4. Electrons in a solid 3. Crystal vibrations 4.2. Free-electron - - PowerPoint PPT Presentation

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4. Electrons in a solid 3. Crystal vibrations 4.2. Free-electron - - PowerPoint PPT Presentation

Jan 07, 2023 106 likes •198 views

4. Electrons in a solid 3. Crystal vibrations 4.2. Free-electron gas 3. Crystal vibrations 90 92 Effective one-electron potential 4.1. Bloch - Theorem 3. Crystal vibrations 4.2. Density of states 3. Crystal vibrations 91 93

slide-1
SLIDE 1

90

  • 4. Electrons in a solid
  • 3. Crystal vibrations

Effective one-electron potential

91

4.1. Bloch - Theorem

  • 3. Crystal vibrations

Translational symmetry yields Bloch-wave: and therefore:

92

4.2. Free-electron gas

  • 3. Crystal vibrations

93

4.2. Density of states

  • 3. Crystal vibrations
slide-2
SLIDE 2

94

4.2. Dispersion relation - bandstructure

  • 3. Crystal vibrations

95

4.3. Fermi distribution-function

  • 4. Electrons in solids

f(E,T) E/k (10 K)

B 4

15.4 2 k T

B

E E

vac kin

EF h e ν

1.0 0.8 0.6 0.4 0.2 0.0

f(E,T)

0.4 0.2 0.0

  • 0.2

E - EF (eV)

T=

750 K 500 K 290 K 100 K

96

4.4. Specific heat

  • 4. Electrons in solids

E D(E)

97

4.4. Specific heat

  • 4. Electrons in solids
slide-3
SLIDE 3

98

4.5. Electrostatic Screening

  • 4. Electrons in solids

for slowly varying potential δU: Poisson equation: Solution: whereby Thomas-Fermi screening length:

99

4.5. Mott insulator

  • 4. Electrons in solids

energy

r

n.n. dist. Evac e /r 1/r

r/rTF

Edeloc Eloc

n metal insulator conductivity

In a shallow potential the number

  • f bound states decreases

increase of n ⇒ decrease of rTF ⇒ free electrons Mott transition: insulator ⇒ metal

100

4.5. Thermionic emission

  • 4. Electrons in solids

Work function

101

4.6. Nearly free electrons

  • 4. Electrons in solids
slide-4
SLIDE 4

102

4.6. Nearly free electrons

  • 4. Electrons in solids

103

4.6. Nearly free electrons

  • 4. Electrons in solids

104

4.6. Nearly free electrons

  • 4. Electrons in solids

K=2π/a

Extended zone scheme Reduced zone scheme Repeated zone scheme

105

4.6. Solution at Brillouin zone-boundary

  • 4. Electrons in solids

Standing waves:

slide-5
SLIDE 5

106

Energie kz G/2 |V| |V|

Bandlücke: 2V z=0 z OF

s-Welle p-Welle

a

4.6 Standing wave at Brillouin zone-boundary

V(z)

  • 4. Electrons in solids

107

4.7 Tight binding bandstructure

Aluminium-Bandstruktur mit 1 s-artigen, 3 p-artigen und 5 d-artigen Basisfunktionen (j = 1 . . . 9) nach: D. A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum Press (New York) 1986.

  • 4. Electrons in solids

108

4.7 s- and pz- waves

  • 4. Electrons in solids

109

4.8 Examples - metall

Cu, fcc: 3s2 3p6 3d10 4s1, s(p) and d bands, metal

  • 4. Electrons in solids
slide-6
SLIDE 6

110

4.8 Examples - semiconductor

Ge, 4s2 4p2, → sp3 hybridized, absolute band gap, semiconductor

  • 4. Electrons in solids

111

4.8 Examples – Fermi surfaces

Fermi surfaces

Al

  • 4. Electrons in solids

112

4.8 Occupied states – photoelectron spectroscopy

vertical transition

  • 4. Electrons in solids

113

4.8 Unoccupied states

Inverse photoemission (IPES) Two-photon photoemission (2PPE)

  • 4. Electrons in solids
slide-7
SLIDE 7

109

4.8 Examples - metall

Cu, fcc: 3s2 3p6 3d10 4s1, s(p) and d bands, metal

  • 4. Electrons in solids

110

4.8 Examples - semiconductor

Ge, 4s2 4p2, → sp3 hybridized, absolute band gap, semiconductor

  • 4. Electrons in solids

111

4.8 Examples – Fermi surfaces

Fermi surfaces

Al

  • 4. Electrons in solids

112

4.8 Occupied states – photoelectron spectroscopy

vertical transition

  • 4. Electrons in solids
slide-8
SLIDE 8

113

4.8 Unoccupied states

Inverse photoemission (IPES) Two-photon photoemission (2PPE)

  • 4. Electrons in solids