solid state physics 07-semiconductors

Solid State PhysicsUNIST, Jungwoo Yoo

1. What holds atoms together - interatomic forces (Ch. 1.6)2. Arrangement of atoms in solid - crystal structure (Ch. 1.1-4) - Elementary crystallography - Typical crystal structures - X-ray Crystallography3. Atomic vibration in solid - lattice vibration (Ch. 2) - Sound waves - Lattice vibrations - Heat capacity from lattice vibration - Thermal conductivity4. Free electron gas - an early look at metals (Ch. 3) - The free electron model, Transport properties of the conduction electrons---------------------------------------------------------------------------------------------------------(Midterm I) 5. Free electron in crystal - the effect of periodic potential (Ch. 4) - Nearly free electron theory - Block's theorem (Ch. 11.3) - The tight binding approach - Insulator, semiconductor, or metal - Band structure and optical properties6. Waves in crystal (Ch. 11) - Elastic scattering of waves by a crystal - Wavelike normal modes - Block's theorem - Normal modes, reciprocal lattice, brillouin zone7. Semiconductors (Ch. 5) - Electrons and holes - Methods of providing electrons and holes - Transport properties - Non-equilibrium carrier densities8. Semiconductor devices (Ch. 6) - The p-n junction - Other devices based on p-n junction - Metal-oxide-semiconductor field-effect transistor (MOSFET)---------------------------------------------------------------------------------------------------------------(Final)

All about atoms

backstage

All about electrons

Main character

Main applications


Semiconductors

- Electrons and holes- Methods of providing electrons and holes- Transport properties- Non-equilibrium carrier densities


Introduction

The name “semiconductor” implies that it conducts somewhere between the two cases (conductors or insulators)

Conductivity :

s metals ~ 1010/Ω-cm

s insulators ~ 10-22/Ω-cm

The conductivity (σ) of asemiconductor (S/C) lies between these two extreme cases.

S/C

Category of materials based on range of conductivity


0a

a

k

Forbidden band

First allowed band

Second allowed band

gEF

If the band is partially filled a metal If bandgap is greater than 3.2 eV a insulator (no visible light can be ab-sorbed) bandgap is less than ~ 2.5 eV a semiconductor

Introduction

Category of materials based on electronic structure


Important semiconductor materials are

I) Group IV: Si, Ge (diamond structure) Si, Eg (at RT) ~ 1.12 eV, Indirect bandgap Ge, Eg (at RT) ~ 0.67 eV SiC (zincblende structure), Eg (at RT) ~ 2.86 eV,

II) Group III-V: GaAs, InSb, GaN, AlP (zincblende structure)

GaAs, Eg (at RT) ~ 1.43 eV, Direct bandgap InSb, , Eg (at RT) ~ 0.17 eV, GaN, Eg (at RT) ~ 3.4 eV, InP, Eg (at RT) ~ 2.45 eV,

Introduction


IntroductionThe properties of semiconductor is dominated by electrons in states close to the top of the valence band and the bottom of the conduction band

gE

eg m

kE

2

22

hm

k

2

22

k

Conduction band

Valence band

The states near the top of the valence band behave like free particles of negative mass hm

a The behavior of a nearly full valence band can be calculated by ignoring the filled states completely and regarding each empty state as being occupied by a particle of positive charge lel positive mass and energy . hm

hmk 2/22

a This fictious particles are referred to as holes.


HolesFrom the conservation of energy, the energy required to create electron in state k1 and hole in state k

heg m

k

m

kE

22

2221

2

Identified as an energy of Electron in state k1

Identified as an energy of hole in state k

heg m

k

m

kE

22

2221

2

hh m

k

2

22

k

Hole dispersion relationa obtained from inverting the dispersion relation of valence band electron

Similarly, the removal of an electron momentum from the valence band corresponds to the addition of momentum to the valence band.

k

k

Hole has

hh m

k

2

22

kph


Holes

The equation of motion for a hole

)( BvEev

dt

vdm h

h

hhh

The scattering of a hole from state to state corresponds to the scattering of an electron from state to state so that the scattering time for holes is directly related to that for electrons.

