References S.M. Sze, Physics of semiconductor devices (Wiley) Chap 1,2 Ashcroft-Mermin, Solid State Physics (Saunders) Chap 4-5, 28-29 Grosso- Parravicini,

References

S.M. Sze, Physics of semiconductor devices(Wiley) Chap 1,2

Ashcroft-Mermin, Solid State Physics(Saunders) Chap 4-5, 28-29

Grosso- Parravicini, Solid State PhysicsChap 13-14

Metodi Sperimentali della Fisica Moderna

P-N junctions

Short review of semiconductor properties

P-n junctions

1

Crystal lattice

R = n1a1 + n2a2 + n3a3

n1 , n2 , n3 = integers

Bravais lattice = set of all points such that

a1 , a2 , a3 = primitive vectors

Simple cubic (SC) Body centered cubic (BCC) Face centered cubic (FCC)

a1 = (-x+y+z)a/2

a2 = (x-y+z)a/2

a3 = (x+y-z)a/2

a1 = (y+z)a/2

a2 = (x+z)a/2

a3 = (x+y)a/2

2

Primitive unit cell = volume of space that can fill up all the available space with no overlapping (not unique)

a1 ; a2 ; a3 = primitive vectors are not unique

Crystal lattice

3

Crystal lattice with basis

Diamond lattice

The crystal can be described bydifferent primitive vectors,adding also the atoms withinthe unit cell using a basis

^

^

^

z

y

x

a

a

a

Primitivevectors

Basis

^^^

4

0

zyxa

Lattice withtwo point basis

4

Directions and planes

[1 0 0]

[1 1 0][1 1 1]

[n1, n2, n3] = crystal direction

(n1, n2, n3) = crystal plane

Miller indexes

different surface atomic density5

Reciprocal lattice

Plane waverk ie

The set of K yelding plane waves with the periodicityof a Bravais lattice is the reciprocal lattice

rKRrK )( ii ee

321

213

321

132

321

321 2 ,2 ,2

aaaaa

baaa

aab

aaaaa

b

Bravais lattice

The vectors defining the reciprocal lattice are

ijji 2ab

332211 bbbk kkk

R = n1a1 + n2a2 + n3a3

3322112 nknknk Rk

1 RKie

If k is a linear combination

To be a k of the reciprocal space the ki must be integers6

Reciprocal lattice

7

Real lattice

Reciprocal lattice

Brillouin zone: region of points closer to a lattice point than to any other lattice point in the reciprocal space.

The main symmetry directions are labelled, K, X, W, L

Periodic potential U(r)p

k Free electronCrystal momentum

bcc fcc

L

X

W K

What is the physical meaning of k?

8

Fermi level

Consider N non-interacting electrons confined in a volume V = L3

Non interacting The ground state is obtained from the single e- levels in V

and filling them up with the N electrons

)()(2

22

rr m1 e- Schroedinger

equation

Represent the electron confinement to V with a boundary condition

(x+L, y, z) = (x, y, z) (x, y +L, z) = (x, y, z) (x, y, z +L) = (x, y, z)

This leads to running waves, while choosing as boundary condition for to vanish at the surface would give rise to standing waves……

= Energy of the level

STANDING WAVES9

http://www.walter-fendt.de/ph14e/stwaverefl.htm

rkk r 1

)( ieV

The solution of the Schroedingerequation is

With energiesmk

2)(

22k

The one-electron level k(r) is aneigenstate of the momentum operator

ii

rP

with eigenvalue that is the electron momentum

rkkrr

rprP

)(

)()(

iei

kp

But k is also a wave vector of the plane wave corresponding to the wavelength

V

rrk d)(1

a) Neglect the boundary condition Normalization condition

k 2

10

)( 1)( Lrk

k r ieV

1 LkiLkiLki xzyx eee

ez = 1 only for z = 2m

zzyyxx mL

kmL

kmL

k 2

;2

;2

the area per point is (2/L)2

b) Consider the boundary condition

The bc are satisfied by the plane wave only if

wavevector components

integers ; ; zyx mmm

The allowed wavevectors in the k space are those with components made by integer multiples of 2/L

11

area per point (2/L)2

How many allowed k are contained in a region of K space large compared to 2/L?

