Solid-State Electronics
‹ Textbook:
“Semiconductor Physics and Devices”
By Donald A. Neamen, 1997
‹ Reference:
“Advanced Semiconductor Fundamentals”
By Robert F. Pierret 1987
“Fundamentals of Solid-State Electronics”
By C.-T. Sah, World Scientific, 1994
‹ Homework: 0%
‹ Midterm Exam: 60%
‹ Final Exam: 40%
Solid-State Electronics
Chap. 1
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Contents
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Chap. 1
Chap. 2
Chap. 3
Chap. 4
Chap. 5
Chap. 6
Chap. 7
Solid-State Electronics
Chap. 1
Solid State Electronics: A General Introduction
Introduction to Quantum Mechanics
Quantum Theory of Solids
Semiconductor at Equilibrium
Carrier Motions:
Nonequilibrium Excess Carriers in Semiconductors
Junction Diodes　2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 1. Solid State Electronics: A General
Introduction
‹
‹
‹
‹
‹
Introduction
Classification of materials
Crystalline and impure semiconductors
Crystal lattices and periodic structure
Reciprocal lattice
Solid-State Electronics
Chap. 1
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Introduction
‹
Solid-state electronic materials:
–
‹
Conductors, semiconductors, and insulators,
A solid contains electrons, ions, and atoms, ~1023/cm3.
⇒ too closely packed to be described by classical Newtonian mechanics.
‹
Extensions of Newtonian mechanics:
–
–
Quantum mechanics to deal with the uncertainties from small distances;
Statistical mechanics to deal with the large number of particles.
Solid-State Electronics
Chap. 1
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Classifications of Materials
‹ According to their viscosity, materials are classified into solids, liquid,
and gas phases.
Solid
Liquid
Gas
Low
Medium
High
Atomic density
High
Medium
Low
Hardness
High
Medium
Low
Diffusivity
‹ Low diffusivity, High density, and High mechanical strength means
that small channel openings and high interparticle force in solids.
Solid-State Electronics
Chap. 1
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Classification Schemes of Solids
‹
‹
‹
‹
Geometry (Crystallinity v.s. Imperfection)
Purity (Pure v.s. Impure)
Electrical Classification (Electrical Conductivity)
Mechanical Classification (Binding Force)
Solid-State Electronics
Chap. 1
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Geometry
‹ Crystallinity
– Single crystalline, polycrystalline, and amorphous
Solid-State Electronics
Chap. 1
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Geometry
‹ Imperfection
– A solid is imperfect when it is not crystalline (e.g., impure) or its atom are
displaced from the positions on a periodic array of points (e.g., physical defect).
– Defect: (Vacancy or Interstitial)
– Impurity:
Solid-State Electronics
Chap. 1
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Purity
‹ Pure v.s. Impure
‹ Impurity:
– chemical impurities:a solid contains a variety of randomly located foreign atoms,
e.g., P in n-Si.
– an array of periodically located foreign atoms is known as an impure crystal with a
superlattice, e.g., GaAs
‹ Distinction between chemical impurities and physical defects.
Solid-State Electronics
Chap. 1
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Electrical Conductivity
Material type
Resistivity
(Ω-cm)
Conduction Electron
density (cm-3)
Examples
Superconductor
0 (low T)
0 (high T)
1023
Good Conductor
10-6 – 10-5
1022 – 1023
metals: K, Na, Cu, Au
Conductor
10-5 – 10-2
1017 – 1022
semi-metal: As, B,
Graphite
Semiconductor
10-2 – 10-9
106 – 1017
Ge, Si, GaAs, InP
Semi-insulator
1010 – 1014
101 – 105
Amorphous Si
Insulator
1014 – 1022
1 – 10
Solid-State Electronics
Chap. 1
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Sn, Pb
Oxides
SiO2, Si3N4,
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mechanical Classification
‹ Based on the atomic forces (binding force) that bind the atom together,
the crystals could be divided into:
– Crystal of Inert Gases (Low-T solid):
Van der Wall Force: dipole-dipole interaction
– Ionic Crystals (8 ~ 10 eV bond energy):
Electrostatic force: Coulomb force, NaCl, etc.
– Metal Crystals
Delocalized electrons of high concentration, (1 e/atom)
– Hydrogen-bonded Crystals ( 0.1 eV bond energy)
H2O, Protein molecules, DNA, etc.
Solid-State Electronics
Chap. 1
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Binding Force
‹ Bond energy is a useful parameter to provide a qualitative gauge on
whether
– The binding force of the atom is strong or weak;
– The bond is easy or hard to be broken by energetic electrons, holes, ions, and
ionizing radiation such as high-energy photons and x-ray.
‹ In semiconductors, bonds are covalent or slightly ionic bonds. Each
bond contains two electrons—electron-pair bond.A bond is broken
when one of its electron is removed by impact collision (energetic
particles) or x-ray radiation, —dangling bond.
Solid-State Electronics
Chap. 1
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Semiconductors for Electronic Device Application
‹
For electronic application, semiconductors must be crystalline and
must contain a well-controlled concentration of specific impurities.
‹ Crystalline semiconductors are needed so the defect density is low.
Since defects are electron and hole traps where e--h+ can recombine
and disappear, short lifetime.
‹ The role of impurities in semiconductors:
1. To provide a wide range of conductivity (III- B or V-P in Si).
2. To provide two types of charge carriers (electrons and holes) to
carry the electrical current , or to provide two conductivity types,
n-type (by electrons) and p-type (by holes)
‹ Group III and V impurities in Si are dopant impurities to provide
conductive electrons and holes. However, group I, II, and VI atoms
in Si are known as recombination impurities (lifetime killers)when
their concentration is low.
Solid-State Electronics
Chap. 1
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Crystal Lattices
‹ A crystal is a material whose atoms are situated periodically on
interpenetrating arrays of points known as crystal lattice or lattice
points.
‹ The following terms are useful to describe the geometry of the
periodicity of crystal atoms:
– Unit cell; Primitive Unit Cell
– Basis vectors a, b, c ; Primitive Basic vectors
– Translation vector of the lattice; Rn = n1a +n2b +n3c
– Miller Indices
Solid-State Electronics
Chap. 1
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Basis Vectors
‹ The simplest means of representing an atomic array is by translation.
Each lattice point can be translated by basis vectors, â, b̂ , ĉ.
‹ Translation vectors: can be mathematically represented by the basis
vectors. Rn = n1 â + n2 + n3 ĉ, where n1, n2, and n3 are integers.
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Unit Cell
‹ Unit cell: is a small volume of the crystal that can be used to represent
the entire crystal. (not unique)
‹ Primitive unit cell: the smallest unit cell that can be repeated to form
the lattice. (not unique) Example: FCC lattice
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Miller Indices
‹ To denote the crystal directions and planes for the 3-d crystals.
Plane (h k l)
Equivalent planes {h k l}
Direction [h k l]
Equivalent directions <h k l>
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Miller Indices
‹ To describe the plane by Miller Indices
– Find the intercepts of the plane with x, y, and z axes.
– Take the reciprocals of the intercepts
– Multiply the lowest common denominator = Mliller indices
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Example Use of Miller Indices
‹ Wafer Specification (Wafer Flats)
Solid-State Electronics
Chap. 1
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
3-D Crystal Structures
‹ In 3-d solids, there are 7 crystal systems (1) triclinic, (2) monoclinic, (3)
orthorhombic, (4) hexagonal, (5) rhombohedral, (6) tetragonal, and (7)
cubic systems.
Solid-State Electronics
Chap. 1
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
3-D Crystal Structures
‹ In 3-d solids, there 14 Bravais or space lattices.
N-fold symmetry:
⇒6-fold symmetry
With 2π/n rotation, the
crystal looks the same!
Solid-State Electronics
Chap. 1
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Basic Cubic Lattice
‹ Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered
Cubic (FCC)
Solid-State Electronics
Chap. 1
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Surface Density
‹ Consider a BCC structure and the (110) plane, the surface density is
found by dividing the number of lattice atoms by the surface area;
Surface density = 2 atoms
(a1 )(a1 2 )
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Diamond Structure (Cubic System)
‹ Most semiconductors are not in the 7 crystal systems mentioned above.
‹ Elemental Semiconductos: (C, Si, Ge, Sn)
lattice constant a
v
1 1 1 v 1 1 1
a = (− , ,− ), b = ( , ,− )
4 4 4v
4 4 4
v
a •b
1
θ = cos −1 v v ,θ = cos −1( ) = 109.4o
3
a || b |
(0,0,0)
θ=109.4o
1 1 1
( − , ,− )
4 4 4
1 1 1
( , ,− )
4 4 4
‹ The space lattice of diamond is fcc. It is composed of two fcc lattices
displaced from each other by ¼ of a body diagonal, (¼, ¼, ¼ )a
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Diamond Structure
‹ Or the diamond could be visualized by a bcc with four of the corner
atoms missing.
Solid-State Electronics
Chap. 1
25
