Calculating Band Structure
Calculating Band Structure Nearly free electron • Assume plane wave solution for electrons • Weak potential V(x) • Brillouin zone edge
Tight binding method • • • •
Electrons in local atomic states (bound states) Interatomic interactions >> lower potential Unbound states for electrons Energy Gap = difference between bound / unbound states
Crystal Field Splitting • Group theory to determine crystalline symmetry • Crystalline symmetry establishes relevant energy levels • Field splitting of energy levels
However all approaches assume a crystal structures. Bands and energy gaps still exist without the need for crystalline structure. For these systems, Molecular Orbital theory is used.
Free Electron Model • Energy bands consist of a large number of closely spaced energy levels. • Free electron model assumes electrons are free to move within the metal but are confined to the metal by potential barriers. • This model is OK for metals, but does not work for semiconductors since the effects of periodic potential have been ignored.
Kronig-Penny Model • This model takes into account the effect of periodic arrangement of electron energy levels as a function of lattice constant a • As the lattice constant is reduced, there is an overlap of electron wavefunctions that leads to splitting of energy levels consistent with Pauli exclusion principle. A further lowering of the lattice constant causes the energy bands to split again
Energy bands for diamond versus lattice constant.
Formation of Bands
Periodic potential Band gap
Inter-atom interactions Many more states
Conduction / valence bands Free electron model
Conduction band states
Valence band states
Bound states
Conduction / valence bands Conduction band states Lowest Unoccupied Molecular Level (LUMO)
Valence band states Highest Occupied Molecular Orbital (HOMO)
Electrons fill from bottom up Semiconductor = filled valence band
Example band structures Ge
Si
GaAs
Find: Valence bands? Conduction bands? Energy Gap? Highest Occupied Molecular Level (HOMO)? Lowest Unoccupied Molecular Level (LUMO)?
Simple Energy Diagram
A simplified energy band diagram used to describe semiconductors. Shown are the valence and conduction band as indicated by the valence band edge, Ev, and the conduction band edge, Ec. The vacuum level, Evacuum, and the electron affinity, , are also indicated on the figure.
Metals, Insulators and Semiconductors
Possible energy band diagrams of a crystal. Shown are: a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band separated by a large bandgap.
Semiconductors • Filled valence band (valence = 4, 3+5, 2+6) • Insulator at zero temperature Metals Free electrons Valence not 4
Binary system III-V: GaAs, InP, GaN, GaP Binary II-VI: CdTe, ZnS,
Semiconductors Si, Ge Filled p shells 4 valence electrons
Eg Temperature Dependence
Eg Doping Dependence Doping, N, introduces impurity bands that lower the bandgap.
Energy bands in Electric Field
Electrons travel down. Holes travel up.
Energy band diagram in the presence of a uniform electric field. Shown are the upper almost-empty band and the lower almost-filled band. The tilt of the bands is caused by an externally applied electric field.
The effective mass The presence of the periodic potential, due to the atoms in the crystal without the valence electrons, changes the properties of the electrons. Therefore, the mass of the electron differs from the free electron mass, m0. Because of the anisotropy of the effective mass and the presence of multiple equivalent band minima, we define two types of effective mass: 1) the effective mass for density of states calculations and 2) the effective mass for conductivity calculations.
Motion of Electrons and Holes in Bands
Electron excited out of valence band Temperature Light Defect …
Electron in conduction band state Empty state in valence band (Hole = empty state)
Electrons - holes Electron in conduction band NOT localized
Hole in valence band Usually less Mobile (higher effective mass), but not always
Electron – hole pairs in different bands large separation
Region Near Gap In the region near the gap, Local maximum / minimum dE/dk = 0
e(k)
Conduction band
effective mass m* = h2/(d2E/dk2) Electrons Minimum energy Bottom of conduction band Holes Opposite E(k) derivative “Opposite effective charge” Top of valence band
kx
Valence band
General Carrier Concentration Probability of hopping into state n0 = (number of states / energy) * energy distribution
Conduction band
Gap
gc (E) = density of states f (E) = energy distribution Valence band
Density of states The density of states in a semiconductor equals the density per unit volume and energy of the number of solutions to Schrödinger's equation.
