Thermoelectric Transport Properties in Magnetically Ordered Crystals

research papers

Thermoelectric transport properties in magnetically ordered crystals

ISSN 2053-2733 Hans Grimmer*

Research with Neutrons and Muons, Paul Scherrer Institut, WLGA/019, Villigen PSI, CH-5232, Switzerland. *Correspondence e-mail: [email protected]

Received 12 January 2017 Accepted 10 April 2017 The forms of the tensors describing thermoelectric transport properties in magnetically ordered crystals are given for frequently used orientations of the 122 space-time point groups up to second order in an applied magnetic ﬁeld. It is Edited by W. F. Kuhs, Georg-August University shown which forms are interchanged for the point groups of the hexagonal Go¨ttingen, Germany crystal family by two different conventions for the connection between the Hermann–Mauguin symbol and the orientation of the Cartesian coordinate Keywords: thermoelectricity; transport properties; magnetic order; galvanomagnetic system. The forms are given in Nye notation, which conspicuously shows how effects; thermomagnetic effects. the forms for different point groups are related. It is shown that the measurable effects in magnetically ordered crystals can be decomposed into an effect occurring in all crystals and one coming from the magnetic ordering. Errors in the literature are pointed out.

1. Introduction Based on general principles of statistical physics (Landau & Lifshitz, 1958) Shtrikman & Thomas (1965a,b) gave the Onsager relations in a form convenient for deriving transport properties in magnetically ordered crystals. They applied the relations to the electric resistivity of crystals in an external magnetic ﬁeld H. Similar results were also obtained by Butzal & Birss (1982), who do not give reference to Shtrikman & Thomas (1965a,b). Grimmer (1993) extended their approach to thermoelectric transport properties. Consider a crystal in which an electric current with density j ﬂows and a temperature gradient grad T exists. These produce a heat current with density h and a gradient of the electrochemical potential ’ ¼ ’ =e, where ’ is the electric potential, is the chemical potential and e is the charge of the electron. Assuming linear relationships and using Cartesian coordinates we can write [equation (4) in Grimmer (1993)]

@’ 0 @T ¼ ikðH; MÞjk þ ikðH; MÞ ; ð1aÞ @xi @xk

00 @T hi ¼ TikðH; MÞjk kikðH; MÞ ; ð1bÞ @xk where the property tensors depend on the external magnetic field H, assumed to be constant or varying slowly as a function of space and time, and on the time-averaged magnetization field M describing the magnetic configuration of the crystal. Time inversion (time reversal) 10 relates crystals with magnetic moments of opposite sign; M and H change sign under 10. Whereas M varies over a (primitive) magnetic unit cell for magnetically ordered crystals and is invariant under space inversion 1 only for centrosymmetric crystals, the vector H is # 2017 International Union of Crystallography approximately constant in a magnetic unit cell; H is a so-called

