prezentacja programu powerpointszczytko/ldsn/2_ldsn_2015_tight_binding.pdfbloch function has a form:...
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![Page 2: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/2.jpg)
A wave packet
2015-11-27 2S. Kryszewski GdaΕsk 2010
![Page 3: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/3.jpg)
A wave packet
2015-11-27 3
Note that
is a fourrier transform of
Gaussian packet:
S. Kryszewski GdaΕsk 2010
![Page 4: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/4.jpg)
A wave packet
2015-11-27 4
Gaussian packet:
S. Kryszewski GdaΕsk 2010
![Page 5: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/5.jpg)
A wave packet
2015-11-27 5
Derivation:
S. Kryszewski GdaΕsk 2010
![Page 6: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/6.jpg)
A wave packet
2015-11-27 6S. Kryszewski GdaΕsk 2010
![Page 7: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/7.jpg)
A wave packet
2015-11-27 7
Ordering expression:
![Page 8: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/8.jpg)
A wave packet
2015-11-27 8
Ordering expression:
S. Kryszewski GdaΕsk 2010
![Page 9: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/9.jpg)
Pakiet falowy
2015-11-27 9
![Page 10: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/10.jpg)
A wave packet
2015-11-27 10
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
![Page 11: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/11.jpg)
A wave packet
2015-11-27 11
![Page 12: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/12.jpg)
A wave packet
2015-11-27 12
foton
elektron
Mass particle (electron) and massless (photon)
![Page 13: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/13.jpg)
The band theory of solids.
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![Page 14: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/14.jpg)
2015-11-27 14
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
![Page 15: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/15.jpg)
2015-11-27 15
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
![Page 16: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/16.jpg)
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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
![Page 17: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/17.jpg)
2015-11-27 17
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
![Page 18: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/18.jpg)
2015-11-27 18
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
![Page 19: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/19.jpg)
2015-11-27 19
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
![Page 20: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/20.jpg)
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.
2015-11-27 20
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
![Page 21: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/21.jpg)
Ibach, Luth
2015-11-27 21
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
![Page 22: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/22.jpg)
PotencjaΕ periodyczny
2015-11-27 22
Yu, Cardona Fundametals of semiconductors
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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 23
π» = ββ2
2πΞ + π
ππ,π Τ¦π = π’π,π Τ¦π πππ Τ¦π
πΈ =β2π2
2π+ π
Bloch theorem
Periodic potential
![Page 24: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/24.jpg)
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
![Page 25: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/25.jpg)
PrzykΕad:Electron motion in a periodic potential.
Thus:
The solution is:
2015-11-27 25
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
![Page 26: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/26.jpg)
Bloch functions whose wave vectors differ by a reciprocal lattice vector, are the same!
Proof:
What about energies?
2015-11-27 26
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
![Page 27: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/27.jpg)
Bloch functions whose wave vectors differ by a reciprocal lattice vector, are the same!
Proof:
What about energies?
2015-11-27 27
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).
![Page 28: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/28.jpg)
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
282015-11-27
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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
292015-11-27
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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
302015-11-27
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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
312015-11-27
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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
322015-11-27
β2π
π
2π
π
![Page 33: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/33.jpg)
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
332015-11-27
Reduced Brilloin zone.On the border of the Brillouin zoneenergies are degenerated
![Page 34: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/34.jpg)
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
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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,
352015-11-27
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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
362015-11-27
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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
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kx
k
ax
probability density
probability density
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
382015-11-27
π =2π
π
![Page 39: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/39.jpg)
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same π-vector π
ππ₯
probability density
probability density
392015-11-27
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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
402015-11-27
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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
412015-11-27
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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
422015-11-27
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kx
k
ax
432015-11-27
The nearly free-electron approximation
Periodic potential
8π
πβπ
π
6π
π
π
π
4π
πβ2π
π
2π
π
π
ππ₯
![Page 44: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/44.jpg)
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
![Page 45: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/45.jpg)
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
![Page 46: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/46.jpg)
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.
