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A wave packet
2015-11-27 2S. Kryszewski Gdańsk 2010
A wave packet
2015-11-27 3
Note that
is a fourrier transform of
Gaussian packet:
S. Kryszewski Gdańsk 2010
A wave packet
2015-11-27 4
Gaussian packet:
S. Kryszewski Gdańsk 2010
A wave packet
2015-11-27 5
Derivation:
S. Kryszewski Gdańsk 2010
A wave packet
2015-11-27 6S. Kryszewski Gdańsk 2010
A wave packet
2015-11-27 7
Ordering expression:
A wave packet
2015-11-27 8
Ordering expression:
S. Kryszewski Gdańsk 2010
Pakiet falowy
2015-11-27 9
A wave packet
2015-11-27 10
𝑚𝑒 = 9.11 10−31kgℎ = 6.626 10−34 Js = 4.136 10−15 eV sℏ = 1.055 10−34 J s = 6.582 10−16 eV s𝑒 = −1.602 10−19 C
Δ𝑥 Δ𝑝 ≥ℏ
2
A wave packet
2015-11-27 11
A wave packet
2015-11-27 12
foton
elektron
Mass particle (electron) and massless (photon)
The band theory of solids.
2015-11-27 13
2015-11-27 14
The solution of the one-electron Schrödinger equation for a periodic potential has a form of modulated plane wave:
𝑢𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 + 𝑅
𝜑𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟
We introduced coefficient 𝑛 for different solutions corresponding to the same 𝑘 (index). 𝑘-vector is an element of the first Brillouin zone.
Bloch wave,Bloch function
Bloch amplitude,Bloch envelope
𝑢𝑛,𝑘 Ԧ𝑟 =
Ԧ𝐺
𝐶𝑘− Ԧ𝐺𝑒𝑖 Ԧ𝐺 Ԧ𝑟
Bloch theorem
Periodic potential
2015-11-27 15
Brillouin zones
Periodic potential
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Ԧ𝑎𝑖∗ Ԧ𝑎𝑗 = 2𝜋𝛿𝑖𝑗
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
2-D square lattice
2015-11-27 16
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Ԧ𝑎𝑖∗ Ԧ𝑎𝑗 = 2𝜋𝛿𝑖𝑗
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D square lattice
2015-11-27 17
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Ԧ𝑎𝑖∗ Ԧ𝑎𝑗 = 2𝜋𝛿𝑖𝑗
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D square lattice
2015-11-27 18
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Ԧ𝑎𝑖∗ Ԧ𝑎𝑗 = 2𝜋𝛿𝑖𝑗
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/zone_construction.php
Brillouin zones
Periodic potential
2-D hexagonal lattice
2015-11-27 19
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Ԧ𝑎𝑖∗ Ԧ𝑎𝑗 = 2𝜋𝛿𝑖𝑗
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.
2015-11-27 20
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Brillouin zones
Periodic potential
Ibach, Luth
2015-11-27 21
bcc lattice
heksagonal lattice
𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤԦ𝐺 = 𝑚1 Ԧ𝑎1
∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3
∗ , 𝑚𝑖 ∈ ℤ
Brillouin zones
Periodic potential
Potencjał periodyczny
2015-11-27 22
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 𝜑𝑛,𝑘 Ԧ𝑟 = 1 𝑒𝑖𝑘 Ԧ𝑟
The solution is
The momentum operator Ƹ𝑝 = −𝑖ℏ𝛻 acting on 𝜑𝑛,𝑘 Ԧ𝑟
Ƹ𝑝𝜑𝑛,𝑘 Ԧ𝑟 = ℏ𝑘 𝜑𝑛,𝑘 Ԧ𝑟 . The solutions of the Schrödinger equation with a constant potential
are eigenfunctions of the momentum operator. The momentum is well defined, the eigenvalue
of the momentum operator is Ƹ𝑝 = ℏ𝑘 (this defines the sense of 𝑘-vector).
2015-11-27 23
𝐻 = −ℏ2
2𝑚Δ + 𝑉
𝜑𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟
𝐸 =ℏ2𝑘2
2𝑚+ 𝑉
Bloch theorem
Periodic potential
Przykład:Electron motion in a periodic potential.
