Low-dimensional systems and nanostructuresszczytko/LDSN/1_LDSN_2015_Bloch.pdflow-dimensional systems...
69
Low-dimensional systems and nanostructures Faculty of Physics, University of Warsaw [email protected] 1100-4INZ`LDSN
Transcript of Low-dimensional systems and nanostructuresszczytko/LDSN/1_LDSN_2015_Bloch.pdflow-dimensional systems...
Low-dimensional systems and nanostructures
Faculty of Physics, University of Warsaw
1100-4INZ`LDSN
Summary of the lecture
2015-11-27 2
1. Introduction β semiconductor heterostructures (revison of solid state physics )2. Nanotechnology 3. Quantum wells (1)4. Quantum wells (2)5. Quantum dots, Quantum wells in 1D, 2D and 3D6. Optical transitions in nanostructures7. Work on the article about quantum dots
[TEST]8. Carriers in heterostructures9. Tunneling transport10. Quantized conductance 11. Work on the article about the tunneling or conductance12. Electric field in low-dimensional systems13. Magnetic field in low-dimensional systems14. Electric and magnetic fields in low-dimensional systems15. Revision
[Final TEST]
Summary of the lecture
2015-11-27 3
1. Introduction β semiconductor heterostructuresRevision of solid state physics: Born-Oppenheimer approximation, Hartree-Fock method and one electron Hamiltonian, periodic potential, Bloch states, band structure, effective mass.
2. Nanotechnology Revision of solid state physics: tight-binding approximation, Linear Combination of Atomic Orbitals (LCAO).Nanotechnology. Semiconductor heterostructures. Technology of low dimensional structures. Bandgap engineering: straddling, staggered and broken gap. Valence band offset.
3. Quantum wells (1)Infinite square quantum well. Finite square quantum well. Quantum well in heterostructures: finite square well with different effective masses in the well and barriers.
4. Quantum wells (2)Harmonic potential (parabolic well). Triangular potential. Wentzel β Krammers β Brillouin (WKB) method.Band structure in 3D, 2D. Coulomb potential in 2D
Summary of the lecture
2015-11-27 4
5. Quantum dots, Quantum wells in 1D, 2D and 3DQuantum wells in 1D, 2D and 3D. Quantum wires and quantum dots. Bottom-up approach for low-dimensional systems and nanostructures. Energy gap as a function of the well width.
6. Optical transitions in nanostructuresTime-dependent perturbation theory, Fermi golden rule, interband and intraband transitions in semiconductor heterostructures
7. Work on the article about quantum dotsStudents have to read the article (Phys. Rev. Lett., Nature, Science, etc.) and answer questions. Discussion.
8. Carriers in heterostructuresDensity of states of low dimensional systems. Doping of semiconductors. Heterojunction, p-n junction, metal-semiconductor junction, Schotky barrier
Summary of the lecture
2015-11-27 5
9. Tunneling transportContinuity equation. Potential step. Tunneling through the barrier. Transfer matrix approach. Resonant tunneling. Quantum unit of conductance.
10. Quantized conductance Quantized conductance. Coulomb blockade, one-electron transistor.
11. Work on the article about the tunneling or conductanceStudents have to read the article (Phys. Rev. Lett., Nature, Science, etc.) and answer questions. Discussion.
12. Electric field in low-dimensional systemsScalar and vector potentials. Carriers in electric field: scalar and vector potential in Schrodinger equation. Schrodinger equation with uniform electric field. Local density of states. Franz-Kieldysh effect.
Summary of the lecture
2015-11-27 6
13. Magnetic field in low-dimensional systemsCarriers in magnetic field. Schrodinger equation with uniform magnetic field β symmetric gauge, Landau gauge. Landau levels, degeneracy of Landau levels.
14. Electric and magnetic fields in low-dimensional systemsSchrodinger equation with uniform electric and magnetic field. Hall effect. Shubnikov-de Haas effect. Quantum Hall effect. Fractional Quantum Hall Effect. Hofstadter butterfly. Fock-Darvinspectra
15. Revision Revision and preparing for the exam.
Summary of the exercises
2015-11-27 7
1. Introduction β semiconductor heterostructuresSchrodinger equation. Wave packet, Gaussian wavepacket .
2. Nanotechnology Tight-binding approximation: graphene bandctructure.
3. Quantum wells (1)Infinite square quantum well. Finite square quantum well. Finite square well with different effective masses in the well and barriers.
