MENA2000 Electrons: Lectures 5-6 (week 19)

Effective mass method; electrons and holes;

• Internal and external forces affecting an electron in crystal

• Dynamics of electrons in a band

• Hole - an electron near the top of an energy band

Effective mass method for calculating energy levels for defects

• Hydrogen-like impurities

• Recap of the hydrogen model

• Calculation of the donor and acceptor levels

MENA2000 : Fundamental structure of solid materials - Lecture Plan calender week

Ti 15/1 14-16 Introduction to the course

Module I – Crystallography and crystal structures (H, Fjellvåg, föreläsningskompendium)

Må 21/1 10-12 Basis and unit cells, Bravais lattices (2D and 3D), density, etc 4

Ti 22/1 14-16 Point symmetry; stereographical projections; crystal planes and directions

Må 28/1 10-12 Symmetry of molecules; introduction to groups; Crystal structures; concepts, digital tools 5

Ti 29/1 14-16 continued

Må 04/2 10-12 Translation symmetry, space groups, international tables for crystallography 6

Ti 05/2 14-16 continued

Må 11/2 10-12 Crystal structures of functional materials II; defects, solid solutions 7

Ti 12/2 14-16 Reciprocal lattice; Diffraction, XRD fingerprint analysis

Module II – Phonons C.Kittel’s Introduction to Solid State Physics; chapters 3 (pp73-85) 4, 5, and 18 (pp.557-561)

Må 18/2 10-12 Analysis of elastic strains in crystals; elastic waves. 8

Ti 19/2 14-16 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D

Må 25/2 10-12 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 9

Ti 26/2 14-16 Effect of temperature - Planck distribution; lattice heat capacity: Dulong-Petit, Einstein, and Debye models

Må 04/3 10-12 Comparison of different lattice heat capacity models 10

Ti 05/3 09-10 Thermal conductivity and thermal expansion

Module III – Molecular orbital therory (H.Sønsteby, Molecular Symmetry in Solid-State Chemistry, föreläsningskompendium)

Må 11/3 10-12 Introduction to symmetry and molecular symmetry 11

Ti 12/3 14-16 Group theory for material scientists

Må 18/3 10-12 Character tables 12

Ti 19/3 14-16 Bonding from a symmetry perspective

lecture-free week 13

Må 01/4 10-12 The role of symmetry in vibrational spectroscopy 14

Ti 02/4 14-16 Crystal field theory and ligand field theory

Må 08/4 10-12 Interplay between MO-theory and band structure in 1D 15

Ti 09/4 14-16 Band structure and density of states (DOS) in 3D from MO-theory

Easter 16

Module IV – Electrons C.Kittel’s Introduction to Solid State Physics; chapters 6, 7, 11 (pp 315-317) 18 (pp. 528-530) , and Appendix D

Ti 23/4 14-16 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 17

Må 29/4 10-12 DOS of FEFG in 1D and 3D at ground state (T=0) 18

Må 06/5 10-12 Effect of temperature – Fermi-Dirac distribution; Fermi energy and Fermi level; heat capacity of FEFG 19

Ti 07/5 14-16 Origin of the band gap and nearly free electron model

Fr 10/5 08-10 Effective mass method; electrons and holes; 20

Må 20/5 10-12 Effective mass method for calculating localized energy levels for defects in crystals 21

Ti 21/5 14-16 Carrier statistics in semiconductors and p-n junctions

Summary

Må 27/5 10-12 Course in a nutshell

Examination

To 06/6 09-13

MENA2000

Faste materialers fundamentale oppbygning

https://www.uio.no/studier/emner/matnat/fys/MENA2000/

krystallers atomære oppbygging, hvordan krystaller klassifiseres, symmetri kjemiske bindinger og molekylorbitalteoriteori, og hvordan en periodisk struktur gir opphav til fenomener som gittervibrasjoner og elektroniske båndstrukturer.

Module I – Crystallography and crystal structures

Module II – Phonons

Module III – Molecular orbital therory

Module IV – Electrons

MENA2000 : Fundamental structure of solid materials - Lecture Plan calender week Ti 15/1 14-16 Introduction to the course

Module I – Crystallography and crystal structures (H, Fjellvåg, föreläsningskompendium)

Må 21/1 10-12 Basis and unit cells, Bravais lattices (2D and 3D), density, etc 4

Ti 22/1 14-16 Point symmetry; stereographical projections; crystal planes and directions

Må 28/1 10-12 Symmetry of molecules; introduction to groups; Crystal structures; concepts, digital tools 5

Ti 29/1 14-16 continued

Må 04/2 10-12 Translation symmetry, space groups, international tables for crystallography 6

Ti 05/2 14-16 continued

Må 11/2 10-12 Crystal structures of functional materials II; defects, solid solutions 7

Ti 12/2 14-16 Reciprocal lattice; Diffraction, XRD fingerprint analysis

Module II – Phonons C.Kittel’s Introduction to Solid State Physics; chapters 3 (pp73-85) 4, 5, and 18 (pp.557-561)

Må 18/2 10-12 Crystals as diffraction grids; Laue condition; Bragg plains and Brillouin zones; 8

Ti 19/2 14-16 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D

Må 25/2 10-12 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 9

Ti 26/2 14-16 Effect of temperature - Planck distribution; lattice heat capacity: Dulong-Petit, Einstein, and Debye models

Må 04/3 10-12 Comparison of different lattice heat capacity models 10

Ti 05/3 14-16 Thermal conductivity and thermal expansion

Module III – Molecular orbital therory (H.Sønsteby, Molecular Symmetry in Solid-State Chemistry, föreläsningskompendium)

