Review 4 on I Physical Electronics Important Slide … Engineering by Prof. Sungsik Lee Carrier...

Semiconductor Engineering by Prof. Sungsik Lee

Carrier Statisticsand State Distributions

(Carriers, Fermi-Dirac Statistics in Solids, Fermi Level,Density of States, etc.)

Review 4 on

Physical ElectronicsI

Important

Slide

to watch

without

doing

anything


Highway

Carpark-Highway Analogy

Carpark

Vacancy movement opposite to parked car’s moving direction

one vacancy

one highway carRoad Gap

Average Line

Band gap

Free holesValence electrons

Hole transport

Electron transportConduction Band

Valence Band

Fermi level

Empty lanes: Density of States for Electrons

Empty places: Density of States for Holes

Free (mobile) electrons

I


Top questions of this lecture

• What makes valence electrons excited into the CB ?

• What is the Fermi level ?

• How does the Density of States look like ?

• How determines Free Electrons and Holes numbers ?

• Are they the same each other all the time ?

• What makes it different ?

• And how does it relate to the Fermi level ?

I


Free Carriers: Free Electrons and HolesWhat makes it free ?

Eg

n0 = 0

p0 = 0

EC

EV

n0 = 1.5x1010 /cm3

p0 = 1.5x1010 /cm3

EC

EV

~ 1.1 eV

Eth = kT (at T = 300 K)

~ 26m eV

Eth = kT (at T = 0 K)

= 0 eV

Both cases at the thermal equilibrium �� No external effect (e.g. bias and Light)

Intrinsic Semiconductor �� n0 = p0 �� EF is exactly in the middle of Eg.

Increasing

Temperature

Free electrons notation: n0

Free holes notation: p0

Maximum n0 ~ 5x1022 /cm3

Maximum p0 ~ 5x1022 /cm3

<< Eg ~ 1.15 eV

1/1012

1/1012

EF EF

For the Example semiconductor of Silicon:

Valence electrons Free holes

Free electrons

�� Temperature

Fermi level Thermal

excitationT = 0 T ≠ 0

E

x

I


Intrinsic Semiconductor (n0 = p0) in Thermal Equilibrium

n0 = 1.5x1010 /cm3

p0 = 1.5x1010 /cm3

EC

EV

Eth = kT (at T = 300 K) ~ 26m eV << Eg ~ 1.15 eV

EF

For the Example semiconductor (bulk = 3D)

of Silicon at T = 300 K:

Average Line

E

gc(E)

gV(E)

EC

EV

EF

Density of States for Electrons

Density of States for Holes

Density of States (Vacancies) ?

How to determine the Carriers Number with EF ?

Carpark-Highway Analogy

with Multi-roads (Band)

Thermal

excitation

CB

VB

Vacancies

Height

Vacancies

I


Physical Distribution of Vacancies (Rooms)for Thermally Excited Carriers: Density of States in Energy (3D Semiconductor)

EC

EV

EF

gc(E)

gV(E)

EC

EV

EF



Thermal

excitation

CB

VB

When T ≠ 0

3D

Semiconductor

��

.�

��

∗

�

�.�

� � ��

��

.�

��

∗

�

�.�

��

E



EC

EV

EF

gc(E)

gV(E)

EC

EV

EF



Thermal

excitation

CB

VB

When T ≠ 0

2D

Semiconductor

��

��∗

�E

��

��∗

�



EC

EV

EF

E

gc(E)

gV(E)

EC

EV

EF



Thermal

excitation

CB

VB

When T ≠ 0

1D

Semiconductor

��

��∗ .�

� �� .�

��

��∗ .�

� � � �� .�


Mathematical Derivation of 3D Density of States

dk

ky

kx

kz

4π π π π k2

Supposing you have a symmetrical 3-D structure

which is a sphere centered at the origin (0,0.,0) in the k-space.

L : length in real space [m]

vk : unit amount in k-space [rad3/m3]

Va : total angular amount in real space [m3/rad3]

��

�

Ntotal : total number of states

�� × ��× � ��

� �!�

�� !�

� ��

∗ ��

��

k : spatial frequency in the conduction band

→ !� ��

∗

��

�

� � ��

!� ��

∗

��

� � ��

!�

� Differential elements

2 spins

• Volume for 3D

• Area for 2D

• Length for 1D

k


Mathematical Derivation of 3D Density of States (continuous)

∴ �� .�

��∗ℏ

�.�� − ��

nu : number of states per unit volume [/cm3]

�$ = �� = �

� !� =

& �

��∗ (� − ��)ℏ

��∗ℏ

� �� − �� !� = �� !�

� = ��∗ (� − ��)ℏ

�

k : spatial frequency in the conduction band

→ !� = ��∗ℏ

� �

�� − �� !� = ��∗

ℏ� �

� − �� !�

�� (�): Density of States [cm-3eV-1]

Same result as the Textbook

�� (�): Density of States [cm-3eV-1]


Summary of Key Equationsfor DoS in CB for 3 different dimensions

vk

Va

Ntotal

nu

k [rad/m]

��

dk = f(dE)

