Representational analysis of magnetic structures Branton J. - - PowerPoint PPT Presentation

Representational analysis of magnetic structures

Branton J. Campbell Harold T. Stokes

Department of Physics & Astronomy Brigham Young University

New Trends in Magnetic Structure Determination Institute Laue Langevin Grenoble France, 12-16 December 2016

Applications of Representation Analysis

Molecular vibrations Hybridized and molecular orbitals Crystal-field splitting Electronic-transition selection rules Crystal band structure Landau theory of phase transitions Parameterize crystal distortions

• ccupational

displacive magnetic lattice strain

Group representations

Representations map group elements onto matrices that

• bey the same multiplication table as the group.

        1 1         1 1         1 1           1 1

I

x

2

y

2

z

2 2x 2y = 2z

                             1 1 1 1 1 1

Point group: 222 I 2x 2y 2z I I 2x 2y 2z 2x 2x I 2z 2y 2y 2y 2z I 2x 2z 2z 2y 2x I

Discovered by Ferdinand Frobenius (Germany, 1897).

1: 1 → 1 2 → 1 2 → 1 2 → 1

• 2: 1 → 1 2 → 1 2 → 1

2 → 1

• 3: 1 → 1

1 2 → 1

• 1 2 → 1

1 2 → 1

• 1
• 3 1 ⊕ 2 1

2 Reducible representation: Irreducible representations can’t be separated into smaller pieces! Irreps are recipes for symmetry breaking!

michaeldepippo.com

Irreducible representations (irreps)

Irreps provide a symmetry-based coordinate system (parameter set) for describing deviations from symmetry.

Wonderful orthogonality theorem (WOT)

Issai Schur

Beautiful Computable

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

Orthogonality and completeness relations (1904-07)

Irrep basis function Irrep mode Order parameter component Symmetry-adapted basis function Symmetry mode

A rose by any other name …

Distortion modes (too vague) Normal modes (superposition of modes of same irrep)

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

z y x

1 +

• z

y x

2

z y x

2

z y x

• Familiar symmetry modes

Under the group operations, a orbital transforms like which irrep?

• Action of point group = 222 on orbital

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

• 1

1 1 1

z y x

1

z y x

2

z y x

2

z y x

• ,

, , ,

Action of point group = 222 on orbital Under the group operations, a orbital transforms like which irrep?

Familiar symmetry modes

Irreps of the symmetry group

• f a sphere: O(3)

Spherical harmonics! Irreps of the translational group of a periodic signal Fourier harmonics!

1 2 3

Familiar symmetry modes

1 1 1 1 1

• 1 1
• 1

1 1 0 0 1

• 1

0 0 1 1

• 0 0

1

• 1
• 1
• 2

̅ 4

• 4
• 2
• ′

1 ̅

• 4
• 4

′

′ ′

1

• 1

1

• 1 1

1 1 1 0 1

• 1
• 0 0

1 1

• 0 0

1

• 1

0 0 1 1

Multiply the matrix of an unprimed operator by 1 to

• btain the matrix of the corresponding primed operator.

Irrep

• f

2D magnetic irrep

• In 3D real space:

1 1

• 1
• ̅
• 4

1 1

• 1

4 1

• 1

In 2D carrier space: 1

• 1
• 4

is a 90° CW rotation

Irrep

• f
• 2D magnetic irrep

1 1 1 1 1

• 1 1
• 1

1 1 0 0 1 1

• 0 0

1

• 1

0 0 1

• 1
• 1

2

• ′

1 2 ,

• ̅

4

• 4
• 4

′

4

• ̅′

′

• ̅
• ̅
• ̅
• ̅
• Irrep
• f

2D magnetic irrep

Distortion space

For a given subgroup symmetry and cell size, the collection

• f all variable structural parameters spans a vector space

that contains all possible distortions. 41’ parent 4 unique atoms in supercell → 8 structural parameters in the xy plane 2 2 supercell general

Symmetry modes: a new parameter set

Symmetry modes yield an orthogonal basis for distortion space. 2′

• 2

2 2 Γ

,

, , , 0,0,0 ½, 0,0 0, ½, 0 ½, 0,0 0, ½, 0 ½, ½, 0

Symmetry modes: a new parameter set

Symmetry modes yield an orthogonal basis for distortion space. 2′

• 2

2 2 Γ

,

, , , 0,0,0 ½, 0,0 0, ½, 0 ½, 0,0 0, ½, 0 ½, ½, 0

Displacive/magnetic/occupancy/strain

La2CoRuO6 Displacive 1/2, 1/2, 0 M

• 1/2, 0, 0 X
• 1/4, 1/4, 1/4 R
• Site order (Co/Ru)

1/2, 1/2, 1/2

• Magnetic (Co)

1/4, 1/4, 1/4 Λ

J.W. Bos and J.P. Attfield, J. Mater. Chem. 15, 715–720 (2005).

