introduction to solid state physics lecture ii

Sonia HaddadLaboratoire de Physique de la Matière Condensée

Faculté des Sciences de Tunis, Université Tunis El Manar

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Introduction to Solid State Physics

Lecture II

Outline

Lecture I: Introduction to Solid State Physics

• Brief story…

• Solid state physics in daily life

• Basics of Solid State Physics

Lecture II: Electronic band structure and electronic transport

• Electronic band structure: Tight binding approach

• Applications to graphene: Dirac electrons

Lecture III: Introduction to Topological materials

• Introduction to topology in Physics

• Quantum Hall effect

• Haldane model

2S. Haddad, ASP2021-26-07-2021-II

Outline

• Lecture I: Introduction to Solid State Physics

• A Brief story…

• Solid state physics in daily life

• Basics of Solid State Physics

Lecture II: Electronic band structure and electronic transport

• Tight binding approach

• Applications to graphene: Dirac electrons

Lecture III: Introduction to Topological materials

• Introduction to topology in Physics

• Quantum Hall effect

• Haldane model

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References

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Solid State Physics Neil Ashcroft and N. Mermin

Band Theory and Electronic Properties of Solids, John Singleton

The Oxford Solid State Basics, Steven Simons

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What are the consequences of the translational symmetry on the electronic properties?

Bloch Theorem

There are symmetries in the crystal! Translations

Lecture I: Introduction to solid state Physics

Bloch Theorem

Approximations

• Born-Oppenheimer: ions fixed compared to electrons𝑇𝑖𝑜𝑛𝑠 ≪ 𝑇𝑒 , 𝑉𝑖𝑜𝑛𝑠−𝑖𝑜𝑛𝑠 ≈ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

• Independent electrons

Crystal Hamiltonian:

𝐻𝑐𝑟𝑦𝑠 = 𝑇𝑖𝑜𝑛𝑠 + 𝑇𝑒 + 𝑉𝑒−𝑖𝑜𝑛𝑠 +𝑉𝑒−𝑒 + 𝑉𝑖𝑜𝑛𝑠−𝑖𝑜𝑛𝑠 Many-body problem!

Electron in crystals: classical or quantum?

Characteristic temperature in metals 𝑇𝐹 ≈ 104K ≫ 𝑇𝑎𝑚𝑏𝑖𝑎𝑛𝑡 Quantum Mechanics

𝐻𝑐𝑟𝑦𝑠 = σ𝑖𝐻 Ԧ𝑟𝑖 , 𝐻 Ԧ𝑟 =𝑝𝑖2

2𝑚+ 𝑉(Ԧ𝑟), one-body problem

𝑉(Ԧ𝑟)= 𝑉(Ԧ𝑟+𝑅) periodic potential


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Bloch Theorem

𝐻 Ԧ𝑟 =𝑝𝑖2

2𝑚+ 𝑉(Ԧ𝑟) with 𝑉(Ԧ𝑟)= 𝑉(Ԧ𝑟+𝑅) 𝐻 Ԧ𝑟 commute with 𝑇𝑅 (translational operator)

𝑇𝑅 is unitary Eigenfuntions of the form 𝑒𝑖𝑘.𝑅, 𝑘 wave vector

𝑇𝑅 , 𝐻 Ԧ𝑟 =0 Eigenfuntions of 𝐻 Ԧ𝑟 the form 𝜓𝑘( Ԧ𝜌, 𝑅) = 𝑓( Ԧ𝜌)𝑒𝑖𝑘.𝑅

Ԧ𝜌: position of the electron in the unit cell𝑓 Ԧ𝜌 defined in te unit cellԦ𝑟: position of the electron in the crystal

Ԧ𝑟

Ԧ𝜌

𝑅

Ԧ𝑟 = Ԧ𝜌 + 𝑅

𝜓𝑘(Ԧ𝑟) = 𝑢𝑘(Ԧ𝑟)𝑒𝑖𝑘. Ԧ𝑟

𝑢𝑘 Ԧ𝑟 + 𝑅 = 𝑢𝑘 Ԧ𝑟 cellular function

Bloch Theorem


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Quiz 5: Prove it.

