Physics PY4118 Physics of Semiconductor Devices › fpetersweb › FrankWeb › courses...PY4118 Physics of Semiconductor Devices At 𝑻= , energy levels below are filled with electrons,

Physics PY4118

Physics of Semiconductor Devices

5. Band Filling

Coláiste na hOllscoile Corcaigh, Éire

University College Cork, Ireland

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Department of Physics 5.1

PY4118 Physics of

Semiconductor Devices

Semiconductor Bands

The lowest filled bands are filled with electrons that are bound near the atomic nucleus.

The electrons in the highest bands are used in covalent bonds. They are called valence electrons. Their bands are valence bands.

The next band, unpopulated at 0𝐾, is called the conduction band.



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PY4118 Physics of


Fermi Level and Energy

◼ Pauli exclusion leads to electrons populating higher states rather than all sitting in the ground state

◼ Fermi statistics govern how electrons move into even higher states due to temperature



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PY4118 Physics of


Fermi Level in Metals

Metal

Energy level at the bottom of the partially filled band

Highest occupied energy level at 𝑇 = 0𝐾

partially

filled

band

Pauli’s Exclusion Principle at Work.Coláiste na hOllscoile Corcaigh, Éire


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𝐸 = 𝐸𝐹

𝐸 = 0

PY4118 Physics of


Fermi Level in Metals

Fermi Velocity:

Copper:

RT: 1000°C:

The amount of thermal energy is much less that the Fermi Energy!Even at high Temperature.

Metals conduct very well even at very low TColáiste na hOllscoile Corcaigh, Éire


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1

2𝑚𝑣𝐹

2 = 𝐸𝐹

𝐸𝐹 = 7𝑒𝑉 𝑣𝐹 = 1.6 × 106𝑚

𝑠

𝑘𝐵𝑇 ≅ 0.025𝑒𝑉 𝑘𝐵𝑇 ≅ 0.11𝑒𝑉

PY4118 Physics of


The Occupation Probability (1)

◼ Or: “What is the probability that an available level will be populated?”

◼ A group of probability distribution functions have been derived using Statistical Mechanics.

◼ Other names are “Energy Distribution Functions”



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PY4118 Physics of



◼ Maxwell - Boltzmann ❑ Identical – no Pauli

❑ distinguishable

◼ Bose-Einstein❑ Identical – no Pauli

❑ indistinguishable

◼ Fermi-Dirac❑ Identical - Pauli

❑ indistinguishable



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𝑃 𝐸 = 𝑒−𝐸−𝐸𝐹𝑘𝐵𝑇

𝑃 𝐸 =1

𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 − 1

𝑃 𝐸 = 𝑓 𝐸 =1

𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1

PY4118 Physics of



If: or:

Remember, at RT:

Fermi - Dirac Maxwell - Boltzmann



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𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 ≫ 1 𝐸 − 𝐸𝐹 > 𝑘𝐵𝑇

𝑘𝐵𝑇 ≅1

40𝑒𝑉

𝑓 𝐸 =1


𝑃 𝐸 = 𝑒−𝐸−𝐸𝐹𝑘𝐵𝑇

PY4118 Physics of


𝒇(𝑬)

𝟏

𝟎𝑬𝑬𝑭

½

If 𝑇 = 0𝐾, 𝑓(𝐸) is a simple step function with an edge at 𝐸𝐹, i.e., all the states with energies below the fermi energy 𝐸𝐹, are completely occupied, and all the states with energies above 𝐸𝐹 are completely vacant.

