applications of representations theory in solid-state...

University of Hamburg, Department of Earth Sciences

Applications of representations theory in Applications of representations theory in solidsolid--state physics and chemistrystate physics and chemistry

Boriana Mihailova

Vibrations in molecules and solidsVibrations in molecules and solids

GoalsGoals

Group theory analysis of vibrational spectra

“User-friendly” approach to group-theory analysis

fun easy absolutely necessary!

without forgetting thatwithout forgetting that

Human beingHuman being Clicking species

= = Thinking species

OutlineOutline

2. Brillouin zone centre spectroscopy. Raman and Infrared the most commonly used methods to study atomic dynamics

1. Atomic dynamics. Phonons

3. Group theory analysis (GTA) :- selection rules for conventional IR and Raman spectra

“by hand” and using on-line tools

- “deviations” from the GTA predictions

- second order phonon processes; compatibility relations;hyper-Raman scattering, a note about resonance Raman spectra

4. Electron states. Selection rules for optical transitions

ReferencesReferences

http://reference.iucr.org/dictionary/

http://www.cryst.ehu.es

http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html

http://minerals.gps.caltech.edu/

http://riodb01.ibase.aist.go.jp/sdbs/cgi-bin/cre_index.cgi?lang=eng

Rousseau, Bauman & Porto, Normal mode determination in crystals.J. Raman Spectrosc. 10, (1981) 253-290

Fateley, McDevitt & Bentley, Infrared and Raman selection rules for lattice vibrations: the Correlation method. Appl. Spectrosc. 25, (1971) 155-173

M. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory. Application to the Physics of Condensed Matter. Springer 2008

Atomic dynamics in crystals Atomic dynamics in crystals

Visualization: UNISOFTVisualization: UNISOFT, Prof. G. Eckold et al., University of Göttingen

KLiSOKLiSO44, hexagonal, hexagonal

6

Crystal normal modes (Crystal normal modes (eigenmodeseigenmodes))

Atomic vibrations in crystals = Superposition of normal modes (eigenmodes)

a mode involving mainly S-Ot bond stretchinge.g.,

a mode involving SO4 translations and Li motions vs K atoms

PhononsPhonons

Atomic vibrations in a periodicperiodic solid

standing elastic waves normal modes (S, {ui}s )

crystals : N atoms in the primitive unit cell vibrating in the 3D space 3N degrees of freedom finite number of normal states

quantization of crystal vibrational energy

N atoms 3 dimensions 3N phonons phonons

phononphonon quantum of crystal vibrational energy

phonons: quasi-particles (elementary excitations in solids)- En = (n+1/2)ħ, - m0 = 0, p = ħK (quasi-momentum), K q RL- integer spin

Bose-Einstein statistics: n(,T)= 1/[exp(ħ/kBT)-1] (equilibrium population of phonons at temperature T)

Harmonic oscillator

2

n=0

n=1

n=2

n=3

Phonon frequencies and atom vector displacementsPhonon frequencies and atom vector displacements

phonon S, {ui}s eigenvalues and eigenvectors of D = f (mi, K({ri}), {ri})

K m2m1

a

Hooke’s low : Kxxm Atomic bonds elastic springs

Equation of motion for a 3D crystal with N atoms in the primitive unit cell :

in a matrix form: qq wqDw )(2

(3N1) (3N1)(3N3N)

)(1)( ''

''' qq ss

ssss mm

D dynamical matrix

second derivativesof the crystal potential

= 1,2,3 i = 1,..., N

atomic vector displacements

''

,''',',2 )(

iiiii wDw qq q

qq ,,

1 i

ii u

mw

0)( 2 qwδqD

mK

phonon S, {ui}s carry essential structural information !

Brillouin zone

chain in 1D

Types of phononsTypes of phonons

phonon dispersion: ac(q) op(q),

diatomic chain

AcousticAcoustic phonon: u1, u2, in-phaseOpticalOptical phonon: u1, u2, out-of-phase

for q 0, op > ac

1 Longitudinal1 Longitudinal: wave polarization (u) || wave propagation (q)22 TransverseTransverse: wave polarization (u) wave propagation (q)

LALA

TATA

LOLO

TOTO

3D crystal with N atoms per cell : 3 acoustic and 3N – 3 optical phonons

chain in 3D

induced dipole moment interact with light

Phonon (Raman and IR) spectroscopyPhonon (Raman and IR) spectroscopy

Infrared absorption: Infrared absorption:

Raman scattering Raman scattering inelastic light scattering from optical phonons

electromagnetic wave as a probe radiation (photon – opt. phonon interaction):

, kground state

excited state

s, k s, ki

Stokes anti-Stokes is is

)()( phononGS

phononESphoton EE

Kkk is Kkk is

anti-Stokes Stokes

Phonon (Raman and IR) spectroscopyPhonon (Raman and IR) spectroscopy

• only optical phonons near the BZ centre are involved Kkk si ikkK 2max (e.g. Raman, 180-scattering geometry)

a

K max

0Kphoton-phonon interaction only for

(a ~ 10 Å)i (IR, vis, UV) ~ 103 – 105 Å ki ~ 10-5 – 10-3 Å Kmax

10 [cm-1] 1.24 [meV] 10 [cm-1] 0.30 [THz] [Å].[cm-1] = 108

cm-1 E= ħck = ħc(2/) = hc(1/)• spectroscopic units:

• IR and Raman spectra are different for the same crystal

different interaction phenomena different selection rules !