1k 2k2k 1k h

+ sign


Energy band structures of GaAs and Si

Band Structure of Semiconductors

-1

-2

0

2

3

1

4GaAs Conduction

band

Valance band

0

Eg

[111] [100] k

Ene

rgy

(eV

)

-1

-2

0

2

3

1

4Si Conduction

band

Valance band

0

Eg

[111] [100] k

Ene

rgy

(eV

)


Energy band structure of GaAs

Band gap is the smallest energy separation between the valence and conduction band edges.

The smallest energy difference occurs at the same momentum value

Direct band gap semiconductor


-1

-2

0

2

3

1

4GaAs Conduction

band

Valance band

0

ΔE=0.31

Eg

[111] [100] k

Ene

rgy

(eV

)


Energy band structure of Si

The smallest energy gap is between the top of the VB at k=0 and one of the CB minima away from k=0

Indirect band gap semiconductor


k

-1

-2

0

2

3

1

4Si Conduction

band

Valance band

0

Eg

[111] [100]

Ene

rgy

(eV

)


Methods of Providing Electrons and Holes

Donor and acceptor impuri-ties

To increase the conductivity, one can dope pure s/c with atoms from column lll or V of periodic table. This process is called as doping and the added atoms are called as dopants impurities.

This impurities takes up a substitutional rather than an interstitial po-sition.After forming four covalent bonds demanded by the structure, there is an extra valence electron (hole) leading occupied states in the con-duction band (empty states in the valence band)

The conductivity of a pure (intrinsic) s/c is low due to the low number of free carriers (The number of carriers are generated by thermally or electromagnetic radiation for a pure s/c)




Si

Si Si

Si

V

Weakly bound electron

Normal bond with two electrons

Si + Column V impurity atoms

• n-type doped semiconductor




Si

Si Si

Si

III


Si + Column III impurity atoms

• p-type doped semiconductor

Weakly bound hole



Donor and acceptor impuri-tiesThe extra electron from dopant is not completely free since escape of electron to large distances leaves the impurity atoms with a net posi-tive charge; at finite separations the positive charge exerts an attrac-tive force on the electron and leads to the existence of a bound state for the electron.

The strength of binding between charged impurity and electron can be estimated by employing standard result for the energy levels of the hydrogen atom

2

eV6.13

nEn

-13.6eV is Redberg constant

202

4

42eV6.13

eme

The energy level of hydrogen atom given by

and Bohr radius is A529.0

42

220

0 em

na

e




In medium with dielectric constant

,

42

*2

0222

4

n

emEn

02

22

4*

em

nrn

For germanium, the effective mass is , dielectric constant is emm 2.0* 8.15

,-0.01eVeV6.13*21

em

mE

oo

1 A04A53.0*

m

mr e

a the combination of small effective mass and large dielectric constant gives very weak binding of the extra electron to the impurity and a very extended wavefunction for the bound state.

Note E1 is less than at room temperature (0.026 eV) TkB

a most of electrons provided by doping free to move through the crystal




gE

k

Conduction band

Valence band

DEDonor level

DG EE

Acceptor levelAE AE

At T = 0, all extra elec-trons Occupy at donor level.

At finite T, thermal en-ergycan ionize extra elec-tronsinto the conduction bands



Thermal excitation of carri-ersThe probability of occupation of a state of energy is given by

1

1),(

/)( TkBe

Tf

)0( TF

At T = 0, electron occupy up to , so that Fermi energy is referred to

At finite T, we define Fermi level as

)(T



Thermal excitation of carri-ersThe density of states for conduction and valence band are given from dispersion relation

2/12/332

)()2(2

)( ge EmV

g

2/12/332

)()2(2

)(

emV

g

Conduction band:

Valence band:

GE0

DG EE

AN)(g

)(g

)(f

AE

DN

The Fermi level is somewhere in the band gap.

The Fermi function is very close to the unity in the valence band and very small in the conduction band.



Thermal excitation of carri-ersThe analytic expressions for the # of electrons in the conduction band the # of holes in the valence band

For the electron energy in the conduction band, TkB TkBef /)()(

The # of electron per unit volume in the conduction band

GE

dgfV

n )()(1

G

B

E

TkGe deEm

/)(2/12/3

32)()2(

2

1

0

/2/1/)(32

2/3

2

)2(

deem TkTkEe BBG

TkE

CBGeN /)(

where

2/3

2

22

h

TkmN Be

C

Nc is the effective number of states per unit volume in the conduction band if we imagine them concentrated at the bottom of the band,GE

