IMPRS band structure pdf.ppt - Max Planck · PDF file1.4 Sommerfeld model in 3D free electrons in solids ... 2.1 Electronic band structure of solids band structure Effective one-electron

IMPRS Block Course, 22.3. – 26.3.2013

Band structure of solids - „a physicist point of view“

Martin WolfFritz-Haber-Institut of the Max-Planck-Society, Berlin, Germany

1. Electrons in solids: the free electron gas

2 Electronic band structure of periodic crystals2. Electronic band structure of periodic crystals

3. Examples & angle-resolved photoemission

4. Surface states

2Solid state physics - concepts

Structure of solid state materials

Introduction

Structure of solid state materials

- model system: single crystals with periodic boundary conditions - but defects play important role doping in semiconductors, optical materials - alloys and amorphous solids technological relevance (steel, glass, ...)

Theoretical description of solids

Schrödinger equation for electrons and nuclei

electron density of Si

- Schrödinger equation for electrons and nuclei

+ Si+ SiBUT: how for ~1023 atoms / cm3 ! ? Born-Oppenheimer approximation

- Maxwell equations for electromagnetic fields

- Statistical physics and thermodynamics

+ Si Density functional theory (DFT): Hohnberg+Kohn: „All physical observablesare uniquely described by (electron) density (r)”

→ distribution functions for electrons (Fermions) and phonons (Bosons)

Solids are many body systemsq y y ( ) y ( ) ab initio calculation of solid state properties

Solids are many body systems

- electron-phonon interaction - exchange and correlation effects

useful concept: single particle picture

EFermi

occupied- useful concept: single particle picture- however simple models (e.g. Drude model)

describe often already essential physics

occupiedstates

31. Electrons in solids: Overview free electrons in solids

Ashcroft / Mermin

41.1 Electrons in solids: Sommerfeld model free electrons in solids

Effective one-electron potential: (rigid lattice)

V - Born-Oppenheimer approximation- Single electron approximation

Sommerfeld model:particle in a box“„particle in a box

51.2 The free electron gas free electrons in solids

Sommerfeld model in 1DSommerfeld model in 1D

V

61.3 Standing waves free electrons in solids

D. Eigler (IBM): STM image of iron atoms on Cu(111)

71.4 Sommerfeld model in 3D free electrons in solids

Sommerfeld model in 3DSommerfeld model in 3D

for periodic boundary conditions !

81.5 The density of states (DOS) free electrons in solids

spin

F

E

EndEEEDUF

53

0

91.6 Fermi gas at T = 0 free electrons in solids

Fermi distribution: Density of states: Fermi sphere:

f (E T 0) EmED

23

22

22

1

Fermi distribution: Density of states: Fermi sphere:

22

2 FF km

E

f (E, T=0)

101.7 Fermi Dirac distribution free electrons in solids

T 5000 KT = 5000 K

112.1 Electronic band structure of solids band structure

Effective one-electron potential:

B O h i i ti- Born-Oppenheimer approximation- Single electron approximation- Periodic potential

- Diffraction of electron waves deviation from free electron

b d t t (b d )

p

band structure (band gaps,…)

12Reminder: Lattice planes and Miller Indexes Diffraction

Lattice planes: 3 lattice vectors 321ian which do not fall ona straight line define a lattice plane, which is characterizedby 3 integers, the Miller indices hkl:

Lattice planes:33an 3 lattice vectors 3,2,1, iiian which do not fall on

1a2a

3a

11an

22an

Zlkhn

mln

mkn

mh ,,,1,1,1

321

The reciprocal lattice vector:

321 gggG lkhhkl Ghkl

is perpendicular to the lattice plane (hkl) and the distance b t dj t l ibetween adjacent planes is:

Gdhkl

2

Ghkl

dhkl

13Reminder: Diffraction and Bragg law Diffraction

2Gk hkl

hkl dk 2sin2 0 GThe Laue condition is equivalent to:

This yields the Bragg law for diffraction sin2 hkldhkl

Bragg reflection

k

Ghkl

Bragg reflectionat crystal planes hkl

k

k0dhkl

2k0 Ghkl

Bragg: Laue:

Diffraction Reminder: The reciprocal lattice

Zlkhlkh GZ3 RF d

Zmm ,2RG

Zlkhlkh ,,,321 gggG

ijji 2agthe condition requires:

Znan ii

ii 1

,RFor and

The resulting basis vectors of the reciprocal lattice are:

lkjlkj ,,aagand therefore

)(2,

)(2,

)(2

321

213

321

132

321

321 aaa

aagaaa

aagaaa

aag

Real space latticeExample: 2D - lattice Reciprocal lattice

units [m] units [m-1]

15Reminder: Wigner-Seitz cell Structure

1/21/2

Goal: primitive unit cell (with minimum volume)

Example: construction in 2 dimensions

bcc fcc

C4

C3

C4

bcc fccC3

C3

C2C2

The Wigner Seitz cell comprises all points closer to a certain lattice point

The Wigner-Seitz cell reflects point group symmetry of the lattice

The Wigner Seitz cell comprises all points closer to a certain lattice point than to any other point of the Bravais lattice

16Reminder: Brillouin zones Structure

bcc lattice fcc lattice hexagonal

The first Brillouin zone is defined as the Wigner Seitz cell of the reciprocal lattice

bcc lattice fcc lattice hexagonallattice

dirrez a

a 4

dirrez a

a 4

dirrez a

a34

dir

rez cc 4

Note: The reciprocal lattice of the bcc lattice is the fcc lattice and vice versa.