4. Quantum wells (2)Harmonic potential (parabolic well). Triangular potential. Wentzel β Krammers β Brillouin (WKB) method.
5. Double quantum wells. Quantum dots.Double quantum wells. Quantum dots (2D and 3D harmonic potential)
Summary of the exercises
2015-11-27 8
6. Optical transitions in nanostructuresInterband and intraband transitions in semiconductor heterostructures. Continuity equation.
7. Carriers in heterostructures (1)Transfer matrix approach. Potential step.
8. Carriers in heterostructures (2)Tunneling through the barrier.
9. Resonant tunnelingResonant tunneling.
10. Quantized conductance Quantized conductance. Coulomb blockade.
11. Local density of states Local density of states.
Summary of the exercises
2015-11-27 9
12. Electric field in low-dimensional systemsCarriers in electric field: scalar and vector potential in Schrodinger equation.
13. Magnetic field in low-dimensional systemsSchrodinger equation with uniform magnetic field β symmetric gauge, Landau gauge. Landau levels, degeneracy of Landau levels.
14. Electric and magnetic fields in low-dimensional systemsSchrodinger equation with uniform electric and magnetic field. Conductivity and resistivity tensors
15. Hall effect. Fock-Darvin spectrumHall effect. Fock-Darvin spectrum.
Assessment criteria:
2015-11-27 10
HomeworksDiscussion of scientific papersTests to check the effective use of the skills acquired during the lectureExam: final test and oral exam
NANO era
2015-11-27 11
NANO era
2015-11-27 12
NANO era
2015-11-27 13
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
Epoka NANO
2015-11-27 14
Motoryzacja (Hummer H2 sport utility truck) BudownictwoSamoczyszczΔ cy siΔ beton
Sport
Ubrania (Nano-Tex)Kosmetyki
www.sts.utexas.edu/projects/nanomodules/
AGDSamoczyszczΔ ca siΔ lodΓ³wka Samsung
Nano SilverSeal
iPod Nano
ElektronikaWyΕwietlacze OLED
NANO era
2015-11-27 15
http://www.haruyama.co.jp/
Antiviral Business Suits Fight H1N1 Swine Flu With Science & Style
The $585 suits that went on sale today (October 8, 2009) are treated with Titanium Dioxide, a chemical compound commonly used in cosmetics and toothpaste. According to company spokes-person Junko Hirohata, TiO2 has photocatalytic properties, meaning that it when exposed to light it breaks down organic materials.
NANO era
2015-11-27 16
Epoka NANO
2015-11-27 17
NANO era
2015-11-27 18
Chemistry Biology
Physics
NANO
NANO era
2015-11-27 19
NANO
Material Engineering Electronics
Medycine
Medical diagnostics
Pharmacy
Fundamental research
Technologia chemiczna
Energetics
Chemistry Biology
Physics
The uncertainty principle
2015-11-27 20
Band theory of solids
2015-11-27 21
conduction
valence
metal semiconductor isolator
Jak zobaczyΔ przerwΔ?
