Prezentacja programu PowerPointszczytko/LDSN/2_LDSN_2015_Tight_Binding.pdfBloch function has a form:...

Tight-Binding Approximation

Faculty of Physics UW

[email protected]

A wave packet

2015-11-27 2S. Kryszewski Gdańsk 2010

A wave packet

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Note that

is a fourrier transform of

Gaussian packet:

S. Kryszewski Gdańsk 2010

A wave packet

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Gaussian packet:


A wave packet

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Derivation:


A wave packet

2015-11-27 6S. Kryszewski Gdańsk 2010

A wave packet

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Ordering expression:

A wave packet

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Ordering expression:


Pakiet falowy

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A wave packet

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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

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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 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

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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

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∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ



Brillouin zones

Periodic potential

2-D square lattice

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∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ



Brillouin zones

Periodic potential

2-D square lattice

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∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ



Brillouin zones

Periodic potential

2-D hexagonal lattice

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∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ



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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∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ

Brillouin zones

Periodic potential

Ibach, Luth

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bcc lattice

heksagonal lattice


∗ +𝑚2 Ԧ𝑎2∗ +𝑚3 Ԧ𝑎3

∗ , 𝑚𝑖 ∈ ℤ

Brillouin zones

Periodic potential

Potencjał 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 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).

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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.

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𝑉 Ԧ𝑟 =

Ԧ𝐺

𝑉Ԧ𝐺 exp 𝑖 Ԧ𝐺 Ԧ𝑟

𝜓𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟


Ԧ𝐺


Bloch theorem

Periodic potential

Przykład:Electron motion in a periodic potential.

Thus:

The solution is:

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𝑉 Ԧ𝑟 =

Ԧ𝐺

𝑉Ԧ𝐺 exp 𝑖 Ԧ𝐺 Ԧ𝑟

𝜓𝑛,𝑘 Ԧ𝑟 = 𝑢𝑛,𝑘 Ԧ𝑟 𝑒𝑖𝑘 Ԧ𝑟


Ԧ𝐺


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?

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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

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Periodic potential


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

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Periodic potential


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

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Periodic potential


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



Periodic potential


2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =

ℏ2 𝑘 + Ԧ𝐺2

2𝑚

−8𝜋

𝑎

8𝜋

𝑎−6𝜋

𝑎

6𝜋

𝑎−4𝜋

𝑎

4𝜋

𝑎

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−2𝜋

𝑎

2𝜋

𝑎



Periodic potential


2𝑚= 𝐸 𝑛, 𝑘 + Ԧ𝐺 =

ℏ2 𝑘 + Ԧ𝐺2

2𝑚

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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

– –


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


kx

– –


Periodic potential

𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 =ℏ2 𝑘 + Ԧ𝐺

2

2𝑚


𝑔𝑖 =2𝜋

𝑎𝑖

𝜋

𝑎𝑥−𝜋

𝑎𝑥

[hkl]=

000,

100,100, 200, 200,

010,010,001,001,

352015-11-27


kx

– –


Periodic potential

𝐸 𝑛, 𝑘 = 𝐸 𝑛, 𝑘 + Ԧ𝐺 =ℏ2 𝑘 + Ԧ𝐺

2

2𝑚


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


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


Periodic potential

Plane waves of the same 𝑘-vector 𝜋

𝑎𝑥

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𝜆 =2𝜋

𝑘

kx

k

ax


Periodic potential


𝑎𝑥

probability density

probability density

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kx

k

ax


Periodic potential


𝑎𝑥

probability density

probability density

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kx

k

ax


Periodic potential


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


Periodic potential


𝑎𝑥

see Ibach, Lutch

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kx

k

ax

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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


Periodic potential

xa

kx

k

ax

452015-11-27

8𝜋

𝑎−𝜋

𝑎

6𝜋

𝑎

𝜋

𝑎

4𝜋

𝑎−2𝜋

𝑎

2𝜋

𝑎


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


2015-11-27 48

W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics



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 = ⋯


2015-11-27 51



𝑛


𝑛







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


2015-11-27 52



𝑛


𝑛







𝐸 𝑘 ≈1


=

𝑛,𝑚



Only diagonal terms 𝑅𝑛 = 𝑅𝑚 in 𝐸𝑗

Only the vicinity of 𝑅_𝑛

2015-11-27 53

𝐸 𝑘 ≈1


=

𝑛,𝑚


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


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.


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𝐵𝑗=−න𝜑𝑗∗Ԧ𝑟−𝑅𝑚

𝑉′Ԧ𝑟−𝑅𝑛

𝜑𝑗Ԧ𝑟−𝑅𝑛

𝑑𝑉


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𝐵𝑗=−න𝜑𝑗∗Ԧ𝑟−𝑅𝑚

𝑉′Ԧ𝑟−𝑅𝑛

𝜑𝑗Ԧ𝑟−𝑅𝑛

𝑑𝑉


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


2015-11-27 58



𝐸 𝑘 = ± 𝛾02 1 + 4 cos2

𝑘𝑦𝑎

2+ 4 cos

𝑘𝑦𝑎

2⋅ cos

𝑘𝑥 3𝑎



2015-11-27 59



𝐸 𝑘 = ± 𝛾02 1 + 4 cos2

𝑘𝑦𝑎

2+ 4 cos

𝑘𝑦𝑎

2⋅ cos

𝑘𝑥 3𝑎



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



𝐸 𝑘 = ± 𝛾02 1 + 4 cos2

𝑘𝑦𝑎

2+ 4 cos

𝑘𝑦𝑎

2⋅ cos

𝑘𝑥 3𝑎



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.


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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.


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Be

C, Si, Ge

Tight-Binding ApproximationThe states can mix: for instance 𝑠𝑝3 hybridization.

2015-11-27 64


2015-11-27 65

Michał Baj


Fermi surfaces of metals

2015-11-27 66

Ashcroft, Mermin

Cu


2015-11-27 67

http://physics.unl.edu/tsymbal/teaching/SSP-927/Section%2010_Metals-Electron_dynamics_and_Fermi_surfaces.pdf


2015-11-27 68

Michał Baj


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).



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!


2015-11-27 72

Michał Baj

Cu: [1s22s22p63s23p6] 3d104s1

[Ar] 3d104s1

d-band


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


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



2015-11-27 76

GaAsDirect bandgap, Eg = 1,42 eV


The minimum of the conduction band in Γ point, constant energy surfaces – spheres, mc=0,065 m0




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




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,




Photoemission spectroscopy

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ℏ𝜔 = 𝜙 + 𝐸𝑘𝑖𝑛 + 𝐸𝑏

work function potential barrier


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ℏ𝜔 = 𝜙 + 𝐸𝑘𝑖𝑛 + 𝐸𝑏

work function potential barrier


http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html

Phys. Rev. B 71, 161403 (2005)

2015-11-27 81

http://link.aps.org/doi/10.1103/PhysRevB.71.161403


http://www.physics.berkeley.edu/research/lanzara/research/Graphite.html

Phys. Rev. B 71, 161403 (2005)

2015-11-27 82

http://link.aps.org/doi/10.1103/PhysRevB.71.161403

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)

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p:/

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Het

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Bandgap engineering

2015-11-27 88

Valence band offset (Anderson’s rule)

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Wei

, Co

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uta

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Mat

eria

ls S

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ce, 3

0, 3

37

–34

8 (

20

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Valence band offset:

(powinowactwo)

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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


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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Sem

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