Bloch Theorem

𝜓𝑘 Ԧ𝑟 = 𝑢𝑘 Ԧ𝑟 𝑒𝑖𝑘. Ԧ𝑟



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Consequences

𝜓𝑘(Ԧ𝑟 + 𝑅) = 𝑢𝑘( Ԧ𝑟 + 𝑅)𝑒𝑖𝑘. Ԧ𝑟+𝑅

𝜓𝑘(Ԧ𝑟 + 𝑅) = 𝜓𝑘(Ԧ𝑟) 𝑒𝑖𝑘.𝑅

𝜓𝑘 Ԧ𝑟 is not periodic in the real space

𝜓𝑘 Ԧ𝑟 is periodic in the real space (OK)

What is the corresponding energy?

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Electrons in crystals are under a periodic potential

Assumptions• crystal is infinite• Periodic Boundary conditions or Born–von Karman boundary conditions• Focus on bulk behavior (not on the surface)

For a free electron described by plane wave

Lecture I: Introduction to Condensed Matter Physics: Bloch Theorem

𝑁𝑗 ≫ 1 number of unit cells in the jth direction

𝑘 =𝑚1

𝑁1Ԧ𝑎1∗ +

𝑚2

𝑁2Ԧ𝑎2∗ +

𝑚3

𝑁3Ԧ𝑎3∗ , 𝑚𝑖 ∈ ℤ point in the reciprocal space

Periodic Boundary conditions

𝐾 = 𝑚1 Ԧ𝑎1∗ +𝑚2 Ԧ𝑎2

∗+ 𝑚3 Ԧ𝑎3∗ , 𝑚𝑖 ∈ ℤ point in the reciprocal Lattice

Real crystal (finite)

𝑁1𝑎1

𝑁2𝑎2

Tight binding approachLecture II: Introduction to solid state Physics

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State of art

Sommerfeld Model (1928):

• Ions are fixed • Conducting electrons described by quantum free electron gas (Fermi-Dirac

statistics)• Explain the contribution of electrons to the heat capacity of metals• BUT did not explain the exitance of insulators, semi-conductors, semi-metals…

Is there another picture to describe the crystal?


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Chemical point of view: crystal is formed by atoms (or group of atoms)

• Electrons are bound to their original atoms (tight binding)• Initial state: isolated atoms: interatomic distance 𝑑 ≫ lattice parameter ≫ 𝑎• Reduce d • Final state: crystal 𝑑 = 𝑎

𝑑 ≫ 𝑎

Reduce d

isolated atoms

crystal


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• Initial state: electrons are localized on their original atom • On the atom, electron wave function= atomic orbital (AO)• By reducing the interatomic distance: AO can overlap

3𝑠

Na Na

Na: 1𝑠22𝑠22𝑝6 3𝑠1


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in the crystal: Electron wave function= Linear combination of AO (LCAO approximation)

𝑑 ≫ 𝑎

Reduce d

isolated atoms

crystal

𝑎


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In tight binding approach (LCAO approximation) :

Could one explain the existence of semiconductors, insulators…?

𝑎


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Latticeparameter 𝑎

N isolated atoms crystal

휀𝑎3

휀𝑎1

휀𝑎2

By reducing the interatomic distance: degeneracy lifting

N x degenerate

N x degenerate

N x degenerate

Energy band


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𝑁 ≫ 1 x degenerate

𝑁 ≫ 1 x degenerate


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In tight binding approach: Could one explain the existence of semiconductors, insulators…?

YES

Depends on band filling: position of Fermi Energy

Fermi Energy: last occupied electronic energy level at T=0 K

atom: energy level fillingcrystal: energy band filling


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In tight binding approach: Could one explain the existence of semiconductors, insulators…?

YES

Depends on band filling: position of Fermi Energy

Fermi Energy: last occupied electronic energy level at T=0 K


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In tight binding approach:

What is the form of the electronic band energy?

• Derive the eigenenergy of the electronic Hamiltonian

𝐻 Ԧ𝑟 = 𝐻0 + ∆𝑉(Ԧ𝑟) with ∆𝑉(Ԧ𝑟) = ∆𝑉(Ԧ𝑟+𝑅)

𝐻0: Hamiltonian of one electron for isolated atoms𝐻0= 𝐻𝑎𝑡𝑜𝑚𝑖𝑐 on each site (atom)∆𝑉(Ԧ𝑟) periodic perturbation describing differences between the true potential in

the crystal and the potential of an isolated atom.