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Zinc Blende Structure (Cubic system)
‹ Compound Semiconductors:
(SiC, SiGe, GaAs, GaP, InP, InAs, InSb, etc)
– Has the same geometry as the diamond structure except that zinc blende crystals
are binary or contains two different kinds of host atoms.
Solid-State Electronics
Chap. 1
26
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Wurzite Structure (Hexagonal system)
‹ Compound Semiconductors
(ZnO, GaN, ALN, ZnS, ZnTe)
– The adjacent tetrahedrons in zinc blende structure are rotated 60o to give the
wurzite structure.
– The distortion changes the symmetry: cubic →hexagonal
– Distortion also increase the energy gap, which offers the potential for optical device
applications.
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Reciprocal Lattice
‹ Every crystal structure has two lattices associated with it, the crystal
lattice (real space) and the reciprocal lattice (momentum space).
‹ The relationship between the crystal lattice vector ( aˆ , bˆ, cˆ ) and
reciprocal lattice vector ( Aˆ , Bˆ , Cˆ ) is
ˆ
ˆ
ˆA = 2π bxcˆ ; Bˆ = 2π cˆxaˆ ; Cˆ = 2π aˆxb
aˆ ⋅ bˆxcˆ
aˆ ⋅ bˆxcˆ
aˆ ⋅ bˆxcˆ
‹ The crystal lattice vectors have the dimensions of [length] and the
vectors in the reciprocal lattice have the dimensions of [1/length],
which means in the momentum space. (k = 2π/λ)
‹ A diffraction pattern of a crystal is a map of the reciprocal lattice of the
crystal.
Solid-State Electronics
Chap. 1
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Example
‹ Consider a BCC lattice and its reciprocal lattice (FCC)
‹ Similarly, the reciprocal lattice of an FCC is BCC lattice.
Solid-State Electronics
Chap. 1
29
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 2. Introduction to Quantum Mechanics
‹ Principles of Quantum Mechanics
‹ Schrödinger’s Wave Equation
‹ Application of Schrödinger’s Wave Equation
‹ Homework
Solid-State Electronics
Chap. 2
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Introduction
‹ In solids, there are about 1023 electrons and ions packed in
a volume of 1 cm3. The consequences of this highly
packing density :
– Interparticle distance is very small: ~2x10-8 cm.
⇒the instantaneous position and velocity of the particle are no longer
deterministic. Thus, the electrons motion in solids must be
analyzed by a probability theory.
Quantum mechanics ⇔Newtonian mechanics
Schrodinger’s equation: to describe the position probability of a
particle.
Solid-State Electronics
Chap. 2
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Introduction
– The force acting on the j-th particle comes from all the other 1023-1
particles.
– The rate of collision between particles is very high, 1013
collisions/sec
⇒average electron motion instead of the motion of each electron at a
given instance of time are interested. (Statistical Mechanics)
equilibrium statistical mechanics:
Fermi-Dirac quantum-distribution ⇔Boltzmann classical distribution
Solid-State Electronics
Chap. 2
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Principles of Quantum Mechanics
‹ Principle of energy quanta
‹ Wave-Particle duality principle
‹ Uncertainty principle
Solid-State Electronics
Chap. 2
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Quanta
‹ Consider a light incident on a surface of a material as shown below:
‹ Classical theory: as long as the intensity of light is strong enough
⇒photoelectrons will be emitted from the material.
‹ Photoelectric Effect: experimental results shows “NOT”.
‹ Observation:
– as the frequency of incident light ν < νo: no electron emitted.
– as ν > νo:at const. frequency, intensity↑, emission rate↑, K.E. unchanged.
at const. intensity, the max. K. E. ∝ the frequency of incident light.
Solid-State Electronics
Chap. 2
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quanta and Photon
‹ Planck postulated that thermal radiation is emitted from a heated
surface in discrete energy called quanta. The energy of these quanta is
given by
E = hν, h = 6.625 x 10-34 J-sec (Planck’s constant)
‹ According to the photoelectric results, Einstein suggested that the
energy in a light wave is also contained in discrete packets called
photon whose energy is also given by E = hν.
The maximum K.E. of the photoelectron is Tmax = ½mv2 = hν - hνo
‹ The momentum of a photon, p = h/λ
Solid-State Electronics
Chap. 2
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Wave-Particle Duality
‹ de Broglie postulated the existence of matter waves. He suggested that
since waves exhibit particle-like behavior, then particles should be
expected to show wave-like properties.
‹ de Broglie suggested that the wavelength of a particle is expressed as
λ = h /p, where p is the momentum of a particle
‹ Davisson-Germer experimentally proved de Broglie postulation of
“Wave Nature of Electrons”.
Solid-State Electronics
Chap. 2
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Davisson-Germer Experiment
‹ Consider the experimental setup below:
‹ Observation:
– the existence of a peak in the density of scattered electrons can be
explained as a constructive interference of waves scattered by the periodic
atoms.
– the angular distribution of the deflected electrons is very similar to an
interference pattern produced by light diffracted from a grating.
Solid-State Electronics
Chap. 2
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Conclusion
‹ In some cases, EM wave behaves like particles (photons) and
sometimes particles behave as if they are waves.
⇒Wave-particle duality principle applies primarily to SMALL particles,
e.g., electrons, protons, neutrons.
For large particles, classical mechanics still apply.
Solid-State Electronics
Chap. 2
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Uncertainty Principle
‹ Heisenberg states that we cannot describe with absolute accuracy the
behavior of the subatomic particles.
1. It is impossible to simultaneously describe with the absolute accuracy
the position and momentum of a particle.
∆p ∆x ≥ ħ. (ħ = h/2π = 1.054x10-34 J-sec)
2. It is impossible to simultaneously describe with the absolute accuracy
the energy of a particle and the instant of time the particle has this
energy.
∆E ∆t ≥ ħ
‹ The uncertainty principle implies that these simultaneous
measurements are in error to a certain extent. However, ħ is very small,
the uncertainty principle is only significant for small particles.
Solid-State Electronics
Chap. 2
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Schrodinger’s Wave Equation
‹ Based on the principle of quanta and the wave-particle duality
principle, Schrodinger’s equation describes the motion of electrons in a
crystal.
‹ 1-D Schrodinger’s equation,
− h 2 ∂ 2 Ψ ( x, t )
∂Ψ ( x, t )
V
x
x
t
j
⋅
+
(
)
Ψ
(
,
)
=
h
∂x 2
∂t
2m
‹ Where Ψ(x,t) is the wave function, which is used to describe the
behavior of the system, and mathematically can be a complex quantity.
‹ V(x) is the potential function.
‹ Assume the wave function Ψ(x,t) = ψ(x)φ(t), then the Schrodinger eq.
Becomes
− h2
∂ 2ψ ( x)
∂φ (t )
2m
Solid-State Electronics
Chap. 2
φ (t )
∂x 2
+ V ( x)ψ ( x)φ (t ) = jhψ ( x)
11
∂t
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Schrodinger’s Wave Equation
− h 2 1 ∂ 2ψ ( x)
1 ∂φ (t )
+
V
(
x
)
=
j
h
=E
2
φ (t ) ∂t
2m ψ ( x) ∂x
‹ where E is the total energy, and the solution of the eq. is
and the time-indep. Schrodinger equation can be written as φ (t ) = e − j ( E / h )t
∂ 2ψ ( x) 2m
+ 2 ( E − V ( x))ψ ( x) = 0
h
∂x 2
‹ The physical meaning of wave function:
– Ψ(x,t) is a complex function, so it can not by itself represent a real
physical quantity.
– |Ψ2(x,t)| is the probability of finding the particle between x and
x+dx at a given time, or is a probability density function.
– |Ψ2(x,t)|= Ψ(x,t) Ψ*(x,t) =ψ(x)* ψ(x) = |ψ(x)|2 -- indep. of time
Solid-State Electronics
Chap. 2
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Boundary Conditions
∫
∞
2
ψ ( x) dx = 1
since |ψ(x)|2 represents the probability density function, then for a
single particle, the probability of finding the particle somewhere is
certain.
If the total energy E and the potential V(x) are finite everywhere,
2. ψ(x) must be finite, single-valued, and continuous.
3. ∂ψ(x)/∂x must be finite, single-valued, and continuous.
1.
−∞
Solid-State Electronics
Chap. 2
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Applications of Schrodinger’s Eq.
‹ The infinite Potential Well
‹ In region I, III, ψ(x) = 0, since E is finite and a particle cannot
penetrate the infinite potential barriers.
‹ In region II, the particle is contained within a finite region of space and
V = 0. 1-D time-indep. Schrodinger’s eq. becomes
∂ 2ψ ( x) 2mE
+ 2 ψ ( x) = 0
h
∂x 2
‹ the solution is given by
ψ ( x) = A1 cos Kx + A2 sin Kx, where K =
Solid-State Electronics
Chap. 2
14
2mE
h2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Infinite Potential Well
‹ Boundary conditions:
1. ψ(x) must be continuous, so that ψ(x = 0) = ψ(x = a) = 0
⇒A1 = A2sinKa ≡ 0 ⇒ K = nπ/a, where n is a positive integer.
2.
∞
2
a
2
2
2
ψ
(
x
)
dx
=
1
sin
1
⇒
=
⇒
=
A
Kxdx
A
2
2
∫−∞
∫
a
0
So the time-indep. Wave equation is given by
ψ ( x) =
2
nπx
sin(
) where n = 1,2,3...
a
a
‹ The solution represents the electron in the infinite potential well is in a
standing waveform. The parameter K is related to the total energy E,
therefore,
h 2 n 2π 2
E = En =
Solid-State Electronics