TkEC

BGeNn /)(

)( GC EfNn


,)1(

10

naxn

a

ndxex

!)1( nn

)2/1(

,2

)12(531)2/1(

m

mm

,....3,2,1m



Thermal excitation of carri-ersThe probability that a state in the valence band is occupied by a hole is , )(1 f

1

1

11

11)(1 /)(/)(

/)(

/)(

TkTk

Tk

Tk BB

B

B ee

e

ef

For ,)( TkB TkBef /)()(1

dgfV

p )()](1[1 0

0 /)(2/1

32

2/3

)(2

)2(

dem Tkh B

0

/2/1/32

2/3

2

)2(

deem TkTkh BB

,/ Tk

VBeN

where

2/3

2

22

h

TkmN Bh

V

Nv is the effective number of states per unit volume in the valence band if they were all concentrated at the top of the band, 0

TkV

BeNp /

)]0(1[ fNp V



Thermal excitation of carri-ers

The product of hole and electron concentration

TkEVC

BGeNNnp /

Independent of and impurity concentration, but it depends on the temperature

TkV

TkEC

B

BG

eNp

eNn/

/)(

Law of mass action



For Intrinsic Semiconductor

For pure semiconductor, the electron and hole concentrations are equal since a hole in the valence band can only be created by the excitation of an elec-tron into the conduction band.

TkEVCii

BGeNNpn 2/2/1

The chemical potential of an intrinsic semiconductor can be expressed

1// //)( TkV

TkEC

BBG eNeNpn

CVTkE NNe BG //)2(

a

a )/ln(4

3

2

1)/ln(

2

1

2

1ehBGCVBG mmTkENNTkE

The intrinsic carrier concentrations are

The typical intrinsic carrier concentration at RT for Si: (2×1016m-3)Ge: (2×1019m-

3)

Typically, the Fermi level (chemical potential) is essentially in the middle of the band gap since

GB ETk

TkV

TkEC

B

BG

eNp

eNn/

/)(



Intrinsic be-havior

)(2 Tnnp i

The product of electron and hole concentrations in any semiconductor is often ex-pressed as

Practically, it is impossible to have ideally pure semiconductor, there will be al-ways some impurities

The criterion for intrinsic semiconductor is

ND < 2×1016m-3 for Si

ND < 2×1019m-3 for Ge

The atomic concentration is 5×1028

1 in 1012 for Si

1 in 109 for Ge

The content of impurities should be less than



Extrinsic be-haviorWhen acceptors and donors are present, chemical potential changes

We determine chemical potential based on the electrical neutrality of the whole crystal.

The condition for electrical neutrality requires the densities of negative and posi-tive charge associated with should be equal.

DA NpNnAN : concentrations of ionized accep-

tors

: concentrations of ionized donors

DN

)( AAA EfNN )](1[ DGDD EEfNN



Extrinsic be-haviorTypically, both types of impurities (donors and acceptors) present in actual semi-conductor materials

Consider n doped semiconductor, a # of donors exceed # of accep-tors

i) At T = 0,

a All acceptor levels are fully occupied by electrons from the donor impuri-tiesa donor levels un-ionizedAD NN

a Fermi energy, DGF EET )0(

ii) At low T ( ), DB ETk

a Donors are started to be ionized but its number have changed much yet.a The Fermi level still close to the Fermi energy DG EET )(

TkV

TkEC

B

BG

eNp

eNn/

/)(

TkEC

BDeNn / TkEVCi

BGeNNn 2/2/1 since GD EE

a The electron concentration at conduction band

a From law of mass action, ipp

a Electron is majority carrier, hole is minority carrier (n-type materi-als)





i) As T increase

a The # of ionized donors becomes comparable to the total # of donorsa The probability of occupation of donor level becomes low

a Fermi level lies below the donor level

a For some range of T, all the donors and acceptors are ionized

TkV

TkEC

B

BG

eNp

eNn/

/)(

AD NNn

a Then the Fermi level can be obtained as

AD

CBG NN

NTkE ln

For a good semiconductor device operation, all the impurities should be ionized at room temperature.





i) As T increase further,

a The # of carriers induced by the thermal activation exceed the doping a Hole concentration increases towards the electron concentration

a Fermi level falls towards the centre of the gap

a Eventually, it behave like intrinsic semiconductors



1/T (K-1)

1/T (K-1)

E

n,p (m-3)




