代写代考 Semiconductor Fundamentals

Semiconductor Fundamentals – (II)
2.3 Energy Bands
2.4 Doping of Semiconductors
Gary , PhD

https://www.xjtlu.edu.cn/en/departments/academic-departments/electrical-and-electronic-engineering/staff/chun-zhao

HW-1: solution
⚫ Si atomic density:
#Atoms = 8(1/8)+6(1/2)+4 = 8
Volume a03 (5.4310−8 cm)3
• Number of atoms in a unit cell: • 4 atoms completely inside cell
• Each of the 8 atoms on corners are shared among cells→count as 1 atom inside cell
• Each of the 6 atoms on the faces are shared among 2 cells→count as 3 atoms inside cell
Total number inside the cell = 4 + 1 + 3 = 8
• Cell volume:
(0.543 nm)3 = 1.6 x 10-22 cm3
=51022cm−3

HW-2: solution
Why the Miller indices of this plane is (010)?
h, k and l are reduced/enlarged to 3 integers having the same ratio. (010)
x-intercept of plane y-intercept of plane z-intercept of plane
h: inverse x-intercept of plane k: inverse y-intercept of plane l: inverse z-intercept of plane

Last lecture:

Ev distance
Simplified version of energy band model, indicating
⚫ bottom edge of the conduction band (Ec)
⚫ top edge of the valence band (Ev)
➢ Ec and Ev are separated by the band gap energy Eg
electron energy

2.3 Energy Bands
⚫ Band theory
➢ What’s a Semiconductor ➢ Fermi Level
➢ Band model of e & h
➢ Bond model of e & h
⚫ Generation and recombination ⚫ Intrinsic semiconductor

What is a Semiconductor?
⚫ Lowresistivity=>“conductor”e.g.Al,Cu
⚫ High resistivity => “insulator” e.g. SiO2
⚫ Intermediateresistivity=> “semiconductor”
➢ conductivity lies between that of conductors and insulators

Band Theory of Solids
⚫ A useful way to visualize the difference between conductors, insulators and semiconductors is to plot the available energies for electrons in the materials. In conductors the valence band overlaps the conduction band, and in semiconductors or insulator there is a small or big gap between the valence and conduction bands.
⚫ An important parameter in the band theory is the Fermi level.

Energy Bands of Silicon & Germanium
⚫ At finite temperatures, the number of electrons which reach the conduction band and contribute to current can be modeled by the Fermi function.
1) Fermi level? 2) “holes”?

Fermi function and Fermi level Probability that a state at energy level, E, is occupied by
one electron is,
Students → electrons
Seat row→energy level, E. Seat → state
Example: Students in a theatre class room.
Every row has different energy level, E. For example, for
row 7, its energy level is E7.
The probability for one student to occupy a seat on row 7
can be calculated by f(E7).
EF is a energy level at which f(E) is 50%.
1+exp(E−EF ) kT

Fermi function and Fermi level Probability that a state at energy level, E, is occupied
by one electron is, f(E)= 1
1+exp(E−EF ) kT
f(E): Fermi-Dirac function
An increase in E will reduce f(E)
EF — Fermi-level
When E = EF, f(E=EF) = 0.5.
textbook P.66
The first order approximation
f(E)=0 for E>EF f(E)=1 for E 0
The energy absorbed from a covalent bond
photon frees an electron from
In Si, the minimum energy required is 1.1eV, which corresponds
to ~1 mm wavelength (infrared region).
1 eV = energy gained by an electron falling through 1 V
potential = qeV = 1.6 x 10-19 CV = 1.6 x 10-19 J.

Light interactions with Semiconductors Absorption of light in a semiconductor:
• For energies of light greater than the bandgap ( absorbed
• For energies less than the bandgap (hν Eg), light is

Recombination
⚫ When a conduction electron and hole meet, each one is eliminated, a process called “recombination”. The energy lost by the conduction electron (when it “falls” back into the covalent bond) can be released in two ways:
to the semiconductor lattice (vibrations)
“thermal recombination”→semiconductor is heated
to photon emission
“optical recombination”→light is emitted
▪ It is the basis for light-emitting diodes and laser diodes.