Thus:
The solution is:
Applying Ƹ𝑝 = −𝑖ℏ𝛻 we get Ƹ𝑝𝜓 Ԧ𝑟 = −𝑖ℏ 𝑖 𝑘 + 𝛻𝑢𝑛,𝑘 𝑒𝑖𝑘 Ԧ𝑟 ≠ ℏ𝑘𝜓 Ԧ𝑟 .
Momentum of the Bloch function is not well defined!
ℏ𝑘 is so-called quasi-momentum or crystal momentum.
2015-11-27 24
𝑉 Ԧ𝑟 =
Ԧ𝐺
𝑉Ԧ𝐺 exp 𝑖 Ԧ𝐺 Ԧ𝑟
𝜓𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟
𝑢𝑛,𝑘 Ԧ𝑟 =
Ԧ𝐺
𝐶𝑘− Ԧ𝐺𝑒𝑖 Ԧ𝐺 Ԧ𝑟
Bloch theorem
Periodic potential
Przykład:Electron motion in a periodic potential.
Thus:
The solution is:
2015-11-27 25
𝑉 Ԧ𝑟 =
Ԧ𝐺
𝑉Ԧ𝐺 exp 𝑖 Ԧ𝐺 Ԧ𝑟
𝜓𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟
𝑢𝑛,𝑘 Ԧ𝑟 =
Ԧ𝐺
𝐶𝑘− Ԧ𝐺𝑒𝑖 Ԧ𝐺 Ԧ𝑟
Bloch theorem
Periodic potential
LxLy
Lz
If our crystal has a finite size set of vectors 𝑘 is finite (though enormous!). for instance, we can assume periodic boundary conditions and then:
𝑘𝑖 = 0,±2𝜋
𝐿𝑖, ±
4𝜋
𝐿𝑖, ±
6𝜋
𝐿𝑖, … , ±
2𝜋𝑛𝑖𝐿𝑖
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
𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝜓𝑛,𝑘 Ԧ𝑟 Ԧ𝐺 = ℎ Ԧ𝑔1 + 𝑘 Ԧ𝑔2 + 𝑙 Ԧ𝑔3
𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝑢𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 𝑒𝑖(𝑘+ Ԧ𝐺) Ԧ𝑟 =
Ԧ𝐺′
𝐶 𝑘 + Ԧ𝐺 − Ԧ𝐺′ 𝑒−𝑖 Ԧ𝐺′ Ԧ𝑟 𝑒𝑖(𝑘+ Ԧ𝐺) Ԧ𝑟 = ⋯
=
Ԧ𝐺′′
𝐶 𝑘 − Ԧ𝐺′′ 𝑒−𝑖 Ԧ𝐺′′ Ԧ𝑟 𝑒𝑖(𝑘) Ԧ𝑟 = 𝜓𝑛,𝑘 Ԧ𝑟
Ԧ𝑝2
2𝑚0+ 𝑉 Ԧ𝑟 𝜓𝑛,𝑘 Ԧ𝑟 = 𝐸 𝑛, 𝑘 𝜓𝑛,𝑘 Ԧ𝑟
Ԧ𝑝2
2𝑚0+ 𝑉 Ԧ𝑟 𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟
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
Ԧ𝑝2
2𝑚0+ 𝑉 Ԧ𝑟 𝜓𝑛,𝑘 Ԧ𝑟 = 𝐸 𝑛, 𝑘 𝜓𝑛,𝑘 Ԧ𝑟
Ԧ𝑝2
2𝑚0+ 𝑉 Ԧ𝑟 𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟
𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝑢𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 𝑒𝑖(𝑘+ Ԧ𝐺) Ԧ𝑟 =
Ԧ𝐺′
𝐶 𝑘 + Ԧ𝐺 − Ԧ𝐺′ 𝑒−𝑖 Ԧ𝐺′ Ԧ𝑟 𝑒𝑖(𝑘+ Ԧ𝐺) Ԧ𝑟 = ⋯
=
Ԧ𝐺′′
𝐶 𝑘 − Ԧ𝐺′′ 𝑒−𝑖 Ԧ𝐺′′ Ԧ𝑟 𝑒𝑖(𝑘) Ԧ𝑟 = 𝜓𝑛,𝑘 Ԧ𝑟
𝜓𝑛,𝑘+ Ԧ𝐺 Ԧ𝑟 = 𝜓𝑛,𝑘 Ԧ𝑟 Ԧ𝐺 = ℎ Ԧ𝑔1 + 𝑘 Ԧ𝑔2 + 𝑙 Ԧ𝑔3
⇒ 𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺
Energy eigenvalues are a periodic function of 𝑘 (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
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚
−8𝜋
𝑎
8𝜋
𝑎−6𝜋
𝑎
6𝜋
𝑎−4𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
282015-11-27
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =
ℏ2 𝑘 + Ԧ𝐺2
2𝑚
−8𝜋
𝑎
8𝜋
𝑎−6𝜋
𝑎
6𝜋
𝑎−4𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
292015-11-27
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =
ℏ2 𝑘 + Ԧ𝐺2
2𝑚
−8𝜋
𝑎
8𝜋
𝑎−6𝜋
𝑎
6𝜋
𝑎−4𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
302015-11-27
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =
ℏ2 𝑘 + Ԧ𝐺2
2𝑚
−8𝜋
𝑎
8𝜋
𝑎−6𝜋
𝑎
6𝜋
𝑎−4𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
312015-11-27
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =
ℏ2 𝑘 + Ԧ𝐺2
2𝑚
−8𝜋
𝑎
8𝜋
𝑎−6𝜋
𝑎
6𝜋
𝑎−4𝜋
𝑎
4𝜋
𝑎
322015-11-27
−2𝜋
𝑎
2𝜋
𝑎
