1

Homogeneous Semiconductors

direct indirect

Ch. 28

filled bands insulator

thermal promotion

photoconductivity

TkE Bge 2/−

semiconductor Eg ~ .2 - 2 eV

Gap varies linearly with T at RT and quadratically at T 0.

Semiconductor Band Structure

( ) νμνμν

μεε kkk C1

2

2)( −Μ+= ∑hr

( ) νμνμν

μεε kkk V1

2

2)( −Μ−= ∑hr

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

3

23

2

22

1

21

2

2)(

mk

mk

mkk C

hrεε

⎟⎟⎠

⎞⎜⎜⎝

⎛++−=

3

23

2

22

1

21

2

2)(

mk

mk

mkk V

hrεε

effective mass theory

SiGe

“cigar-shaped” CBM

2

Semiconductor Bands

Sze, Phys. of Semicond. Dev.

indirect bandgap

free electrons

Cyclotron Resonance

Hvce

dtvd rr

mr

×=Μ

( )tievv ω−= 0Re rr

cmeH

*=ω

2/1* det

⎟⎟⎠

⎞⎜⎜⎝

⎛ΜΜ

=zz

m

2/1

3232

221

21

321*

ˆˆˆ ⎟⎟⎠

⎞⎜⎜⎝

⎛

++=

mHmHmHmmmm

zzHH Μ=⋅Μ⋅= ˆˆ

Μ=det

)(80.2)10(2

9 kgaussHHzfcc ×==πω

2/16/1

,2/1,2/3)001()1,,(

1212 =

=⇒

⎩⎨⎧

=⋅+ n

nGe

Sinnn

Ge Si

3

Carrier Mobility

How does the temperature-dependence of conductivity change from metals to semiconductors?

mne /2τσ =

The concept of mobility: μ

Evd μ=

Enenevj d μ==

μσ ne=

T=300 K

Resistivity Anisotropy?

SiGe

[ ]12 −Μ= τσ ne

[ ]iipocketthi

ne 12 −

−

Μ= ∑τσ

For ellipsoids: three different effective masses for cyclotron oscillation, density of states, and conductivity.

4

p-type

Number Of Carriers In Thermal Equilibrium

⎥⎦⎤

⎢⎣⎡

+−= −∞−∫ 1

11)()( /)( Tkvv B

V

egdTp με

εεε

1)(

/)( += −∞−∫ Tk

vB

V

egd εμ

ε εε

Tkee B

TkTk

B

B>>−≈

+−

− εμμεεμ ,

11 /)(/)(Tke

e BTk

TkB

B>>−≈

+−

− μεεμμε ,

11 /)(/)(

property of intrinsic semiconductor

Tkcc

BC

C

egdTN /)()()( εε

εεε −−∞

∫=

Tkcc

BCeTNTn /)()()( με −−=

1)()( /)( +

= −

∞

∫ Tkc

c BC egdTn μεε

εεbulk semiconductor, far from surfaces

Tkvv

BVeTPTp /)()()( εμ −−=

Tkvv

BVV egdTP /)()()( εεε

εε −−

∞−∫=

Carrier Concentration

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

3

23

2

22

1

21

2

2)(

mk

mk

mkk C

hrεε

1/)(2/)(2/)(2 2

2

23

23

22

21

21 =

−+

−+

− hhh CCC mk

mk

mk

εεεεεε

23

2/3,

,, ||2)(π

εεεh

vcvcvc

mg −=

2/3

22

41)( ⎟

⎠⎞

⎜⎝⎛=

hπTkmTN Bc

c

2/3

22

41)( ⎟

⎠⎞

⎜⎝⎛=

hπTkmTP Bv

v

323

22

21

41)(2)(2)(2

34)(

πεεεεεεπε

hhhCCC

pocketsmmmNN −−−

=

5

Carrier Concentration

Tkcc

BCeTNTn /)()()( με −−=

TkEvc

Tkvcvc

BgBVC eTPTNeTPTNTpTn //)( )()()()()()( −−− == εε

TkEvcivc

BgeTPTNTnTpTn 2/)()()()()( −===

( ) TkEvc

Bi

BgemmTkTn 2/4/32/3

22

41)( −

⎟⎠⎞

⎜⎝⎛=

hπ

Tkvv

BVeTPTp /)()()( εμ −−=

In equilibrium, always holds.

