fys3400 - vår 2020 (kondenserte fasers fysikk)€¦ · on 5/2 10-12 mechanical properties of...

FYS3400 - Vår 2020 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v17/index.html

Pensum: Introduction to Solid State Physics

by Charles Kittel (Chapters 1-9 and 17 - 20)

Andrej Kuznetsov

delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO

Tel: +47-22857762,

e-post: [email protected]

visiting address: MiNaLab, Gaustadaleen 23a

http://www.uio.no/studier/emner/matnat/fys/FYS3410/v16/index.html

http://www.uio.no/studier/emner/matnat/fys/FYS3410/v16/index.html

2020 FYS3400 Lecture Plan (based on C.Kittel’s Introduction to SSP, Chapters 1-9, 17-20 + guest lectutes)

Module I – Periodity and Disorder (Chapters 1-3, 19, 20) calender week

To 16/1 12-13 Introduction.

On 22/1 10-12 Crystal bonding. Periodicity and lattices. Lattice planes and Miller indices. Reciprocal space. 4

To 23/1 12-13 Bragg diffraction and Laue condition

On 29/1 10-12 Ewald construction, interpretation of a diffraction experiment, Bragg planes and Brillouin zones 5

To 30/1 12-13 Surfaces and interfaces. Disorder. Defects crystals. Equilibrium concentration of vacancies

On 5/2 10-12 Mechanical properties of solids. Diffusion phenomena in solids; Summary of Module I 6

Module II – Phonons (Chapters 4, 5, and 18 pp.557-561)

To 6/2 12-13 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D

On 12/2 10-12 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 7

To 13/2 12-13 Effect of temperature - Planck distribution;

On 19/2 10-12 Lattice heat capacity: Dulong-Petit, Einstein, and Debye models 8

To 20/2 12-13 Comparison of different lattice heat capacity models

On 26/2 10-12 Thermal conductivity and thermal expansion 9

To 27/2 12-13 Summary of Module II

Module III – Electrons (Chapters 6, 7, 11 - pp 315-317, 18 - pp.528-530, 19, and Appendix D)

On 4/3 10-12 Free electron gas (FEG) versus free electron Fermi gas (FEFG); DOS of FEFG in 3D 10

To 5/3 12-13 Effect of temperature – Fermi-Dirac distribution; Heat capacity of FEFG in 3D

On 11/3 10-12 DOS of FEFG in 2D - quantum wells, DOS in 1D – quantum wires, and in 0D – quantum dots 11

To 12/3 12-13 Transport properties of electrons; Summary of module III.

teaching free week 12

Module IV – Disordered systems (guest lecture slides)

On 25/3 10-12 Thermal properties of glasses: Model of two level systems (Joakim Bergli) 13

To 26/3 12-13 Experiments in porous media (Gaute Linga)

On 1/4 10-12 Electron transport in disordered solids: wave localization and hopping (Joakim Bergli) 14

To 2/4 12-13 Theory of porous media (Gaute Linga)

Easter 15

On 15/4 10-12 Advanced theory of disordered systems (??) 16

To 16/4 12-13 Advanced theory of disordered systems (??)

Module V – Semiconductors (Chapters 8, 9 pp 223-231, and 17, 19)

On 22/4 10-12 Origin of the band gap; Nearly free electron model; Kronig-Penney model 1

To 23/4 12-13 Effective mass method for calculating localized energy levels for defects in crystals

On 29/4 10-12 Intrinsic and extrinsic electrons and holes in semiconductors 18

To 30/4 12-13 Carrier statistics in semiconductors

On 6/5 10-12 p-n junctions 19

To 07/5 12-13 Optical properties of semiconductors (Inhwan Lee)

On 13/5 10-12 Advanced optoelectronic devices (Inhwan Lee) 20

Summary and repetition

To 14/5 12-13 Repetition - course in a nutshell

Exam: oral examination

May 28th – 29th

Condensed Matter Physics

Solid State Physics of Crystals

Properties of Waves in Periodic Lattices

Electron waves in lattices

Free electrons

Electron DOS

Fermi-Dirac distribution

Elastic waves in lattices

Vibrations

Phonon DOS

Planck distribution

Elecronic properties: Electron concentration and transport,

contribution to the heat capacity

Thermal properties: heat capacity and conductance,

thermal expansion

Advanced theory and novel materials properties

Condensed Matter Physics

Solid State Physics of Crystals

Properties of Waves in Periodic Lattices

Electron waves in lattices

Free electrons

Electron DOS

Fermi-Dirac distribution

Elastic waves in lattices

Vibrations

Phonon DOS

Planck distribution

Elecronic properties: Electron concentration and transport,

contribution to the heat capacity

Thermal properties: heat capacity and conductance,

thermal expansion

Advanced theory and novel materials properties

Disordered

systems

Disordered

systems

Disordered

systems

Disordered

systems

Lectures 5-6: Vibrations and phonons

• Examples of phonon-assisted processes

• Infinite 1D lattice with one or two atoms in the basis;

