01-Jan-24

PYL 102: Principles of Electronic Materials

Amartya Sengupta
amartya@physics.iitd.ac.in
Schedule: Mon, Wed (11-11.50 AM); Thurs(12:00 – 12:50 PM)
Room #: LH 325
Office: VI-415B
Phone #: x 1382
Office Hours: By appointment only

: +91 95827 33597 (For Immediate/Urgent Correspondence)

Course Objectives
To present the principles of behaviour of electrons in metals, semiconductors, dielectric and magnetic materials and
thus develop understanding of the basis properties of electronic materials. The course lays foundation for advanced
courses in engineering aspects of materials and their applications.

Concepts of Brillouin Zone, Concept of effective Structure of Solids like Copper, Silicon, Diamond and
mass and hole, Density of states in metal and Graphene in real & Reciprocal space, Interaction of
semiconductor in 3D to 1D, The Fermi–Dirac Xray and neutron with solids for determination of
Probability, Semiconductors in Equilibrium, Charge crystal and magnetic structures. Classical and
Carriers in Semiconductors, The Intrinsic Carrier quantum free electron theory, Drude Model, DC and
Concentration, The Intrinsic Fermi-Level Position, AC conductivity of metal, Hall effect, Fermi Sphere
Equilibrium Distribution of Electrons & Holes. The and their temperature correlation. Energy bands
Extrinsic Semiconductor, Degenerate and (Bloch & K-P), Finding distinction of electronic
Nondegenerate Semiconductors, M-S junction, p-n materials: Basics of Magnetism, types of
junction, direct and indirect band gap interactions, Ordering temperature, Magnetic
semiconductors. domains, Anisotropies etc.

AMARTYA SENGUPTA SUJIT MANNA

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Course References

Attendance & Grading Policy

Quiz: 10 marks Quiz: 10 marks
Assignments: 10 marks Assignments: 10 marks
MS Exams: 30 marks ES Exams: 30 marks

Examinations - OPEN – BOOK/OPEN NOTE Examinations – CLOSED BOOK/CLOSED NOTE

Syllabus – All topics covered till Mid-Semester Syllabus – All topics covered from Mid-Sem exams –
exams End Sem Exams

 NO electronic devices are allowed during the lecture (EXCEPT When Directed)

 To Obtain Pass grade, you must score at least 30% marks overall AND you have to obtain at least 25% marks
in each individual component.

 No Re-Quiz or Re-Exam will be given.

 The Grading Policy is same for Audit students

 Institute Attendance policy will be followed (for each session individually): if your attendance is < 75%, you
will get one grade less, < 60%, you will get 2 grades less. (ONLY PAPER BASED ATTENDANCE IN CLASS WILL
BE CONSIDERED)

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Periodic Table in Microelectronics

Purity & Performance

The fundamental property that most clearly links electronic materials is purity. In many cases extreme measures are
taken to prevent contamination.

Example: One trillion Si atoms fill a cube ~ 3 μm on a side. No more than 1 Fe atom can be allowed in such a volume of typical Si.
This is more than 1000 times the purity requirement of other applications.

Performance can have many aspects including electronic properties of the material such as conductivity, free carrier
mobility, etc., and physical and chemical properties such as mechanical strength, stability against diffusional mixing or
reaction with adjacent materials, and many more.

Example: (a) Metal Contacts: It has taken years to switch from Al to Cu as the metal connecting devices in integrated circuits.
Copper diffuses rapidly and causes very large problems if it gets into the active device regions.

(b) Semiconductors: Electrons can be more easily accelerated in GaAs than in Si and live a shorter time. Both of these
contribute to faster device speeds. However, GaAs is frail and also, there is lack of a good insulator and good contacts. These
problems have never been solved, while all of the major problems facing applications of Si, except its inability to emit light, have
been overcome.

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Semiconducting Materials

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Band Theory of Solids

• In order to account for decreasing resistivity with increasing temperature as well as other
properties of semiconductors, a new theory known as the band theory is introduced.

• The essential feature of the band theory is that the allowed energy states for electrons are
nearly continuous over certain ranges, called energy bands, with forbidden energy gaps
between the bands.

Bonding and Anti-Bonding Orbitals

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Hydrogen Molecule & LC Circuits

(a) Energy vs. the interatomic separation R.

(b) Schematic diagram showing the changes in the electron energy as two isolated H atoms, far left and
far right, come together to form a hydrogen molecule.