Acta Cryst. (2017). A73, 333–345 https://doi.org/10.1107/S2053273317005368 333 research papers magnetic vector, i.e. it changes sign under 10 and is invariant possible and that the Hall effect must vanish identically if under space inversion 1. H ¼ 0. 0þ þ We introduce a 6 6 matrix R by defining Equations (7a), (7b) show that ik , ikl and iklm are 0 0 þ 0 00 R ¼ ; R ¼ ; R ¼ ; R ¼ k ; ð2Þ invariant under time reversal 1 whereas ik , ikl and iklm ik ik ikþ3 ik iþ3 k ik iþ3 kþ3 ik change sign. The tensors symmetric in i and k describe electric where i; k ¼ 1; 2; 3. Defining 0þ þ resistance [ik ðM0Þ], linear magnetoresistance [iklðM0Þ] and þ R ðÞ¼H; M 1 ½R ðÞH; M R ðÞH; M ; ð3Þ (quadratic) magnetoresistance [iklmðM0Þ], the tensors anti- 2 symmetric in i and k describe the spontaneous Hall effect R Rþ 0 we split the matrix into a part that is invariant and a [ik ðM0Þ], (ordinary) Hall effect [iklðM0Þ] and (quadratic) R 0 part that changes sign under time reversal 1 . The Onsager anomalous Hall effect [iklmðM0Þ]. relations can then be written as (Shtrikman & Thomas, Similarly, the thermal transport properties are described by a b 1965 , ) kijðH; MÞ,(i, j = 1, 2, 3), i.e.byRiþ3 jþ3ðH; MÞ according to equation (2). For ¼ i þ 3and ¼ j þ 3 we can replace R by RðH; MÞ¼RðH; MÞ: ð4Þ k in equations (3)–(7a), (7b). The part invariant under 10, þ It follows that R can also be written as ð Þ ð Þ ÂÃkij H; M , is symmetric, the part changing sign, kij H; M , R ðÞ¼H; M 1 R ðÞH; M R ðÞH; M ; ð5Þ is antisymmetric in i and j. Equations (7a), (7b) show that 2 0þ þ 0 kij , kijl and kijlm are invariant under time reversal 1 i.e. Rþ is symmetric and R antisymmetric in and . 0 þ whereas kij , kijl and kijlm change sign. The tensors symmetric For small magnetic fields the matrices R can be expanded 0þ in i and j describe heat conductivity [kij ðM0Þ], linear magneto- in powers of H, þ heat-conductivity [kijlðM0Þ] and (quadratic) magneto-heat- 0 conductivity [þ ðM Þ], the tensors antisymmetric in i and j RðÞ¼H; M R ðM ÞþRlðÞM Hl þ RlmðÞM HlHm iklm 0 0 0 0 0 0 describe the spontaneous Righi–Leduc effect [kij ðM0Þ, þ RlmnðÞM0 HlHmHn þ ...; ð6Þ (ordinary) Righi–Leduc effect [kijlðM0Þ] and (quadratic) where M0 is the magnetization of the crystal for H ¼ 0. Note anomalous Righi–Leduc effect [kiklmðM0Þ]. that M0 ¼ 0 for dia- and paramagnetic crystals. It follows from The thermoelectric transport properties are described by 00 equation (3) that ijðH; MÞ,(i, j =1,2,3),i.e. by Riþ3 jðH; MÞ according to ÂÃ 0 0 1 0 0 0 equation (2) and by jiðH; MÞ,(i, j = 1, 2, 3), i.e. by R ðM0Þ¼2 RðM0ÞRðM0Þ ¼R ðM0Þ; ð7aÞ 00 Rjiþ3ðH; MÞ.For ¼ i þ 3 and ¼ j we can replace R by ÂÃ 1 in equations (3)–(7a), (7b); for ¼ j and ¼ i þ 3 we can R ðM Þ¼ R ðM ÞR ðM Þ ¼R ðM Þ: ð7bÞ 0 l 0 2 l 0 l 0 l 0 replace R by in equations (3)–(7a), (7b). The part invariant 0 00þ 0þ Relations similar to equation (7a) hold for all terms under 1 satisfies ij ðH; MÞ¼ji ðH; MÞ, the part changing 00 0 connected with even powers of H, relations similar to equation sign satisfies ij ðH; MÞ¼ji ðH; MÞ. Equations (7a), (7b) 00þ 000þ 0 00 0þ 00þ (7b) for all terms connected with odd powers of H. Because H show that ij , ij , ijl , ijl , ijlm and ijlm are invariant 0 0þ 0 00 000 0þ 00þ 0 changes sign under time reversal 1 it follows that R and under time reversal 1 whereas ij , ij , ijl , ijl , ijlm and 0 þ 00 R are invariant under 1 . More generally, R is ijlm change sign. Note that these tensors have no internal l l1...ln 0 0 00þ invariant under 1 if n is even, Rl ...l is invariant under 1 symmetry. The effects described are the Seebeck effect (ij ), 1 n 000þ 0þ if n is odd. Peltier effect (Tij ), linear magneto-Seebeck effect (ijl ), 00þ Consider first the electric transport properties of a crystal linear magneto-Peltier effect (Tijl ), (quadratic) magneto- 0þ described by ikðH; MÞ,(i, k =1,2,3),i.e.byRikðH; MÞ Seebeck effect (ijlm), (quadratic) magneto-Peltier effect 00þ 00 according to equation (2). According to equation (1a) the (Tijlm), spontaneous Nernst effect (ij ), spontaneous 000 0 tensor ikðH; MÞ gives grad ’ in a crystal in which an electric Ettingshausen effect (Tij ), (ordinary) Nernst effect (ijl ), 00 current j flows and in which the temperature is constant (ordinary) Ettingshausen effect (Tijl ), (quadratic) anom- 0 throughout the crystal. For ¼ i and ¼ k we can replace R alous Nernst effect (ijlm) and (quadratic) anomalous 0 00 by in equations (3)–(7a), (7b). The part invariant under 1 , Ettingshausen effect (T ijlm). þ ikðH; MÞ, is symmetric, the part changing sign, ikðH; MÞ,is The above-mentioned 12 effects invariant under time antisymmetric in i and k. inversion 10 are also invariant under space inversion 1 (and, 0 As shown in Shtrikman & Thomas (1965a,b), only the therefore, also under space-time inversion 1 ). The tensors þ symmetric part ikðH; MÞ contributes to energy dissipation, describing such effects are called even (Sirotin & Shaskols- @’ kaya, 1982); they can occur in crystals with any of the 122 Q ¼j ¼ j j ¼ j j ¼ j þ j : ð8Þ space-time point groups. The 12 effects changing sign under i @x i ik k i ki k i ik k i time inversion 10 are invariant under space inversion 1 (and, þ 0 The symmetric part ikðH; MÞ describes the electric resistance, therefore, change sign under space-time inversion 1 ). The the antisymmetric part ikðH; MÞ the Hall effect, and their tensors describing such effects are called magnetic (Sirotin & dependence on H.IfM ¼ 0, it follows from equation (6) that Shaskolskaya, 1982); they cannot occur in crystals with any of 0 þ contains only terms connected with even powers and the 53 space-time point groups containing 10 or 1 as a separate only terms connected with odd powers of H. For crystals with element. Magnetically ordered crystals have one of the 90 M ¼ 0 it follows that magnetoresistance linear in H is not magnetic point groups (MPGs) that do not contain time

334 Hans Grimmer Thermoelectric transport properties Acta Cryst. (2017). A73, 333–345 research papers

Table 1 The tensors describing thermoelectric transport properties considered in x3. Even tensors (invariant under 1 and 10) Magnetic tensors (invariant under 1, Number of point groups describing effects allowed in all 122 point groups change sign under 10) describing allowing the effect

