Solid State Electronics
Chpt 3. Energy Bands and Charge
Carriers in Semiconductors
서울 시립대학교 공과대학
전자전기컴퓨터공학부
이주한
E-mail : jhl@uos.ac.kr
태초(太初)에 하나님이 천지(天地)를 창조(創造)하시니라. 땅이 혼돈(混沌)하고 공허(空虛)하며 흑암(黑暗)이
깊음 위에 있고 하나님의 신(神)은 수면(水面)에 운행(運行)하시니라. 하나님이 가라사대 빛이 있으라 하시매
빛이 있었고 그 빛이 하나님의 보시기에 좋았더라.
---창세기 1:1 ~4 중에서
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Solid State Electronics
Objectives
• Understand conduction and valence energy bands, and how band gaps are
formed
• Appreciate the idea of doping in semiconductors
• Use the density of states and Fermi Dirac statistics to calculate carrier
concentrations
• Calculate drift currents in an electric field in terms of carrier mobility
• Discuss the idea of “effective” mass
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Solid State Electronics
3.1 Bonding Forces and Energy Bands in Solids
The influence of neighboring atoms on the energy levels of a particular atom can be treated as a small
perturbation, giving rise to shifting and splitting of energy states into energy bands
Bonding Forces in Solids
• Ionic Bonding
- NaCl
- Each Na atom is surrounded by six nearest neighbor Cl
atoms
- Na : [Ne]3s1
- Cl : [Ne] 3s23p5
- Na+ ion has a net positive charge
- Cl- ion has a net negative charge
- Coulombic forces pull the lattice together
- All electrons are tightly bound
- Good Isolator
Figure 3–1 Different types of chemical bonding in
solids: (a) an example of ionic bonding in NaCl
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Solid State Electronics
• Metal Bonding
- In a metal atom the outer electronic shell is only partially filled
- The metal is made up of ions with closed shells immersed in a sea of free electrons
- An interaction between positive ion cores and the surrounding free electrons induce the forces
holding the lattice
- Electrons are free to move around the crystal under the influence of an electric field
• Covalent Bonding
- Each atom in the Ge, Si, or C diamond lattice is surrounded by four nearest neighbors,
each with four electrons in the outer orbit
-The bonding forces arise from quantum mechanical interaction between the shared electrons.
- Each electron pair constitutes a covalent bonding
- Two electrons must have opposite spin to satisfy
the Pauli’s exclusion principle
- No Free electrons
- Insulators
Figure 3–1 (b) covalent bonding in the Si crystal,
viewed along a <100> direction (see also Figs. 1–8
and 1–9).
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Solid State Electronics
Energy Bands
- As isolated atoms are brought together to form a solid, various interactions occur between
neighboring atoms
- The coulombic potential wells of two atoms close to each other
- By solving the Schrodinger Eq, the composite two-electron wavefunctions are linear combinations
of the individual atomic orbitals (LCAO)
- Two electrons in the bonding orbital have opposite spins, while those in the anti-bonding state have
parallel spins
-In the region between the two nuclei the Coulombic potential energy V(r) is lowered because an
electron would be attracted by two nuclei
- The original isolated atomic energy level would be split into two, a lower bonding energy level and a
higher antibonding energy.
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Solid State Electronics
Figure 3–2 Linear combinations of atomic orbitals (LCAO): The LCAO when two atoms
are brought together leads to two distinct “normal” modes—a higher energy antibonding
orbital, and a lower energy bonding orbital. Note that the electron probability density is
high in the region between the ion cores (covalent “bond”), leading to lowering of the
bonding energy level and the cohesion of the crystal. If instead of two atoms, one brings
together N atoms, there will be N distinct LCAO, and N closely spaced energy levels in a
band.
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Solid State Electronics
- As atoms are brought together, the application of the Pauli’s exclusion principle becomes important
- As the spacing between the two atoms becomes smaller, the electron wavefunctions begin to
overlap
- Due to the exclusion principle, there must one electron per level after energy level splitting
- In solid, many atoms are brought together, so that the split energy levels form essentially
continuous bands of energies.
-Valence Band: Lower band
-Conduction Band: Upper Band
-At 0 K, Valence band is filled, whereas Conduction band is completely empty.