Må 11/3 10-12 Introduction to symmetry and molecular symmetry 11

Ti 12/3 14-16 Group theory for material scientists

Må 18/3 10-12 Character tables 12

Ti 19/3 14-16 Bonding from a symmetry perspective

lecture-free week 13

Må 01/4 10-12 The role of symmetry in vibrational spectroscopy 14

Ti 02/4 14-16 Crystal field theory and ligand field theory

Må 08/4 10-12 Interplay between MO-theory and band structure in 1D 15

Ti 09/4 14-16 Band structure and density of states (DOS) in 3D from MO-theory

Easter 16

Module IV – Electrons C.Kittel’s Introduction to Solid State Physics; chapters 6, 7, 11 (pp 315-317) 18 (pp. 528-530) , and Appendix D

Ti 23/4 14-16 Free electron gas (FEG) versus free electron Fermi gas (FEFG) 17

Må 29/4 10-12 DOS of FEFG in 1D and 3D at ground state (T=0) 18

Må 06/5 10-12 Effect of temperature – Fermi-Dirac distribution; Fermi energy and Fermi level; heat capacity of FEFG 19

Ti 07/5 14-16 Origin of the band gap and nearly free electron model

Fr 10/5 08-10 Effective mass method; electrons and holes; 20

Må 20/5 10-12 Effective mass method for calculating localized energy levels for defects in crystals 21

Ti 21/5 14-16 Carrier statistics in semiconductors and p-n junctions

Summary

Må 27/5 10-12 Course in a nutshell

Examination

To 06/6 09-13

MENA2000 Electrons: Lectures 5-6 (week 19)

Effective mass method; electrons and holes;

• Internal and external forces affecting electrons in crystal

• Dynamics of electrons in the band

• Hole - an electron near the top of the energy band

Effective mass method for calculating energy levels for defects

• Hydrogen-like impurities

• Recap of the hydrogen model

• Calculation of the donor and acceptor levels

• The electron is subject to internal forces from the lattice (ions and core

electrons) AND external forces such as electric fields

• In a crystal lattice, the net force may be opposite the external force,

however:

+ + + + +

Ep(x)

-

Fext =-qE

Fint =-dEp/dx

Internal and external forces affecting an electron in crystal

• electron acceleration is not equal to Fext/me, but rather…

• a = (Fext + Fint)/me == Fext/m*

• The dispersion relation E(K) compensates for the internal forces due to the

crystal and allows us to use classical concepts for the electron as long as its

mass is taken as m*

+ + + + +

Ep(x)

-

Fext =-qE

Fint =-dEp/dx

Internal and external forces affecting an electron in crystal

Dynamics of electrons in a band

The external electric field causes a change in the k vectors of all electrons:

Eedt

kdF

If the electrons are in a partially filled band, this will break the

symmetry of electron states in the 1st BZ and produce a net current.

But if they are in a filled band, even though all electrons change k

vectors, the symmetry remains, so J = 0.

E

kx

a

a

v

kx

When an electron reaches the 1st BZ edge (at k = /a) it immediately

reappears at the opposite edge (k = -/a) and continues to increase

its k value.

As an electron’s k value increases, its velocity increases, then

decreases to zero and then becomes negative when it re-emerges at k

= -/a!!

E e

dt

k d

Dynamics of electrons in a band

Hole - an electron near the top of an energy band

• The hole can be understood as an electron with negative effective mass

• An electron near the top of an energy band will have a negative effective mass

• A negatively charged particle with a negative mass will be accelerated like a positive particle with a positive mass (a hole!)

/a

E(K)

K

F = m*a = QE

Without the crystal lattice, the hole would not exist!

The hole is a pure consequence of the periodic potential operating in the

crystal!!!

E(K) and E(x)

/a

E(K)

K

conduction band

valence band

EC

EV +

-

x

E(x)

Eg

Generation and Recombination of electron-hole pairs

conduction band

valence band

EC

EV +

-

x

E(x)

+

-

a

b

x

y

/a

E(Kx)

Kx

/b

E(Ky)

Ky

Different lattice spacings lead to different curvatures for E(K)

and effective masses that depend on the direction of motion.

Real lattices

light m*

(larger d2E/dK2)

heavy m*

(smaller d2E/dK2)

Real lattices

21

, 2

1c ij

i j

Em

k k

MENA2000 Electrons: Lectures 5-6 (week 19)

Effective mass method; electrons and holes;

• Internal and external forces affecting an electron in crystal

• Dynamics of electrons in a band

• Hole - an electron near the top of an energy band

Effective mass method for calculating energy levels for defects

• Hydrogen-like impurities

• Recap of the hydrogen model

• Calculation of the donor and acceptor levels

Hydrogen like impurities in semiconductors

d

Hydrogen-like donor Hydrogen atom

P donor in Si can be modeled as hydrogen-like atom

Hydrogen atom - Bohr model

222

0

42

2

0

2

0

22

0

2

2

0

2

2

0

2

22

0

22

0

2

0

2

0

2

2

2

2

0

1

2)4(

2)4( :energy Total

242

1 :energy Kinetic

44 :energy Potential

4

1

4

4)(44

...3,2,1 ,for 4

1

n

emZEK

r

ZeVKE

r

ZemvK

r

Zedr

r

ZeV

n

Ze

mr

nv

mZe

nr

mr

n

mr

nmrrmvZe

nnmvrLr

vm

r

Ze

r

Hydrogen atom - Bohr model

eV613)4(2 2

0

40

H .qm

E

eV0.05

2

0s

0

0

*neV613

) π(42

q*n

20s

4

d

Km

m.

K

mE

eV613)4(2 2

0

40

H .qm

E

Instead of m0, we have to use mn*.

Instead of o, we have to use Ks o.

Ks is the relative dielectric constant

of Si (Ks, Si = 11.8).

Hydrogen-like donor

Hydrogen like impurities in semiconductors