Energy of the plane wave in empty space as the function of wave vector:
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 =ℏ2𝑘2
2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =
ℏ2 𝑘 + Ԧ𝐺2
2𝑚
332015-11-27
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
– –
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 =ℏ2 𝑘 + Ԧ𝐺
2
2𝑚
Ԧ𝐺 = ℎ Ԧ𝑔1 + 𝑘 Ԧ𝑔2 + 𝑙 Ԧ𝑔3
𝑔𝑖 =2𝜋
𝑎𝑖
𝜋
𝑎𝑥−𝜋
𝑎𝑥342015-11-27
The band structure of nearly free-electron cubic lattice
kx
– –
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 =ℏ2 𝑘 + Ԧ𝐺
2
2𝑚
Ԧ𝐺 = ℎ Ԧ𝑔1 + 𝑘 Ԧ𝑔2 + 𝑙 Ԧ𝑔3
𝑔𝑖 =2𝜋
𝑎𝑖
𝜋
𝑎𝑥−𝜋
𝑎𝑥
[hkl]=
000,
100,100, 200, 200,
010,010,001,001,
352015-11-27
The band structure of nearly free-electron cubic lattice
kx
– –
The nearly free-electron approximation
Periodic potential
𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 =ℏ2 𝑘 + Ԧ𝐺
2
2𝑚
Ԧ𝐺 = ℎ Ԧ𝑔1 + 𝑘 Ԧ𝑔2 + 𝑙 Ԧ𝑔3
𝑔𝑖 =2𝜋
𝑎𝑖
𝜋
𝑎𝑥−𝜋
𝑎𝑥
[hkl]=
000,
100,100, 200, 200,
010,010,001,001,
110,101,110,101,101,110,101,110– –– –– – – –
362015-11-27
kx
The appropriate expresions for a perturbation calculation of the influence of a small potential
„small potetntial”
Small potential inluence on the borders of the Brilloun zone:
hkl = 000, 100,100, 200, 200,– –
(1D)
kx
The nearly free-electron approximation
Periodic potential
𝑉 𝑥 = 𝑉0 cos2𝜋
𝑎𝑥
𝑉 𝑥 = 𝑉0 cos2𝜋
𝑎𝑥 =
𝑉02
𝑒𝑖2𝜋𝑎 𝑥 + 𝑒−𝑖
2𝜋𝑎 𝑥
𝜋
𝑎𝑥−𝜋
𝑎𝑥
𝜋
𝑎𝑥372015-11-27
kx
k
ax
probability density
probability density
The nearly free-electron approximation
Periodic potential
Plane waves of the same 𝑘-vector 𝜋
𝑎𝑥
382015-11-27
𝜆 =2𝜋
𝑘
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same 𝑘-vector 𝜋
𝑎𝑥
probability density
probability density
392015-11-27
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same 𝑘-vector 𝜋
𝑎𝑥
probability density
probability density
402015-11-27
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same 𝑘-vector 𝜋
𝑎𝑥
probability density
probability density
𝐸± =1
2
ℏ2
2𝑚0
𝐺
2− 𝜅
2
+𝐺
2+ 𝜅
2
±
ℏ2
2𝑚0
𝐺
2− 𝜅
2
+𝐺
2+ 𝜅
22
+ 4𝑉0
412015-11-27
kx
k
ax
The nearly free-electron approximation
Periodic potential
Plane waves of the same 𝑘-vector 𝜋
𝑎𝑥
see Ibach, Lutch
422015-11-27
kx
k
ax
432015-11-27
The nearly free-electron approximation
Periodic potential
8𝜋
𝑎−𝜋
𝑎
6𝜋
𝑎
𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
𝜋
𝑎𝑥
xa
kx
k
ax
442015-11-27
8𝜋
𝑎−𝜋
𝑎
6𝜋
𝑎
𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
The nearly free-electron approximation
Periodic potential
xa
kx
k
ax
452015-11-27
8𝜋
𝑎−𝜋
𝑎
6𝜋
𝑎
𝜋
𝑎
4𝜋
𝑎−2𝜋
𝑎
2𝜋
𝑎
The nearly free-electron approximation
Periodic potential
band
band
band
462015-11-27
The electronic band structure