Intrinsic Semiconductors

law of mass action

indep. of impurity level!

2i

npn vc =

i

Bi

i

Bi

nepnen

Tkv

Tkc

/)(

/)(

μμ

μμ

−−

−

==

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

c

vB

gvi m

mTkE

ln43

2εμ

6

Extrinsic Semiconductors

i

Bi

i

Bi

nepnen

Tkv

Tkc

/)(

/)(

μμ

μμ

−−

−

==

0≠Δ=− npn vc

2i

npn vc =

[ ] nnnpn

iv

c Δ±+Δ=⎭⎬⎫

⎩⎨⎧

214)(

21 2/122

[ ]Tknn

Bii

/)(sinh2 μμ −=Δ

{

Impurity Levels

0*0 ammr sε=

eVmm

sB 6.131

2

*

εε =

2

2

0 mea h

=

rerVSε

2

)( −=r

7

Effective Mass Theory Of Ionization Energy

Start with filled valence band in a semiconductor. Solution of states in conduction band are plane waves with effective mass m*.

Bring in a proton. Treat the screened (due to valence band electrons) Coulomb interaction as perturbation to derive lowest unoccupied state.

Expand in plane waves and apply perturbation theory. Every term is smaller than the free electron case by the same factor. What is this factor?

Start with vacuum. Solutions of states in vacuum are plane waves.

Bring in a proton. Treat Coulomb interaction as perturbation to derive lowest unoccupied state.

Expand bound state in plane waves and apply perturbation theory. Eventually an lowest energy of –13.6 eV is found.

2*

2

2)( k

mk C

hr+=εε

22

2)( k

mk hr=ε

rerVSε

2

)( −=r

rerV

2

)( −=r

Hydrogen Atom Impurity Atom

Scaling In Effective Mass Theory

Use the perturbed k=0 state as example:

∑≠ −

><+=′

0

2

)()0(

||0)0()0(

k k

kVr

r

r

εεεε ∑

≠ −

><+=′

0

2

)()0(

||0)0(

kCBM k

kVr

r

r

εεεε

factor of εs smallerHydrogen Atom Impurity Atom

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

mk

2

22h⎟⎟⎠

⎞⎜⎜⎝

⎛−= *

22

2mkh

scaling factor = 2

* 1

smm

ε

larger than free electrons for same Δk

8

Population Of Impurity Levels

)(

)(

jj

jj

NE

NEj

eeN

n μβ

μβ

−−

−−

∑∑=><

)(21)(

)(

11

212

μεβμεβ

μεβ

−−−

−−

+=

+=><

dd

d

eeen

11

222

)(21

)(

)2(

)2(

++

=++

=>< −

−

−−

−−

a

a

a

a

ee

eeeen εμβ

εμβ

μεββμ

μεββμ

)(211 aeNp a

a εμβ −+=

1)(21 +

= −μεβ deNn d

d

Why don’t we use simple Fermi distribution?

donors

acceptors

><−=>< np 2

1)2ln( += −− μεβ Tk

dd Bde

Nn

electrons on donors

holes on acceptors

Carrier Densities of Doped Semiconductors

avaddc ppNNnn ++−=+

advc NNpnn −≈−=Δ

[ ] )(214)(

21 2/122

adiadv

c NNnNNpn

−±+−=⎭⎬⎫

⎩⎨⎧

)(21

adiv

c NNnpn

−±≈⎭⎬⎫

⎩⎨⎧

)(sinh2 ii

ad

nNN μμβ −=

−

low doping conc. or high T

graphical method)()( ddvaac nNppNn −+=−+negative charge positive charge

determines μ

9

Extrinsic Semiconductors

ad

ad

iv

adc

NN

NNnp

NNn>

⎪⎪

⎭

⎪⎪

⎬

⎫

−≈

−≈

2

da

dav

da

ic

NNNNp

NNnn

>

⎪⎪

⎭

⎪⎪

⎬

⎫

−≈

−≈

2

intrinsic

extrinsic freeze-out

ad NN >

Screening In Doped Semiconductors

d

BSn Ne

Tk2

ελ =

a

BSp Ne

Tk2

ελ =

Debye Length is the distance over which mobile charge can screen out electric field. It is the distance over which significant charge separation can occur.