Monoelectronic Hamiltonian

𝐻𝑎𝑡𝑜𝑚𝑖𝑐𝐻𝑎𝑡𝑜𝑚𝑖𝑐 + ∆𝑉(Ԧ𝑟)


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In tight binding approach:


• Derive the eigenenergy of the electronic Hamiltonian

𝜓𝑘(Ԧ𝑟) = 𝑢𝑘(Ԧ𝑟)𝑒𝑖𝑘. Ԧ𝑟


𝐻 Ԧ𝑟 = 𝐻0 + ∆𝑉(Ԧ𝑟) with ∆𝑉(Ԧ𝑟) = ∆𝑉(Ԧ𝑟+𝑅)

• Keeping in mind: Electron wave function= Linear combination of AO

• Electron wave function is a Bloch function

Monoelectronic Hamiltonian


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Solve 𝐻| ൿ𝜓𝑘 = 𝐸 𝑘 | ൿ𝜓𝑘

• Variational method

Focus on one AO 𝜑𝑎 Ԧ𝑟 of energy 휀𝑎 : 𝐻𝑎𝑡𝑜𝑚𝑖𝑐 | ۧ𝜑𝑎 = 휀𝑎| ۧ𝜑𝑎

Example: as in alkaline crystals with valence electron in s orbital

Trial state: | ൿ𝜓𝑘 =1

𝑁σ𝑅𝑖𝐶𝑅𝑖(𝑘)| 𝜑𝑎,𝑅𝑖

Electron wave function= Linear combination of AO

Bloch function

𝐶𝑅𝑖 𝑘 = 𝑒𝑖𝑘.𝑅𝑖

𝜓𝑘(Ԧ𝑟 + 𝑅) = 𝜓𝑘(Ԧ𝑟) 𝑒𝑖𝑘.𝑅

𝐸 𝑘 =𝜓𝑘 𝐻 𝜓𝑘

𝜓𝑘 𝜓𝑘

Approximations: | ۧ𝜑𝑎 orthonormal basis

Quiz: Prove it.


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𝐸 𝑘 = 휀𝑎 + 𝛼 +

𝑅𝑖≠0

𝑡𝑅𝑖 𝑒𝑖𝑘.𝑅𝑖

𝑡𝑅𝑖 = 𝜑𝑎,0 𝐻 𝜑𝑎,𝑅𝑖: hopping integral between atom at the origin and the atom on site 𝑅𝑖

𝛼 = 𝜑𝑎,𝑅 𝐻 𝜑𝑎,𝑅 onsite overlap integral

𝐸 𝑘 = 휀𝑎 + 𝛼 +

𝑅𝛿

𝑡𝛿

𝑅𝑖≠0

𝑒𝑖𝑘.𝑅𝛿

𝑡𝑎 = 𝑡1𝑡−𝑎 = 𝑡1

𝑡2𝑎 = 𝑡2𝑡−2𝑎 = 𝑡2

Quiz: what should be the signs of 𝛼 and 𝑡𝑅𝑖 ?

electronic band structure


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𝐸 𝑘 = 휀𝑎 + 𝛼 +

𝑅𝛿

𝑡𝛿

𝑅𝑖≠0


𝐸 𝑘 + 𝐾 = 𝐸 𝑘 , 𝐾 ⊂ 𝑅𝐿, 𝑠𝑖𝑛𝑐𝑒 𝑒𝑖𝐾.𝑅= 1, 𝑅 ⊂ 𝐷𝐿

𝑅 = 𝑛1 Ԧ𝑎1 + 𝑛2 Ԧ𝑎2 + 𝑛3 Ԧ𝑎3, 𝑛𝑖 ∈ ℤ, point in the real (direct) Lattice

𝐾 = 𝑚1 Ԧ𝑎1∗ +𝑚2 Ԧ𝑎2

∗+ 𝑚3 Ԧ𝑎3∗ , 𝑚𝑖 ∈ ℤ point in the reciprocal Lattice

𝐸 𝑘 is periodic could be limited to the 1st Brillouin zone


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𝐸 𝑘 = 휀𝑎 + 𝛼 +

𝑅𝛿

𝑡𝛿

𝑅𝑖≠0

𝑒𝑖𝑘.𝑅𝛿𝑡1𝑡1

Example: 1D crystal with first neighbor hopping parameters

𝐸 𝑘 = 휀𝑎 − 2𝑡 cos 𝑘𝑥𝑎 (𝑡>0, 𝛼=0)

Which energies correspond to filled electronic states?Where is the Fermi level?