Calculation of the number of states with wavenumber less than k
Fermi-surface (3-D) ky
• K-space
Allowed state for k-vector
– Set of allowed k vectors
• Fermi surface – Electrons occupy all kf2 states less than Ef*2m/h – kF ~ wavelength of electron wavefunction
kx 2p/L
Volume in lattice Area of sphere / k states in spheres
4pk F 3 1 3 (2p / L)3
kF 3 6p
2
L3 N
Density of states http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm
Number of states:
kF 3
N 2 6p 2 L3
Density in energy:
Kinetic energy of electron: Density of states / energy: In conduction band, Nc:
Different m* in conduction and valence band
Density of States in 1, 2 and 3D
Probability density functions The distribution or probability density functions describe the probability that particles occupy the available energy levels in a given system. Of particular interest is the probability density function of electrons, called the Fermi function. The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are halfinteger spin particles, which obey the Pauli exclusion principle.
Fermi-Dirac vs other distributions
Maxwell-Boltzmann: Noninteracting particles Bose-Einstein: Bosons
Intrinsic: Ec – Ef = ½ Eg High temperature: Fermi ~ Boltzmann
Carrier Densities The density of occupied states per unit volume and energy, n(E), ), is simply the product of the density of states in the conduction band, gc(E) and the Fermi-Dirac probability function, f(E).
Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy, p(E), equals:
Carrier Densities
Product of density of states and distribution -- defines accessible bands -- within kT of Ef
Carrier Densities Electrons
Holes
Limiting Cases 0 K:
Non-degenerate semiconductors: semiconductors for which the Fermi energy is at least 3kT away from either band edge.
Intrinsic Semiconductor Intrinsic semiconductors are usually non-degenerate
Mass Action Law The product of the electron and hole density equals the square of the intrinsic carrier density for any non-degenerate semiconductor.
The mass action law is a powerful relation which enables to quickly find the hole density if the electron density is known or vice versa
Doped Semiconductor Add alternative element for electron/holes
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Si = Si = Si = Si =
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Si = Si = Si = Si = =
Si = Si = Si = Si =
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B valence=3
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P valence = 5
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Si valence = 4
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Si = Si -- B = Si =
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Si = Si = P = Si =
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Si = Si = Si = Si =
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Si = Si = Si = Si = =
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Phosphorous n-doped Electron added to conduction band
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All electron paired Insulator at T=0
Si = Si = Si = Si = =
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Pure Si
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Si = Si = Si = Si =
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Si = Si = Si = Si =
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Si = Si = Si = Si = e-
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Si = Si = Si = Si =
Boron p-doped Positive hole added to conduction band
Hole
Dopant Energy levels P 0.046eV
As 0.054eV
Energy required to donate electron
Easily ionized = easily donate electrons Si Eg=1.2eV Cu 0.53eV Cu 0.40eV
Cu 0.24eV
Au 0.54eV
Large energy bad. Add scattering Donate no carriers
Au 0. 35eV Au 0. 29eV
B 0.044eV
Energy required to donate hole
Carrier concentration in thermal equilibrium • Carrier concentration vs. inverse temperature Thermally activated Intrinsic carriers
Region of Functional device
ne N(carriers) = N(dopants) Activation of dopants
1/T(K)
Dopants and Fermi Level 2 2 kF 3 kF • Free electron metal: ne , e 3p 2 F 2m
• Intrinsic semiconductor – –
n(electrons) = n(holes) Fermi energy = middle
Ec Ef Ev
• n-doped material – –
n(electrons) >> n(holes) Fermi level near conduction band
• p-doped materials – –
n(electrons) >> n(holes) Fermi level near conduction band
Ec Ef Ev Ec Ef Ev
Fermi Energy is not material specific but depends on doping level and type
Mobility and Dopants • Dopants destroy periodicity
e
– Scattering, lower mobility
10000 Mobility (cm2/V-s) 1000
e
GaAs
e h
Si
100 1E14
1E15
1E16
1E17
Dopant Concentration (cm-3)
1E18
Doping / Implantation
Implants: (1)NBL (isolation) (2) Deep n (Collector) (3) Base well (p) (4) Emitter (n) (5) Base contact
• Simple bipolar transistor = 5 implants • Complicated CMOS circuit >12
Tight binding method • • • •
Electrons in local atomic states (bound states) Interatomic interactions >> lower potential Unbound states for electrons Energy Gap = difference between bound / unbound states
Crystal Field Splitting • Group theory to determine crystalline symmetry • Crystalline symmetry establishes relevant energy levels • Field splitting of energy levels
However all approaches assume a crystal structures. Bands and energy gaps still exist without the need for crystalline structure. For these systems, Molecular Orbital theory is used.