Parameters

.�

��∗ℏ

�.�� − ��

8(2( *

2+,∗ (- − -.)ℏ/

+,∗2ℏ/

� 1- − -.� 1-

12 = +,∗2ℏ/

� 1- − -.� 1-

2+,∗ (- − -.)ℏ/

�

2 3*2( * 4(2/12

3*2( *

4(2/12

��∗ ℏ

4(2( /

2+,∗ (- − -.)ℏ/

� +,∗2ℏ/

� 1- − -.� 1-

12 = +,∗2ℏ/

� 1- − -.� 1-

2+,∗ (- − -.)ℏ/

�

2 3/2( / 2(212

3/2( /

2(212[rad3/m3]

[cm-3]

[cm-3eV-1]

[m3/rad3] [m2/rad2]

[rad2/m2]

[cm-2]

[cm-2eV-1] ��∗ .�

ℏ � − �5 .�

42(

+,∗2ℏ/

� 1- − -.� 1-

12 = +,∗2ℏ/

� 1- − -.� 1-

2+,∗ (- − -.)ℏ/

�

2 32( 212

32(

212

[m/rad]

[rad/m]

[cm-1]

[cm-1eV-1]

3D (d=3) 2D (d=2) 1D (d=1)

Derivation of DoS for 2D or 1D �� Midterm Exam ?

= 2 362( 6 78

= 9:;:<=36

= >?1-

= 362( 6


Density of States in 3D Semiconductorand Ambient Temperature �� Unchanged with Temp.

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

When T = 0: No Carrier

�� = .�

��∗ℏ

�.��− ��

�� = .�

��∗ℏ

�.�� − ��

E

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

When T ≠ 0: Free Carriers excited

�� = .�

��∗

�

�.�

��

��

.�

��

∗

�

�.�

��

E

Density of States (Vacancies) is unchanged with changing Temperature.(but the effective mass may be changed)

n0 = p0 ≠ 0 n0 = p0 = 0

I


Carrier Numbers in 3D Semiconductorand Ambient Temperature �� Changed with Temp.

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

When T = 300 K: More Carriers

�� = .�

��∗

�

�.�

��

��

.�

��

∗

�

�.�

��

E

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

When T = 100 K: Free Carriers excited

��

.�

��

∗

�

�.�

��

��

.�

��

∗

�

�.�

��

E

�� Who determines the carrier numbers as a function of Temperature ?

n0 = p0 ≠ 0 n0 = p0 ≠ 0

n0 = p0 ≠ 0 at T = 100 K n0 = p0 ≠ 0 at T = 300 K<

I


Fermi Function

@ � ��

� A BC�� D�E

Enrico Fermi (Italian)

1901-1954

Probability of Occupancy


Derivation of Fermi-Dirac Statistics (Appendix)

� �

� �≡ @ �

��


� �

� �

Possible # of electrons

distributed (occupied)

at an Energy level (E)

Degeneracy (total places = rooms)

at an Energy level (E)


Fermi-Dirac Integraland Carrier density (concentration) in volume

��

.�

��

∗

�

�.�

��

��

.�

��

∗

�

�.�

��

@� � ��


@� � � � � @� �

��@��

��@��

� � G ��@��

H

��dE

� � G ��@��

��IH

dE

Area underneath

Area underneath

Outline

Outline

Fermi function

Fermi function

DoS

DoS

Carrier density [cm-3] = (DoS)(Fermi Function)

Energy of DoS begins

Energy of DoS finishs

dE

I


Physical Indicator of Probabilityof Existence of Thermally Excited Carriers: Fermi Function with Fermi Level

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

��

.�

��

∗

�

�.�

��

��

.�

��

∗

�

�.�

��

En0 E

EF

EC

EV

@� � � @D��

@��

@� � ��


@� � � � � @� �

1

0.5

��@��

��@��

� � G ��@��

H

��dE

p0 = n0

� � G ��@��

��IH

dE

Area underneath

Area underneath

Outline

Outline

Fermi function

Fermi function

DoS

DoS

I


What happens to Fermi Function and Fermi Levelin Intrinsic Semiconductor with increasing Temperature

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

��

.�

��

∗

�

�.�

��

��

.�

��

∗

�

�.�

��

En0 E

EF

EC

EV

@� � � @D��

@��

@� � ��


1

0.5

��@��

��@��

� � G ��@��

H

��dE

p0 = n0

� � G ��@��

��IH

dE

Area underneath

Area underneath

Outline

Outline

increasing Temperature

@� � � � � @� �

Textbook: Figure 4.1

Fermi function

Fermi function

DoS

DoS

I


Fermi-Dirac Statisticsand Carrier Density Formula with Boltzmann’s approximation

� � G ��@��

��IH

dE

� � G ��@��

H

��dE

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

Analytical form

Analytical form

Boltzmann’s Approximation

� ≈ G ��@��D�

H

��dE

� � G ��@��D�

��IH

dE

@� � ��


@� � � � � @� �

≈ BC� �� D�E

� @� �D

@� �D � � � @� �D

K@� ≪ BC�� D�E

⇒�� D�E

≫ �


Boltzmann’s approximationand Effective Density of States

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

Analytical form

Analytical form

Boltzmann’s Approximation ��

� ≈ G ��@��D�

H

��dE

� � G ��@��D�

��IH

dE

K@� ≪ BC�� D�E

⇒�� D�E

≫ � ⇒ O��B�P�, �� D ≫ �E

��

∗�E

R

�.�

��

∗�E

R

�.�

Effective DoS for CB

Effective DoS for VB

@�S�D ≪ ��

@�S�D ≫ ��


Physical Meaning of Fermi Level (EF)and Charge Neutrality with Carrier Density

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

Electron density

Hole density

EF

gc(E)

gV(E)