172.1 Electronic band structure of solids band structure

Effective one-electron potential:

B O h i i tiE(k) = ħk2/2m

- Born-Oppenheimer approximation- Single electron approximation- Periodic potential

- Bragg reflexion at Brillouin zoneboundary standing waves

p

- Opening of band gaps

Experimental observation by photoelectron spectroscopy

Courtesy of Eli Rotenberg ‐ ALS

192.2 Origin of the band gap band structure

band gap

202.3 Bloch Theorem band structure

U t ti S h ödi tiUse stationary Schrödinger equation:

for a periodic potential in a crystal: with lattice vector

This implies (non-degenerate wave functions):

Solution Bloch theorem:

The wave function in a periodic potential is given by aThe wave function in a periodic potential is given by aa periodic function uk(r) modulated by a plane wave:

212.4 Bloch Theorem band structure

Translational symmetryTranslational symmetry

Bloch-wave:

Symmetry properties:

222.5 Consequences of the Bloch Theorem band structure

Th Bl h th i ld f th f ti d Ei t t

)Gk()k( nn

The Bloch theorem yields for the wave functions and Eigenstates:

)Gk(E)k(E nn

where the index n counts the Eigenstates En of electrons in the periodic potential.B d t t b d d t th 1 B ill iBand structure can be reduced to the 1. Brillouin zone

Repeated zonescheme:

Extended zonescheme:

Reduced zonescheme:

G

232.6 Consequences of the Bloch Theorem band structure

ExtendedExtended zone sheme

-G

Reduced zone sheme o e s e e

242.7 Nearly free electron approximation band structure

252.9 Tight binding approximation band structure

“perturbation”HA i = Ei i

p

kk HkE

)(

i h 1

kk

kE

)(

with m = n ± 1

m = n ± 1


px x

s

px x

s

s-orbital is orbital i


E2

E1

Egap

282.12 Electronic band structure in 3D band structure

292.13 Tight binding approximation: Examples band structure

1.BZ 2.BZ

302.14 Tight binding approximation: Examples band structure

Tight binding Bloch wave:

Band structure of Aluminium using 1 s, 3 p and 5 d basis set (j = 1 . . . 9) for tight binding calc.

from: D. A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum Press (New York) 1986.

313.1 Classification of solid state materials band structure

EF

EF

kT

323.2 Photoemission spectroscopy band structure

e-

verticaltransition

transition

333.2 Photoemission spectroscopy band structure

verticaltransitiontransition

Determination of band structure:

Assume free electron final state:

Courtesy of Eli Rotenberg ‐ ALS

353.3 Band structure and density of states band structure

V

363.4 Examples of band structures band structure

Egap

point

ener

gy K point

Fermilevel

Graphene

374.1 Surface states surface states

B k t l ti l t t f l d t l ti f• Broken translational symmetry at surface can lead to solutions ofthe Schrödinger equation which are localized at the surface.

• Surface state exists in band gaps of the bulk projected band structured f tl di l ll l t b lk b d dand frequently disperse nearly parallel to bulk band edges

• Surface states do not disperse normal to the surface

6 Elid

F n=1

6

4

EvacCu(111)solid vacuum

E

E -

EF

n=02

E

sp-inverted Band gap

Surface t t

projected band gap

Evac

0

2

EFstate

0.0 0.5 1.0-2

k (Å )||-1 M

zsurface



• Surface state exists in band gaps of the bulk projected band structured f tl di l ll l t b lk b d dand frequently disperse nearly parallel to bulk band edges

• Surface states do not disperse normal to the surface

Shockley state: 6 EShockley state:- contribution of s/p-orbitals dominate- delocalized parallel to surface- pronounced dispersion E

F n=14

EvacCu(111)

p p

E - n=0

0

2

EFTamm state- contribution of d-orbitals dominate (or pz)- more localized character

0.0 0.5 1.0-2

k (Å )||-1 M

more localized character- small dispersion

Surface resonance:- Resonant with bulk bands (& strong mixing)Resonant with bulk bands (& strong mixing)- Penetration into the bulk (weak localization at surface)



• Example: two-band model for the Cu(111) surface

~e-z

404.3 Surfacs states surface states


• Example: two-band model for the Cu(111) & Cu(001) surface

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IMPRS band structure pdf.ppt - Max Planck · PDF file1.4 Sommerfeld model in 3D free electrons in solids ... 2.1 Electronic band structure of solids band structure Effective one-electron