Raman and IR intensitiesRaman and IR intensities

IR activityIR activity: induced dipole moment due to the change in the atomic positions

Qk – configurational coordinate

0, IR activity

),,( zyx μ

IR: “asymmetrical”, “one-directional”

Raman: “symmetrical”, “two-directional”

N.B.! simultaneous IR and Raman activity – only in non-centrosymmetric structures

...)(0

0 kk

QQ

Q μμμ

2

0

k

IR QI μ

Raman activityRaman activity: induced dipole moment due to deformation of the e- shell

Polarizability tensor:

0, Raman activity

zzyzxz

yzyyxy

xzxyxx

α P = .E

induced polarization (dipole moment per unit cell)

...)(0

0 kk

QQ

Q ααα

2

0i

kSRaman Q

I eαe

Raman and IR activity in moleculesRaman and IR activity in molecules

sym. stretching

bending

anti-sym. stretching

H2O

IRIR : change in RamanRaman : change in pol. ellips. size (Tr()), shape () and/or orientation (ij, ij)

1

2

3

Raman and IR activity in crystalsRaman and IR activity in crystals

Pb2

Pb1

Op

Ot

P

ba

c

Pb2

Pb1

Op

Ot

P

ba

c

Raman-active

Pb2

Pb1

Op

Ot

P

ba

c

IR-active

Raman-active IR-active

Pb2

Pb1

Op

Ot

P

ba

c

Isolated TO4 group

IR-active

Raman-active

R3mCrystal: Pb3(PO4)2,

Methods for normal phonon mode determinationMethods for normal phonon mode determination

N.B.! Tabulated information for:first-order, linear-response, non-resonance interaction processes

(one phonon only) (one photon only) (ħi < EESelectron-EGS

electron)

Three techniques of selection rule determination at the Brillouin zone centre:

• Factor group analysis (Bhagavantum & Venkatarayudu)

• Molecular site group analysis (Correlation method)

the effect of each symmetry operation in the factor group on each type of atom in the unit cell

Rousseau, Bauman & Porto, J. Raman Spectrosc. 10, (1981) 253-290

Bilbao Server, SAM, Bilbao Server, SAM, http://www.cryst.ehu.es/rep/sam.html

symmetry analysis of the ionic group (molecule) site symmetry of the central atom + factor group symmetry

• Nuclear site group analysisNuclear site group analysis

site symmetry analysis is carried out on every atom in the primitive unit cell set of tables ensuring a great ease in selection rule determinationpreliminary info required: space group and occupied Wyckoff positions

Spectroscopy and Group theory. The PhilosophySpectroscopy and Group theory. The Philosophy

normal phonon modes irreducible representations

Lattice (molecular) dynamics

obey the symmetry restrictions from all symmetry operations in the group

= normal phonon modes

symmetry-specific transformation upon different symmetry operations

Crystals: Space groups specificpoint symmetry at each k

Raman and IR k = 0 (-point)Crystals, molecules: Point group Irreducible representations

n3

4

3

2

1

Dynamical matrix in a diagonal formDynamical matrix in a diagonal form

jiij uu

UD

2

ii wu s 2

nnzznxz

xxxzxyxx

xzzzyzxz

xyyzyyxy

nnxzxxxzxyxx

DD

DDDDDDDDDDDD

DDDDD

1

22121212

12111111

12111111

12111111

41

41

41

31

31

21

11

43

43

43

43

42

42

42

42

41

41

41

41

32

32

32

32

31

31

31

31

2111

22111

2111

111222

1111

zyxzyxzyx

nznynxzyxzyx

uuuuuuuuuuuuuuuuuu

(eigenvalues)(eigenvectors)

singlesingle

doublydegenerate

triplydegenerate

atomic displacement vectors for a given normal mode (symmetry of irrep)

2 symmetry-equivalent sets of a.d.v.

3 symmetry-equivalent sets of a.d.v.

1 symmetry-allowed set of a.d.v.another symmetry-allowed set

Dynamical matrix and Group TheoryDynamical matrix and Group Theory

D commutates with all matrix forms of the mechanical representation m

m D = mDm block-diag. form via irreps, i.e. S: S-1mS = m = S S-1

D S S-1 = D S S-1 S-1()S

S-1DS = S-1DS

block-diagonalized form of D !

D has two parts “force-field” F and “geometry” G

F peak positions (exact values of eigenfrequencies) and intensities (exact values of the atomic displacement amplitudes)

G mode degeneracy (relative values of eigen frequencies)mode symmetry (sets of relative atomic amplitudes)

S-1DS = S-1DSS-1S S-1S

Character tablesCharacter tables

normal phonon modes irreducible representations

Symmetry element: matrix representation A

C3v (3m)

001100010

r3

r2

r1

C3 (3)

v (m)

Point group

1 3 m

3:

Character: i

iiA)(Tr A

Symmetry elements

100010001

1:

100010001

m:

010100001

reducible irreducible (block-diagonal)

3 0 1reducible

characters

21

230

23

210

001

100010001

irreducible A1

E1 1 1

2 -1 0

A1 + E 3 0 1

Mulliken symbols

Reminder: (1 231)

(232)

MullikenMulliken symbolssymbols

A, B : 1D representations non-degenerate (single) modeonly one set of atom vector displacements (u1, u2,…,uN) for a given wavenumber

A: symmetric with respect to the principle rotation axis n (Cn) B: anti-symmetric with respect to the principle rotation axis n (Cn) E: 2D representation doubly degenerate mode

two sets of atom vector displacements (u1, u2,…,uN) for a given wavenumber

T (F): 3D representation triply degenerate modethree sets of atom vector displacements (u1, u2,…,uN) for a given wavenumber

E mode

2D system

X

Y

T mode

3D system

X

Y

Z

subscripts g, u (Xg, Xu) : symmetric or anti-symmetric to inversion 1superscripts ’,” (X’, X”) : symmetric or anti-symmetric to a mirror plane msubscripts 1,2 (X1, X2) : symmetric or anti-symmetric to add. m or n

• biaxial crystals (tricl, monocl, orth) : only A , B• uniaxial crystals (tetra, rhombo, hexa): A,B, E• cubic: A, E, T

Symbols and notationsSymbols and notations

Triclinic Monoclinic Trigonal (Rhombohedral)