• Examples of dispersion relations in 3D;

• Finite chain of atoms, Born – von Karman boundary conditions;

• Phonon density of states in 1-D;

• Collective crystal vibrations – phonons;

Ghkl

k′

-k

k

Diffraction

k K k G

Photoluminescence

CB

VB

ED

EA hn

hn

hn

EXCITATION •Photo generation •Electrical injection

Eg

Photons

Photoluminescence

longitudinal wave transverse wave

Vibrations of crystals with monatomic basis


a

Spring constant, g Mass, m

xn xn+1xn-1

Equilibrium Position

Deformed Position

us: displacement of the sth atom from its equilibrium position

us-1 us us+1

M

1 1s s s s sF C u u C u u Force on sth plane =

Equation of motion: 2

1 122s

s s s

d uM C u u u

dt

i t

s su t u e → 2

1 1 2s s s sM u C u u u

0

iK as

su u e → 2 2i K a i K aM C e e 2 21 cos

CKa

M

Dispersion relation

2 24 1sin

2

CKa

M

4 1sin

2

CKa

M

(only neighboring planes interact )


G

dv

d K

vG = 0 at zone boundaries

2 1cos

2

CaKa

M

g K

v

1-D:

Group velocity:

4 1sin

2

CKa

M


2

1 12

2

2 12

2

2

ss s s

ss s s

d uM C v v u

dt

d vM C u u v

dt

i sK a i t

su ue

i sK a i t

sv ve →

2

1

2

2

1 2

1 2

i K a

i K a

M u Cv e Cu

M v Cu e Cv

2

1

2

2

2 10

1 2

i K a

i K a

C M C e

C e C M

Vibrations of crystals with two atoms per basis

Ka → π:

(M1 >M2 )

22

1

2 /

2 /

C M optical

acousticalC M

4 2 2

1 2 1 22 2 1 cos 0M M C M M C Ka

=

=

1 22

2 2

1 2

1 12

2

C opticalM M

CK a acoustical

M M

Ka → 0:







• Phonon density of states in 1-D


p atoms in primitive cell → d p branches of dispersion.

d = 3 → 3 acoustical : 1 LA + 2 TA

(3p –3) optical: (p–1) LO + 2(p–1) TO

E.g., Ge or KBr: p = 2 → 1 LA + 2 TA + 1 LO + 2 TO branches

Ge KBr

Number of allowed K in 1st BZ = N


Phonon dispersion in real crystals: aluminium FCC lattice with 1

atom in the basis

In a 3-D atomic lattice we

expect to observe 3 different

branches of the dispersion

relation, since there are two

mutually perpendicular

transverse wave patterns in

addition to the longitudinal

pattern we have considered.

Along different directions in

the reciprocal lattice the

shape of the dispersion

relation is different. But

note the resemblance to the

simple 1-D result we found.

Phonon dispersion in real crystals: FCC lattice with 1 (Al) and 2

(Diamond) atoms in the basis

Characteristic points of the reciprocal space – Γ, X, K, and L points are

introduced at the center and bounduries of the first Brillouin zone

Calculating phonon density of states – DOS – in 1-D

A vibrational mode is a vibration of a given wave vector (and thus ),

frequency , and energy . How many modes are found in the

interval between and ?

E

k

),,( kE

),,( kdkdEEd

# modes kdkNdEENdNdN

3)()()(

We will first find N(k) by examining allowed values of k. Then we will be

able to calculate N() and evaluate CV in the Debye model.

First step: simplify problem by using periodic boundary conditions for the

linear chain of atoms:

x = sa x = (s+N)a

L = Na

s

s+N-1

s+1

s+2

We assume atoms s

and s+N have the

same displacement—

the lattice has periodic

behavior, where N is

very large.

Lecture 5-6: Vibrations and phonons







• Thermal equilibrium occupancy of phonons – Planck distribution.

Calculating phonon density of states – DOS – in 1-D

This sets a condition on

allowed k values: ...,3,2,12

2 nNa

nknkNa

So the separation between

allowed solutions (k values) is:

independent of k, so

the density of modes

in k-space is uniform

Since atoms s and s+N have the same displacement, we can write:

Nss uu ))(()( taNskitksai ueue ikNae1

Nan

Nak

22

Thus, in 1-D: 22

1 LNa

kspacekofinterval

modesof#

Lecture 5-6: Vibrations and phonons





• Phonon density of states in 1-, 2-, and 3-D;


Energy level diagram for a chain of

atoms with one atom per unit cell and a

lengt of N unit cells

Energy level

diagram for one

harmonic oscillator

fys3400 - vår 2020 (kondenserte fasers fysikk)€¦ · on 5/2 10-12 mechanical properties of...

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