Bakir Chapter2 Final

8/9/2019 Bakir Chapter2 Final

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August 18, 2014 ECE 3040: Chapter 2.1-2.2

Semiconductor Fundamentals

Chapter 2 Semiconductor

Device Physics Fundamentals

2.1 Semiconductor Materials

2.2 Crystal Structure

2.3 Semiconductor Models

2.4 Semiconductor Doping

2.5 Carrier Statistic

2.6 Carrier Transport

2.7 Carrier Generation/Recombination

Literature:

Pierret, Chapter 1-3, page 1-132

Acknowledgement Oliver Brand for slides

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2.1 Semiconductor Materials

Material Classification by Resistivity:

Literature: Pierret, Chapter 1.1, page 3-6

MaterialResistivity

[!"cm]

Insulators 105< !

Semiconductors 10-3< < 105 (108)

Conductors !< 10-3

Jaeger, Blalock, Table 2.1

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Semiconductor Materials

Semiconductors haveconductivities between thoseof insulators and those ofconductors

Semiconductors are materialswhose electric properties

(e.g. conductivity & resistivity)can be controlledover a widerange by doping, i.e. additionof controlled amounts ofspecific impurity atoms

Semiconductor materials are

the base of all semiconductordevices, such as diodes andtransistors (BJT andMOSFET)

Pierret, Fig. 3.8

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Resistivity/Conductivity of Insulators,

Semiconductor and Conductors

Sze, Fig. 2.1

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What Elements are Semiconductors based on?

Elemental Semiconductors:Silicon (Si), Germanium (Ge)

Compound Semiconductors:

III-V: e.g. Gallium Arsenide (GaAs)II-VI: e.g. Zinc Selenide (ZnSe)IV-IV: Silicon Carbide (SiC)

Alloy Semiconductors:Binary: Si1-xGexTernary: AlxGa1-xAs

AlxIn1-xAsGaAs1-xPx

Quaternary: GaxIn1-xAs1-yPy

Because of its well-developedfabrication technology, SILICONis by far the most importantsemiconductor material

Jaeger, Blalock, Table 2.2

Shaded = most important semiconductor elements

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Element and Compound

Semiconductors

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The OtherSemiconductor Materials

Silicon is the dominatingsemiconductor material The othersemiconductor

materials are utilized inapplications requiring e.g.

High speed

Optoelectronic properties High temperature operation

Example: Ternary alloys, suchas AlxGa1-xAs, have a tunablebandgap, enabling LEDs andlaser diodes with engineeredoutput spectrum

Substantial research isperformed (also at GaTech!) onboth Si and non-Sisemiconductor devices

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2.2 Crystal Structure

2.2.1 Unit Cell

Unit Cell & Primitive Unit Cell

Cubic Crystal Structure

2.2.2 Semiconductor Lattice

Diamond lattice Zincblende lattice

2.2.3 Miller Indices

Crystal Directions

Crystal Planes

Literature: Pierret, Chapter 1.2, page 6-16

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Classification of Solids

Solids can be classified based on their degree of atomic order

Most semicon. devices are based on crystalline semiconductors!

Amorphous

No recognizablelong-range order

Polycrystalline

Solid is made up ofcrystallites, i.e.

segments which arecompletely ordered

Crystalline

Entire solid is madeup of atoms in an

orderly array

Pierret, Fig. 1.1

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2.2.1 Crystal Unit Cell

Crystal= Periodic 3-D arrangement ofatoms

Unit cell= small portion of the crystal

that can be used to reproduce thecrystal

Primitive (unit) cell= smallest unit cell

possible

Unit cells (including primitive unit cells)are not unique

A unit cell does not need to be primitive;often it is more convenient to have aslightly larger unit cell with orthogonal

sides (instead of primitive cell with non-orthogonal sides)

WWW-page on crystal structures:http://departments.kings.edu/chemlab/chemlab_v2/

Which cells are unit cells?Which cells are primitive cells?

Which cell is not a unit cell?

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Cubic Crystal Structure

Simple cubic (sc) is the simplestof all 3-D unit cells:

Cube with an atompositioned at each corner

Only 1/8 of each corneratom is inside the unit cell: 1

atom per unit cell; eachatom has 6 nearestneighbors

The side length is called thelattice constant a

Very few materials crystallize ina simple-cubic lattice

How can we seethe crystalstructure of a crystal?