Consider p doped semiconductor, a # of acceptors exceed # of donors

i) At T = 0,

a All donor levels are empty by acceptor impurities

a Empty acceptor levelsDA NN

a Fermi energy, AF ET )0(

ii) At low T ( ), DB ETk

a Acceptors are started to be ionized but its number haven’t changed much yet.a The Fermi level still close to the Fermi energy AET )(

TkV

TkEC

B

BG

eNp

eNn/

/)(

TkEC

BAeNp / TkEVCi

BGeNNp 2/2/1 since GA EE

a The hole concentration at valence band

a From law of mass action, inn

a Hole is majority carrier, electron is minority carrier (p-type materi-als)




Consider p doped semiconductor,

i) As T increase

a The # of occupied acceptors becomes comparable to the total # of ac-ceptorsa The probability of occupation of acceptor level becomes high

a Fermi level lies above the acceptor level a For some range of T, all the acceptors are ionized

TkV

TkEC

B

BG

eNp

eNn/

/)(

DA NNp

a Then the Fermi level can be obtained as

DA

VB NN

NTk ln

For a good semiconductor device operation, all the impurities should be ionized at room temperature.

a # of acceptors exceed # of donors




Consider p doped semiconductor,

i) As T increase further,

a The # of carriers induced by the thermal activation exceed the doping a electron concentration increases towards the hole concentration

a Fermi level rise towards the centre of the gap

a Eventually, it behave like intrinsic semiconductors

a # of acceptors exceed # of donors


Absorption of Electromagnetic Radiation

The absorption coefficient for electromagnetic radiation of germanium vs photon energy at temperature 77 and 300 K.

The increase in absorption occur when the photons have sufficient energy to ex-cite a valence band electron into the conduction band

a the photon energies at the onset of absorption therefore provide a measure of the energy gap in semiconductors

Sharp increase of absorp-tion occur at two place

a this is due to the fact that germanium has indirect band-gap structure

Abso

rpti

on c

oeffi

cient

(cm

-1)

Photon energy (eV)



gE

]111[k

'gE

The absorption and emission of photon (or phonon) should satisfy i) conservation of energy ii) conservation of momentum

ph phkk

Photon has sufficient energy to overcome bandgap but has small momentum

For conduction electron, R~

Phonon has low energy but has momentum comparable to conduction

Bphonon k~ Rhk phonon /

eV32 ph /hk nm10~ 3For visible light,



]111[k


ph phkk





pho-ton

'gE

0k





ph phkk





]111[k

'gEpho-

ton 0kphonon 0~

0kk



Transport Properties

)(

)(

BvEev

dt

vdm

BvEev

dt

vdm

hh

hhh

ee

eee

The equation of motion for the carriers in the presence of electric and magnetic fields

EEm

ev

EEm

ev

hh

hh

ee

ee

Em

pe

m

nevpevnej

h

h

e

ehe

22

EEpene he

)(

he pene

h

hh

e

ee

m

e

m

e

In the intrinsic region, contribution of electrons and holes to the conductivity are usually similar

In the extrinsic region, conductivity is normally dominated by the majority carrier

Electrical conductiv-ity

In steady state,


Conductivities for arsenic-doped n-type germa-nium.Arsenic donor impurities with the approximate concentrations indicated.

Thermal activation of donor

Conduct

ivit

y (

W-1m

-1)

The dashed line indicate steep increase associated with the transition to the intrinsic behavior

1/T (K)Scattering with phonon

Extrinsic region: all donors are ex-cited a 113 m10

C10 19ea from 112 sVm1 e

a from s10 12e gme k10 31

Transport Properties


Hall effect

The Hall Effect

In steady state, the Lorentz force on the electrons is just balanced by the force due to the Hall field

Bve

HEe

HE

jBRE HH

BveEev

me

z

x

B

v

kji

e

00

00

xxe eEvm /

)(0 BvEe xy

a

neRH /1 HRand

BjRneBjBvE xHxxy )/(

a


Hall effectThe Hall Effect in Semiconductor

peR

neR

H

H

1

1

in n-type semiconductor(n>>p)

in p-type semiconductor(p>>n)

The Hall measurement deter-mine the majority carrier, car-rier concentration, and mobil-ity

HR

Consider arsenic-doped n-type germanium in previous sec-tion 13

1922mT10

1010

11

neRH

Since resistivity is m10/1 3

a the Ohmic and Hall electric fields are equal in a field of 1Ta the total electric field is 45 degree to the current flow, (Hall angle = 45 de-gree)a We define this magnetic field as Hall field B0 (Hall field is useful measure of the strength of the Hall effect)

11

0 HR

B

The Hall effect in semiconductors is much larger than in metals due to smaller carrier density


Non-Equilibrium Carrier DensitiesWhen we operate semiconductor devices, the carrier concentrations are dis-turbed from their thermal equilibrium values.

a to understand the behavior of such devices we need the equations that de-scribe the variation in space and time of the disturbances.

