prezentacja programu powerpointszczytko/ldsn/2_ldsn_2015_tight_binding.pdfbloch function has a form:...
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A wave packet
2015-11-27 2S. Kryszewski GdaΕsk 2010
A wave packet
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Note that
is a fourrier transform of
Gaussian packet:
S. Kryszewski GdaΕsk 2010
A wave packet
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Gaussian packet:
S. Kryszewski GdaΕsk 2010
A wave packet
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Derivation:
S. Kryszewski GdaΕsk 2010
A wave packet
2015-11-27 6S. Kryszewski GdaΕsk 2010
A wave packet
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Ordering expression:
A wave packet
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Ordering expression:
S. Kryszewski GdaΕsk 2010
Pakiet falowy
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A wave packet
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ππ = 9.11 10β31kgβ = 6.626 10β34 Js = 4.136 10β15 eV sβ = 1.055 10β34 J s = 6.582 10β16 eV sπ = β1.602 10β19 C
Ξπ₯ Ξπ β₯β
2
A wave packet
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A wave packet
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foton
elektron
Mass particle (electron) and massless (photon)
The band theory of solids.
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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
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Brillouin zones
Periodic potential
π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Τ¦ππβ Τ¦ππ = 2ππΏππ
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
2-D square lattice
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π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Τ¦ππβ Τ¦ππ = 2ππΏππ
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D square lattice
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π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Τ¦ππβ Τ¦ππ = 2ππΏππ
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D square lattice
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π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Τ¦ππβ Τ¦ππ = 2ππΏππ
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D hexagonal lattice
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π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Τ¦ππβ Τ¦ππ = 2ππΏππ
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
http://oen.dydaktyka.agh.edu.pl/dydaktyka/fizyka/c_teoria_pasmowa/2.php
Brillouin zone for face centered cubic (fcc) lattice. The limiting zone walls comes from reciprocal latticepoints (2,0,0) square and (1,1,1) hexagonal.
Brillouin zone in 1D
Brillouin zone in 2D, oblique lattice.
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π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Brillouin zones
Periodic potential
Ibach, Luth
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bcc lattice
heksagonal lattice
π = π1 Τ¦π1 + π2 Τ¦π2 + π3 Τ¦π3, ππ β β€Τ¦πΊ = π1 Τ¦π1
β +π2 Τ¦π2β +π3 Τ¦π3
β , ππ β β€
Brillouin zones
Periodic potential
PotencjaΕ periodyczny
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Yu, Cardona Fundametals of semiconductors
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).
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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.
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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
PrzykΕad:Electron motion in a periodic potential.
Thus:
The solution is:
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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
LxLy
Lz
If our crystal has a finite size set of vectors π is finite (though enormous!). for instance, we can assume periodic boundary conditions and then:
ππ = 0,Β±2π
πΏπ, Β±
4π
πΏπ, Β±
6π
πΏπ, β¦ , Β±
2ππππΏπ
Bloch functions whose wave vectors differ by a reciprocal lattice vector, are the same!
Proof:
What about energies?
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Bloch theorem
Periodic potential
ππ,π+ Τ¦πΊ Τ¦π = ππ,π Τ¦π Τ¦πΊ = β Τ¦π1 + π Τ¦π2 + π Τ¦π3
ππ,π+ Τ¦πΊ Τ¦π = π’π,π+ Τ¦πΊ Τ¦π ππ(π+ Τ¦πΊ) Τ¦π =
Τ¦πΊβ²
πΆ π + Τ¦πΊ β Τ¦πΊβ² πβπ Τ¦πΊβ² Τ¦π ππ(π+ Τ¦πΊ) Τ¦π = β―
=
Τ¦πΊβ²β²
πΆ π β Τ¦πΊβ²β² πβπ Τ¦πΊβ²β² Τ¦π ππ(π) Τ¦π = ππ,π Τ¦π
Τ¦π2
2π0+ π Τ¦π ππ,π Τ¦π = πΈ π, π ππ,π Τ¦π
Τ¦π2
2π0+ π Τ¦π ππ,π+ Τ¦πΊ Τ¦π = πΈ π, π + Τ¦πΊ ππ,π+ Τ¦πΊ Τ¦π
Bloch functions whose wave vectors differ by a reciprocal lattice vector, are the same!