Free Electron Model • Energy bands consist of a large number of closely spaced energy levels. • Free electron model assumes electrons are free to move within the metal but are confined to the metal by potential barriers. • This model is OK for metals, but does not work for semiconductors since the effects of periodic potential have been ignored.
Kronig-Penny Model • This model takes into account the effect of periodic arrangement of electron energy levels as a function of lattice constant a • As the lattice constant is reduced, there is an overlap of electron wavefunctions that leads to splitting of energy levels consistent with Pauli exclusion principle. A further lowering of the lattice constant causes the energy bands to split again
Energy bands for diamond versus lattice constant.
Formation of Bands
Periodic potential Band gap
Inter-atom interactions Many more states
Conduction / valence bands Free electron model
Conduction band states
Valence band states
Bound states
Conduction / valence bands Conduction band states Lowest Unoccupied Molecular Level (LUMO)
Valence band states Highest Occupied Molecular Orbital (HOMO)
Electrons fill from bottom up Semiconductor = filled valence band
Example band structures Ge
Si
GaAs
Find: Valence bands? Conduction bands? Energy Gap? Highest Occupied Molecular Level (HOMO)? Lowest Unoccupied Molecular Level (LUMO)?
Simple Energy Diagram
A simplified energy band diagram used to describe semiconductors. Shown are the valence and conduction band as indicated by the valence band edge, Ev, and the conduction band edge, Ec. The vacuum level, Evacuum, and the electron affinity, , are also indicated on the figure.
Metals, Insulators and Semiconductors
Possible energy band diagrams of a crystal. Shown are: a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band separated by a large bandgap.
Semiconductors • Filled valence band (valence = 4, 3+5, 2+6) • Insulator at zero temperature Metals Free electrons Valence not 4
Binary system III-V: GaAs, InP, GaN, GaP Binary II-VI: CdTe, ZnS,
Semiconductors Si, Ge Filled p shells 4 valence electrons
Eg Temperature Dependence
Eg Doping Dependence Doping, N, introduces impurity bands that lower the bandgap.
Energy bands in Electric Field
Electrons travel down. Holes travel up.
Energy band diagram in the presence of a uniform electric field. Shown are the upper almost-empty band and the lower almost-filled band. The tilt of the bands is caused by an externally applied electric field.
The effective mass The presence of the periodic potential, due to the atoms in the crystal without the valence electrons, changes the properties of the electrons. Therefore, the mass of the electron differs from the free electron mass, m0. Because of the anisotropy of the effective mass and the presence of multiple equivalent band minima, we define two types of effective mass: 1) the effective mass for density of states calculations and 2) the effective mass for conductivity calculations.
Motion of Electrons and Holes in Bands
Electron excited out of valence band Temperature Light Defect …
Electron in conduction band state Empty state in valence band (Hole = empty state)
Electrons - holes Electron in conduction band NOT localized
Hole in valence band Usually less Mobile (higher effective mass), but not always
Electron – hole pairs in different bands large separation
Region Near Gap In the region near the gap, Local maximum / minimum dE/dk = 0
e(k)
Conduction band
effective mass m* = h2/(d2E/dk2) Electrons Minimum energy Bottom of conduction band Holes Opposite E(k) derivative “Opposite effective charge” Top of valence band
kx
Valence band
General Carrier Concentration Probability of hopping into state n0 = (number of states / energy) * energy distribution
Conduction band
Gap
gc (E) = density of states f (E) = energy distribution Valence band
Density of states The density of states in a semiconductor equals the density per unit volume and energy of the number of solutions to Schrödinger's equation.