EN
ER
GIA
EL
EK
TR
ON
ΓW
conduction
conduction
valence valence
Band theory of solids
2015-11-27 22
HOMO
LUMO
for molecules:
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band2.html
small distance - bands large distance - levels
Band theory of solids
2015-11-27 23
H. I
bac
h
H. Ibach
J. G
inte
r, H
. Ib
ach
MaΕa odlegΕoΕΔ miΔdzy atomamipasma
DuΕΌa odlegΕoΕΔ miΔdzy atomamipoziomy
Molecules
Strong chemical bond when:β’ Large value of overlap integral S and porportional to it HAB.β’ Small difference between atomic orbitals Eat, A, Eat,B
Eat,B
Eat,A
e1
e2
heteronuclear diatomic molecules e.g. CO, NO, HCl, HF
HAA β Eat,A HBB β Eat,B
Assuming Eat,A < Eat,B
AatBat
BatAB
Bat
AatBat
AatAB
Aat
EE
SEHE
EE
SEHE
,,
2
,
,2
,,
2
,
,1
)(
)(
e
e
Band theory of solids
2015-11-27 25
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
Band theory of solids
2015-11-27 26
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
Theoretical description of condensed matter
Born β Oppenheimer approximation
Max Born
(1882-1970)
Jacob R. Oppenheimer
(1904-1967)
2015-11-27 27
Theoretical description of condensed matter
Full non-relativistic Hamiltonian of the system of nuclei and electrons:
π» Τ¦π, π Ξ¨ Τ¦π, π = πΈΞ¨ Τ¦π, π
π» Τ¦π, π
= ββ2
2π
π
π»π2 β
π
β2
2πππ»π2 β β
1
4πν0
π,π
πππ2
Τ¦ππ β π π+
+1
4πν0
π<πΎ
ππππΎπ2
π π β π πΎ+
1
4πν0
π<π
π2
Τ¦ππ β Τ¦ππ=
= ππ + ππ + π Τ¦π, π + ππ Τ¦π + πΊ π
Electron and nuclear (ions) subsystems coordinates are intermixed, separation of electronic and nuclear variables is impossible
Assumption: motion of atomic nuclei and electrons in a molecule can be separatedBorn-Oppenheimer adiabatic approximation
2015-11-27 28
Born β Oppenheimer approximation
LCAO method
11/27/2015 29
The solution of the equation of electron states requires numerical methods
π»ππ Τ¦π, π Ξ¨πππ Τ¦π, π = ππ + π Τ¦π, π + ππ Τ¦π Ξ¨ππ
π Τ¦π, π = πΈπππ π Ξ¨ππ
π Τ¦π, π
One of methods: LCAO-MO with Hartree-Fock approximation β self-consistent field method (iterative method), π-electron wave function as Slater determinant, trivially satisfies the antisymmetric property of the exact solution:
Ξ¨πππ Τ¦π1, Τ¦π2, Τ¦π3, β¦ π 1, π 2, π 3, β¦ =
1
π!
π1π π
Τ¦π1, π 1 π1π π
Τ¦π2, π 2
π2π π
Τ¦π1, π 1 π2π π
Τ¦π2, π 2
β¦ π1π π
Τ¦ππ, π π
β¦ π2π π
Τ¦ππ, π 2β¦ β¦
πππ π
Τ¦π1, π 1 πππ π
Τ¦π2, π 2β¦
β¦ πππ π
Τ¦ππ, π π
Each of the single-electron spin-orbital πππ π
Τ¦ππ, π π must be different β two spin-orbital can for instance share the same orbital function, but then theirs spins are different
πππ π
Τ¦ππ, π π = πππ π
Τ¦ππ01
or πππ π
Τ¦ππ10
Theoretical description of condensed matter
DFT method
2015-11-27 30
Theoretical description of condensed matter
Opis teoretyczny materii skondensowanejHartree approximation (one-electron)
Ξ¨πππ Τ¦π1, Τ¦π2, Τ¦π3, β¦ = π1 Τ¦π1 β π2 Τ¦π2 β π3 Τ¦π3 β β¦ β ππ Τ¦ππ
We assume that an average potential from other ions and electrons acts on each electron:
π
ππ2
2π+
π
ππ Τ¦ππ Ξ¨πππ Τ¦π1, Τ¦π2, Τ¦π3, β¦ = πΈπ‘ππ‘
π Ξ¨πππ Τ¦π1, Τ¦π2, Τ¦π3, β¦
Thus
ππ2
2π+ ππ Τ¦ππ ππ Τ¦ππ = πΈπππ Τ¦ππ
If every potential is the same π1 Τ¦π1 β π2 Τ¦π2 β β― β ππ Τ¦ππ β π Τ¦π we getOne-electron SchrΓΆdinger equation:
π2
2π+ π Τ¦π ππ Τ¦ππ = πΈπππ Τ¦ππ
This time π is the set of quantum numbers of one-electron quantum states ππ Τ¦ππ of energies πΈπ. One-electron states are subject to the Pauli exclusion principle. A significant change in the number of electrons in a given band, leads to the change of π Τ¦πand of the one βparticle spectra! (for instance energy gap renormalization )
π
πΈπ = πΈπ‘ππ‘
2015-11-27 31
11/27/2015 32
Bloch theoremAssumptions:Motionless atoms, crystal (periodic) lattice .One-electron Hartree approximation
βOne-electronβ SchrΓΆdinger equation
Effective potential, periodic potential of the crystal lattice, the same for all electrons.
Self-consistent field method β the multi-electron issue is reduced to the solution of one-electron problem in a potential of all other electrons and atoms
or Hartree-Fock approximation (Slater determinant).
