korea university chapter 2. carrier...

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Photonics Laboratory

Chapter 2. Carrier Modelling

: Carriers are the entities that transport charge from place to place inside a material and

hence give rise to electrical currents. In everyday life the most commonly encountered

type of carrier is the electron, the subatomic particle responsible for charge transport in

metallic wires. Within semiconductors one again encounters the familiar electron, but

there is also a second equally important type of carrier – the hole.

1. The quantization concept

Hydrogen atom:

𝐸𝐻=−𝑚0𝑞4

2 4𝜋𝜀0 ℎ/2𝜋 𝑛 2 = −13.6

𝑛2 𝑒𝑉,

where EH is the electron binding energy within the hydrogen atom, m0 is the mass of a

free electron, q is the magnitude of the electronic charge, 𝜀0 is the permittivity of free

space, h is Planck’s constant, and n is the energy quantum number (n=1, 2, 3, …).

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Chapter 2. Carrier Modelling

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2. Valence bonds

Relative to the hydrogen atom, the energy level scheme in a multi-electron atom like silicon is, as

one might intuitively expect, decidedly more complex. As pictured in Fig. 2.2, ten of the 14 Si-

atom electrons occupy very deep-lying energy levels and are tightly bound to the nucleus of the

atom. The binding is so strong, in fact, that these ten electrons remain essentially unperturbed

during chemical reactions or normal atom-atom interactions, with the ten-electron-plus-nucleus

combination often being referred to as the “core” of the atom. The remaining four Si-atom

electrons, on the other hand, are rather weakly bound and are collectively called the valence

electrons because of their strong participation in chemical reactions.

2

2. Valence bonds

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The isolated Si atom, or a Si atom not interacting with other atoms, was found to

contain four valence electrons. Si atoms incorporated in the diamond lattice, on the

other hand, exhibit a bonding that involves an attraction between each atom and its

four nearest neighbors.

3

2. Valence bonds

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Photonics Laboratory 4

2. Valence bonds

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3. Energy bands

5

3. Energy bands

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3. Energy bands

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(3-1) Carriers in energy bands

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(3-1) Carriers in energy bands

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(3.2) Band gap and material classification

At room temperature (T=300 K), EG=1.42 eV in GaAs, EG=1.12 eV in Si, and EG=0.66 eV in Ge.

8

(3.2) Band gap and material classification

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4. Carrier properties

(4.1) Charge

Both electrons and holes are charged entities. Electrons are negatively charged, holes

are positively charged: q=1.6×10-19 coul. The electron and hole charges are –q and +q,

respectively.

(4.2) Effective mass

The effective mass of electrons within a crystal is a function of the semiconductor

material (Si, Ge, etc.) and is different from the mass of electrons within a vacuum. An

electron of rest mass m0 is moving in a vacuum between two parallel plates under the

influence of an electric field

where v is the electron velocity and t is time. Next consider electrons moving in a

semiconductor crystal under the influence of an applied electric field.

where mn* is the electron effective mass.

(2.2) 0dt

vdmqF

(2.3) *

dt

vdmqF n

4. Carrier properties

9

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*

p

*

n mm and qq- holesor F

Gein /102

Siin /101

GaAsin /102

re temperaturoomAt

.npn

,conditions mequilibriuunder tor semiconduc intrinsican Given

mcarriers/c intrinsic ofnumber n

holes/cm ofnumber p

cmelectrons/ ofnumber n

tor.semiconduc pureextremely an toreferstor semiconduc intrinsic The

material intrinsicin numbersCarrier )34(

313

310

36

i

3

i

3

3

cm

cm

cmni

(4-3) Carrier numbers in intrinsic material

10

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(4-4) Manipulation of carrier numbers-doping

: The intrinsic carrier concentration is relatively small compared with the number of

bonds that could be broken.

Ex) In Si, There are 5×1022atoms/cm3 and four bonds per atom, making a grand total of

2×1023 bonds or valence band electrons per cm3.


Doping is the addition of controlled amounts of specific impurity atoms with the express

purpose of increasing either the electron or the hole concentration.

! 1010

10

electrons alence #

13

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10

vof

ni

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12

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.E-EE

is levelenergy donor The 11.8).(Ksconstant dielectric Si theis K here

1.01

42

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w

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13

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14

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15

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?(Si)E (d)

Si?in donors and acceptors ofenergy ionization theis What (c)

300K/at eVmany how toequal is kT (b)

energy? of joulesmany how toequal is 1eV (a)

2.1)Ex

g


16

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joules.101.61eV

energy? of joulesmany how toequal is 1eV (a)

2.1)Ex

19-


16

300Kat 1.12eV

?(Si)E (d) g

0.0259eV

.)(300)J101.38(

or )(300)eV10(8.617kT

300K/at eVmany how toequal is kT (b)

23-

5-

J

eV191061

1

0.1eV.E

Si?in donors and acceptors ofenergy ionization theis What (c)

B

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5. Density of states

(5-1) Density of states

We are now interested in the energy distribution of states, or density of states, because the state

distribution is an essential component in determining carrier distributions and concentrations.