Calculation of the number of states with wavenumber less than k
Fermi-surface (3-D) ky
• K-space
Allowed state for k-vector
– Set of allowed k vectors
• Fermi surface – Electrons occupy all kf2 states less than Ef*2m/h – kF ~ wavelength of electron wavefunction
kx 2p/L
Volume in lattice Area of sphere / k states in spheres
4pk F 3 1 3 (2p / L)3
kF 3 6p
2
L3 N
Density of states http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm
Number of states:
kF 3
N 2 6p 2 L3
Density in energy:
Kinetic energy of electron: Density of states / energy: In conduction band, Nc:
Different m* in conduction and valence band
Density of States in 1, 2 and 3D
Probability density functions The distribution or probability density functions describe the probability that particles occupy the available energy levels in a given system. Of particular interest is the probability density function of electrons, called the Fermi function. The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are halfinteger spin particles, which obey the Pauli exclusion principle.
Fermi-Dirac vs other distributions
Maxwell-Boltzmann: Noninteracting particles Bose-Einstein: Bosons
Intrinsic: Ec – Ef = ½ Eg High temperature: Fermi ~ Boltzmann
Carrier Densities The density of occupied states per unit volume and energy, n(E), ), is simply the product of the density of states in the conduction band, gc(E) and the Fermi-Dirac probability function, f(E).
Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy, p(E), equals:
Carrier Densities
Product of density of states and distribution -- defines accessible bands -- within kT of Ef
Carrier Densities Electrons
Holes
Limiting Cases 0 K:
Non-degenerate semiconductors: semiconductors for which the Fermi energy is at least 3kT away from either band edge.
Intrinsic Semiconductor Intrinsic semiconductors are usually non-degenerate
Mass Action Law The product of the electron and hole density equals the square of the intrinsic carrier density for any non-degenerate semiconductor.
The mass action law is a powerful relation which enables to quickly find the hole density if the electron density is known or vice versa
Doped Semiconductor Add alternative element for electron/holes
=
=
=
=
=
=
=
=
=
=
=
=
=
Si = Si = Si = Si =
=
=
=
=
=
Si = Si = Si = Si = =
Si = Si = Si = Si =
=
=
=
B valence=3
=
P valence = 5
=
Si valence = 4
=
=
=
=
=
=
=
=
=
=
Si = Si -- B = Si =
=
Si = Si = P = Si =
=
Si = Si = Si = Si =
=
=
Si = Si = Si = Si = =
=
=
=
=
Phosphorous n-doped Electron added to conduction band
=
=
=
=
=
All electron paired Insulator at T=0
Si = Si = Si = Si = =
=
=
=
=
Pure Si
=
=
Si = Si = Si = Si =
=
=
=
Si = Si = Si = Si =
=
Si = Si = Si = Si = e-
=
Si = Si = Si = Si =
Boron p-doped Positive hole added to conduction band
Hole
Dopant Energy levels P 0.046eV
As 0.054eV
Energy required to donate electron
Easily ionized = easily donate electrons Si Eg=1.2eV Cu 0.53eV Cu 0.40eV
Cu 0.24eV
Au 0.54eV
Large energy bad. Add scattering Donate no carriers
Au 0. 35eV Au 0. 29eV
B 0.044eV
Energy required to donate hole
Carrier concentration in thermal equilibrium • Carrier concentration vs. inverse temperature Thermally activated Intrinsic carriers
Region of Functional device
ne N(carriers) = N(dopants) Activation of dopants
1/T(K)
Dopants and Fermi Level 2 2 kF 3 kF • Free electron metal: ne , e 3p 2 F 2m
• Intrinsic semiconductor – –
n(electrons) = n(holes) Fermi energy = middle
Ec Ef Ev
• n-doped material – –
n(electrons) >> n(holes) Fermi level near conduction band
• p-doped materials – –
n(electrons) >> n(holes) Fermi level near conduction band
Ec Ef Ev Ec Ef Ev
Fermi Energy is not material specific but depends on doping level and type
Mobility and Dopants • Dopants destroy periodicity
e
– Scattering, lower mobility
10000 Mobility (cm2/V-s) 1000
e
GaAs
e h
Si
100 1E14
1E15
1E16
1E17
Dopant Concentration (cm-3)
1E18
Doping / Implantation
Implants: (1)NBL (isolation) (2) Deep n (Collector) (3) Base well (p) (4) Emitter (n) (5) Base contact
• Simple bipolar transistor = 5 implants • Complicated CMOS circuit >12