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Solid State Electronics
Figure 3–3 Energy levels in Si as a function of interatomic spacing. The core levels (n
= 1, 2) in Si are completely filled with electrons. At the actual atomic spacing of the
crystal, the 2N electrons in the 3s subshell and the 2N electrons in the 3p subshell
undergo sp3 hybridization, and all end up in the lower 4N states (valence band), while
the higher-lying 4N states (conduction band) are empty, separated by a band gap.
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Solid State Electronics
Metals, Semiconductors, and Insulators
- Each solid has its own characteristic energy band structure
- The variation in band structure is responsible for the wide range of electrical characteristics
- Semiconductor materials at 0 K have basically the same structure as insulators
- Filled valence band separated from an empty conduction band by a bandgap
• Difference between Insulator and Semiconductor
- Size of Bandgap Eg
- Much smaller in Semiconductor
- Si : 1.1 eV
- Diamond : 5 eV
-At room temperature a semiconductor with a 1-eV band gap will have a significant number of electrons
excited thermally across the energy gap into the conduction band
• Metal
- Bands either overlap or are only partially filled
- The electrons and empty energy states are intermixed within the bands
so that electrons move freely under the influence of an electric field.
University of Seoul, Prof. Ju Han Lee
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Solid State Electronics
Figure 3–4
University of Seoul, Prof. Ju Han Lee
Typical band structures at 0 K.
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Solid State Electronics
Direct and Indirect Semicondcutors
- In a band structure calculation a single electron is assumed to travel through a perfectly periodic
lattice
- The space-dependent wave function for the electron is
k: wavevector
- U(kx,x) : Bloch Function
• E-k Diagram
- Allowed values of energy can be plotted vs. the propagation constant k
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Solid State Electronics
Figure 3–5 Direct and indirect electron transitions in semiconductors: (a) direct
transition with accompanying photon emission; (b) indirect transition via a defect level.
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Solid State Electronics
Figure 3–6 Variation of direct and indirect conduction bands in AIGaAs as a function of
composition: (a) the (E, k) diagram for GaAs, showing three minima in the conduction
band; (b) AIAs band diagram.
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Solid State Electronics
3.2 Charge Carriers in Semiconductors
- Current in Metal : Free electrons can move as a group under the influence of an electric field
- Current in Semiconductors : The increase in conduction band electrons by thermal excitation
and the empty states left in the valence band as the temperature is raised
Electrons and Holes
- As the temperature of a semiconductor is raised from 0 K, some electrons in the valence band
receive enough thermal energy to be excited across the bandgap to the conduction band
- As a result, some electrons in an otherwise empty conduction band and some unoccupied states in
the valence band
Empty State: Hole
Electron-hole pairs : EHPs
Figure 3–7
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Electron–hole pairs in a semiconductor.
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Solid State Electronics
Effective Mass
- The electrons in a crystal are not completely free but instead interact with the periodic potential
of the lattice
- Their wave-particle motion can not be expected to be the same as for electrons in free space
-In applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered
values of particle mass
Effective Mass of an electron
- The Curvature of d2E/dk2 is positive at the conduction band minima
- The Curvature of d2E/dk2 is negative at the valence band maxima
- Negative Effective Mass : electrons near the top of the valence band
- Valence band electrons with negative charge and negative mass move in an electric field in the same
direction as holes with positive charge and positive mass
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Solid State Electronics
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Solid State Electronics
Figure 3–10 Realistic band structures in semiconductors: (a) conduction
and valence bands in Si and GaAs along [111] and [100]; (b) ellipsoidal
constant energy surface for Si, near the six conduction band minima along
the X-directions. (From Chelikowsky and Cohen, Phys. Rev. B14, 556, 1976.)
k = 0 at  point
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Solid State Electronics
Intrinsic Material
- A perfect semiconductor crystal with no impurities
- No charge carriers at 0 K
- At higher temperature EHPs are generated
- EHPs are the only charge carriers
Intrinsic concentration
Figure 3–11
Electron–hole pairs in the covalent bonding model of the Si crystal.
