ch1 1 carrier_modeling

8/11/2019 Ch1 1 Carrier_modeling

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Basic Theory of Semiconductors

and Junctions

References:(1) RF Pierret, “Semiconductor Device Fundamentals”, Addison Wesley, 1996.(2) RS Muller & TI Kamins, “Device Electronics for Integrated Circuits,” John Wiley &

Sons, 1986, 2nd Edition

Carrier Modeling

Carrier Action

PN Junction

MS Junction



Carrier Modeling

Quantization Concept

Hydrogen atom model, isolated Si atom

Semiconductor Models

Bonding model, energy band model, carriers

Distribution of CarriersCarrier numbers in intrinsic semiconductor, doping,carrier distribution

Carrier Concentration

Carrier concentration calculation, Fermi levelcalculation



ps

+14

The quantization concept

Isolated Si atom

Hydrogen atom model

- Niels Bohr, 1903

- Explained why the emitted light was observed at onlycertain discrete wavelength

- Assume orbiting electron could take only certain valuesof angular momentum

- Electron limited to certain energies inside H atom

n=1

2 Electrons

n=2

8 Electrons

n=3

Two allowed levelsat same energy

Six allowed levelsat same energy

- 10 electrons occupy very deep energy levels,

and these 10 electrons are treated as “core” ofatom along with nucleus due to tight bound

- Remaining 4 electrons, weakly bond,collectively called “valence electrons”

...3,2,1,6.13

)4(2 22

0

4

0 neV

nn

qm E H

2/h

+

n=2

-3.4eV

n=3

EH=-1.51eV

n=1

-13.6eV



Semiconductor Models: Bonding model

Line: a sharedvalence electron

Circle: the coreof a Si atom

Bonding model: Si atoms share one of valenceelectrons with the four nearest neighbors

Line refers to a shared valence electron

Circle refers to the core of a Si atom

Applications: can explain

A missing of atom

Breaking of atom-to-atom bond and freeing ofan electron

Drawback: cannot give more information, such asenergy-related aspects of events

Physical picture of a missingatom using bonding model

Physical picture of breaking of atom-to-atom

bond and freeing of an electron usingbonding model

-

Bonding model



Mostly filled

Mostly empty

Semiconductor Models: Energy band model

N IsolatedSi atom

ps n=3

4N electrons total8N electron states total

(6N p-states total)

(2N s-states total)

E

Crystal Si with N atoms

4N allowed states(conduction band)

4N allowed states(valence band)

No statesE

E (electron energy)

ETOP

EBOTTOM

EC

EVEg

Decreasing atom spacingIsolated Si atom Si lattice spacing

EC

EV

4N emptystates

2N + 2Nfilled states

Silicon semiconductor

E (electron energy)



Semiconductor Models: Applications

Electron and hole carrier

EC

EV

Empty

Completelyfilled

EC

EV

EC

EV

No carriersOne electron carrier One hole carrier

-

Bonding model

Energy band model



Semiconductor Models: Applications

Band gap and material classification

Insulator:

wide band-gap, difficult to excite at room temperature, poorconductor

Metal:

either very small or no band gap exists due to an overlap of thevalence and conduction bands

Semiconductor:

between insulator and metal

EC

EV

Eg

Insulator

EC

EV

Eg

Semiconductor

EC

EC

EV

EV

Metal



-e-

Carrier Distributions: Carrier properties

Carrier Properties

Charge:

One electron carries charge of “–q”; one hole carriescharge of “+q”

Effective mass: Physical meaning

Have multiple components due to the variety ofcarrier acceleration with different direction of travel

in a crystal Some examples of effective mass: density of

states effective mass, conductivity effectivemass, …

Material mn*/m0 Mp*/m0

Si 1.18 0.81

Ge 0.55 0.36

GaAs 0.066 0.52

Density of states effective masses at 300K

- a =Fext

mn

Fext

Ex

x

Vacuum

a =Fext

mn*

Fint

Ex

x

Crystal

++

+ +

+

+



Carrier Distribution: Intrinsic semiconductor

Carrier numbers in intrinsic semiconductor

Intrinsic:

Pure material, no impurity atom

n=p=ni (why n=p?)