0þ 0 ij Electric resistance (H =0) ij Spontaneous Hall effect (H =0) 31 þ ijl (Ordinary) Hall effect ijl Linear magnetoresistance 66 þ ijlm (Quadratic) magnetoresistance ijlm (Quadratic) anomalous Hall effect 66 0þ 0 kij Heat conductivity (H =0) kij Spontaneous Righi–Leduc effect (H =0) 31 þ kijl (Ordinary) Righi–Leduc effect kijl Linear magneto-heat-conductivity 66 þ kijlm (Quadratic) magneto-heat-conductivity kijlm (Quadratic) anomalous Righi–Leduc effect 66 00þ 00 ij Seebeck effect (H =0) ij Spontaneous Nernst effect (H =0) 58 0 0þ ijl (Ordinary) Nernst effect ijl Linear magneto-Seebeck effect 69 0þ 0 ijlm (Quadratic) magneto-Seebeck effect ijlm (Quadratic) anomalous Nernst effect 69 000þ 000 Tij Peltier effect (H =0) Tij Spontaneous Ettingshausen effect (H =0) 58 00 00þ Tijl (Ordinary) Ettingshausen effect Tijl Linear magneto-Peltier effect 69 00þ 00 Tijlm (Quadratic) magneto-Peltier effect Tijlm (Quadratic) anomalous Ettingshausen effect 69

inversion 10 as a separate element. If H ¼ 0, spontaneous 2. Symmetry restrictions on tensors describing crystal Hall and Righi–Leduc effects are possible for the 31 properties MPGs allowing ferromagnetism, spontaneous Nernst and Ettingshausen effects are allowed for 58 MPGs. Whereas This section discusses basic facts and conventions important magneto-resistance, magneto-heat-conductivity, magneto- for the unambiguous interpretation of the restrictions that Seebeck and magneto-Peltier effect are of even order follow from the Neumann principle on the form of property in H in crystals without magnetic order, such effects linear tensors for different choices of the Cartesian coordinate in H are possible in the case of magnetoresistance and system. magneto-heat-conductivity for the 66 MPGs allowing piezo- Neglecting magnetic order, crystals can be classiﬁed magnetism, and in the case of the magneto-Seebeck and according to their symmetry into 230 ‘crystallographic space magneto-Peltier effects for all 122 53 = 69 MPGs that do groups’ [‘Fedorov groups’ in Russian literature, see e.g. 0 not contain space-time inversion 1 as a separate element. Also Vainshtein (1996)]; neglecting also translational symmetry the (quadratic) anomalous Hall and Righi–Leduc effects they fall into 32 crystal classes corresponding to the 32 crys- occur in the 66 MPGs allowing piezomagnetism, whereas the tallographic point groups. Two kinds of tensors can be (quadratic) anomalous Nernst and Ettingshausen effects distinguished according to whether they are invariant or occur for all 69 MPGs that do not contain space-time inversion change sign under space inversion 1. They have been referred 0 1 as a separate element. These effects, which occur only in to as (plus or) p tensors and (minus or) m tensors (Grimmer, magnetically ordered crystals belonging to certain MPGs, 1991) or as even and odd tensors (Sirotin & Shaskolskaya, are described by magnetic tensors. They are summarized in 1982). Adding 1 to the generators of a point group, one of the Table 1. 11 centrosymmetric point groups is obtained. Point groups Kleiner (1966), Malinowski (1986), Seemann et al. (2015) associated in this way to the same centrosymmetric point and the Bilbao Crystallographic Server (2017) consider the group are said to belong to the same Laue class. Each Laue restrictions on the sum of the even and the magnetic tensor class contains exactly one point group containing only rota- given in the same row of Table 1. It will be shown that the two tions; it is convenient to use its symbol to denote the Laue tensors can be determined separately. The restrictions that class. follow from the Neumann principle [see e.g. Nye (1985) or Whereas the restrictions imposed by the Neumann principle section 3.2.2.1 in International Tables for Crystallography on tensors describing crystal properties depend for even (2016)] for the form of the tensors describing the effects tensors of given rank and internal symmetry only on the Laue occurring already at H ¼ 0 or linear in H have been derived class to which the point group belongs, odd tensors vanish for by Grimmer (1993), where the results are given in Table 1 in a the 11 centrosymmetric crystal classes and satisfy generally very compact form using graphic symbols that allow one to see different restrictions for the remaining 21 classes. Odd tensors immediately the components with equal or opposite values of low, non-zero rank vanish also for some of the 21 non- and the number of independent components. These results will centrosymmetric crystal classes, e.g. polar vectors (i.e. odd be presented here in a more user-friendly form, for additional tensors of rank 1) describing crystal properties can exist only orientations of the Cartesian coordinate system, and for the ten polar point groups 1, 2, 3, 4, 6, m, mm2, 3m,4mm augmented by the restrictions for the tensors describing the and 6mm. [See Table 3.2.2.1 in International Tables for Crys- effects quadratic in H. tallography (2016)]. Crystals belonging to one of these ten

Acta Cryst. (2017). A73, 333–345 Hans Grimmer Thermoelectric transport properties 335 research papers

Table 2 The four kinds of tensors deﬁned by their behaviour under inversions. Invariance of the tensor is denoted by +, sign change by . Space Time Space-time Name used by 0 inversion 1 inversion 10 inversion 1 Name used by Grimmer (1991) Sirotin & Shaskolskaya (1982)

++ + i tensor, invariant under all inversions Even tensor + s tensor, invariant under space inversion Magnetic tensor + t tensor, invariant under time inversion Electric tensor + u tensor, invariant under space-time inversion Magnetoelectric tensor