33 8/2 V

L

Volume occupied by one point

The k space density of levels is38

V

Fermi wavevector

12

build the N-electron (non interacting) ground state by filling up the one-electron levels

Each level can contain 2 electrons (one for each spin)mk

2)(

22k

For N large, we get a sphere

3

4 3Fk

Number of allowed k within the sphere VkVk FF

2

3

3

3

683

4

Vk

Vk

N FF2

3

2

3

362

Occupied for k < kF

Unoccupied for k > kF 2

3

3Fk

nVN

FERMI WAVE VECTOR

the sphere radius is defined kF

volume

For N non interacting e- in a volume V

The ground state is formed by occupying all single-particle levels

13

14

Electronic structure of a solid

We start by assuming the positions of atoms RThe Hamiltonian describing the electronic structure is

N

ji ji

N

i

N

i i

i eZem

p

1,

2

1 1

22

21

2 H

rrRrR

ElectronKinetic energy

Electron-ion interaction

Electron-electron interaction

Impossible to calculate solutions

many-body problem

Hartree-Fock is not good for solids, it dumps the many bodyproblem in the correction term

15

1-D band theory: Tight-binding model

Semiconductors

The lattice potential is constructed from a superposition of N free atoms potentials Va(r) arranged on a chain with lattice constant a(but it is not the true lattice potential)

N

naL nVV

1

) ()( arr

0)( )()(2 rrr aa EV

V along the ion line

V atomic

V between Atomic planes

Potential

Atomic Schroedinger equation

(r) is the eigenstate of the isolated atom16

)()( )()()(2 rrrrr EVVV aLa

The Schroedinger equation for the crystal is

(r) are Wannier functions,similar to the solutions of the atomicSchroedinger equations

N

nn nc

1

) ()( arr

The solution can be constructed using a linear combination of the type

cn : coefficient to be found

Insert the solution into the Schroedinger equation matrix

Only the on-site and first neighbor coefficients are retainedthus defining the matrix elements ,

1,, mlmlaL mVVl ml , Eigenvector of two generic sites

Recalling that the must be of the type eikr 17

Reducing the matrix elements allows to obtain the expansion coefficients cn

),,( KEE

Energy band

Hence the eigen energy corresponding to a solution of the Schroedinger equation is

Dispersion relation for energy bands

Wavevector along the Brillouin directions

It is important how much the (r)overlaps between neighboring sites.

The bandwidth depends on the overlap.

The potential breaks the degeneracy18

The potential is also affecting the bands at the high symmetry pointsof the brillouin zone giving rise to prohibited energies, i.e. energy gaps

19

Ge Si GaAs

Energy bands for electrons T dependence of energy gap

20

Fermi level in metals

The ground state of N electrons in a solid is made similarly to the free electron case

The electron levels are identified by quantum numbers n and k,and the structure of the solid gives rise to the electronic bands

When all levels are filled with electrons,in a metal the last band is only partially filled

Fermi level is the energy below which the one-electron levelsare occupied and above which are unoccupied

21

Semiconductors

The semiconductors are characterized by a gap between the last occupied band, defined as VALENCE BAND and the first unoccupied band, defined as CONDUCTION BAND

At T 0 there is a finite probability that some electrons will be thermally excited across the gap, and there will be conduction of ELECTRONS and HOLES

TK

E

excB

G

eN 2el. .

GaAs: EG = 1.4 eV at T = 300 K, KBT 0.025 eV and N 7 1013

InSb: EG = 0.16 eV at T = 300 K, KBT 0.025 eV and N 4 1016

Hence the definition of Fermi level as "the energy below which the one-electron levels are occupied and above which are unoccupied" is valid for all energies in the gap.

EF is not univocally determined

One has to define a number telling the energy position of the electrons with the lowest binding energy

CB

VB

will see that

22

SemiconductorsThermally excited electrons will be mostly close to CBM and holes close to VBM

so the band dispersion can be approximated by a parabolic relation

kk

kk

V

C

12

12

2)(

2)(

Mk

Mk

The tensor M can be diagonalized so

3

23

2

22

1

212

3

23

2

22

1

212

222)(

222)(

mk

mk

mk

mk

mk

mk

V

C

k

k

Effective masses

0.49m0.16m

0.28m0.044m

SiGe

Constant energy surfaces for e- Constant energy surfaces for h23

Chemical potential & Fermi-Dirac function

Gibbs energy for a system containing N particles

1

)()()()(

TKBe

gFgN

g() = DOS

dNVdPSdTdG

At constant T and P dNdG dN

dG Chemical potential

If the system loses 1 particle, the free energy changes

System of identical fermionsThe average number of fermions in a single-particle

state i 1

1

TK

i

B

i

e

n

Average number of fermions with energy i =

F–D distribution ni X the degeneracy gi (i.e. the number of states with energy i)

