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Physics PY4118
Physics of Semiconductor Devices
5. Band Filling
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
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.
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.2
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.3
PY4118 Physics of
Semiconductor Devices
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
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.4
𝐸 = 𝐸𝐹
𝐸 = 0
PY4118 Physics of
Semiconductor Devices
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
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.5
1
2𝑚𝑣𝐹
2 = 𝐸𝐹
𝐸𝐹 = 7𝑒𝑉 𝑣𝐹 = 1.6 × 106𝑚
𝑠
𝑘𝐵𝑇 ≅ 0.025𝑒𝑉 𝑘𝐵𝑇 ≅ 0.11𝑒𝑉
PY4118 Physics of
Semiconductor Devices
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”
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.6
PY4118 Physics of
Semiconductor Devices
The Occupation Probability (2)
◼ Maxwell - Boltzmann ❑ Identical – no Pauli
❑ distinguishable
◼ Bose-Einstein❑ Identical – no Pauli
❑ indistinguishable
◼ Fermi-Dirac❑ Identical - Pauli
❑ indistinguishable
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.7
𝑃 𝐸 = 𝑒−𝐸−𝐸𝐹𝑘𝐵𝑇
𝑃 𝐸 =1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 − 1
𝑃 𝐸 = 𝑓 𝐸 =1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1
PY4118 Physics of
Semiconductor Devices
The Occupation Probability (3)
If: or:
Remember, at RT:
Fermi - Dirac Maxwell - Boltzmann
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.8
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 ≫ 1 𝐸 − 𝐸𝐹 > 𝑘𝐵𝑇
𝑘𝐵𝑇 ≅1
40𝑒𝑉
𝑓 𝐸 =1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1
𝑃 𝐸 = 𝑒−𝐸−𝐸𝐹𝑘𝐵𝑇
PY4118 Physics of
Semiconductor Devices
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)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.9
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
Semiconductor Devices
The Fermi Function (2)
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.10
PY4118 Physics of
Semiconductor Devices
The Fermi Function (3)
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.11
PY4118 Physics of
Semiconductor Devices
The Fermi Function (4)
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.12
PY4118 Physics of
Semiconductor Devices
The Fermi Function (5)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.13
𝑓 𝐸 =1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1
𝑓𝐸
𝐸 − 𝐸𝐹(𝑒𝑉)
PY4118 Physics of
Semiconductor Devices
◼ 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.
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.14
𝐸𝐶,𝑉 𝐸𝐶,𝑉
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.15
PY4118 Physics of
Semiconductor Devices
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
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.16
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.17
𝑓 𝐸
1 − 𝑓 𝐸
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.18
𝑓 𝐸 − 𝐸𝐹 =1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1
=1
𝑒Δ𝐸𝑘𝐵𝑇 + 1
1 − 𝑓 𝐸 − 𝐸𝐹 = 1 −1
𝑒Δ𝐸𝑘𝐵𝑇 + 1
=𝑒Δ𝐸𝑘𝐵𝑇
𝑒Δ𝐸𝑘𝐵𝑇 + 1
=1
1 + 𝑒−Δ𝐸𝑘𝐵𝑇
= 𝑓 𝐸𝐹 − 𝐸
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.19
5 1 × 10−87 3 × 10−32
IR
Red
Blue
𝑓 𝐸 =𝑁𝑥𝑁0
=1
𝑒𝐸−𝐸𝐹𝑘𝐵𝑇 + 1
PY4118 Physics of
Semiconductor Devices
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?
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.20
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.21
= ×
𝑛 = ÷
PY4118 Physics of
Semiconductor Devices
Conduction Electrons?How many are there?
Number of
Atoms
In Sample=
Sample
Volume
Material
Density
Molar Mass
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.22
×
/𝑁𝐴𝑁𝐴 = Avogadro's Number
= 6.022 × 1023/𝑚𝑜𝑙
PY4118 Physics of
Semiconductor Devices
Number Density
Material
Density
Molar Mass
Number of
Valence electrons
In Sample
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.23
×𝑛 =
× 𝑁𝐴
PY4118 Physics of
Semiconductor Devices
Number Density
8.96 g/cm3
63.54 g
1
Copper
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.24
× × 𝑁𝐴𝑛 =
= 9 × 1022𝑐𝑚−3 = 9 × 1028𝑚−3
PY4118 Physics of
Semiconductor Devices
Number Density
2.65 g/cm3
28.08 g
0
Silicon
This calculation was for metals.For semiconductors we need Fermi statistics
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.25
× × 𝑁𝐴𝑛 =
= 0 𝑚−3
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.26
PY4118 Physics of
Semiconductor Devices
Infinite barrier 3D box (1)
Divide by:
3x 1D solutions
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.27
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
Semiconductor Devices
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:
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.28
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
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.29
𝑘𝑥
𝑘𝑦
𝑘𝑧
𝑉𝑘 =𝜋
𝐴
𝜋
𝐵
𝜋
𝐶=𝜋3
𝑉
𝑉𝐹 =4
3𝜋𝑘𝐹
3
𝑘𝑖 > 0
PY4118 Physics of
Semiconductor Devices
𝑁 = 2 ×1
2
3𝑉𝐹𝑉𝑘
= 2 ×1
8
43𝜋𝑘𝐹
3
𝜋3
𝑉
=1
3
𝑘𝐹3𝑉
𝜋2
Density of States (2)
So, the number of filled states is:spin
Thus:
And:
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.30
3D
𝑘𝐹 =3𝜋2𝑁
𝑉
13 𝑛 =
𝑁
𝑉
𝐸𝐹 =ℏ2𝑘𝐹
2
2𝑚=
ℏ2
2𝑚
3𝜋2𝑁
𝑉
23
=ℏ2 3𝜋2𝑛
23
2𝑚
PY4118 Physics of
Semiconductor Devices
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.
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.31
𝐸𝐹 =ℏ2 3𝜋2𝑛
23
2𝑚
𝑛 = 8.5 × 1028𝑚−3
𝐸𝐹 = 7𝑒𝑉
PY4118 Physics of
Semiconductor Devices
Silicon:
Fermi Energy Semiconductor
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.32
𝐸𝐹 =ℏ2 3𝜋2𝑛
23
2𝑚
𝑛 = 0 → 5 × 1021𝑚−3
𝐸𝐹 = 0 → 1.1𝑒𝑉
PY4118 Physics of
Semiconductor Devices
Density of States (3)
Inverting we get the number density:
And can then calculate the density of states:
For a semiconductor:
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.33
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
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.34
PY4118 Physics of
Semiconductor Devices
Population of Bands
To actually predict the distribution of carriers in the band we need both Fermi statistics and the density of states:
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.35
PY4118 Physics of
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.36
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
Semiconductor Devices
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
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.37
PY4118 Physics of
Semiconductor Devices
Density of Semiconductor States
Intrinsic Semiconductor
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland
ROINN NA FISICE
Department of Physics 5.38
𝜌 𝐸 𝑓 𝐸 𝑛 𝐸
𝑝 𝐸