crystal classes are pyro-, ferro- or ferri-electric (Schmid, Also the forms of odd tensors for some of the other ten non- 1973). centrosymmetric point groups are given for several orienta- To formulate the restrictions imposed by the Neumann tions: 1m1 and 11m; mm2, m2m and 2mm; 42m and 4m2; 62m principle, a Cartesian coordinate system (CCS) is introduced, and 6m2, leading to a total of 12 + 15 = 27 forms for odd the axes of which are given by the international (Hermann– tensors. Note that the forms given in Grimmer (1991, 1993) for Mauguin) symbol of the point group. An entry in the 32 and 3m hold for 321 and 3m1, respectively. In the present Hermann–Mauguin (HM) symbol consists (apart from paper, also the forms for 312 and 31m will be given. possible primes and bars) of a number N = 1, 2, 3, 4 or 6 or the Crystals without magnetic order (dia- and paramagnetic letter m or N/m. The usual conventions will be followed: in the crystals) are invariant under the inversion of magnetic monoclinic and orthorhombic crystal systems, the x, y and z moments, usually referred to as time inversion 10 (Grimmer, axes of the CCS are parallel to the symmetry axes given in the 1993). Such crystals belong to one of the 230 space-time first, second and third entries, respectively. In the monoclinic groups obtained from the ordinary space groups by adding 10 system, there is only one symmetry axis, which is usually as additional generator. Interpreting 10 for graphical repre- chosen as y, and a short HM symbol with only one entry is sentations as exchanging black and white, the 230 space most often used, e.g. 2/m instead of 1 2/m 1. In the trigonal and groups containing the element 10 are called ‘grey’ (category I), hexagonal systems, the z, x and y axes are parallel to the and the 230 ordinary space groups ‘monochrome’ (category symmetry axes given in the first, second and third entries, II). In addition, there are 1191 ‘black–white’ space groups respectively. In the tetragonal system, the z axis is parallel to containing 10 only in combination with other operations, in the symmetry axis given in the first entry; the x and y axes are total 1651 ‘magnetic space groups’ (see Kleiner, 1966, or parallel to the symmetry axes given in the second entry, which Borovik-Romanov et al., 2014), referred to as ‘Shubnikov appear in two mutually perpendicular directions. In the cubic groups’ in Russian literature (see e.g. Vainshtein, 1996). system, the symmetry axes given in the first entry appear in Among the 1191 black–white space groups 517 contain 10 in three mutually perpendicular directions; the x, y and z axes of combination with translations (category IIIb), the other 674 do the CCS are chosen parallel to these directions (Nye, 1985; not contain 10 combined with translations (category IIIa). Grimmer, 1991; Borovik-Romanov et al., 2014). For certain According to Kleiner (1966), category IIIa can be further split 0 point groups, different HM symbols are possible, e.g. 6m2or into IIIa1 consisting of the 422 groups that do not contain 1 0 62m, corresponding to inequivalent choices of the CCS. Only and IIIa2 consisting of the 252 groups containing 1 . the standard symbol 6m2 is used in Nye (1985) and it is stated Neglecting translational symmetry, the 1651 magnetic space explicitly if a CCS is used that deviates from the conventions groups fall into 122 crystal classes corresponding to the 122 given above. Confusion may arise from the conventions space-time point groups, 32 of which are grey, 32 are mono- proposed in the IEEE Standard on Piezoelectricity (1988), chrome and 58 black–white. If the space group is grey the where the standard HM symbol 6m2 is used together with a point group is also grey, if the space group is monochrome the CCS corresponding to 62m according to the conventions given point group is also monochrome; space groups of category IIIa above. have black–white point groups whereas space groups of Grimmer (1991) has shown how 9 3 = 27 different forms category IIIb have grey point groups [see also Fig. 1 of Kleiner of an odd property tensor of given rank and internal symmetry (1966)]. can be derived from 6 2 = 12 forms for the corresponding Four kinds of tensors can be distinguished according to even property tensor. This is illustrated in his Fig. 1 for tensors whether they are invariant or change sign under space inver- 0 of third rank, symmetric in two of its three indices (Jahn sion 1, time inversion 10 and space-time inversion 1 , as shown symbol V[V2], see Jahn, 1949) and in his Fig. 2 for the fourth- in Table 2 (see also Table 3 in Grimmer, 1991). Note that i and rank tensor [V 2]2. Similarly, the forms of [V2], {V2}, V2, V [V2], s tensors are ‘even’, t and u tensors ‘odd’ as far as their V{V2} and V 3 are given in Table 1 of Grimmer (1993). Even behaviour under 1 is concerned. In the following, we shall use and odd property tensors of given rank and internal symmetry the term ‘even tensor’ in the sense of Sirotin & Shaskolskaya satisfy the same restrictions for pure rotation groups; these are (1982). given for standard orientations, where the monoclinic group 2 Considering that the 11 Laue classes were originally defined appears in two forms 121 and 112, leading to 11 + 1 = 12 forms. as determining the symmetry of X-ray photographs if Friedel’s