Ξ¨πππ Τ¦π1, Τ¦π2, Τ¦π3, β¦ = π1 Τ¦π1 β π2 Τ¦π2 β π3 Τ¦π3 β β¦ β ππ Τ¦ππ
π2
2π+ π Τ¦π ππ Τ¦ππ = πΈπππ Τ¦ππ
π Τ¦π = π Τ¦π + π
Periodic potential
2015-11-27 33
Crystal lattice:
π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€
For periodic functions with the lattice period π
π Τ¦π = π Τ¦π + π a good base in the Fourier series
expansion are functions π Τ¦π = exp π Τ¦πΊ Τ¦π which depend
on the reciprocal lattice vectors:Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
exp π Τ¦πΊ Τ¦π + π =
= exp π Τ¦πΊ Τ¦π β exp π Τ¦πΊπ = exp π Τ¦πΊ Τ¦π exp 2π π1π1 + π2π2 + π3π3 =exp π Τ¦πΊ Τ¦π
therefore π Τ¦π = π Τ¦π + π and finally we get:
π Τ¦π =
Τ¦πΊ
πΤ¦πΊ exp π Τ¦πΊ Τ¦π
Τ¦ππ Τ¦ππβ = 2ππΏππ
Bloch theorem
Periodic potential
2015-11-27 34
Periodic potenetial we can expand as a Fourier series:
π Τ¦π =
Τ¦πΊ
πΤ¦πΊ exp π Τ¦πΊ Τ¦π
The wavefunction can be represented as a sum of plane waves of different wavelengths satisfying periodic boundary conditions :
π Τ¦π =
π
πΆπ exp ππ Τ¦π
SchrΓΆdinger equation:
ΖΈπ2
2π+ π Τ¦π π Τ¦π = πΈ π Τ¦π
π
β2π2
2ππΆπ exp ππ Τ¦π +
π, Τ¦πΊ
πΆπ πΤ¦πΊ exp π π + Τ¦πΊ Τ¦π = πΈ
π
πΆπ exp ππ Τ¦π
This is an equation for πΈ and πΆπ for all vectors π, Τ¦π and Τ¦πΊ.
Bloch theorem
Periodic potential
See also: Ibach, Luth βSolid State Physicsβ
2015-11-27 35
π
β2π2
2ππΆπ exp ππ Τ¦π +
π, Τ¦πΊ
πΆπ πΤ¦πΊ exp π π + Τ¦πΊ Τ¦π = πΈ
π
πΆπ exp ππ Τ¦π
We get SchrΓΆdinger equation in a form:
π
exp ππ Τ¦πβ2π2
2πβ πΈ πΆπ +
Τ¦πΊ
πΆπβ Τ¦πΊπΤ¦πΊ = 0
That must be met for each vector Τ¦π.
The sum is over all π, Τ¦πΊ ,therefore:
π, Τ¦πΊ
πΆπ πΤ¦πΊ exp π π + Τ¦πΊ Τ¦π = β¦ π + Τ¦πΊ β πβ¦
=
π, Τ¦πΊ
πΆπβ Τ¦πΊ πΤ¦πΊ exp ππ Τ¦π
Bloch theorem
Periodic potential
2015-11-27 36
π
exp ππ Τ¦πβ2π2
2πβ πΈ πΆπ +
Τ¦πΊ
πΆπβ Τ¦πΊπΤ¦πΊ = 0
for each vector Τ¦π.
Thus, for each vector π we got equation for coefficients πΆπand πΈ:
β2π2
2πβ πΈ πΆπ +
Τ¦πΊ
πΆπβ Τ¦πΊπΤ¦πΊ = 0
In this equation for πΆπ also coefficients shifted by Τ¦πΊ like πΆπβ Τ¦πΊ1, πΆπβ Τ¦πΊ2
, πΆπβ Τ¦πΊ3appear
(but others do not, even when we started for any π!).
This equation couples those expansion coefficients π Τ¦π = ΟππΆπ exp ππ Τ¦π , whose π - values
differ from one another by a reciprocal lattice vector Τ¦πΊ.πΆπ1πΆπ2πΆπ2β¦πΆππTry to plot the mattrix of this equation
Bloch theorem
Periodic potential
11/27/2015 37
β2π2
2πβ πΈ πΆπ +
Τ¦πΊ
πΆπβ Τ¦πΊπΤ¦πΊ = 0
We do not need to solve these equations for all vectors Τ¦πΊ βwe can find a solution in one unit cell of the reciprocal lattice and copy it π times (π β number of unit cells)!