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Solid State Electronics
• Steady State (No time variation)
- Steady State : A system that is in a steady state remains constant over time, but that constant
state requires continual work
- Recombination occurs when an electron in the conduction band makes a transition (direct or
indirect) to an any empty state (hole) in the valence band, and thus annihilating the pair
- Recombination of EHPs at the same rate at which they are generated
Generation
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Recombination
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Solid State Electronics
• At equilibrium
Recombination Rate
Generation Rate
- Equilibrium : A system at equilibrium, is stable over time, but no energy or work is required to
maintain that condition
• Rate of Recombination
n0, p0: Equilibrium Concentration
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Solid State Electronics
Extrinsic Material
- It is possible to create carriers in semiconductors by purposely introducing impurities into the
crystal.
- Doping process : Implantation or Diffusion
- By doping a crystal can be altered so that its has a predominance of either electrons or holes
-This sort of material is called “Extrinsic
• n-type
- Create electrons
-Impurities
- Column V : P, As, Sb
- Donor
• p-type
- Create holes
-Impurities
- Column III : B, Al, Ga, In
- Acceptor
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Solid State Electronics
Figure 3–12 Energy band model and chemical bond model of dopants in
semiconductors: (a) donation of electrons from donor level to conduction band; (b)
acceptance of valence band electrons by an acceptor level, and the resulting creation of
holes; (c) donor and acceptor atoms in the covalent bonding model of a Si crystal.
University of Seoul, Prof. Ju Han Lee
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Solid State Electronics
- When impurities or lattice defects are introduced into an otherwise perfect crystal, additional
levels are created in the energy band structure.
•-n-type
- Column V (P, As, Sb)
- Donor Level
- n0 >> ni, p0
- Majority Carrier : Electron
- Minority Carrier: Hole
• p-type
- Column III (B, Al, Ga, In)
- Acceptor Level
-p0 >> ni, n0
- Majority Carrier : Hole
- Minority Carrier: Electron
- In Si, the usual donor and acceptor levels lie about 0.03 ~ 0.06 eV from a band edge
University of Seoul, Prof. Ju Han Lee
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Solid State Electronics
3.3 Carrier Concentrations
- In calculating semiconductor electrical properties and analyzing device behavior,
it is necessary to know the number of charge carriers per cm 3 in the material
- The distribution of carriers over the available energy states
Fermi Level
- Electrons in solids obey Fermi-Dirac statistics
- Distribution of electrons over a range of allowed energy levels at thermal equilibrium
k : Boltzmann’s constant 8.62 x 10-5 eV/K = 1.38 x 10-23 J/K
-f(E) : Fermi-Dirac distribution function -> the probability that an available energy state at E will be
occupied by an electron at absolute temperature T.
- EF : Fermi Level
- an energy state at the Fermi Level has a probability of ½ of being occupied by an electron
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Solid State Electronics
Figure 3–14
The Fermi–Dirac distribution function. (Derivation in Appendix V)
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Solid State Electronics
- f(E) is the probability of occupancy of an available state at E
- If there is no available state , there is no possibility of finding an electron.
- In the bandgap there is no available state -> no electron
- Intrinsic Semiconductor : EF must lie at the middle of the bandgap
- n-type : EF moves closer to Ec (Conduction band edge)
- p-type: EF moves closer to Ev (Valence band edge)
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Solid State Electronics
Figure 3–15
The Fermi distribution function applied to semiconductors: (a) intrinsic
material; (b) n-type material; (c) p-type material.
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Solid State Electronics
Electron and Hole Concentrations at Equilibrium
- Concentration of electrons in the conduction band
N(E)dE : Density of States (cm-3) in the energy range dE
0 represents equilibrium
-N(E) can be calculated by using quantum mechanics and Pauli exclusion principle
- N(E) is proportional to E1/2
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Solid State Electronics
Figure 3–16 Schematic band diagram, density of states, Fermi–Dirac distribution, and
the carrier concentrations for (a) intrinsic, (b) n-type, and (c) p-type semiconductors at
thermal equilibrium.