Sze, Fig. 2.3

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Cubic Crystal Structures

fcc

andbcc

Lattice Body-Centered-Cubic (bcc) unit cell

Additional atom in the center of the cube

2 atoms per unit cell8 nearest neighbors for each atom

Examples: Cr, W, Na, Fe

Face-Centered-Cubic (fcc) unit cell

Additional atoms in the center of eachface of the cube

Also known as cubic-close-packed (ccp) 4 atoms per unit cell

12 nearest neighbors for each atom

Examples: Ni, Ag, Au, Cu, AlSze, Fig. 2.3

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2.2.2 Diamond LatticeSilicon Crystal Structure

Silicon and germanium have a

diamond lattice, a special cubiccrystal structure

The diamond lattice consists oftwo interpenetrating face-

centered-cubic (fcc) latticesshifted by one-quarter of thecube body diagonal

Each silicon atom has 4 nearestneighbors forming a tetraederstructure

The silicon unit cell has a lattice

constant a = 5.43 (T = 300 K)(1 = 10-10m = 10-8cm = 0.1 nm)

Each silicon atom forms fourcovalent bonds with its nearestneighbors (see Chapter 2.3)

Sze, Fig. 2.4

Calculate:How many Si atoms dowe have per cm-3?

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Diamond Lattice along Axis

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Zincblende LatticeGaAs Crystal Structure

Most III-V semiconductors (e.g.

GaAs) have a zincblende lattice,again a cubic crystal structure

Similar to the diamond lattice,the zincblende lattice consists of

two interpenetrating face-centered-cubic (fcc) latticesshifted by one-quarter of thecube body diagonal, but withdifferent atoms occupying eachfcc sublattice

The GaAs unit cell has a lattice

constant a = 5.65 (T = 300 K)(1 = 10-10m = 10-8cm = 0.1 nm)

Sze, Fig. 2.4

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2.2.3 Miller Indices

Miller indices are used to specifycrystallographic planes anddirections

How do I find the Miller index for aplane? Record where the plane

intersects coordinate axes inunits of a: e.g. 1,2,3

Invert intercept values:e.g. 1, 1/2, 1/3

Multiply with constant to getsmallest possible set of wholenumbers: e.g. 6,3,2

Enclose set with curvilinearbracket: e.g. (632)

Note: What are the other planes indices?

(6 " 32)# (63 2)

(001)

(632)

(22 1)

Pierret, Fig. 1.6

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More on Miller Indices

Miller indices for directions are found analogous to thecomponents of a vector

In a cubic crystal, the planes

are equivalentbecause of symmetry and summarized as {100}

For cubic crystals, the direction [hkl] is normal to the plane (hkl)

(hkl) Particular crystal plane

{hkl} Equivalent planes

[hkl] Particular crystal direction

Equivalent directions

(100),(010),(001),(1 00),(01 0),(001)" {100}

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Crystal Planes and Directions Cubic Crystal Structure

Pierret, Fig. 1.7

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Wafer Flats

Silicon wafers used for device processing have flats (or a notch)indicating crystal direction and doping type

A {100} waferhas{100} surface

For {100} wafer,the surface directionperpendicularto primary flatis [011]

How is a waferfabricated?

[011]

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August 18, 2014 ECE 3040: Chapter 2.1-2.2 Semiconductor

Fundamentals

2.3 Semiconductor Models

2.3.1 Electron States in Atoms2.3.2 Semiconductor Bond Model

Covalent Bonds2.3.3 Semiconductor Band Model

Energy Bands

Band Gap Electron and Holes

Band Structure & Effective Mass

Simplified Semiconductor Band Model

Literature: Pierret, Chapter 2.1-2.3, page 23-40

Pierret, Appendix A, page 733-748

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Fundamentals

Silicon Atomic Structure

Silicon with its 14 electronshas (in its ground state) filledn = 1 (2 electrons) and n = 2(8 electrons) levels; theseare deep-lying energy levels,tightly bound to the nucleus

In case of the n = 3, thepossible states (2 s- and 6 p-states) are filled with the 4remaining electrons; these 4rather weakly boundelectrons are called valence

electronsand participate inchemical reactions andatom-atom interaction

Pierret, Fig. 2.2

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Fundamentals

2.3.2 Semiconductor Bond Model

Bonding to next neighbors isachieved by sharing valenceelectrons (electrons inoutermost shell), formingcovalent bonds

In case of silicon: Atomic number

= number of electrons= 14

Number of valenceelectrons = 4

Covalent tetraederbonding (diamond lattice)

Sze, Fig. 2.11

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Fundamentals

Semiconductor Bond Model

Valence electrons areweakly bound (compared toelectrons in inner shells)

Thermal energy at roomtemperature can break

covalent bonds andfree

electrons which now

contribute to the materialsconduction

Remaining are missingbonds/electrons, which are

called holes (and alsocontribute to the conduction,but why?)