• It is convenient to present the energies only in the 1st Brillouin zone.• The electron state in the solid state is given by the wave vector of the 1st Brillouin zone, band number and a spin.
2015-11-27 47
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 – Linear Combination of Atomic Orbitals) 𝐻 = 𝐻𝐴 + 𝑉′:
𝐻𝐴 Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 = 𝐸𝑗𝜑𝑗 Ԧ𝑟 − 𝑅𝑛
𝐻 = 𝐻𝐴 + 𝑉′ = −ℏ2
2𝑚Δ + 𝑉𝐴 Ԧ𝑟 − 𝑅𝑛 +
𝑚≠𝑛
𝑉𝐴 Ԧ𝑟 − 𝑅𝑚
Perturbation: the influence of atoms in the neighborhood of 𝑅𝑚 :
𝑉′ Ԧ𝑟 − 𝑅𝑛 =
𝑚≠𝑛
𝑉𝐴 Ԧ𝑟 − 𝑅𝑚
Good for valence band of covalent crystals, 𝑑-orbital bands etc.
𝑗-th state 𝑒 Atom in position 𝑅𝑛Equation for the free atoms thatform the crystal
𝐻𝐴 = −ℏ2
2𝑚Δ + 𝑉𝐴 Ԧ𝑟 − 𝑅𝑛
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Approximate solution in the form of the Bloch function:
Φ𝑗,𝑘 Ԧ𝑟 =
𝑛
𝑎𝑛𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 =
𝑛
exp 𝑖 𝑘𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛
Check:Φ𝑗,𝑘+ Ԧ𝐺 Ԧ𝑟 = Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 + 𝑇 = exp 𝑖 𝑘𝑇 Φ𝑗,𝑘 Ԧ𝑟
Energies determined by the variational method:
𝐸 𝑘 ≤Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 Φ𝑗,𝑘 Ԧ𝑟
Expression
Φ𝑗,𝑘 Ԧ𝑟 Φ𝑗,𝑘 Ԧ𝑟 =
𝑛,𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚 න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
can be easily simplify assuming a small overlap of wave functions for 𝑛 ≠ 𝑚
Φ𝑗,𝑘 Ԧ𝑟 Φ𝑗,𝑘 Ԧ𝑟 =
𝑛
න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉 = ⋯
Tight-Binding Approximation
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Approximate solution in the form of the Bloch function:
Φ𝑗,𝑘 Ԧ𝑟 =
𝑛
𝑎𝑛𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 =
𝑛
exp 𝑖 𝑘𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛
Check:Φ𝑗,𝑘+ Ԧ𝐺 Ԧ𝑟 = Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 + 𝑇 = exp 𝑖 𝑘𝑇 Φ𝑗,𝑘 Ԧ𝑟
Energies determined by the variational method:
𝐸 𝑘 ≤Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 Φ𝑗,𝑘 Ԧ𝑟
𝐸 𝑘 ≈1
𝑁Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟 =
=
𝑛,𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚 න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝐸𝑗 + 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
Tight-Binding Approximation
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Approximate solution in the form of the Bloch function:
Φ𝑗,𝑘 Ԧ𝑟 =
𝑛
𝑎𝑛𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 =
𝑛
exp 𝑖 𝑘𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛
Check:Φ𝑗,𝑘+ Ԧ𝐺 Ԧ𝑟 = Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 + 𝑇 = exp 𝑖 𝑘𝑇 Φ𝑗,𝑘 Ԧ𝑟
Energies determined by the variational method:
𝐸 𝑘 ≤Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟
Φ𝑗,𝑘 Ԧ𝑟 Φ𝑗,𝑘 Ԧ𝑟
𝐸 𝑘 ≈1
𝑁Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟 =
=
𝑛,𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚 න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝐸𝑗 + 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
Tight-Binding Approximation
Only diagonal terms 𝑅𝑛 = 𝑅𝑚 in 𝐸𝑗
Only the vicinity of 𝑅_𝑛
2015-11-27 53
𝐸 𝑘 ≈1
𝑁Φ𝑗,𝑘 Ԧ𝑟 𝐻 Φ𝑗,𝑘 Ԧ𝑟 =
=
𝑛,𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚 න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝐸𝑗 + 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
When the atomic states 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 are spherically symmetric (𝑠-states), then overlap
integrals depend only on the distance between atoms:
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 𝐵𝑗
𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚
𝐴𝑗 = −න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑛 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
𝐵𝑗 = −න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
Restricted to only the nearest neighbours of 𝑅𝑛
Tight-Binding Approximation
Only diagonal terms 𝑅𝑛 = 𝑅𝑚 in 𝐸𝑗
Only the vicinity of 𝑅_𝑛
2015-11-27 54
When the atomic states 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 are spherically symmetric (𝑠-states), then overlap
integrals depend only on the distance between atoms:
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 𝐵𝑗
𝑚
exp 𝑖𝑘 𝑅𝑛 − 𝑅𝑚
𝐴𝑗 = −න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑛 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
𝐵𝑗 = −න𝜑𝑗∗ Ԧ𝑟 − 𝑅𝑚 𝑉′ Ԧ𝑟 − 𝑅𝑛 𝜑𝑗 Ԧ𝑟 − 𝑅𝑛 𝑑𝑉
The result of the summation depends on the symmetry of the lattice:
For 𝑠𝑐 structure: 𝑅𝑛 − 𝑅𝑚 = ±𝑎, 0,0 ; 0, ±𝑎, 0 ; 0,0, ±𝑎 ;
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 2𝐵𝑗 cos 𝑘𝑥𝑎 + cos 𝑘𝑦𝑎 + cos 𝑘𝑧𝑎
For 𝑏𝑐𝑐 structure :
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 8𝐵𝑗 cos𝑘𝑥𝑎
2cos
𝑘𝑦𝑎
2cos
𝑘𝑧𝑎
2
For𝑓𝑐𝑐 structure :
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 4𝐵𝑗 cos𝑘𝑥𝑎
2cos
𝑘𝑦𝑎
2+ 𝑐. 𝑝.
Tight-Binding Approximation
2015-11-27 55
For 𝑠𝑐 structure: 𝑅𝑛 − 𝑅𝑚 = ±𝑎, 0,0 ; 0, ±𝑎, 0 ; 0,0, ±𝑎 ;
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 2𝐵𝑗 cos 𝑘𝑥𝑎 + cos 𝑘𝑦𝑎 + cos 𝑘𝑧𝑎
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𝐵𝑗=−න𝜑𝑗∗Ԧ𝑟−𝑅𝑚
𝑉′Ԧ𝑟−𝑅𝑛
𝜑𝑗Ԧ𝑟−𝑅𝑛
𝑑𝑉
Tight-Binding Approximation
2015-11-27 56
For 𝑠𝑐 structure: 𝑅𝑛 − 𝑅𝑚 = ±𝑎, 0,0 ; 0, ±𝑎, 0 ; 0,0, ±𝑎 ;
𝐸𝑛 𝑘 ≈ 𝐸𝑗 − 𝐴𝑗 − 2𝐵𝑗 cos 𝑘𝑥𝑎 + cos 𝑘𝑦𝑎 + cos 𝑘𝑧𝑎
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𝐵𝑗=−න𝜑𝑗∗Ԧ𝑟−𝑅𝑚
𝑉′Ԧ𝑟−𝑅𝑛
𝜑𝑗Ԧ𝑟−𝑅𝑛
𝑑𝑉
Tight-Binding Approximation
2015-11-27 57