Fermi–Dirac (F–D) distribution

1

TK

iiii

B

i

e

gngn

For solids

1

1)(

TKBe

F

24

Carrier concentration in SemiconductorsTo understand the equivalent of Fermi level in semiconductors, we have to evaluate the number of carriers in thermal equilibrium

1

1)( )(

TK

CC

BC

e

gdTn

g() = DOS

The carrier density depends on the chemical potential

TK

TK

BV

BC

TK

TK

TK

TK

B

B

B

B

e

e

e

e

1

1

1

1

TKV

TKV

TKC

TKC

B

VV

B

V

B

C

C

B

C

egdeTp

egdeTn

)()(

)()(

Suppose

1

1)(

1

11 )()(

TKV

TKVV

B

V

B

V

e

gd

e

gdTp

electrons

holes

1

1)( )(

TK

VV

B

V

e

gdTp

25

Since only the carriers within KBT of the band edges contribute, theeffective mass approximation is good and the DOS g() is

2/3,23

,

,

2)( VC

VC

VC mg

)(2

41

)(

)(2

41

)(

2/3

2

2/3

2

TPeTKm

eTp

TNeTKm

eTn

VTKBvTK

V

CTKBcTK

C

B

V

B

V

B

C

B

C

(mC)3, (mV)3 effective masses

Therefore it is the chemical potential that sets the density of carriers n,p and also the energy position of the states within the gap.

This is usually referred to Fermi level in semiconductors but it is NOT a Fermi level

TKV

TKC

B

V

B

C

eTPTp

eTNTn

)()(

)()(Carrier density

NC, PV

Effective density of statesintegrated over all energies

26

3192/32/3

10300

5.2)(

cmx

KT

mm

TN cC

2/3

2

2/3

2

241

)(

241

)(

TKmTP

TKmTN

BvV

BcC

NC, PV effective DOS for carriers

TKV

TKC

B

V

B

C

eTPTp

eTNTn

)()(

)()(

Carrier density

27

At finite temperature thermal excitation of e- leaves an equal number of holesin the VB so that

n = p = ni intrinsic carrier density

TK

E

VCTK

VC

VTK

CTK

VC

B

g

B

CV

B

V

B

C

eTPTNeTPTN

TPeTNeTpTnTpTn

)()()()(

)()()()()()(

TK

E

VCiB

g

eTPTNTnTpTn

)()()()()( 2

depends only on the gap energy

Mass action lawTK

E

VCiB

g

eTPTNTn 2)()()(

Intrinsic semiconductors

28


)()()( TnTpTn iVC

C

VBgVFi m

mTKEE ln

43

2

TKVVi

TKCci

B

Vi

B

iC

ePpn

eNnn

TKV

TKC

B

Vi

B

iC

eTPeTN

)()(

What is the chemical potential for the intrinsic case i?

TKTKTK

TK

TK

C

V B

ViC

B

Vi

B

iC

B

Vi

B

iC

ee

e

eNP

2

TK

E

TK

E

TKNP

B

iVg

B

ViVg

B

ViC

C

V 2222

ln

C

VBgVi N

PTKEln

22

29


C

VBgVFi m

mTKEE ln

43

2

So for T 0, the chemical potential i, is in the middle of the energy gap

Since (mV/mC) 1, the chemical potential i

does not change more than ~ KBT

TK

TK

BV

BC

Are valid

30


Considering (mV/m) (mC/m) 1, at RT

Silicon: EG= 1.1 eV

31922/34/34/3

10300

5.2)(

cmxe

KT

mm

mm

Tn TKE

VCi

B

G

3192 10 5.2)(

cmxeTn TK

E

iB

G

102 108.2

xe TKE

B

G

132 109.6

xe TKE

B

G

GaAs: EG= 1.4 eV

InSb: EG= 0.17 eV 99.02

TKE

B

G

e

But mC and mV are 0.01

310107)( cmxTni

37107.1)( cmxTni

319104.2)( cmxTni316101.2)( cmxTni

31

C

VBgVFi N

PTKEE ln

22

Fermi level for intrinsic semiconductors

band diagram density of states carrier concentrations

TKV

TKC

B

V

B

C

ePp

eNn

2innp

32

C

VV

C

Donors and acceptors

)()()( TnTpTn iVC

n-type: a Si atom is replaced by P atom with an extra e-. The P atom is called donor

p-type: a Si atom is replaced by B atom with an extra hole. The B atom is called acceptor