Sze, Fig. 2.12

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August 18, 2014 ECE 3040: Chapter 2.1-2.2 SemiconductorFundamentals

2.3.3 Semiconductor Band Model

Single atom:according to Bohrsatom model, theelectrons occupywell defined,discrete energy

levels

Crystal: due to the interaction of neighboring atoms, thediscrete energy levels split up and a bandstructureisdeveloping featuring allowed and forbidden energy bands

Sze, Fig. 2.15

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Semiconductor crystal has valence

bandand conduction bandseparated by bandgap Eg

At T = 0 K:

Completely filled valence band

Completely empty conductionband

At elevated T, some electrons canbe excited from the valence to theconduction band

Note: Electrons in bands are notassociated with particular atom

From

Isolated

Atomsto a Crystal

Pierret, Fig. 2.5

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Semiconductor Band Model

Reason for energy level splitting is the Pauli principlestatingthat no 2 electrons can occupy the same state (with respect toenergy, momentum, spin), i.e. have the same quantum numbersn, l, m, and s

Simplified Model: valence and conduction band are separatedby a band gap; ECis the lowest energy of the conduction band,

EVthe highest energy of the valence band

EC

EV

Band GapEg

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Carriers: Electrons and Holes

Completely filled band and empty band do not contribute to current

conduction

If an electron is excitedfrom the valence into the conduction band,the additional electronin the conduction band contributes to thecurrent conduction; similarly, the missing electron in the valence band(like a bubble in a liquid), a so-called hole, contributes to the currentconduction; the (thermal) excitation has created an electron/hole pair

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Band Gap & Material Classification

Pierret, Fig. 2.8

Insulator Semiconductor

Metal

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Indirect vs. Direct Bandgap Semiconductor

Simplified Si & GaAs Band Structure

Sze, Fig. 2.18

Indirect Bandgap Direct Bandgap

#p $0 #p = 0

Using quantummechanicsbysolving the 3-DSchrdingerequationin theperiodic potential

of the nuclii Numerical

calculations deliverthe band diagrams,i.e. the E(k), E(p)relationships with

wavenumber kandcrystal momentump

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Characteristics of Si and GaAs

Silicon Si

Element semiconductor Diamond lattice structure Indirect Semiconductor Generation of electron-hole

pairs requires !E and !p Bandgap: Eg= 1.12 eV at

room temperature Lattice constant: a = 5.43

Gallium Arsenide GaAs

III-V semiconductor Zincblende lattice structure Direct Semiconductor Generation of electron-hole

pairs requires !E only,making GaAs suitable forphotonic applications

Bandgap: Eg= 1.42 eV atroom temperature

Lattice constant: a = 5.65

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Simplified Semiconductor

Bandstructure

EC

EV

Band GapEg

mn*

mp*

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Effective Mass meff (mn*, mp*)

Definition 1-D:

Definition 3-D:

Electron kinetics in single-crystal material is similar tofree electron behavior but with effective mass meff

meff

"1=

1

!2

#2E(k)

#

k

2

1

meff

"

#$

%

&'

ij

=

1

!2

(2E(k)

(k i(kj

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Effective Mass in Si, Ge & GaAs

Material mn*/m0 mp

*/m0

Si 1.18 0.81

Ge 0.55 0.36

GaAs 0.066 0.52

Free electron mass: m0= 9.1 10-31kg

Shown are average electron and hole effective masses asneeded for the density of statesat 300 K (see Chapter 2.5),but keep in mind that mn* and mp* are tensors!