The carrier densities can be expressed as

),('),( 0 txnntxn

),('),( 0 txpptxp

00 , pn are the thermal equilibrium concentrations and independent of posi-tion.

',' pn are departures of the concentrations, these disturbances are small compared to the majority carrier concentrations.


The process that can change the carrier concentrations for a region of semi-conductor between x and x + dx

i) Recombination

ii) Generation

iii) Diffusion

iv) Drift

r : recombination rate per unit volume

g : recombination rate per unit volume

,x

nDJ ee

When the carrier concentration depends on position, the diffusion will occur

x

nDJ hh

The net rate per unit area at which electrons enter the region between x and x+dx as a result of diffusion

Current density

,2

2

xx

nDx

x

Je

e

In the presence of an electric field, Current density ,EnJ ee ,EnJ pp

The net rate per unit area at which electrons enter the region between x and x+dx as a result of drift

,)(

xx

nEx

x

Je

e

Non-Equilibrium Carrier Densities


Non-Equilibrium Carrier DensitiesThe Continuity Equation

x

nE

x

nDrg

t

nee

)(2

x

pE

x

pDrg

t

phh

)(2

Addition of all the contribution that induce change in carrier concentration leads

Generationg/vol

Recombinationr/vol

x x+dx

Diffusion Diffusion

x

n x

nD

2

xx

n x

nD

2

Drift Drift

xnEn xxnEn


Non-Equilibrium Carrier DensitiesElectrical Neutrality

When there is momentary departure from electrical neutrality at some place of a homogeneous semiconductor, electrical neutrality will be achieved by the redis-tribution of the majority carrier resulting from the electric field associated with the charged region.

DA NpNn 00via the condition for electrical neutral-ity

The charge density at a point in the semiconductor is

AD NnNpe

We obtain '' npe i) electrical neutrality is attained when the disturbances of the electron and hole densities are equal

ii) The electric field is generated by the departure from electrical neu-tralityThe generated electric field is given by the Gauss law

'' np

'' np

0

E

0

)''(

npe

dx

dE


Non-Equilibrium Carrier DensitiesElectrical NeutralityConsider n–type homogeneously doped semiconductors, the effect of this electric field on the distribution of the majority carrier is given

02

2 )''('''

npe

nx

nE

x

nDrg

t

neee

second orderslow process

2

22 '

)''('

x

nnp

t

nDD

ee

D ne

00 : dielectric relaxation time

2/1

0

e

eD ne

D

: Debye length

For a finite value of (p’-n’) at any point in space or time disappears by a redis-tribution of the electrons on a time scale and a length scale .

D D

For , 11m100 e s10100

1010 1211

0

e

D

02/1

20 A400

ne

TkBD

e

Be e

TkD From , K300T

,10 322m10 n


Non-Equilibrium Carrier DensitiesElectrical NeutralitySince the motion of the majority carrier is such as to make everywhere and it is only necessary to solve the continuity equation for the minority carriers:

x

nE

x

nDrg

t

nee

''' 2

'' pn

x

pE

x

pDrg

t

pee

''' 2

in p-type semiconductor,

in n-type semiconductor,


Non-Equilibrium Carrier DensitiesGeneration and RecombinationGeneration: the rate g at which valence band electrons make transitions to the con-duction band depends on the number of electrons in the valence band and the probability that at any time one of them acquires enough energy to make the transi-tion. a the generation rate remains at its equilibrium value )(0 Tg

Recombination: the recombination rate for direct recombination depends linearly on both carrier concentrations.

a the recombination rate npTkr )(

)')(')(()()()( 0000 ppnnTkTgnpTrTgrg

)'''')(( 00 pnnppnTk

since in thermal equilibrium, the recombination rate and generation rate should be equal

