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Complementi di Fisica - Lecture 27 26-11-2012 L.Lanceri - Complementi di Fisica 1 “Complementi di Fisica” Lecture 27 Livio Lanceri Università di Trieste Trieste, 01-12-2010 26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 2 In this lecture Contents Doped (“extrinsic”) semiconductors at equilibrium Concentrations of charge carriers (electrons and holes) Fermi level Reference textbooks D.A. Neamen, Semiconductor Physics and Devices, McGraw- Hill, 3 rd ed., 2003, p.83-96 (Density of states, Fermi-Dirac), p. 115-145 (4.2 Dopant atoms and energy levels, 4.3 extrinsic semiconductor, 4.4 Statistics of donors and acceptors, 4.5 Charge neutrality, 4.6 Position of the Fermi level) R.F.Pierret, Advanced Semiconductor Fundamentals, Prentice Hall, 2003, 2 nd ed., p. 96-128. S.M.Sze, Semiconductor Devices - Physics and Technology, J.Wiley & Sons, 2 nd ed., 1985, p. 21-28.

Transcript of “Complementi di Fisica” - [email protected]/.../content/1/CF2012_L27_v03_ho.pdf ·...

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Complementi di Fisica - Lecture 27 26-11-2012

L.Lanceri - Complementi di Fisica 1

“Complementi di Fisica”Lecture 27

Livio LanceriUniversità di Trieste

Trieste, 01-12-2010

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 2

In this lecture• Contents

– Doped (“extrinsic”) semiconductors at equilibrium• Concentrations of charge carriers (electrons and

holes)• Fermi level

• Reference textbooks– D.A. Neamen, Semiconductor Physics and Devices, McGraw-

Hill, 3rd ed., 2003, p.83-96 (Density of states, Fermi-Dirac), p.115-145 (4.2 Dopant atoms and energy levels, 4.3 extrinsicsemiconductor, 4.4 Statistics of donors and acceptors, 4.5Charge neutrality, 4.6 Position of the Fermi level)

– R.F.Pierret, Advanced Semiconductor Fundamentals, PrenticeHall, 2003, 2nd ed., p. 96-128.

– S.M.Sze, Semiconductor Devices - Physics and Technology,J.Wiley & Sons, 2nd ed., 1985, p. 21-28.

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“extrinsic” (doped)semiconductors

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Donors and acceptors• Bond model:

n-type Si with “donor” impurities(As: 5 valence electrons)The fifth electron is “donated”to the conduction band; the remaining positive ion is fixed

p-type Si with “acceptor” impurities(B: 3 valence electrons)An additional electron is “accepted”from the valence band, creating a holeThe resulting negative ion is fixed

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Donors and acceptors• Energy band model

n-type Si with “donor ” energy levelsvery close to the conduction band;ionization energy is very small,most donor atoms are ionizedalready at room temperature !

p-type Si with “acceptor ” energy levelsvery close to the valence band;most acceptor atoms capture an electron, leaving a free holealready at room temperature !

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 6

Impurities and ionization energies - 1• Measured ionization energies (eV)

Sienergy gapEg = 1.12eV

GaAsenergy gapEg =1.42eV

EC EC - E (eV)

EC EC - E (eV)

E - EV (eV)EV

E - EV (eV)EV

A = Acceptor D = Donor

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Impurities and ionization energies - 2• Doping with donors (n-type)

– For instance: P or As in Si– Level close to EC

• Doping with acceptors (p-type)– For instance: B in Si– Level close to EV

• Unwanted impurities– Many impurities (unavoidable, to some extent) contribute

levels close to the center of the gap– Also these levels are important, as “traps” or “recombination-

generation centers” (more on this later)

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 8

Impurities and ionization energies - 3• Donor ionization energies (eV): can we understand or at

least guess the order of magnitude in simple terms? Bohrmodel…

isolated H atom, lowest level:Bohr model, n=1

−q, m0+q

vacuum: ε0

eV6.13

eV16.1318

1

22220

40

!=

!=!=

Ennh

qmEn "ionization energy

−q, mn*

+q

Si: ε/ε0 = 12

Donor atom in Si crystal:Lowest level for the external electron

eV085.09.0121eV6.13

18

21,

,0

20

222

4

,

!""!

##$

%&&'

(#$%&

'(=)=

**

D

nHnn

nD

E

Emm

nhqmE

++

+

ionization energy

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Impurities and ionization energies - 4• Acceptor ionization energies (eV): similar reasoning

isolated H atom, lowest level:Bohr model, n=1

eV6.13

eV16.1318

1

22220

40

!=

!=!=

Ennh

qmEn "

−q, m0+q

vacuum: ε0

ionization energy

+q, mp*

−q

Si: ε/ε0 = 12

Acceptor atom in Si crystal:Lowest level for the hole

eV018.019.0121eV6.13

18

21,

,0

20

222

4

,

!""!

