Finding Number of Electrons

Question of Interest

• How many electrons are there in the conduction band? • How many holes are there in the valence band?

• It depends on:

- Temperature of the crystal - Size of the bandgap

- Number of available states in the conduction band and valence band

Finding Number of Electrons

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Number of states in an energy band

electron and it becomes a hole when there is no electron there

Density of States

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E_C

E_V Density of states

D(E)

• The number of states in a band is a function of energy given by the function D(E)

E_C Energy at the bottom of

the conduction band

Energy at the top of the valence band

E_V

• The total number available states in a band is obtained by

integration of the entire band

! " =

8% 2 ℎ( )*

∗(_,

" − "_. /01 " ≥ "_. 8% 2

ℎ( )*∗(, "3 − " /01 " ≤ "3

No need to know

Fermi-Dirac Statistics [1]

Carrier distribution

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

• the carrier distribution follows a Fermi-Dirac function f_e(E) that define the probably of a state being filled by a electron and

Without supply of energy

With surrounding energy source (e.g. room temperature)

kT E E

e _F

e E

f

-+ =

1

1 )

(

f_e(E)

E E_F

1 0 fe(E)

E

1 0

E_F

• E_F is defined as the Fermi energy level where the probability to find a electron in a state is ½

Fermi-Dirac Statistics [2]

For material with a bandgap

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

E_C

E_V

e

-0 1

E_F states

states

f_e(E) • Due to symmetry, E_F is located

somewhere in the middle of the bandgap

• at 0K, all electrons stay below E_F (or E_V) • above 0K, electrons have finite

probability to stay at the conduction band

• The probably to find a “hole” in a state is given by f_h(E) where

f_h(E) =1− f_e(E) = 1

1+e

E_f−E kT

E_C

E_V

0 1

E_F states

states

Fermi-Dirac Statistics [3]

Number of Electrons/Holes

• the total number of electron is given by the number of available states multiplied by the probability that a state is filled

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

E_C

E_V

E_C

E_V e

-0

1 _h+

E_C

E_V Density of states Fermi-Dirac # of e- _{and h}+

D(E)

f_e(E)

f_e(E)

D(E)

n_e

_{( )}

_{( )}

_{( )}

#e− = n_i = D E

_{( )}

1+e

E−E_F kT E_Conduction

∫

dE #h+ = #e− = n_i = D E

_{( )}

1+e E_F−E

kT E_Valence

∫

dE

Equivalent density of states

Equivalent Density of States [1]

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

• We only care about the total density of states, but do not care where they are located

• To simplify the discussion, we assume all states are located at the edge of the band, E_C and E_V

E_C E_V

E_C E_V

N_C

N_V

•The equivalent density of states at the conduction band and valence band are denoted by N_C and N_V respectively

• the “equivalent” density of states at the conduction and valence band edges are

Equivalent Density of States [2]

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering 2 / 3 2 2 2 _÷ ø ö ç è æ = h kT m N e C

p 3/2

2 2 2 ÷ ø ö ç è æ = h kT m N h V p ; -m_e: effective mass of e- _(1.02x10-30_kg)

-m_h: effective mass of h+ _(5.25x10-31_kg)

• once the temperature is know, the number of states at the edge of the conduction and valence band become more or less constant

• For silicon at room temperature, they are

3 19_cm 10

8 .

2 ´

-=

C

N N_V =1.09×1019cm−3

-h: Planck constant (6.63x10-34_Js)

-k: Boltzmann constant (8.62x10-5_eV/K)

-T: absolute temperature

Simplified Carrier Statistics [1]

Simplification with the equivalent density of state

available states multiplied by the probability that a state is filled

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

E_C

E_V

E_C

E_V

0

1 _h+

E_C

E_V

Fermi-Dirac # of e- _{and h}+

N_C

f_e(E)

f_e(E_C) N_C

#e− = n_i = N_C f_e

_{( )}

E_C−E_F kT

#h+ = n_i = N_V f_h

_{( )}

1+e

E_F−E_V kT

E_C E_V

Density of states

D_e(E)

N_V

Simplified Carrier Statistics [2]

Boltzmann Approximation

• the Fermi-Dirac function is not easy to use and the Boltzmann approximation is often applied

f_e(E)= 1 1+e

E−E_F kT

≈ e− E−E_F

kT

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering E −E_F >> kT

for

kT ≈ 0.025eV at room temperature

and

E_C

E_V

e

-0 1

E_F

€

f_e

( )

E−E_F kT

f_e(E) = 1

1+ e E−E_F

kT

f_h

_{( )}

E_F−E kT

similarly

note: _{fh(E)=1-fe(E)}

Simplified Carrier Statistics [3]

Number of Electrons and Holes

• After Boltzmann’s approximation, the number of elections and holes in intrinsic silicon can be found by

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

n_i ≈ N_C exp− EC − Ei kT ⎛

⎝

⎜ ⎞ ⎠

⎟ p_i = n_i ≈ N_V exp− Ei −EV

kT

⎛ ⎝

⎜ ⎞ ⎠ ⎟

;

np = n_i2 ≈ N_CN_V exp− EG

kT

# $

% & ' (

• At room temperature, n_i=1.45x1010_cm-3 _{(or simply 10}10_cm-3₎

-n: number of carriers in the conduction band

-p: number of carriers in the valence band

-n_i/p_i: intrinsic number of carriers in conduction/valence band

-E_F: Fermi-level of a material

-E_i: Fermi-level of intrinsic silicon

Meaning of Fermi-Dirac Statistics [1]

Visualizing the Fermi-Dirac Statistics

• To understand the concept of Fermi-level (E_F), we can borrow the water analogy again

• At thermal equilibrium (no battery voltage), the Fermi-level outside the semiconductor is the same

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Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering

conduction band

electron as water vapor

valence band holes as bubbles

E_C

E_V

E_F E_G=1.1eV

water

water table

• Take the forbidden gap as a soft impermeable material • Take the Fermi energy level (E_F), as the water table that

represent the water pressure Level depends

on battery

semiconductor

external reference (battery)

external reference (battery)

Meaning of Fermi-Dirac Statistics [2]

Under external voltage

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• Along the channel, the “water level” gradually become lower causing the “soft material” floating on it to slope downward • The carriers will move from one end to the other causing

conduction

water

semiconductor

V

-+

• You may consider a battery define the Fermi-level at the two sides of the

semiconductor

• In static system, you may imagine the system is

further divided into many

small sections to be lowered by different amount