~1E10 cm-3 in Si at room temperature

Carrier numbers is very low, poor conductor

for example,

Si atom ~ 5E22 cm-3

Total bonds (or total valence electrons) ~ 2E23 cm-3

Ratio of breaking bonds / total ~ 1E10/2E23

~ 1E-13 very small value !!!



Carrier Distribution: Doping

Doping: manipulation of carrier numbers n- and p-type:

Physical picture (bonding model)

N-type: P+ atom replace Si atom the 5th

electron weakly bonded easily to be free P-type: B+ atom replace Si atom

Energy band model (to answer the question: how

easy to make electron free) Binding energy

Can be modeled using pseudo-hydrogenatom model

-

P+

Ec

Ev

ED0.045eV(P) (Si) ~ 1.12eV

Donors Bindingenergy

Acceptor Bindingenergy

Sb 0.039 eV B 0.045 eV

P 0.045 eV Al 0.067 eV

As 0.054 eV Ga 0.072 eV

Dopant-site binding energy

+

B-



Carrier Distribution: DOS

Density of States (DOS)

Carrier Distribution

How the carriers are distributed with energy levelsWhat we should know:

How many states are available at a given energy E Probability that an available state at energy E will be occupied by an electron

Analogy: The density of states can be likened to the description of the seating in a

football stadium, Number of seats in the stadium a given distance from the playing field Probability of seats to be occupied

gC(E)dE means no. of conduction band states in dE

gV(E)dE means no. of valence band states in dE

Calculated with quantum physics

Only valid for energies not too far from the band energies

E

Ev

Ec gC(E)

gV(E)

32

** )(2)(

C nn

c

E E mm E g

for E EC

32

** )(2)(

E E mm E g

V p p

v

for E EV



EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11

Carrier Distribution: Fermi-Dirac Function

Fermi-Dirac Function

T=0 K

1/2

1

EF

E

T>0 K

E

1

EF

Some discussions

T=0K

T>0K

if E=EF, F(EF)=1/2

if EEF+3kT, f(E)exp[-(E-EF)/kT]

if EEF-3kT, f(E)1-exp[(E-EF)/kT]

A note on the meaning of “Fermi level”

For energy level below EF, it will beoccupied by electron

In thermal equilibrium, EF constantthrough the system

kT E E F e E f

/)(1

1)(

Where, EF is Fermi energy, k is Boltzmanconstant, T is Temperature in Kelvin (K)

f(E) represents probability of one state atenergy E to be occupied by electron




Carrier Distribution: How carrier concentration looks like





Carrier Distribution: Dopant Effect

gc(E)

gv(E)

Ec

Ev

EF

1-f(E)

f(E)

gc(E)

gv(E)

Ec

Ev

EF

gc(E)f(E)

gv(E)[1-f(E)]

1-f(E)

f(E)

gc(E)

gv(E)

Ec

EvEF

1-f(E)

f(E)

Energy band Density of states Occupancy factors Carrier distributions

N-type

Intrinsic

P-type




Since gc(E)dE represents the number of

conduction band states/cm3 lying in the E to E+dEenergy range

f(E) specifies the probability that an availablestate at an every E will be occupied by an

electron then, gc(E)f(E)dE gives the number of conduction

band electrons/cm3 lying in the E to E+dE energyrange

So, integration of gc(E)f(E)dE over all conduction

band energies must yield the total number ofelectrons in the conduction band.