Donors and acceptors introduce extra energy levels, whose energy can be estimatedusing the H atom model

Si: 4 valence e-

P: 5 valence e-

B: 3 valence e-

33

Donors and acceptors

1) Neglect the ion core of the inserted P atom 2) The P is represented by a Si atom with 1 hole (e+) fixed in the site + 1 e-

Why hydrogen model?

e- binding energy = 10.486 eV (P ionization potential)

Charge field reduced by the macroscopic dielectric constant (13 for Si)

Isolated atom Atom inside crystal

The e- moving in the lattice has energies of the type E=Ec(k) (k =crystal wavevector)

The allowed energy levels has to be closeto conduction band minimum to minimize energy

Assume parabolic bands with effective masses

The electron of the donor impurity is represented as a particle of charge –e and and mass m*moving in the presence of attractive charge e/

H atom

2

4

0

2

2

0

meE

mea

nm

mm

amm

r 05.0 x*0*0

22 e

e

*mmeV

mm

Emm

E 6.13 x1

2

*

02

*

Impurity

r0 up to 10 nm

34

Extrinsic semiconductorsThe density of donors or acceptors is above 1012/cm3

The level energies are very small compared to the energy gapso it is very easy to excite an electron from a donor level or a hole from an acceptor

level

35

TK

EDD

B

FD

eg

NN 1

1

11

g = ground state degeneracy = 2The donor level are localized hence cannot accommodate 2 electrons due to charge repulsion

DONOR

Density of ionized donors

TK

ED

B

FD

eg

P

1

1

1)(

Probability of occupationof donor levels

Probability of findingionized donor levels

)(1 DP

Fermi level for extrinsic semiconductors

Introduce impuritiesnot all dopants are necessarily ionized:

depends on the impurity energy level and T

ND (cm-3) = donor concentrationD = energy of donor level

TK

ED

D

B

DF

ge

NN

1

36

TK

EDA

B

AF

eg

NN 1

1

11

ACCEPTOR

Density of ionized acceptor

TKEA

B

AF

eg

P

1

1

1)(Probability of occupation

of acceptor levels

Probability of findingionized acceptor levels

)(1 AP


Introduce impuritiesnot all dopants are necessarily ionized:

depends on the impurity energy level and T

NA (cm-3) = acceptor concentrationA = energy of acceptor level

g = ground state degeneracy = 4each acceptor impurity level can

accept one hole of either spin The impurity level is doubly degenerate at K =0

TK

EA

A

B

FA

ge

NN

1

37

pND

preserve charge neutrality

nNA intrinsicextrinsicintrinsicextrinsic


Total negative charges(electrons and ionized acceptors)

With impurity atoms introduced

Total positive charges(holes and ionized donors)

pNnN DA

2innp Mass action law still valid

Add DONORS only

DD NpNn

TK

EDTK

E

C

B

DF

B

FC

e

NeN

21

1

38

if I add both donors and acceptors

e density in CB

Ionized donor density

h density in VB


DONORS

Graphically solved todetermine EF

Plot for two different values of D

39

TK

EDTKE

C

B

DF

B

FC

e

NeN

21

1

V

2D

C1D


DONORS n-type semiconductor


40

modificare le E della figura in epsilon

CD

V

C

V


ACCEPTORS p-type semiconductor


41

modificare le E della figura in epsilon

CC

AV V

For a set of dopant concentrations it is possible to estimate the Fermi level

TKED

TKE

C

B

DF

B

FC

e

NeN

21

1

TKEA

TKE

V

B

FA

B

FV

e

NeN

21

1

42

D CV

pNn D Charge neutrality

Carrier concentration temperature dependence

TK

E

V

TK

EDTK

E

CB

VF

B

DF

B

FC

eN

e

NeN

21D = energy level of the donor impurity

TDKB = (C-D) ionization impurity temperature

1) T<< TD = D/KB freezing out region

D < EF < C

C

DBCDF N

NTKE

2ln

22

TKDC B

DC

eNN

Tn 2

2)(

Carrier density in the conduction band

neglect p

TK

E

D

TK

E

TK

E

DTK

E

CB

FD

B

FD

B

FD

B

FC

eN

e

eNeN

2

21

12

43

TKB

DC

eTn 2)(

At RT almost all electron (donors) and holes (acceptors ) are excited = saturation condition