![Page 47: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/47.jpg)
2015-11-27 47
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
The electronic band structure
![Page 48: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/48.jpg)
2015-11-27 48
W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics
The electronic band structure
![Page 49: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/49.jpg)
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
![Page 50: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/50.jpg)
2015-11-27 50
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
![Page 51: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/51.jpg)
2015-11-27 51
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
![Page 52: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/52.jpg)
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
![Page 53: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/53.jpg)
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
![Page 54: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/54.jpg)
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
![Page 55: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/55.jpg)
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
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π΅π=βΰΆ±ππβΤ¦πβπ π
πβ²Τ¦πβπ π
ππΤ¦πβπ π
ππ
Tight-Binding Approximation
![Page 56: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/56.jpg)
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
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π΅π=βΰΆ±ππβΤ¦πβπ π
πβ²Τ¦πβπ π
ππΤ¦πβπ π
ππ
Tight-Binding Approximation
![Page 57: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/57.jpg)
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
![Page 58: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/58.jpg)
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
![Page 59: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/59.jpg)
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
![Page 60: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/60.jpg)
2015-11-27 60
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
![Page 61: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/61.jpg)
2015-11-27 61
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
![Page 62: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/62.jpg)
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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
![Page 63: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/63.jpg)
2015-11-27 63
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Be
C, Si, Ge
Tight-Binding ApproximationThe states can mix: for instance π π3 hybridization.
![Page 64: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/64.jpg)
2015-11-27 64
Tight-Binding Approximation
![Page 65: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/65.jpg)
2015-11-27 65
MichaΕ Baj
Tight-Binding Approximation
![Page 66: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/66.jpg)
Fermi surfaces of metals
2015-11-27 66
Ashcroft, Mermin
Cu
![Page 67: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/67.jpg)
Fermi surfaces of metals
2015-11-27 67
http://physics.unl.edu/tsymbal/teaching/SSP-927/Section%2010_Metals-Electron_dynamics_and_Fermi_surfaces.pdf
![Page 68: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/68.jpg)
Fermi surfaces of metals
2015-11-27 68
MichaΕ Baj
![Page 69: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/69.jpg)
Tight-Binding Approximation
2015-11-27 69
![Page 70: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/70.jpg)
2015-11-27 70
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Tight-Binding Approximation
Tight-Binding Approximation
![Page 71: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/71.jpg)
2015-11-27 71
MichaΕ BajSzm
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Density of stateslike for freeelectrons!
Tight-Binding Approximation
![Page 72: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/72.jpg)
2015-11-27 72
MichaΕ Baj
Cu: [1s22s22p63s23p6] 3d104s1
[Ar] 3d104s1
d-band
Tight-Binding Approximation
![Page 73: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/73.jpg)
2015-11-27 73
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
![Page 74: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/74.jpg)
Semiconductors
2015-11-27 74
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
![Page 75: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/75.jpg)
2015-11-27 75
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
![Page 76: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/76.jpg)
2015-11-27 76
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
![Page 77: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/77.jpg)
2015-11-27 77
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
![Page 78: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/78.jpg)
2015-11-27 78
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
![Page 79: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/79.jpg)
Photoemission spectroscopy
2015-11-27 79
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work function potential barrier
![Page 80: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/80.jpg)
Photoemission spectroscopy
2015-11-27 80
H. I
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βπ = π + πΈπππ + πΈπ
work function potential barrier
![Page 81: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/81.jpg)
Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
2015-11-27 81
![Page 82: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/82.jpg)
Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
2015-11-27 82
![Page 83: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/83.jpg)
Semiconductor heterostructures
2015-11-27 83
![Page 84: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/84.jpg)
Semiconductor heterostructures
2015-11-27 84
Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
![Page 85: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/85.jpg)
Bandgap engineering
2015-11-27 85
How can we change the heterostructure band structure:β’ selecting a material (eg., GaAs / AlAs)β’ controlling the compositionβ’ controlling the stress
![Page 86: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/86.jpg)
Heterostruktury pΓ³Εprzewodnikowe
2015-11-27 86
Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
![Page 87: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/87.jpg)
Bandgap engineering
2015-11-27
87
Valence band offset (Andersonβs rule)
Valence band offset:
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Bandgap engineering
2015-11-27 88
Valence band offset (Andersonβs rule)
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Bandgap engineering
2015-11-27 89
Valence band offset
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Bandgap engineering
2015-11-27 90
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₯
![Page 91: Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form: Periodic function, so-called Bloch factor Generally non-periodic function Example:](https://reader034.vdocuments.mx/reader034/viewer/2022050117/5f4e44edb39a3d7fc26e29b9/html5/thumbnails/91.jpg)
Bandgap engineering
2015-11-27 91
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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