Non-magnetic case: WO3

m Pm3 1 P

P4/nmm P4/ncc Pbcn P21/n P21/c Pc

Symmetry relationships Phase transitions

1173 K 993 K 623 K 290 K

1 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Irreps/OPDs

• , 0,0
• , 0,0
• , 0,0
• , 0,0
• , ,
• , 0,0
• , ,

Γ

• , ,
• , , 0

, 0,0

• 0,0,0,0, ,
• , , 0

, , 0

• 0,0, , , ,
• , , 0

, ,

• 0,0, , , ,

Describe the structures of each of the phases with a common parameter set. Demonstrates generality.

• One mode can affect many symmetry-distinct atoms. One atom can

be affected by many modes.

• Symmetry-modes span the same configurational space as traditional

coordinates if all relevant k-points, irreps, and OPD components are considered simultaneously. Number of free variables is conserved!

• The relationship between traditional and symmetry-mode coordinate

systems is linear! Related by a square numerical invertible matrix derived from group representation theory.

• Symmetry modes often provide the more natural/efficient basis.

Even complicated magnetic structures are often described by the modes of a single irrep!

Traditional vs symmetry-mode parameters

A recipe for symmetry breaking

1 2 3 4 5 6 7 8 Find the group elements whose matrices leave some vector invariant. The vector used is called the order parameter direction or OPD. The resulting symmetry is called an isotropy subgroup of the parent.

• 0 ⇒ ,
• ⇒ ,
• ⇒

1 1 1 1 1

• 1 1
• 1

1 1 0 0 1 1

• 0 0

1

• 1

0 0 1

• 1
• Example: irrep of 41

1 2

• ′

1 2 ,

• ̅

4

• 4
• 4

′

4

• ̅′

′

• 2

2

• 2

A recipe for symmetry breaking

1 2 3 4 5 6 7 8 Find the group elements whose matrices leave some vector invariant. The vector used is called the order parameter direction or OPD. The resulting symmetry is called an isotropy subgroup of the parent.

• 0 ⇒ ,
• ⇒ ,
• ⇒

1 1 1 1 1

• 1 1
• 1

1 1 0 0 1 1

• 0 0

1

• 1

0 0 1

• 1
• Example: irrep of 41

1 2

• , ,
• ′

1 2

• ̅

4

• 4
• 4

′

4

• ̅′

′

• 2′

′2′ 2′

(a,a,0) Cmm'm' (a,0,0) P4mm'm' (a,a,a) R-3m' (a,b,0) P2'/m' (a,a,b) C2'/m' (a,b,c) P1

• Pm3

m1′

Order Parameter Directions

kernal = 5 epikernals m

irrep of

′

Each SG irrep defined at a specific k vector

(¼,¼,0) (½,½,0) (½,0,0)

2 2  2 2

(½,0,0) Cell doubling along axis: Distinct k in the Brillouin zone have distinct irreps (matrices depend

• n k). But two distinct k separated by a reciprocal-lattice vector are

equivalent (have identical irreps). Just consider 1st Brillioun zone.

Primitive Cubic Face-Centered Cubic Primitive Tetragonal Face-Centered Ortho Rhombohedral Body-Centered Cubic

k-vector labels (Miller & Love)

Miller & Love vs Kovalev symbols

primitive cubic

The star of a k-vector

220 222

1,0,0 0,1,0 0,0,1 ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ 1 , 0,0 0, 1 , 0 0,0, 1

• 11

Full set of inequivalent k vectors related by symmetry

The star of a k-vector

220 222

1,0,0 0,1,0 0,0,1 ½, ½, ½ ½, ½, ½ ½, ½, ½ ½, ½, ½ 11 ½, ½, ½ ½, ½, ½ 1,1,1 1,0,0 1 , 0,0 2,0,0 Full set of inequivalent k vectors related by symmetry

Complete (full-star) space-group irreps

2 , ½, 0,0 , 0, ½, 0

1 1 1 1 1

• 1 1
• 1

1 1 0 0 1 1

• 0 0

1

• 1

0 0 1

• 1
• 1

2

• ′

1 2 ,

• ̅

4

• 4
• 4

′

4

• ̅′

′

• ̅
• ̅
• ̅
• ̅
• Here, the two irrep dimensions correspond to the k-vectors of the star.

Selecting an isotropy subgroup

parent space-group symmetry irrep

finite number for each k-star (e.g. X1

+, X3 +, 5 )

• rder-parameter direction (OPD)

finite number for each irrep; special points/lines/planes in abstract carrier space

k-star

finite number of types (e.g. , , ), but  number of points

isotropy subgroup

[1] space-group or superspace-group type [2] supercell basis (relative to parent) [3] origin of supercell (relative to parent).

{(1,0,0),(0,1,0),(0,0,1)}

P2

a b

Same SG type but different bases

{(2,0,0),(0,1,0),(0,0,1)} {(2,0,0),(0,2,0),(0,0,1)} {(1,0,0),(0,2,0),(0,0,1)}

Importance of basis is common knowledge.

P2

Two distinct cases with same basis = {(2,0,0),(0,2,0),(0,0,1)}

a b

• rigin: (0,0,0)
• rigin: (½,0,0)

Same basis and SG type but different origins

Different supercell origins

• ften result in entirely different

isotropy subgroups -- not common knowledge!