000 )()( pnTkTg

n-type p-type

n

prg

'

p

nrg

'

n-type

p-typen is minority carrier lifetime

Typically ~ 10-7s


Non-Equilibrium Carrier DensitiesGeneration and RecombinationThe continuity equation for minority carriers

x

nE

x

nD

n

t

nee

n

'''' 2

x

pE

x

pD

p

t

pee

e

'''' 2

for electrons in p-type

for holes in n-type


Non-Equilibrium Carrier DensitiesInjection of minority carriers at a steady rateConsider a long thin rod of p-type semiconductors with cross sectional area A

If we inject electrons with a steady rate of N per unit area per second at one end, in a steady state ( ) and in the absence of an applied field, the continuity equation becomes

0/' tn

x

nE

x

nD

n

t

nee

n

'''' 2

22

2 ''

eL

n

x

n

a where diffusion length

2/1)( pee DL

aThe solution of continuity equation eLxCen /'

From boundary condition at x=0,

x

nDJ ee

a

ee

xe L

CD

x

nDN

0

Therefore, the excess electron concentration is given by

eLx

e

e eD

NLn /'

AInjection of N electron per unit area per sec-ond


Non-Equilibrium Carrier DensitiesInjection of minority carriers at a steady rate

eLx

e

e eD

NLn /'

'n

eL eL2x

Excess concentration of electrons as a func-tion of position within the rod

eL eL2x

Ne

Current density

Electron contribution

Holecontribution

Constant total current den-sity

The electric current den-sity


Non-Equilibrium Carrier DensitiesInjection of a pulse of minority carriers

A Bn-type

Closing the switch for a short period of time injecs a pulse of holes at point A and a pulse of electrons at point B

p’ n’ The excess carrier densities immedi-ately after the injection of the pulses

p’, n’ The excess carrier concentration af-ter a few dielectric relaxation time

p’, n’The pulse of holes has broadened due to diffusion. The area de-creases due to recombination

p’, n’

Eth

In the presence of addition steady electric field, the pulse of holes also drifts at a velocity Eh

tD

xt

tD

Pp

hnh 4exp

)4('

2

2/1 The area decrease as

The width of pulse increases as

nt /exp

2/1tDh


Summary

0a

a

k

Forbidden band

First allowed band

Second allowed band

gEF

If the band is partially filled a metal If bandgap is greater than 3.2 eV a insulator (no visible light can be ab-sorbed) bandgap is less than ~ 2.5 eV a semiconductor

Category of materials based on electronic structure


Summary

Important semiconductor materials are

I) Group IV: Si, Ge (diamond structure) Si, Eg (at RT) ~ 1.12 eV, Indirect bandgap Ge, Eg (at RT) ~ 0.67 eV SiC (zincblende structure), Eg (at RT) ~ 2.86 eV,

II) Group III-V: GaAs, InSb, GaN, AlP (zincblende structure)

GaAs, Eg (at RT) ~ 1.43 eV, Direct bandgap InSb, , Eg (at RT) ~ 0.17 eV, GaN, Eg (at RT) ~ 3.4 eV, InP, Eg (at RT) ~ 2.45 eV,

-1

-2

0

2

3

1

4GaAs Conduction

band

Valance band

0

Eg

[111] [100]k

Ene

rgy

(eV

)

-1

-2

0

2

3

1

4Si Conduction

band

Valance band

0

Eg

[111] [100]k

Ene

rgy

(eV

)


gE

eg m

kE

2

22

hm

k

2

22

k

Conduction band

Valence band

Hole has

hh m

k

2

22

kph

Summary

The properties of semiconductor is dominated by electron at the bottom of con-duction band and holes at the top of valence bands.

We introduce concept of hole to describe a nearly full valence band. Hole has posi-tive charge lel positive mass and energy . hm hmk 2/22

Electron and hole dispersion relation


The conductivity of a pure (intrinsic) s/c is low due to the low number of free carriers. (To increase the conductivity, one can dope pure s/c with atoms from column lll or V of periodic table. This process is called as doping and the added atoms are called as dopants impurities. This impurities takes up a substitutional rather than an interstitial position.

Summary

p-type doped semiconductor (Column IV with Column III impurity atoms): deficiency of electron

n-type doped semiconductor (Column IV with Column V impurity atoms): excess of electron

Si

Si Si

Si

III


Weakly bound hole

Si

Si Si

Si

V

Weakly bound electron


Doping


gE

k

Conduction band

Valence band

DEDonor level

DG EE

Acceptor levelAE AE

At T = 0, all extra electrons Occupy at donor level.