Proof:
What about energies?
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Bloch theorem
Periodic potential
Τ¦π2
2π0+ π Τ¦π ππ,π Τ¦π = πΈ π, π ππ,π Τ¦π
Τ¦π2
2π0+ π Τ¦π ππ,π+ Τ¦πΊ Τ¦π = πΈ π, π + Τ¦πΊ ππ,π+ Τ¦πΊ Τ¦π
ππ,π+ Τ¦πΊ Τ¦π = π’π,π+ Τ¦πΊ Τ¦π ππ(π+ Τ¦πΊ) Τ¦π =
Τ¦πΊβ²
πΆ π + Τ¦πΊ β Τ¦πΊβ² πβπ Τ¦πΊβ² Τ¦π ππ(π+ Τ¦πΊ) Τ¦π = β―
=
Τ¦πΊβ²β²
πΆ π β Τ¦πΊβ²β² πβπ Τ¦πΊβ²β² Τ¦π ππ(π) Τ¦π = ππ,π Τ¦π
ππ,π+ Τ¦πΊ Τ¦π = ππ,π Τ¦π Τ¦πΊ = β Τ¦π1 + π Τ¦π2 + π Τ¦π3
β πΈ π, π = πΈ π, π + Τ¦πΊ
Energy eigenvalues are a periodic function of π (wave vectors of Bloch function).
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π
β8π
π
8π
πβ6π
π
6π
πβ4π
π
4π
πβ2π
π
2π
π
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Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π= πΈ π, π + Τ¦πΊ =
β2 π + Τ¦πΊ2
2π
β8π
π
8π
πβ6π
π
6π
πβ4π
π
4π
πβ2π
π
2π
π
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Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π= πΈ π, π + Τ¦πΊ =
β2 π + Τ¦πΊ2
2π
β8π
π
8π
πβ6π
π
6π
πβ4π
π
4π
πβ2π
π
2π
π
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Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π= πΈ π, π + Τ¦πΊ =
β2 π + Τ¦πΊ2
2π
β8π
π
8π
πβ6π
π
6π
πβ4π
π
4π
πβ2π
π
2π
π
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Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π= πΈ π, π + Τ¦πΊ =
β2 π + Τ¦πΊ2
2π
β8π
π
8π
πβ6π
π
6π
πβ4π
π
4π
π
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β2π
π
2π
π
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
πΈ π, π =β2π2
2π= πΈ π, π + Τ¦πΊ =
β2 π + Τ¦πΊ2
2π
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Reduced Brilloin zone.On the border of the Brillouin zoneenergies are degenerated
The band structure of nearly free-electron cubic lattice
[hkl]=
000,
100,100, 200, 200,
kx
β β
The nearly free-electron approximation
Periodic potential
πΈ π, π = πΈ π, π + Τ¦πΊ =β2 π + Τ¦πΊ
2
2π
Τ¦πΊ = β Τ¦π1 + π Τ¦π2 + π Τ¦π3
ππ =2π
ππ
π
ππ₯βπ
ππ₯342015-11-27
The band structure of nearly free-electron cubic lattice
kx
β β
The nearly free-electron approximation
Periodic potential
πΈ π, π = πΈ π, π + Τ¦πΊ =β2 π + Τ¦πΊ
2
2π
Τ¦πΊ = β Τ¦π1 + π Τ¦π2 + π Τ¦π3
ππ =2π
ππ
π
ππ₯βπ
ππ₯
[hkl]=
000,
100,100, 200, 200,
010,010,001,001,
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The band structure of nearly free-electron cubic lattice
kx
β β
The nearly free-electron approximation
Periodic potential
πΈ π, π = πΈ π, π + Τ¦πΊ =β2 π + Τ¦πΊ
2
2π
Τ¦πΊ = β Τ¦π1 + π Τ¦π2 + π Τ¦π3
ππ =2π
ππ
π
ππ₯βπ
ππ₯
[hkl]=
000,
100,100, 200, 200,
010,010,001,001,
110,101,110,101,101,110,101,110β ββ ββ β β β
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kx
The appropriate expresions for a perturbation calculation of the influence of a small potential