Tetragonal Hexagonal Cubic

C1 1 C2 2 C3 3 C4 4 C6 6 T 23 Ci 1 CS m C3i 3 S4 4 C3h 6 C2h 2/m C4h 4/m C6h 6/m Th 3m C2v mm2 C3v 3m C4v 4mm C6v 6mm D3d m3 D2d 42m D3h 6m2 Td m34 D2 222 D3 32 D4 422 D6 622 O 432 D2h mmm D4h 4/mmm D6h 6/mmm Oh mm3

Dn: E, Cn; nC2 to Cn; T: tetrahedral symmetry; O: octahedral (cubic) symmetry

Point groups:

Symmetry element Schönflies notation International (Hermann-Mauguin)

Identity E 1 Rotation axes Cn n = 1, 2, 3, 4, 6 Mirror planes m to n-fold axis || to n-fold axis bisecting (2,2)

hvd

m, mz mv, md, m’

Inversion I 1 Rotoinversion axes Sn 6,4,3,2,1n Translation tn tn Screw axes k

nC nk Glide planes g a, b, c, n, d

(on the Bilbao Server: POINT)

Spectroscopy and Group theory. General outlinesSpectroscopy and Group theory. General outlines

Identify the symmetry operations defining the point group G in equilibrium

Find the irrep decomposition of equivalence rep eq (atoms remaining invariant under the sym. operations of G); (eq) for each SO is the number of inv. atoms vibrations transformation properties of a radial vector r (x,y,z)

mol.vibr. = eq vec – trans – rot

latt.vibr. = eq vec,

different degrees of freedom

trans: simple translation

rot : simple rotation

about the centre of mass

irreps of polar vector, e.g. r (x,y,z)irreps of axial vector, e.g. r p

)1(iC

)1(iC

Infrared activity: transformation properties of a vector (1st-rank tensor) Gvec irrep symtotally initialfinal x,y,z remain positive

• GTA of the point group the corresponding k-point (N.B. ac(k0) 0)• compatibility relations between neighbouring k-points to form a phonon branch

Lattice vibrations for k 0:

opt.latt.vibr. = eq vec – acous (ac(k=0) = 0)

Raman activity: transformation properties of a 2nd-rank symmetric tensor

x2, y2, z2, xy, zy, zx remain positive finalinitial Ramanirrep'

Working example for simple point groups, Cs (m),

Constructing character tablesConstructing character tables

X

Z

Y ),,( zyx

),,( zyx

Cs (m)

sym. operations

E (1) h (m) functions

Point group selection rules

irreps(symmetry types of modes)

1

1

Two single irreps: totally sym A’, antisym to reflection A’’

A’

A”

1

1

x, y

z

change in sign

)0,,()'( yxA μ

),0,0()''( zA μIR di

pole

momen

ts

, x2, y2, z2, xy

, xz, yz small rot. about Z does not change the symmetry

, Jz

, Jx, Jy

Raman

tens

ors

c

bdda

A )'(α

fe

fe

A )''(α

ExerciseExercise: construct the character table for D2 (222)

Constructing character tablesConstructing character tables

D2 (222)

sym. operations

functions

Point group selection rules

irreps(modes)

A

B2

B1

B3

1

1

1

1 1 1 1

11

1

1

1

11

1

1

E (1) C2x (2x) C2z (2z)C2y (2y)

C2x (2x)

),,( zyx

C2z (2z)

C2y (2y)

),,( zyx ),,( zyx

),,( zyx

x2, y2, z2

x, Jx, yzy, Jy, xz

z, Jz, xy

),0,0()( 1 zB μ )0,,0()( 2 yB μ )0,0,()( 3 xB μ

c

ba

A)(α

dd

B )( 1α

e

eB )( 2α

ffB )( 3α

A : IR-inactive

Dipole moments

Polarizability tensors

),,( zyx ),,( zyx ),,( zyx ),,( zyx

1:2x:2y:2z:

ExerciseExercise: construct the character table for H2O

Normal modes in molecules. HNormal modes in molecules. H22OO

sym. operationsPoint group

selection rules

C2 (mm2) E (1) C2 (2) y (my) x (mx) functions

A1

B1

A2

B2

1

11

1

1 11

11

11

1 11

1 1irreps

Jz, xyx, Jy, xz

y, Jx, yz

z, x2, y2, z2

mol.vibr. = eq vec – trans – rot

x (mx)

y (my)C2 (2)

),,( zyx ),,( zyx ),,( zyx ),,( zyx

E (1):C2 (2):y (my):x (mx):

Z

XY

Characters for atomic site transformations1 2 my mx

eq(H2O) 3

atomsremaining invariant

OHH

O OHH

O

11 3 2A1 + B2

either 2A1 + B2or A1 + 2B2

only 2A1 + B2

mol.vibr. = (2A1+B2) (A1+B1+B2) –(A1+B1+B2) –(A2+B1+B2) =

= 2A1+2B1 +2B2+B2+A2+A1–– A1 – B1 – B2 –A2 – B1 – B2

H2O = 2A1 (Raman, IR)+B2 (Raman, IR)

3 irreps

Correlation method. Example HCorrelation method. Example H22OO

x (mx)

y (my)C2 (2)

Z

XY

Using tabulated info: Character tables Correlation tables

O: site symmetry C2

H: site symmetry CS = C1h

H2O: Point group C2

e.g. http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html

Correlation method. Example HCorrelation method. Example H22OO

x (mx)

y (my)C2 (2)

Z

XY

O: site symmetry C2

H: site symmetry CS = C1h

H2O: Point group C2

Oxygen

Hydrogen

Identify the translation modes: (vibration translation)

O (C2) : A1 + B1 + B2

2H (Cs) : 2(2A’ + A”)= 4A’ + 2A”

Correlation method. Example HCorrelation method. Example H22O O (C2)