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Formation of Bands

The formation of 2s energy band from the 2s orbitals when N Li atoms come together to form the Li solid.
There are N 2s electrons, but 2N states in the band. The 2s band is therefore only half full.
The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid. Thus, each
Li atom has a closed K shell (full 1s orbital).

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Formation of Bands

As Li atoms are brought

together from infinity, the
atomic orbitals overlap and
give rise to bands.

Outer orbitals overlap first.

The 3s orbitals give rise to the
3s band, 2p orbitals to the 2p
band, and so on. The various
bands overlap to produce a
single band in which the
energy is nearly
continuous.

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Band Theory of Solids

In our analysis through all of the earlier lectures, we do not have a solid to begin with. Instead the atoms are
independent to begin with and are brought together to build the solid. All of the electrons are bound to their
respective individual atoms to begin with. In this case the atoms are free to begin with while the “electrons are
tightly bound” to begin with. In view of the focus on the electronic properties of the materials, this approach is
referred to as the Tight binding approximation – highlighting the status of the electrons at the start of the
model.

There is another approach to modeling materials which starts from a diametrically opposite position. In this
approach, we adopt a picture of the solid that says that there are ionic cores at fixed lattice locations and that
there is a free electron gas enveloping these ionic cores. In other words we assume that the solid already exists
and that the ionic cores are tightly bound to their lattice locations while the “electrons are free” to run through
the extent of the solid. This is called the Free electron approximation.

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As seen in this example and in Figure 3.35 , the E - EF >> kT notation is somewhat misleading. The Maxwell–Boltzmann and
Fermi–Dirac functions are within 5 percent of each other
when E - EF == 3 kT .

Electron Statistics

The Boltzmann energy

distribution describes the
statistics of particles, such as
electrons, when there are
many more available states
than the number of particles.

The Fermi-Dirac f(E) describes the statistics

E-Ef >> kT
of electrons in a solid. The electrons interact with
each other and the environment,
obeying the Pauli exclusion principle.  E 
P ( E )  A exp   
 kT 
N2  E  E1 
 exp  2 
N1  kT 
1
f (E) 
 E  EF  The function f F ( E ) is called the Fermi–Dirac distribution or probability function and gives
1  exp  the probability that a quantum state at the energy E will be occupied by an electron. The
 kT  energy E F is called the Fermi energy. Another interpretation of the distribution function is
that f F ( E ) is the ratio of filled to total quantum states at
any energy E .

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Practice Example
Approximately what is the probability that, at room temperature (300 K), an electron at the top of
the highest filled band in diamond (an insulator) will jump the energy gap Eg? For diamond, Eg is 5.5
eV.

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Current Flow in Semiconductors

Mobility ∗

Electric Field

Carrier Concentration Electronic Charge Chemical composition, crystal structure, temperature,

doping, etc. – SOLID STATE PHYSICS

Density of States, Fermi function – QUANTUM

MECHANICS & STATISTICAL MECHANICS
Carrier Density

Mobility, Diffusion – ELECTRODYNAMICS

Density of States Fermi Function

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Practice Example
Approximately what is the probability that, at room temperature (300 K), an electron at the top of
the highest filled band in diamond (an insulator) will jump the energy gap Eg? For diamond, Eg is 5.5
eV.

Boltzmann relation relates the population Nx of atoms at energy level Ex to the population N0 at energy level E0, where
the atoms are part of a system at temperature T.

We use the above expression to approximate the probability P that an electron in an insulator will jump the energy gap
Eg.

This means that approximately 3 electrons out of 1093 electrons would jump across the energy
gap. Because any diamond stone has fewer than 1023 electrons, we see that the probability of the
jump is vanishingly small. No wonder diamond is such a good insulator.

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Current Flow in Semiconductors

Mobility ∗

Electric Field

Carrier Concentration Electronic Charge Chemical composition, crystal structure, temperature,

doping, etc. – SOLID STATE PHYSICS

Density of States, Fermi function – QUANTUM

MECHANICS & STATISTICAL MECHANICS
Carrier Density

Mobility, Diffusion – ELECTRODYNAMICS

Density of States Fermi Function

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Band Theory of Solids

• Electrons interact with each other and in a real solid atoms are
vibrating causing time dependent variations in the potential
energy.
ℎ
− ∇ + 𝑉 𝜓 𝑟⃑ = 𝐸𝜓 𝑟⃑
2𝑚
• The wavefunctions for the electron must satisfy Bloch’s theorem
.⃑
𝜓 𝑟⃑ = 𝜓 , 𝑟⃑ = 𝑒 𝑢 , 𝑟⃑
𝜓 𝑟⃑ + 𝑅 = 𝑒 . 𝜓 𝑟⃑
, ,
𝑢 , 𝑟⃑ + 𝑅 = 𝑢 , 𝑟⃑
• There are two main categories of realistic band structure
calculation for semiconductors:
– Methods which describe the entire valence and conduction bands. Tight Binding Methods,
– Methods which describe near band-edge band structures. pseudopotential methods

K.P
methods

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KrÖnig-Penney Model
sin 𝛼𝑎
𝑃 + cos 𝛼𝑎 = cos 𝑘𝑎
𝛼𝑎
𝑚𝑎𝑉 𝑏 2𝑚𝐸
𝑃= ;𝛼 =
ℏ ℏ

𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸𝜓 𝑟⃑ = 0 Region 1
𝑑𝑥 ℏ
Allowed bands
𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸 − 𝑉 𝜓 𝑟⃑ = 0 Region 2
𝑑𝑥 ℏ
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KrÖnig-Penney Model
sin 𝛼𝑎
𝑃 + cos 𝛼𝑎 = cos 𝑘𝑎
𝛼𝑎
𝑚𝑎𝑉 𝑏 2𝑚𝐸
𝑃= ;𝛼 =
ℏ ℏ

𝑚𝑎𝑉 𝑏 3
𝑃= = 𝜋 Large Potential
ℏ 2
Barrier strength

𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸𝜓 𝑟⃑ = 0
𝑑𝑥 ℏ
𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸 − 𝑉 𝜓 𝑟⃑ = 0 Allowed bands are narrow
𝑑𝑥 ℏ
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KrÖnig-Penney Model
sin 𝛼𝑎
𝑃 + cos 𝛼𝑎 = cos 𝑘𝑎
𝛼𝑎
𝑚𝑎𝑉 𝑏 2𝑚𝐸
𝑃= ;𝛼 =
ℏ ℏ 𝑚𝑎𝑉 𝑏 𝜋
𝑃= =
ℏ 10
Small Potential
Barrier strength

𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸𝜓 𝑟⃑ = 0
𝑑𝑥 ℏ
𝑑 𝜓 𝑟⃑ 2𝑚
+ 𝐸 − 𝑉 𝜓 𝑟⃑ = 0 Allowed bands are wide
𝑑𝑥 ℏ
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KrÖnig-Penney Model
RECALL • The parabola is repeated periodically in
sin 𝛼𝑎 intervals of 𝑛
𝑃 + cos 𝛼𝑎 = cos 𝑘𝑎 • Energy is a periodic function of 𝑘𝑥 with
𝛼𝑎 the periodicity
𝑚𝑎𝑉 𝑏 2𝑚𝐸
𝑃= ;𝛼 =
ℏ ℏ

ℏ 𝑘
Free-particle energies, 𝑃 =0&𝐸 =
2𝑚
cos 𝛼𝑎 = cos 𝑘 𝑎 = cos 𝑘 𝑎 + 𝑛 ⋅ 2𝜋

2𝜋 2𝑚𝐸
⇒ 𝑘 +𝑛 =
𝑎 ℏ
6𝜋 4𝜋 2𝜋 2𝜋 4𝜋 6𝜋
− − −
𝑎 𝑎 𝑎 𝑎 𝑎 𝑎
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E-k diagrams
ℏ 𝑘 1st Brillouin Zone
𝐸=
2𝑚

3𝜋 2𝜋 𝜋 0 𝜋 2𝜋 3𝜋
− − −
𝑎 𝑎 𝑎 Γ 𝑝𝑜𝑖𝑛𝑡 𝑎 𝑎 𝑎
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Near-parabolic approximation

• For electrical phenomena, only the electrons located near the

maximum of the valence band and the minimum of the conduction
band are of interest.

• These are the energy levels where free moving electrons and
missing valence electrons are found.

• The energy dependence on momentum can be approximated by a

square parabolic function

𝐸 𝑘 =𝐸 +𝐴 𝑘−𝑘

𝐸 𝑘 =𝐸 +𝐴 𝑘−𝑘

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Effective Mass of electrons

• If the same magnitude of electric field is applied to both

electrons in vacuum and inside the crystal, the electrons
will accelerate at a different rate from each other due to
the existence of different potentials inside the crystal.

• The electron inside the crystal has to try to make its own
way.

• So the electrons inside the crystal will have a different

mass than that of the electron in vacuum.

• This altered mass is called as an effective-mass.

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Effective Mass of carriers

GaAs

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Effective Mass of carriers

GaAs

35

Effective Mass
• Effective mass in a crystal varies as a function of 𝑘

• Effective mass will be different for different energy bands.

• Upper half of an energy band, effective mass < 0 (holes)

• Lower half of energy band, effective mass > 0 1 d 𝐸

(conduction electrons) 𝑚∗ =
ℏ d𝑘
• Middle of an energy band, effective mass  (core
electrons)

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Electronic Properties

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Density of States

The Density of States (DOS) is essentially the number of different states at a particular
energy level that electrons are allowed to occupy, i.e. the number of electron states per unit
volume per unit energy.

Bulk properties such as specific heat, paramagnetic susceptibility, and other transport
phenomena of conductive solids depend on this function.

DOS calculations allow one to determine the general distribution of states as a function of
energy and can also determine the spacing between energy bands in semiconductors

1 2𝑚∗
𝑔 𝐸 = 𝐸−𝐸
2𝜋 ℏ
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Example Calculation for DOS

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Density of States (GaAs)

• Both Conduction & Valence band structures are approximately

spherical
• Electrons within the CB are characterized by a single isotropic
effective mass

1 2𝑚∗
𝑔 𝐸 = 𝐸−𝐸
2𝜋 ℏ

• Electrons within the VB are in two separate bands, i.e. in

two k = 0 degenerate sub-bands
• SO band is neglected

1 2𝑚∗ ,
𝐸 −𝐸 =𝑔 𝐸
2𝜋 ℏ
𝑔 𝐸 =
1 2𝑚∗ ,
𝑚∗ = 𝑚∗ , + 𝑚∗ , 𝐸 −𝐸 =𝑔 𝐸
2𝜋 ℏ
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VB Density of States (Si, Ge)

• Electrons within the VB are in two separate bands, i.e. in

two k = 0 degenerate sub-bands
• SO band is neglected

1 2𝑚∗ ,
𝐸 −𝐸 =𝑔 𝐸
2𝜋 ℏ
𝑔 𝐸 =
1 2𝑚∗ ,
𝑚∗ = 𝑚∗ , + 𝑚∗ , 𝐸 −𝐸 =𝑔 𝐸
2𝜋 ℏ
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Band Structure of Silicon

Near Band-edge ℏ 𝑘 ℏ 𝑘 +𝑘
constant energy 𝐸 𝒌 = ∗ +
surface ellipsoids 2𝑚 2𝑚∗
0.98𝑚 0.19𝑚

• L-point valley, ~1.1 eV above the band edge.

• Above this is the Γ-point edge. The direct bandgap of Si is ∼ 3.4 eV – has
very strong absorption coefficient

• Due to the 6-fold degeneracy, the electron transport in Si is quite poor -

very large density of states leading to a high scattering rate

• The split-off (SO) band is also very close for Si since the split-off energy is
only 44 meV.

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Fermi function
1 • The function 𝑓(𝐸), the Fermi–Dirac distribution function, gives the probability that
𝑓 𝐸 =
an available energy state at 𝐸 will be occupied by an electron at absolute
1+𝑒
temperature 𝑇.
1
𝑓 𝐸 = • An energy state at the Fermi level has a probability of 1/2 of being occupied by
2
an electron

• The Fermi function is symmetrical about 𝐸 for all temperatures

• At conditions, 𝐸~𝐸 ≫ 𝑘𝑇, FD distribution can be approximated by

using the MB distribution.

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Carrier Concentration
1
𝑓 𝐸 =
1+𝑒

𝒏= 𝝆 𝑬 𝒇 𝑬 𝐝𝑬
𝑬𝑪

1 2𝑚∗
𝑔 𝐸 = 𝐸−𝐸
2𝜋 ℏ
2
𝑛=𝑁 ℱ (𝜂 )
𝜋 Defining,
Effective CB Density of States 𝑚∗ 𝑘𝑇
𝑁 =2
Effective VB Density of States
2𝜋ℏ
2
𝑝=𝑁 ℱ (𝜂 ) 𝑚∗ 𝑘𝑇
𝜋 𝑁 =2
2𝜋ℏ
Rigorously correct expression for
Carrier Concentration

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Approximate Carrier Concentrations

Approximation Condition:
𝐸~𝐸 ≫ 𝑘𝑇

𝜂 , ≤ −3
𝐸 − 𝐸 ≥ 3𝑘𝑇
𝐸 − 𝐸 ≥ 3𝑘𝑇
1
≈𝑒 Applicable for
RECALL 𝑛=𝑁 𝑒
1+𝑒 NON-DEGENERATE
𝑚∗ 𝑘𝑇 Semiconductors
𝑁 =2
2𝜋ℏ 𝑝=𝑁 𝑒

𝑚∗ 𝑘𝑇 Approximate
𝑁 =2 expression for
2𝜋ℏ
Carrier concentration

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Rigorous vs Approximate relations

2
𝑛=𝑁 ℱ (𝜂 )
𝜋
Rigorously correct expression for
Carrier Concentration

𝑛=𝑁 𝑒
ℱ 𝜂 ,𝑒

Approximate
expression for
Carrier Concentration

𝐸 − 𝐸 ≥ 3𝑘𝑇 (𝜂 ≤ −3)
𝐸 − 𝐸 ≥ 3𝑘𝑇 (𝜂 ≤ −3)

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Semiconductors in Thermal Equilibrium

2
𝑛=𝑁 ℱ (𝜂 )
𝜋
Effective CB Density of States

Effective VB Density of States

2
𝑝=𝑁 ℱ (𝜂 )
𝜋
Rigorously correct expression for
Carrier Concentration

𝑛=𝑁 𝑒

𝑝=𝑁 𝑒

Approximate
expression for
Carrier Concentration

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Donors & Acceptors

Ionization Energy,
𝑚∗ 𝑞 13.6 𝑚∗
𝐸 = =
2 4𝜋𝜖 𝜖 ℏ 𝜖 𝑚

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Carrier Concentration and Fermi Levels

RECALL
For an intrinsic semiconductor, 𝑛 = 𝑝 = 𝑛 ;𝐸 = 𝐸
𝑛=𝑁 𝑒

𝑝=𝑁 𝑒

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Carrier Concentration with temperature

RECALL 𝑁 𝑁
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛
𝑛 𝑛
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Determination of Fermi Levels

RECALL For an intrinsic semiconductor, 𝑛 = 𝑝 = 𝑛 ;𝐸 = 𝐸
𝑛=𝑁 𝑒
𝑁𝑒 =𝑁 𝑒
𝑝=𝑁 𝑒
RECALL
𝐸 is at the bandgap center in
intrinsic semiconductors when
𝑚∗ 𝑘𝑇
𝑁 =2 𝑚∗ = 𝑚∗ or when 𝑇 = 0𝐾.
2𝜋ℏ
𝑚∗ 𝑘𝑇
𝑁 =2
2𝜋ℏ

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Determination of Fermi Levels

RECALL For an intrinsic semiconductor, 𝑛 = 𝑝 = 𝑛 ;𝐸 = 𝐸
𝑛=𝑁 𝑒
𝑁𝑒 =𝑁 𝑒
𝑝=𝑁 𝑒
RECALL
𝐸 is at the bandgap center in
intrinsic semiconductors when
𝑚∗ 𝑘𝑇
𝑁 =2 𝑚∗ = 𝑚∗ or when 𝑇 = 0𝐾.
2𝜋ℏ
𝑚∗ 𝑘𝑇 For extrinsic semiconductors, 𝑛≅𝑁 in donor-doped & 𝑝≅𝑁 in acceptor doped
𝑁 =2 semiconductors
2𝜋ℏ 𝑛 𝑝
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 = −𝑘𝑇𝑙𝑛
𝑛 𝑛 𝐸 moves systematically upward in
𝑛=𝑛𝑒 𝑁 energy from 𝐸 with increasing donor
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 doping and systematically downward in
𝑝=𝑛𝑒 𝑛
energy from 𝐸 with increasing
𝑁
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 acceptor doping.
𝑛
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Charge Neutrality in Semiconductors

In a uniformly doped semiconductor, local charge density, 𝜌 = 𝑞(𝑝 − 𝑛 + 𝑁 − 𝑁 ) = 0
𝑝−𝑛+𝑁 −𝑁 = 0 Charge-Neutrality
At room temperature, 𝑁 =𝑁 ; 𝑁 =𝑁 Relationship

RECALL
For a nondegenerate uniformly doped semiconductor with complete ionization, 𝑛𝑝 = 𝑛
−𝑛+𝑁 −𝑁 =0 ⇒𝑛 −𝑛 𝑁 −𝑁 −𝑛 =0

𝑁 −𝑁 𝑁 −𝑁
𝑛= + +𝑛
2 2
+
𝑁 −𝑁 𝑁 −𝑁
𝑝= + +𝑛
2 2

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Practice Example

A Silicon sample is doped with 10 boron atoms per 𝑐𝑚 .

(a) Calculate the carrier concentrations in the Si sample at 300 K.
(b) What are the carrier concentrations at 470 K?

75

Practice Example

A Silicon sample is doped with 10 boron atoms per 𝑐𝑚 .

(a) Calculate the carrier concentrations in the Si sample at 300 K.
(b) What are the carrier concentrations at 470 K?

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Carrier Action – Mobility

Scattering

Lattice Impurity

1 1 1
= + +. .
𝜇 𝜇 𝜇

Mobility (cm2/V-s)

79

Resistivity

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Carrier Action – Drift velocity

a) A Si bar 1 µm long and 100 µm2 in cross-sectional area is doped with
1017 cm-3 phosphorus. Find the current at 300 K with 10 V applied.

a) How long does it take an average electron to drift 1 µm in pure Si at

an electric field of 100 V/cm? Repeat for 105 V/cm.

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Practice Example

A Silicon sample is doped with 10 boron atoms per 𝑐𝑚 .

(a) Calculate the carrier concentrations in the Si sample at 300 K.
(b) What are the carrier concentrations at 470 K?

Draw the energy band diagrams of the sample for the above cases.
𝐸 ∗
𝐸 = − 0.0073 𝑒𝑉 Given, 𝐸 = 1.08 𝑒𝑉 and ∗ = 0.69 @ 300𝐾 & 0.71 @ 470𝐾.
2
𝑁
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 = 0.239 𝑒𝑉
𝑛

𝐸
𝐸 =− 0.0104 𝑒𝑉
2
𝑝
𝐸 − 𝐸 = −𝑘𝑇𝑙𝑛 = 0.0195𝑒𝑉
𝑛

78

Drift Velocity

83

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Semiconductor Electrostatics
• An (E, k) diagram is a plot of the total electron energy
as a function of the crystal-direction–dependent
electron wavevector at some point in space.

• Bottom of the conduction band corresponds to zero

electron velocity or kinetic energy, and simply gives
the potential energy at that point in space.

Only PE, No KE The slopes of the (𝐸, 𝑥) band edges at different points in
space reflect the local electric fields at those points.
Gains KE at the
expense of PE If the electric field between A and B were not constant,
the slope of the band edge would also not be constant
Scattering, lattice
but vary at each point reflecting the magnitude and
relaxation processes
direction of the local electric field.

Only PE, No KE In practice, the electron may lose its kinetic energy in
stages by a series of scattering events.

84

Practice Example
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = ±𝐿
4
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = 0
4

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Practice Example
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = ±𝐿
4
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = 0
4

88

Practice Example
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = ±𝐿
4
𝐸
𝐸 −𝐸 = 𝑎𝑡 𝑥 = 0
4

89

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Carriers in Semiconductors under equilibrium

• Generation is the process whereby electrons

and holes are created

• Recombination is the process whereby electrons

and holes are annihilated

• In thermal equilibrium, carrier concentrations

are independent of time, and therefore, the
generation and recombination rates are equal

𝐺 =𝐺 =𝑅 =𝑅

Thermal Generation Thermal Generation Recombination rate Recombination rate

rate of electrons rate of holes of electrons of holes

91

Excess Carriers in Semiconductors

For the direct band-to band generation,
the excess electrons and holes are also
created in pairs:
𝑔 =𝑔
On external influence, excess electrons
and holes are generated:
𝑛 = 𝑛 + 𝛿𝑛
𝑝 = 𝑝 + 𝛿𝑝

𝑛𝑝 ≠ 𝑛 𝑝 ≠ 𝑛

𝑅 =𝑅

92

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 02-Feb-24

Excess Carriers in Semiconductors

For the direct band-to band generation,
the excess electrons and holes are also
created in pairs:
𝑔 =𝑔
On external influence, excess electrons
and holes are generated:
𝑛 = 𝑛 + 𝛿𝑛
𝑝 = 𝑝 + 𝛿𝑝

𝑛𝑝 ≠ 𝑛 𝑝 ≠ 𝑛

𝑅 =𝑅

92

Fermi levels with temperature

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Fermi Level variation with temperature

RECALL 𝑁 𝑁
𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛 𝐸 − 𝐸 = 𝑘𝑇𝑙𝑛
𝑛 𝑛
94

2
 07-Feb-24

Practice Example
GaAs is doped with 1015
acceptors/cm3. The
intrinsic carrier
concentration is
approximately 106 cm-3,
carrier recombination
lifetime of 10 ns.