The Fermi Function (1)



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𝑓 𝐸 < 𝐸𝐹 , 0 =1

𝑒−∞ + 1=1

1⇒ 1

𝑓 𝐸 > 𝐸𝐹 , 0 =1

𝑒∞ + 1=1

∞⇒ 0

𝑓 𝐸 = 𝐸𝐹 , 0 =1

𝑒0 + 1=1

2

PY4118 Physics of



0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

Energy (eV)

P(E

)

Maxwell - Boltzmann

Fermi - Dirac



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PY4118 Physics of



0

0.2

0.4

0.6

0.8

1

1.2

6.5 6.7 6.9 7.1 7.3 7.5

Energy (eV)

P(E

)

Maxwell - Boltzmann

Fermi - Dirac



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PY4118 Physics of



0

0.2

0.4

0.6

0.8

1

1.2

6.5 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5

Energy (eV)

P(E

)

300K

500K

1000K



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PY4118 Physics of





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𝑓 𝐸 =1


𝑓𝐸

𝐸 − 𝐸𝐹(𝑒𝑉)

PY4118 Physics of


◼ At 𝑻 = 𝟎, energy levels below 𝑬𝑭 are filled with electrons, while all levels above 𝑬𝑭 are empty.

◼ Electrons are free to move into “empty” states of conduction band with only a small electric field 𝐸, leading to high electrical conductivity!

◼ At 𝑻 > 𝟎, electrons have a probability to be thermally “excited” from below the Fermi energy to above it.

Band Diagram: Metal

𝐸𝐹 𝐸𝐹

Fermi “filling” function

Energy band to be

“filled”

Moderate T𝑇 = 0 𝐾

“Fill” the energy band

with electrons.



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𝐸𝐶,𝑉 𝐸𝐶,𝑉

PY4118 Physics of


Band Diagram: Insulator

◼ At 𝑻 = 𝟎, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.❑ Fermi energy 𝐸𝐹 is at midpoint of large energy gap (2 − 10 𝑒𝑉) between conduction

and valence bands.

◼ At 𝑻 > 𝟎, electrons are NOT thermally “excited” from valence to conduction band, leading to zero conductivity.

𝐸𝐹

𝐸𝐶

𝐸𝑉

Conduction band(Empty)

Valence band(Filled)

𝐸𝑔𝑎𝑝

𝑇 > 0



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PY4118 Physics of


Band Diagram: Semiconductor

◼ At 𝑻 = 𝟎, valence band is filled with electrons and conduction band is empty, leading to zero conductivity.

◼ At 𝑻 > 𝟎, electrons thermally “excited” from valence to conduction band, leading to partially empty valence and partially filled conduction bands.

𝐸𝐹𝐸𝐶

𝐸𝑉

Conduction band(Partially Filled)

Valence band(Partially Empty)

𝑇 > 0

Thus: SemiconductorColáiste na hOllscoile Corcaigh, Éire


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PY4118 Physics of


Re-examine the Semiconductor

𝐸𝐹𝐸𝐶

𝐸𝑉

Conduction band(Partially Filled)

Valence band(Partially Empty)

𝑇 > 0

It is the absence of an electron that makes a hole



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𝑓 𝐸

1 − 𝑓 𝐸

PY4118 Physics of


Symmetry of f(E)

The function is symmetric around the Fermi energy.That is: the distribution of electrons above 𝐸𝐹 equals the distribution of holes below 𝐸𝐹



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𝑓 𝐸 − 𝐸𝐹 =1


=1

𝑒Δ𝐸𝑘𝐵𝑇 + 1

1 − 𝑓 𝐸 − 𝐸𝐹 = 1 −1


=𝑒Δ𝐸𝑘𝐵𝑇


=1

1 + 𝑒−Δ𝐸𝑘𝐵𝑇

= 𝑓 𝐸𝐹 − 𝐸

PY4118 Physics of


Examples

𝐸–𝐸𝐹(𝑒𝑉)

0.1

𝑓(𝐸) 𝑇 = 290𝐶

2 × 10−2

𝑓(𝐸) 𝑇 = 800𝐶

2 × 10−1

0.5 2 × 10−9 7 × 10−4

1 4 × 10−18 5 × 10−7

1.5 9 × 10−27 4 × 10−10



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5 1 × 10−87 3 × 10−32

IR

Red

Blue

𝑓 𝐸 =𝑁𝑥𝑁0

=1


PY4118 Physics of


Focus

◼ The Pauli exclusion principle leads to high energy electron states being filled even at low temperature.