EC

EV

EF

CB

VB

En0 E

EF

EC

EV

@� � � @D��

@��1

0.5

p0 = n0

To be p0 = n0 �� the Intrinsic SC.

|�D � ��| � |�� D|= (Ec-Ev) / 2 = Eg / 2

�D � ��

�� DEg

Charge Neutrality

Fermi level is exactly centered in the BG

U� � V � � � �

I


N-type Semiconductor (example)and Fermi Level with Charge Neutrality

EF

gc(E)

gV(E)

EC

EV

CB

VB

Eg

E

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

@��

@��

p0 = n0 : Intrinsic Semiconductor

EF

gc(E)

gV(E)

EC

EV

CB

VB

Eg

E

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

@��

@��

p0 < n0 : n-type Semiconductor

p0 = n0 = ni : Intrinsic Concentration

n0 p0 = ni2: Mass action Law

p0 < n0

U� � V � � � �

U� � V � � � W

Is this Correct ?

I


Mass Action Law (Appendix)

n0 p0 = ni2: Mass action Law

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

�K ≈ ��BC� ��E

� �K� ��BC� ��E

@�S|�D � ��| � |�� D|= (Ec-Ev) / 2 = Eg / 2

�� BC��D � ��E

BC�� D�E

� ��BC�� E

� ��BC� ��E

� ��BC� ��E

BC� ��E

� ��BC� ��E

��BC� ��E

� �K�K � �K�K � �K

�� n0 p0 = ni2


Extrinsic Semiconductor (doped semiconductor)and Fermi Level with Charge Neutrality

EF

gc(E)

gV(E)

EC

EV

CB

VB

Eg

E

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

@�� @��

p0 > n0 : p-type Semiconductor

EF

gc(E)

gV(E)

EC

EV

CB

VB

Eg

E

� ≈ ��BC��D � ��E

� ≈ ��BC�� D�E

@��

@��

p0 < n0 : n-type Semiconductor

ND+

NA-

U� � V � � �AND+− N

A−

=

It is still zero !Ionized donor density

Ionized accepter density

Donor levelAcceptor level

I

EF is the reference energy

level at which the charge

neutrality is satisfied.


Intrinsic vs. Extrinsic Semiconductors

p0≠ n0 : Extrinsic Semiconductorp0 = n0 : Intrinsic Semiconductor

Volume Charge Density Volume Charge Density Volume Charge Density Volume Charge Density ---- NeutralityNeutralityNeutralityNeutrality

U� � V � � � A ND+− N

A−

=

where � = �5k A �O��B��S, NA

−= �O��B��S� = ��k A ��S, N

D+=��S,�5k = ��k

U� = V � − �AND+ − N

A−

=

where � = �5k A �O��B��SN

A−= �O��B��S =

� = ��k A ��SN

D+=��S=

⇒ U�= V � − � =

U� = V � − �AND+ − N

A−

=

where � = �5k A �O��B��SN

A−= �O��B��S≠(� − �l�B)� = ��k A ��S

ND+=��S≠(� − �l�B)

⇒ U�= V �5k A �O��B��S − ��k − ��SA��S − �O��B��S =

I


Donor level and Fermi functionin n-type semiconductor (example)

I

U� = V � − �AND+ =

EF

gc(E)

gV(E)

EC

EV

CB

VB

Eg

E

� J ��BC� �D − ��E

� J ��BC� �� − �D�E

@�(�)@�(�)

ND+

Donor level

E

@�(�)ND+

ND

(neutral)

profile: nD

10.5

occupied

empty (ionized = unoccupied)

= m ��(� − @� � )!��5��

ND+ = −� A �

ED

Donor

Level: ED

�� = 9nexp(− - − -q /2r/

�

��

EED

9n

ND

ND+ empty: Positive as it lost(donated) electron

Occupied: Neutral as it is

Nature of electron donor


Top questions of this lecture

• What makes valence electrons excited into the CB ?

• What is the Fermi level ?

• How does the Density of States look like ?

• What determines Free carriers and Holes numbers ?

• Are they the same each other all the time ?

• What makes it different ?

• And how does it relate to the Fermi level ?

• What happens to the charge neutrality ?

I


Next Lecture

Review on

Physical

Electronics

(물리전자물리전자물리전자물리전자복습복습복습복습)

Review 4 on I Physical Electronics Important Slide … Engineering by Prof. Sungsik Lee Carrier...

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Transcript of Review 4 on I Physical Electronics Important Slide … Engineering by Prof. Sungsik Lee Carrier...