Thus we can find eigenvalues πΈ β πΈπ β πΈ π corresponding
to the wave-function ππ Τ¦π represented as a superposition of
plane waves whose wave vectors π differ only by reciprocal
lattice vectors Τ¦πΊ.
Wave vector π is a good quantum number according to which the energy eigenvalues and quantum states may be indexed. Thus the function π Τ¦π is the superposition of ππ Τ¦π of
energies πΈ π
π Τ¦π =
π
πΆπ exp ππ Τ¦π = β― =
π
ππ Τ¦π
(later on we introduce coefficient π for different solutions of πΈπ corresponding to the same π)
Bloch theorem
Periodic potential
2015-11-27 38
Wave-function which is the solution of the Schrodinger equation ππ Τ¦π is represented as a superposition of plane
waves whose wave vectors π differ only by reciprocal lattice
vectors Τ¦πΊ and it has energies πΈπ = πΈ π :
ππ Τ¦π =
Τ¦πΊ
πΆπβ Τ¦πΊ exp π π β Τ¦πΊ Τ¦π
Each vector π β Τ¦πΊ can enumerate states; it is convenient to choose the shortest vector (which belongs to the first Brillouin zone).
ππ Τ¦π =
Τ¦πΊ
πΆπβ Τ¦πΊ ππ πβ Τ¦πΊ Τ¦π =
Τ¦πΊ
πΆπβ Τ¦πΊ πβπ Τ¦πΊ Τ¦π πππ Τ¦π = π’π Τ¦π πππ Τ¦π
The function π’π Τ¦π is a Fourier series over reciprocal lattice points Τ¦πΊ, and thus has the
periodicity of the lattice.
Bloch theorem
Periodic potential
2015-11-27 39
Bloch theorem
Periodic potential
π1 β π2 = Τ¦πΊ
ππ Τ¦π = π’π Τ¦π πππ Τ¦π
Bloch waves whose wave vectors differ by a reciprocal lattice vector are IDENTICAL!
2015-11-27 40
The solution of the one-electron SchrΓΆdinger equation for a periodic potential has a form of modulated plane wave:
π’π,π Τ¦π = π’π,π Τ¦π + π
ππ,π Τ¦π = π’π,π Τ¦π πππ Τ¦π
We introduced coefficient π for different solutions corresponding to the same π (index). π-vector is an element of the first Brillouin zone.
Bloch wave,Bloch function
Bloch amplitude,Bloch envelope
π’π,π Τ¦π =
Τ¦πΊ
πΆπβ Τ¦πΊππ Τ¦πΊ Τ¦π
Bloch theorem
Periodic potential
Bloch function has a form:
Periodic function, so-called Bloch factor
Generally non-periodic function
Example: electron in a constant potential
substituting ππ,π Τ¦π = 1 πππ Τ¦π
The solution is
The momentum operator ΖΈπ = βπβπ» acting on ππ,π Τ¦π
ΖΈπππ,π Τ¦π = βπ ππ,π Τ¦π . The solutions of the SchrΓΆdinger equation with a constant potential
are eigenfunctions of the momentum operator. The momentum is well defined, the eigenvalue
of the momentum operator is ΖΈπ = βπ (this defines the sense of π-vector).
2015-11-27 41
π» = ββ2
2πΞ + π
ππ,π Τ¦π = π’π,π Τ¦π πππ Τ¦π
πΈ =β2π2
2π+ π
Bloch theorem
Periodic potential
PrzykΕad:Electron motion in a periodic potential.
Thus:
The solution is:
Applying ΖΈπ = βπβπ» we get ΖΈππ Τ¦π = βπβ π π + π»π’π,π πππ Τ¦π β βππ Τ¦π .
Momentum of the Bloch function is not well defined!
βπ is so-called quasi-momentum or crystal momentum.
2015-11-27 42
π Τ¦π =
Τ¦πΊ
πΤ¦πΊ exp π Τ¦πΊ Τ¦π
ππ,π Τ¦π = π’π,π Τ¦π πππ Τ¦π
π’π,π Τ¦π =
Τ¦πΊ
πΆπβ Τ¦πΊππ Τ¦πΊ Τ¦π
Bloch theorem
Periodic potential
ΖΈππ Τ¦π = βπβ π π + π»π’π,π πππ Τ¦π β βππ Τ¦π .