336 Hans Grimmer Thermoelectric transport properties Acta Cryst. (2017). A73, 333–345 research papers

Table 3 With this definition of (space-time) Laue class it remains The 69 space-time point groups for which magnetic tensors can exist and true that the restrictions imposed by the Neumann principle their subdivision into 21 magnetic form classes and two categories. on tensors describing crystal properties depend for even The point groups in the first category are monochrome; they correspond to tensors of given rank and internal symmetry only on the Laue space-group category II. The point groups in the second category are black– 0 a1 class to which the point group belongs. Magnetic (s) tensors white and do not contain 1 ; they correspond to space-group category III . 0 The third column begins with the point group containing only rotations vanish for the 53 point groups that contain 10 and/or 1 , electric (possibly combined with time inversion) and ends with the centrosymmetric (t) tensors vanish for the 53 point groups that contain 1 and/or point group. The magnetic form classes and the corresponding point groups 0 are listed not only for the standard orientation of the Cartesian coordinate 1 , magnetoelectric (u) tensors vanish for the 53 point groups system but, in parentheses, also for the other orientations that will be that contain 1 and/or 10. In each of these three cases there considered. remain 69 point groups, which may be grouped into 21 form Category Magnetic form class Point groups classes according to the restrictions imposed by the Neumann principle. For magnetic tensors this grouping has been given II 1 1, 1 explicitly in Table II of Kleiner (1966) and in Table 1 of (211) (211), (m11), (2/m11) Grimmer (1993) and for all three cases in Fig. 1 of Grimmer 121 121, 1m1, 12/m1 (1994). (112) (112), (11m), (112/m) 11 of the 21 magnetic form classes contain only mono- 222 222, mm2, (2mm), (m2m), mmm chrome point groups. Each of these 11 classes contains exactly 33,3 one group consisting of rotations only; it is convenient to use 321 321, 3m1, 3m1 its symbol to denote the magnetic form class. (Note that these (312) (312), (31m), (31m) 11 magnetic form classes look like the Laue classes of ordinary 44,4, 4/m point groups. However, only magnetically ordered structures 0 422 422, 4mm, 42m,(4m2), 4/mmm that have no symmetry operations involving 1 belong to one 66,6, 6/m of these 11 magnetic form classes.) The other ten magnetic form classes consist of black–white point groups. Each of these 622 622, 6mm, 6m2, (62m), 6/mmm ten classes contains exactly one group consisting only of 23 23, m3 rotations and rotations combined with 10; it is convenient to 432 432, 43m, m3m use its symbol to denote the magnetic form class. We list in Table 3 the 21 magnetic form classes and the 69 corresponding a1 0 0 0 0 0 III (2 11) (2 11), (m 11), (2 /m 11) point groups not only for the standard orientation of the CCS 12011201, 1m01, 120/m01 but, in parentheses, also for the other orientations that will be (1120) (1120), (11m0), (1120/m0) considered in this paper. For each of the 69 point groups one (22020) (22020), (2m0m0), (mm020), (m20m0), (mm0m0) HM symbol is considered as standard, other HM symbols (20220)(20220), (m02m0), (20mm0), (m0m20), (m0mm0) designating the same point group appear in parentheses, e.g. 0 0 0 0 0 0 0 0 0 0 0 0 2020220202, m0m02, m020m,(20m0m), m0m0m m 2 m,(2m m), (mm 2 ), (m2 m ), (2 mm ), (m m2 ). Note that 0 40 40, 4 ,40/m the restrictions on the form of magnetic property tensors for 0 0 40220 40220,40mm0, 4 2m0, 4 m20,40/mmm0 the 21 magnetic form classes coincide with the restrictions for odd (m) tensors for the 21 non-centrosymmetric crystal classes (40202) (40202), (40m0m), (40m02), (4020m), (40/mm0m) (Grimmer, 1991). 32013201, 3m01, 3m01 0 0 0 0 In the next section thermoelectric transport properties will (312 ) (312 ), (31m ), (31m ) be considered, distinguishing those described by even tensors 0 0 0 0 0 0 0 0 0 0 0 0 42 2 42 2 ,4m m , 42 m ,(4m 2 ), 4/mm m from those described by magnetic tensors. The spontaneous 0 0 0 0 0 0 0 0 0 0 0 0 62 2 62 2 ,6m m , 6m 2 ,(62 m ), 6/mm m Nernst and Ettingshausen effects are allowed in all the 0 60 60, 6 ,60/m0 magnetic form classes of Table 3 except the last three [60,60220 60220 60220,60mm0, 60m20, 602m0,60/m0mm0 (60202), 40320], i.e. in 69 11 = 58 point groups; linear 0 0 (60202) (60202), (60m0m), (6 20m), (6 m02), (60/m0m0m) magnetoresistance and linear magneto-heat-conductivity are 0 40320 40320, 4 3m0, m3m0 allowed in all magnetic form classes except 432, i.e. in the 69 3 = 66 point groups allowing the piezomagnetic effect (see Borovik-Romanov et al., 2014). law is valid, it is natural to extend their definition as follows to The monochrome magnetic form classes 1, 2, 3, 4 and 6, the 122 space-time point groups: adding 1 and 10 to the which contain 13 point groups, and the black–white magnetic generators of a point group, one of the 11 point groups form classes 20,20202, 320,42020 and 62020, which contain 18 0 containing 1, 10 and 1 is obtained. Point groups associated in point groups, allow physical properties described by magnetic this way to the same grey, centrosymmetric point group are vectors. Crystals having one of these 13 + 18 = 31 point groups said to belong to the same (space-time) Laue class. Each of are pyro-, ferro- or ferrimagnetic (Schmid, 1973) and allow the these 11 Laue classes contains exactly one group consisting of spontaneous Hall and Righi–Leduc effects. 44 among the 230 rotations only. We denote each Laue class by adding 10 to the space groups of category II have one of those 13 monochrome HM symbol of its pure rotation. point groups and 231 among the 422 space groups of category

Acta Cryst. (2017). A73, 333–345 Hans Grimmer Thermoelectric transport properties 337 research papers

IIIa1 have one of those 18 black–white point groups. The 3.1. Thermoelectric transport properties described by even remaining 186 space groups of category II, the remaining 191 tensors a1 space groups of category III , the 252 space groups of cate- Consider ﬁrst the effects occurring in the absence of an a2 b gory III and the 517 space groups of category III corre- external magnetic ﬁeld: electrical resistivity 0þ and heat spond to antiferromagnetic crystals [see section 1.5.2.3 of conductivity k0þ are described by symmetric tensors of rank 2; Borovik-Romanov et al. (2014)]. Crystals with space groups of the Seebeck effect 00þ and Peltier effect T000þ are described b category I (dia- and paramagnets), category III and category by general tensors of rank 2, where 000þ ¼þ00þ, as follows a2 ij ji III are not compatible with properties described by magnetic from equation (5). The form of these tensors is given in Fig. 1. tensors.

3. Thermoelectric transport properties The restrictions that follow from the Neumann principle on the form of the tensors mentioned in Table 1 will be given in this section. The even tensors (listed on the left side of Table 1) will be considered in x3.1, the magnetic ones (listed on the right side of Table 1) will be considered in x3.2.