Carrier concentration temperature dependence

2) TD < T << EG/KB saturation region

Intrinsic electrons negligible

DTKE

C NeNTn B

FC

)( Majority carriers

Minority carriers p(T)

D

ii

NTn

TnTn

Tp)(

)()(

)(22

Intrinsic Silicon 31010)( cmTni

N-type Silicon 314101 cmxND

36101)( cmxTp

3) TD < EG/KB < T intrinsic region44

D < EF < C

TKB

DC

eTn 2)(

Mobility

m depends on carrier interactionwithin the lattice and on temperature

Enem

neJ

c

nn

2

)( pne pn For e- and holes

- Carrier transport for a semiconductor in the presence of an electric field E- 2 parabolic bands (valence and conduction) and effective masses mv and mc

Current density

mobility

t = scattering timen = carrier density

Epem

peJ

m

e

v

pp

v

pp

2

Why? We need to know what is happening to carriers for concentrations out of equilibrium

Evdrift c

nn m

e

45

Consider a concentration gradient in the solid

Dn is the diffusion coefficient

Diffusion coefficient

neDJ nn

Suppose the flux goes from regions of high concentration to regions of low concentrationwith a magnitude that is proportional to the concentration gradient

neDEneJ nnn

For a doped semiconductor with a carrier concentration gradient andapplied electric field the total current density is

Drift Diffusion

46

Mobility-diffusion coefficient relation

)()( xneTNn TKE

CB

FC

eTk

D Bnn

The mobility p, and the diffusion coefficient D, are not independent

Consider an n-type semiconductor with nonuniform (n=n(x))doping concentration and without an external applied field

0

xn

eDneJ nnn

Drift current balances the diffusion currentxn

eDne nn

xexC

)(

)()()( xex CC

)()(1

xnTKe

xnxTKx

n

B

C

B

n

TKe

eDneB

nn

the nonuniform doping generates a potential (x)that rigidly shifts the energy levels of the semiconductor

internal electric field

47

p-type semiconductor n-type semiconductor

All donors and acceptors ionized (saturation condition)

DD NN

ND,A (cm-3) = donor,acceptor concentration

AA NN

The p-n junction

TKE

AA

B

FA

ge

NN

1

48

TKE

DD

B

DF

ge

NN

1

The p-n junction

p-type semiconductor

n-type semiconductor

Bring together the two regions

A region depleted of majority carriers is formed at the interface (space charge region)

1) Some e- move from n-type to p and recombine with h2) Some h move from p-type to n and recombine with e-

3) In the n region close to x=0 remains non neutralized donors (+)4) In the p region close to x=0 remains non neutralized acceptors (-)

A strong electric field is built up opposing further diffusion of majority carriers, reaching equilibrium

49

The Fermi level must remain constant throughout the depletion layer and the samplesince the system is at equilibrium

0

0

xn

ekT

e

xn

eDneJ

n

nnn

At equilibrium and no external field E applied , no current flows so

xE

nxn

eTNn

F

TKE

CB

FC

)(

eTk

D Bnn

0

x

Ene F

n

This means that the band in the two regions bend to adjust across the p-n junction

be

50

How much is the potential?

Depends on carrier concentration

TK

E

VA

TK

E

CD

B

VF

B

FC

eTPNTp

eTNNTn

)()(

)()(

pn FFb EEe

V

A

B

VF

C

D

B

FC

PN

TK

E

NN

TK

E

p

n

ln

ln

A

VBVF

C

DBCF

NP

TKE

NN

TKE

p

n

ln

ln

A

V

C

DBVCFFb N

PNN

TKEEepn

lnln

VC

ADBGb PN

NNTKEe ln

Contact potential

51

TKxe

DTKxe

TK

E

C

TK

Exe

CTK

Ex

C

BBB

FC

B

FC

B

FC

eNeeTN

eTNeTNxn)()(

)()(

)(

)()()( xex CC For electrons

TKEx

CB

FC

eTNxn

TK

xxe

Bexnxn)()(

21

21

)()(

At equilibrium, free carrier concentrations (e- and holes) depends on the position across the junction

Free electron density is high when Fermi level is close to bottom of CB

52

TK

xE

VB

VF

eNxp)(

)(

TK

xxe

Bexpxp)()(

21

21

)()(

At equilibrium, free carrier concentrations (e- and holes) depends on the position across the junction