The k-SUBGROUPSMAG program of the Bilbao Crystallographic Server calculates the kernel and the epikernels of an irrep and presents them in graphical

• format. The k-maximal subgroups correspond to the

simple OPDs (i.e. OPDs with a single variable). The lower epikernels correspond to more complicated OPDs that are superpositions of simple OPDs. The kernel arises from the most general OPD, and includes all of the freedoms of all of the simple OPDs. Subgroup listing from ISODISTORT (includes OPD, group number, basis, and origin): Subgroup listing from k‐SUBGROUPSMAG

OPDs and ISGs of mGM4+ of

½ ½ 0 1 0 0 0 0 1

Complete space-group irreps at special- points Simultaneous action of entire k star. 8 cases worked manually (1968-1984). Little- group irreps Faddeyev; Kovalev; Zak, Casher, Glück & Gur; Bradley, Cracknell, Davies, Miller, Love (1964-1979)

reciprocal space

Tables of Stokes & Hatch (1984, 1987): all 4777 space groups irreps at special ; 15239 isotropy subgroups [green book]. Complete space-group irreps at any commensurate point Karep (1992), ISOTROPY (1998) real-time calculations

¼ ¾ 0

Complete space-group irreps (any commensurate or incommensurate ) ISO-IR, Stokes & Campbell (2014) Tabulated – not real-time

00

Space-group irrep calculations

Analogy: integers → rational numbers → real numbers

Multi-ferroic TbMnO3

Cycloidal magnetic structure. The transverse Σ and longitudinal Σ order parameters are superposed 90° out of phase. Major historical controversy unnecessary. Just consider the whole k-star () to get the symmetry and physical properties right! This combination breaks the inversion symmetry and can therefore couple to ferroelectric Γ

• irrep, making it a multi-ferroic material.

Incommensurate example: Skyrmion lattice

2, , 0 , 2, 0 , , 0 In this image, we used 0.11 Lattice of clockwise magnetic whirlpools. Λ , 0, , 0, , 0 of 61′

Irreps vs Pirreps

A physically-irreducible representation (pirrep) can always be brought to real form, and has order parameters equivalent to those of complex irrep. Bring to real form via similarity transform with

• 1

1

• Γ

Γ

Γ∗

• ∗
• Γ

Γ ∗ ∗

• ∗

Γ Γ Γ

• Γ
• Γ

′ ′ ′

• 2
• Γ

Γ

Γ∗

• ∗

Complex irrep dimension = Pirrep dimension = number of arms in the star of number of arm-pairs in the star of ,

• 1 or 2

dimension of the little group of 1 : 1 or 2 complex: type 2 or 3

Irrep/Pirrep dimension

Irrep analysis depends on parent origin

• rigin

shift

• Mn at 0,0,0

Mn at ½, ½, ½ Example: cubic perovskite ( )

Magnetic modes (single-k point)

MODY (W. Sikora, F. Bialas, L. Pytlik, 1992)

• Treats scalar, vector, axial vector and higher-order tensor OPs.

BASIREPS (J. Rodriguez-Carvajal, 2003)

• Development version (1996).
• Integrated within Fullprof for magnetic & displacive refinements.

SARAh (A. Wills)

• Magnetic and displacive modes (1999).
• Interfaced to GSAS and Fullprof for magnetic modes (2001).
• Global-search algorithm explores all modes of all irreps at one k vector.

Applications were primarily for magnetic cases because many magnetic structures can be described with a single sinusoidal magnetic wave. This proved not to be the case with other types of order parameters.

ISOTROPY (Stokes, Hatch, 1998)

• Project strain, displacive, or magnetic symmetry modes
• Full star at special and non-special commensurate k points

General OP types, full k star, multi-k

ISODISTORT (Stokes, Campbell, Hatch, 2005)

• Simultaneous mode parameterization of an entire structure
• General full--star, multi-k, multi-irrep, multi-OP distortions
• Filtered search, interactive visualization, crystallographic I/O
• Automated mode decomposition (2006)
• Mode types: displacive (2005), strain (2007), occupational (2007),

magnetic (2010), rotational (2014), and ellipsoidal ADP modes (?)

• Full use of space-group symmetry (or superspace symmetry 2014)

REPRES/AMPLIMODES (Bilbao Cryst. Server team, 2006/2009)

• Full--star, multi-, multi-irrep displacive decompositions

JANA (V. Petricek, M. Dusek, L. Palatinus, 2011)

• Integrated calculation of displacive and magnetic modes.
• Allows refinements involving entire k star.

Dorian Hatch, David Tanner Brigham Young University Manuel Perez-Mato, Mois Aroyo and the Bilbao Crystallographic Server team EPV/EHU, Bilbao, Spain Vaclav Petricek (JANA) Institute of Physics, Praha, Czech Republic Juan Rodriguez-Carvajal (FULLPROF) Institut Laue-Langevin, France

Acknowledgments