Figure 2 The restrictions satisﬁed by the property tensors of ranks 3 and 4 listed on Figure 1 the left-hand side of Table 1. The notation used for the matrix elements is The restrictions satisﬁed by the property tensors of rank 2 listed on the explained at the bottom of Fig. 1. Laue class symbols in parentheses refer left-hand side of Table 1. Laue class symbols between parentheses refer to to alternative choices of the Cartesian coordinate system. (a) Anorthic, alternative choices of the Cartesian coordinate system. monoclinic and orthorhombic Laue classes.

338 Hans Grimmer Thermoelectric transport properties Acta Cryst. (2017). A73, 333–345 research papers

In the special cases of cubic crystals (Laue classes 2310 and similar for the Righi–Leduc tensor. Numerical values for the 0 00þ 4321 ) the Seebeck tensor has only one independent Hall constant R ¼ 213 have been given e.g. in Table 20.2 of 00þ 00þ 00þ 0 element 11 ¼ 22 ¼ 33 (see Fig. 1). It follows that Newnham (2005). For Laue class 4321 the three independent components of þ , i.e. þ , þ and þ , give rise to grad ’ k grad T if j ¼ 0: ijlm 1111 1122 1212 longitudinal magnetoresistance, transverse magnetoresistance Numerical values for the absolute Seebeck coefﬁcient and the planar Hall effect, respectively. Similarly, the three 00þ þ þ þ þ S ¼ 11 (also called ‘absolute thermopower’) have been independent components of kijlm, i.e. k1111, k1122 and k1212, give given e.g. in Table XVI of White & Minges (1997). rise to longitudinal thermal magnetoresistance, transverse Similarly, the Peltier tensor T000þ has only one indepen- thermal magnetoresistance (also called the Maggi–Righi– 000þ 000þ 000þ dent element T11 ¼ T22 ¼ T33 . It follows that Leduc effect) and the planar thermal Hall effect, respectively h k j if grad T ¼ 0: (see ch. 20 in Newnham, 2005). 00þ 000þ The two tensors are related by 11 = 11 . The form of even tensors describing effects occurring in an external magnetic ﬁeld H are given in Fig. 2. The tensors of rank 3 describe effects linear in H, those of rank 4 effects quadratic in H. In the special cases of cubic crystals (Laue classes 2310 and 0 4321 ) the Hall tensor ijl has only one independent element 231 ¼ 312 ¼ 123 ¼321 ¼132 ¼213. The situation is

Figure 2 (continued) Figure 2 (continued) (b) Tetragonal and cubic Laue classes. (c) Trigonal and hexagonal Laue classes.

Acta Cryst. (2017). A73, 333–345 Hans Grimmer Thermoelectric transport properties 339 research papers

þ þ 0 Note in Fig. 2(c) that the property tensors ijkl and kijkl Fig. 3 shows that there is no spontaneous Hall effect and satisfy an additional restriction, which reduces the number of no spontaneous Righi–Leduc effect k0 for point groups with þ independent components by 1. Similarly, ijkl satisﬁes an more than one rotation axis. additional restriction for the Laue classes 32110, (31210), 62210 Let me show that the even and magnetic tensors can be and two additional restrictions for the Laue classes 310 and 610. determined individually by experiment. 0 0 0 0þ The components that are exchanged in 3211 and 3121 are If has been measured, the components ij of the elec- 0 shown in different colours. trical resistivity and the components ij of the spontaneous Hall effect are obtained as follows:

3.2. Thermoelectric transport properties described by magnetic tensors For certain magnetically ordered crystals there may occur, in addition to the even property tensors, magnetic property tensors. Effects occurring in the absence of an external magnetic ﬁeld, the spontaneous Hall effect 0 and spontaneous Righi–Leduc effect k0, are described by antisymmetric tensors of rank 2; the spontaneous Nernst effect 00 and spontaneous Ettingshausen effect T000 are described by 000 00 general tensors of rank 2, where ij ¼ji , as follows from equation (5). The form of these tensors is given in Fig. 3 for the point groups in category II.

Figure 4 Figure 3 The restrictions for the point groups in category II satisﬁed by the The restrictions for the point groups in category II satisﬁed by the property tensors of ranks 3 and 4 listed on the right-hand side of Table 1. property tensors of rank 2 listed on the right-hand side of Table 1. The The notation used for the matrix elements is explained at the bottom of notation used for the matrix elements is explained at the bottom of Fig. 1. Fig. 1. Magnetic form class symbols in parentheses refer to alternative Magnetic form class symbols in parentheses refer to alternative choices of choices of the Cartesian coordinate system. (a) Anorthic, monoclinic and the Cartesian coordinate system. orthorhombic magnetic form classes in category II.

340 Hans Grimmer Thermoelectric transport properties Acta Cryst. (2017). A73, 333–345 research papers ÀÁ ÀÁ 0þ 1 0 0 0þ 000 1 000 00 00 ij ¼ 2 ij þ ji ¼ ji ; ð8aÞ ij ¼ 2 ij ji ¼ji : ð10bÞ ÀÁ 0 1 0 0 0 ij ¼ 2 ij ji ¼ji : ð8bÞ The form of the even tensors has been given in Fig. 1, the form Replacing by k, the analogous results for heat conductivity of the magnetic tensors in Fig. 3. 0þ 0 kij and the spontaneous Righi–Leduc effect kij are obtained. The tensors of rank 3 describing effects linear in H and the 00 000 00þ If and have been measured, the components ij of tensors of rank 4 describing effects quadratic in H can simi- 00 the Seebeck effect and the components ij of the sponta- larly be split into an even and a magnetic tensor. neous Nernst effect are obtained as follows: The form of magnetic tensors describing effects occurring in ÀÁ 00þ ¼ 1 00 þ 000 ; ð9aÞ an external magnetic ﬁeld H are given in Fig. 4 for the point ij 2 ij ji groups in category II. ÀÁ 00 1 00 000 ij ¼ 2 ij ji : ð9bÞ

000þ Similarly, the components ij of the Peltier effect and the 000 components ij of the spontaneous Ettingshausen effect are obtained as ÀÁ 000þ 1 000 00 00þ ij ¼ 2 ij þ ji ¼ ji ; ð10aÞ

Figure 4 (continued) Figure 4 (continued) (b) Tetragonal and cubic magnetic form classes in category II. (c) Trigonal and hexagonal magnetic form classes in category II.