2.3 Energy Bands
⚫ Band theory
➢ What’s a Semiconductor ➢ Fermi Level
➢ Band model of e & h
➢ Bond model of e & h
⚫ Generation and recombination ⚫ Intrinsic semiconductor

Intrinsic semiconductors
Intrinsic: pure semiconductor
A hole is created simultaneously with a free
n(free electron density) = p(hole density) EF is in the middle of the bandgap
Ec: Bottom edge of conduction band
Free electron
Ev: Top edge of valence band

Carrier Concentrations in Intrinsic Si
⚫ The “band-gap energy” Eg is the amount of energy needed to remove an electron from a covalent bond. Eg=1.12eV
⚫ The concentration of conduction electrons in intrinsic silicon, ni, depends exponentially on Eg and the absolute temperature (T):
n =5.21015T3/2 exp −Eg electrons/cm3
Boltzmann constant
n 11010 electrons/cm3 at300K i
n 11015 electrons/cm3 at600K i
8.62E-5 eV/K

n =5.21015T3/2 exp −Eg electrons/cm3
conduction
ni  1×1010 cm-3 at room temperature

Intrinsic Semiconductor
Silicon has four valence electrons
⚫ It covalently bonds with 4 adjacent atoms in the crystal lattice
⚫ Increasing Temperature Causes Creation of Free Carriers. 1010cm-3 free carriers at 23oC (out of 2x1023cm-3): Intrinsic Conductivity.
⚫ Si atomic density: 5 ×1022 cm-3

Semiconductor Current
⚫ Both electrons and holes contribute to current flow in an intrinsic semiconductor.

2.4 The doping of semiconductors
➢ Doping elements ➢ Doping: N type
➢ Doping: P type
➢ Counter doping

The Doping
⚫ The addition of a small percentage of foreign atoms in the regular crystal lattice of silicon or germanium produces dramatic changes in their electrical properties, producing n-type and p- type semiconductors.
⚫ Definition of Terms:
n = number of electrons/cm3
p = number of holes/cm3
ni = intrinsic carrier concentration In a pure semiconductor,
n = p = ni

Valence Electrons
⚫ The electrons in the outermost shell of an atom are called valence electrons; they dictate the nature of the chemical reactions of the atom and largely determine the electrical nature of solid matter. The electrical properties of matter are pictured in the band theory of solids in terms of how much energy it takes to free a valence electron.

The Doping of Semiconductors Pentavalent impurities (donor impurities = donors)
Impurity atom with 5 valence electrons produce semiconductors by contributing extra electrons.
Trivalent impurities (acceptor impurities = acceptors)
Impurity atoms with 3 valence electrons produce semiconductors by producing a “hole” or electron deficiency.

2.4 The doping of semiconductors
➢ Doping elements ➢ Doping: N type ➢ Doping: P type
➢ Counter doping

Column V elements are Doping (N type) donors, e.g. P, As, Sb

Column V elements are Doping (N type) donors, e.g. P, As, Sb
By substituting a Si atom with a special impurity atom (Column V element), a conduction electron is created.
Donors: P, As, Sb

Phosphorus has 5 valence electrons
⚫ ‘Donates’ one conduction electron to lattice
⚫ Our substrate has 1015cm-3 phosphorus

Doping (N type)
⚫ If Si is doped with phosphorus (P), each P atom can contribute a conduction electron, so that the Si lattice has more electrons than holes, i.e. it becomes “N type”:
ND = Concentration of donors
n = electron concentration ND+ = Concentration of ionized donors
Immobile ion: ND+ Ionized donor
niD ? n=ND+ +niD
Ionization energy < 50meV: ≈ ND+ >>niD

Doped by impurities of 5 valence electrons (donors) At room temperature, one donor will create one free
Holes are not created
niD<1014 cm-3 and ni1010 cm-3
Since ND>>niD, The density of free electron can
be controlled through doping.
Ionized donor
Free electron

