The structural distortion in antiferromagnetic LaFeAsO investigated by a group-theoretical approach
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r The structural distortion in antiferromagnetic LaFeAsO investigated by agroup-theoretical approach
Ekkehard Kr¨uger and Horst P. Strunk
Institut f¨ur Materialwissenschaft, Materialphysik,Universit¨at Stuttgart, D-70569 Stuttgart, Germany (Dated: November 16, 2018)As experimentally well established, undoped LaFeAsO is antiferromagnetic below 137K with themagnetic moments lying on the Fe sites. We determine the orthorhombic body-centered group
Imma (74) as the space group of the experimentally observed magnetic structure in the undistortedlattice, i.e., in a lattice possessing no structural distortions in addition to the magnetostriction.We show that LaFeAsO possesses a partly filled “magnetic band” with Bloch functions that canbe unitarily transformed into optimally localized Wannier functions adapted to the space group
Imma . This finding is interpreted in the framework of a nonadiabatic extension of the Heisenbergmodel of magnetism, the nonadiabatic Heisenberg model. Within this model, however, the magneticstructure with the space group
Imma is not stable but can be stabilized by a (slight) distortion ofthe crystal turning the space group
Imma into the space group
P nn
Keywords: magnetism, nonadiabatic Heisenberg model, group theory
I. INTRODUCTION
Undoped LaFeAsO undergoes an abrupt structural dis-tortion from tetragonal to orthorhombic [1–3] or to mono-clinic [4] symmetry at ∼ K as well as an antiferromag-netic spin ordering transition at ∼ K [1–4]. Clarinade la Cruz et al. [4] studied with high accuracy the struc-tural distortion of LaFeAsO at 4 K by neutron-scatteringexperiments and found the Fe and O atoms to be slightlyshifted from their positions in the tetragonal phase. Inagreement with the space group P /nmm , the z coordi-nates of iron and oxygen are exactly z = 1 / z = 0,respectively, in the tetragonal phase. The values ∆ z ofthe displacements at 4 K in ± z direction are reported to∆ z = 0 . z = 0 . K and137 K shall be discussed in a following paper in the con-text of the superconducting state in the doped material,the present paper investigates the magnetic structure andthe associated structural distortions below 137 K .In Sec. III we shall determine the space group S of theexperimentally observed magnetic structure in LaFeAsOunder the assumption that there are no distortions in ad-dition to the magnetostriction. We shall show that S iscompatible with the symmetry of the Bloch functions ofthe energy band of LaFeAsO denoted in Fig. 1 by the boldline. In the following Sec. IV we will ask whether S maybe the space group of a stable magnetic structure, andin Sec. V we shall present the concept of “allowed space groups”. Finally, in Sec. VI we shall propose that theexperimentally observed distortions of the crystal corre-spond to a change from the non-allowed space group S of the undistorted crystal to an allowed space group.The NHM does not distinguished between orbital andspin moments. Therefore, we always speak of “magneticmoments” which may consist of both orbital and spinmoments. II. COORDINATE SYSTEMS
The coordinates systems used in this paper are de-picted in Figs. 2 and 3. First, we use the x, y, z co-ordinate systems as depicted in Fig. 2 which coincideswith the coordinates normally used in the literature onLaFeAsO, as, for instance, in the tables of Ref. [4]. Thiscoordinate system defines the point group operations inthe way described in Fig. 2.In the following we shall consider three structures inpure LaFeAsO whose unit cells together with the basictranslations T , T , and T of the corresponding Bravaislattices are given in Fig. 3. Fig. 3 (a) shows the tetragonalprimitive Bravais lattice Γ q of paramagnetic LaFeAsOabove 155 K which has the space group P /nmm [4, 11–15]. In Fig. 3 (b) the experimentally observed anti-ferromagnetic structure in a hypothetically undistortedcrystal is depicted, which has the orthorhombic body-centered Bravais lattice Γ vo . Finally, Fig. 3 (c) shows theantiferromagnetic structure in distorted LaFeAsO below137 K with the orthorhombic primitive Bravais lattice Γ o .For the description of these three structures we needto resort to three coordinate systems with unusual rela-tionships. While the basic translations T , T , and T are adapted to the respective structure, the origin of the x, y, z coordinate systems and the directions of its axes -2-101 Γ X M A Z R X R A Z Γ M Γ _ Γ _ Γ + Γ + Γ + M M M A R A A Z + Z + Z _ Z _ R X X A A M M Z _ Z + E n e r gy ( e V ) E F Γ _ Γ + Γ _ FIG. 1. Band structure of tetragonal LaFeAsO as calculated by the FHI-aims program [9, 10], with symmetry labels determinedby the authors. The symmetry labels can be identified from Table I, see Appendix. The bold line shows the magnetic band(as defined in Sec. III.2) consisting of two branches. relative to the atoms are fixed for all the three structuresin Figs. 3. Therefore, we may compare directly the spacegroups of the structures because the same point groupoperation in any structure is notated by the same sym-bol. As a consequence, the A and B axes (as given inFig. 2), but not the x and y axes, are perpendicular toeach other in the orthorhombic structures. Furthermore,this choice of the x, y, z axes requires a renaming of thepoint group operations given in the tables in the text-book of Bradley and Cracknell [16], as described in thenotes to Tables I, II, and III. III. THE UNDISTORTED MAGNETICSTRUCTURE
In this section we show that the experimentally ob-served magnetic structure in LaFeAsO is compatible withthe symmetry of the Bloch functions of the magneticband denoted in Fig. 1 by the bold line. First, in the fol-lowing subsection we shall determine the magnetic groupof the experimentally observed magnetic structure underthe assumption that there are no structural distortionsin the material in addition to the magnetostriction.