Acta Cryst. (2017). A73, 333–345 Hans Grimmer Thermoelectric transport properties 341 research papers

We now deal with the magnetic form classes in category neous effects. Numerical values of this component for Fe, Cu IIIa1. The form of the property tensors of rank 2 is given in and Ni have been given for the spontaneous Hall effect by Fig. 5. Miyasato et al. (2007), Nagaosa et al. (2010) and Ko¨dderitzsch Fe (body-centred cubic), Co (hexagonal close-packed) and Ni (face-centred cubic) have in the paramagnetic phase (i.e. above the Curie point) point groups m3m10 ,6/mmm10 and m3m10 , respectively; in the ferromagnetic phase they have point groups 4/mm0m0,6/mm0m0 and 3m0 with spontaneous magnetization parallel to the principal symmetry axis [see Table 16.3 and Fig. 16.8 in Newnham (2005)]. Fig. 5 shows that the corresponding magnetic form classes 42020,62020 and 320 have only one independent component describing the spontaneous Hall effect, and similarly for the three other sponta-

Figure 6 Figure 5 The restrictions for the point groups in category IIIa1 satisﬁed by the The restrictions for the point groups in category IIIa1 satisﬁed by the property tensors of ranks 3 and 4 listed on the right-hand side of Table 1. property tensors of rank 2 listed on the right-hand side of Table 1. The The notation used for the matrix elements is explained at the bottom of notation used for the matrix elements is explained at the bottom of Fig. 1. Fig. 1. Magnetic form class symbols in parentheses refer to alternative Magnetic form class symbols in parentheses refer to alternative choices of choices of the Cartesian coordinate system. (a) Anorthic, monoclinic and the Cartesian coordinate system. orthorhombic magnetic form classes in category IIIa1.

342 Hans Grimmer Thermoelectric transport properties Acta Cryst. (2017). A73, 333–345 research papers et al. (2015), and for the spontaneous Nernst effect by Miya- 4. Remarks and corrections to the literature sato et al. (2007). The forms of (0þ, k0þ) and 000þ given in Fig. 1 correspond to The form of magnetic tensors describing effects occurring in those given for r and s0 in Table V of Kleiner (1966); the form an external magnetic ﬁeld H is given in Fig. 6 for the point of 00þ corresponds to the form of s given in his Table IV. A groups in category IIIa1. The tensors of rank 3 describe effects linear in H, the tensors of rank 4 describe effects quadratic in H. For category IIIa1, the experimental separation of the even and magnetic property tensors is particularly easy because for standard coordinate systems each non-zero component is due either exclusively to the even or exclusively to the magnetic tensor, with the exception of the magnetic form classes 40,40220 and 40320.

Figure 6 (continued) Figure 6 (continued) (b) Tetragonal and cubic magnetic form classes in category IIIa1. (c) Trigonal and hexagonal magnetic form classes in category IIIa1.

Acta Cryst. (2017). A73, 333–345 Hans Grimmer Thermoelectric transport properties 343 research papers comparison shows that the class 210 in his Table V corresponds thermoelectric transport properties independent, linear and to our (11210), i.e. the monoclinic axis has been chosen along z. quadratic in an external magnetic field H, respectively. The Note also that wrong results are given in Table V for the form results have been illustrated with experimental values of of s0 in the Laue classes 30 (= 310), 410,610 and the limiting class tensor components. 110. A number of errors in the literature concerning the form of Instead of the magnetic tensor 0, Kleiner (1966) considers tensors describing thermoelectric transport properties have the sum 0 ¼ 0þ þ 0, and similarly in the three other cases. been corrected. For the point groups in category II he finds that r ¼ 0þ þ 0 Kleiner (1966), Malinowski (1986), Seemann et al. (2015) (or k0þ þ k0), s ¼ 00þ þ 00 and s0 ¼ 000þ þ 000 all and the Bilbao Crystallographic Server (2017) consider the have the form of s given in his Table IV, i.e. r; s and s0 are restrictions on the sum of an even tensor occurring in crystals independent tensors of rank 2 without internal symmetry; we belonging to any of the 122 space-time point groups and a obtained in Figs. 1 and 3 that 0þ and k0þ are symmetric, 0 magnetic tensor occurring only in crystals belonging to point and k0 antisymmetric in their two indices and that groups allowing certain types of magnetic order, as indicated 000þ 00þ 000 00 ij ¼ ji , ij ¼ji . in Table 3. Similarly, Wimmer et al. (2016) consider the Comparing ijl inFig.2withr in Table VII of Seemann et restrictions on the sum of an electric and a magnetoelectric al. (2015), we find that the same components are zero and the property tensor in their consideration of spin–orbit torques. same components are equal or opposite, except that the results The main message of the present paper is the proof that for for 32110 and 31210 are interchanged. This tells us that each of the pairs given in the 12 rows of Table 1, the even and Seemann et al. (2015) do not adhere to the usual convention the magnetic tensor can be determined individually by concerning the orientation of the Cartesian coordinate system experiment. The thermoelectric tensors 0 and 00 are related in the case of the hexagonal crystal family. The components differently for even and magnetic tensors; the electric and that are exchanged in 32110 and 31210 are shown in different thermal tensors, and k, have different internal symmetries colours in Fig. 2(c). for even and magnetic tensors. Malinowski (1986) considers the Hall tensor and presents in In other words, treating the even and magnetic tensors as þ his Table IA results related to the results given for ijl and ijl distinct tensors means: in our Figs. 2, 4 and 6. Similarly as Kleiner (1966) and (i) Making it possible (as in the case of tensors describing Seemann et al. (2015) do for tensors of rank 2, he does not equilibrium properties) to associate to each property tensor þ separate the magnetic tensor ijl (which describes linear one of the characteristics even, magnetic, electric or magnetoresistance) and the even tensor ijl (which describes magnetoelectric, describing its behaviour under inversions. the ordinary Hall effect). His results generally agree with ours; (ii) Distinguishing the energy dissipative and the lossless however, a number of mistakes should be noted in his Table part of .

