Introduction to Magnetic Symmetry. I. Magnetic space groups

J. Manuel Perez-Mato Facultad de Ciencia y Tecnología Universidad del País Vasco, UPV-EHU BILBAO, SPAIN WHAT IS SYMMETRY?

A symmetry operation in a solid IS NOT only a more or less complex transformation leaving the system invariant…. But it MUST fulfill that the resulting constraints can only be broken through a .

A well defined symmetry operation (in a thermodynamic system) must be maintained when scalar fields (temperature, pressure,…) are changed, except if a phase transition takes place.

“symmetry-forced” means : “forced for a thermodynamic phase “symmetry-allowed” means : “allowed within a thermodynamic phase”

Symmetry-dictated properties can be considered symmetry “protected” Reminder of symmetry in non-magnetic structures

Space Group: LaMnO3 Pnma

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270 Reminder of symmetry in non-magnetic structures

Space Group: LaMnO3 Pnma

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270

Space Group: set of operations {R|t} for all atoms: atom {R|t} atom' {R|t}: R - rotation or rotation+plus inversion x' x t - translation (x,y,z) y' = R y + t z' z Reminder of symmetry in non-magnetic structures

Space Group: LaMnO3 Pnma

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270

Space Group: set of operations {R|t} for all atoms: atom {R|t} atom' {R|t}: R - rotation or rotation+plus inversion x' x t - translation (x,y,z) y' = R y + t z' z

Seitz Notation Pnma: 8 related positions for a general position: (x,y,z) (-x+1/2,-y,z+1/2) (-x,y+1/2,-z) (x+1/2,-y+1/2,-z+1/2) == {2x| ½ ½ ½ }

(-x,-y,-z) (x+1/2,y,-z+1/2) (x,-y+1/2,z) (-x+1/2,y+1/2,z+1/2) == {mx| ½ ½ ½ } 4 related positions for a special position of type (x, ¼, z): special positions are tabulated: (x,1/4,z) (-x+1/2,3/4,z+1/2) (-x,3/4,-z) (x+1/2,1/4,-z+1/2) Wyckoff positions or orbits Reminder of symmetry in non-magnetic structures

Space Group: LaMnO3 Pnma

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270

Relations among atoms from the space group: more than "geometrical", they are "thermodynamic" properties

they may be zero within experimental resolution but this is NOT symmetry forced. La1 ( ≈ 0.0 0.25000 ≈0.0)

¼ rigorously fulfilled – if broken, it means a different phase Reminder of symmetry in non-magnetic structures

Space Group: LaMnO3 Pnma

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270

Relations among atoms from the space group: more than "geometrical", they are "thermodynamic" properties

they may be zero within experimental resolution but this is NOT symmetry forced. La1 ( ≈ 0.0 0.25000 ≈0.0)

¼ rigorously fulfilled – if broken, it means a different phase

Whatever microscopic model of atomic forces, if consistently applied, it will yield:

Fy (La1)= 0.000000 (exact!) Symmetry is only detected when it does not exist! Magnetic Symmetry: We do not add but substract symmetry operations !

The LOST ymmetry operation: (always present in non-magnetic structures but ABSENT in magnetically ordered ones!)

Time inversion/reversal: {1’|0,0,0}

- Does not change nuclear variables - Changes sign of ALL atomic magnetic moments {1’|000}

(x,y,z,-m) == (x,y,z,-1)

Magnetic structures only have symmetry operations where time reversal 1’ is combined with other transformations, or is not present at all: {1’|t} = {1’|0,0,0} {1|t} {m’| t}= {1’|0,0,0} {m|t} {2’|t} = {1’|0,0,0}{2|t} {3’+|t} = {1’|0,0,0}{3+|t}, etc. But {1’|0,0,0} alone is never a symmetry operation of a magn. struct. All NON-magnetic structures have time inversion symmetry

If all atomic magnetic moments are zero, time inversion is a (trivial) symmetry operation of the structure:

Actual symmetry of the non-magnetic phase:

Pnma1' = Pnma + {1’|000}x Pnma (grey group)