Chap. 2
2ma
2
where n is a positive integer
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Infinite Potential Well
‹ That means that the energy of the particle in the infinite potential well
is “quantized”. That is, the energy of the particle can only have
particular discrete values.
Solid-State Electronics
Chap. 2
16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Step Potential Function
‹ Consider a particle being incident on a step potential barrier:
‹ In region I, V = 0,
‹ And the general solution of this equation is
∂ 2ψ 1 ( x) 2mE
+ 2 ψ 1 ( x) = 0
2
∂x
h
ψ 1 ( x) = A1e jK x + B1e − jK x ( x ≤ 0) where K1 =
1
1
2mE
h2
‹ In region II, V = Vo, if we assume E < Vo, then
∂ 2ψ 2 ( x)
∂x
Solid-State Electronics
Chap. 2
2
−
2m
h
2
(Vo − E )ψ 2 ( x) = 0
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Step Potential Function
‹ The general solution is in the form
ψ 2 ( x) = A2e − K 2 x + B2e + K 2 x ( x ≥ 0) where K 2 =
‹ Boundary Conditions:
2m(Vo − E )
h2
ψ 2 ( x) = A2e − K 2 x ( x ≥ 0)
– ψ2(x) must remain finite, ⇒B2 ≡ 0 ⇒
– ψ(x) must be continuous, i.e., ψ1(x = 0) = ψ2(x = 0) ⇒A1+B1 = A2
– ∂ψ(x)/ ∂x must be continuous, i.e., ∂ψ 1
∂x
=
x =0
∂ψ 21
∂x
⇒ jK1 A1 − jK1 B1 = − K 2 A2
x =0
‹ A1, B1, and A2 could be solved from the above equations.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Potential Barrier
‹ Consider the potential barrier function as shown:
‹ Assume the total energy of an incident particle
E < Vo, as before, we could solve the
Schrodinger’s equations in each region, and obtain
ψ 1 ( x) = A1e jK x + B1e − jK x
1
1
ψ 2 ( x) = A2e K 2 x + B2e − K 2 x
where K1 =
ψ 3 ( x) = A3e jK x + B3e − jK x
1
1
2m(Vo − E )
2mE
and
K
=
2
h2
h2
‹ We can solve B1, A2, B2, and A3 in terms of A1 from boundary
conditions:
– B3 = 0 , once a particle enters in region III, there is no potential
changes to cause a reflection, therefore, B3 must be zero.
– At x = 0 and x = a, the corresponding wave function and its first
derivative must be continuous.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Potential Barrier
‹ The results implies that there is a finite probability that a particle will
penetrate the barrier, that is so called “tunneling”.
*
A
⋅
A
3
3
‹ The transmission coefficient is defined by T =
A1 ⋅ A1*
‹ If E<<Vo,
 E 
E
T ≅ 16 1 −  exp(− 2 K 2 a )
 Vo 
Vo 
‹ This phenomenon is called “tunneling” and it violates classical
mechanics.
Solid-State Electronics
Chap. 2
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
One-Electron Atom
‹ Consider the one-electron atom potential function due to the coulomb
2
attraction between the proton and electron: V (r ) = − e
4πε o r
‹ Then we can generalize the Schrodinger’s eq. to 3-D in spherical
coordinates:
2mo
1 ∂ 2 ∂ψ
1
∂ 2ψ
1
∂
∂ψ
r
⋅
(
)
+
⋅
+
⋅
(sin
θ
⋅
)
+
( E − V (r ))ψ = 0
r 2 ∂r
r 2 sin 2 θ ∂φ 2 r 2 sin 2 θ ∂θ
h2
∂r
∂θ
‹ Assume the solution to the equation can be written as
ψ (r ,θ , φ ) = R(r ) ⋅ Θ(θ ) ⋅ Φ(φ )
‹ Then the solution Φ is of the form, Φ = ejmφ, where m is an integer.
Solid-State Electronics
Chap. 2
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
One-Electron Atom
‹ Similarly, we can generate two additional constants n and l for the
variables θ and r. n, l, and m are known as quantum numbers (integers)
n = 1,2,3,...
‹ l = n − 1, n − 2, n − 3,...,0 , each set of quantum numbers corresponds to a
m = l , l − 1,...,0
quantum state which the electron may occupy.
‹ The solution of the wave equation is designated by ψnlm. For the lowest
energy state (n=1, l=0, m=0),
ψ 100
1 1
=
⋅  
π  ao 
3/ 2
e − r / ao where ao = 0.529 angstrom
‹ The electron energy E is quantized,
Solid-State Electronics
Chap. 2
22
− mo e 4
En =
(4πε o )2 2h 2 n 2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
One Electron Atom
‹ The probability density function, or the probability of finding the
electron at a particular distance form the nucleus, is proportional to
ψ100ψ*100 and also to the differential volume of the shell around the
nucleus.
‹ The electron is not localized at a given radius.
Solid-State Electronics
Chap. 2
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
‹ 2.1
‹ 2.15
‹ 2.23
Solid-State Electronics
Chap. 2
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 3. Introduction to Quantum Theory of
Solids
‹ Allowed and Forbidden Energy Bands
‹ k-space Diagrams
‹ Electrical Conduction in Solids
‹ Density of State Functions
‹ Statistical Mechanics
‹ Homework
Solid-State Electronics
Chap. 3
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Preview
‹ Recall from the previous analysis that the energy of a bound electron is
quantized. And for the one-electron atom, the probability of finding the
electron at a particular distance from the nucleus is not localized at a
given radius.
‹ Consider two atoms that are in close proximity to each other. The
wave functions of the two atom electrons overlap, which means that
the two electrons will interact. This interaction results in the discrete
quantized energy level splitting into two discrete energy levels.
Solid-State Electronics
Chap. 3
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Formation of Energy Bands
‹ Consider a regular periodic arrangement of atoms in which each atoms
contains more than one electron. If the atoms are initially far apart, the
electrons in adjacent atoms will not interact and will occupy the
discrete energy levels.
‹ If the atoms are brought closer enough, the outmost electrons will
interact and the energy levels will split into a band of allowed energies.
Solid-State Electronics
Chap. 3
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Formation of Energy Bands
Solid-State Electronics
Chap. 3
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Kronig-Penny Model
‹ The concept of allowed and forbidden energy levels can be developed
by considering Schrodinger’s equation.
Kronig-Penny Model
Solid-State Electronics
Chap. 3
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Kronig-Penny Model
‹ The Kronig-Penny model is an idealized periodic potential
representing a 1-D single crystal.
‹ We need to solve Schrodinger’s equation in each region.
‹ To obtain the solution to the Schrodinger’s equation, we make use of
Bloch theorem. Bloch states that all one-electron wave functions,
involving periodically varying potential energy functions, must be of
the form, ψ(x) = u(x)ejkx, u(x) is a periodic function with period (a+b)
and k is called a constant of the motion.
‹ The total wave function Ψ(x,t) may be written as Ψ(x,t) = u(x)ej(kx-(E/ħ)t).
‹ In region I (0 < x < a), V(x) = 0, then Schrodinger’s equation becomes
d 2u1 ( x)
du ( x)
2mE
+ 2 jk 1
− (k 2 − α 2 )u1 ( x) = 0 , α 2 ≡ 2
dx
dx
h
Solid-State Electronics
Chap. 3
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Kronig-Penny Model
‹ The solution in region I is of the form,
u1 ( x) = Ae j (α − k ) x + Be − j (α + k ) x for 0 < x < a
‹ In region II (-b < x < 0), V(x) = Vo, and apply Schrodinger’s eq.
2m V
d 2u 2 ( x)
du ( x)
+ 2 jk 2
− (k 2 − β 2 )u2 ( x) = 0 , β 2 ≡ α 2 − o2 o
h
dx
dx
The solution for region II is of the form,
u2 ( x) = Ce j ( β − k ) x + De − j ( β + k ) x for -b < x < 0
‹ Boundary conditions:
u1 (0) = u 2 (0) ⇒ A + B − C − D = 0
du1
dx
du1
dx
=
x =0
du2
dx
⇒ (α − k )A − (α + k )B − (β − k )C + (β + k )D = 0
x =0
u1 (a) = u2 (−b) ⇒ Ae j (α − k ) a + Be − j (α + k ) a − Ce − j ( β − k ) b − De j ( β + k ) b = 0
=
x=a
du2
dx
⇒ (α − k )Ae j (α − k ) a − (α + k )Be − j (α + k ) a − (β − k )Ce − j ( β − k )b + (β + k )De j ( β + k )b = 0
x=−b
Solid-State Electronics
Chap. 3
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Kronig-Penny Model
‹ There is a nontrivial solution if, and only if, the determinant of the
coefficients is zero. This result is
(
)
− α2 + β2
− 1 ≤ f (ξ ≡ E / Vo ) =
(sin αa )(sin βb) + (cos αa )(cos βb) = cos k (a + b) ≤ 1
2αβ
‹ The above equation relates k to the total energy E (through α) and the
potential function Vo (through β). The allowed values of E can be
determined by graphical or numerical methods.
Solid-State Electronics
Chap. 3
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Kronig-Penny Model
‹ Recall -1≤cosk(a+b)≤1, so E-values which cause f(ξ) to lie in the range
-1≤ f(ξ) ≤1 are the allowed system energies.—
‹ The ranges of allowed energies are called energy bands; the excluded
energy ranges (|f(ξ)|≥1) are called the forbidden gaps or bandgaps .
‹ The energy bands in a crystal can be visualized by
‹
Energy
4
3
2
1
Solid-State Electronics
Chap. 3
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k Diagram
Solid-State Electronics
Chap. 3
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
k-space Diagram
‹ Consider the special case for which Vo = 0, (free particle case)
⇒ cosα(a+b) = cosk(a+b), i.e., α = k,
⇒α =
2mE
h2
=
1
2m( mv 2 )
p
2
= =k
h
h2
,where p is the particle momentum
and k is referred as a wave number.
‹ We can also relate the energy and momentum as E = k2ħ2/2m
Solid-State Electronics
Chap. 3
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k diagram
‹ More interesting solution occur for E < Vo (β = jγ), which applies to
the electron bound within the crystal. The result could be written as
γ 2 −α 2
(sin αa )(sinh γb) + (cos αa )(cosh γb) = cos k (a + b)
2αγ
‹ Consider a special case, b→0, Vo →∞, but bVo is finite, the above eq.
becomes
sin αa
mV ba
P'

