

Donor Acceptor


Ionization of donors and acceptors

E

Ec

Ev

Ed

Ei

(a)

Ec

Ev

Ei

Ea

(b)

Figure 2.6.5 Ionization of a) a shallow donor and b) a shallow acceptor

dd NNn =≅ +0

aa NNp =≅ −0

Shallow impurities


Ionization energy model

+q

Donor ion

electron

r

-q

Figure 2.6.6 Trajectory of an electron bound to a donor ion within a semiconductor crystal. A 2-D square lattice is used for ease of illustration.

eV 6.132

0

*

r

conddc

m

mEE

ε=−

15


Compensated material

Ec

Ed

Ea

Ev

−+ −=− ad NNpn


Range of doping densities

intrinsic low high silicondensity density densityni (300K) nSi(cm-3) (cm-3) (cm-3) (cm-3)

1010 1013←⎯⎯⎯→ 1020 1023


Analysis for non-degenerate semiconductors

0)( =−+−= −+adoo NNnpqρ

−+ −+= adi

o NNn

nn

2

22)2

(2 i

adado n

NNNNn +

−+

−=

−+−+ 22)2

(2 i

dadao n

NNNNp +

−+

−=

+−+−

i

oiF n

nkTEE ln+=i

oiF n

pkTEE ln−=

Charge neutrality

Mass action law

Solution to the quadratic equation

Calculation of the Fermi energy


n-type material

For Nd+ - Na

- >> ni

p = ni2 /(Nd

+ - Na- )

n = Nd+ - Na

-

ExampleNd = 1017 cm-3

Na = 5 x 1016 cm-3

ni = 1010 cm-3

n = 5 x 1016 cm-3

p = 2 x 103 cm-3

22)2

(2 i

adad nNNNNn +−

+−

=−+−+

16


General analysis

1.0E+10

1.0E+12

1.0E+14

1.0E+16

1.0E+18

1.0E+20

1.0E+22

-0.5 0 0.5 1 1.5

Fermi energy (eV)

Car

rier

den

sity

(cm

-3)

p n p + N d+n + N a

-

E cE v

E F

1010

1012

1014

1016

1018

1020

1022

Figure 2.6.8 Graphical solution of the Fermi energy based on the general analysis.


Temperature dependence

1E+14

1E+15

1E+16

1E+17

1E+18

0 5 10 15 20

1000/T (1000/K)

Ele

ctro

n de

nsity

(cm

-3) Intrinsic

Freeze-out

Extrinsic

1014

1015

1016

1017

1018

Figure 2.6.9 Electron density as a function of temperature


Non-equilibrium carrier densities

)exp(kT

EFnnnn in

io−

=+= δ

)exp(kT

FEnppp pi

io−

=+= δ

Quasi-Fermi energies: Fp and Fp


2.7 Carrier Transport

Carrier driftMobility

Doping and temperature dependenceResistivityVelocity saturation

DiffusionDiffusion current derivationEinstein relation

Total currentHall effect

17


Carrier drift

Ev

V -+

I

A

L

Figure 2.7.1 Drift of a carrier due to an applied electric field.

vLQ

tQIr /

==

vvLA

QAIJ rrr

rρ===

vqnJ rr−=

vqpJ rr=

Electrons: Holes:


Random carrier motion

x

= 0E = 0 /E

Figure 2.7.2 Random motion of carriers in a semiconductor with and without an applied electric field.


Mobility vs. doping density

0200400600800

1000120014001600

1.0E+14 1.0E+16 1.0E+18 1.0E+20

Doping density (cm-3)

Mob

ility

(cm

2 /V-s

)

Electrons

Holes

1014 1016 1018 1020

Figure 2.7.3 Electron and hole mobility versus doping density for silicon.

αμμμμ

minmaxmin

)(1rN

N+

−+=

Arsenic Phosphorous Boron

μmin (cm2/V-s) 52.2 68.5 44.9

μmax (cm2/V-s) 1417 1414 470.5

Nr (cm-3) 9.68 x 1016 9.20 x 1016 2.23 x 1017

α 0.68 0.711 0.719


Doping dependent mobility

−+ += ad NNNwithα

μμμμ)(1minmax

min

rNN

+

−+=

18


Resistivity of silicon

0.0001

0.001

0.01

0.1

1

10

100

1000

1.E+14 1.E+16 1.E+18 1.E+20


Res

istiv

ity ( Ω

−cm

)

1014 1015 1018 1020101910171016

n-typep-type

)(11

pnq pn μμσρ

+==

Figure 2.7.4 Resistivity of n-type and p-type silicon versus doping density.