16 operations: (x,y,z,+1) (-x+1/2,-y,z+1/2,+1) (-x,y+1/2,-z,+1) (x+1/2,-y+1/2,-z+1/2,+1)

(-x,-y,-z,+1) (x+1/2,y,-z+1/2,+1) (x,-y+1/2,z,+1) (-x+1/2,y+1/2,z+1/2,+1)

(x,y,z,-1) (-x+1/2,-y,z+1/2,-1) (-x,y+1/2,-z,-1) (x+1/2,-y+1/2,-z+1/2,-1)

(-x,-y,-z,-1) (x+1/2,y,-z+1/2,-1) (x,-y+1/2,z,-1) (-x+1/2,y+1/2,z+1/2,-1)

{R|t} θ=1 Notation: (x+1/2,-y+1/2,-z+1/2,+1) == {2x| ½ ½ ½ } {R,θ|t} (x+1/2,-y+1/2,-z+1/2,-1) == {2x'| ½ ½ ½ } {R’|t} θ=-1 magnetic ordering breaks symmetry of time inversion

Magnetic ordered phases:

LaMnO3

Pnma1' (x,y,z,+1) (-x+1/2,-y,z+1/2,+1) (-x,y+1/2,-z,+1) (x+1/2,-y+1/2,-z+1/2,+1)

(-x,-y,-z,+1) (x+1/2,y,-z+1/2,+1) (x,-y+1/2,z,+1) (-x+1/2,y+1/2,z+1/2,+1) (x,y,z,-1) (-x+1/2,-y,z+1/2,-1) (-x,y+1/2,-z,-1) (x+1/2,-y+1/2,-z+1/2,-1)

(-x,-y,-z,-1) (x+1/2,y,-z+1/2,-1) (x,-y+1/2,z,-1) (-x+1/2,y+1/2,z+1/2,-1) magnetic ordering breaks symmetry of time inversion

Magnetic ordered phases:

Time inversion {1'|0 0 0} is NOT a symmetry operation of a magnetic phase

LaMnO3

Pnma1' (x,y,z,+1) (-x+1/2,-y,z+1/2,+1) (-x,y+1/2,-z,+1) (x+1/2,-y+1/2,-z+1/2,+1)

(-x,-y,-z,+1) (x+1/2,y,-z+1/2,+1) (x,-y+1/2,z,+1) (-x+1/2,y+1/2,z+1/2,+1) (x,y,z,-1) (-x+1/2,-y,z+1/2,-1) (-x,y+1/2,-z,-1) (x+1/2,-y+1/2,-z+1/2,-1)

(-x,-y,-z,-1) (x+1/2,y,-z+1/2,-1) (x,-y+1/2,z,-1) (-x+1/2,y+1/2,z+1/2,-1) magnetic ordering breaks symmetry of time inversion

Magnetic ordered phases:

Time inversion {1'|0 0 0} is NOT a symmetry operation of a magnetic phase

LaMnO3

Pnma1' Pn'ma' (x,y,z,+1) (-x+1/2,-y,z+1/2,+1) (-x,y+1/2,-z,+1) (x+1/2,-y+1/2,-z+1/2,+1)

(-x,-y,-z,+1) (x+1/2,y,-z+1/2,+1) (x,-y+1/2,z,+1) (-x+1/2,y+1/2,z+1/2,+1) (x,y,z,-1) (-x+1/2,-y,z+1/2,-1) (-x,y+1/2,-z,-1) (x+1/2,-y+1/2,-z+1/2,-1)

(-x,-y,-z,-1) (x+1/2,y,-z+1/2,-1) (x,-y+1/2,z,-1) (-x+1/2,y+1/2,z+1/2,-1) For space operations, the magnetic moments transform as pseudovectors or axial vectors:

Taxial(R)= det[R] R

m m

m

atom {R,θ|t}

x' x (x,y,z) (for positions: the same y' = R y + t as with Pnma) z' z

mx' mx (mx,my,mz) my' = θ det( R) R my mz' mz

θ=-1 if time inversion MAGNETIC SYMMETRY IN COMMENSURATE CRYSTALS

A symmetry operation fullfills:

• the operation belongs to the set of transformations that keep the energy invariant: rotations translations space inversion time reversal

• the system is undistinguishable after the transformation

Symmetry operations in commensurate magnetic crystals:

magnetic space group: { {Ri| ti} , {R'j|tj} } θ = +1 without time reversal or { {Ri , θ| ti}} θ = -1 with time reversal Description of a in a crystallographic

form:

Magnetic space Group: LaMnO3 Pn'ma'

Lattice parameters: 5.7461 7.6637 5.5333 90.000 90.000 90.000

Atomic positions of asymmetric unit: La1 0.05130 0.25000 -0.00950 Mn1 0.00000 0.00000 0.50000 O1 0.48490 0.25000 0.07770 O2 0.30850 0.04080 0.72270 Mn1 Magnetic moments of the asymmetric unit (µB): Mn1 3.87 0.0 0.0

for all atoms:

Pn’ma’: atom {R,θ|t} 1 x,y,z,+1 Symmetry operations 2 -x,y+1/2,-z,+1 are relevant both for 3 -x,-y,-z,+1 positions and moments (x,y,z) x' x (for positions: the same 4 x,-y+1/2,z,+1 y' = R y + t as with Pnma) z' z 5 x+1/2,-y+1/2,-z+1/2,-1 6 -x+1/2,-y,z+1/2,-1 mx' mx 7 -x+1/2,y+1/2,z+1/2,-1 (mx,my,mz) my' = θ det( R) R my 8 x+1/2,y,-z+1/2,-1 mz' mz

θ=-1 if time inversion Parent symmetry Pnma1’ Possible symmetries for a k=0 magnetic ordering:

Possible maximal symmetries obtained with k-SUBGROUPSMAG for a k=0 magnetic ordering: Output of MGENPOS in BCS

Magnetic point group: m’mm’

Pn’ma’ = P121/m1 + {2’100|1/2,1/2,1/2} P121/m1 Wyckoff positions:

Space Group: Output of Pn'ma' MWYCKPOS in BCS

La

Mn

mode along z mode along x (A ) x mode along y (Fy) (Gz) weak ferromagnet Types of magnetic space groups: (for a commensurate magnetic structure resulting from a paramagnetic phase having a grey magnetic group G1’) F subgroup of G Time inversion {1’|0 0 0} is NOT a symmetry operation of F ≤ G magnetic structure, but combined with a translation it can be… magn. point groups: nuclear space group: magn. space group: (space group) Type I F PF F some may allow ferromagnetic order

black and white group

Type III F +{R’|t}F PF + R’PF F + {R|t}F =H some may allow ferromagnetic order grey group (lattice duplicated) Type IV F + {1’ |t}F PF + 1’ PF F + {1|t}F = H antiferromagnetic order ( not allowed) antitranslation / anticentering

(Type II are the grey groups ……) Output of MGENPOS

Example of type IV MSG

Propagation vector k≠0

Pbmn21 = Pmn21 + {1’|0,1/2,0} Pmn21 The MSG in a magCIF file :

These files permit the different alternative models to be analyzed, refined, shown graphically, transported to ab-initio codes etc., with programs as ISODISTORT, JANA2006, FULLPROF, STRCONVERT, etc. A controlled descent to lower symmetries is also possible. Type of MSG depends on the propagation vector of the magnetic ordering: • Most magn. orderings are 1k-magnetic structures. • 1k-magnetic structures: moment changes from one unit cell to another according to a single wave vector or propagation vector k. • Phase factor for unit cell T: exp(-i2πk.T) • The lattice translations such that exp(-i2πk.T)=1 define the lattice mantained by the magnetic structure. • The lattice translations such that exp(-i2πk.T)= -1, are kept as antitranslations (type IV MSG). Only occur if nk=recipr. lattice vector with n=even

type IV MSG (1’|1 0 0) k=(1/2,0,0) (1’|1/2 0 0)

x a x 2a magnetic unit cell k=(1/3,0,0) type I or III MSG

a x x 3a magnetic unit cell multiple k structures: analogous situation … Possible magnetic symmetries for a magnetic phase with propagation vector (1/2,0,0) and parent space group Pnma

obtained with k-SUBGROUPSMAG k=(1/2,0,0)