III.1. The experimentally determined magneticstructure and its magnetic group
We show that the group
Imma = Γ vo D h (74) with theorthorhombic body-centered Bravais lattice Γ vo depictedin Fig. 3 (b) is the space group of the experimentally de-termined magnetic structure as it is given by Fig. 4 ofRef. [4]. In Fig. 3 (b) the unit cell of antiferromagneticLaFeAsO is depicted where only the Fe atoms are indi-cated since the magnetic moments are localized on theFe sites [1, 4]. This Fig. 3 (b) is identical with Fig. 4 ofRef. [4] because by application of the basic translations T , T , T to the four Fe atoms in the unit cell the mag-netic structure in Fig. 4 of Ref. [4] can be constructed.In a first step we show that the positions of the atomsof LaFeAsO are invariant under the symmetry operationsof Imma . They are clearly invariant under the transla-tions of
Imma since its basic translations T , T , T asgiven in Fig. 3 (b) are also translations of the paramag-netic structure depicted in Fig. 3 (a) [cf. the followingrelation (2)]. In order to show that the atoms of LaFeAsOare also invariant under the rotations and reflections of xz y Fe B A
FIG. 2. Cartesian coordinate system with the A and B axesdefining the point group operations as used in this paper. The x, y, z coordinate system is identical with the x, y, z coordi-nate systems in Fig. 3. The A and B axes lie in the planespanned by the x and y axes and, in all the structures inFigs. 3 (a), (b), and (c), they have the direction of a latticetranslation. Hence, in all the structures they are perpendicu-lar to one another, while the x axis is perpendicular to the y axis only in the tetragonal structure. The z , A , and B axeshave the same positions in all the three structures in Figs. 3(a), (b), and (c). The point group operations with the indices x , y , z , a , and b are related to the depicted x , y , z , A , and B axes, respectively. For instance, C − z is a clockwise rotationof the lattice through π radians about the z axis, and C a isa rotation through π radians about the A axis. σ da = IC a ( I is the inversion) is a reflection in the plain containing theorigin and being perpendicular to the A axis. Imma , it is sufficient to consider the generating elements { C z |
12 12 } , { C a | } , { I |
12 12 } (1)of Imma as they are given in Table 3.7 of Ref. [16].However, in this paper they are written in the coordi-nate system defined by Figs. 2 and 3 (b), cf. the notesto Table II. We may easily compare the space group
Imma with the paramagnetic space group P /nmm bywriting the translations of the generating elements (1) interms of the basic translations of the Bravais lattice Γ q of P /nmm as depicted in Fig. 3 (a). Using the relation T → T + T T → − T + T T → − T − T . Γ vo → Γ q (2)[as derived from Figs. 3 (a) and (b)] we get the symme-try operations { C z |
12 12 } , { C a |
12 12 } , and { I | } , all ofthem being in the tetragonal space group P /nmm , seethe symmetry operations belonging to point Γ in Table I.Consequently, within the coordinate systems defined inFigs. 3 (a) and (b), the group Imma is a subgroup of P /nmm . The important implication is that the posi-tions of the atoms of LaFeAsO are invariant under thesymmetry operations of Imma .Now we can show that also the magnetic structure inFig. 3 (b) is invariant under the generating elements (1) of
Imma : One of the four Fe atoms in the unit celllies at the position p = T + T + T . Applyingon this atom, e.g., the rotation C z , we get an atom atthe position p = T − T with the false direction ofthe magnetic moment. Then the associated translation t = T + T , however, moves this atom to the position p = T + T of an Fe atom with the correct direc-tion of the magnetic moment. In the same way, it canbe shown that the symmetry operation { C z |
12 12 } leavesinvariant the directions of the magnetic moments of thethree other Fe atoms in the unit cell. We get the same re-sult for the other two generating elements { C a | } and { I |
12 12 } . Consequently, the group Imma is the spacegroup of the experimentally determined magnetic struc-ture in LaFeAsO.In addition, the magnetic structure depicted in Fig. 3(b) is invariant under the anti-unitary operator { K |
12 12 } ,where K denotes the operator of time inversion. First, K reverses all the magnetic moments (and leaves invariantthe positions of the atoms). Then the associated transla-tion t = T + T moves the Fe atoms to positions withthe original directions of the magnetic moments. Also t leaves invariant the positions of the atoms because itis a lattice translation (namely T ) in the paramagneticlattice depicted in Fig. 3 (a). Hence, the magnetic group M of the experimentally determined magnetic structurein undistorted LaFeAsO as depicted in Fig. 3 (b) may bewritten as M = Imma + { K |
12 12 } Imma. (3)
III.2. The symmetry of the Bloch functions of themagnetic band
The energy band of LaFeAsO denoted in Fig. 1 by thebold line is characterized by the representationsΓ − , Γ +5 ; X , X ; M , M ; A , A ; Z +1 , Z − ; R , R . (4)Folding this energy band into the Brillouin zone of thespace group Imma of the antiferromagnetic structure inthe undistorted crystal depicted in Fig. 3 (b), the repre-sentations (4) of the Bloch functions transform asΓ − → Γ − Γ +5 → Γ +2 + Γ +4 M → X +3 + X − M → X +4 + X − A → Γ +3 + Γ − A → Γ +2 + Γ − Z +1 → X +1 Z − → X − + X − R , X → T + T R , X → T + T (5)see Table IV. The underlined representations form aband listed in Table V, namely band 2 in Table V (b).Hence, the Bloch functions of this band can be unitarilytransformed into Wannier functions that are – as well localized as possible, x T T T c/2a z y Fe a (a) x T T T yz magnetic moments (b) Fe T T z y T O xc the displacementsdirections of (c) FIG. 3. Coordinate systems and unit cells ofthree (magnetic) structures in LaFeAsO. For rea-sons of clarity, only the Fe atoms and in structure(c) also the O atoms are shown. a and c denotethe lengths of the sides in the tetragonal unit cell.The coordinate systems define the symmetry op-erations { R | pqr } as used in this paper. They arewritten in the Seitz notation detailed in the text-book of Bradley and Cracknell [16]: R stands fora point group operation and pqr denotes the sub-sequent translation. The point group operation R is related to the x, y, z coordinate system as de-fined in Fig. 2, and pqr stands for the translation t = p T + q T + r T with the basic translations T i being different for the structures (a), (b), and(c). The origin of the coordinate systems is fixedfor all the three structures (a), (b), and (c).(a) The paramagnetic structure of LaFeAsO withthe tetragonal space group P /nmm (129).(b) The antiferromagnetic structure in the undis-torted material with the orthorhombic spacegroup Imma (74).(c) The antiferromagnetic structure in distortedLaFeAsO with the “allowed” orthorhombic spacegroup