97

Quasi-Fermi Levels
Quasi-Fermi levels are conceptual constructs, defined energy levels that can be used in conjunction with the energy
band diagram to specify the carrier concentrations inside a semiconductor under nonequilibrium conditions.
RECALL • Two energies, 𝐸 , the quasi Fermi level for electrons, & 𝐸 , the quasi-Fermi level for
𝑛=𝑛𝑒 holes
• These energies are related to the nonequilibrium carrier concentrations in the same way
𝑝=𝑛𝑒 𝐸 is related to the equilibrium carrier concentrations

𝑛 For 𝑡 ≥ 0, non- Equilibrium 𝑛 = 𝑛 + Δ𝑛 = 10 /𝑐𝑚

𝑛=𝑛𝑒 ⇒𝐸 = 𝐸 + 𝑘𝑇𝑙𝑛 State:
𝑛 𝑝 = 𝑝 + Δ𝑝 = 10 /𝑐𝑚
𝑝 𝐸 − 𝐸 = 𝐸 − 𝐸 = 0.25𝑒𝑉
𝑝=𝑛𝑒 ⇒𝐸 = 𝐸 − 𝑘𝑇𝑙𝑛
𝑛 𝐸 − 𝐸 = −0.060𝑒𝑉
A uniformly donor-doped silicon wafer at room
temperature is suddenly photo-illuminated at
𝑡 = 0. Assuming 𝑁 = 10 /𝑐𝑚 , and excess
electrons and holes, Δ𝑛 = Δ𝑝 = 10 /𝑐𝑚
throughout the semiconductor, find the non-
equilibrium positions of the Fermi levels.

100

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Quasi-Fermi Levels
Quasi-Fermi levels are conceptual constructs, defined energy levels that can be used in conjunction with the energy
band diagram to specify the carrier concentrations inside a semiconductor under nonequilibrium conditions.
RECALL • Two energies, 𝐸 , the quasi Fermi level for electrons, & 𝐸 , the quasi-Fermi level for
𝑛=𝑛𝑒 holes
• These energies are related to the nonequilibrium carrier concentrations in the same way
𝑝=𝑛𝑒 𝐸 is related to the equilibrium carrier concentrations

𝑛
𝑛=𝑛𝑒 ⇒𝐸 = 𝐸 + 𝑘𝑇𝑙𝑛
𝑛
𝑝
𝑝=𝑛𝑒 ⇒𝐸 = 𝐸 − 𝑘𝑇𝑙𝑛
𝑛

𝑝 = 𝑝 + Δ𝑝 𝑒

102

Quasi-Fermi Levels
Quasi-Fermi levels are conceptual constructs, defined energy levels that can be used in conjunction with the energy
band diagram to specify the carrier concentrations inside a semiconductor under nonequilibrium conditions.
RECALL

𝑛=𝑁 𝑒
𝑝=𝑁 𝑒
𝑛𝑝 = 𝑛 1
𝑓 𝐸 =
𝑛𝑝 = 𝑁 𝑁 𝑒
1+𝑒
RECALL
1
1 𝑓 𝐸 =
𝑓 𝐸 = 1+𝑒
1+𝑒

Equilibrium Non-Equilibrium

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p-n junctions

108

Electrostatics of p-n junctions

RECALL

𝐷 𝐷 𝑘𝑇
= =
𝜇 𝜇 𝑒

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 14-Feb-24

Junctions in practical devices

Electrode
Oxide insulation
p+-AlGaAs (Contacting layer)
p-AlGaAs (Confining layer)
n-AlGaAs
p-GaAs (Active layer)
n-AlGaAs (Confining layer)
n-GaAs (Substrate)

Schematic illustration of the cross sectional structure of a buried

heterostructure laser diode.

114

I-V characteristics
𝑝 − 𝑛 diode
Schottky diode
‘Near Ohmic’
contact
Ohmic contact

115