IA: in the fourth entry on p. 37 (i.e. K: 4, ...) P321 in the (iii) Distinguishing between electrical resistivity and the second row of the matrix should be replaced by P231. Five of spontaneous Hall effect and, similarly, distinguishing between 0 the 122 space-time point groups, i.e. 4/m0,40/m0, 3 ,6/m0 and the two effects in each of the other 11 rows of Table 1. 60/m are missing in Table IA. For these ﬁve groups, r has the (iv) Distinguishing between the tensors related by 0 000 00 000 00 form given in the fourth entry on p. 38 (i.e. K: 41 , ...). Note ij ¼ ji and those related by ij ¼ji , between the 00 0 00 0 that ‘32 (when 2||x), 3m, 3m (when m?x)’ in Table IA corre- tensors related by ijl ¼ jil and those related by ijl ¼jil, 00 0 sponds to our magnetic form class 321. Similarly ‘32 (when between the tensors related by ijlm ¼ jilm and those related 0 0 00 0 2||y), 3m, 3m (when m?y)’ corresponds to our (312), ‘6 m2 , by ijlm ¼jilm. 0 6 m02, 60202, 60m0m,60/m0m0m (when 2||x or m?x)’ corresponds 0 0 to 60220, and ‘6 m20, 6 m02, 60202, 60m0m,60/m0m0m (when 2||y or m?y)’ corresponds to (60202). It follows from the results in Fig. 2 that the column ijkl in References Table 20.1 of Newnham (2005) should be corrected as follows: Bilbao Crystallographic Server (2017). MTENSOR: symmetry- 45(12) in the row for Laue class 3, 25(8) in the rows for 32 and adapted form of crystal tensors in magnetic phases. http:// 6. Similarly, in his Table 21.5 the column mnpq should be www.cryst.ehu.es. corrected as follows: 27(12) in the row for Laue class 6 and Borovik-Romanov, A. S., Grimmer, H. & Kenzelmann, M. (2014). In International Tables for Crystallography, Vol. D, Physical Proper- 15(5) in the row for 23; also column mnp needs a correction: 6(2) in the row for 23. ties of Crystals, 2nd ed., edited by A. Authier. Chichester: Wiley. Butzal, H. D. & Birss, R. R. (1982). Physica (Utrecht) A, 114, 518–521. Grimmer, H. (1991). Acta Cryst. A47, 226–232. Grimmer, H. (1993). Acta Cryst. A49, 763–771. 5. Summary Grimmer, H. (1994). Ferroelectrics, 161, 181–189. Taking also magnetic order into account, crystals can be IEEE Standard on Piezoelectricity (1988). ANSI/IEEE Std 176–1987. assigned to one of 122 classes according to their space-time New York: IEEE. International Tables for Crystallography (2016). Vol. A, Space-Group point-group symmetry. For all these classes, we derived the Symmetry, 6th ed., edited by M. I. Aroyo. Chichester: Wiley. restrictions following from the Neumann principle on the form Jahn, H. A. (1949). Acta Cryst. 2, 30–33. of even and of magnetic tensors of ranks 2, 3 and 4 describing Kleiner, W. H. (1966). Phys. Rev. 142, 318–326.

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Ko¨dderitzsch, D., Chadova, K. & Ebert, H. (2015). Phys. Rev. B, 92, Schmid, H. (1973). Int. J. Magn. 4, 337–361. 184415. Seemann, M., Ko¨dderitzsch, D., Wimmer, S. & Ebert, H. (2015). Phys. Landau, L. D. & Lifshitz, E. M. (1958). Statistical Physics, section 119. Rev. B, 92, 155138. London: Pergamon Press. Shtrikman, S. & Thomas, H. (1965a). Solid State Commun. 3, 147–150. Malinowski, S. (1986). Acta Phys. Pol. A, 69, 33–43. Shtrikman, S. & Thomas, H. (1965b). Solid State Commun. 3, civ. Miyasato, T., Abe, N., Fujii, T., Asamitsu, A., Onoda, S., Onose, Y., Sirotin, Yu. I. & Shaskolskaya, M. P. (1982). Fundamentals of Crystal Nagaosa, N. & Tokura, Y. (2007). Phys. Rev. Lett. 99, 086602. Physics. Moscow: Mir Publishers. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Vainshtein, B. K. (1996). Fundamentals of Crystals: Symmetry and (2010). Rev. Mod. Phys. 82, 1539–1592. Methods of Structural Crystallography. Berlin: Springer-Verlag. Newnham, R. E. (2005). Properties of Materials. Oxford University White, G. K. & Minges, M. L. (1997). Int. J. Thermophys. 18, 1269– Press. 1327. Nye, J. F. (1985). Physical Properties of Crystals, 2nd ed. Oxford: Wimmer, S., Chadova, K., Seemann, M., Ko¨dderitzsch, D. & Ebert, H. Clarendon Press. (2016). Phys. Rev. B, 94, 054415.

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