P nn z direction which are invariant under the mag-netic group M = P nn { K |
12 12 } P nn
2. Since
P nn
Imma , the magnetic mo-ments may have the same directions as in theundistorted structure (b). However, in the mag-netic group M the moments may also be cantedwithin a plain perpendicular to the z axis as de-picted in Fig. 4. – centered at the Fe atoms, – and symmetry-adapted to the magnetic group M in Eq. (3),see the notes to Table V. For this reason, this band iscalled “magnetic band” related to the magnetic group M .The NHM predicts that the electrons of this partly filledband may lower their Coulomb correlation energy by ac-tivating an exchange mechanism producing a magneticstructure with the magnetic group M and with the mag-netic moments lying on the Fe sites, i.e., by producing theexperimentally determined magnetic structure [6, 17, 18].In this sense we say that the symmetry of the Bloch func-tions is “compatible” with the experimentally determinedmagnetic structure.Table V lists all the possible magnetic bands related tothe magnetic group M in Eq. (3). By this table one canmake sure that the symmetry of the Bloch functions inrelation (5) corresponds only to band 2 in Table V (b).That is, the magnetic moments in LaFeAsO may only besituated at the Fe sites. IV. STABILITY OF A MAGNETICSTRUCTURE
The magnetic band denoted in Fig. 1 by the bold lineis related to the magnetic group M in Eq. (3). However,this structure cannot exist in the undistorted materialbecause the space group Imma does not possess suit-able representations. This shall be substantiated in thefollowing.Let be M = S + { K | t } S (6)the magnetic group of a given antiferromagnetic struc-ture, with S denoting the space group of this structureand { K | t } being an anti-unitary operator leaving invari-ant the magnetic structure. K still denotes the operatorof time inversion.The operator K reverses the magnetic moments in theantiferromagnetic ground state | G i , so | G i = K | G i (7)is the state with the opposite directions of the magneticmoments which clearly is different from | G i . Within theNHM, both states | G i and | G i are eigenstates of a Hamil-tonian denoted in Ref. [17] by e H which commutes with K . Hence, | G i and | G i belong to an irreducible two-dimensional corepresentation e D M of the group f M = M + KM (8)of the Hamiltonian e H , see Ref. [17], Sec. III.C.However, when we restrict ourselves to the symmetryoperations P of the subgroup M of f M , then e D M must subduce a one-dimensional corepresentation D M of M ,that is, P | G i = c · | G i for P ∈ M, (9)where | c | = 1. In particular, this Eq. (9) is satisfied forthe anti-unitary operator { K | t } , { K | t }| G i = c · | G i , (10)where still | c | = 1.A stable magnetic state | G i may exist if f M possessesat least one suitable corepresentation e D M . From thiscondition it follows the Theorem IV.1
A stable magnetic state with the spacegroup S can exist if S has at least one one-dimensionalsingle-valued representation D (i) following case (a) with respect to the magneticgroup S + { K | t } S and(ii) following case (c) with respect to the magneticgroup S + KS . The cases (a) and (c) are defined in Eqs. (7.3.45) and(7.3.47), respectively, of Ref. [16]. They determine the di-mension of the corepresentations of the magnetic groups S + { K | t } S and S + KS , respectively, which are de-rived [16] from the representation D of S . The one-dimensional representation D may satisfy the second con-dition (ii) only if D has non-real characters.This Theorem IV.1 was proposed in Ref. [17] and writ-ten down in the present form in Ref. [19]. It can beunderstood following the theory of corepresentations asgiven in Sec. 7.3. of the textbook of Bradley and Crack-nell [16]: Let D be a representation of S satisfying thefirst condition (i) of the theorem, then we may derivefrom D the one-dimensional corepresentation D M of M given by Eq. (7.3.45) of Ref. [16] (with A = { K | t } and N = 1).Then we may derive from D M the two-dimensionalcorepresentation e D M of f M as it is given in Eq. (7.3.17) ofRef. [16], where now A = K . [Bradley and Cracknell as-sume that the subgroup G in their Eq. (7.3.11) does notcontain anti-unitary elements; for the derivation of theirEq. (7.3.17), however, these authors do not make use ofthis assumption.] If D satisfies additionally the secondcondition (ii) of the theorem, then the corepresentation e D M has just the required properties, cf. the AuxiliaryPublication, citation 9, of Ref. [17].In the present paper the cases (a) and (c) are deter-mined (in Tables II and III) by the relatively straightfor-ward equation (7.3.51) of Ref. [16], X B ∈ C χ ( B ) = + | S | in case (a) −| S | in case (b)0 in case (c) . (11)The sum runs over the symmetry operations B in theleft cosets C = { K | t } S and C = KS , respectively, ofthe magnetic groups. χ ( B ) is the character of B in therepresentation D and | S | denotes, as usual, the order of S .Table II shows that the space group Imma does notpossess one-dimensional representations with non-realcharacters. Consequently, all the one-dimensional rep-resentations of
Imma follow case (a) with respect to themagnetic group
Imma + KImma and hence, stable mag-netic structures with the space group
Imma do not exist.
