University of Hamburg, Department of Earth Sciences Applications of representations theory in Applications of representations theory in solid solid - - state physics and chemistry state physics and chemistry Boriana Mihailova Vibrations in molecules and solids Vibrations in molecules and solids
87
Embed
Applications of representations theory in solid-state ...cloud.crm2.univ-lorraine.fr/pdf/nancy2010/Mihailova.pdf · normal phonon modes irreducible representations Lattice (molecular)
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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
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
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
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
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
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 )(
Experimental geometryExperimental geometry
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
Experimental geometryExperimental geometry
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)
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
SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server
SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server
• 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
Compatibility relationsCompatibility relations
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
Compatibility relationsCompatibility relations
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)
Compatibility relationsCompatibility relations
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
Compatibility relationsCompatibility relations
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
HyperHyper--Raman ScatteringRaman Scattering
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