βsmall potetntialβ
Small potential inluence on the borders of the Brilloun zone:
hkl = 000, 100,100, 200, 200,β β
(1D)
kx
The nearly free-electron approximation
Periodic potential
π π₯ = π0 cos2π
ππ₯
π π₯ = π0 cos2π
ππ₯ =
π02
ππ2ππ π₯ + πβπ
2ππ π₯
π
ππ₯βπ
ππ₯
π
ππ₯372015-11-27
kx
k
ax
probability density
probability density
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
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π =2π
π
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
probability density
probability density
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kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
probability density
probability density
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kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
probability density
probability density
πΈΒ± =1
2
β2
2π0
πΊ
2β π
2
+πΊ
2+ π
2
Β±
β2
2π0
πΊ
2β π
2
+πΊ
2+ π
22
+ 4π0
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kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
see Ibach, Lutch
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kx
k
ax
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The nearly free-electron approximation
Periodic potential
8π
πβπ
π
6π
π
π
π
4π
πβ2π
π
2π
π
π
ππ₯
xa
kx
k
ax
442015-11-27
8π
πβπ
π
6π
π
π
π
4π
πβ2π
π
2π
π
The nearly free-electron approximation
Periodic potential
xa
kx
k
ax
452015-11-27
8π
πβπ
π
6π
π
π
π
4π
πβ2π
π
2π
π
The nearly free-electron approximation
Periodic potential
band
band
band
462015-11-27
The electronic band structure
β’ It is convenient to present the energies only in the 1st Brillouin zone.β’ The electron state in the solid state is given by the wave vector of the 1st Brillouin zone, band number and a spin.
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W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
The electronic band structure
2015-11-27 48
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
The electronic band structure
Tight-Binding Approximation
2015-11-27 49
We describe the crystal electrons in terms of a linear superposition of atomic eigenfunctions(LCAO β Linear Combination of Atomic Orbitals) π» = π»π΄ + πβ²:
π»π΄ Τ¦π β π π ππ Τ¦π β π π = πΈπππ Τ¦π β π π
π» = π»π΄ + πβ² = ββ2
2πΞ + ππ΄ Τ¦π β π π +
πβ π
ππ΄ Τ¦π β π π
Perturbation: the influence of atoms in the neighborhood of π π :
πβ² Τ¦π β π π =
πβ π
ππ΄ Τ¦π β π π
Good for valence band of covalent crystals, π-orbital bands etc.
π-th state π Atom in position π πEquation for the free atoms thatform the crystal
π»π΄ = ββ2
2πΞ + ππ΄ Τ¦π β π π
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Approximate solution in the form of the Bloch function:
Ξ¦π,π Τ¦π =
π
ππππ Τ¦π β π π =
π
exp π ππ π ππ Τ¦π β π π