Correlate the site group irreps with the point group irrep

O (C2) : A1 + B1 + B2 the same

2H (Cs) : 4A’ + 2A” 2(A1 + B2) + A2 + B1

H2O = A1+B1+B2 + 2A1+A2+B1+ 2B2 – (A1+B1+B2) – (A2+B1+B2) = 2A1 + B2

O atom H atoms translation rotation

Normal modes in molecules. Bilbao Server: SAMNormal modes in molecules. Bilbao Server: SAM

O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z) x (mx)

y (my)C2 (2)


O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z)

H2O C2 (mm2)

Pmm2


H2O C2 (mm2)O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z)

OH

Point group

normal modes

symmetry operations

characters

selection rules

H2O : A1+B1+B2 + 2A1+A2+1B1+2B2 – (A1+B1+B2) – (A2+B1+B2) = 2A1 + B2

O atom H atoms H2O translation H2O rotation

sym. stretching

bending

anti-sym. stretching

A1 A1 B2

Working example:

CaCO3, calcitecalcite, R3c (167) D3d6

Ca: (6b) 0 0 0C : (6a) 0 0 0.25O : (18e) 0.25682 0 0.25

(0,0,0)+ (2/3,1/3,1/3)+ (1/3,2/3,2/3)+

Structural information onhttps://icsd.fiz-karlsruhe.de/icsd/

Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM

Ca: (6b) 0 0 0C : (6a) 0 0 0.25O : (18e) 0.25682 0 0.25

or input, thenclick

CalciteCalcite



CalciteCalcite

rotation (inactive)

Ca: (6b) : C : (6a) : O : (18a) :

A1u + A2u + 2EuA2g + A2u + Eg + EuA1g + A1u + 2A2g + 2A2u + 3Eg + 3Eu

IR-active z 0x , y 0 + acoustic

(N = 6:3 +6:3+18:3 = 10)Total: 10A + 10E = 30 3N = 30 opt = A1g(R) + 2A1u(ina) + 3A2g(ina) + 3A2u(IR)+ 4Eg(R) + 5Eu(IR)

5 Raman peaks and 8 IR peaks are expected

xx = -yy xyxzyz

Raman-activenon-zero componentsxx = yy zz

acoustic: A2u+Eu (the heaviest atom)

silent (inactive)

number of operation of each class

Point group

normal modes

symmetry operations

characters

selection rules

Spectra fromSpectra from

134

5

2

Practical exercisePractical exercise: number of expected Raman and IR peaks of aragonite

CaCO3, aragonite, Pnma (62) D2h16

Ca : (4c) 0.24046 0.25 0.4150C : (4c) 0.08518 0.25 0.76211O1 : (4c) 0.09557 0.25 0.92224O2 : (8d) 0.08726 0.47347 0.68065

SolutionSolution: opt = 9Ag(R) + 6Au(ina) + 6B1g (R) + 8B1u (IR) + 9B2g (R) + 5B2u (IR) + 6B3g (R) + 8B3u (IR)

30 Raman peaks and 21 IR peaks are expected


Spectra of CaCOSpectra of CaCO33

Calcite: Calcite: Raman-active = A1g + 4Eg

Aragonite:Aragonite: Raman-active = 9Ag + 6B1g + 9B2g + 6B3g

22 observed

Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod

The perovskite structure type ABO3

mPm3 1hO

A: (1b): 0.5 0.5 0.5B: (1a): 0 0 0O: (3d): 0.5 0 0

Working exampleWorking example::

1000cossin0sincos

R+ for proper rotation at = 360/n- for improper rotation (rotation + reflection in h)

Magic formula

– number of modes for each symmetry species

)(1 )()( Rgg

n crystalRR

R

R

Rgg – order of the point group G

gR – number of elements for each class R

)(R – the character for R and irreducible representation

– character for a certain crystal

wR – number of atoms remaining invariant under operation R

)cos21()( Rcrystal wR

(E is proper rot. at 0, I improper rot at 180, h improper rot. at 0)


1. Determine wR (number of atoms invariant under R) and crystal(R)

• Number of atoms per unit cell: A: 1; B: 8 1/8 = 1; O: 12 1/4 = 3; total = 5; Z = 1

ABO3• Identity E (1), gR = 1 :

wR = 5, cr= 5(+1+2cos0)=5(1+2)=53= 15

• four-fold axis C4 (4), gR = 6 :1C4

2C43C4

4C4 5C4N.B. we consider only one axis from all 6 equivalent as well as all axes parallel to this certain axis, which go through the unit cell

wR = 1 1A + 4(2 1/8 B + 1 1/4 O) = 3

wR = 1 1A + 2 1/8B + 6 1/8 B = 1 + 1 =2

• three-fold axis C3 (3), gR = 8 :1C3

2C33C3

5C3

4C3

6C3

7C3

cr= 3(+1+2cos90)=3(1+20)= 31 = 3

cr= 2(+1+2cos120)=3(1-21/2)= 30 = 0


• two-fold axis C2 (2), gR = 3 :

wR = 1 1A + 4(21/8B+11/4O) + 4( 2 1/4O) = 1+2+2=5

1C22C2

3C24C2

5C2

6C2

7C28C29C2

wR = (1 1A + 21/4O) + 2( 2 1/8B) + +4(11/8B) + 2(11/4O) = 1+1/2+1/2+1/2+1/2=3

• two-fold axis C2’ (2’), gR = 8 :

1C2’

2C2’

3C2’

4C2’5C2’

6C2’

7C2’

8C2’

9C2’

• centre of inversion I ( ), gR = 1 :1wR = 5 (the same as for identity operation)

cr= 5(+1+2cos180)=5(1-21)= 5(-1) = -5

cr= 3(+1+2cos180)=3(1-21)= 3(-1) = -3

cr= 5(-1+2cos180)=5(-1-2)= 5(-3) = -15


wR = 3 (the same as for proper four-fold rotation C4)• improper four-fold axis S4 ( ), gR = 6 :4

wR = 2 (the same as for proper three-fold rotation C3) • improper three-fold axis S6 ( ), gR = 8 :3

wR = 3(11A+81/8B+121/4O) = 5 (all atoms invariant) 2h

1h

3h

• mirror plane h (m), gR = 3 :

wR = 11A+81/8B+41/4O = 3• mirror plane d (m’), gR = 6 :