Free hole density is high when Fermi level is close to top of VB

TKxe

ATK

Ex

VBB

FC

eNeTPxp)()(

)(

53

0nn0pnCarrier concentrations

TK

xxe

Bexnxn)()(

21

21

)()(

TK

e

npB

b

enn

00TK

e

npB

b

epp

00

Equilibrium concentration of e- in neutral bulk n material

Equilibrium concentration of e- in neutral bulk p material01)( pnxn

x2 inside n

02)( nnxn

TK

xxe

Bexpxp)()(

21

21

)()(

x1 inside p

Equilibrium concentration of holes in neutral bulk n material

Equilibrium concentration ofholes in neutral bulk p material01)( ppxp

x2 inside n

02)( npxp

x1 inside p

x1 x2

0np0pp

54

Approaching the depletion regionthe number of holes decreases and the layer depleted of holes is created.

The fixed charge density is

DNn ANp

DeNAeN

Approaching the depletion region the number of e- decreases and the layer depleted of e-

is created.

The fixed charge density is

The electrostatic potential follows the Poisson equation

pd nd

nD

pA

dxeN

x-deN

0

0

02

2 )(

r

xdxd

Space-charge region and internal electric field

Depletion layer approximation

55

Homogeneous semiconductorat equilibrium, no diffusion and Jd = 0 E = 0

DNn ANp

Boundary conditions:

0)(

0)('

p

p

d

d

Boundary conditions:

bn

n

d

d

)(

0)('

)(0

pr

A dxeN

dxd

E

20

x2

)( pr

A deN

x

Integrate the Poisson equation

)(0

nr

D dxeN

dxd

E

20

x2

)( nr

Db d

eNx

–dp < x < 0 0 < x < dp

Homogeneous semiconductorat equilibrium,no diffusion and Jd = 0 E = 0

Integrate the Poisson equation

Choose theConstant = 0

56

The continuity conditions for electric field at x = 0 gives

nnr

Db

ppr

A

dxdeN

x-ddeN

x0 x

2

0 x2

)(2

0

2

0

nDpA dNdN Like charge neutrality

nr

Dp

r

A deN

deN

00

The continuity conditions for electric potential at x = 0 gives

202 2nDb

rpA dN

edN

22

02 nDpAr

b dNdNe

57

Solving for dp & dn

Depends on dopant concentration

The total width of the depletion layer is

eNNNN

d

eNNN

Nd

br

ADD

An

br

ADA

Dp

0

0

21

21

DAD

A

A

DAbr

A

Dnnp NNN

NN

NN

eN

Ndddw

121

2

0

mF

mcmNN

V

r

DA

b

/10

1010

1

100

322316

nmmxw 500105 7

eNN

NNddw br

AD

DAnp

02

14 102 Vcmxw AV

b58

Current-voltage behavior of the p-n junction

Apply a voltage V < b to the junction

Total voltage drop is in the depletion layer

eV

NNNN

d

eV

NNN

Nd

br

ADD

An

br

ADA

Dp

)(21

)(21

0

0

eV

NN

NNddw br

AD

DAnp

)(2 0

V is > 0 if the barrier is decreased

V is < 0 if the barrier is increased

The total current through the junction is given by the changes in the minority carrierconcentration, since the majority carriers are absent in the depletion layer

Forward bias

Reverse bias

resistivity of space charge region >> bulk p,n regions

V < b

No V restriction

59

Equilibrium concentration of e- in bulk n material

Low injection conditions =majority carrier conc. have negligible

change at the boundaries of depletion layer

TK

Ve

npB

b

edndn)(

)()(

0)( nn ndn

TKeV

pTK

Ve

npBB

b

enendn 0

)(

0)(

0nn0pn

Equilibrium concentration of e- in bulk p material(minority carriers)

The minority carrier concentration in –dp

(p-type region/depl. region boundary) depends on V through the exponential

Applying the potential + V andassuming a quasi-equilibrium state

60

the e current is in theopposite direction of the e flow

the h current is in thesame direction of the h flow

TKeV

ppBendn 0)(

0nn0pn

Changes in minority carrier concentrations at the ends of the depletion layer

1)( 0

TKeV

ppBendn

TKeV

nnBepdp 0)(

1)( 0

TKeV

nnBepdp

The e- (minority carrier) concentration in –dp

(p-type region/depl. region boundary) depends on V through the exponential

The hole (minority carrier) concentration in dp

(n-type region/depl. region boundary) depends on V through the exponential

61

0nn0pn

The total electron current is the diffusion current (drift due to minority carriers is negligible outside depletion region)