◼ Fermi Dirac statistics provide the probability that available electron states will be populated.

◼ But how many electrons, and how many electron states are there?



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PY4118 Physics of


Conduction Electrons in Metals?How many are there?

Number of

Conduction Electrons

In Sample

Number of

Atoms

In Sample

Number of

Valence electrons

In Sample

Sample

Volume

Number of

Conduction Electrons

In Sample



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= ×

𝑛 = ÷

PY4118 Physics of


Conduction Electrons?How many are there?

Number of

Atoms

In Sample=

Sample

Volume

Material

Density

Molar Mass



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×

/𝑁𝐴𝑁𝐴 = Avogadro's Number

= 6.022 × 1023/𝑚𝑜𝑙

PY4118 Physics of


Number Density

Material

Density

Molar Mass

Number of

Valence electrons

In Sample



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×𝑛 =

× 𝑁𝐴

PY4118 Physics of


Number Density

8.96 g/cm3

63.54 g

1

Copper



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× × 𝑁𝐴𝑛 =

= 9 × 1022𝑐𝑚−3 = 9 × 1028𝑚−3

PY4118 Physics of


Number Density

2.65 g/cm3

28.08 g

0

Silicon

This calculation was for metals.For semiconductors we need Fermi statistics



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× × 𝑁𝐴𝑛 =

= 0 𝑚−3

PY4118 Physics of


Focus

◼ We know the probability that a state is filled.

◼ We know the number of electrons available.

◼ How many electron states are there?

Density of States



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PY4118 Physics of


Infinite barrier 3D box (1)

Divide by:

3x 1D solutions



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𝑑2𝜓

𝑑𝑥2+𝑑2𝜓

𝑑𝑦2+𝑑2𝜓

𝑑𝑧2+2𝑚𝐸

ℏ2𝜓 = 0 𝑘2 =

2𝑚𝐸

ℏ2

𝜓 𝑥, 𝑦, 𝑧 = 𝜓𝑥 𝑥 𝜓𝑦 𝑦 𝜓𝑧 𝑧

1

𝜓𝑥

𝑑2𝜓𝑥𝑑𝑥2

+1

𝜓𝑦

𝑑2𝜓𝑦

𝑑𝑦2+

1

𝜓𝑧

𝑑2𝜓𝑧𝑑𝑧2

+ 𝑘2 = 0

𝑘2 = 𝑘𝑥2 + 𝑘𝑦

2 + 𝑘𝑧2

1

𝜓𝑖

𝑑2𝜓𝑖

𝑑𝑥2+ 𝑘𝑖

2 = 0, 𝑖 = 𝑥, 𝑦, 𝑧

PY4118 Physics of


Infinite barrier 3D box (2)

With a box of dimensions: 𝐴, 𝐵, 𝐶 with a corner at (0,0,0)

We are interested in the number of states with an energy less than the Fermi Energy:



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𝜓 𝑥, 𝑦, 𝑧 = 𝐷 sin 𝑘𝑥𝑥 sin 𝑘𝑦𝑦 sin 𝑘𝑧𝑧

𝑘𝑥 =𝑛𝑥𝜋

𝐴, 𝑘𝑦 =

𝑛𝑦𝜋

𝐵, 𝑘𝑧 =

𝑛𝑧𝜋

𝐶

𝐸 =ℏ2𝑘2

2𝑚=ℏ2𝜋2

2𝑚

𝑛𝑥𝐴

2

+𝑛𝑦

𝐵

2

+𝑛𝑧𝐶

2

𝐸 =ℏ2𝑘𝐹

2

2𝑚→ 𝑘𝐹

2 =2𝑚

ℏ2𝐸𝐹

PY4118 Physics of


Density of States (1)

The volume (in k-space) of one state is:

The volume (in k-space) of the Fermi sphere is:

BUT, we are only interested in the positive quadrant:

AND, there are 2 spin states



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𝑘𝑥

𝑘𝑦

𝑘𝑧

𝑉𝑘 =𝜋

𝐴

𝜋

𝐵

𝜋

𝐶=𝜋3

𝑉

𝑉𝐹 =4

3𝜋𝑘𝐹

3

𝑘𝑖 > 0

PY4118 Physics of


𝑁 = 2 ×1

2

3𝑉𝐹𝑉𝑘

= 2 ×1

8

43𝜋𝑘𝐹

3

𝜋3

𝑉

=1

3

𝑘𝐹3𝑉

𝜋2


So, the number of filled states is:spin

Thus:

And:



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3D

𝑘𝐹 =3𝜋2𝑁

𝑉

13 𝑛 =

𝑁

𝑉

𝐸𝐹 =ℏ2𝑘𝐹

2

2𝑚=

ℏ2

2𝑚

3𝜋2𝑁

𝑉

23

=ℏ2 3𝜋2𝑛

23

2𝑚

PY4118 Physics of


Copper:

Fermi Energy Revisited – Metal

…as we had assumed

Note, that this calculation was only based on theproperties of the material. Our previous assumptionwas not used.



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𝐸𝐹 =ℏ2 3𝜋2𝑛

23

2𝑚

𝑛 = 8.5 × 1028𝑚−3

𝐸𝐹 = 7𝑒𝑉

PY4118 Physics of


Silicon:

Fermi Energy Semiconductor



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𝐸𝐹 =ℏ2 3𝜋2𝑛

23

2𝑚

𝑛 = 0 → 5 × 1021𝑚−3

𝐸𝐹 = 0 → 1.1𝑒𝑉

PY4118 Physics of



Inverting we get the number density:

And can then calculate the density of states:

For a semiconductor:



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𝑛 =1

3𝜋22𝑚𝐸

ℏ2

32

𝜌 𝐸 =𝑑𝑛

𝑑𝐸

𝜌 𝐸 =𝑑𝑛

𝑑𝐸=3

2

1

3𝜋22𝑚𝐸

ℏ2

12 2𝑚

ℏ2=

1

2𝜋22𝑚

ℏ2

32

𝐸

𝜌 𝐸 =1

2𝜋22𝑚∗

ℏ2

32

𝐸 − 𝐸𝐶

PY4118 Physics of


Aside

We now have the tools we need to talk about semiconductors:

◼ Bands: limit the allowable 𝑘, 𝐸 values

◼ Fermi statistics: provide the proportion of filled states

◼ Density of states provides the number of available states



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PY4118 Physics of


Population of Bands

To actually predict the distribution of carriers in the band we need both Fermi statistics and the density of states:



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PY4118 Physics of


Density of Occupied States

Copper @ 8𝑒𝑉:

Above the Fermi Level the number of occupied states

Decreases exponentially.

Fermi-Dirac

Statistics

The number of carriers within



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𝜌𝑜 𝐸 Δ𝐸 = 𝜌 𝐸 𝑓 𝐸 Δ𝐸 =𝑑𝑛 𝐸

𝑑𝐸𝑓 𝐸 Δ𝐸

Δ𝐸

𝑓 𝐸 = 5 × 10−18

𝜌 𝐸 = 1.9 × 1028

𝜌𝑜 𝐸 = 9.7 × 1010

PY4118 Physics of


Density of Occupied States

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0 2 4 6 8 10

Energy (eV)

De

ns

ity

of

Oc

cu

pie

d S

tate

s (

x1

028)

eV

-1m

-3

0K

1000K



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PY4118 Physics of


Density of Semiconductor States

Intrinsic Semiconductor



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𝜌 𝐸 𝑓 𝐸 𝑛 𝐸

𝑝 𝐸

Physics PY4118 Physics of Semiconductor Devices › fpetersweb › FrankWeb › courses...PY4118 Physics of Semiconductor Devices At 𝑻= , energy levels below are filled with electrons,

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