1d

2d3d

cr= 3(-1+2cos90)=3(-1+0)= 3(-1) = -3

cr= 2(-1+2cos60)=2(-1+1)= 20 = 0

cr= 5(-1+2cos0)=5(-1+2)= 51 = 5

cr= 3(-1+2cos0)=3(-1+2)= 31 = 3


Oh ( mm3 )

E (1)

C4 (4)

C2 (2)

C3 (3)

C2’ (2’)

I ( 1 )

S4 ( 4 )

S6 ( 3 )

h (m)

d (m’)

gR 1 6 3 8 6 1 6 8 3 6

A1g 1 1 1 1 1 1 1 1 1 1 A1u 1 1 1 1 1 -1 -1 -1 -1 -1 A2g 1 -1 1 1 -1 1 -1 1 1 -1 A2u 1 -1 1 1 -1 -1 1 -1 -1 1 Eg 2 0 2 -1 0 2 0 2 -1 0 Eu 2 0 2 -1 0 -2 0 -2 1 0 T1g 3 1 -1 0 -1 3 1 -1 0 -1 T1u 3 1 -1 0 -1 -3 -1 1 0 1 T2g 3 -1 -1 0 1 3 -1 -1 0 1 T2u 3 -1 -1 0 1 -3 1 1 0 -1

cr(R) 15 3 -5 0 -3 -15 -3 0 5 3

)(1 )()( Rg

gn crystalR

RR

g = 48

ABO3

A1g: n = (1511+613-15+0-18-15-18+0+15+18)/48 = 0 no such modeA1u: n = (1511+613-15+0-18+15+18+0-15-18)/48 = 0 no such mode

likewise A2g, A2u, Eg, Eu, T1g, T2u: n() = 0 no such modes T1u: n = 4 T2u: n = 1

latt.vibr.(ABO3) = 4T1u + T2u opt.vibr.(ABO3) = 3T1u + T2u

2. Apply the magic formula

PerovskitePerovskite--type structure type structure ABABOO33

A: (1b): 0.5 0.5 0.5B’/B’’: (1a): 0 0 0

O: (3d): 0.5 0 0

A : (8c) : 0.25 0.25 0.25B’ : (4a) : 0 0 0B”: (4b) : 0.5 0 0O : (24e): 0.255 0 0

(0,0,0)+ (0,1/2,1/2)+ (1/2,0,1/2)+ (1/2,1/2,0)+

single perovskite-type

chemical B-site disorder

double perovskite-type

chemical 1:1 B-site order

mPm3 1O h(221)A(B’,B”)O3 mFm3AB’0.5B”0.5O3 5O h(225)


A: (1b):B: (1a): O: (3d):

T1u

T1u

2T1u + T2u

Total: 5T = 15 3N (5 atoms)opt = 3T1u(IR) + T2u(ina)

acoustic

xyz

BO6 stretching

mPm3 1O h(221)A(B’,B”)O3

BO6 bending (3 sets)(3 sets)

B-c. transl. (3 sets)


A : (8c):B’: (4a): B”: (4b): O: (24e):

T1u + T2g

T1u

T1uA1g + Eg + T1g + 2T1u + T2g + T2u

Total: 1A+1E+ 9T = 30 3N (N=8:4+4:4+4:4+24:4)

opt = A1g + Eg + T1g(ina) + 3T1u(IR) + T2u(ina)

T1u acoustic

ExerciseExercise: determine the atom vector displacements for A1g, Eg, T2g, and add. T1u

mFm3AB’0.5B”0.5O3 5O h(225)

A1g Eg

xy

z

T2g

T2g

T1u


Ba: (1a): 0.0 0.0 0.0Ti: (1b): 0.5 0.5 0.595

O1: (1b): 0.5 0.5 -0.025O2: (2c): 0.5 0.0 0.489

mPm3 1O h(221)BaTiO3

PerovskitePerovskite--type structure type structure ABABOO33 : ferroelectric phases: ferroelectric phases

orthorhombic

cubic

rhombohedraltetragonal

Ba: (1a): 0.0 0.0 0.0Ti: (1b): 0.5 0.5 0.595O: (3c): 0.5 0 0.5

P4mm (99) 14vC

Ba: (2a): 0.0 0.0 0.0Ti: (2b): 0.5 0.5 0.515

O1: (2b): 0.5 0.5 0.009O2: (4c): 0.5 0.264 0.248

Ba: (1a): 0.0 0.0 0.0Ti: (1a): 0.4853 0.4853 0.4853O: (1b): 0.5088 0.5088 0.0185

Amm2 (38) 142vC

(0,0,0)+ (0,1/2,1/2)+R3m (160) 5

3vC


Ba:Ti:O:

BaTiO3

PerovskitePerovskite--type structure type structure ABABOO33 : ferroelectric phases: ferroelectric phases

mPm3 P4mm Amm2 R3m

A1 + E A1 + E A1 + EA1 + EB1 + E

T1u

T1u

T1u

T1u

T2u

Ba:Ti:

O1:O2:

A1 + B1 + B2

A1 + B1 + B2

A1 + B1 + B2

A1 + B1 + B2

A1 + A2 + B2

Ba:Ti:

O1:O2:

A1 + E A1 + E A1 + EA1 + EA2 + E

Ba:Ti:O:

Polar modesPolar modes:simultaneously Raman and IR active

xxz, yy

z, zzz

mode polarization along (~ u)

LO: q || TO: q


Experimental geometryExperimental geometry

IIR 2 Iinc Itrans Imeas

z

or

y

Infrared transmission (only TO are detectible)

polarizerX

Y

Z

IRaman 2

back-scatteringgeometry

Porto’s notation: A(BC)DA, D - directions of the propagation of incident (ki) and scattered (ks) light, B, C – directions of the polarization of the incident (Ei) and scattered (Es) light

ki

ks

ki

ks

(qx,qy,0)