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Effective Mass Concept

The movement of carriers (electrons and holes)inside the semiconductor crystal can be described byclassical (Newtonian) mechanics relations just byreplacing the free-electron-mass m0with the effectivemass m

n* (or m

p* for holes)

The effective masses are however tensors, i.e. thecarrier acceleration generally varies with direction oftravel in the crystal

F =mn* dvn

dtand F =mp

* dvp

dt

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2.4 Semiconductor Doping

Example: Silicon Doping: Doping with donoratoms,

phosphorous Por arsenicAs,having both 5 valenceelectrons, yielding N-typesilicon

Doping with acceptoratoms, boron B,having 3valence electrons, yieldingP-type silicon

With doping, the electricalconductivity of thesemiconductor material isadjusted over a wide range

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Semiconductor Doping

Doping in Bond Model: N-Type Doping:fifths valence electron does notparticipate in covalent bonds tonext neighbors; only small energyis required to freethis electron,thus contributing to the materialsconductivity

P-Type Doping:due to doping atom with threevalence electrons, one electron ismissing for the covalent bonds tothe four next neighbors; thismissing electron(called a hole)can hop from covalent bond tobond, thus also contributing to thematerials conductivity

Sze, Fig. 2.23

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Semiconductor Doping

Doping in Band Model: Doping atoms distort the periodicity of

the single-crystal crystal structure, thuscreating allowed (localized) energystates in the band gap

N-Type Doping: additional electron

states EDclose to the bottom of theconduction band EC; at roomtemperature, the electrons from thesestates are excited into the conductionband, increasing the conductivity

P-Type Doping: additional electronstates EAclose to the top of the valence

band EV; at room temperature, thesestates are occupied by electrons fromthe valence band, increasing thenumber of holes in the valence bandand, thus, the conductivity

EC

EiEV

ED

EC

EiEV

EA

P in Si: EC- ED= 0.045 eV

B in Si: EA- EV= 0.045 eV

EC- ED resp. EA- EV"kT = 0.026 eVat room temperature

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Carrier Concentration

Intrinsic Semiconductor(no doping: NA, ND= 0)

Carrier Concentration: n = p = n i

Extrinsic Semiconductor(doping concentration NA, ND#0)

Assumption: Complete Ionization (see Chapter 2.5) Majority Carrier Concentrations:

N-Type Semiconductor:Density of donor atoms: ND[cm

-3]Density of electrons (ND ni) : n"ND[cm

-3]

P-Type Semiconductor:Density of acceptor atoms: NA[cm-3]Density of holes (NA ni) : p"NA[cm

-3]

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2.5 Carrier Statistic

2.5.1 Density of States2.5.2 Fermi Function & Fermi Energy

Physical Interpretation Characteristics

2.5.3 Carrier Densities Intrinsic/Extrinsic Semiconductor Intrinsic Fermi Energy Mass Action Law Temperature Dependence

2.5.4 Charge Neutrality Relationship2.5.5 Non-Complete Ionization

Pierret, Chapter 2.4-2.6, page 40-68

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2.5.1 Density of States

From quantum mechanics, we not only obtain the bandstructure, i.e., the E(k) relations, but also the density ofstates g(E)dE, i.e., how many allowed states are in therange E$E+dE:

gC(E)dE =mn

* 2mn* E "EC( )

#2!3

dE $ E "EC( )1/2

gV(E)dE =mp

*

2mp*

EV "E( )#2!3

dE $EV "E( )1/2

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Density of States

Band Gap

EC

EV

EgC(E)

gV(E)

g(E)

Units of g(E)dE are [cm-3]

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2.5.2 Fermi Function & Fermi Energy

What determines whether anallowed state is occupied byan electron or not?

Fermi Function f(E):

f(E) is a probability functionwhich gives the probabilitywhether a state is occupiedor not

EFis the Fermi Energy

f(E) =1

1+ e(E"EF)/kT

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Characteristics of Fermi Function

Because of Pauli principle:0 %f(E) %1

f(E = EF) = 0.5The probability that a state is

occupied at the Fermienergy is 50%

f(E) is symmetric around EF:f(EF + E) = 1 f(EF E)

For T = 0 the Fermi functionbecomes a step function, i.e.all states below EFareoccupied, all states above EFempty

0

0.5

1

0 1 2

Energy E [eV]

FermiFunctionf(E)

EF= 1 eVkT = 0.0259 eV

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Physical Interpretation of

Fermi Energy EF

The Fermi energy EFhas the function of athermodynamic potential

In thermodynamic equilibrium, EFis constantacross the device!!!