1st Brillouin zone


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Example: 1D crystal with first neighbor hopping parameter

Assume: 1 valence electron/atomUnit cell contains 1 atomCrystals: N unit cells

PBC: 𝑘 =𝑚1

𝑁1Ԧ𝑎1∗ +

𝑚2

𝑁2Ԧ𝑎2∗ +

𝑚3

𝑁3Ԧ𝑎3∗ , 𝑚𝑖 ∈ ℤ point in the reciprocal space

𝑁 = 𝑁1𝑁2𝑁3

𝑁1𝑎1

𝑁2𝑎2

Ԧ𝑎1∗ =

2𝜋

𝑎Ԧ𝑒𝑥 , Ԧ𝑎2

∗ =2𝜋

𝑎Ԧ𝑒𝑦 , Ԧ𝑎3

∗ =2𝜋

𝑎Ԧ𝑒𝑧

Cubic lattice

𝑘 in 1st BZ, −𝜋

𝑎< 𝑘𝑥≤

𝜋

𝑎, −

𝜋

𝑎< 𝑘𝑦≤

𝜋

𝑎, −

𝜋

𝑎< 𝑘𝑧≤

𝜋

𝑎

𝑘 in 1st BZ, , −𝑁1

2< 𝑚1 ≤

𝑁1

2, −

𝑁2

2< 𝑚3 ≤

𝑁2

2, −

𝑁3

2< 𝑚3 ≤

𝑁3

2

in 1st BZ there are N possible states 𝑘 satisfying the PBC

For each state 𝑘: 2 electronic states | ↑, ൿ𝑘 and | ↓, ൿ𝑘

Each band 𝐸 𝑘 can contain 2N electronic states


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Example 1: 1D crystal with first neighbor hopping parameters

Assume: 1 valence electron/atomUnit cell contains 1 atomCrystals: N unit cells

N electrons to distribute on electronic bands containing each 2N electronic states

Nb of filled band=N/2N=1/2 band

𝐸𝐹

filled

empty


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𝐸 𝑘 = 휀𝑎 + 𝛼 +

𝑅𝛿

𝑡𝛿

𝑅𝑖≠0


𝐸 𝑘 = 휀0 − 2t ቀcos 𝑎𝑘𝑥 + cos ൯𝑎𝑘𝑦

Irreducible BZ

High symmetry directions in IBZ

Example 2: square crystal with first neighbor hopping parameter

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Graphene, the wonder material


Mechanical exfoliation (scotch)

Chemical Vapor Deposition (CVD) Methane heated and deposit on substrate graphene

Epitaxy :

Silicon carbide (SiC) heated (>1100 °C) graphene

LPMC, FST, Université Tunis El Manar 2929


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Part II: graphene

Graphene: 2D crystalCarbon atoms arranged in a honeycomb lattice

Is honeycomb lattice a Bravais lattice?

?

NO

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31Mosaic in El Jem (Roman city in Tunisia) Photo by Gilles Montambaux

Honeycomb lattice in a Roman city (Tunisia)

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Graphene:

Unit cell: 2 first neighboring carbon atoms (chemically inequivalent)

Bravais lattice: hexagonal (triangular)

Lattice parameter 𝒂 = 𝒂𝟎 𝟑, 𝒂𝟎 distance between 1st neighboring atoms

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Part II: graphene: crystalline structure

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Graphene:

C: 1𝑠22𝑠22𝑝2 4 valence electrons

hybridization sp2

s Bond (inplane 120°)orbital p perpendicular to the plane: responsible of electronic conduction

33


C: 1𝑠2 2𝑠2 2𝑝𝑥1 2𝑝𝑦

1 2𝑝𝑧

↑↓↑↓ ↑ ↑

sp2

↓↑↓ ↑ ↑

C: 1𝑠2 2𝑠2 2𝑝𝑥1 2𝑝𝑦

1 2𝑝𝑧1

↑

Mix 1 AO s and 2 AO p

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Zinc Blende crystalstructure

Carbon C: [He] 2s22p2

hybridation sp3


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Graphene: electronic band structure

