Thermoelectric Transport Properties in Magnetically Ordered Crystals
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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 tempera- ture gradient grad T exists. These produce a heat current with density h and a gradient of the electrochemical potential ’ ¼ ’ À =e, where ’ is the electric potential, is the chemical potential and Àe is the charge of the electron. Assuming linear relationships and using Cartesian coordinates we can write [equation (4) in Grimmer (1993)] @’ 0 @T À ¼ ikðH; MÞjk þ ÆikðH; MÞ ; ð1aÞ @xi @xk 00 @T hi ¼ TÆikðH; MÞjk À 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 À þ We introduce a 6  6 matrix R by defining Equations (7a), (7b) show that ik , ikl and iklm are 0 0À þ À 0 00 R ¼ ; R ¼ Æ ; R ¼ Æ ; 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 RÆ ðÞ¼H; M 1 ½R ðÞÆH; M R ðÞÀH; ÀM ; ð3Þ (quadratic) magnetoresistance [iklmðM0Þ], the tensors anti- 2 symmetric in i and k describe the spontaneous Hall effect R Rþ 0À À we split the matrix into a part that is invariant and a [ik ðM0Þ], (ordinary) Hall effect [iklðM0Þ] and (quadratic) RÀ 0 À 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; ÀMÞ: ð4Þ k in equations (3)–(7a), (7b). The part invariant under 10, Æ þ À 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 , RÆ ðÞ¼H; M 1 R ðÞÆH; M R ðÞH; M ; ð5Þ is antisymmetric in i and j. Equations (7a), (7b) show that 2 0þ À þ 0 kij , kijl and kijlm are invariant under time reversal 1 i.e. Rþ is symmetric and RÀ antisymmetric in and . 0À þ À whereas kij , kijl and kijlm change sign. The tensors symmetric For small magnetic fields the matrices RÆ 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- Æ 0Æ Æ Æ 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 0À Æ describe the spontaneous Righi–Leduc effect [kij ðM0Þ, þ RlmnðÞM0 HlHmHn þ ...; ð6Þ À (ordinary) Righi–Leduc effect [kijlðM0Þ] and (quadratic) À 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 ÆijðH; MÞ,(i, j =1,2,3),i.e. by Riþ3 jðH; MÞ according to Âà 0 0Æ 1 0 0 0Æ equation (2) and by ÆjiðH; 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 Æ ÂÃ Æ 1 Æ 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 Æ in equations (3)–(7a), (7b). The part invariant 0 00þ 0þ Relations similar to equation (7a) hold for all terms under 1 satisfies Æij ðH; MÞ¼Æji ðH; MÞ, the part changing 00 0 connected with even powers of H, relations similar to equation sign satisfies Æij ðH; MÞ¼ÀÆji 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 Æij , Æij , Æijl , Æijl , Æijlm and Æijlm are invariant 0 0þ 0 00À 000À 0þ 00þ 0 changes sign under time reversal 1 it follows that R and under time reversal 1 whereas Æij , Æij , Æijl , Æijl , Æijlm and À 0 þ 00 R are invariant under 1 . More generally, R is Æijlm change sign. Note that these tensors have no internal l l1...ln 0 À 0 00þ invariant under 1 if n is even, Rl ...l is invariant under 1 symmetry. The effects described are the Seebeck effect (Æij ), 1 n 000þ 0þ if n is odd. Peltier effect (TÆij ), linear magneto-Seebeck effect (Æijl ), 00þ Consider first the electric transport properties of a crystal linear magneto-Peltier effect (TÆijl ), (quadratic) magneto- 0þ described by ikðH; MÞ,(i, k =1,2,3),i.e.byRikðH; MÞ Seebeck effect (Æijlm), (quadratic) magneto-Peltier effect 00þ 00À according to equation (2). According to equation (1a) the (TÆijlm), spontaneous Nernst effect (Æij ), spontaneous 000À 0 tensor ikðH; MÞ gives grad ’ in a crystal in which an electric Ettingshausen effect (TÆij ), (ordinary) Nernst effect (Æijl ), 00 current j flows and in which the temperature is constant (ordinary) Ettingshausen effect (TÆijl ), (quadratic) anom- 0 throughout the crystal. For ¼ i and ¼ k we can replace R alous Nernst effect (Æijlm) and (quadratic) anomalous 0 00 by in equations (3)–(7a), (7b). The part invariant under 1 , Ettingshausen effect (T Æijlm). þ À 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 À 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 À 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.