A gradient in EFindicates non-equilibrium, resulting ina net flow of carriers

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Boltzmann/Classical

Approximation

Away from E = EF, i.e. for |E EF| > 3kT, the Fermi function canbe approximated by exponential functions:

This approximation is called the Boltzmannor Classicalapproximation

E "EF( ) >3kT # f(E) $e"(E"EF)/kT

EF "E( ) >3kT # f(E) $ 1" e"(E"EF)/kT

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2.5.3 Carrier Density n and p

The carrier densities of electrons n [cm-3] and holes p [cm-3] canbe calculated from the density of states g(E) and the Fermifunction f(E):

The solution of the integral generally requires numericalmethods; in case of the Boltzmann approximation however, theintegrals can be solved analytically!

n = gC(E) f(E) dE

EC

Top

"

p = gV(E) 1# f(E)( )!"# $#

dE

Bottom

EV

"a hole is a non-occupied electron state

%

-%

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Carrier Concentration

Intrinsic

ExtrinsicN-Type

ExtrinsicP-Type

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Carrier Density Boltzmann

Approximation

For electrons: (EC EF) &3kT

For holes: (EF EV) &3kT

n = NCe

(EF!E

C)/kT

NC =2m

n

*kT

2"!2

#

$%%

&

'((

3/2

)2.86 1019 cm!3

Si @ 300K

" #$$ %$$

p = NVe

(EV!E

F)/kT

NV =2

mp

*kT

2"!2

#

$%%

&

'((

3/ 2

)2.66 1019 cm!3

Si @ 300K

" #$$ %$$

EC

Ei

EV

EC3kT&EF&EV+3kT

NC,V

=2.51!1019 m

n,p

*

m0

"

#$$

%

&''

3/2

cm(3

NC)3.2!1019 cm(3

NV) 1.8 !1019 cm(3

for Si@ 300K

From Pierret:

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Carrier Density General Solution

For electrons:

For holes:

F1/2("c) is the tabulatedFermi-Dirac integralof order 1/2:

n =NC2

"

F1/2 #C( ) with #C = EF $EC( ) /kT

p =NV

2

"F1/2 #V( ) with #V = EV $EF( ) /kT

ECEi

EV

Range

for EF

F1/2 "c( ) ="1/2

1+ e"#"c

0

$

% d"

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Intrinsic Semiconductor

No doping Carrier concentration n = p = ni (for every electron in theconduction band, we have a hole, i.e. missing electron,in the valence band)

Fermi energy EF= Eiclose to center of band gap

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Intrinsic Semiconductor Carrier Concentration ni(T)

Pierret, Fig. 2.20

Ge: ni,(300K) = 2.5 1013cm-3GaAs: ni,(300K) = 2.25 10

6cm-3

Si: ni,(300K) = 1.0 1010cm-3

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Location of Ei

From the intrinsic carrier concentrations

With n = p, the intrinsic Fermi level E ican be extracted

The intrinsic Fermi level is only in the center of the bandgap if mn* = mp* !!

p = ni =NV e(EV "Ei)/kT

n = ni =NC e(Ei"EC)/kT

NV e(EV "Ei)/kT

=NC e(Ei"EC)/kT

" Ei =EC +EV

2+

3

4kT ln

mp*

mn*

#

$

%%

&

'

((

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Extrinsic Semiconductor

From Boltzmann approximation

Analogous for the holes

n =NCe(EF "EC)/kT

=NCe(Ei"EC)/kT

=

ni

! "## $##

e(EF "Ei)/kT

n =nie(EF "Ei)/kT

p = ni e(Ei"EF)/kT

n p = ni2

Mass-Action-Law

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Fermi Energy EF Ei[eV] as a

Function of Doping and Temp.

Sze, Fig. 2.28

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2.5.4 Charge Neutrality Relation

In equilibrium, charge neutralityis fulfilled:

Assuming complete ionization NA= NAand ND

+= ND:

p

holes

!

" n

electrons

!+ ND

+

ionized

donors

!

" NA"

ionized

acceptors

!