Check:Ξ¦π,π+ Τ¦πΊ Τ¦π = Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π + π = exp π ππ Ξ¦π,π Τ¦π
Energies determined by the variational method:
πΈ π β€Ξ¦π,π Τ¦π π» Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π Ξ¦π,π Τ¦π
Expression
Ξ¦π,π Τ¦π Ξ¦π,π Τ¦π =
π,π
exp ππ π π β π π ΰΆ±ππβ Τ¦π β π π ππ Τ¦π β π π ππ
can be easily simplify assuming a small overlap of wave functions for π β π
Ξ¦π,π Τ¦π Ξ¦π,π Τ¦π =
π
ΰΆ±ππβ Τ¦π β π π ππ Τ¦π β π π ππ = β―
Tight-Binding Approximation
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Approximate solution in the form of the Bloch function:
Ξ¦π,π Τ¦π =
π
ππππ Τ¦π β π π =
π
exp π ππ π ππ Τ¦π β π π
Check:Ξ¦π,π+ Τ¦πΊ Τ¦π = Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π + π = exp π ππ Ξ¦π,π Τ¦π
Energies determined by the variational method:
πΈ π β€Ξ¦π,π Τ¦π π» Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π Ξ¦π,π Τ¦π
πΈ π β1
πΞ¦π,π Τ¦π π» Ξ¦π,π Τ¦π =
=
π,π
exp ππ π π β π π ΰΆ±ππβ Τ¦π β π π πΈπ + πβ² Τ¦π β π π ππ Τ¦π β π π ππ
Tight-Binding Approximation
2015-11-27 52
Approximate solution in the form of the Bloch function:
Ξ¦π,π Τ¦π =
π
ππππ Τ¦π β π π =
π
exp π ππ π ππ Τ¦π β π π
Check:Ξ¦π,π+ Τ¦πΊ Τ¦π = Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π + π = exp π ππ Ξ¦π,π Τ¦π
Energies determined by the variational method:
πΈ π β€Ξ¦π,π Τ¦π π» Ξ¦π,π Τ¦π
Ξ¦π,π Τ¦π Ξ¦π,π Τ¦π
πΈ π β1
πΞ¦π,π Τ¦π π» Ξ¦π,π Τ¦π =
=
π,π
exp ππ π π β π π ΰΆ±ππβ Τ¦π β π π πΈπ + πβ² Τ¦π β π π ππ Τ¦π β π π ππ
Tight-Binding Approximation
Only diagonal terms π π = π π in πΈπ
Only the vicinity of π _π
2015-11-27 53
πΈ π β1
πΞ¦π,π Τ¦π π» Ξ¦π,π Τ¦π =
=
π,π
exp ππ π π β π π ΰΆ±ππβ Τ¦π β π π πΈπ + πβ² Τ¦π β π π ππ Τ¦π β π π ππ
When the atomic states ππ Τ¦π β π π are spherically symmetric (π -states), then overlap
integrals depend only on the distance between atoms:
πΈπ π β πΈπ β π΄π β π΅π
π
exp ππ π π β π π
π΄π = βΰΆ±ππβ Τ¦π β π π πβ² Τ¦π β π π ππ Τ¦π β π π ππ
π΅π = βΰΆ±ππβ Τ¦π β π π πβ² Τ¦π β π π ππ Τ¦π β π π ππ
Restricted to only the nearest neighbours of π π
Tight-Binding Approximation
Only diagonal terms π π = π π in πΈπ
Only the vicinity of π _π
2015-11-27 54
When the atomic states ππ Τ¦π β π π are spherically symmetric (π -states), then overlap
integrals depend only on the distance between atoms:
πΈπ π β πΈπ β π΄π β π΅π
π
exp ππ π π β π π
π΄π = βΰΆ±ππβ Τ¦π β π π πβ² Τ¦π β π π ππ Τ¦π β π π ππ
π΅π = βΰΆ±ππβ Τ¦π β π π πβ² Τ¦π β π π ππ Τ¦π β π π ππ
The result of the summation depends on the symmetry of the lattice:
For π π structure: π π β π π = Β±π, 0,0 ; 0, Β±π, 0 ; 0,0, Β±π ;
πΈπ π β πΈπ β π΄π β 2π΅π cos ππ₯π + cos ππ¦π + cos ππ§π
For πππ structure :
πΈπ π β πΈπ β π΄π β 8π΅π cosππ₯π
2cos
ππ¦π
2cos
ππ§π
2
Forπππ structure :
πΈπ π β πΈπ β π΄π β 4π΅π cosππ₯π
2cos
ππ¦π
2+ π. π.