Sheet resistance: Uniform doping

WtLR ρ

=

Resistance of a uniformly doped semiconductor

Units: Ohm/square

Sheet resistance Lt

W

WLRR s=

Material withresistivity ρ

tRs

ρ=

L

W= Rs x number of squares


Sheet resistance

0.01

0.1

1

10

100

1000

10000

1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20


Shee

t res

istan

ce (

Ω/sq

)

1014 1015 1018 1020101910171016

n-typep-type

tRs

ρ=

WLRR s=

Figure 2.7.5 Sheet resistance of a 14 mil thick n-type and p-type silicon wafer versus doping density.

t = 14 mil


Non-uniform doping

Nd

Na(x)

xj x

Nd,Na

x+dx

x

Figure 2.7.6 Sheet resistance calculation of a non-uniform doping distribution.

∫∫≅=

jj x

ap

x

p

s

dxxNqdxxpxq

R

00

)(

1

)()(

1

μμ

dxxpxqdxdR

ps )()(

1μ

ρ==

19


E

v

(a)

vsat

E

v

(b)

vsat

v =μ Evpeak

Epeak

Velocity saturation

Figure 2.7.7 Velocity-field relation for a) materials without accessible higher bands and b) materials with an accessible higher band.

satv

vE

EE

μμ

+=

1)(


Diffusion current derivation

n

x-l l0

n(-l )n(l )

Figure 2.7.8 Carrier density profile used to derive the diffusion current expression

)]()([21

,, lxnlxnvthleftrightnrightleftnn =−−==Φ−Φ=Φ →→

dxdnvl

llxnlxnvl ththn −=

−=−=−=Φ

2)()(

dxdnvlqqJ thnn =Φ−=

dxdnDqJ nn =


Diffusion current derivation

dxdnvlqqJ thnn =Φ−= dx

dnDqJ nn =

dxdpDqJ pp −=

cth

lvτ

=22

2*thvmkT

=

μτ

qkT

mq

qvm

vl cthth == *

2*

tnnn VqkTD μμ ==

tppp VqkTD μμ ==

Einstein relation Derivation


Total current

dxdnDqnqJ nnn += Eμ

dxdpDqpqJ ppp −= Eμ

)( pntotal JJAI +=

Total carrier current = Drift current + Diffusion current

Total current = electron current + hole current

20


L

W

t

Vx

zy

x

vxEx

Bz

Ey

Fyvx

Ey

Fy

-

++

-

VH

L

W

t

VxEx

Bz

-

+

VH+ -

IxIx

L

W

t

Vx

zy

x

vx

Ex

Bz

Ey

vx

Ey

-

+

VH

L

W

t

VxEx

Bz

-

+

VH+ -

IxIx

+ -

+

-

The Hall effect

Figure 2.7.9 Hall setup and carrier motion for a) holes and b) electrons.

a) b)

0

1

pzx

yH qpBJ

==Δ E

R


2.8 Carrier Recombination and Generation

Recombination/generation mechanismsOther generation mechanisms

Simple R/G modelBand-to-band recombinationSHR recombinationSurface recombinationAuger recombination


Carrier recombination and generation

Figure 2.8.1 Carrier recombination mechanisms in semiconductors

E

Et

Ec

Ev

Band-to-bandrecombination

Trap-assistedrecombination

Augerrecombination


Light absorption and ionization

Figure 2.8.2 Carrier generation due to: a) light absorption and b) ionization due to high-energy particle beams

E

Ec

Ev

generation due tolight absorption

ionization due to chargedhigh-energy particles

Eph > EgEg QEph

21


Avalanche multiplicationE

Ec

Ev

E

Figure 2.8.3 Impact ionization and avalanche multiplication of electrons and holes in the presence of a large electric field.