+ cos αa = cos ka, P ' ≡ 
αa

o
2
h



‹ The solution of the above equation results in a band of allowed
energies.
Solid-State Electronics
Chap. 3
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k diagram
‹ Consider the function of f (αa) = P'
Solid-State Electronics
Chap. 3
13
sin αa
+ cos αa graphically,
αa
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k diagram
‹ E-k diagram could be generated from the above figure.
‹ This shows the concept of the allowed energy bands for the particle
propagating in the crystal.
Solid-State Electronics
Chap. 3
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Reduced k-space
Solid-State Electronics
Chap. 3
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Electrical Conduction in Solids
‹ the Bond Model
‹ Energy Band
E-K diagram of a semiconductor
Solid-State Electronics
Chap. 3
16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Drift Current
‹ If an external force is applied to the electrons in the conduction band
and there are empty energy states into which the electrons can move,
electrons can gain energy and a net momentum.
n
‹ The drift current due to the motion of electrons is J = −e∑ vi
i =1
where n is the number of electrons per volume and vi is the electron
velocity in the crystal.
Solid-State Electronics
Chap. 3
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Electron Effective Mass
‹ The movement of an electron in a lattice will be different than that of
an electron in free space. There are internal forces in the crystal due to
the positively charged ions or protons and electrons, which will
influence the motion of electrons in the crystal. We can write
Ftotal = Fext + Fint = ma
‹ Since it is difficult to take into account of all of the internal forces, we
can write
F = m* a
ext
‹ m* is called the effective mass which takes into account the particle
mass and the effect of the internal forces.
Solid-State Electronics
Chap. 3
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective mass, E-k diagram
‹ Recall for a free electron, the energy and momentum are related by
1 dE p
p 2 h 2k 2
dE h 2 k hp
E=
=
⇒
=
=
⇒
= =v
2m 2m
h dk m
dk
m
m
– So the first derivative of E w.r.t. k is related to the velocity of the particle.
‹ In addition,
d 2E h2
1 d 2E 1
=
⇒ 2
=
h dk 2 m
dk 2 m
– So the second derivative of E w.r.t. k is inversely proportional to the mass
of the particle.
‹ In general, the effective
mass could be related to
1
1 d 2E
=
m* h 2 dk 2
Solid-State Electronics
Chap. 3
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective mass, E-k diagram
‹ m* >0 near the bottoms of all band; m* <0 near the tops of all bands
‹ m* <0 means that, in response to an applied force, the electron will accelerate
in a direction opposite to that expected from purely classical consideration.
‹ In general, carriers are populated near the top or bottom band edge in a
semiconductor—the E-k relationship is typically parabolic and, therefore,
d 2E
= constant ...E near E edge
dk 2
thus carriers with energies near the top or bottom of an energy band typically
exhibit a CONSTANT effective mass
Solid-State Electronics
Chap. 3
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Concept of Hole
Solid-State Electronics
Chap. 3
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Extrapolation of Concepts to 3-D
‹ Brilliouin Zones: is defined as a Wigner-Seitz cell in the reciprocal lattice.
‹ Γ point: Zone center (k = 0) ⇒ (0 0 0 )
‹ X point: Zone-boundary along a <1 0 0 >
2π
direction ⇒ a (1,0,0) 6 symmetric points
(1 0 0) (-1 0 0) (0 1 0) (0 -1 0) (0 0 1) (0 0 -1)
‹ L point: Zone-boundary along a <1 1 1>
direction ⇒ 2π ( 1 , 1 , 1 ) 8 symmetric points
a 2 2 2
‹ Γ, X, and L points are highly symmetric ⇒ energy stable states ⇒ carriers
accumulate near these points in the k-space.
Solid-State Electronics
Chap. 3
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k diagram of Si, Ge, GaAs
Solid-State Electronics
Chap. 3
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Band
‹ Valence Band:
– In all cases the valence-band maximum occurs at the zone center, at k = 0
– is actually composed of three subbands. Two are degenerate at k = 0,
while the third band maximizes at a slightly reduced energy.
The k = 0 degenerate band with the smaller curvature about k = 0 is called
“heavy-hole” band, and the k = 0 degenerate band with the larger
curvature is called “light-hole” band. The subband maximizing at a
slightly reduced energy is the “split-off” band.
– Near k = 0 the shape and the curvature of the subbands is essentially
orientation independent.
Solid-State Electronics
Chap. 3
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Band
‹ Conduction band:
– is composed of a number of subbands. The various subbands exhibit
localized and abssolute minima at the zone center or along one of the
high-symmetry diirections.
– In Ge the conduction-band minimum occurs right at the zone boundary
along <111> direction. ( there are 8 equivalent conduction-band minima.)
– The Si conduction-band minimum occurs at k~0.9(2π/a) from the zone
center along <100> direction. (6 equivalent conduction-band minima)
– GaAs has the conduction-band minimum at the zone center directly over
the valence-band maximum. Morever, the L-valley at the zone boundary
<111> direction lies only 0.29 eV above the conduction-band minimum.
Even under equilibrium, the L-valley contains a non-negligible electron
population at elevated temp. The intervalley transition should be taken
into account.
Solid-State Electronics
Chap. 3
25
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Metal, Semiconductor, and Insulator
Insulator
Solid-State Electronics
Chap. 3
Semiconductor
26
Metal
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The k-space of Si and GaAs
‹ Direct bandgap: the valence band maximum and the conduction band
minimum both occur at k = 0. Therefore, the transition between the
two allowed bands can take place without change in crystal momentum.
Solid-State Electronics
Chap. 3
27
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Constant-Energy Surfaces
‹ A 3-D k-space plot of all the allowed k-values associated with a given
energy E. The geometrical shapes, being associated with a given
energy, are called constant-energy surfaces (CES).
‹ Consider the CES’s characterizing the conduction-band structures near
Ec in Ge, Si, and GaAs.
(a) Constant-energy surfaces
Solid-State Electronics
Chap. 3
(b) Ge surface at the
Brillouin-zone boundaries.
28
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Constant-Energy Surfaces of Ec
‹ For Ge, Ec occurs along each of the 8 equivalent <111> directions; a
Si conduction band minimum, along each of 6 equivalent <100>
directions. For GaAs, Ec is positioned at the zone center, giving rise to
a single constant-energy surface.
‹ For energy slightly removed from Ec:
E-Ec ≈ Ak12+Bk22+Ck32,
where k1, k2, k3 are k-space coordinates measured from the center of a band
minimum along principle axes.
For example: Ge, the k1, k2, k3 coordinate system would be centered at the
[111] L-point and one of the coordinate axes, say k1-axis, would be directed
along the kx-ky-kz [111] direction.
‹ For GaAs, A = B = C, exhibits spherical CES;
‹ For Ge and Si, B=C, the CES’s are ellipsoids of revolution.
Solid-State Electronics
Chap. 3
29
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
‹ In 3-D crystals the electron acceleration arising from an applied force
is analogously by
dv
1
= * ⋅F
dt m
where
 mxx−1 mxy−1 mxz−1 
 −1

1
−1
−1
=  m yx m yy m yz 
m *  −1
−1
−1 
 mzx mzy mzz 
1 ∂2E
m = 2
h ∂ki ∂k j
−1
ij
..i, j = x, y , z
‹ For GaAs, E − Ec = A(k x2 + k y2 + k z2 ) , so mij = 0 if i≠j, and mxx−1 = m −yy1 = mzz−1 =
2A
h2
therefore, we can define mii=me*, that is the the effective mass tensor reduces
to a scalar, giving rise to an orientation-indep. equation of motion like that of a
classical particle.
h2
2
2
2
⇒ E − Ec =
Solid-State Electronics
Chap. 3
30
2m
2
e
(k x + k y + k z )
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
‹ For Si and Ge:
E-Ec = Ak12+B(k22+k32)
so mij = 0 if i≠j, and mxx−1 = 2 A2 , myy−1 = mzz−1 = 2 B2
h
h
‹ Because m11 is associated with the k-space direction lying along the
axis of revolution, it is called the longitudinal effective mass ml*.
Similarly, m22 = m33, being associated with a direction perpendicular to
the axis of revolution, is called the transverse effective mass mt*.
h2 2 h2
2
2
⇒ E − Ec =
k
+
(
k
+
k
)
1
2
3
2
2
2ml
2mt
Solid-State Electronics
Chap. 3
31
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
‹ The relative sizes of ml* and mt* can be deduced by inspection of the Si
and Ge constant-energy plots.


length of the elliosoid


along the axis of revolution
m 

=

max. width of the ellipsoid
m


perpendicu
lar
to
the
axis
of
revolution


*
l
*
t
‹ For both Ge and Si, ml* > mt*. Further, ml*/mt* of Ge > ml*/mt* of Si.
‹ The valence-band structure of Si, Ge, and GaAs are approximately
spherical and composed of three subbands. Thus, the holes in a given
subband can be characterized by a single effective mass parameter, but
three effective mass (mhh*, mlh*, and mso*) are required to characterize
the entire hole population. The split-off band, being depressed in
energy, is only sparsely populated and is often ignored.
Solid-State Electronics
Chap. 3
32
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass measurement
‹ The near-extrema point band structure, multiplicity and orientation of
band minima, etc. were all originally confirmed by cyclotron
resonance measurement.
‹ Resonance experiment is performed in a microwave resonance
cavity at temperature 4K. A static B field and an rf E-field
oriented normal to B are applied across the sample. The carriers
in the sample will move in an orbit-like path about the direction
of B and the cyclotron frequency ωc = qB/mc. When the B-field
strength is adjusted such that ωc = the ω of the rf E-field, the carriers
absorb energy from the E-field (in resonance). ⇒m= qB/ωc
‹ Repeating the different B-field orientations allows one to separate out the
effective mass factors (ml* and mt*)
Solid-State Electronics
Chap. 3
33
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass of Si, Ge, and GaAs
Solid-State Electronics
Chap. 3
34
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
‹ To calculate the electron and hole concentrations in a material, we
must determine the density of these allowed energy states as a function
of energy.
‹ Electrons are allowed to move relatively freely in the conduction band
of a semiconductor but are confined to the crystal.
‹ To simulate the density of allowed states, consider an appropriate
model: A free electron confined to a 3-D infinite potential well, where
the potential well represents the crystal.
‹ The potential of the well is defined as
V(x,y,z) = 0 for 0<x<a, 0<y<a, 0<z<a, and V(x,y,z) = ∞ elsewhere
‹ Solving the Schrodinger’s equation, we can obtain
2
2mE
2
2
2
2
2
2
2 π 
⇒
h2
Solid-State Electronics
Chap. 3
= k = k x + k y + k z = (nx + n y + nz ) 2 
a 
35
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
‹
‹ The volume of a single quantum state is Vk =(π/a)3, and the differential
volume in k-space is 4πk2dk
‹ Therefore, we can determine the density of quantum states in k-space
2
2
as
 1  4πk dk k dk 3
= 2 ⋅a
gT (k )dk = 2 
3
π
8
  π 
 