=0

p"n+

ND"NA

=

0n p = ni

2

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Electron and Hole Concentration

From mass-action-law and charge neutrality:

Special Cases:

n =ND"NA

2+

ND"NA2

#

$%

&

'(

2

+ ni2

p =NA"ND

2 +NA"ND

2

#

$%

&

'(

2

+ ni2

ND >>NA and ND >>ni: n =ND

p=

ni2

/NDNA >>ND and NA >>ni: p =NA

n =ni2 /NA

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Measurement of Carrier Concentration Hall Effect

The carrier concentration n or pcan be measured using the Halleffect

Assume: p-type semiconductorplate with L > W t (thin plate)with current I applied in x-directionand a magnetic induction B

z

applied along z-direction

Holes experience a Lorentz forcein y-direction:

Hole accumulation creates electric field in y-direction (electrostatic force =

Lorentz force, because no current flow in y-direction), resulting in a Hallvoltage VH:

x

y

zBz

+

+VH

VI

L

t

W x

y

P-Type Semiconductor

F = q (!

v "!

B) = (0,#qvxBz,0)

VH = "yW = vxBzW =Jxqp

BzW =1

qp

#RH

!

IBzt

Hall Coefficient

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2.5.5 Temperature Dependence of

Carrier Density

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Temperature

Dependence

of Carrier

Density

Pierret, Fig. 2.22

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Carrier Statistics Formula Summary

Charge Neutrality & Mass-Action Law:

Intrinsic Semiconductor

Extrinsic Semiconductor

Linking Doping and Carrier Concentration

Linking Band Structure and Carrier Concentration

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2.6 Carrier Transport

2.6.1 Carrier Drift Drift Current Mobility Resistivity

Band Structure under Applied Field2.6.2 Carrier Diffusion

Diffusion Current2.6.3 Total Current Equations

2.6.4 Einstein Relations

Pierret, Chapter 3.1-3.2, page 75-104

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2.6.1 Carrier Drift

Drift = Charged particle motion in response to an

applied electric field

Macroscopic Definition: Carriers of a given type(electrons or holes) move along at a constantvelocity, the drift velocity, parallel or antiparallel to theapplied electric field

Drift Current Densities[A/cm2]:!

Jp,drift=

q p

!

vd,p!

Jn,drift = "q n !

vd,n

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Carrier Drift

For small electric fields, the drift velocity isproportional to the applied electric field with themobility 'as proportionality factor:

The mobility is the central parameter characterizingthe carrier drift, resulting in the following currentdensities:

vd,p = p "

vd,n = #n "

!

Jp,drift = q p p!

"!

Jn,drift = q n n!

"!

Jdrift =!

Jn +!

Jp = q nn+ pp[ ] !

"

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Drift Velocity in Undoped Si at

Room Temperature

Pierret, Fig. 3.4

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Carrier Mobility

Definition of the carrier mobility [cm2/Vs]:

Room-temperature mobility of Si and GaAs:

p "vd,p

#n " $

vd,n

#

Silicon (low-doped)'n"1360 cm

2/Vs

'p"460 cm2/Vs

Gallium Arsenide (low-doped)'n"8500 cm2/Vs'p"400 cm

2/Vs

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Scattering

Mechanisms I (a) Impurityand (b) lattice

scatteringlimit the carriermobility

The mobility decreases withincreasing totaldopingconcentration (NA+ ND)

= q "

m*

mean free time

between collisions

effective mass

Pierret, Fig. 3.5

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Room Temperature Mobility in Si as a

Function of Doping Concentration

Pierret, Fig. 3.5a

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Scattering

Mechanisms IIT-dependence of the mobility:

Lattice Scattering:in Si: &~ T(2.2'2.3)(experiment)Reducing the temperature meansless thermal lattice vibration, i.e.less interaction with carriers, i.e.

higher mobility

Impurity Scattering:

in Si: &~ T+1.5(theory)Dominant scattering mechanism atlow temperatures; interaction isreduced at higher temperatures,

i.e. higher thermal velocities,because carrier is less time inclose proximity to impurity

&= &(T, NA+ND)

Pierret, Fig. 3.7

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Resistivity

Definition of conductivity $and resistivity ![%cm]tensor:

From the drift current, theresistivity is given by:

!

J=

"

!

#=

1

$

!

#

" =1

q nn + pp( )Pierret, Fig. 3.8

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Four-Point Probe

The resistivity of asemiconductor can bedetermined by a current-voltage measurement usinga four-point probe

W is the substrate thickness,CF a correction factor (seegraph)

" =V

IW CF

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Band Bending by Electric Field

The presence of an electricfield results in a bandbending, i.e. ECand EVareno longer constant

Electrostatic Potential V:

Note: V and E are arbitraryto within a constant!