Doping levels of 1016 cm-3: λ ~ 30 nm at RT.

10

Transport In Nondegenerate Semiconductors

kkk

kv Ckk

rh

rrh

hh

rrrr ⋅Μ=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⋅Μ⋅

∇=∇= −−

112

211)( εε vk r

h

r⋅Μ=

1Ellipsoids are your friends!

222)(

12 vvkvkkk CCC

rrrrh

rrhr ⋅Μ⋅

+=⋅

+=⋅Μ⋅

+=−

εεεε

31221 4

exp1

1)(πμεkd

Tkkk

kdkf

B

C

r

rrh

rr

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −+⋅Μ⋅

+

=−

( )2121212

2)( zzzyyyxxxC kkkk −−− Μ+Μ+Μ+=

hrεεFirst, choose x,y,z along

principal axes of ellipsoid

( ) 2/332

2/12/3

31113

2/32/3

)(3det2

41)(2

34)( C

zzyyxx

CN εεππ

εεπε −Μ

=⎟⎠⎞

⎜⎝⎛

ΜΜΜ

−=

−−− hh

( ) 2/132

2/12/1

)(det2)( Cg εεπ

ε −Μ

=h

What is the velocity distribution of the carriers of a semiconductor?

Transport In Nondeg. Semicond. (cont.)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ ⋅Μ⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −≈=

−

∫∫ Tkkkkd

Tkkdkfn

BB

C

2exp

4exp)(

12

3

rrh

rrr

πεμ

( )⎭⎬⎫

⎩⎨⎧ ′−′Μ′⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

⎭⎬⎫

⎩⎨⎧ ′−′′⎟⎟

⎠

⎞⎜⎜⎝

⎛ −≈ ∫∫

∞∞

Tkd

TkTkgd

Tkn

BB

C

BB

C εεπ

εεμεεε

εμexp)(det2expexp)(exp 2/1

32

2/12/1

00 h

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

Μ TkTkn

B

C

B

εμπ expdet)(

22/12/3

32/3 h

20

π=−∞

∫ xedxx

3

12

42expexp)(

πεμ kd

Tkkk

Tkkdkf

BB

C

rrrhrr

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ ⋅Μ⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −≈

−

( )vdvfvd

Tkvv

Tknkdkf

BB

rr

h

rrrhrr

)(det2

expdet)(4

2)( 32/12/33

32/3

=⎟⎠⎞

⎜⎝⎛ Μ⎭⎬⎫

⎩⎨⎧ ⋅Μ⋅−

Μ=

ππ

volume zyx dkdkdk=

zyxzzzyyyxxx

zyx dvdvdvdvdvdv

dkdkdk 3

dethhhh

Μ=

ΜΜΜ=

1>>−TkB

C με

11

Transport In Nondeg. Semicond. (cont.)

⎭⎬⎫

⎩⎨⎧

Μ−Μ

= ∑μν

νμνμβ

πvv

Tknvf

B 2exp

)2(det

)( 2/3

2/1r

Same as molecular velocity distribution of ideal gas!

n in classical gas is specified. Here it is not.

In a classical gas, M is diagonal.

Many classical theories can be applied directly to semiconductors.

( )3*m

Degenerate Semiconductors: Impurity band conduction.

Inhomogeneous SemiconductorsCh. 29

⎭⎬⎫

⎩⎨⎧

<>

=0,00,

)(xxN

xN dd

⎭⎬⎫

⎩⎨⎧

<>

=0,0,0

)(xNx

xNa

a

abrupt junction

⎭⎬⎫

⎩⎨⎧ +−−=

TkxeTPxp

B

vvv

)(exp)()( φεμ

⎭⎬⎫

⎩⎨⎧ −−−=

Tkxe

TNxnB

vcc

μφε )(exp)()(

semiclassical model (weak fields)