HoMnO3

exp(i2πk.a) = -1

Symmetry operation {1’|1/2,0,0} is present in any case: all MSGs are type IV

(magnetic cell= (2ap,bp,cp)) BNS setting: a BNS BNS unit cell = magnetic unit cell

BNS lattice = actual lattice describing the periodicity

{1’|1/2,0,0}

OG setting:

aOG OG pr. unit cell = 1/2 magnetic pr. unit cell

OG lattice = “black &white” lattice with half of the lattice translations being actually “antitranslations”

The set of antitranslations in the lattice can be defined by a wave vector k : {1’|1,0,0} OG exp(i2πkOG.T) = -1 à T antitranslation exp(i2 k .T) = 1 à T translation kOG=(1/2,0,0) π OG If the propagation vector k of the OG setting: magnetic ordering with respect to the parent phase is such that aOG 2k = reciprocal lattice vector of parent phase, then:

OG lattice = parent lattice

k= kOG {1’|1,0,0}

( kOG≠ k in any other case!) kOG=(1/2,0,0)

{1’|1,0,0} is equivalent to {1|1,0,0} for non-magnetic degrees of freedom: the OG lattice/unit cell is the actual lattice/unit cell of the atomic positions. OG group labels vs. BNS group labels: Example: Space group resulting from deleting the time reversal part of the symmetry operations of the MSG Pana21:

result if all operations with +1 and -1 in are included: Pmn21

result if only operations with +1 are included: Pna21 Unambiguous description of a MSG as subgroup a parent gray group: (P,p) transformation to standard Pnma1’ ! Pbmn21 (-b, 2a , c; 1/4, 1/4, 0) of the MSG

p = (p1, p2, p3) p = (p1, p2, p3) s s s s (a ,b ,c )= (ap,bp,cp).P , O = Op + p1 ap + p2 bp + p3 cp

MSG standard unit cell parent unit cell origin shlft

Transformation to standard setting:

symmetry operation:

1 1 positions: magnetic moment (absolute) components: -1 -1 One should not confuse:

transformation to standard Pnma1’ ! Pbmn21 (-b, 2a , c; 1/4, 1/4, 0) from the parent setting of Pnma

P mn2 (-b, a , c; 1/8, 1/4, 0) transformation to standard from b 1 the setting being used in the application of the MSG.

(in this case (2ap, bp, cp;0,0,0) ) Listing of Magnetic Space Groups Acta Cryst. A (2008) http://www.iucr.org/books

Daniel T. Litvin

29 D.T. Litvin Tables of the 1651 MAGNETIC SPACE GROUPS (MSGs)

2008

OG setting! Towards the systematic application of Magnetic Symmetry….

1. Computer readable listings of MSGs (Shubnikov groups) 2. Extension of some of the programs of the ISOTROPY suite to magnetic systems (combined application of irreps and magnetic symmetry groups) H. T. Stokes B. J. Campbell http://stokes.byu.edu/iso/isotropy.php Magnetic symmetry application tools and databases in the Bilbao Crystallographic Server

main developers: Samuel V. Gallego, Luis Elcoro, Emre S. Tasci, Mois. I. Aroyo, J. Manuel Perez-Mato Consequences of symmetry

Von Neumann principle:

• all variables/parameters/degrees of freedom compatible with the symmetry can be present in the total distortion • Tensor crystal properties are constrained by the point group symmetry of the crystal.