Direct lattice Reciprocal lattice

Ԧ𝑎𝑗 . 𝑏𝑗 = 2𝜋 𝛿𝑖𝑗

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𝐀𝐦−𝟏,𝐧+𝟏 𝐀𝐦+𝟏,𝐧−𝟏

3636

Tight binding approach: 2 valence electrons / unit cell

𝑡1

𝑅𝑚,𝑛 = 𝑚 Ԧ𝑎1 + 𝑛 Ԧ𝑎2

Ԧ𝑎1 = 𝑎1

2,3

2et Ԧ𝑎2 = 𝑎 −

1

2,3

2

φm,nA |φm′,n′

A = φm,nB |φm′,n′

B = 𝛿𝑚𝑚′𝛿𝑛𝑛′ et φm,nA |φm′,n′

B = 0

φm,nB |H|φm′,n′

A = φm,nA |H|φm′,n′

B = −𝑡1 if 𝐵𝑚,𝑛 and 𝐴𝑚′,𝑛′ are <1>

LCAO: two AO/unit cell | φm,nۄA et | φm,nۄ

B

𝐶 𝑅𝑚+1,𝑛 = 𝑒𝑖𝑘.𝑎1𝐶 𝑅𝑚,𝑛

𝐶 𝑅𝑚,𝑛+1 = 𝑒𝑖𝑘.𝑎2𝐶 𝑅𝑚,𝑛 ,

𝑑 𝑅𝑚+1,𝑛 = 𝑒𝑖𝑘.𝑎1𝑑 𝑅𝑚,𝑛 ,

𝑑 𝑅𝑚,𝑛+1 = 𝑒𝑖𝑘.𝑎2𝑑 𝑅𝑚,𝑛


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| ψۄ𝑘=

𝑅𝑚,𝑛

𝐶 𝑅𝑚,𝑛 φm,nۄA + 𝑑 𝑅𝑚,𝑛 φm,nۄ

B

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Project 𝐻| ψۄ 𝑘 = 휀(𝑘)| ψۄ 𝑘 on | φm,nۄA and | φm,nۄ

B

ቐ휀(𝑘)𝐶 𝑅𝑚,𝑛 − 𝑡1 𝑓

∗ 𝑘 𝑑 𝑅𝑚,𝑛 = 0

−𝑡1𝑓 𝑘 𝐶 𝑅𝑚,𝑛 + 휀 𝑘 𝑑 𝑅𝑚,𝑛 = 0

𝑓 𝑘 = 1 + 𝑒𝑖𝑘.𝑎1+ 𝑒𝑖𝑘.𝑎2

Ԧ𝑎1 = 𝑎1

2,3

2et Ԧ𝑎2 = 𝑎 −

1

2,3

2

Band index

휀𝜆 (𝑘)


E(k)

38

K’K

CB2 bands

Fermi level: at K et K’

Graphene is a semi-metal

VB

38


1 valence electron / atom

2 valence electrons / unit cell

Quiz: How many bands to fill?

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N : nb of unit cells

2N electrons to distribute on bands

containing each 2N electronic states

One filled band

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Graphène: structure de bandes?

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Second quantization approach


𝐻 = −𝑡

𝑅𝐴

𝑐𝑅𝐴

+

𝑖=1

3

𝑐𝑅𝐴+ Ԧ𝑑𝑖

+ ℎ. 𝑐.

𝑐𝐴+ 𝑘 =

1

𝑁

𝑅𝐴

𝑒𝑖𝑘.𝑅𝐴 𝑐𝑅𝐴

+ , 𝑐𝐵+ 𝑘 =

1

𝑁

𝑅𝐴

𝑒𝑖𝑘.𝑅𝐴 𝑐𝑅𝐴+ Ԧ𝑑3

+

𝐻 =

𝑘⊂1 𝑍𝐵

Ψ+ 𝑘 ℎ 𝑘 Ψ 𝑘

Ψ 𝑘 =𝑐𝐴 𝑘

𝑐𝐵 𝑘Ψ+ 𝑘 = 𝑐𝐴

+ 𝑘 𝑐𝐵+ 𝑘 ℎ 𝑘 =

0 𝑓 𝑘

𝑓∗ 𝑘 0

Developing 𝑓 𝑘 around K and K’ points (at low energy)

𝑓 𝑘 = 1 + 𝑒𝑖𝑘.𝑎1+ 𝑒𝑖𝑘.𝑎2

Relation de dispersion

Linear dispersion relation around the points K et K’

41

around K and K’ points, the electrons behave as massless fermions

Dirac electrons

Electron has zero

effective mass

41

K and K’: Dirac points

Dirac cone

in valley K, K’

Hamiltonian at low energy (around Fermi level):


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𝐻𝜉 𝑘 = ℏ𝑣𝐹 𝜉𝜎𝑥𝑘𝑥 + 𝜎𝑦𝑘𝑦

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Thank you for your attention

Any questions?

introduction to solid state physics lecture ii

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