a
– The factor, 2, takes into account the two spin states allowed for each
quantum state; the next factor, 1/8, takes into account that we are
considering only the quantum states for positive values of kx, ky, and kz.
Solid-State Electronics
Chap. 3
36
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
‹ Recall that
k2 =
2mE
h2
‹ We can determine the density of states as a function of energy E by
gT ( E )dE =
3
4π
2
⋅
(
2
m
)
⋅ E ⋅ dE ⋅ a 3
3
h
‹ Therefore, the density of states per unit volume is given by
gT ( E )dE =
3
4π
2
⋅
(
2
m
)
⋅ E ⋅ dE
3
h
‹ Extension to semiconductors, the density of states in conduction band
3
is modified as
4π
* 2
gc (E) =
h
3
⋅ (2mn ) ⋅ E − Ec
and the density of states in valence band is modified as
gv (E) =
Solid-State Electronics
Chap. 3
3
4π
*
2
⋅
(
2
m
)
⋅ E − Ev
p
3
h
37
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
‹ mn* and mp* are the electron and hole density of states effective masses.
In general, the effective mass used in the density of states expression
must be an average of the band-structure effective masses.
Solid-State Electronics
Chap. 3
38
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of States Effective Mass
‹ Conduction Band--GaAs: the GaAs conduction band structure is
approximately spherical and the electronss within the band are
characterized by a single isotropic effective mass, me*, ⇒ mn* = me*...GaAs
‹ Conduction Band--Si, Ge: the conduction band structure in Si and Ge
is characterized by ellipsoidal energy surfaces centered, respectively,
at points along the <100> and <111> directions in k-space.
mn* = 6 3 (ml*mt*2 ) 3 ...Si
1
2
mn* = 4 3 (ml*mt*2 ) 3 ...Ge
1
2
‹ Valence Band--Si, Ge, GaAs: the valence band structures are al
characterized by approximately spherical constant-energy surfaces
(degenerate).
*
*
*
[
m p = (mhh ) + (mlh )
Solid-State Electronics
Chap. 3
3
39
2
3
]
2
2
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of States Effective Mass
Solid-State Electronics
Chap. 3
40
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Statistics Mechanics
‹ In dealing with large numbers of particles, we are interested only in the
statistical behavior of the whole group rather than in the behavior of
each individual particle.
‹ There are three distribution laws determining the distribution of
particles among available energy states.
‹ Maxwell-Boltzmann probability function:
– Particles are considered to be distinguishable by being numbered for 1 to
N with no limit to the number of particles allowed in each energy state.
‹ Bose-Einstein probability function:
– Particles are considered to be indistinguishable and there is no limit to the
number of particles permitted in each quantum state. (e.g., photons)
‹ Fermi-Dirac probability function:
– Particles are indistinguishable but only one particle is permitted in each
quantum state. (e.g., electrons in a crystal)
Solid-State Electronics
Chap. 3
41
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Fermi-Dirac Distribution
‹ Fermi-Dirac distribution function gives the probability that a quantum
state at the energy E will be occupied by an electron.
f (E) =
1
E −E F
1 + exp(
)
kT
‹ the Fermi energy (EF) determine the statistical distribution of electrons
and does not have to correspond to an allowed energy level.
‹ At T = 0K, f(E < EF) = 1 and f(E >EF ) = 0, electrons are in the lowest
possible energy states so that all states below EF are filled and all states
above EF are empty.
Solid-State Electronics
Chap. 3
42
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Fermi-Dirac Distribution, at T=0K
Solid-State Electronics
Chap. 3
43
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Fermi-Dirac Distribution
‹ For T > 0K, electrons gain a certain amount of thermal energy so that
some electrons can jump to higher energy levels, which means that the
distribution of electrons among the available energy states will change.
‹ For T > 0K, f(E = EF) = ½
Solid-State Electronics
Chap. 3
44
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Boltamann Approximation
‹ Consider T >> 0K, the Fermi-Dirac function could be approximated by
f (E) =
1
 − ( E − EF ) 
≈ exp 

E −E F
kT


1 + exp(
)
kT
which is known as the Maxwell-Boltzmann approximation.
Solid-State Electronics
Chap. 3
45
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
‹ 3.5
‹ 3.8
‹ 3.16
Solid-State Electronics
Chap. 3
46
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 4. Semiconductor in Equilibrium
‹ Carriers in Semiconductors
‹ Dopant Atoms and Energy Levels
‹ Extrinsic Semiconductor
‹ Statistics of Donors and Acceptors
‹ Charge Neutrality
‹ Position of Fermi Energy
Solid-State Electronics
Chap. 4
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Equilibrium Distribution of Electrons and Holes
‹ The distribution of electrons in the conduction band is given by the
density of allowed quantum states times the probability that a state will
be occupied.
n( E ) = g c ( E ) f ( E )
The thermal equilibrium conc. of electrons no is given by
∞
no = ∫ g c ( E ) f ( E )
Ec
‹ Similarly, the distribution of holes in the valence band is given by the
density of allowed quantum states times the probability that a state will
not be occupied by an electron.
p ( E ) = g v ( E )[1 − f ( E )]
‹ And the thermal equilibrium conc. Of holes po is given by
Ev
po = ∫ g v ( E )[1 − f ( E )]
−∞
Solid-State Electronics
Chap. 4
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Equilibrium Distribution of Electrons and Holes
Solid-State Electronics
Chap. 4
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The no and po eqs.
‹ Recall the thermal equilibrium conc. of electrons
∞
no = ∫ g c ( E ) f ( E )
Ec
‹ Assume that the Fermi energy is within the bandgap. For electrons in
the conduction band, if Ec-EF >>kT, then E-EF>>kT, so the Fermi
probability function reduces to the Boltzmann approximation,
f ( E ) ≅ exp
‹ Then
no = ∫
∞
Ec
(
4π 2mn*
h3
)
32
[−( E − EF )]
kT
 2πmn* kT 
 − ( E − EF ) 

E − Ec exp 
dE = 2
2

kT
h




 2πmn* kT 
‹ We may define N c = 2 h 2 


32
 − ( Ec − E F ) 
exp 

kT


32
, (at T =300K, Nc ~1019 cm-3), which
is called the effective density of states function in the conduction band
Solid-State Electronics
Chap. 4
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The no and po eqs.
‹ The thermal equilibrium conc. of holes in the valence band is given by
Ev
po = ∫ g v ( E )[1 − f ( E )]
−∞
‹ For energy states in the valence band, E<Ev. If (EF-Ev)>>kT,
‹ Then,
po = ∫
Ev
−∞
1 − f ( E ) ≅ exp
(
4π 2m
)
* 32
p
h3
[−( E F − E )]
kT
 2πm*p kT 
 − ( EF − E ) 

dE = 2
Ev − E exp 
2



kT


 h

 2πm*p kT 
‹ We may define N v = 2 h 2 


32
 − ( E F − Ev ) 
exp 

kT


32
, (at T =300K, Nv ~1019 cm-3), which
is called the effective density of states function in the valence band
Solid-State Electronics
Chap. 4
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
nopo product
‹ The product of the general expressions for no and po are given by
 − Eg 
 − ( Ec − E F ) 
 − ( E F − Ev ) 
N
N
N
no po = N c exp 
⋅
exp
exp
=
c v
 kT 
 v


kT
kT






⇒ for a semiconductor in thermal equilibrium, the product of no and po is
always a constant for a given material and at a given temp.
‹ Effective Density of States Function
Solid-State Electronics
Chap. 4
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Intrinsic Carrier Concentration
‹ For an intrinsic semiconductor, the conc. of electrons in the conduction
band, ni, is equal to the conc. of holes in the valence band, pi.
‹ The Fermi energy level for the intrinsic semiconductor is called the
intrinsic Fermi energy, EFi.
‹ For an intrinsic semiconductor,
32
 2πmn* kT 
 − ( Ec − E Fi ) 
 − ( Ec − E Fi ) 

no = ni = 2
N
=
exp
exp
c
2





h
kT
kT






32
 2πm*p kT 
 − ( E Fi − Ev ) 
 − ( E Fi − Ev ) 

=
exp
exp
po = pi = 2
N
v




 h2 
kT
kT






 − Eg 
⇒ n = N c N v exp 
, where E g is the bandgap energy
kT


2
i
‹ For an given semiconductor at a constant temperature, the value of ni is
constant, and independent of the Fermi energy.
Solid-State Electronics
Chap. 4
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Intrinsic Carrier Conc.
‹ Commonly accepted values
of ni at T = 300 K
Silicon
GaAs
Germanium
Solid-State Electronics
Chap. 4
ni = 1.5x1010 cm-3
ni = 1.8x106 cm-3
ni = 1.4x1013 cm-3
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Intrinsic Fermi-Level Position
‹ For an intrinsic semiconductor, ni = pi,
− ( Ec − E Fi )
− ( EFi − Ev )
⇒ N c exp[
] = N v exp[
]
kT
kT
m*p
Nv
1
3
1
3
⇒ E Fi = ( Ec + Ev ) + kT ln( ) = ( Ec + Ev ) + kT ln( * )
2
4
2
4
mn
Nc
‹
‹
‹
‹
Emidgap =(Ec+Ev)/2: is called the midgap energy.
If mp* = mn*, then EFi = Emidgap (exactly in the center of the bandgap)
If mp* > mn*, then EFi > Emidgap (above the center of the bandgap)
If mp* < mn*, then EFi < Emidgap (below the center of the bandgap)
Solid-State Electronics
Chap. 4
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Dopant and Energy Levels
Solid-State Electronics
Chap. 4
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Acceptors and Energy Levels
Solid-State Electronics
Chap. 4
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ionization Energy
‹ Ionization energy is the energy required to elevate the donor electron
into the conduction band.
Solid-State Electronics
Chap. 4
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Extrinsic Semiconductor
‹ Adding donor or acceptor impurity atoms to a semiconductor will
change the distribution of electrons and holes in the material, and
therefore, the Fermi energy position will change correspondingly.
‹ Recall
− (E − E )
− (E − E )

ni = N c exp 

c
kT
Fi


N
=
exp
v




Fi
kT
v



 − ( Ec − E F ) 
 − ( Ec − EFi ) + ( EF − EFi ) 
no = N c exp 
N
=
exp
c



kT
kT




 − ( E F − Ev ) 
 − ( E F − E Fi ) + ( Ev − EFi ) 
po = N v exp 
N
=
exp
v



kT
kT




 E − E Fi 
 − ( E F − E Fi ) 
⇒ no = ni exp  F
p
n
=
and
exp
o
i



kT
 kT 


Solid-State Electronics
Chap. 4
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Extrinsic Semiconductor
‹ When the donor impurity atoms are added, the density of electrons is
greater than the density of holes, (no > po) ⇒ n-type; EF > EFi
‹ When the acceptor impurity atoms are added, the density of electrons
is less than the density of holes, (no < po) ⇒ p-type; EF < EFi
Solid-State Electronics
Chap. 4
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Degenerate and Nondegenerate
‹ If the conc. of dopant atoms added is small compared to the density of
the host atoms, then the impurity are far apart so that there is no
interaction between donor electrons, for example, in an n-material.
⇒nondegenerate semiconductor
‹ If the conc. of dopant atoms added increases such that the distance
between the impurity atoms decreases and the donor electrons begin to
interact with each other, then the single discrete donor energy will split
into a band of energies. ⇒EF move toward Ec
‹ The widen of the band of donor states may overlap the bottom of the
conduction band. This occurs when the donor conc. becomes
comparable with the effective density of states, EF ≥ Ec
⇒degenerate semiconductor
Solid-State Electronics
Chap. 4
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Degenerate and Nondegenerate
Solid-State Electronics
Chap. 4
16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Statistics of Donors and Acceptors
‹ The probability of electrons occupying the donor energy state was
given by
N
nd =
d
1
E − EF
1 + exp( d
)
g
kT
, g : degeneracy factor
where Nd is the conc. of donor atoms, nd is the density of electrons
occupying the donor level and Ed is the energy of the donor level. g =2
since each donor level has two spin orientation, thus each donor level
has two quantum states.
‹ Therefore the conc. of ionized donors Nd+ = Nd –nd
‹ Similarly, the conc. of ionized acceptors Na- = Na –pa, where
pa =
Na
, g = 4 for the acceptor level in Si and GaAs
1
E F − Ea
1 + exp(
)
g
kT
Solid-State Electronics
Chap. 4
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Complete Ionization
‹ If we assume Ed-EF>> kT or EF-Ea >> kT (e.g. T= 300 K), then
 − ( Ed − E F ) 
nd ≈ 2 N d exp 
⇒ N d+ = N d − nd ≅ N d

kT


 − ( E F − Ea ) 
pa ≈ 4 N a exp 
⇒ N a− = N a − pa ≅ N a

kT


that is, the donor/acceptor states are almost
completely ionized and all the donor/acceptor
impurity atoms have donated an electron/hole
to the conduction/valence band.
Solid-State Electronics
Chap. 4
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Freeze-out
‹ At T = 0K, no electrons from the donor state are thermally elevated
into the conduction band; this effect is called freeze-out.
‹ At T = 0K, all electrons are in their lowest possible energy state; that is
for an n-type semiconductor, each donor state must contain an electron,
therefore, nd = Nd or Nd+ = 0, which means that the Fermi level must be
above the donor level.
Solid-State Electronics
Chap. 4
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Charge Neutrality
‹ In thermal equilibrium, the semiconductor is electrically neutral. The
electrons distributing among the various energy states creating
negative and positive charges, but the net charge density is zero.
‹ Compensated Semiconductors: is one that contains both donor and
acceptor impurity atoms in the same region. A n-type compensated
semiconductor occurs when Nd > Na and a p-type semiconductor
occurs when Na > Nd.
‹ The charge neutrality condition is expressed by
no + N a− = po + N d+
where no and po are the thermal equilibrium conc. of e- and h+ in the
conduction band and valence band, respectively. Nd+ is the conc. Of
positively charged donor states and Na- is the conc. of negatively
charged acceptor states.
Solid-State Electronics
Chap. 4
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Compensated Semiconductor
Solid-State Electronics
Chap. 4
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Compensated Semiconductor
‹ If we assume complete ionization, Nd+ = Nd and Na- = Na, then
ni2
no + N a = po + N d , recall po =
no
ni2
2
no + N a =
+ N d ⇒ no − ( N d − N a )no − ni2 = 0
nn
⇒ no
(N − N a ) +
= d
2
 Nd − Na 
2
 + ni

2


2
‹ If Na = Nd = 0, (for the intrinsic case), ⇒no = po
‹ If Nd >> Na, ⇒no = Nd
‹ If Na > Nd, ⇒ po
(N − N d ) +
= a
2
 Na − Nd 
2

 + ni
2


2
is used to
calculate the conc. of holes in valence band
Solid-State Electronics
Chap. 4
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Compensated Semiconductor
Solid-State Electronics
Chap. 4
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Position of Fermi Level
‹ The position of Fermi level is a function of the doping concentration
and a function of temperature, EF(n, p, T).
‹ Assume Boltzmann approximation is valid, we have
 and p = N exp − (E F − Ev ) 
o
v
kT 
kT 

N 
N 
⇒ Ec − E F = kT ln c 
and EF − Ev = kT ln v 
 no 
 po 
no = N c exp 

− ( Ec − E F )
n 
or E F − E Fi = kT ln o 
 ni 
Solid-State Electronics
Chap. 4
p 
and EFi − Ev = kT ln o 
 ni 
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU
EF(n, p, T)
Solid-State Electronics
Chap. 4
25
Instructor: Pei-Wen Li
Dept. of E. E. NCU
EF(n, p, T)
Solid-State Electronics
Chap. 4
26
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
‹ 4.18
‹ 4.20
‹ 4.24
Solid-State Electronics
Chap. 4
27
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 5. Carrier Motion
‹
‹
‹
‹
‹
Carrier Drift
Carrier Diffusion
Graded Impurity Distribution
Hall Effect
Homework
Solid-State Electronics
Chap. 5
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Carrier Drift
‹ When an E-field (force) applied to a semiconductor, electrons and
holes will experience a net acceleration and net movement, if there are
available energy states in the conduction band and valence band. The
net movement of charge due to an electric field (force) is called “drift”.
‹ Mobility: the acceleration of a hole due to an E-field is related by
* dv
= qE
F = mp
dt
If we assume the effective mass and E-field are constants, the we can
obtain the drift velocity of the hole by
eEt
vd = * + vi ∝ t , E
mp
where vi is the initial velocity (e.g. thermal velocity) of the hole and t is
the acceleration time.
Solid-State Electronics
Chap. 5
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
E=0
‹ In semiconductors, holes/electrons are involved in collisions with
ionized impurity atoms and with thermally vibration lattice atoms. As
the hole accelerates in a crystal due to the E-field, the velocity/kinetic
energy increases. When it collides with an atom in the crystal, it lose s
most of its energy. The hole will again accelerate/gain energy until is
again involved in a scattering process.
Solid-State Electronics
Chap. 5
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
‹ If the mean time between collisions is denoted by τcp, then the average
drift velocity between collisions is
vdp eτ cp
 eτ cp 
µp =
= *
vd =  *  E ≡ µ p E
 m 
E
mp
p


where µp (cm2/V-sec) is called the hole mobility which is an important
parameter of the semiconductor since it describes how well a particle
will move due to an E-field.
‹ Two collision mechanisms dominate in a semiconductor:
– Phonon or lattice scattering: related to the thermal motion of atoms; µL ∝T-3/2
– Ionized impurity scattering: coulomb interaction between the electron/hole and the
+
−
ionized impurities; µI ∝T3/2/NI., N I = N d + N a : total ionized impurity conc. ↑, µI ↓
If T↑, the thermal velocity of hole/electron ↑⇒carrier spends less time in the
vicinity of the impurity. ⇒ less scattering effect ⇒ µI ↑
Solid-State Electronics
Chap. 5
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Mobility
Electron mobility
Solid-State Electronics
Chap. 5
Hole mobility
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Drift Current Density
‹ If the volume charge density of holes, qp, moves at an average drift
velocity vdp, the drift current density is given by
Jdrfp = (ep) vdp = eµppE.
Similarly, the drift current density due to electrons is given by
Jdrfn = (-en) vdp = (-en)(-µnE)=eµnnE
‹ The total drift current density is given by Jdrf = e(µnn+µpp) E
Solid-State Electronics
Chap. 5
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Conductivity
‹ The conductivity σ of a semiconductor material is defined by Jdrf ≡ σ E,
so σ= e(µnn+µpp) in units of (ohm-cm)-1
‹ The resistivity ρ of a semiconductor is defined by ρ ≡ 1/ σ
Solid-State Electronics
Chap. 5
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Resistivity Measurement
‹ Four-point probe measurement
ρ = 2πs
Solid-State Electronics
Chap. 5
V
Fc ; Fc : correction factor
I
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation
‹ So far we assumed that mobility is indep. of E-field, that is the drift
velocity is in proportion with the E-field. This holds for low E-filed. In
reality, the drift velocity saturates at ~107 cm/sec at an E-field ~30
kV/cm. So the drift current density will also saturate and becomes
indep. of the applied E-field.
Solid-State Electronics
Chap. 5
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation of GaAs
‹ For GaAs, the electron drift velocity reaches a peak and then decreases
as the E-field increases. ⇒negative differential mobility/resistivity,
which could be used in the design of oscillators.
‹ This could be understood by considering the E-k diagram of GaAs.
Solid-State Electronics
Chap. 5
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Velocity Saturation of GaAs
‹ In the lower valley, the density of state effective mass of the electron
mn* = 0.067mo. The small effective mass leads to a large mobility. As
the E-field increases, the energy of the electron increases and can be
scattered into the upper valley, where the density of states effective
mass is 0.55mo. The large effective mass yields a smaller mobility.
‹ The intervalley transfer mechanism results in a decreasing average
drift velocity of electrons with E-field, or the negative differential
mobility characteristic.
Solid-State Electronics
Chap. 5
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Carrier Diffusion
‹ Diffusion is the process whereby particles flow from a region of high
concentration toward a region of low concentration. The net flow of
charge would result in a diffusion current.
Solid-State Electronics
Chap. 5
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Diffusion Current Density
‹ The electron diffusion current density is given by Jndif = eDndn/dx,
where Dn is called the electron diffusion coefficient, has units of cm2/s.
‹ The hole diffusion current density is given by Jpdif = -eDpdp/dx,
where Dp is called the hole diffusion coefficient, has units of cm2/s.
‹ The total current density composed of the drift and the diffusion
current density.
1-D J = enµ n E x + epµ p E x + eDn dn − eD p dp
dx
or
3-D
Solid-State Electronics
Chap. 5
dx
J = enµ n E x + epµ p E x + eDn∇n − eD p ∇p
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Graded Impurity Distribution
‹ In some cases, a semiconductors is not doped uniformly. If the
semiconductor reaches thermal equilibrium, the Fermi level is constant
through the crystal so the energy-band diagram may qualitatively look
like:
‹
‹ Since the doping concentration decreases as x increases, there will be a
diffusion of majority carrier electrons in the +x direction.
‹ The flow of electrons leave behind positive donor ions. The separation
of positive ions and negative electrons induces an E-field in +x
direction to oppose the diffusion process.
Solid-State Electronics
Chap. 5
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Induced E-Field
dφ
d ( E /( −e))
1 dE
Fi
Fi
=
‹ The induced E-field is defined as E x = − = −
dx
dx
e dx
that is, if the intrinsic Fermi level changes as a function of distance
through a semiconductor in thermal equilibrium, an E-field exists.
‹ If we assume a quasi-neutrality condition in which the electron
concentration is almost equal to the donor impurity concentration, then
 N d ( x) 
 E − Ei 