At finite T, thermal energycan ionize extra electronsinto the conduction bands

Summary

Donor and Acceptor level

Rough estimation of donor electron binding energy ~ 0.01 eVand its radius ~ 40 Å


The probability of occupation of a state of energy is given by

1

1),( /)( TkBe

Tf

)0( TF

)(T

Summary

Carrier concentration

Fermi energy: chemical potential at T =0:

Fermi level: chemical potential at finite T:

The analytic expressions for the # of electrons in the conduction band the # of holes in the valence band

TkV

TkEC

B

BG

eNp

eNn/

/)(

,

22

2/3

2

Tkm

N BeC

2/3

2

22

Tkm

N BhV

Law of mass action:TkE

VCBGeNNnp /

For the intrinsic semiconduc-tor, carrier concentrations are

TkEVCii

BGeNNpn 2/2/1

)/ln(4

3

2

1ehBG mmTkE

)(2 Tnnp i

1/ pn a


1/T (K-1)

1/T (K-1)

E

n,p (m-3)

SummaryTemperature dependence of carrier concentration


Abso

rpti

on c

oeffi

cient

(cm

-1)

Photon energy (eV)

]111[k

'gEpho-

ton 0kphonon 0~

0kk

Summary


The absorption and emission of photon (or phonon) should sat-isfy i) conservation of energy ii) conservation of momentumPhoton : large energy, small momentum ,/hk nm10~ 3Electron: large momentum R~Phonon: low energy, large momentum Bphonon k~ Rhk phonon /


)(

)(

BvEev

dt

vdm

BvEev

dt

vdm

hh

hhh

ee

eee

EEm

ev

EEm

ev

hh

hh

ee

ee

Em

pe

m

nevpevnej

h

h

e

ehe

22

EEpene he

)(

he pene

h

hh

e

ee

m

e

m

e

peR

neR

H

H

1

1

in n-type semiconductor(n>>p)

in p-type semiconductor(p>>n)

The Hall measurement deter-mine the majority carrier, car-rier concentration, and mobil-ity

HR

SummaryTransport properties

Steady state sol.

Ohm’s law

Hall effects

Eqn. of motion


x

nE

x

nDrg

t

nee

)(2

x

pE

x

pDrg

t

phh

)(2

Addition of all the contribution that induce change in carrier concentration leads

Generationg/vol

Recombinationr/vol

x x+dx

Diffusion Diffusion

x

n x

nD

2

xx

n x

nD

2

Drift Drift

xnEn xxnEn

SummaryContinuity equation


0

2 )''('''

npe

nx

nE

x

nDrg

t

neee

second orderslow process

2

22 '

)''('

x

nnp

t

nDD

ee

D ne

00 : dielectric relaxation time

2/1

0

e

eD ne

D

: Debye length

Summary

Redistribution of electrons(holes)

x

nE

x

nDrg

t

nee

)(2

For n-type

Non-equilibrium charge induces electric field0

)''(

npe

dx

dE

For , 11m100 e s10~ 12D

0

A400~D

For a finite value of (p’-n’) at any point in space or time disappears by a redis-tribution of the electrons on a time scale and a length scale .

D D


x

nE

x

nD

n

t

nee

n

'''' 2

2

2 ''

eL

n

x

n

a where diffusion length

2/1)( pee DL

Therefore, the excess electron concentration is given by

eLx

e

e eD

NLn /'

SummaryGeneration and Recombination

Generation rate remains at its equilibrium value )(0 Tg

Recombination: the recombination rate for direct recombination depends linearly on both carrier concentrations.

n

prg

'

p

nrg

'

n-type

p-type

n is minority carrier lifetime

Typically ~ 10-7s

x

nE

x

nDrg

t

nee

''' 2

For electron concentration in p-type semiconductor,

Injection of minority carriers at a steady rate


eLx

e

e eD

NLn /'

'n

eL eL2x

eL eL2x

Ne

Current density

Electron contribution

Holecontribution

Constant total current den-sity

Summary

solid state physics 07-semiconductors

Science

conduction band valence

solid state physics

metal band structure

conduction band edges

state k1

jungwoo yoo holes

energy of electron

solid crystal structure