Electrostatic Field #:

V = " 1q

EC "Eref( )

"= #$V

" =1

q

dECdx

=

1

q

dEVdx

=

1

q

dEidx

x

V

electronmovement

x

acceleration

EC

Ei

EV

Ecollisionloss of energy

holemovement

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Band Bending and Electric Field

Pierret, Fig. 3.10

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Diffusion Current

In case of diffusion of charged particles, i.e. electronand holes in our case, a diffusion current is resulting(equaling the product of flux density and carriercharge)

with diffusion coefficients Dnand Dp[cm2/s]

Why has the electron diffusion current no

sign?

Jp,diff = "qDp #pJn,diff = qDn #n

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2.6.3 Total Current Equations

Summing up drift and diffusion current densities, weobtain the total current density equations:

!

Jp = q p p

!

"#qDp

!

$p!

Jn = q n n!

"+ qDn!

$n

!

J =!

Jn +!

Jp

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2.6.4 Einstein Relations

Einstein Relationsrelate drift(') and diffusion (D)

Simplified Derivation:

Assume non-uniformly dopedn-type semiconductor in

equilibrium (i.e. EF= const.across the material)

In equilibrium, the total electronand hole current densities arezero:!

Jp = q p p

!

"#qDp!

$p = 0!

Jn = q n n!

"+ qDn!

$n = 0

Pierret, Fig. 3.14

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Einstein Relations

Fromwe obtain

knowing that

Inserting into the current density equation for theelectrons yields theEinstein Relationsrelating drift('n) and diffusion (Dn)

(similar derivationfor Dpand 'p)

Dn =kT

qn and Dp =

kT

qp

n =nie(EF"Ei)/kT

dn

dx= "

nikT

e(EF"Ei)/kTdEidx

= "q#

kTnie

(EF "Ei)/kT

=n

! "## $##

# =1

q

dEi

dx

and dEF

dx

=0

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Drift & Diffusion in Si (T = 300 K)

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2.7 Carrier Generation & Recombination

Perturbation of semiconductor, i.e. an excess or deficit incarrier concentrationwith respect to the equilibrium values iscreated, resulting in non-equilibrium conditions

If the perturbation is removed, recombination/generation (R-G)processes will restore equilibrium conditions; if perturbation ismaintained, R-G processes will stabilize the (non-equilibrium)carrier concentrations

2.7.1 Generation and Recombination Processes

2.7.2 R-G Statistics

2.7.3 Continuity Equations

Literature: Pierret, Chapter 3.3-3.5.1, page 105-132

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2.7.1 Generation/

Recombination

Processes Band-to-band R-G

processes onlyinvolveelectron in conduction bandand hole in valence band

R-G center generation/recombinationrequires

R-G center (lattice defect orimpurity which generatesstates in the bandgap)

Auger recombinationrequires 3 carriers (either 2holes and 1 electron or 2electrons and 1 hole)

All processes occur at alltimes (even in equilibrium),with the process having thehighest rate dominating

Reco

mbination G

enerat

ion

Pierret, Fig. 3.15

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Perturbation:

Carrier Injection/Generation

Processes:

Photogeneration

Operation of diode in forward direction

Impact ionization

p n > ni2

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Perturbation:

Carrier Extraction/Recombination

Process:

Operation of diode in reverse direction

p n < ni

2

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Carrier Generation/Recombination

After switching off the distortion, the system returns tothe equilibrium state (p n = ni

2) with a characteristictime constant, i.e. excess carriers will recombine (in casep n > ni

2) or additional carriers will be generated (in case

p n < ni2

)Notation: nn electron density in n-type semiconductor

pn hole density in n-type semiconductornn0, pn0 equilibrium carrier densities

np electron density in p-type semiconductor

pp hole density in p-type semiconductornp0, pp0 equilibrium carrier densities

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Low-Level/High-Level Injection

Low-Level Injection: High-Level Injection:

"n = "p nn0

pn = pn0 + "p $ "p > nn0

Example: Injection by photogeneration (&n = &p) in n-typematerial

"n = "p = 1012cm#3

nn $ 1015cm#3

pn $ 1012cm#3

"n = "p = 1017cm#3

nn $ 1017cm#3

pn $ 1017cm#3

Example: n-type Si, ND= nn0= 1015cm-3, p = ni

2/n = 105cm-3

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Low-Level/High-Level Injection