17

5. Density of states

).E E (if dEE and Ebetween rangeenergy the

in lying states/cm band valenceofnumber therepresents :(E)dEg

),E E (if dEE and Ebetween rangeenergy the

in lying states/cm band conduction ofnumber therepresents :(E)dEg

ly.respective bands, valenceand

conduction in the Eenergy an at states ofdensity theare (E)g and (E)g

v

3

v

c

3

c

vc

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02

2

22

2

22

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2222

82422En

mL

hn

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n

m

h

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One-dimensional infinite-potential well

: A model of electrons in semiconductor

(5-1) Density of states: Modelling

18

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21

3

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(5-1) Density of states: Calculation

19

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v

/

/*

c

//*

EE ,)(

)(

EE , )(

)(

by defined is states ofdensity

electons ofnumber total:)(

)()()(

bygiven are dEE and Ebetween electrons ofnumber The

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20


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21

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(5-2) The Fermi function

Whereas the density of states tells one how many states exist at a given energy E, the

Fermi function f(E) specifies how many of the existing states at the energy E will be

filled with an electron, or equivalently,

𝑓 𝐸 =1

1+𝑒(𝐸−𝐸𝐹)/𝑘𝑇 (2.7)

F(E): specifies, under equilibrium conditions, the probability that an available sate at an

energy E will be occupied by an electron.

EF = Fermi energy or Fermi level

k = Boltzmann constant (𝑘 = 8.617 × 10−5 eV/K or 1.38× 10−23 J/K)

T = temperature in Kelvin (K)


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𝑓 𝐸 =1

1+𝑒(𝐸−𝐸𝐹)/𝑘𝑇 (2.7)

(1) T→ 0

𝐸 < 𝐸𝐹: (𝐸 − 𝐸𝐹)/kT → −∞ → 𝑓(𝐸 < 𝐸𝐹)=1

𝐸 > 𝐸𝐹: (𝐸 − 𝐸𝐹)/kT → +∞ → 𝑓(𝐸 > 𝐸𝐹)=0

All states at energies above EF will be empty for temperatures T → 0 K. In other words, there is a

sharp cutoff in the filling of allowed energy states at the Fermi energy EF when the system

temperature approaches absolute zero.

(2) T > 0 𝐾

empty. be willE above moreor 3kT energiesat statesmost Moreover,

energy. increasing with zero lly toexponentia

decaysy probabilit state-filledor function Fermi theE Above

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F

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kT

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3

]./)1]/),3)(

.2/1)(,)(

23


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small. quite typicallyis formalism0K T in the

y prominentl appears that intervalenergy 3kT thegap, band Si the toCompared

(Si).E 0.0777eV3kT and eV 0.0259kT 300K),(T re temperaturoomAt (iV)

filled. be willE below moreor 3kT

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kT

kT

kTkTiii

24


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Ex 2.2) The probability that a state is filled at the conduction band edge (Ec) is

precisely equal to the probability that a state is empty at the valence band edge (Ev).

Where is the Fermi level located?

midgap.at positioned is level Fermi :

2

EE E-E2E

1

1

1

1

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The

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ee

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vvFFc

kTEEkTEE

kTEE

kTEE

kTEE

kTEE

kTEEv

kTEEc

vc

vFFv

Fv

Fv

Fv

Fv

Fv

Fc

26


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(5-3) Equilibrium distribution of carriers

The desired distribution is obtained by simply multiplying the appropriate density of states by the

appropriate occupancy factor – gc(E)f(E) yields the distribution of electrons in the conduction

band and gv(E)[1-f(E)] yields the distribution of holes (unfilled states) in the valence band.

(5-3) Carrier densities

.)/exp(

,)(

)(

by obtained is electrons ofnumber

electons ofnumber total:)(

)()()(

bygiven are dEE and Ebetween electrons ofnumber The

//

f

ee

ee

eee

EEf(E)E

h

mEg

The

dEEnN

EfEgEn

1

124 21

3

23

0

27

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particles ofnumber total:)(

)()()(

)()()(

)()()(

bygiven are dEE and Ebetween particles ofnumber The

0

dEEnN

EfEgEn

EfEgEn

EfEgEn

FDFDFD

BEBEBE

BBB

: The desired distribution is obtained by simply multiplying the appropriate density of states by

the appropriate occupancy factor – gc(E)f(E) yields the distribution of electrons in the

conduction band and gv(E)[1-f(E)] yields the distribution of holes (unfilled states) in the valence

band.

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)](exp[~

)exp(

)(

,)(

and )()()( where

)(

, and between intervalenergy an for , energy,given ahan smaller tor

toequalenergy as have that electrons ofnumber thebe to Defined

/

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32

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particles ofnumber total:

)(

)()()(

0

dEEnN

EfEgEn FDFDFD

30

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)(

)()()(

0

dEEnN

EfEgEn FDFDFD

31

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)(

)()()(

0

dEEnN

EfEgEn FDFDFD

32

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1) In general, all carrier distributions are zero at the band edges, reach a peak value very close to

Ec or Ev, and then decay very rapidly toward zero as one moves upward into the conduction

band or downward into the valence band.

2) When Ef is positioned in the upper half of the band gap (or higher), the electron distribution

greatly outweighs the hole distribution. Likewise, a predominace of holes results when Ef lies

below the middle of the gap.

3) The greatest number of electrons or holes are close to Ec and Ev, reflecting the peak in the

carrier concentrations near the band edges.

33

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.00

0.25

0.50

0.75

1.00

f(E

)

Energy (eV)

10K

100K

200K

300K

400K

600K

T

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0.4 0.6

0.00

0.25

0.50

0.75

1.00

f(E

)

Energy (eV)

10K

100K

200K

300K

400K

600K

T

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.00

0.25

0.50

0.75

1.00

f(E

)

Energy (eV)

Ef=0.2eV

Ef=0.36eV

Ef=0.56eV

Ef=0.76

Ef=0.92

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-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00E+000

1.00E+018

2.00E+018

3.00E+018

4.00E+018

5.00E+018

D

en

sity o

f S

tate

s(c

m-3)

Energy(eV)

valence band

conduction band

band gap

korea university chapter 2. carrier...

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