βπ is so-called quasi-momentum or crystal momentum.
If we consider interactions with other quasi-particles (electrons, phonons, magnons etc.) existing in the crystal and real particles penetrating through the crystal (e.g. photons, neutron) the momentum conservation law must be replaced by the quasi-momentum conservation law :
π
βππ +
π
Τ¦ππ =
π
βππβ² +
π
Τ¦ππβ² + β Τ¦πΊ
The energy conservation is always the same:
π
πΈπ =
π
πΈπβ²
2015-11-27 43
Bloch theorem
Periodic potential
kΒ·p perturbation theory β effective mass
2015-11-27 44
ππ,π Τ¦π = πππ Τ¦ππ’π,π Τ¦π
π-vector is not the momentum (momentum operator ΖΈπ = βπβπ»)
ΖΈπππ,π Τ¦π = βπβ ππ + π»π’π,π Τ¦π πππ Τ¦π β βπππ,π Τ¦π
Bloch function in the SchrΓΆdinger equation:
Ξππ,π Τ¦π = β― = Ξπ’π,π Τ¦π + 2πππ»π’π,π Τ¦π β π2π’π,π Τ¦π πππ Τ¦π
By substitution of this expression and simplification by πππ Τ¦π we got equation for π’π,π Τ¦π :
ββ2
2πΞ β
β
ππππ» +
β2
2ππ2 π’π,π Τ¦π =
ΖΈπ2
2π+β
ππ ΖΈπ +
β2π2
2ππ’π,π Τ¦π
The Schrodinger equation for the envelope π’π,π Τ¦π :
ΖΈπ2
2π+β
ππ ΖΈπ + π Τ¦π π’π,π Τ¦π = πΈπ β
β2π2
2ππ’π,π Τ¦π
2015-11-27 45
The Schrodinger equation for the envelope π’π,π Τ¦π :
ΖΈπ2
2π+β
ππ ΖΈπ + π Τ¦π π’π,π Τ¦π = πΈ β
β2π2
2ππ’π,π Τ¦π
This is so-called ππ perturbation theory used for the calculations of the energies and
wavefunctions at some π = π0.
The full Hamiltonian
π»ππ’π,π Τ¦π = π»π0+ π»β² π’π,π Τ¦π = πΈπ π π’π,π Τ¦π
Perturbation:
π»β² =β
ππ β π0 ΖΈπ
By the perturbation theory we find the function π’π,π Τ¦π and the energy πΈπ π .
kΒ·p perturbation theory β effective mass
2015-11-27 46
Landolt-Boernstein
Expanding πΈπ π = πΈπ ββ2π2
2πaround an extreme point, e.g. π = 0:
close bands
kΒ·p perturbation theory β effective mass
2015-11-27 47
We expand πΈπ π = πΈπ ββ2π2
2πaround an extreme point, e.g. π = 0:
πΈπ π = πΈπ 0 + π»ππβ² +
πβ π
π»ππβ² 2
πΈπ 0 β πΈπ 0+β―
For
π»ππβ² = ΰΆ±π’π,0 Τ¦π π»β² π’π,0 Τ¦π π3π = β
πβ
ππΰΆ±π’π,0 Τ¦π π»π’π,0 Τ¦π π3π =
π=1
3
ππππ
πΈπ π = πΈπ 0 +
π=1
3
ππππ +
π=1
3
π=1
3β2
2ππΏππ + πππ ππππ +β―
Linear in π
the linear terms in extremum vanish
πΈπ π = πΈπ 0 +
π=1
3
π=1
31
πβ
β2ππππ
2+ β―
kΒ·p perturbation theory β effective mass
2015-11-27 48
πΈπ π = πΈπ 0 +
π=1
3
π=1
31
πππβ
β2ππππ
2+ β―
The inverse effective-mass tensor:
1
πππβ =
πΏππ
π+2β2
π2
πβ π
π’π,0πππ₯π
π’π,0 π3π β π’π,0
πππ₯π
π’π,0 π3π
πΈπ 0 β πΈπ 0
This tensor is symmetric (πππ = πππ). If the energy extremum is in G(k=0) we obtain constant
energy ellipsoid in π-space, with principal axis 1
ππ:
πΈπ π β πΈπ 0 +β2
2
π12
π1β +
π22
π2β +
π32
π3β
where ππβ are the inertial effective masses along these different axes.
kΒ·p perturbation theory β effective mass
The energy En(k) around extremum for the uniaxial crystal (np. GaN):
Around the extremum (e.g. point G(k=0)) we can restrict the solution to approximate parabolic band
For a cubic crystal:
so-called spherical band
In general, depending on the wave vector, also higher order perturbation terms exist. The energy of the electron generally depends on k=(k1,k2,k3). The energy isosurface can be very complex, and its shape depends on all bands.