(qx,0,0)

yy

Raman scattering

right-anglegeometry

Ei Es

zzn = x LOn = y, z TO

yz

yyn zzn yzn

Ei Es Ei Es

xy xz zy

xyn xzn zyn

zz

zzn n = x,y LO+TOn = z TO

X

Y

Z

X

Y

Z

XYYX )( XZZX )( XYZX )(

(ki = ks+q , E is always to k) XXYY )( XXZY )( XZYY )( XZZY )(


cubic system

Ei EsA1, E

T2(LO)Ei Es

ZYYZ )(ki

ks

Z

X

Y

)q,0,0( zqki = ks + q μq ||

ki

ks

)q,0,q( zxq

Z

X

Y Ei Es ZZYX )(

Ei Es

ZZXX )(

xyz

)0,0,μ( xμ

yxz

)0,μ,0( yμ

μq ||x

μq zT2(LO+TO)

μq T2(TO)

yy

ZYXZ )(

e.g., Td ( ) m34

xy

)μ,0,0( zμ

z

non-cubic systeme.g., rhombohedral, C3v (3m)

ki

ks

Z

X

Y z

yy )μ,0,0( zμ

)q,0,0( zq ZYYZ )(Ei Es

Ei Es

ZYXZ )( E(TO)

YZ

X ki

ksA1(TO) z

zzEi Es

YZZY )( )μ,0,0( zμ)0,q,0( yq

E(LO)

)q,0,0( zq )0,μ,0( yμ


yxy

(hexagonal setting)

Ei Es

Y’ Y Z

X ki

ks XYYX )''(

contribution from

xyy

xyy )0,0,μ( xμ

A1(LO)

E(TO)

zyy )μ,0,0( zμ A1(TO)

)0,0,μ( xμ )0,0,q( xq

LOLO--TO splitting TO splitting

Cubic systemsCubic systems: LO-TO splitting of T modes: T(LO) + T(TO)

NonNon--cubic systemscubic systems: {A(LO) ,A(TO)}; {B(LO),B(TO)}; {E(LO),E(TO)}

)()( TOLO general rule: (the potential for LO: U+E; for TO: U)

Cubic crystalsCubic crystals: LO-TO splitting covalency of atomic bonding LO-TO: larger in ionic crystals, smaller in covalent crystals

More peaks than predicted by GTA may be observedMore peaks than predicted by GTA may be observed (info is tabulated)

UniaxialUniaxial crystals:crystals:if short-range forces dominate:

Raman shift (cm-1)

TO TOLO LO

EA

if long-range forces dominate :

Raman shift (cm-1)

A AE ELOTO

under certain propagation and polarization conditions quasi-LO and quasi-TO phonons of mixedmixed A-E character

LOLO--TO splitting TO splitting

LO-TO splitting: sensitive to local polarization fields induced by point defects

a change in IILOLO//IITOTO depending on the type and concentration of dopant

BiBi1212SiOSiO2020:X:X

Raman shift / cm-1

0.094

0.476

0.8991018 cm-3

undoped

Raman shift / cm-1

BiBi44GeGe33OO1212:Mn:MnI23 324IBi(24f), Si(2a),O1(24f),O2(8c), O3(8c)

Bi(16c),Ge(12a),O(48e)

(Td6)(T0

3)

OneOne--mode / twomode / two--mode behaviour in solid solutions mode behaviour in solid solutions

different types of atoms in the same crystallogr. position,

covalent character of chemical bonding : short correlation length relatively large difference in f(B’/B”-O) and/or m(B’/B”)

• two peaks corresponding to “pure” B’-O and B”-O phonon modes • intensity ratio I(B’-O )/I(B”-O ) depends on x

e.g. (B’1-xB”x)Oy

• one peak corresponding to the mixed B’-O/B”-O phonon mode • ~ lineal dependence of the peak position on dopant concentration x

ionic character of chemical bonding : long correlation length similarity in ri(B’/B”), f(B’/B”-O) and m(B’/B”)

twotwo--mode behaviour:mode behaviour:

oneone--mode behaviour:mode behaviour:

intermediate classesintermediate classes of materials: two-mode over x (0, xm) and one-mode over x (xm, 1)

OneOne--mode / twomode / two--mode behaviour in solid solutions mode behaviour in solid solutions

one-mode behaviour

PbSc0.5(Ta1-xNbx)0.5O3

mf

two-mode behaviour

Pb3[(P1-xAsx)O4]2

T > TC

NonNon--centrosymmetriccentrosymmetric crystals with compositional disordercrystals with compositional disorder

LOLO--TOTO splitting + one-mode/twotwo--modemode behaviour: we may observe fourfour peaks instead of oneone !

aa

a

000000

bb

b

2000000

000003030

bb

022

200

200

dd

d

d

022

200

200

dd

d

d

0000000

dd

aa

a

000000

bb

b

2000000

0000000

dd

00000

00

d

d

0000

000

dd

000030003

bb

X

Y

ZX’

Y’

Z

rotZ(

’ = UTU

100 200 300 400 500 600 700 800 9000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Inte

nsity

/ a.

u.

Raman shift / cm-1

PSTcpp 0 deg, p-pol. PSTdpp 45 deg, p-pol.

100 200 300 400 500 600 700 800 9000.00

0.05

0.10

0.15

0.20

Inte

nsity

/ a.

u.

Raman shift / cm-1

PSTccp 0 deg, c-pol. PSTdcp 45 deg, c-pol.