Tight-Binding Approximation
2015-11-27 55
For π π structure: π π β π π = Β±π, 0,0 ; 0, Β±π, 0 ; 0,0, Β±π ;
πΈπ π β πΈπ β π΄π β 2π΅π cos ππ₯π + cos ππ¦π + cos ππ§π
H. I
bac
h, H
. LΓΌ
th, S
olid
-Sta
te P
hys
ics
π΅π=βΰΆ±ππβΤ¦πβπ π
πβ²Τ¦πβπ π
ππΤ¦πβπ π
ππ
Tight-Binding Approximation
2015-11-27 56
For π π structure: π π β π π = Β±π, 0,0 ; 0, Β±π, 0 ; 0,0, Β±π ;
πΈπ π β πΈπ β π΄π β 2π΅π cos ππ₯π + cos ππ¦π + cos ππ§π
H. I
bac
h, H
. LΓΌ
th, S
olid
-Sta
te P
hys
ics
π΅π=βΰΆ±ππβΤ¦πβπ π
πβ²Τ¦πβπ π
ππΤ¦πβπ π
ππ
Tight-Binding Approximation
2015-11-27 57
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, βThe Band Theory of Graphiteβ, Physical Review 71, 622 (1947).] gives:
πΈ π = Β± πΎ02 1 + 4 cos2
ππ¦π
2+ 4 cos
ππ¦π
2β cos
ππ₯ 3π
2β β Ηπ π β ππ
Tight-Binding Approximation
2015-11-27 58
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, βThe Band Theory of Graphiteβ, Physical Review 71, 622 (1947).] gives:
πΈ π = Β± πΎ02 1 + 4 cos2
ππ¦π
2+ 4 cos
ππ¦π
2β cos
ππ₯ 3π
2β β Ηπ π β ππ
Tight-Binding Approximation
2015-11-27 59
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, βThe Band Theory of Graphiteβ, Physical Review 71, 622 (1947).] gives:
πΈ π = Β± πΎ02 1 + 4 cos2
ππ¦π
2+ 4 cos
ππ¦π
2β cos
ππ₯ 3π
2β β Ηπ π β ππ
Tight-Binding Approximation
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The number of states in the volume of ππ2:
π πΈ =2
2π 2 ππ2 =2
2π 2 π π β ππ2=
2
2π 2 ππΈ
β Ηπ
2
π πΈ =ππ πΈ
ππΈ=
πΈ
π β Ηπ 2
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, βThe Band Theory of Graphiteβ, Physical Review 71, 622 (1947).] gives:
πΈ π = Β± πΎ02 1 + 4 cos2
ππ¦π
2+ 4 cos
ππ¦π
2β cos
ππ₯ 3π
2β β Ηπ π β ππ
Tight-Binding Approximation
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The existence of the band structure arising from the discrete energy levels of isolated atoms due to the interaction between them. We can classify the electronic states as belonging to the electronic shells π , π, π etc.
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The existence of a forbidden gap is not tied to the periodicity of the lattice! Amorphous materials can also display a band gap.
If a crystal with a primitive cubic lattice contains π atoms and thus π primitive unit cells, then an atomic energy level π¬π of the free atom will split into π states (due to the interaction with the rest of π β 1 atoms).
Each band can be occupied by 2π electrons.
Tight-Binding Approximation
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An odd number of electrons per cell(metal)
An even number of electrons per cell(non-metal)
An even number of electrons per cell but overlapping bands(metals of the II group, e.g. Be β next slide!)
If a crystal with a primitive cubic lattice contains π atoms and thus π primitive unit cells, then an atomic energy level π¬π of the free atom will split into π states (due to the interaction with the rest of π β 1 atoms). Each band can be occupied by ππ΅ electrons.
Tight-Binding Approximation
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Be
C, Si, Ge
Tight-Binding ApproximationThe states can mix: for instance π π3 hybridization.
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Tight-Binding Approximation
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MichaΕ Baj
Tight-Binding Approximation
Fermi surfaces of metals
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Ashcroft, Mermin
Cu
Fermi surfaces of metals
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http://physics.unl.edu/tsymbal/teaching/SSP-927/Section%2010_Metals-Electron_dynamics_and_Fermi_surfaces.pdf
Fermi surfaces of metals
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MichaΕ Baj
Tight-Binding Approximation
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., S
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Tight-Binding Approximation
Tight-Binding Approximation
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., S
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.: P
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. Rev
. B2
1,
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28
(1
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Density of stateslike for freeelectrons!