##$

%&&'

(#$%&

'(=)=

**

D

nHpp

nD

E

Emm

nhqm

E++

+

ionization energy

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 10

Impurities and ionization energies - 5• The simple Bohr model for donor and acceptor

ionization energies:– Describes the impurities in an oversimplified way (pseudo-

hydrogen atoms with outer electron or hole orbiting throughthe semiconductor material):

• Effective mass mn*, mp*• Dielectric constant ε

– Gives the correct orders of magnitude for ionization energies(EI < 0.1eV): OK because many semiconductor atoms areincluded in the Bohr orbit corresponding to the lowest level(Exercise: easily checked by computing the Bohr radius)

– Is not supposed to be able to reproduce the details !• Small ionization energy ⇒ most donor/acceptor atoms

are ionized at room temperature

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n-type in more detailExpect:

n > ni mobile electronsND

+

fixed positive ions(out of ND donors)

p < ni mobile holes

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Intrinsic carrier concentration

gC E( ) =mn

! 2mn! E " Ec( )

# 2!3

F E( ) ! e" E"EF( ) kT

gV E( ) =mp

! 2mp! EV " E( )

# 2!3

1! F E( ) " e! EF !E( ) kT

n ! NCe" EC "EF( ) kT

p ! NVe" EF "EV( ) kT

NC ! 2 2"mn#kT

h2$

% &

'

( ) 3 2

NV ! 22"mp

#kTh2

$

% &

'

( ) 3 2

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Extrinsic carrier concentration

gC E( ) =mn

! 2mn! E " Ec( )

# 2!3

F E( ) ! e" E"EF( ) kT

gV E( ) =mp

! 2mp! EV " E( )

# 2!3

1! F E( ) " e! EF !E( ) kT

n ! NCe" EC "EF( ) kT

p ! NVe" EF "EV( ) kT

NC ! 2 2"mn#kT

h2$

% &

'

( ) 3 2

NV ! 22"mp

#kTh2

$

% &

'

( ) 3 2

The same ingredients…

…the same (formal) results ! But now n ≠ p : why ??

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 14

Carrier concentrations: intrinsic• (approximate) equations valid for both intrinsic and extrinsic:

• Intrinsic case (EF ≡ Ei)

n ! NCe" EC "EF( ) kT

NC ! 2 2"mn#kT

h2$

% &

'

( ) 3 2

NV ! 22"mp

#kTh2

$

% &

'

( ) 3 2

p ! NVe" EF "EV( ) kT

( ) ( )

( ) kTEVCi

kTEEVCi

VCiF

kTEEV

kTEECi

gVC

ViiC

eNNneNNnnp

EEEE

eNeNnpn

22

2...

!!!

!!!!

="==

+#==

====

NC = nieEC !Ei( ) kT

NV = nieEi !EV( ) kT

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Carrier concentrations: extrinsic• (approximate) equations valid for both intrinsic and extrinsic:

• Extrinsic case: n ≠ p ⇔ EF ≠ Ei

( ) kTEEC

FCeNn !!"23

2

22 !!"

#$$%

&'

(

hkTmN n

C)

23

2

22 !

!"

#$$%

&'

(

hkTm

N pV

)( ) kTEEV

VFeNp !!"

( ) ( ) ( )

( ) ( ) ( )

AiA

DiDi

kTEEi

kTEEkTEEi

kTEEi

kTEEkTEEi

NnnNppNnpNnnnnp

eneenp

eneennFiVFVi

iFFCiC

2

22

:type-

:type-

!!

!!=

==

==""""

""""

( ) kTEEiC

iCenN !=

( ) kTEEiV

VienN !=

approximatefor

completeionization

EF moves away from Ei

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Fermi level EF for complete ionization• Complete ionization of donors (n-type):

• Complete ionization of acceptors (p-type):

( ) ( )

!!"

#$$%

&='(

=()= '''

D

CFC

kTEE

D

CD

kTEEC

NNkTEE

eNNNeNn FCFC

ln

( ) ( )

!!"

#$$%

&='(

=()= '''

A

VVF

kTEE

A

VA

kTEEV

NNkTEE

eNNNeNp FCVF

ln

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EF : dependence on n, p at fixed T

n-type

p-type

n ! ND " EC # EF $ kT lnNC

ND

%

& '

(

) *

p ! NA " EF # EV = kT ln NV

NA

$

% &

'

( )

NA cm-3[ ]

ND cm-3[ ]

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Concentrations and EF : an example• A silicon ingot is doped with As (ND≈1016 atoms/cm3). Find the carrier

concentration and the Fermi level at room temperature (T=300K)Assuming complete ionization of donors:

n ! ND =1016cm"3

ni = NCe" EC "Ei( ) kT !