Changes in minority carrier concentrations

1)()( 0

TKeV

pnpnpDiff

nBeneDdneDdJ

1)()( 0

TKeV

npnpnDiffp

BepeDdpeDdJ

The total hole current is the diffusion current (drift is negligible outside depletion region)

100

TKeV

pnnpBeneDpeDJ

Narrow space-charge region:neglect generation and recombination processes within itthe total current is given by the sum of the contributions

62

1TK

eV

sBeJJ

Ideal diode equation

00 pnnps neDpeDJ reverse saturation current

Js is made by minority carrier contribution only:Holes are created thermally in the n-type region;

those close to the barrier region reach the depletion layer by diffusion and are swept by the

internal electric field to the p-sideSame hold for e-

63

Ideal I-V characteristic

1TK

eV

sBeJJ

The impedance is low in forward bias 1- 10

The impedance is high in reverse bias 1 105

Differential resistanceVI

R

1

Real I-V characteristic

(a) Generation-recombination (b) Diffusion-current (c) High-injection (d) Series-resistance effect(e) Reverse leakage current due to generation-recombination and surface effects.

64

Junction breakdown

breakdown mechanisms(1) thermal instability(2) tunneling(3) Avalanche multiplication

For high fields applied to a p-n junction, the junction breaks down and conducts a very large

current only in reverse-bias

reverse current at high reverse voltage heat dissipation Increase in junction temperature increases the reverse current Js

breakdown

Thermal instability

Large fields can induce tunneling through the barriers

Tunneling

65

e

TkD Bn

n

00 pnnps neDpeDJ

V~ 106 V/cm

Junction breakdown

Avalanche multiplication

In the presence of high field, reverse carrierscan induce ionization (new electron – holes pairs)

within the depletion region by scattering withthe material atoms

Breakdown voltage depends on impurity con. and gradient

Zener diode = well controlled VBD

43

316

23

101.160

cmN

eVE

V GBD 66

Electronic devicesLight emitting diodesSolar cells Photodetectors

Emitter: higly doped p region ~ 1018 cm-3

Base: weakly doped n region ~ 1014 cm-3

Collector: medium doped p region ~ 1016 cm-3

Base region width << holes diffusion length in n type

holes holese-

Bipolar junction TRANSISTOR

67


Base-Emitter:forward bias

Base-Collector:reverse bias

Setting VB to ground = 0 VC << 0 < VE

base region width << holes diffusion length in n type All holes are collected in C

TRAN(sfer)-(re)SISTOR

E to B current: holes injection in the base + e injection in emitter

e injection in E small: E is strongly doped, while B is weakly doped

we expect IC to be only slightly smaller than IE

1018 cm-3 1014 cm-3 1016 cm-3

Name coined to address the "tran(sfer)-(res)istor" action from low-impedance EB junction to high-impedance CB junction

68

VC << 0 < VE

Current transfer parameter99.0E

C

II

Collector current

Emitter current

1001

CE

C

B

C

III

II

Current gain

CEB III Base current

three-terminal device whose resistance between two terminals is controlled by the