Transformation of Transformation of polarizabilitypolarizability tensors tensors

A1g(O) + Eg(O)

I(F2g )

F2g(O) + F2g(Pb)

I(Eg )

How GTA helps in complex systems How GTA helps in complex systems

chemical B-site order/disorder ferro-/paraelectric

+

B’

B’’

(A’,A’’)(B’,B’’)O3

Perovskite-type relaxor ferroelectrics

Diffraction methods:

dynamic PNR

mPm3

average structure over

time and space

bold: Ei || [100]thin: Ei || [110]

PSTPST

PSNPSN

p-pol c-pol

Pb-O bo

nd st

r.

Pb-O bo

nd st

r.

F2uF1uF1u

BO 3tra

nsl. v

s Pb

BO 3tra

nsl. v

s PbF1u

B-catio

n locali

sed

B-loca

lised

F1u

BO 6str

etchin

g

BO 6str

.

A1gEg

BO 6be

nding

BO 6be

nding

F2g

Pb-loc

alise

d

Pb-loc

alise

d

F2g

prototype structuremFm3

Pb : F2g (R)

O : A1g(R) + Eg (R) + F1g(s) + 2F1u (IR) + F2g(R) + F2u(s)

B-cations : F1u(IR) + F1u(IR)

A-siteB-siteO

GTA of GTA of RelaxorsRelaxors. Peak assignment. Peak assignment

m(Ta) > m(Nb)

f(Ta-O) > f(Nb-O)

X-ray diffraction Raman scattering

hk0

F1u

F1u F2u

F2g

phonon modes

GTA of GTA of RelaxorsRelaxors. Temperature as variable . Temperature as variable

BB--localised mode ~ 240 cmlocalised mode ~ 240 cm--11

TmTa

PbPb--localised mode ~ 53 cmlocalised mode ~ 53 cm--11

allowed

anomalous

TB T*

“intra-cluster” “inter-cluster”

T* Tm

GTA of GTA of RelaxorsRelaxors. Temperature as variable . Temperature as variable

X-ray diffractionRaman scattering

F1u

F2uF1u

phonon modes

F2g

006 046

GTA of GTA of RelaxorsRelaxors. Pressure as variable . Pressure as variable

SecondSecond--order phonon processes order phonon processes

, kground state

excited state

one photon ↔ two phonons

low probability low intensity

• overtones: 21, 22, 23, …

• combinational modes: 1 2, 1 3, 2 3, …slight -deviation due to mode-mode interactions

General rules:General rules: the symmetry of the II-order state = direct product of the I-order states

overtones: i i combinations: i i

(high-order states : further multiplication) modes away from the BZ-centre may contribute to II-order spectra

e.g. phonon(-k) + phonon(+k) = state (k=0)N.B.! acoustic(k0) 0, i.e. non- acoustic modes may also participate

Benefits:Benefits: inactive -point modes may be observed in IR and Raman spectra non--point modes may be observed in IR and Raman spectra

SecondSecond--order phonon processes order phonon processes

Reminder about the direct product rules: Reminder about the direct product rules:

Product tables:

SecondSecond--order phonon processes. Bilbao Serverorder phonon processes. Bilbao Server

e.g. A1 A2 = A2 inactive; EE = A1+A2+E Raman-active; IR-inactive

SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server

Dresselhaus et al, Springer, 2008

Td (43m)

RamanRaman

Raman, IRRaman, IR

SecondSecond--order phonon processes. k order phonon processes. k 00

• Determine the point group at the corresponding k-point • Find the corresponding normal modes and overtones and combinations

e.g. overtones and combinations at R are the same as at

SecondSecond--order phonon processes. k order phonon processes. k 00

In some cases, the determination of non- overtones and combination modes may be far from straightforward, this holds especially for non-symmorphic groups (having glide plane or screw axis)

(Special Thanks to Alexander Litvinchuk, University of Houston)Bilbao Server, DIRPRO can help


• Scroll down to the end of the resulting file (the most important info)

• Different nomenclature is confusing consider the matrices of S to recognize the symmetrical type, compare with matrix representation in POINT/SAM; be patientbe patient!

The phase

Compatibility relationsCompatibility relations

(the associated characters cannot “jump”)

Going away from Going away from --point (applied to phonon and electron states):point (applied to phonon and electron states):

• Symmetry of the wave-vector group may change

- fewer elements of symmetry at k 0 (if no glide planes or screw axes )- more elements of symmetry at k 0 (glide planes or screw axes )

• The effect of a given symmetry element on the wave function must be continuous between the centre and the boundary of the Brillouin zone

• k-branches must follow the compatibility relation principle

Pb

Kuzmany, Springer 1998


Working example:

Consider the compatibility relations along the line X for Pm3m

1. Compare the character tables of (m-3m) and (4mm)

2. Compare the character tables of (4mm) and X (4/mmm)

3. Find which modes can develop along the X-direction


Find the normal modes having the same characters for the corresponding symmetry elements in the two point groups

m-3m 4mmA1g A1

Fully compatible modesFully compatible modes:

A1u A2A2g B1A1g B2

For degenerate modes possible combinations (sums of characters) must be considered

e.g. E in 4mm is not compatible with modes in m-3m ((2) for 4mm for m-3m)


Fully compatible modesFully compatible modes:

X

4/mmm4mmA1

A2B1

B2

Eg and EuE

A1g and A2u

A2g and A1u

B1g and B2u

B2g and B1u


Dresselhaus et al, Springer, 2008

mPm3

Some modes split following the compatibility relations along the k-line

When phonons (e- levels) of differentsymmetry approach one another: • crossing: retain their original symmetry after crossing

• anticrossing: admixed wave functions in an appropriate linear combination phonon-phonon (e--e-) interaction

When phonons (e- levels) of the samesymmetry approach one another:

HyperHyper--Raman ScatteringRaman Scattering

NonNon--linear processlinear process

...61

21 lkjijklkjijkjiji EEEEEEP

ijk - 3rd-rank tensor

...)(0

0 kk

QQ

Q βββ

0, hyper-Raman activity

ijk= ikj

HR spectra of a perovskite-type relaxor

Hehlen et al, PRB 2007

If Ej || Ek || crystallographic axis

ijk= ijj 33 non-symmetric matrix

zzzzyyzxx

yzzyyyyxx

xzzxyyxxx

ijj

hyperpolarizability tensor

s, k s, ki)