2015-11-27 49
πΈπ π = πΈπ 0 +β2
2
π12 + π2
2
πβ₯β +
π32
πβ₯β
πΈπ π = πΈπ 0 +β2π2
2πβ
kΒ·p perturbation theory β effective mass
The energy En(k) around extremum
R. StΔpniewski
2015-11-27 50
Non-parabolic band
Non-spherical band
kΒ·p perturbation theory β effective mass
The band theory of solids
Examples:
D. Wasik.
2015-11-27 51
kΒ·p perturbation theory β effective mass
The band theory of solids.
2015-11-27 52
Energy bands
2015-11-27 53
Examples:
E
kEg
G
The band theory of solids
The band theory of solids.
2015-11-27 54
https://nanohub.org/resources/10751
2015-11-27 55
Energy bands
Energy bands
2015-11-27 56
Examples:
D. Wasik.
The band theory of solids
Energy bands
2015-11-27 57
Opto-electronics requires direct gap materials.
Infinite well β particle in the box
2015-11-27 58
π π₯, π‘ =2
πΏsin πππ₯ πβπππ‘
Inside the quantum well :
ππ =ππ
πΏ
νπ =β2ππ
2
2π=β2π2π2
2ππΏ2
Infinite well
2015-11-27 59
π π₯, π‘ =2
πΏsin πππ₯ πβπππ‘
Inside the quantum well :
ππ =ππ
πΏ
νπ =β2ππ
2
2π=β2π2π2
2ππΏ2
Infinite well
2015-11-27 60
π π₯, π‘ =2
πΏsin πππ₯ πβπππ‘
Inside the quantum well :
νπ = πΈπ +β2ππ
2
2π= πΈπ +
β2π2π2
2π0πβπΏ2
ππ =ππ
πΏ
πΈπ
ν1 = πΈπ +β2π2
2π0πβπΏ2
ν2 = πΈπ +2β2π2
π0πβπΏ2
ν3 = πΈπ +9β2π2
2π0πβπΏ2
Infinite well
2015-11-27 61
π π₯, π‘ =2
πΏsin πππ₯ πβπππ‘
WewnΔ trz studni:
νπ = πΈπ£ +β2ππ
2
2π= πΈπ£ β
β2π2π2
2π0πβπΏ2
ππ =ππ
πΏ
πΈπ£ν1 = πΈπ£ β
β2π2
2π0πβπΏ2
ν2 = πΈπ£ β2β2π2
π0πβπΏ2
ν3 = πΈπ£ β9β2π2
2π0πβπΏ2
Infinite well
2015-11-27 62
WewnΔ trz studni:
ππ =ππ
πΏπ π§, π‘ = αsin πππ§
cos πππ§πβπππ‘
βπ
2< π§ <
π
2
ββ2
2π
π2
ππ§2π π§ + V0 z π π§ = νπ π§
ν < π0
π π§ = π· exp(Β±π π§)
β2π 2
2π= π0 β ν = π΅
Finite well
2015-11-27 63
WewnΔ trz studni:
Energy bands
2015-11-27 64
Examples:
E
kEg
G
The band theory of solids
Electrones and holes
Quite often it is more conveninent to know the Density of States - number of states in the range of the energies (E, E+d E). For the spherical and parabolic energy band:
kx
ky
kx
-1 -0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Energia (eV)
Gesto
sc s
tanΓ³w
2D case
2015-11-27 65
Density of states (DOS)
Optical transitions
2015-11-27 66
Semiconductor heterostructures
2015-11-27 67
Bandgap engineering
2015-11-27 68
How can we change the heterostructuresβ band structure:β’ By choosing the material (np. GaAs/AlAs)β’ By the control of the composition (binary, ternary, quaternary, quiternary alloys)β’ By the control of the tension
2015-11-27 69