third


69

11

11

2221

1211

KT

eV

KT

eV

C

KT

eV

KT

eV

E

CE

CE

eaeaJ

eaeaJ

equilibrium concentrationsof minority carriers

Cp

Bn

Ep

n

p

n

0

0

0

100

TKeV

pnnpBeneDpeDJ

The current between E-B and B-C will be of the type


70

11

11

2221

1211

KT

eV

KT

eV

C

KT

eV

KT

eV

E

CE

CE

eaeaJ

eaeaJ

B

B

B

nBC

C

pCC

B

B

B

nBB

B

B

B

nBB

E

pEE

L

w

L

peD

L

neDa

L

w

L

peDaa

L

w

L

peD

L

neDa

coth

sinh

coth

0022

1

02112

0011


For wB >> LB

B

nBC

C

pCC

B

nBB

E

pEE

L

peD

L

neDa

aa

L

peD

L

neDa

0022

2112

0011

0

1

1

22

11

KT

eV

C

KT

eV

E

C

E

eaJ

eaJ

the pnp device is nothing more than the sum of two independent junctions

71

11

11

2221

1211

KT

eV

KT

eV

C

KT

eV

KT

eV

E

CE

CE

eaeaJ

eaeaJ

B

nBC

C

pCC

B

nBB

B

nBB

E

pEE

w

peD

L

neDa

w

peDaa

w

peD

L

neDa

0022

02112

0011

eVE >> KBT VC << 0

B

E

dB

aE

E

B

w

L

N

N

D

D

For large , a small variation of IB gives large IC

1

1

21

11

KT

eV

C

KT

eV

E

E

E

eaJ

eaJ

2111

21

aaa

JJJ

CE

C

B

C

II


base region width << holes diffusion length in n typeFor wB << LB

B

E

DBpE

AEnB

B

E

i

i

pE

nB

E

B

E

pEE

B

nBB

w

L

Nn

Np

w

L

nn

n

p

D

D

L

neD

wp

eD

0

02

2

0

0

0

0

20 ipEaE nnN

Minority carriers conc

D

ii

Nn

nn

Tp22

)(

72


Collector and base currents as a function of base-emitter voltage

B

C

II

B

E

dB

aE

E

B

w

L

N

N

D

D

73

A bipolar transistor can be connected in three circuit configurationsdepending on which lead is common to the input and output circuits


common-base common-emitter

common-collector

74

forward-biased semiconductor p-n junction emitting spontaneous radiation in the UV, vis, IR

Light Emitting Diode (LED)

Excited electron

Electroluminescence

(a) intrinsic emission(b) higher-energy emission

Interbandtransitions

Chemical impuritiestransitions

Intrabandtransitions

75


forward bias p-n junction: injection of minority carriers across the junction Holes AND electrons efficient radiative recombination inside

The material determines the

color of the emitted light

Basic Principle

76


The material determines the

color of the emitted light

Efficiency improved by a layer with lower EG

White lightcombine LEDs of different colors: red, green, and blue costly

Single LED covered with a color converter.absorbs the original LED light and emits light of different frequency.

The converter material can be phosphor, organic dye,

77

SOLAR CELLS

() = absorption coefficient() = photon fluxR() = reflected fraction

xeRxG 1,

• incident on the front surface• Generation rate of e- hole pairs at x from the

semiconductor surface

x

Solar spectrum for different relative atmospheric path lenghts

Silicon p-n junction solar cell:reference device for all solar cells

x

78

SOLAR CELLS

Assumed abrupt doping profiles ND >> NA

• Abrupt p-n junction solar cell• Constant doping on each side• ND >> NA Depletion region at the n-side can

be neglected• No electric fields outside the depletion region • Low injection condition

),(,, pneDJ pnpn

Photogenerated carriers collection

n,p regions: diffusion depletion region: drift

010

dxdJ

e

nnG n

n

ppn

EneJ pnpn ,,

electrons in p-type substrate

010

dx

dJ

epp

G p

p

nnp

holes in n-type substrate

79

SOLAR CELLS

Current density equations across depletion region boundaries

n side

dxdp

eDEepJ

dx

dneDEenJ

nppnp

pnnpn

Continuity equation 01 0

2

2

p

nnxnp

ppeR

dxpd

D

Solved with appropriate boundary conditions

80

SOLAR CELLS

Current density equations

p side

dxdp

eDEepJ

dx

dneDEenJ

nppnp

pnnpn

Continuity equation 01 0

2

2

n

ppxpn

nneR

dx

ndD

Solved with appropriate boundary conditions

81

SOLAR CELLS

Total photocurrent

Spectral response

drpnL JJJJ

SR for differentrecombination

velocities

R

JJJSR drpn

1

Total current IL is obtained by integration on

82

SOLAR CELLS

Equivalent circuit for an ideal pn diode solar cell equivalent circuit

IL = Constant photocurrent source

in parallel with the junction

IS = diode saturation current

LKTqV

S IeII

1

open-circuit voltage setting I = 0

S

L

S

LOC I

I

qKT

I

I

qKT

V ln1ln

KTeE

KT

e

pnnps

GB

eeneDpeDJ

00

VOC increases for decreasing Js Js is decreasing for large EG

Maximum absorption for small EG 83

References S.M. Sze, Physics of semiconductor devices (Wiley) Chap 1,2 Ashcroft-Mermin, Solid State Physics (Saunders) Chap 4-5, 28-29 Grosso- Parravicini,

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