Electron and Hole Concentrations
No E field, no B field, no light
⚫ Under thermal equilibrium conditions, the product of the conduction-electron density and the hole density is ALWAYS equal to the square of ni:
N-type material at RT
np = n 2 i
=(1010)2/cm3 at RT
Example: at RT ND=1015/cm3 n=1015/cm3 ND+=1015/cm3 p=105/cm3

N-Type Semiconductor
⚫ The addition of pentavalent impurities such as Sb, As or P contributes free electrons, greatly increasing the conductivity of the intrinsic semiconductor.
⚫ Phosphorus may be added by diffusion of phosphine gas (PH3).
⚫ EF is shifted to the up-half of the bandgap for n- type.

Properties of N-type
n>>p, so “n-type”.
Electrons are ‘majority’ charge carriers and
holes are ‘minority’ charge carriers.
Space charge: when an electron is freed, it left a positively charged atom behind, which is fixed in space
Fermi potential: f How ‘strong’ the n-type is
Space charge

2.4 The doping of semiconductors
➢ Doping elements ➢ Doping: N type
➢ Doping: P type ➢ Counter doping

Column III elements are Doping (P type) acceptors, e.g. B, Al, Ga
By substituting a Si atom with a special impurity atom (Column III element), a conduction hole is created.
Acceptors: B, Al, Ga, In

Boron has 3 valence electrons
⚫ ‘Accepts’ one electron from lattice
⚫ Creates a ‘hole’

Doping (P type)
⚫ If Si is doped with Boron (B), each B atom can contribute a hole, so that the Si lattice has more holes than electrons, i.e. it becomes “P type”:
Hole is created when a neighboring valence electron moves to the B atom.
Column III elements are acceptors, e.g. B
NA = concentration of acceptors
p = hole concentration NA- = concentration of ionized acceptors

Doping (P type)
Column III elements are acceptors, e.g. B
⚫ If Si is doped with Boron (B), each B atom can contribute a hole, so that the Si lattice has more holes than electrons, i.e. it becomes “P type”:
NA = concentration of acceptors
p = hole concentration NA- = concentration of ionized acceptors
Immobile ion: ionized acceptor
p = NA- + piA
Ionization energy < 50meV: ≈ NA- >>piA

Doped by impurities of 3 valence electrons
(acceptors)
At room temperature, one acceptor will create one hole.
Free electrons are not created
piA<>n, so “p-type”.
Holes are ‘majority’ charge carriers and
electrons are ‘minority’ charge carriers. Space charge: negative charges bonded to
Boron atoms
Fermi potential: f How ‘strong’ the p-type is

Summary of doping
elements contribute conduction electrons, and are called donors.
Column-III
elements contribute holes, and are called acceptors

2.4 The doping of semiconductors
➢ Doping elements ➢ Doping: N type
➢ Doping: P type
➢ Counter doping

Counter Doping
This is a n-type Si, n = ND + ni. Nomally ND >> ni，so n = ND

Counter Doping
Adding the same B as P causes the doping type to change. n = p = ND – NA+ ni = ni.

Counter Doping
i pN−N,nn2
A D NA −ND The addition of one more B than P causes the doping
type to change from n-type to p-type
P-type material (NA > ND)

Counter Doping
i nN−N,p n2
D A ND −NA The addition of one more P than B causes the doping
type to change from p-type to n-type
N-type material (ND > NA)

Dopant Compensation
⚫ An N-type semiconductor can be converted into P-type material by counter-doping it with acceptors such that NA > ND.
⚫ A compensated semiconductor material has both acceptors and donors.
N-type material P-type material (ND > NA) “net doping” (NA > ND)
pNA −ND pi ni
ND −NA NA −ND What is the relationship between EF and n/p?

Counter Doping Process
⚫ To form pn junction
Concentration
1017 cm-3 103 cm-3

Next week:
Semiconductor Fundamentals – (III)
2.5 Boltzmann approximation & E , n, p F
2.6 Carrier drift and diffusion

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