Stokes anti-Stokes is 2 Kkk is 2

is 2 Kkk is 2


Selection rulesSelection rules: according to the transformation properties of a 3rd-rank tensor

Outlines:Outlines: IR-active modes are always HR-active, no matter of the crystal symmetry In centrosymmetric systems only the odd modes (‘u’) may be HR-active Raman-active modes are not hyper-Raman-active and vice versa in non-centrosymmtrical systems some types of modes may be both Raman and HR-active modes inactive in both Raman and IR spectra may be HR-active the nonsymmetrical part of the HP tensor which is symmetrical in two indices gives rise to additional HR-active modes as compared to the completely symmetrical tensor

Denisov et al., Phys.Rep. 151 (1987) 1-92


V.N.Denisov , B.N. Mavrin, V.P. Podobedov, Phys.Rep. 151 (1987) 1-92

Resonance Raman ScatteringResonance Raman Scattering

s, k s, ki

Stokes anti-Stokes is is Kkk is Kkk is

e- ES

• Selection rules for both e- and phonon states ()• Fundamental and II-order harmonics may be enhanced• LO are strongly enhanced

200 400 600 800 10000

1000

2000

parallel-polarised

Inte

nsity

/ a.

u.Raman shift / cm-1

457 nm 413 nm 350 nm 334 nm

2.0 2.5 3.0 3.5 4.0 4.50

2

4

6

8

Eg(PST)

RRS

3.20eV(387.5nm)

2.71eV(457nm)

3.00eV(413nm)

3.55eV(350nm)

3.71eV(334nm)

diel

ectri

c co

nsta

nt

Energy / eV

r

i

valence

conductionEg

Electron statesElectron states

GTA conceptGTA concept : the same for phonon and electron states s

2 ↔ Enus ↔ n

Atomic Atomic orbitalsorbitals: : imm

lmlm

l ePNY )(cos),( ,spherical harmonics:

Normalization coefficient Legendre polynomial

),( ml

lYr linear combination of simple Cartesian coordinate functions

l = 0 : 1;l = 1 : x, y, z;l = 2 : xy, yz, zx, x2-y2, 3z2-r2, …

s-likep-liked-like, …

1s 2p

3d

4f

D ↔


Cubic crystal field environment for a paramagnetic transition Cubic crystal field environment for a paramagnetic transition cationcation

Crystal field theory:Crystal field theory: how the atomic surroundings affects the electron energy levels

3d dx2-y2

dxy

dz2

dxz dyz ),(2 mY

five-fold level

three-fold level

two-fold level


o: c: d: t = 1 : -8/9 : -1/2 : -4/9

octahedron (6-fold polyhedron): t2g – stabilized, eg – destabilized;cube (8-fold), dodecahedron (12-fold), tetrahedron (4-fold): opposite

dx2-y2

dxy

dz2

dxz dyz

the electron lobes are toward the anions add. repulsive forces destabilization

vice versa


basic functions for e- states ↔ irreducible representationsCharacter tables:

e.g.

• splitting of levels: according to transformation rules of irreps

Optical absorption processes

, k

• electric dipole transitions ↔ IR selection rulestransformation properties of a polar vector

Gvec irrep symtotally initialfinal

dipole operator

Optical transitions. Symmetry selection rulesOptical transitions. Symmetry selection rules

Working example: Determine the allowed electric dipole transitions in Oh from T2g

• check for each irrep if contains A1g

• irrep for the dipole operator: T1u

T1u T2g = A2u + Eu + T1u + T2u (see product tables)

A1g (A2u+Eu+T1u+T2u) = (A2u+Eu+T1u+T2u) no

A1u (A2u+Eu+T1u+T2u) = (A2g+Eg+T1g+T2g) noA2g (A2u+Eu+T1u+T2u) = (A1u+Eg+T2u+T1u) no

A2u (A2u+Eu+T1u+T2u) = (A1g+Eg+T2g+T1g) yes

Eg (A2u+Eu+T1u+T2u) = (Eu+A1u+A2u+Eu+T2u+T1u+T2u+T1u) noEu (A2u+Eu+T1u+T2u) = (Eg+A1g+A2g+Eg+T2g+T1g+T2g+T1g) yes

T1g (A2u+Eu+T1u+T2u) = (T2u+T1u+T2u+A1u+Eu+T2u+T1u+A2u+Eu+T2u+T1u) no

T2g (A2u+Eu+T1u+T2u) = (T1u+T1u+T2u+A2u+Eu+T2u+T1u+A1u+Eu+T2u+T1u) noT1u (A2u+Eu+T1u+T2u) = (T2g+T1g+T2g+A1g+Eg+T2g+T1g+A2g+Eg+T2g+T1g) yes

T2u (A2u+Eu+T1u+T2u) = (T1g+T1g+T2u+A2g+Eg+T2g+T1g+A1g+Eg+T2g+T1g) yes

Tabulated info on transition products

Optical transitions. Symmetry selection rulesOptical transitions. Symmetry selection rules

Group theory:Group theory: predicts the number of expected IR and Raman peaksone needs to know: crystal space symmetry + occupied Wyckoff positions

Deviations Deviations from the predictions of the groupfrom the predictions of the group--theory analysis: theory analysis:

• LO-TO splitting if no centre of inversion (info included in the tables)• one-mode – two-mode behaviour in solid solutions

• local structural distortions (length scale ~ 2-3 nm, time scale ~ 10-12 s)

• Experimental difficulties (low-intensity peaks, hardly resolved peaks)

ConclusionsConclusions

What should we do before performing a Raman or IR experiment?What should we do before performing a Raman or IR experiment?

Do groupDo group--theory analysistheory analysis

applications of representations theory in solid-state...

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