Tight-Binding Approximation
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MichaΕ Baj
Cu: [1s22s22p63s23p6] 3d104s1
[Ar] 3d104s1
d-band
Tight-Binding Approximation
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Ni: [1s22s22p63s23p6] 3d94s1 [Ar] 3d94s1
ferromagnetic ordering, exchange interaction, different energies for different spins, two different densities of states for two spins β and β
Ξ β Stoner gap
Semiconductors
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Semiconductors of the group IV (Si, Ge) βdiamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) β zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) βNaCl structure
SiIndirect bandgap, Eg = 1,1 eV
Conduction band minima in Ξ point, constant energy surfaces β ellipsoids (6 pieces), m||=0,92 m0, mβ₯=0,19 m0,
The maximum of the valence band in Ξ point, mlh=0,153 m0, mhh=0,537 m0, mso=0,234 m0, Ξso= 0,043 eV
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GeIndirect bandgap, Eg = 0,66 eV
The maximum of the valence band in Ξ point, mlh=0,04 m0, mhh=0,3 m0, mso=0,09 m0, Ξso= 0,29 eV
Conduction band minima in Ξ point, constant energy surfaces β ellipsoids (8 pieces), m||=1,6 m0, mβ₯=0,08 m0,
SemiconductorsSemiconductors of the group IV (Si, Ge) βdiamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) β zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) βNaCl structure
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GaAsDirect bandgap, Eg = 1,42 eV
The maximum of the valence band in Ξ point, mlh=0,076 m0, mhh=0,5 m0, mso=0,145 m0, Ξso= 0,34 eV
The minimum of the conduction band in Ξ point, constant energy surfaces β spheres, mc=0,065 m0
SemiconductorsSemiconductors of the group IV (Si, Ge) βdiamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) β zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) βNaCl structure
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a-SnDiamond structureZero energy gap, Eg = 0 eV (inverted band-structure) The maximum of the valence band in Ξ point, mv=0,195 m0, mv2=0,058 m0, Ξso=0,8 eV The minimum of the conduction band in Ξ point, constant energy surfaces β spheres, mc=0,024 m0
SemiconductorsSemiconductors of the group IV (Si, Ge) βdiamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) β zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) βNaCl structure
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PbSeDirect bandgap in L-point, Eg = 0,28 eV
The maximum of the valence band in L point, constant energy surfaces β ellipsoids , m||=0,068 m0, mβ₯=0,034 m0,
The minimum of the conduction band in L point, constant energy surfaces β ellipsoids m||=0,07 m0, mβ₯=0,04 m0,
SemiconductorsSemiconductors of the group IV (Si, Ge) βdiamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) β zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) βNaCl structure
Photoemission spectroscopy
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βπ = π + πΈπππ + πΈπ
work function potential barrier
Photoemission spectroscopy
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βπ = π + πΈπππ + πΈπ
work function potential barrier
Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
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Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
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Semiconductor heterostructures
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Semiconductor heterostructures
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Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
Bandgap engineering
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How can we change the heterostructure band structure:β’ selecting a material (eg., GaAs / AlAs)β’ controlling the compositionβ’ controlling the stress
Heterostruktury pΓ³Εprzewodnikowe
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Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
Bandgap engineering
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87
Valence band offset (Andersonβs rule)
Valence band offset:
(powinowactwo)
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Bandgap engineering
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Valence band offset (Andersonβs rule)
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Valence band offset:
(powinowactwo)
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Bandgap engineering
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Valence band offset
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Bandgap engineering
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How can we change the heterostructure band structure:β’ selecting a material (eg., GaAs / AlAs)β’ controlling the compositionβ’ controlling the stress
Vegardβs law:the empirical heuristic that the lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituents' (A and B) lattice parameters at the same temperature:
π = ππ΄ 1 β π₯ + ππ΅π₯
Bandgap engineering
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Vegardβs law:Relationship to band gaps of βbinarycompoundβ:
πΈ = πΈπ΄ 1 β π₯ + πΈπ΅π₯ β ππ₯(1 β π₯)
b β so-called βbowingβ (curvature) of the energy gap
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How can we change the heterostructure band structure:β’ selecting a material (eg., GaAs / AlAs)β’ controlling the compositionβ’ controlling the stress