! 2.8 #1019( ) # 5.2 #10"10( ) !!1.45 #1010cm"3

p ! ni2 ND =

1.45 #1010( )21016

=

= 2.1#104cm"3

EC ! EF = kT ln NC

ND

"

# $

%

& ' = 0.0259ln 2.8 (10

19

1016"

# $

%

& ' =

= 0.206 eV

n = nieEF !Ei( ) kT " EF ! Ei = kT ln n

ni

#

$ %

&

' (

EF ! Ei = 0.0259ln 1016

1.45 )1010#

$ %

&

' ( = 0.354 eV

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Carrier concentrations: general case• Both donor and acceptor impurities present

simultaneously– Concentrations: ND and NA respectively– ND

+ and NA− : concentrations of ionized donors and acceptors

• Starting point: overall charge neutrality(only “unbalanced” charges appear explicitely),and mass action law

n + NA! = p + ND

+ np = ni2

ND+ = ND ! nD

NA! = NA ! pA

nD electrons in donor states (non-ionized donors)

pA holes in acceptor states (non-ionized acceptors)

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Carrier concentrations: general case

ND+ = ND ! nD

NA! = NA ! pA

nDpA

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Carrier concentrations

• Assuming complete ionization of donors and acceptors

NA! " NA ND

+ " ND

n + NA = p + ND np = ni2

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Carrier concentrations• Solving for n, p:

2iDA nnpNpNn =+=+

n-type: ND > NA p-type: NA > ND

nn = 12ND ! NA + ND ! NA( )2 + 4ni

2" # $

% & '

pn = ni2 nn

if ND ! NA >> ninn ( ND ! NA

pp = 12NA ! ND + NA ! ND( )2 + 4ni

2" # $

% & '

np = ni2 pp

if ND ! NA >> nipp ( NA ! ND

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Extrinsic Fermi level vs temperature• Fermi level computation in

the general case, taking Tinto account:– from ND and NA compute n, p– from n, p extract EF

• ni increases with T !– Low T: extrinsic behaviour– High T: intrinsic behaviour

n = nieEF !Ei( ) kT " EF ! Ei = kT ln n

ni

#

$ %

&

' (

p = nieEi !EF( ) kT " Ei ! EF = kT ln p

ni

#

$ %

&

' (

26-11-2012 L.Lanceri - Complementi di Fisica - Lecture 27 24

Electron density vs temperature• Another point of view:

electron concentrationin n-type semiconductor– Very low T:

• “freeze-out” region• donors not ionized

– Medium T:• “extrinsic” region• n ≈ ND

– High T:• “intrinsic” region• n ≈ ni

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Non-degenerate and degenerate s.c.• All the results in this lecture assumed the Boltzmann

approximation, that is EC - EF > 3kT, EF - EV > 3kT(“non-degenerate” semiconductors)– Relatively small doping concentrations: donor and acceptor atoms

can be considered as “isolated” (energy levels)

• For very large doping concentrations (ND ≈ NC , NA ≈ NV)(“degenerate” semiconductors)– Donor (acceptor) energy levels become bands, overlapping with the

conduction (valence) band– The effective forbidden gap becomes smaller– The Fermi level is inside the enlarged conduction (valence) band,

with a very large concentration of electrons (holes) available forconduction

– The Boltzmann approximation is no longer valid in this case

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Lecture 27 - summary• The “mass action law” (np = ni

2) is valid also for extrinsic s.c.

• The intrinsic concentration ni increases with T

• We found relations between n, p, ni and EF

• In the intrinsic case n = p = ni

• We have learned how to compute n, p, and EF in the generalextrinsic case (depending on ND, NA)

• Specifying EF is a very convenient way of parameterizing n, p

• Intrinsic/extrinsic behaviour depends on T !

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Lecture 27 - exercises• Exercise 1: A silicon sample at T=300K contains an acceptor impurity

concentration of NA=1016 cm-3. Determine the concentration of donorimpurity atoms that must be added so that the silicon is n-type and theFermi energy is 0.20 eV below the conduction band edge.

• Exercise 2: Find the electron and hole concentrations and Fermi levelin silicon at 300K (a) for 1x1015 boron atoms/cm3 and (b) for 3x1016 boronatoms /cm3 together with 2.9x1016 arsenic atoms/cm3.

• Exercise 3: Calculate the Fermi level of silicon doped with 1015, 1017

and 1019 phosphorus atoms/cm3, assuming complete ionization. From thecalculated Fermi level, check if the assumption of complete ionization isjustified for each doping. Assume that the ionized donors density isgiven by ND

+ = ND(1-F(ED)).