Example:N-type semiconductorND= 10

15cm-3 Equilibrium:

nn0= ND= 1015cm-3

pn0= ni2/n = 105cm-3

Low-Level Injection:&n = &p = 1012cm-3nn"nn0= 10

15cm-3pn"&p = 10

12cm-3< nn0 High-Level Injection:

&n = &p = 1017cm-3

nn" 1017cm-3> nn0pn"&p = 1017cm-3 > nn0

Equilibrium LL Injection HL Injection

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2.7.2 R-G StatisticsBand-to-Band Recombination

Likely in direct(band gap)semiconductors, such as GaAs

Unlikely in indirect(band gap)semiconductors, such as Si,because of required momentumconservation

In thermal equilibrium, thethermal generation rate Gthequals the recombination rate Rth:

The thermal recombination rate isproportional to the electron andhole carrier densities with'being the proportionality factor

Equilibrium:

Non-Equilibrium:

Gth =Rth =" nn0 pn0

Sze, Fig. 3.10

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Band-to-Band Recombination

In non-equilibrium, the change in minority carrier concentrationis:

Assuming steady state, dpn/dt = 0, we obtain

with the net-recombination rate U

The recombination rate and the thermal generation rate areproportional to the available carrier densities (see equilibrium):

Assuming low-level injection, i.e. &p and pn0


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Band-to-Band Recombination

The minority carrier lifetime (pcontrols the recombination velocityafter the distortion is switched off

The minority carrier lifetime in case of the direct recombination isinversely proportional to the majority carrier equilibrium density nn0

Excess carriers recombine after switching off GL(GL= 0 in DE) with thetime constant (p:

How can we measure the minority carrier lifetime?

U =" nn0 #p =pn$ pn0

1

" nn0

=

pn$ pn0%p

"p #

1

$ nn0

pn(t) = pn0 + "p GL e

#t / "p

dpn(t)

dt=

GL!U=

GL!pn!p

n0

"p

=

GL!

#pn

"p

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R-G Center Recombination

Dominant recombinationprocess in indirectsemiconductors, such as Si

Indirect recombination &generation of electron-hole

pairs via localized energystates in the band gap, socalled recombinationcentersor R-G centers

The R-G centers arecharacterized by their

energy Etand their densityNt

Sze, Fig. 3.12

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Location of Impurity

Atom Energy States

Sze, Fig. 2.24

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R G Center Recombination


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R-G Center Recombination

In a more involved derivation, an expression similar to that of the band-

to-band R/G case can be obtained for R-G center recombination/generation (again assuming minority carriers and low-level injection) inresponse to a perturbation:

The minority carrier lifetimes are now inversely proportional the R-Gcenter concentration NT:

In case of arbitrary injection levels and for both carrier types in a non-degenerate semiconductor, we find:

dpn(t)

dt=G

L!U =G

L!

"pn

#p

anddn

p(t)

dt=G

L!

"np

#n

!n"

1

NT

and !p"

1

NT

U =p

nn

n!ni2

( )"

pp

n+n

ie

(Ei!E

t)/kT#

$%&+"n nn +nie

(Et!E

i)/kT#

$%&

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2 7 3 C ti it E ti


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2.7.3 Continuity Equationsputting it all together

!n

!t=

!n

!tdrift

+

!n

!tdiffusion

=

1

q"

!

Jn

" #$$ %$$

+

!n

!tthermalR#G

=#$n

%n

" #$ %$

+

!n

!tother processes

e.g.GL

" #$$ %$$

!p

!t=

!p

!tdrift

+

!p

!tdiffusion

=#1q"

!

Jp

" #$$ %$$

+

!p

!tthermalR#G

=#$p%

p

" #$ %$

+

!p

!tother processes

e.g.GL

" #$$ %$$

Net change of carrier concentration due to currents and R/G:

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A t 18 2014 ECE 3040 Ch t 2 1 2 2 S i d t

1-D Continuity Equations

!np(x,t)

!t= +n

p

n

!"!x

+n"!np

!x+D

n

!2np

!x2 +G

L#np #np0

$n

!pn(x,t)

!t= #p

n

p

!"

!x#

p"!p

n

!xDrift

! "### $###

+Dp

!2p

n

!x2

Diffusion

! "# $#

+GL

Perturb.

%

#pn#p

n0

$p

thermalR#G

! "# $#

For minority carriers in 1-D case: Jp = q p p " # qDp $p

Jn = q n n "+ qDn $n

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