Carrier recombination and generation

E

Ec

band-to-bandrecombination

trap-assistedrecombination

Augerrecombination

Ev

Et

Figure 2.8.1 Carrier recombination mechanisms in semiconductors

σtht

tii

iSHR vN

kTEEnnp

npnU)cosh(2

2

−++

−=

Trap-assisted recombinationShockley-Hall-Reed recombination

Band-to-band recombination )( 2

ibb nnpbU −=−

)()( 22ipinAuger nnppnnpnU −Γ+−Γ=

Auger recombination


Simple recombination model

n

ppnnn

nnRGU

τ0−

=−=

p

nnppp

ppRGUτ

0−=−=

p-type

n-type


Light absorption

E

Eph > Eg

generation due to

light absorption

Eg

Figure 2.8.2 Carrier generation due to: a) light absorption

AExP

GGph

optlightnlightp

)(,, α==

Generation rate due to light

)(

)(xP

dxxPd

optopt α−=

Light absorption

22


2.9 Continuity Equation

Derivation of the continuity equationDiffusion equation

Solution for quasi-neutral semiconductors


Derivation of the continuity equation

Gn

Ec)(xJn )( dxxJn +

Evx

Rn

x x+dxFigure 2.9.1 Electron currents and possible recombination and generation processes


Derivation of the continuity equation

Gn

Ec)(xJn )( dxxJn +

Evx

Rn

x x+dx

Figure 2.9.1 Electron currents and possible recombination and generation processes

dxAtxRtxGAdxAt

txnnnq

dxxJqxJ nn )),(),(()(),( )()( −+−=

−+

−∂∂

dxxJdxxJ dxxJd

nnn )()()( +=+

),(),(),(1),( txRtxG

xtxJ

qttxn

nnn −+∂

∂=

∂∂

),(),(),(1),( txRtxG

xtxJ

qttxp

ppp −+∂

∂−=

∂∂


Alternate continuity equations

),(),(),(),(),( ),(2

2

txRtxGx

txnDx

txnx

txnt

txnnnnnn −+

∂∂

+∂

∂+

∂∂

= EE μμ

∂∂

),(),(),(),(),( ),(2

2

txRtxGx

txpDx

txpx

txpt

txpppppp −+

∂∂

+∂

∂−

∂∂

−= EE μμ

∂∂

),,,(),,,(),,,(),,,( 1 tzyxRtzyxGtzyxJt

tzyxnnnnq −+∇=

r

∂∂

),,,(),,,(),,,(),,,( 1 tzyxRtzyxGtzyxJt

tzyxppppq −+∇−=

r

∂∂

3-Dimensional continuity equation

Substitution of the current density

23


Diffusion equations

n

pppn

p ntxn

x

txnD

ttxn

τ∂

∂∂

∂ 02

2 ),(),(),( −−=

p

nnnp

n ptxpx

txpD

ttxp

τ∂∂

∂∂ 0

2

2 ),(),(),( −−=

Continuity equations applied to a quasi-neutral region

Assumptions:Zero electric fieldSimple recombination/generation modelValid for minority carriers only

n-type material

p-type material


Steady state solution to the diffusion equations

n

pppn

nxn

xd

xndD

τ0

2

2 )()(0

−−=

p

nnnp

pxpxd

xpdDτ

02

2 )()(0 −−=

nn LxLxpp eDeCnxn //

0)( ++= −pp LxLxnn eBeApxp //

0)( ++= −

nnn DL τ=ppp DL τ=

Steady state equations

General solutions


Formulation as a function of excess carrier densities

Excess minority carrier densities

Diffusion equations

nnn δ+= 0 ppp δ+= 0

22

2 )(0

n

pp

L

n

xd

nd δδ−= 22

2 )(0

p

nn

Lp

xdpd δδ

−=


2.10 The Drift-Diffusion Model

Assumptions10 equations, 10 unknowns

24


Drift-diffusion model assumptions

Full ionization:all dopants are assumed to be ionized (shallow dopants)

Non-degenerate: the Fermi energy is assumed to be at least 3 kTbelow/above the conduction/valence band edge.

Steady state:All variables are independent of time

Constant temperature:The temperature is constant throughout the device.


Drift-diffusion model variables

ρ the charge densityn the electron densityp the hole densityE the electric fieldφ the potentialEi the intrinsic energyFn the electron quasi-Fermi energyFp the hole quasi-Fermi energyJn the electron current densityJp the hole current density


Drift-diffusion model equations)( −+ −+−= ad NNnpqρ

ερ

=dxdE

E−=dxdφ

Eqdx

dEi =

kTEFi inenn /)( −=

kTFEi

pienp /)( −=

dxdnDqnqJ nnn += Eμ

dxdpDqpqJ ppp −= Eμ

τ1

)cosh(20

21

kTEE

npn

nnpxJ

iti

inq −

++

−−

∂∂

=

τ1

)cosh(20

21

kTEE

npn

nnpx

J

iti

ipq −

++

−−

∂

∂−=

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