)(xepH nn φε −⎟⎠⎞

⎜⎝⎛=h

r

1D problem

φ and n (p) solved self-consistently

12

electrochemical potential

p-n Junction In Equilibrium

⎭⎬⎫

⎩⎨⎧ −∞−−=∞=

TkeTNnN

B

cccd

μφε )(exp)()(

⎭⎬⎫

⎩⎨⎧ −∞+−−=−∞=

Tke

TPpNB

vvva

)(exp)()(

φεμ⎥⎦

⎤⎢⎣

⎡+−=−∞−∞

vc

adBvc PN

NNTkee ln)()( εεφφ

⎥⎦

⎤⎢⎣

⎡+=Δ

vc

adBg PN

NNTkEe lnφ

)()( xexe φμμ +=

⎭⎬⎫

⎩⎨⎧ −−=

Tkx

TNxnB

eccc

)(exp)()(

με

⎭⎬⎫

⎩⎨⎧ −−=

Tkx

TPxpB

vevv

εμ )(exp)()(

)()( −∞−∞=Δ eee μμφ

Abrupt p-n Junction

επρφφ )(4

2

22 x

dxd

=−=∇−

[ ])()()()()( xpxnxNxNex vcad +−−=ρ

⎭⎬⎫

⎩⎨⎧ −∞−−=

⎭⎬⎫

⎩⎨⎧ −∞−=

TkxNxp

TkxNxn

Bav

Bdc

)()(exp)(

)()(exp)(

φφ

φφ

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−<

<<−

<<−

>

=′′

p

pa

nd

n

dx

xdeN

dxeN

dx

x

,0

0,4

0,4,0

)(

επεπ

φevery kT, drops 1/e

13

Abrupt p-n Junction

επ padeN

E4

max =

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−<−∞

<<−++−∞

<<−−∞

>∞

=

p

ppa

nnd

n

dx

xddxeN

dxdxeN

dx

x

),(

0,)(2

)(

0,)(2

)(

),(

)(2

2

φε

πφ

επ

φ

φ

φ

pand dNdN =

( ) φεπ

Δ=+⎟⎠⎞

⎜⎝⎛ 222

pand dNdNe

2/11

, 2)()/(

⎭⎬⎫

⎩⎨⎧

+Δ

=±

eNNNN

dda

dapn π

φε

[ ]o

eVda

dapn Ae

NNNNd

2/1

18

1

, )(10)/(33

⎭⎬⎫

⎩⎨⎧

Δ+

= −

±

φε

From continuity of φ and φ’ at x=0

in equilibrium

Elementary Picture: Rectification At p-n Junction

2/1

0,, )(

1)0()( ⎥⎦

⎤⎢⎣

⎡Δ

−=φVdVd pnpn

V−Δ=Δ 0)( φφ

hhee eJjeJj =−= ;

TkVerech

BeJ /])[( 0 −Δ−∝ φ

genhV

rech JJ =

=0TkeVgen

hrech

BeJJ /=

0)( φΔ

applied bias+ forward bias- reverse bias

recombination current:majority carrier current entering the depletion region

generation current: minority carrier current entering the depletion region. Independent of V.

14

Elementary Picture: Rectification At p-n Junction

( )1/ −=−= TkeVgenh

genh

rechh

BeJJJJ( )1/ −= TnkeV

SBejj

( )( )1/ −+= TkeVgenh

gene

BeJJej

( ) TkEgenh

gene

BgeJJe /−∝+

saturation current

T dependence

Ideality Factor

( )1/ −= TnkeVS

Bejj

⎟⎠⎞

⎜⎝⎛=−

dVjd

eTkn B ln1

Nonequilibrium p-n Junction: General Aspects

dxdpDpEJdxdnDnEJ

pph

nne

−=

−−=

μ

μnEeeJjE ne μσ =−==

n

colln

mne τ

σ2

=

p

collp

pn

colln

n me

me τ

μτ

μ == ;