• Reversely: any tensor property allowed by the point group symmetry can exist (large or small, but not forced to be zero) Magnec space group: (31.129) HoMnO Pbmn21 3 in non-standard seng.

unit cell: 2ap, bp, cp to transform to convenonal seng : (-b, a, c; 3/8,1/4,0)

WP + (1’|1/2 0 0)

8b (x, y, z | mx, my, mz), (-x+1/4, -y, z+1/2 | -mx, -my, mz), (x, -y+1/2, z | -mx, my, -mz), (-x+1/4, y+1/2, z+1/2 | mx, -my, -mz)

4a (x, 1/4, z| 0, my, 0), (-x+1/4, 3/4, z+1/2 | 0, -my, 0)

Equivalent to the use of space group Pnm21(31) with half cell along a:

Atomic positions of asymmetric unit: Magnetic moments of the asymmetric unit (µB): Ho1 4a 0.04195 0.25000 0.98250 Mn1 3.87 ≈0.0 ≈0.0 Ho2 4a 0.95805 0.75000 0.01750 Mn1 8b 0.00000 0.00000 0.50000 Split independent O1 4a 0.23110 0.25000 0.11130 positions in the lower zero values are not O12 4a 0.76890 0.75000 0.88870 symmetry symmetry “protected” O2 8b 0.16405 0.05340 0.70130 O22 8b 0.83595 0.55340 0.29870

General position: x, y, z not restricted Magnetic Point Group: mm21’ by symmetry! Consequences of symmetry

Effect of the magnetic ordering on the nuclear/lattice structure:

- case 1: no symmetry break for “nuclear structure”

(1’|1 0 0) P2/m Pc2/m

for the nuclear/lattice structure: a x P2/m P2/m

- case 2: symmetry break for “nuclear structure”

Pnma Pbmn21 (-2bp, ap, cp; 1/4,1/4,0)

for the nuclear/lattice structure :

Pnma Pmn21 (-bp, ap, cp; 1/4,1/4,0) Magnetic Space Group and Physical properties

EuZrO3 magndata #0.146 & 0.147

Pnm’a Pn’m’a’

obtained with MTENSOR Magnetoelectric tensor: Ferroic properties A "multiferroic": improper ferroelectric

HoMnO3 param. phase antiferrom. phase 4 total number Pnma 1' P nm2 index 4 a 1 of domains point groups 2 ferroic S mmm1' mm21' index 2 2 domains

Secondary symmetry-allowed effect: spontaneous polarization: Pz

Pnma1' = Panm21 + (1'|000)Panm21+(-1|000) Panm21+ (-1’|000) Panm21

S1 generators of the four domain configurations: {gn} ={(1|000), (-1|000) ,(1'|000) , (-1’|000)}

{-1|000}

-Pz Pz (S ,S )=(0,S) (S1,S2)=(S,0) 1 2 domains: equivalent energy minima Their MSG are equivalent, but not equal in general The importance of non-magnetic atoms: The same spin arrangement can produce different MSGs (and different ferroic properties) depending on the symmetry of the parent structure

I4/mmm, k=(1/2,1/2,0) Cmce, k=(0,0,0) I-42m, k=(1/2,1/2,0)

Pz

Hypothetical spin Pr2CuO4 Gd2CuO4 configuration on a

structure of type GaMnSe4 Polar magnetic symmetry of Lu2MnCoO6 due to the non-magnetic atoms:

chains + + - -

Py

Polar symmetry when Non-polar symmetry without non-magnetic atoms are non-magnetic atoms ! considered.

The polar direction is NOT along the chains! obtained with k=(1/3,1/3,0) k-SUBGROUPSMAG & MAGMODELIZE magnetic site 1b

Ba3MnNb2O9 Polar along c- type II multiferroic

FM along c Conclusion:

• Whatever method one has employed to determine or calculate a magnetic structure, the final model should include its magnetic symmetry! Acknowledgements:

The past and present team in Bilbao of the…

present: past: • M. I. Aroyo • D. Orobengoa • E. Tasci • C. Capillas • G. de la Flor • E. Kroumova • S. V. Gallego • S. Ivantchev • L. Elcoro • G. Madariaga

• J.L. Ribeiro (Braga, Portugal) • H. Stokes & B. Campbell (Provo, USA) – program ISODISTORT • V. Petricek (Prague) - program JANA2006 • J. Rodriguez-Carvajal (Grenoble) - program FullProf