Similarly, the total number of holes in the valenceband can be calculated as

dE E f E gnTOP

C

E

E

c )()(

dE E f E g p

V

BOTTOM

E

E

v )](1[)(

Formulas for n and p

Carrier Concentration: Calculation



where

NC and NV are the effective density of conduction and valence band states

Some discussions

Detailed calculation is very complex, simplified calculation equations areoften used

Only valid for non-degenerate semiconductor (Ev+3kT<EF<Ec-3kT)

Carrier Concentration: Simplified Calculation

kT E E

V

kT E E

C

F V

C F

e N p

e N n/)(

/)(

Simplified calculation for n and p

2/3

2

*

]2

[2

kT m N n

C 2/3

2

*

]2

[2

kT m N

p

V

Degeneratesemiconductor

Degeneratesemiconductor

3kT

3kT

EC

EV

Non-degeneratesemiconductor



Alternative expression for n and p

Considering intrinsic semiconductor, where n=p=ni and EF=Ei. we have

Then, Nc and NV can be calculated as

Then, n and p can be solved and expressed as a function of ni and Ei, so

Carrier Concentration: Alternative expression for n and p

kT E E C i

C ie N n /)( kT E E

V iiV e N n /)(

kT E E iC

iC en N /)( kT E E

iV V ien N /)(

kT E E i

iF enn /)(

kT E E

iF ien p

/)(




Carrier Concentration: Mass action law

kT E

V C

kT E E

V C iGV C e N N e N N n

//)(2

kT E

V C iGe N N n

2/

ni

2

innp

np product (mass action law)

kT E E

C iC ie N n

/)(

kT E E

V iiV e N n

/)(

kT E E

iiF enn

/)(

kT E E

iF ien p

/)(

Some discussions

Very important equations

Equations for n (p) and np product are valid fornon-degenerate semiconductor in equilibrium




Charge neutrality relationship

For an uniformly doped semiconductor to be everywhere charge-neutral clearlyrequires

where

At room, it is very reasonable to assume all dopants ionized, so we have

Finally, one then obtains

03

A D qN qN qnqpcm

charge

0

A D N N n p

D D N N A A N N

0 A D

N N n p

D N = number of ionized (positively charged) donors/cm3

A N =number of ionized (negatively charged) acceptors/cm3

Carrier Concentration: Charge Neutrality

OR



Carrier Concentration: Two general equations

Carrier concentration calculation

2

innp 0 A D N N n p

2/122 ])2

[(2

i A D A D n N N N N n

2/1222

])2

[(2 i

D A D Ai n N N N N

n

n p



Carrier Concentration: Some special cases

Some special cases

Intrinsic semiconductor (N A=0, ND=0)

n = p = ni

Doped semiconductor where either ND - N A ~ ND>> ni or N A - ND ~N A >> ni

n=ND, p=ni2/ ND

p=N A, n=ni

2

/ N A

Doped semiconductor where ni >> |ND - N A|

n = p =ni

Compensated semiconductor (ND-N

A= 0)

like intrinsic material, n = p = ni




Case I: Intrinsic semiconductor

In an intrinsic material

Carrier Concentration: Determination of Fermi level

pn kT Ei E V kT E EiC

V C e N e N /)(/)(

)ln(22 C

V V C i N

N kT E E E

2/3*

*

)(n

p

C

V

m

m

N

N

)ln(4

3

2 *

*

n

pV C i

m

mkT

E E E

EF=Ei in an intrinsic material



Doped semiconductors (non-degenerate, dopants totally ionized) N-type semiconductor

P-type semiconductor

Carrier Concentration: Determination of Fermi level

)/ln( iiF nnkT E E

)/ln( i DiF n N kT E E

)/ln( i AF i n N kT E E

n=N DkT E E

iiF enn

/)(



Carrier Concentration: Temperature Dependence

n=0 n=ND+ n=ND n=ni

0 K Low T Moderate T High T



Summary of “ Carrier Modeling”

Semiconductor Models Bonding Model

Energy Band Model

Carrier Distribution Density of States

Fermi-Dirac Functions

Carrier Concentration Some important equations Mass action law

ch1 1 carrier_modeling

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