(
)
ln
no ≈ ni exp  F
N
x
E
E
kT
≈
⇒
−
=
d
F
i

 kT 
 ni 
d ( EF − Ei ) d (− Ei )
kT dN d ( x)
⇒
=
=
dx
dx
N d ( x) dx
 kT  1 dN d ( x)
⇒ E x = − 
 e  N d ( x) dx
‹ So an E-field is induced due to the nonuniform doping.
Solid-State Electronics
Chap. 5
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Einstein Relation
‹ Assuming there are no electrical connections between the
nonuniformly doped semiconducotr, so that the semiconductor is in
thermal equilibrium, then the individual electron and hole currents
must be zero.
dn
⇒ J n = 0 = enµ n E x + eDn
dx
‹ Assuming quasi-neutrality so that n ≈ Nd(x) and
dN d ( x)
dx
dN d ( x)
 kT  1 dN d ( x)
⇒ 0 = −en µ n N d ( x) 
+ eDn
dx
 e  N d ( x) dx
J n = 0 = eN d ( x) µ n E x + eDn
Dn
kT
- - - -Einstein relation
µn
e
D p kT
⇒
=
‹ Similarly, the hole current Jp = 0
µp
e
⇒
Solid-State Electronics
Chap. 5
=
16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Einstein Relation
‹ Einstein relation says that the diffusion coefficient and mobility are not
independent parameters.
Typical mobility and diffusion coefficient values at T=300K
(µ = cm2/V-sec and D = cm2/sec)
Silicon
GaAs
Germaium
Solid-State Electronics
Chap. 5
µn
Dn
µp
Dp
1350
8500
3900
35
220
101
480
400
1900
12.4
10.4
49.2
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
‹ The hall effect is a consequence of the forces that are exerted on
moving charges by electric and magnetic fields.
‹ We can use Hall measurement to
– Distinguish whether a semiconductor is n or p type
– To measure the majority carrier concentration
– To measure the majority carrier mobility
Solid-State Electronics
Chap. 5
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
‹ A semiconductor is electrically connected to Vx and in turn a current Ix
flows through. If a magnetic field Bz is applied, the electrons/holes
flowing in the semiconductor will experience a force F = q vx x Bz in
the (-y) direction.
‹ If this semiconductor is p-type/n-type, there will be a buildup of
positive/negative charge on the y = 0 surface. The net charge will
induce an E-field EH in the +y-direction for p-type and -y-direction for
n-type. EH is called the Hall field.
‹ In steady state, the magnetic force will be exactly balanced by the
induced E-field force. F = q[E + v x B] = 0 ⇒ EH = vx Bz and the Hall
voltage across the semiconductor is VH = EHW
‹ VH >0 ⇒ p-type, VH < 0 ⇒ n-type
Solid-State Electronics
Chap. 5
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
‹ VH = vx W Bz, for a p-type semiconductor, the drift velocity of hole is
vdx =
‹
Jx
Ix
I B
I B
=
⇒ VH = x z ⇒ p = x z
ep (ep )(Wd )
epd
edVH
for a n-type,
n=−
I x Bz
edVH
‹ Once the majority carrier concentration has been determined, we can
calculate the low-field majority carrier mobility.
IxL
epVxWd
IxL
⇒ µn =
enVxWd
‹ For a p-semiconductor, Jx = epµpEx. ⇒ µ p =
‹ For a n-semiconductor,
Solid-State Electronics
Chap. 5
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hall Effect
Hall-bar with “ear”
Solid-State Electronics
Chap. 5
van deer Parw configuration
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
‹ 5.14
‹ 5.20
Solid-State Electronics
Chap. 5
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Solid-State Electronics
Chap. 5
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 6. Nonequilibrium Excess Carriers in
Semiconductor
‹
‹
‹
‹
‹
‹
Carrier Generation and Recombination
Continuity Equation
Ambipolar Transport
Quasi-Fermi Energy Levels
Excess-Carrier Lifertime
Surface Effects
Solid-State Electronics
Chap. 6
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Nonequilibrium
‹ When a voltage is applied or a current exists in a semiconductor device,
the semiconductor is operating under nonequilibrium conditions.
‹ Excess electrons/holes in the conduction/valence bands may be
generated and recombined in addition to the thermal equilibrium
concentrations if an external excitation is applied to the semiconductor.
‹ Examples:
1. A sudden increase in temperature will increase the thermal
generation rate of electrons and holes so that their concentration will
change with time until new equilibrium reaches.
2. A light illumination on the semiconductor (a flux of photons) can
also generate electron-hole pairs, creating a nonequilibrium condition.
Solid-State Electronics
Chap. 6
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Generation and Recombination
‹ In thermal equilibrium, the electrons are continually being thermal
generated from the valence band (hereby holes are generated) to
conduction band by the random thermal process.
‹ At the same time, electrons moving randomly through the crystal may
come in close proximity to holes and recombine. The rate of
generation and recombination of electrons/holes are equal so the net
electron and hole concentrations are constant (independent of time).
Solid-State Electronics
Chap. 6
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
‹ When high-energy photons are incident on a semiconductor, electronhole pairs are generated (excess electrons/holes) ⇒ the concentration
of electrons in the conduction band and of holes in the valence band
increase above their thermal-equilibrium value. n = no +δn, p = po+ δp
where no/po are thermal–equilibrium concentrations, and δn/δp are the
excess electron/hole concentrations. np ≠ nopo = ni2 ( nonequilibrium)
‹ For the direct band-to-band generation, the generation rates (in the unit
of #/cm3-sec) of electrons and holes are equal; gn’ = gp’ (may be
functions of the space coordinates and time)
Solid-State Electronics
Chap. 6
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
‹ An electron in conduction band may “fall down” into the valence band
and leads to the excess electron-hole recombination process.
‹ Since the excess electrons and holes recombine in pairs so the
recombination rates for excess electrons and holes are equal, Rn’ = Rp’.
(in the unit of #/cm3-sec). ⇒ δn(t) = δp(t)
‹ The direct band-to-band recombination is spontaneous, thus the
probability of an electron and hole recombination is constant with time.
‹ Rn’ = Rp’ ∝ the electron and hole concentration.
Solid-State Electronics
Chap. 6
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
‹ Band-to-Band: direct thermal recombination.
This process is typically radiative,
with the excess energy released
during the process going into the
production of a photon (light)
‹ R-G Center: Induced by certain impurity atoms or crystal defects.
Electron and hole are attracted to the
R-G center and lead to the annihilation
of the electron-hole pair.
Or a carrier is first captured at the R-G
site and then makes an annihilating
transition to the opposite carrier band.
This process is indirect thermal recombination (nonradiative).
Thermal energy (heat) is released during the process (lattice
vibrations, phonons are produced)
Solid-State Electronics
Chap. 6
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
‹ Recombination via Shallow Levels:
—induced by donor or acceptor sites.
At RT, if an electron is captured at a donor site,
however, it has a high probability of being re-emitted into
the conduction band before completing the recombination
process. Therefore, the probability of recombination via
shallow levels is quite low at RT.
It should be noted that the probability of observing shallowlevel processes increases with decreasing system temperature.
‹ Recombination involving Excitons:
It is possible for an electron and a hole to become bound
together into a hydrogen-atom-like arrangement which
moves as a unit in response to applied forces. This coupled
e-h pair is called an “exciton”. The formation of an exciton
can be viewed as introducing a temporary level into the bandgap slightly
above or below the band edge.
Solid-State Electronics
Chap. 6
7
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Recombination Process
‹ Recombination involving Excitons: Recombination involving excitons is a
very important mechanism at low temperatures and is the major lightproducing mechanism in LED’s.
‹ Auger Recombinations:
In a Auger process, band-to-band recombination
at a bandgap center occurs simultaneously with
the collision between two like carriers. The
energy released by the recombination or trapping
subprocess is transferred during the collision to
the surviving carrier. Subsequently, this high
energetic carrier “thermalizes”-loses energy
through collisions with the semiconductor lattice.
Auger recombination increases with carrier concentration, becoming very
important at high carrier concentration. Therefore, Auger recombination
mmust be considered in treating degenerately doped regions (like solar cell,
junction lasers, and LED’s)
Solid-State Electronics
Chap. 6
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Generation Process
‹ Band-to-Band generation:
‹ R-G center generation:
‹ Photoemission from band gap centers:
Solid-State Electronics
Chap. 6
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Generation Process
‹ Impact-Ionization:
An e-h pair is produced as a result of the
energy released when a highly energetic
carrier collides with the crystal lattice. The
generation of carriers through impact ionization
routinely occurs in the high e-filed regions of
devices and is responsible for the avalanche
breakdown in pn junctions.
Solid-State Electronics
Chap. 6
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Momentum Consideration
‹ In a direct semiconductor where the kvalues of electrons and holes are all
bunched near k = 0, little change is
required for the recombination process
to proceed. The conservation of both
energy and crystal momentum is readily
met by the emission of a photon.
‹ In a indirect semiconductor, there is
a large change in crystal momentum
associated with the recombination
process. The emission of a photon
will conserve energy but cannot
simultaneously conserve momentum.
Thus for band-to-band recombination
to proceed in an indirect semiconductor a phonon must be emitted coincident
with the emission of a photon.
Solid-State Electronics
Chap. 6
11
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess Carrier Generation and Recombination
‹ Low-level injection: the excess carrier concentration is much less than
the thermal equilibrium majority carrier concentration, e.g., for a ntype semiconductor, δn = δp << no.
‹ High-level injection: δn ≈ no or δn >> no
‹ For a p-type material (po >> no) under low-level injection, the excess
carrier will decay from the initial excess concentration with time;
δn(t ) = δn(t = 0)e −t /τ
n0
where τn0 is referred to as the excess minority carrier lifetime (τn0 ∝1/p0)
δn(t )
and the recombination rate of excess carriers Rn’ = Rp’=
τ n0
‹ For a n-type material (no >> po) under low-level injection,
Rn’ = Rp’= δp (t )
− t / τ pn 0
δp (t ) = δp (t = 0)e
τ p0
Solid-State Electronics
Chap. 6
12
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Continuity Equations
‹ Consider a differential volume element in which a 1-D hole flux, Fp+ (#
of holes/cm2-sec), is entering this element at x and is leaving at x+dx.
‹ So the net change in hole concentration per unit time is
+
∂Fp
∂p
p
=−
+ gp −
∂t
∂x
τ pt
----continuity equation for holes
‹ Similarly, the continuity equation for electron flux is
∂Fn−
∂n
n
=−
+ gn −
∂t
∂x
τ nt
Solid-State Electronics
Chap. 6
13
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
‹ If a pulse of excess electrons and holes are created at a particular point
due to an applied E-field, the excess e-s and h+s will tend to drift in
opposite directions. However, any separation of e-s and h+s will induce
an internal E-field and create a force attracting the e-s and h+s back.
‹ The internal E-field will hold the pulses of excess e -s and h+s together,
then the electrons and holes will drift or diffuse together with a single
effective mobility or diffusion coefficient. This is so called “ambipolar
diffusion” or “ambipolar transport”.
‹ Fig. Show the above situation
Solid-State Electronics
Chap. 6
14
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
Solid-State Electronics
Chap. 6
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Ambipolar Transport
Solid-State Electronics
Chap. 6
16
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quasi-Fermi Levels
‹ At thermal-equilibrium, the electron and hole concentrations are
functions of the Fermi level by
 E − E Fi 
 E Fi − E F 
p
n
no = ni exp  F
=
and
exp
o
i