Einstein relations

TkeD

TkeD

B

pp

B

nn == μμ ;

in thermal equilibrium, J=0

⎭⎬⎫

⎩⎨⎧ −−−=

Tkxe

TNxnB

vcc

μφε )(exp)()(substitute in

dxdeN

TkeD

dxdeN xe

cB

nxecn

ccφφμ φμεβφμεβ ))(())((0 −−−−−− −=

15

Nonequilibrium p-n Junction: General Aspects

continuity equations

rg

vhv

rg

cec

dtdp

xJ

tp

dtdn

xJ

tn

−

−

⎟⎠

⎞⎜⎝

⎛+∂∂

−=∂∂

⎟⎠

⎞⎜⎝

⎛+∂∂

−=∂∂

h

vv

rg

v

n

cc

rg

c

ppdt

dp

nndt

dn

τ

τ0

0

−−=⎟

⎠⎞

⎜⎝⎛

−−=⎟

⎠⎞

⎜⎝⎛

−

−

0)(1)( cn

cn

c ndttndtdttnττ

+⎟⎟⎠

⎞⎜⎜⎝

⎛−=+

0

0

0

0

=−

+∂∂

=−

+∂∂

h

vvh

n

cce

ppxJ

nnxJ

τ

τ

Nonequilibrium p-n Junction: General Aspects

diffusion region 00

2

2

0

2

2

≈−

=

−=

Epp

dxpd

D

nndx

ndD

h

vvvp

n

cccn

τ

τ

nn LxLxcc eCeCnxn /

2/

10)( ++= −

pppnnn DLDL ττ == ;

[ ] pLxxvvvv epxppxp /)(

00)()()()( −−∞−+∞=

16

Nonequilibrium p-n Junction: General Aspects

n

n

a

igene

LNn

Jτ⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

p

p

d

igenh

LNn

Jτ⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

collnthnBth

n

colln

B

nn

vTkmvm

eTk

eD

τ

τμ

==

==

l;

;

232

21

pcollp

ppncoll

n

nn LL ll

2/12/1

3;

3 ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ττ

ττ

lNrandom walk

Minority carriers generated within a diffusion length of the edge of the depletion region have an opportunity of being swept by the electric field.

generation rate: n0/τ

Nonequilibrium p-n Junction: Detailed Theory

17

Nonequilibrium p-n Junction: Detailed Theory

)()( nhpe deJdeJj +−−=

p

p

dx

vpnh

dx

cnpe

dxdp

DdJ

dxdn

DdJ

=

−=

−=

−=−

)(

)(

nLdx

d

inv

d

iv dxe

Nn

dpNn

xp pn ≥⎥⎦

⎤⎢⎣

⎡−+= −− ,)()( /)(

22

pLdx

a

ipc

a

ic dxe

Nn

dnNn

xn np −≤⎥⎦

⎤⎢⎣

⎡−−+= + ,)()( /)(

22

⎥⎦

⎤⎢⎣

⎡−=

⎥⎦

⎤⎢⎣

⎡−−−=−

d

inv

p

pnh

a

ipc

n

npe

Nn

dpLD

dJ

Nn

dnLD

dJ

2

2

)()(

)()()(−∞cn

Assume that electron (hole) current is constant in depletion region and, therefore, it can be evaluated at any convenient position.

(E is still 0.)

Nonequilibrium p-n Junction: Detailed Theory

⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡−−=

d

inv

p

p

a

ipc

n

n

Nn

dpL

eDNn

dnL

eDj

22

)()(

TkeV

Bnc

Tkencpc

BB eTk

ednedndn /0/ )(

exp)()()( ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−==− Δ− φφ

TkeV

Bpv

Tkepvnv

BB eTk

edpedpdp /0/ )(

exp)()()( ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−−=−= Δ− φφ

dnvnc Ndpdn =− )()(

apcpv Ndndp =−−− )()(

eV<<EgTkeV

Banv

TkeV

Bdpc

B

B

eTk

eNdp

eTk

eNdn

/0

/0

)(exp)(

)(exp)(

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=−

φ

φ

⎟⎟⎠

⎞⎜⎜⎝

⎛+

vc

daBg PN

NNTkE ln

18

Nonequilibrium p-n Junction: Detailed Theory

n

nLτ

2

n

nn

LDτ

2

=

equivalently,TkeV

d

inv

TkeV

a

ipc

B

B

eNn

dp

eNn

dn

/2

/2

)(

)(

=

=−

( )1/2 −⎟⎟⎠

⎞⎜⎜⎝

⎛+= TkeV

dp

p

an

ni

BeNL

DNL

Denj

pd

pigenh

na

nigene

LNDn

J

LNDn

J2

2

=

=

n

n

a

igene

LNnJ

τ⎟⎟⎠

⎞⎜⎜⎝

⎛=

2