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, „The Band Theory of Graphite”, Physical Review 71, 622 (1947).] gives:
𝐸 𝑘 = ± 𝛾02 1 + 4 cos2
𝑘𝑦𝑎
2+ 4 cos
𝑘𝑦𝑎
2⋅ cos
𝑘𝑥 3𝑎
2≈ ℏ ǁ𝑐 𝑘 − 𝑘𝑖
Tight-Binding Approximation
2015-11-27 58
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, „The Band Theory of Graphite”, Physical Review 71, 622 (1947).] gives:
𝐸 𝑘 = ± 𝛾02 1 + 4 cos2
𝑘𝑦𝑎
2+ 4 cos
𝑘𝑦𝑎
2⋅ cos
𝑘𝑥 3𝑎
2≈ ℏ ǁ𝑐 𝑘 − 𝑘𝑖
Tight-Binding Approximation
2015-11-27 59
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, „The Band Theory of Graphite”, Physical Review 71, 622 (1947).] gives:
𝐸 𝑘 = ± 𝛾02 1 + 4 cos2
𝑘𝑦𝑎
2+ 4 cos
𝑘𝑦𝑎
2⋅ cos
𝑘𝑥 3𝑎
2≈ ℏ ǁ𝑐 𝑘 − 𝑘𝑖
Tight-Binding Approximation
2015-11-27 60
The number of states in the volume of 𝜋𝑘2:
𝑁 𝐸 =2
2𝜋 2 𝜋𝑘2 =2
2𝜋 2 𝜋 𝑘 − 𝑘𝑖2=
2
2𝜋 2 𝜋𝐸
ℏ ǁ𝑐
2
𝜌 𝐸 =𝜕𝑁 𝐸
𝜕𝐸=
𝐸
𝜋 ℏ ǁ𝑐 2
Linear dispersion relation in graphene:
Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, „The Band Theory of Graphite”, Physical Review 71, 622 (1947).] gives:
𝐸 𝑘 = ± 𝛾02 1 + 4 cos2
𝑘𝑦𝑎
2+ 4 cos
𝑘𝑦𝑎
2⋅ cos
𝑘𝑥 3𝑎
2≈ ℏ ǁ𝑐 𝑘 − 𝑘𝑖
Tight-Binding Approximation
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 𝑠, 𝑝, 𝑑 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 𝑁 atoms and thus 𝑁 primitive unit cells, then an atomic energy level 𝑬𝒊 of the free atom will split into 𝑁 states (due to the interaction with the rest of 𝑁 − 1 atoms).
Each band can be occupied by 2𝑁 electrons.
Tight-Binding Approximation
2015-11-27 62
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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 → next slide!)
If a crystal with a primitive cubic lattice contains 𝑁 atoms and thus 𝑁 primitive unit cells, then an atomic energy level 𝑬𝒊 of the free atom will split into 𝑁 states (due to the interaction with the rest of 𝑁 − 1 atoms). Each band can be occupied by 𝟐𝑵 electrons.
Tight-Binding Approximation
2015-11-27 63
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Be
C, Si, Ge
Tight-Binding ApproximationThe states can mix: for instance 𝑠𝑝3 hybridization.
2015-11-27 64
Tight-Binding Approximation
2015-11-27 65
Michał Baj
Tight-Binding Approximation
Fermi surfaces of metals
2015-11-27 66
Ashcroft, Mermin
Cu
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
Fermi surfaces of metals
2015-11-27 68
Michał Baj
Tight-Binding Approximation
2015-11-27 69
2015-11-27 70
Michał BajSzm
ulo
wic
z, F
., S
egal
l, B
.: P
hys
. Rev
. B2
1,
56
28
(1
98
0).
Tight-Binding Approximation
Tight-Binding Approximation
2015-11-27 71
Michał BajSzm
ulo
wic
z, F
., S
egal
l, B
.: P
hys
. Rev
. B2
1,
56
28
(1
98
0).
Density of stateslike for freeelectrons!