V. ALLOWED SPACE GROUPS INANTIFERROMAGNETIC LaFeAsO
As shown in the preceding Sec. IV, the magnetic struc-ture depicted in Fig. 3 (b) is not stable because its spacegroup
Imma does not possess suitable representations.However, this structure may be stabilized by a (small)distortion of the crystal that turns the space group
Imma into a new space group possessing representations that al-low the formation of a stable magnetic structure. Thisnew space group shall be named an “allowed” spacegroup in LaFeAsO. In this context we assume that thedistortions of the crystal are so small that the magneticstructure in the distorted material differs hardly from thestructure depicted in Fig. 3 (b) because the symmetry ofthe Bloch functions of the magnetic band is compatiblewith the magnetic group M in Eq. 3. For that reason weassume that the magnetic structure depicted in Fig. 3 (b)is invariant under an allowed space group, i.e., we assumethat an allowed space group is a subgroup of Imma .In this section we look for all the allowed space groupsin antiferromagnetic LaFeAsO by considering all theorthorhombic primitive and monoclinic primitive spacegroups in Table 3.7 of Ref. [16] with at least four pointgroup elements. This is a laborious task because all theconceivable translations and rotations of the coordinatesystems given in Table 3.7 of Ref. [16] should be regarded.The coordinate system of the magnetic structure inLaFeAsO in an orthorhombic primitive or monoclinicprimitive Bravais lattice is depicted in Fig. 3 (c). Thepositions of the atoms and the magnetic structure areinvariant under two anti-unitary operations of the form { K | t } , namely under { K |
12 12 } and { K | } . Hence, wemust look for space groups S possessing one-dimensionalrepresentations following case (c) with respect to themagnetic group S + KS and case (a) with respect eitherto S + { K |
12 12 } S or to S + { K | } S , see Theorem IV.1.We found the two allowed space groups P nn o C v (34) and P P P = Γ o D (19) and shall present them inthe following two Secs. V.1 and V.2. In Sec. VI we willshow that P nn
V.1. The space group
P nn The space group
P nn o and can be defined by the two gener- ating elements { σ da |
12 12 } and { σ db |
12 12 12 } , (12)cf. Table 3.7 of Ref. [16] and the notes to Table III.The symmetry operations (12) are given in the coordinatesystem defined in Figs. 2 and 3 (c). In order to show that P nn
Imma (with the Bravaislattice Γ vo ) we write the translations of the generatingelements (12) in terms of the basic translations of thegroup Imma as given in Fig. 3 (b). Using the relation T → T + T T → T + T T → T + T , Γ o → Γ vo (13)[as deduced from Figs. 3 (b) and (c)], we get the oper-ations { σ da |
12 12 } and { σ db | } , both belonging to thegroup Imma , see the symmetry operations belonging topoint Γ in Table II. Consequently, within the coordinatesystems of Fig. 3,
P nn
Imma .In contrast to the group
Imma , the group
P nn S one-dimensional representations satis-fying the conditions (i) and (ii) of Theorem IV.1 for themagnetic groups M = P nn { K |
12 12 } P nn
P nn
KP nn
2, respectively, see Table III. Fur-ther, the little group of S comprises the whole spacegroup. Hence, P nn stable magnetic structure. A stable antiferromagnetic groundstate | G i is basis function of a one-dimensional corep-resentation of M and | G i and the time inverted state | G i = K | G i form basis functions of an irreducible two-dimensional corepresentation of the group f M = M + KM which might be determined by Eq. (7.3.17) of Ref. [16].[As an example, the analogous corepresentation for thespin-density-wave state in chromium is explicitly given inRef. [17].] V.2. The space group P P P A second allowed space group is the group P P P withthe generating elements { C a | } and { C b |
12 12 } , (15)written again within the coordinate systems of Figs. 2and 3 (c). We do not describe this group but only reportthe result that a magnetic structure with the magneticgroup M = P P P + { K |
12 12 } P P P (16)could be stable in LaFeAsO. However, antiferromagneticLaFeAsO does not possess this space group, see the fol-lowing Sec. VI. VI. DISTORTION OF THE CRYSTAL
The antiferromagnetic structure in undistortedLaFeAsO has the space group
Imma . Hence, in theundistorted antiferromagnetic material the nonadiabaticHamiltonian H n [as defined in Eq. (2.15) of Ref. [6]]would commute with all the symmetry operations of Imma . However, as shown in the preceding sections,
Imma is not an allowed space group but has two allowedsubgroups, namely
P nn P P P . Thus, either P nn P P P , but not Imma may be the spacegroup of the antiferromagnetic structure in LaFeAsO.Consequently, the nonadiabatic Hamiltonian H n com-mutes with the symmetry operations P of one of theallowed space groups S , but does not commute withthose symmetry operations that belong to Imma , butdo not belong to S ,[ H n , P ] = 0 for P ∈ S [ H n , P ] = 0 for P ∈ ( Imma − S ) . (17)For that reason, the Fe and O atoms in LaFeAsO areslightly shifted from their positions in the space group Imma .In order to show this, we write the eight symmetryoperations of
Imma as given in Table II for point Γ interms of the basic translations T , T , and T of theorthorhombic primitive lattice Γ o in Fig. 3 (c), { E | } , { E |
12 12 12 } , { C z | } , { C z |
12 12 } , { C a | } , { C a | } , { C b |
12 12 } , { C b | } , { I |
12 12 } , { I | } , { σ z | } , { σ z | } , { σ da | } , { σ da |
12 12 } , { σ db | } , and { σ db |
12 12 12 } , (18)cf. relation (13). For each symmetry operation fromΓ vo we get two operations in Γ o because the operation { E |
12 12 12 } in Γ o is a lattice translation in Γ vo ( E is theidentity).The antiferromagnetic structure with the space group Imma is invariant under the sixteen symmetry opera-tions (18) now written within the Bravais lattice Γ o ofthe structure in Fig. 3 (c). By inspection of Fig. 3 (c)we find under these sixteen operations four operations,namely { E | } , { C z | } , { σ da |
12 12 } , and { σ db |
12 12 12 } , (19)leaving invariant the distorted crystal as depicted in Fig. 3(c). These four operations form (together with the trans-lations) a group, namely the allowed space group P nn
P nn y xxy z = 3c/2z = c/2 Fe β αββα βαα FIG. 4. The two layers of the Fe atoms in the unit cell ofthe distorted antiferromagnetic structure depicted in Fig. 3(c). The z axis is normal to the paper. For clarity, the smalldisplacements of the Fe atoms (in z direction) are not indi-cated. The x, y, z coordinate system coincides with the x, y, z coordinates in Fig. 3 (c). Within distorted LaFeAsO, themagnetic moments as depicted in Fig. 3 (c) may be rotatedby arbitrary angles α and β within a plain perpendicular tothe z axis because also this canted structure is invariant un-der the magnetic group M = P nn { K |
12 12 } P nn α and β may be different and both angles are likely small becausethe symmetry of the Bloch functions of the magnetic bandis compatible with the magnetic group M of the magneticstructure depicted in Fig. 3 (b). nian H n and stabilize in this way the magnetic struc-ture in LaFeAsO. Together with the anti-unitary oper-ator { K |
12 12 } (which also leaves invariant the distortedstructure), the symmetry operations (19) form the mag-netic group M in Eq. (14).In the magnetic group M the magnetic moments maybe canted by two arbitrary angles α and β as depictedin Fig. 4 (while such canted moments are not invariantunder Imma ). Both angles are likely small because thesymmetry of the Bloch functions of the magnetic bandis compatible with the magnetic group M [in Eq. (3)] ofthe structure depicted in Fig. 3 (b).For the second allowed space group P P P , on theother hand, we have not found any simple distortion ofthe lattice realizing this group. VII. CONCLUSIONS
Within the nonadiabatic Heisenberg model (NHM),the situation in the magnetic band of LaFeAsO (denotedin Fig. 1 by the bold line) is characterized by two dif-ferent group-theoretical phenomena. On the one hand,the magnetic band (i.e., the symmetry of the Bloch func-tions of this band) is related to the magnetic group M of the experimentally observed [1–4] magnetic structurein LaFeAsO and, on the other hand, any structure withthe space group M is unstable. These two phenomenaare discussed in separated subsections VII.1 and VII.2. VII.1. Nonadiabatic condensation energy
The nonadiabatic Heisenberg model (NHM) is definedby three postulates on the Coulomb correlation energy innarrow, partly filled bands [6]. A direct consequence ofthese postulates is the existence of the “nonadiabatic con-densation energy” ∆ E defined by Eq. (2.20) of Ref. [6].The electrons at the Fermi level may lower their Coulombcorrelation energy by ∆ E by occupying an atomic-likestate as defined by Mott [20] and Hubbard [21]: the elec-trons occupy localized states as long as possible and per-form their band motion by hopping from one atom toanother. In the present approach, the atomic-like stateis consistently described in terms of symmetry-adaptedand optimally localized Wannier functions and is pre-cisely defined by an equation in the nonadiabatic system,namely by Eq. (2.19) of Ref. [6]. In case the consideredpartly filled band is a magnetic band related to a mag-netic group M (as considered in this paper), the electronsof this band may gain the energy ∆ E by activating an ex-change mechanism producing a magnetic structure withthe magnetic group M [6, 17, 18].In former papers we could show that evidently themagnetic states in Cr [17], Fe [18], La CuO [19], andYBa Cu O [7] are connected with narrow, partly filledmagnetic bands in the band structures of the respectivematerials. That is, there is evidence that the nonadia-batic condensation energy ∆ E is responsible for the oc-currence of the magnetic states in these materials.Also LaFeAsO possesses a magnetic band related tothe magnetic group M = Imma + { K |
12 12 } of the ex-perimentally observed magnetic structure. This findingsuggests that, first, the electrons of the magnetic bandof LaFeAsO perform the atomic-like motion as definedwithin the NHM by Eq. (2.19) of Ref. [6] and that, sec-ondly, the nonadiabatic condensation energy ∆ E is re-sponsible for the magnetic state. VII.2. Stable magnetic structures
According to Theorem IV.1, not all the magnetic struc-tures may be stable within the NHM. This theorem led al-ready to an understanding of the really existing magnetic groups in Cr [17], La CuO [19], and YBa Cu O [7]. Inthe present paper, this theorem is the key for an un-derstanding of the distortion (in addition to the magne-tostriction) of antiferromagnetic LaFeAsO.Theorem IV.1 defines “allowed” space groups belong-ing to stable magnetic structures. Though the mag-netic band of LaFeAsO is related to the magnetic group M = Imma + { K |
12 12 } , the space group Imma is notallowed. On the other hand, allowed space groups inLaFeAsO are the two groups
P nn P P P . Con-sequently, the magnetic structure may be stabilized bya (slight) change of the atomic sites in such a way thatthe space group Imma is turned into one of the allowedspace groups. Briefly speaking, the structural distortion“realizes” the space group allowed in LaFeAsO.The displacements of the Fe and O atoms as proposedin this paper are depicted in Fig. 3 (c). They have thespace group
P nn
Cmma and P n as they have been reported for thespace group of antiferromagnetic LaFeAsO in the undis-torted [1] and the distorted [4] crystal, respectively. Nev-ertheless, using the space group P n , Clarina de laCruz et al. [4] discovered the important displacements ofthe Fe and O atoms in z direction. The directions of thedisplacements depicted in Fig. 3 (c) do not essentiallydiffer from those given in Table 2 of Ref. [4]: while theycoincide absolutely within the Fe or O layers, Table 2of Ref. [4] suggests that the displacements are periodicwith T /
2. As depicted in Fig. 3 (c), the displacementsproposed in this paper alternate their directions at aninterval of T /
2. However, in the space group
P nn
ACKNOWLEDGMENTS
We wish to thank Volker Blum from the aims teamof the Fritz-Haber-Institut der Max-Planck-Gesellschaftin Berlin for extending the aims program by an outputof the eigenvectors which enabled us to determine thesymmetry of the Bloch functions in the band structureof LaFeAsO. We are indebted to Franz-Werner Gergenfrom the EDV group of the Max-Planck-Institut f¨ur Met-allforschung in Stuttgart for his assistance in getting torun the computer programs needed for this work, and wethank Ernst Helmut Brandt for valuable discussion andOve Jepsen for initiating this paper.
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12 12 } { C +4 z | } { C x | } { C a |
12 12 } { I | } { σ z |
12 12 } { S − z | } { σ x | } { σ da |
12 12 } Γ +1 , Z +1 +2 , Z +2 +3 , Z +3 +4 , Z +4 +5 , Z +5 − , Z − − , Z − − , Z − − , Z − − , Z − M (
12 12 { σ da |
12 12 } { σ da |
12 32 } { I | }{ E | } { E | } { C z |
12 12 } { C z |
12 32 } { σ db | } { σ db | } { I | } M M M M M (
12 12
0) ( continued ) { σ y | } { C +4 z | } { C y | } { S − z | }{ σ y | } { C − z | } { C y | } { S +4 z | }{ σ z |
12 12 } { C b | } { C b | } { σ x | } { C − z | } { C x | } { S +4 z | }{ σ z |
12 32 } { C a |
12 32 } { C a |
12 12 } { σ x | } { C +4 z | } { C x | } { S − z | } M M M M A (
12 12 12 ) { σ da |
12 12 } { σ da |
12 12 } { I | }{ E | } { E | } { C z |
12 12 } { C z |
12 12 } { σ db | } { σ db | } { I | } A A A A A (
12 12 12 ) ( continued ) { σ y | } { C +4 z | } { C y | } { S − z | }{ σ y | } { C − z | } { C y | } { S +4 z | }{ σ z |
12 12 } { C b | } { C b | } { σ x | } { C − z | } { C x | } { S +4 z | }{ σ z |
12 12 } { C a |
12 12 } { C a |
12 12 } { σ x | } { C +4 z | } { C x | } { S − z | } A A A A R (0
12 12 ) { C y | } { C x | } { C z |
12 12 } { σ z |
12 12 } { I | } { σ y | }{ E | } { E | } { C y | } { C x | } { C z |
12 12 } { σ x | } { σ x | } { σ z |
12 12 } { I | } { σ y | } R R X (0 { C y | } { C x | } { C z |
12 12 } { σ z |
12 32 } { I | } { σ y | }{ E | } { E | } { C y | } { C x | } { C z |
12 32 } { σ x | } { σ x | } { σ z |
12 12 } { I | } { σ y | } X X t = − T − T . That is, the symmetry operations P bc given in Table5.7 of Ref. [16] are changed into the operations P p used in this paper by the equation P p = { E | t } P bc { E | − t } , where E is the identity operation, cf. Eq. (3.5.11) of Ref. [16] (which is related to the opposite translation − t ). TABLE II. Character tables of the single-valued irreducible representations of the orthorhombic space group
Imma = Γ vo D h (74) of the antiferromagnetic structure depicted in Fig. 3 (b).Γ(000) K { E | } { C z |