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Solid State Electronics
- Effective Density of States Nc located at the conduction band edge Ec
Boltzmann Distribution
- Assuming Fermi Level EF lies at least several kT below the conduction band edg
𝑓 𝐸𝑐 =
1
1+ 𝑒 (𝐸 𝑐 − 𝐸 𝐹 )/𝑘𝑇
≃ 𝑒 −(𝐸𝑐 − 𝐸𝐹 )/𝑘𝑇
- kT at room temperature is only 0.0259 eV
- At this condition, the concentration of electrons in the conduction band is
𝑛0 = 𝑁𝑐 𝑒 −(𝐸𝑐 − 𝐸𝐹 )/𝑘𝑇
- Effective Density of States Nc
2π𝑚 𝑛 ∗ 𝑘𝑇 3
𝑁𝑐 = 2(
ℎ2
)2
- The electron concentration increases as EF moves closer to the conduction band
- mn* : Density of States Effective Mass for electrons
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Solid State Electronics
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Solid State Electronics
Hole Concentration
- By similar arguments the concentration of holes in the valence band is
- The probability of finding an empty state at Ev is
- For EF larger than Ev by several kT, the concentration of holes in the valence band is
- The effective density of states in the valence band reduced to the band edge is
Mp* : Density of States Effective Mass for holes
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Solid State Electronics
For intrinsic material
- EF lies at some intrinsic level Ei near the middle of the bandgap
- The intrinsic electron and hole concentration are
- The intrinsic electron and hole concentration are equal, n i=pi
For extrinsic material
- The product of n0 and p0 at equilibrium is a constant for a particular material and temperature,
even if the doping is varied
- The constant product of electron and hole concentrations
Mass Action Law
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Carrier concentration
- Relation between doping and intrinsic materials
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Solid State Electronics
Figure 3–17 Intrinsic
carrier concentration for
Ge, Si, and GaAs as a
function of inverse
temperature. The room
temperature values are
marked for reference.
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Solid State Electronics
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Solid State Electronics
3.4 Drift of Carriers in Electric Fields
- Knowledge of carrier concentrations in a solid is necessary for calculating current flow in the presence
of an electric field
- We must take into account the collisions of the charge carriers with the lattice and with the impurities
Conductivity and Mobility
- Chare carries in a solid are in constant motion, even at thermal equilibrium.
Figure 3–20a
Random thermal motion of an electron in a solid.
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Solid State Electronics
Figure 3–20b
Well-directed drift velocity with an applied electric field.
- If an electric field Ex is applied in the
x-direction, each electron experiences
a net force –qEx from the field
- This force may be insufficient to alter
appreciably the random path of an
individual electron
- The effect when averaged over all the
electrons is a net motion of the group in
the –x direction
- n electrons /cm3
Px : Momentum
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Solid State Electronics
- The net acceleration is balanced in steady-state by the decelerations of the collision processes
- For steady state the electrons have on the average a constant net velocity in the negative x-direction
- The individual electrons move in many directions by thermal motion during a given time period, but
the net drift of an average electron responds to the electric field
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Solid State Electronics
• Drift Current
- The number of electrons crossing a unit area per unit time (n<vx>) multiplied by the charge on the
electron (-q)
𝐽𝑥 = −𝑞𝑛 < 𝑣𝑥 > (3-37)
𝑎𝑚𝑝𝑒𝑟𝑒
𝑐𝑜𝑢𝑙𝑜𝑚𝑏 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑐𝑚
=
∙
∙
𝑐𝑚2
𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛
𝑐𝑚3
𝑠
Conductivity
Mobility
mn* : Conductivity effective Mass
Different from Density of States Effective Mass
for electrons
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Total Current due to electrons and holes
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Hall Effect
- If a magnetic field is applied perpendicular to the direction in which holes drifts in a p-type bar,
the path of the holes tends to be deflected.
Hall Coefficient
Hall Concentration
𝐼𝑥
(𝑤𝑡
)ℬ𝑧
1
𝐽𝑥 ℬ𝑧
𝐼𝑥 ℬ𝑧
𝑝0 =
=
=
=
𝑉𝐴𝐵
𝑞𝑅𝐻
𝑞ℰ𝑦
𝑞𝑡𝑉𝐴𝐵
𝑞( 𝑤 )
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Figure 3–25
The Hall effect.
Solid State Electronics
Invariance of the Fermi Level at Equilibrium
- At thermal equilibrium, no current, and therefore not net charge transport
- no net energy transfer
-There is no discontinuity in the equilibrium Fermi Level.
-No Gradient
𝑑𝐸𝐹
𝑑𝑥
=0
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Solid State Electronics
Figure 3–26 Two materials in intimate contact at equilibrium. Since the net motion of electrons is
zero, the equilibrium Fermi level must be constant throughout.
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Solid State Electronics
Homework
Problems 3.3, 3.8, 3.11, 3.24
Self Quiz: Question : Q1, Q3, Q5, Q6
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Solid State Electronics