 kT 
 kT 


‹ Under nonequilibrium conditions, excess carriers are created in a
semiconductor, the Fermi energy is strictly no longer defined. We may
define a quasi-Fermi level, EFn, for electrons and a quasi-Fermi level,
EFp, for holes that apply for nonequilibrium. So that the total electron
and hole concentrations are functions of the quasi-Fermi levels.
 E − EFi 
 E Fi − EF 
p
p
n
no + δn = ni exp  Fn
+
=
and
exp
o
i

 kT 
 kT 


Solid-State Electronics
Chap. 6
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quasi-Fermi Levels
‹ For a n-type semiconductor under thermal equilibrium, the band
diagram is
‹ Under low-level injection, excess carriers are created and the quasiFermi level for holes (minority), EFp, is significantly different from EF.
Solid-State Electronics
Chap. 6
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Excess-Carrier Lifetime
‹ An allowed energy state, also called a trap, within the forbidden
bandgap may act as a recombination center, capturing both electrons
and holes with almost equal probability. (it means that the capture
cross sections for electrons and holes are approximately equal)
‹ Acceptor-type trap:
– it is negatively charged when it contains an electron and it is neutrall when it does
not contain an electron.
‹ Donor-type trap:
– it is positively charged when empty and neutral when filled with an electron
Solid-State Electronics
Chap. 6
19
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
‹ Assume that a single recombination center exists at an energy Et within
the bandgap. And there are four basic processes that may occur at this
single trap.
‹ Process 1: electron from the
conduction band captured by an
initially neutral empty trap.
‹ Process 2: electron emission from a
trap into the conduction band.
‹ Process 3: capture of a hole from the
valence band by a trap containing an
electron.
‹ Process 4: emission of a hole from a
neutral trap into the valence band.
Solid-State Electronics
Chap. 6
20
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
‹ In Process 1: the electron capture rate (#/cm3-sec):
Rcn = CnNt(1-fF(Et))n
Cn=constant proportional to electron-capture cross section
Nt = total concentration in the conduction band
n = electron concentration in the conduction band
fF(Et)= Fermi function at the trap energy
‹ For Process 2: the electron emission rate (#/cm3-sec):
Ren = EnNtfF(Et)
En=constant proportional to electron-capture cross section Cn
‹ In thermal equilibrium, Rcn = Ren, using the Boltzmann approximation
for the Fermi function,
 − ( E c − Et ) 
C
En = n 'Cn = N c exp 
‹ In nonequilibrium, excess electrons exist, kT  n
[
]
Rn = Rcn − Ren = Cn N t n (1 − f F ( Et ) − n ' f F ( Et ) )
Solid-State Electronics
Chap. 6
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Shockley-Read-Hall Theory of Recombination
‹ In Process 3 and 4, the net rate at which holes are captured from the
valence band is given by R p = C p N t [pf F ( Et ) − p ' (1 − f F ( Et )) ]
 − ( Et − E v ) 
p ' = N v exp 

kT

‹ In semiconductor, if the trap density is not too large, the excess
electron and hole concentrations are equal and the recombination rates
of electrons and holes are equal.
⇒ f F ( Et ) =
Cn n + C p p '
Cn ( n + n ' ) + C p ( p + p ' )
and Rn = R p =
CnC p N t (np − ni2 )
Cn ( n + n ' ) + C p ( p + p ' )
≡R
‹ In thermal equilibrium, np = ni2 ⇒ Rn = Rp = 0
Solid-State Electronics
Chap. 6
22
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Surface Effects
‹ Surface states are functionally equivalent to R-G centers localized at
the surface of a material. However, the surface states (or interfacial
traps) are typically found to be continuously distributed in energy
throughout the semiconductor bandgap.
Solid-State Electronics
Chap. 6
23
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Surface Recombination Velocity
‹ As the excess concentration at the surface becomes smaller than that in
the bulk, excess carriers from the bulk region diffuse toward the
surface where they recombine, and the surface recombination velocity
increases.
‹ An infinite surface recombination velocity implies that the excess
minority carrier concentration and lifetime are zero.
Solid-State Electronics
Chap. 6
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
‹ 6.14
‹ 6.17
‹ 6.19
Solid-State Electronics
Chap. 6
25
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Chap 7. P-N junction
‹
‹
‹
‹
P-N junction Formation
Fermi Level Alignment
Built-in E-field (cut-in voltage)
Homework
Solid-State Electronics
Chap. 7
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
P-N junction
P-Semiconductor
N-Semiconductor
EC
EC
EFi
EFi
EF
EV
EF
EV
P-Semiconductor
EC
N-Semiconductor
EC
EFi
EF
EV
EFi
EF
EV
Solid-State Electronics
Chap. 7
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
P-N Junction
P-Semiconductor
EC
N-Semiconductor
EC
EFi
EF
EV
EFi
EV
Depletion region
Solid-State Electronics
Chap. 7
EF
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Hetero-Junction
Semiconductor B
Semiconductor A
EC
EC
EFi
EF
EFi
EF
EV
EV
∆EC < 20 meV
∆EC ~0.15 eV
EC
Relaxed Si0.7Ge0.3
Strained Si0.8Ge0.2
bulk Si
Eg = 1.17 eV
Eg ~ 1.08 eV
Eg ~ 1.0 eV
∆EV ~ 0.15 eV
EV
Eg ~ 0.88 eV
∆EV ~ 0.05 eV
Type II Alignment
Type I Alignment
Solid-State Electronics
Chap. 7
Strained Si
4
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quantum Well
Electron
Confinement
∆EC ~ 0.02 eV
relaxed Si0.7Ge0.3
Eg = 1.08 eV
∆EC ~0.18 eV
Strained Si0.3Ge0.7
Strained Si
Eg ~ 0.72 eV
Eg = 0.88 eV
∆EV ~0.34 eV
∆EV ~ 0.48 eV
Hole
Confinement
Solid-State Electronics
Chap. 7
5
Instructor: Pei-Wen Li
Dept. of E. E. NCU
.. .. .. .. .. .. .. ..
Strained Si1-xGex
Relaxed Si1-xGex
Relaxed Si1-xGex
misfit dislocation
misfit dislocation
bulk Si
Solid-State Electronics
Chap. 7
.. .. .. .. .. .. .. ..
Strained Si
bulk Si
6
Instructor: Pei-Wen Li
Dept. of E. E. NCU