12 12 } { C a | } { C b |
12 12 } { I |
12 12 } { σ z | } { σ da |
12 12 } { σ db | } Γ +1 (a) 1 1 1 1 1 1 1 1Γ +2 (a) 1 -1 1 -1 1 -1 1 -1Γ +3 (a) 1 1 -1 -1 1 1 -1 -1Γ +4 (a) 1 -1 -1 1 1 -1 -1 1Γ − (a) 1 1 1 1 -1 -1 -1 -1Γ − (a) 1 -1 1 -1 -1 1 -1 1Γ − (a) 1 1 -1 -1 -1 -1 1 1Γ − (a) 1 -1 -1 1 -1 1 1 -1 X (
12 12 12 ) K { E | } { C z |
12 12 } { C a | } { C b |
12 12 } { I |
12 12 } { σ z | } { σ da |
12 12 } { σ db | } X +1 (a) 1 1 1 1 1 1 1 1 X +2 (a) 1 -1 1 -1 1 -1 1 -1 X +3 (a) 1 1 -1 -1 1 1 -1 -1 X +4 (a) 1 -1 -1 1 1 -1 -1 1 X − (a) 1 1 1 1 -1 -1 -1 -1 X − (a) 1 -1 1 -1 -1 1 -1 1 X − (a) 1 1 -1 -1 -1 -1 1 1 X − (a) 1 -1 -1 1 -1 1 1 -1 R ( K { E | } { C a | } { I |
12 12 } { σ da |
12 12 } R +1 (a) 1 1 1 1 R − (a) 1 1 -1 -1 R +2 (a) 1 -1 1 -1 R − (a) 1 -1 -1 1 S ( ) K { E | } { C b |
12 12 } { I |
12 12 } { σ db | } S +1 (a) 1 1 1 1 S − (a) 1 1 -1 -1 S +2 (a) 1 -1 1 -1 S − (a) 1 -1 -1 1Notes to Table II(i) The points T and W are not listed because they possess only one two-dimensional representation T and W , respectively.(ii) The character tables are determined from Table 5.7 in Ref. [16]. The origin of the coordinate system used in Ref. [16]is translated into the origin used in this paper by t = T . That is, the symmetry operations P bc given in Table 5.7 ofRef. [16] are changed into the operations P p used in this paper by the equation P p = { E | t } P bc { E | − t } , where E is the identity operation, cf. Eq. (3.5.11) of Ref. [16] (which is related to the opposite translation − t ).(iii) The space group operations are related to the coordinate systems in Figs. 2 and 3 (b). The x , y , and z axes have thesame orientations as in the tetragonal structure in Fig. 3 (a). In this way, Imma becomes a subgroup of the tetragonalgroup P /nmm . As a consequence, the xyz coordinate system used in Ref. [16] is rotated anti-clockwise with respectto the basic translations through π radians about the z axis. Thus, the x and y axes in Ref. [16] become the B and A axes, respectively, in this paper, see Fig. 2. Consequently, the point group operations belonging to the space groupoperations P p calculated by the above equation are renamed in a second step. For instance, the operation C x in Table5.7 of Ref. [16] turns into the operation C b .(iv) K stands for the operator of time-inversion. The entries below K specify whether the related representation follows, withrespect to the magnetic group Imma + KImma , case ( a ), case ( b ), or case ( c ) when they are given by Eqs. (7.3.45-47)of Ref. [16].(v) The cases ( a ), ( b ), and ( c ) are determined by Eq. (11).(vi) All the one-dimensional representations of the space group Imma are real and, hence, follow case (a) with respect to
Imma + KImma . Therefore, stable magnetic structures with this space group do not exist, see Theorem IV.1. TABLE III. Character tables of the single-valued irreducible representations of the orthorhombic space group
P nn o C v (34) of the antiferromagnetic structure depicted in Fig. 3 (c).Γ(000) K { K |
12 12 } { K | } { E | } { C z | } { σ da |
12 12 } { σ db |
12 12 12 } Γ (a) (a) (a) 1 1 1 1Γ (a) (a) (a) 1 1 -1 -1Γ (a) (a) (a) 1 -1 1 -1Γ (a) (a) (a) 1 -1 -1 1 Z (00 ) K { K |
12 12 } { K | } { E | } { σ da |
12 12 } { E | } { σ da |
12 32 } { C z | } { σ db |
12 12 12 } { C z | } { σ db |
12 12 32 } Z (c) (c) (c) 1 i -1 -i 1 i -1 -i Z (c) (c) (c) 1 -i -1 i 1 -i -1 i Z (c) (c) (c) 1 i -1 -i -1 -i 1 i Z (c) (c) (c) 1 -i -1 i -1 i 1 -i S (
12 12 K { K |
12 12 } { K | } { E | } { σ da |
12 12 } { E | } { σ da |
32 12 } { C z | } { σ db |
12 12 12 } { C z | } { σ db |
12 32 12 } S (c) (a) (c) 1 i -1 -i 1 i -1 -i S (c) (a) (c) 1 -i -1 i 1 -i -1 i S (c) (a) (c) 1 i -1 -i -1 -i 1 i S (c) (a) (c) 1 -i -1 i -1 i 1 -iNotes to Table III(i) In addition to Γ only the points Z and S are listed because only Z and S possess one-dimensional non-real representations.(ii) The character tables are determined from Table 5.7 in Ref. [16]. The origin of the coordinate system used in Ref. [16]is translated into the origin used in this paper by t = T . That is, the symmetry operations P bc given in Table 5.7 ofRef. [16] are changed into the operations P p used in this paper by the equation P p = { E | t } P bc { E | − t } , where E is the identity operation, cf. Eq. (3.5.11) of Ref. [16] (which is related to the opposite translation − t ).(iii) The space group operations are related to the coordinate systems in Figs. 2 and 3 (c). The x , y , and z axes have thesame orientations as in the tetragonal structure in Fig. 3 (a). In this way, P nn P /nmm . As a consequence, the xyz coordinate system used in Ref. [16] is rotated anti-clockwise with respectto the basic translations through π radians about the z axis. Thus, the x and y axes in Ref. [16] become the B and A axes, respectively, in this paper, see Fig. 2. Consequently, the point group operations belonging to the space groupoperations P p calculated by the above equation are renamed in a second step. For instance, the operation σ x in Table5.7 of Ref. [16] turns into the operation σ db .(iv) K stands for the operator of time inversion. The entries below K and the operators { K | t } specify whether the relatedrepresentation follows, with respect to the magnetic groups P nn
KP nn
P nn { K | t } P nn
2, respectively, case( a ), case ( b ), or case ( c ) when they are given by Eqs. (7.3.45-47) of Ref. [16].(v) The cases ( a ), ( b ), and ( c ) are determined by Eq. (11).(vi) The little group of point S comprises the whole space group P nn S possesses one-dimensional representationsfollowing case ( c ) and case ( a ) with respect to P nn
KP nn
P nn { K |
12 12 } P nn
2, respectively. Consequently,a magnetic structure with the space group
P nn TABLE IV. Compatibility relations between the Brillouin zone for tetragonal paramagnetic LaFeAsO and the Brillouin zonefor the orthorhombic antiferromagnetic structure depicted in Fig. 3 (b).ΓΓ +1 Γ +2 Γ +3 Γ +4 Γ +5 Γ − Γ − Γ − Γ − Γ − Γ +1 Γ +3 Γ +3 Γ +1 Γ +2 + Γ +4 Γ − Γ − Γ − Γ − Γ − + Γ − XM M M M X +1 + X − X +2 + X − X +3 + X − X +4 + X − XZ +1 Z +2 Z +3 Z +4 Z +5 Z − Z − Z − Z − Z − X +1 X +3 X +3 X +1 X +2 + X +4 X − X − X − X − X − + X − Γ A A A A Γ +1 + Γ − Γ +2 + Γ − Γ +3 + Γ − Γ +4 + Γ − TR R T T TX X T T Notes to Table IV(i) The antiferromagnetic structure in Fig. 3 (b) has the space group
Imma .(ii) The Brillouin zone for the orthorhombic space group
Imma lies within the Brillouin zone for the tetragonal space group P /nmm . The points Γ and A in the tetragonal Brillouin zone are equivalent to the point Γ in the orthorhombic Brillouinzone; M and Z are equivalent to X , and R and X are equivalent to T .(iii) The upper rows list the representations of the little groups of the points of symmetry in the Brillouin zone for thetetragonal paramagnetic phase. The lower rows list representations of the little groups of the related points of symmetryin the Brillouin zone for the antiferromagnetic structure.The representations in the same column are compatible in the following sense: Bloch functions that are basis functionsof a representation D i in the upper row can be unitarily transformed into the basis functions of the representation givenbelow D i .(iv) The compatibility relations are determined in the way described in great detail in Ref. [23].(v) The representations are labeled as given in Tables I and II, respectively. TABLE V. Single-valued representations of all the magnetic bands related to the space group
Imma = Γ vo D h (74) with theWannier functions being centered (a) at the La atoms, (b) at the Fe atoms, (c) at the As atoms, and (d) at the O atoms.(a) LaΓ X R S T W
Band 1 2Γ +1 + 2Γ − X +1 + 2 X − R +1 + 2 R − S − + 2 S +2 T W Band 2 2Γ +2 + 2Γ − X +2 + 2 X − R +1 + 2 R − S +1 + 2 S − T W Band 3 2Γ +3 + 2Γ − X +3 + 2 X − R − + 2 R +2 S +1 + 2 S − T W Band 4 2Γ +4 + 2Γ − X +4 + 2 X − R − + 2 R +2 S − + 2 S +2 T W (b) FeΓ X R S T W
Band 1 Γ +1 + Γ +4 + Γ − + Γ − X +2 + X +3 + X − + X − R +1 + R − + R +2 + R − S +1 + S − + S +2 + S − T W Band 2 Γ +2 + Γ +3 + Γ − + Γ − X +1 + X +4 + X − + X − R +1 + R − + R +2 + R − S +1 + S − + S +2 + S − T W (c) AsΓ X R S T W
Band 1 2Γ +1 + 2Γ − X +1 + 2 X − R +1 + 2 R − S − + 2 S +2 T W Band 2 2Γ +2 + 2Γ − X +2 + 2 X − R +1 + 2 R − S +1 + 2 S − T W Band 3 2Γ +3 + 2Γ − X +3 + 2 X − R − + 2 R +2 S +1 + 2 S − T W Band 4 2Γ +4 + 2Γ − X +4 + 2 X − R − + 2 R +2 S − + 2 S +2 T W (d) OΓ X R S T W
Band 1 Γ +1 + Γ +2 + Γ − + Γ − X +1 + X +2 + X − + X − R +1 + R − + R +2 + R − S +1 + S − + S +2 + S − T W Band 2 Γ +3 + Γ +4 + Γ − + Γ − X +3 + X +4 + X − + X − R +1 + R − + R +2 + R − S +1 + S − + S +2 + S − T W Notes to Table V(i) The antiferromagnetic structure of LaFeAsO depicted in Fig. 3 (b) has the space group
Imma and the magnetic group M = Imma + { K |
12 12 } Imma with K denoting the operator of time-inversion.(ii) Each row defines one band consisting of four branches, because in each case there are four atoms in the unit cell.(iii) The representations are given in Table II.(iv) The bands are determined by Eq. (23) of Ref. [19].(v) Assume a band of the symmetry in any row of theses Tables V (a), (b), (c), or (d) to exist in the band structure ofLaFeAsO. Then the Bloch functions of this band can be unitarily transformed into Wannier functions that are – as well localized as possible; – centered at the assigned (La, Fe, As, or O) atoms; – and symmetry-adapted to the space group Imma .(vi) Eq. (23) of Ref. [19] makes sure that the Wannier function may be chosen to be symmetry-adapted to the space group
Imma . In addition, there exists a Matrix N satisfying both Eqs. (26) (with { K |
12 12 } ) and (32) of Ref. [19] for allthe bands listed in this table. Hence, the Wannier functions may be chosen symmetry adapted to the magnetic group M = Imma + { K |