Tight-Binding Approximation
2015-11-27 72
Michał Baj
Cu: [1s22s22p63s23p6] 3d104s1
[Ar] 3d104s1
d-band
Tight-Binding Approximation
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 ↑ and ↓
Δ – Stoner gap
Semiconductors
2015-11-27 74
Semiconductors of the group IV (Si, Ge) –diamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) – zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) –NaCl structure
SiIndirect bandgap, Eg = 1,1 eV
Conduction band minima in Δ point, constant energy surfaces – ellipsoids (6 pieces), m||=0,92 m0, m⊥=0,19 m0,
The maximum of the valence band in Γ point, mlh=0,153 m0, mhh=0,537 m0, mso=0,234 m0, Δso= 0,043 eV
2015-11-27 75
GeIndirect bandgap, Eg = 0,66 eV
The maximum of the valence band in Γ point, mlh=0,04 m0, mhh=0,3 m0, mso=0,09 m0, Δso= 0,29 eV
Conduction band minima in Δ point, constant energy surfaces – ellipsoids (8 pieces), m||=1,6 m0, m⊥=0,08 m0,
SemiconductorsSemiconductors of the group IV (Si, Ge) –diamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) – zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) –NaCl structure
2015-11-27 76
GaAsDirect bandgap, Eg = 1,42 eV
The maximum of the valence band in Γ point, mlh=0,076 m0, mhh=0,5 m0, mso=0,145 m0, Δso= 0,34 eV
The minimum of the conduction band in Γ point, constant energy surfaces – spheres, mc=0,065 m0
SemiconductorsSemiconductors of the group IV (Si, Ge) –diamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) – zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) –NaCl structure
2015-11-27 77
a-SnDiamond structureZero energy gap, Eg = 0 eV (inverted band-structure) The maximum of the valence band in Γ point, mv=0,195 m0, mv2=0,058 m0, Δso=0,8 eV The minimum of the conduction band in Γ point, constant energy surfaces – spheres, mc=0,024 m0
SemiconductorsSemiconductors of the group IV (Si, Ge) –diamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) – zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) –NaCl structure
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 – ellipsoids , m||=0,068 m0, m⊥=0,034 m0,
The minimum of the conduction band in L point, constant energy surfaces – ellipsoids m||=0,07 m0, m⊥=0,04 m0,
SemiconductorsSemiconductors of the group IV (Si, Ge) –diamond structure
Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) – zinc-blend or wurtzitestructure
Semiconductors AIVBVI (np. SnTe, PbSe) –NaCl structure
Photoemission spectroscopy
2015-11-27 79
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ℏ𝜔 = 𝜙 + 𝐸𝑘𝑖𝑛 + 𝐸𝑏
work function potential barrier
Photoemission spectroscopy
2015-11-27 80
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ℏ𝜔 = 𝜙 + 𝐸𝑘𝑖𝑛 + 𝐸𝑏
work function potential barrier
Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
2015-11-27 81
Photoemission spectroscopy
http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html
Phys. Rev. B 71, 161403 (2005)
2015-11-27 82
Semiconductor heterostructures
2015-11-27 83
Semiconductor heterostructures
2015-11-27 84
Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
Bandgap engineering
2015-11-27 85
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
Heterostruktury półprzewodnikowe
2015-11-27 86
Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
Bandgap engineering
2015-11-27
87
Valence band offset (Anderson’s rule)
Valence band offset:
(powinowactwo)
htt
p:/
/en
.wik
iped
ia.o
rg/w
iki/
Het
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jun
ctio
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Bandgap engineering
2015-11-27 88
Valence band offset (Anderson’s rule)
Su-H
uai
Wei
, Co
mp
uta
tio
nal
Mat
eria
ls S
cien
ce, 3
0, 3
37
–34
8 (
20
04
)
Valence band offset:
(powinowactwo)
htt
p:/
/en
.wik
iped
ia.o
rg/w
iki/
Het
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jun
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Bandgap engineering
2015-11-27 89
Valence band offset
Wal
uki
ewic
zP
hys
ica
B 3
02
–30
3 (
20
01
) 1
23
–13
4
Bandgap engineering
2015-11-27 90
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
Vegard’s 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:
𝑎 = 𝑎𝐴 1 − 𝑥 + 𝑎𝐵𝑥
Bandgap engineering
2015-11-27 91
Vegard’s law:Relationship to band gaps of „binarycompound”:
𝐸 = 𝐸𝐴 1 − 𝑥 + 𝐸𝐵𝑥 − 𝑏𝑥(1 − 𝑥)
b – so-called „bowing” (curvature) of the energy gap
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress