PDF CH02 Carrier modeling [호환 모드] - Seoul National University

Chapter 2.

Carrier Modeling

Sung June Kim g [email protected] http://helios.snu.ac.kr p

1

Subject j

1. 양자역학은 반도체 이론에서 어떻게 쓰이는가?

2. 반도체는 어떤 결합을 하고 있으며 에너지 대역은 어떻게 형성되는가?

3. 정공(hole)은 어떻게 생성되고 그의 전자와의 관계는? 또 이들의 유효 질량이란 무엇인가?

4. 진성 반도체에서 보다 많은 전하 캐리어가 필요할 때는 어떻게 하는가

??

5. 전자의 에너지에 대한 분포는 어떤 모양인가?

Contents

‰ The Quantization Concept

‰ Semiconductor Models

‰ Carrier Properties

‰ Carrier Properties

‰ State and Carrier Distributions

‰ Equilibrium Carrier Concentrations

3

9Carriers: electron, hole

9 Equilibrium: no external voltages magnetic fields stresses or 9 Equilibrium: no external voltages, magnetic fields, stresses, or other perturbing forces acting on the semiconductor. All observables are invariant with time.

• The Quantization Concept

9 In 1913 Niels Bohr hypothesized that “quantization” of the 9 In 1913 Niels Bohr hypothesized that quantization of the electron’s angular momentum was coupled directly to energy quantization

h 13 . 6 , 1 , 2 , 3 , L )

4 (

2

₀ ^{2 2}

4

0

= − =

−

= eV n

n n

q E

_n

m

πε h ) 4

(

2 πε

₀

n n

The hydrogen atom – idealized representation showing the first three allowed electron orbits and the associated energy quantization

and the associated energy quantization

5

‰ Semiconductor Models

‰ Semiconductor Models

• Bonding Model

9 Si i d i h di d l i hibi b di h 9 Si atoms incorporated in the diamond lattice exhibit a bonding that involves an attraction between each atom and its four nearest

neighborsg

9 Any given atom not only contributes four shared electrons but must

Visualization of a missing atom or point defect

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

• Energy Band Model

9 Starting with N isolated Si atoms and conceptually bringing the 9 Starting with N-isolated Si atoms, and conceptually bringing the atoms closer and closer together, Pauli exclusion principle makes a progressive spread in the allowed energies

7

9 E b d th d i i i i t l l d 9 Energy bands: the spread in energies gives rise to closely spaced sets of allowed states Æ conduction band, valence band, band gap (or forbidden gap)

( g p)

9 In filling the allowed energy band states, electrons tend to gravitate to the lowest possible energies

9 Th l b d i l t l t l fill d ith l t d th 9 The valence band is almost completely filled with electrons and the conduction band is all but devoid of electrons

9 The valence band is completely filled and the conduction band p y completely empty at temperatures approaching T=0 K

9

• Carriers

9 When a Si-Si bond is broken and the associated electron is free to wander the released electron is a carrier

wander, the released electron is a carrier

9 In terms of the band model, excitation of valence band electrons into the conduction band creates carriers; electrons in the conduction band are carriers

9Completely filled valence band : no current

9The breaking of a Si-Si bond creates a missing bond or voidg g 9 Missing bond in the bonding scheme, the empty state in the valence band, is the second type of carrier– the hole

9 Both electrons and holes participate in the operation of most 9 Both electrons and holes participate in the operation of most semiconductor devices

11

• Band Gap and Material Classification

9 The major difference between materials lies not in the nature of the energy bands but rather in the magnitude of the energy gap

energy bands, but rather in the magnitude of the energy gap

9 Insulators: wide band gap. The thermal energy available at room temperature excites very few electrons from the valence band into the

f

conduction band; thus very few carriers exist inside the material

9 Metals: very small or no band gap exists at all due to an overlap of the valence and conduction bands. An abundance of carriers Æ

excellent conductors

9 Semiconductors; intermediate case

9 At 300K E =1 42 eV in GaAs E =1 12 eV in Si E =0 66 eV in Ge 9 At 300K, E_G=1.42 eV in GaAs, E_G=1.12 eV in Si, E_G=0.66 eV in Ge.

9 Thermal energy, by exciting electrons from the valence band into the conduction band, creates a moderate number of carriers

13

‰ Carrier Properties

‰ Carrier Properties

• Charge

9 Electron (-q), hole (+q), q=1.6×10( q), ( q), q ^-19 C

• Effective Mass

9 An electron of rest massAn electron of rest mass mm₀₀ is moving in ais moving in a vacuum between two parallel plates under the influence of E,

0

q m d

= − = dt

F E v

9 Conduction band electrons moving between the two parallel end faces of a semiconductor crystal under the influence of an applied faces of a semiconductor crystal under the influence of an applied electric field

9 I dditi t th li d l t i fi ld l t i t l

9 In addition to the applied electric field, electrons in a crystal are also subject to complex crystalline fields

9The motion of carriers in a crystal can be described by Quantum y y Mechanics.

9 If the dimensions of the crystal are large compared to atomic dimensions the complex quantum mechanical formulation for the dimensions, the complex quantum mechanical formulation for the carrier motion between collisions simplifies to yield an equation of motion identical to Newton’s 2^nd equation, except that m₀ is replaced by an effective carrier mass

n

qE m d dt

= − =

∗

F v

Electron effective mass

dt

15

9 F h l ith Æ d ^* Æ ^* 9 For holes with –q Æ q and Æ

9 The carrier acceleration can vary with the direction of travel in a crystal Æ multiple components

m

n

m

_p

y p p

9 Depending on how a macroscopic observable is related to the carrier motion, there are, for example, cyclotron resonance effective masses conductivity effective masses density of state effective mass masses, conductivity effective masses, density of state effective mass, among others.

Material

m

^∗

/ m m

^∗

/ m

Density of State Effective Masses at 300 K Material

m

_n

/ m

₀

m

_p

/ m

₀

Si 1.18 0.81 Ge 0.55 0.36 GaAs 0.066 0.52

• Carrier Numbers in Intrinsic Material

9 Intrinsic semiconductor = pure (undoped) semiconductor 9 An intrinsic semiconductor under equilibrium conditions 9 An intrinsic semiconductor under equilibrium conditions

n = = p n

i

e temperatur room

at Si in cm

n

_i

= 1 × 10

^{10 −3}

9 The electron and hole concentrations in an intrinsic semiconductor are equal:

are equal:

9 Si atom density: 5×10²² cm^-3, total bonds: 2×10²³ cm^-3 and n_i~ 10¹⁰ cm^-3 Æ 1/10¹³ broken in Si @ 300 K

17

• Manipulation of Carrier Numbers - Doping

9 The addition of specific impurity atoms with the purpose of increasing the carrier concentration

increasing the carrier concentration

Common Si dopants.

Donors (Electron-increasing dopants) Acceptors (Hole-increasing dopants)

P Å B Å

Column V Column III

As Å Ga

Sb In Al Column V

elements

Column III elements

9 Column V element with five valence electrons is substituted for a Si atom four of the five valence electrons fits snugly into the bonding atom , four of the five valence electrons fits snugly into the bonding structure

9 At 300 K th d l t i dil f d t d h

9 At 300 K, the donor electron is readily freed to wander hence becomes a carrier (!! no hole generation)

9 The positively charged donor ion left behind (cannot move)p y g ( )

9 The Column III acceptors have three valence electrons and cannot p complete one of the bonds.

19

9 Th C l III t dil t l t f b 9 The Column III atom readily accepts an electron from a nearby

bond, thereby completing its own bonding scheme and in the process creating a hole that can wanderg

9 The negatively charged acceptor ion cannot move, and no electrons are released in the hole-creation process

9 Weakly bound? : a binding energy ~0 1 eV or less 9 Weakly bound? : a binding energy ~0.1 eV or less

The positively charged donor-core-plus-fifth electron may be likened y to a hydrogen atom

11 8 ε = K =

∗ 4 ∗

r

K

S

11.8

ε = =

Donors lE

^B

l Accepters lE

^B

l Donors lE

^B

l Accepters lE

^B

l

Sb 0.039eV B 0.045eV P 0.045eV Al 0.067eV As 0.054eV Ga 0.072eV

In 0.16eV

Table 2.3

Dopant-Site Binding Energies

9 The bound electron occupies an allowed electronic levels at an

E E |E |

Energies

energy E_D=E_c-|E_B|

(1/ 20) (Si)

c D B G

E − E = E ≅ E

9 All donor sites filled with bound electrons at TÆ 0 K

9 As the temperature is increased, with more and more of the weakly bound electrons being donated to the conduction band

9 At 300 K, the ionization is all but total

9 The situation for acceptors is completely analogous

21

The situation for acceptors is completely analogous

Visualization of (a) donor and (b) acceptor action using the energy

E_C E_D 0.045 eV

(P) D

Donor site

(Si) 1.12 eV T = 0 K

T = 300 K

E_V T = 0 K

T = 300 K

23

x

• Carrier-Related Terminology

9 Dopants: specific impurity atoms that are added to semiconductors in controlled amounts for the purpose of increasing either the electron in controlled amounts for the purpose of increasing either the electron or the hole concentration

9 Intrinsic semiconductor: undoped semiconductor 9

9 Extrinsic semiconductor: doped semiconductor

9 Donor: impurity atom that increases the electron concentration 9 Acceptor: impurity atom that increases the hole concentrationp p y 9 n-type material: a donor-doped material

9 p-type material: a acceptor-doped material

9 Majority carrier: the most abundant carrier in a given 9 Majority carrier: the most abundant carrier in a given

semiconductor sample; electron in an n-type material and hole in a p- type material

9 Minority carrier: the least abundant carrier in a given

‰ State and Carrier Distributions

‰ State and Carrier Distributions

• Density of States y

9 How many states are to be found at any given energy in the energy bands?

9 The state distribution is an essential component in determining 9 The state distribution is an essential component in determining carrier distributions and concentrations

9 The results of an analysis based on quantum mechanical consideration: density of states at an energy E

n n c

c 2 3 c

2 ( )

( ) m m E E ,

g E E E

∗ ∗

−

= ≥

c 2 3 c

p p v

( ) ,

2 ( )

( ) ,

g

m m E E

g E E E

π

∗ ∗

−

= ≤

h

25

v

( )

2 3

,

v

g E E E

π h ≤

Difference between and stem from differences in the masses.

) ( E

g

_c

g

_v

(E )

c

( )

g E dE

represents the number of conduction band states/cm³ lying in the energy range between E and E+dE

v

( )

g E dE

represents the number of valence band states/cm³ lying in the energy range between E and E+dE

≤

• The Fermi Function

9 tells one how many states exist at a given energy E

9 Fermi function f(E) specifies the probability that an available state

) ( E g

9 Fermi function f(E) specifies the probability that an available state will be occupied by an electron

( F) /

( ) 1

1

^{E E kT}

f E = e

⁻

+

E_F=Fermi energy

k=Boltzmann constant (k=8.617×10-5 eV/K)

T t t

1 + e

F _T = temperature

27

(i) f(E_F)=1/2 (ii) If

(iii) If

( ) / ( ) /

3 , ^{E EF kT} 1 and ( ) ^{E EF kT} E ≥ EF + kT e ⁻ f E e^{− −}

( ) / ( ) /

3 , ^{E EF kT} 1 and ( ) 1 ^{E EF kT} : 1 ( ) empty E ≤ EF − kT e ⁻ f E −e ⁻ − f E

>>

<<

≅ ( ) ≅

(iv) At T=300 K, kT=0.0259 eV and 3kT=0.0777 eV << E_G

, ( ) ( ) p y

F f f

• Equilibrium Distribution of Carriers

• Equilibrium Distribution of Carriers

)]

( 1

)[

(

&

) ( )

( E f E g E f E

g

_{c v}

−

9 All carrier distributions are zero at the band edges, reach a peak value very close to E_c or E_v

9Wh E i iti d i th h lf f th b d ( hi h )

9When E_F is positioned in the upper half of the band gap (or higher), the electron distribution greatly outweighs the hole distribution

9 Equal number of carriers when E_F is at the middle

29

9 Abb i t d f hi Th t t b f i l d t

9 Abbreviated fashion; The greatest number of circles or dots are drawn close to E_c and E_v, reflecting the peak in the carrier

concentrations near the band edges

‰ Equilibrium Carrier Concentrations

‰ Equilibrium Carrier Concentrations

• Formulas for

n

and p

9 ( ) ( )E f E dE h b f d i b d l / ³ l i 9 : the number of conduction band electrons/cm³ lying in the E to E+dE range

( ) ( ) g E f E dEc

[ ]

top

c v

c

( ) ( )

E E

E

n = ∫ g E f E dE

∫

E_botton^{v v}

( ) 1 [ ( ) ]

p = ∫ g E − f E dE

9 For n-type material

2

^E

E E dE

m

^∗

m

^∗ top

c F

c

n n

( ) /

2 3

2

1

E

E E kT E

E E dE

m m

n π e

⁻

= −

∫ + h

31

3/ 2 n

C

2

2

, the "effective" density of conduction band states 2

N m kT

π

⎡

∗

⎤

= ⎢ ⎥

⎣ h ⎦

3/ 2 p

V 2

2

2 , the "effective" density of valence band states 2

N m kT

π π

∗

⎣ ⎦

⎡ ⎤

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

h h 2 π

⎢ ⎥

⎣ h ⎦

9 At 300 K

Degenerate

2 / 3 0

* , 3

19

,

( 2 . 510 10 cm )( m / m )

N

_{C V}

= ×

⁻ _{n p}

E_c E_F E_F

Degenerate semiconductor Nondegenerate semiconductor 9 If

E

_v

+ 3 kT ≤ E

_F

≤ E

_c

− 3 kT

F c

( ) /

C

E E kT

n = N e

^{C −} _EF semiconductor

(E E ) /kT

n N e

=

−

* Alternative Expressions for n and p

9 For intrinsic semiconductor, E_i=E_F

(E E ) /kT

Alternative Expressions for n and p

kT E E _i

N

^{( − )/}

i c

v i

( ) /

i C

( ) /

i V

E E kT E E kT

n N e n N e

−

−

=

=

V i ^{E E kT}

kT E E i C

v i

i c

e n N

e n N

/ ) (

/ ) (

=

−

=

F i

( ) /

i

( ) /

E E kT E E kT

n = n e

⁻

i F

( ) /

i

E E kT

p = n e

⁻

•

n_i

and the np Product

c v G

( ) / /

2

i C V C V

E E kT E kT

n = N N e

^{− −}

= N N e

⁻

G / 2

i C V

E kT

n = N N e

⁻

np = n

i² 33

Intrinsic carrier concentrations in Ge, Si, and GaAs as a function of

• Charge Neutrality Relationship

9 For the uniformly doped material to be everywhere charge-neutral clearly requires

clearly requires

D A

3

charge

cm = qp − qn + qN

⁺

− qN

⁻

= 0 cm

D A

0

p n − + N

⁺

− N

⁻

=

3 3

cm acceptors/

ionized of

number

donors/cm ionized

of number

=

=

− +

A D

N N

3 3

total number of donors/cm total number of acceptors/cm

D A

N N

=

=

A

p

D A

0

p − + n N − N =

assumes total ionization f d

A

p

35

D A

p

of dopant atoms

9 Assumptions: nondegeneracy and total ionization of dopant atoms Two equations and two unknowns

2 2

n

i

p = n

2 i

D A

0

n n N N n − + − =

2 2

D A i

( ) 0

n − n N − N − n =

1/ 2 2 1/ 2

D A D A 2

2 2

i

N N N N

n = − + ⎡ ⎢ ⎛ ⎜ − ⎞ ⎟ + n ⎤ ⎥

⎝ ⎠

⎢ ⎥

⎣ ⎦

2 1/ 2

⎡ ⎤

− −

⎢ ⎥

⎣ ⎦

9 I t i i i d t (N 0 N 0) Æ

9 Intrinsic semiconductor (N_A=0, N_D=0) Æ n=p=n_i

9 Doped semiconductor where either N_D-N_A≈N_D >> n_i or N_D-N_A≈N_A >> n_i (usual)

D A

,

D i

N

_>>

N N

_>>

n

N

D

n ≅

(nondegenerate, total ionization)

D

i

N

n p ≅

²

/

A D, A i

(nondegenerate, total ionization) N N N n

>> >>

A i

A

N n

n

N p

2

/

≅

≅

37

• Determination of E

_F

9 one-to-one correspondence between E^F and the n & p 9Exact positioning of E_i

9Exact positioning of E_i

n = p p

i c v i

( ) / ( ) /

C V

E E kT E E kT

N e

⁻

= N e

⁻

c v V

i

C

2 2 ln

E E kT N

E N

⎛ ⎞

= + + ⎜ ⎟

⎝

^C

⎠

2 2 ⎝ N ⎠

m

3/ 2

N ⎛

^∗

⎞

c v

3

^p

l m

E E

E kT

⎛

∗

⎞

+ + ⎜ ⎟

V

m

p

N

N m

^∗

⎛ ⎞

= ⎜ ⎜ ⎟ ⎟

c v p

i

ln

2 4

E kT

m

^∗

= + ⎜ ⎜ ⎟ ⎟

⎝ ⎠

9 D d i d t 9 Doped semiconductors

kT E

E i

i

e

F

n

n =

^{( − )/}

F i

ln( /

i

) ln( /

i

) E − E = kT n n = − kT p n

i

tors semiconduc

type -

p for

tors semiconduc

type -

n for

A D

N p

N n

≅

≅

F i

ln(

D

/

i

)

E − E = kT N n

tors semiconduc

type p

A

for N

p ≅

F i D i

i F A i

ln( / ) ln( / ) E E kT N n E − E = kT N n

39

The maximum nondegenerated doping concentrations

3 18

/

10 6

1

N 1 . 6 10

¹⁸

/ cm

³

N

_D

≅ ×