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Determination of magnetic symmetry by optical second-harmonic generation
Journal of Magnetism and Magnetic Materials 226}230 (2001) 961}962
Determination of magnetic symmetry by optical second-harmonic generation M. Fiebig *, D. FroK hlich , St. Leute , Th. Lottermoser , V.V. Pavlov , R.V. Pisarev Institut fu( r Physik, Universita( t Dortmund, 44221 Dortmund, Germany Iowe Physical Technical Institute, St. Petersburg 194021, Russia
Abstract Optical second-harmonic (SH) spectroscopy is introduced as a powerful supplement for the determination of complex magnetic structures. Experimental e!orts are simpli"ed and new degrees of freedom are opened. Thereby, some principal or technical restrictions of neutron or magnetic X-ray di!raction experiments are overcome. As an example, various antiferromagnetic compounds of the Mn> and Cr> ions are discussed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic symmetry; Second-harmonic generation; Magneto-optics; Antiferromagnetism
The determination of symmetry is a prerequisite for any detailed evaluation of phononic, electronic, and magnetic properties of matter. For magnetic crystals, the structure analysis is usually accomplished by applying neutron di!raction experiments. In practice, however, these experiments are subject to several restrictions [1,2], and auxiliary techniques are required for a complete determination of the crystallographic point and space groups. Here, nonlinear optical spectroscopy is introduced as an important supplement to neutron experiments for the study of magnetically ordered materials. Recent experiments have shown [3] that in magnetic materials new types of nonlinear susceptibilities may exist: The second-harmonic polarization PS has to be written as a sum of two contributions of crystallographic or magnetic origin, respectively; PS" ( (i)# (c))ESES, G GHI GHI H I
* Corresponding author. Tel: #39-231-755-5160; Fax: #49231-755-3674. E-mail address: "[email protected] (M. Fiebig).
where ES and ES are the electric "eld components of the H I fundamental light. The tensor (i) does not couple to GHI the magnetic ordering. The tensor (c) depends on the GHI magnetic order parameter and vanishes at the ordering temperature. Its components can be identi"ed experimentally by measuring the intensity of the SH signal I(2)JP (2) for various directions and polarizations G of the incident and emitted light. When the non-zero components of (c) are known, the magnetic structure GHI can be derived [4]. Two examples will be discussed to demonstrate the advantages of our technique. Hexagonal Mn> compounds: RMnO with R"Sc, Y, Ho, Er, Tm, Yb, Lu is hexagonal. Below 70}130 K, frustrated antiferromagnetic ordering of the Mn> spins occurs, which may lead to a multitude of magnetic structures and highly complex phase diagrams [2,5]. Restricting ourselves to planar triangular arrangements of the Mn> spins, six di!erent magnetic structures exist, which are related to an alignment of spins along the x axes (S x), the y axes (S y) or some in-between axes (S ) with parallel ( model) or antiparallel ( model) coupling between adjacent layers at z"0 and z"1/2 in the magnetic unit cell. The nonzero components ijk of the magnetic tensor (c) are di!erent for the six correGHI sponding space groups, which thus can be distinguished by studying magnetic second-harmonic generation
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 7 3 2 - 0
962
M. Fiebig et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 961}962
Fig. 1. MSHG spectra of YMnO (a), ErMnO (b), and HoMnO at 6K (c) and 50K (d). Closed and open circles denote the MSHG intensities for the yyy and xxx polarization con"guration, respectively.
(MSHG) [2]. Fig. 1 shows the MSHG spectra of YMnO , ErMnO , and HoMnO which were measured with the setup described in Ref. [2]. In YMnO , a strong signal was observed in the yyy, but not in the xxx polarization mode, whereas in ErMnO MSHG was observed for the xxx, but not for the yyy con"guration. HoMnO behaves like YMnO at temperatures below 42K. Between ¹ "42K, where a spin reorientation 0 occurs, and the NeH el temperature, it behaves like ErMnO . The corresponding magnetic point symmetries are therefore 6mm with Sx ( type) for YMnO and
) and 6mm with S y ( type) HoMnO (¹(¹ for 0
ErMnO and HoMnO (¹'¹ ). This assignment was 0 not possible before since neutron di!raction experiments could not discriminate between and structures [1]. Note that the spectra in Fig. 1 are not determined by the ion R, but rather by the magnetic symmetry of the compound. Obviously, spectroscopy can be regarded as an additional degree of freedom which may further support the symmetry analysis. Cr O in the spin-yop phase: Cr O is trigonal and antiferromagnetically ordered below ¹ "307.6 K. In , a magnetic "eld B "5.8 T, the spins #op from the 1$ trigonal z-axis into the xy plane, in which they may be oriented along the x, the y, or some in-between axes, each case corresponding to a di!erent magnetic symmetry. Since there are three sets of x and y axes for the trigonal crystal, six possible orientational domains can be realized for each of the three di!erent magnetic symmetries. The analysis of selection rules shows that with MSHG all the 18 resulting magnetic structures can be distinguished [6]. Fig. 2 shows the SH spectra of Cr O for B"0 and for B'B . The polarization dependence 1$ of the spectra for B"0 is due to the interference of crystalline and magnetic contributions to PS (Eq. (1)). For B'B , the splitting is absent which, according to 1$ the MSHG selection rules, can only be expected if the Cr> spins are aligned along the y axes of the crystal, corresponding to a 2/m point symmetry [6]. The image
Fig. 2. SH spectra of Cr O at 1.8 K in zero magnetic "eld (a) and in the spin-#op state at B"6.6 T (b). Closed and open circles denote right and left circular polarized incident light, respectively. A spin-#opped Cr O sample is shown in (c). All six possible orientational antiferromagnetic domains (l> \ ) could be identi"ed by SH polarization analysis.
of a sample in the spin-#op state shows that by further polarization analysis all six orientational domains for the S y case could be identi"ed. In preceding publications, domains were always neglected which lead to contradictory results. Spatial resolution is therefore yet another degree of freedom which is accessible to the MSHG experiments and in the present case indispensable for a correct symmetry analysis. Future experiments will be dedicated to nonlinear magneto-optical investigations of centrosymmetric or opaque samples, which is more involved since three-photon techniques and a re#ection geometry have to be applied. This work was supported by the Deutsche Forschungsgemeinschaft and the Alexander-von-Humboldt-Stiftung. References [1] G.E. Bacon, Neutron Di!raction, Clarendon Press, Oxford, 1975. [2] M. Fiebig, D. FroK hlich, St. Leute, Th. Lottermoser, K. Kohn, V.V. Pavlov, R.V. Pisarev, Phys. Rev. Lett. 84 (2000) 5620. [3] K.H. Bennemann, (Ed.), Nonlinear Optics in Metals, (Clarendon Press, Oxford, 1998). [4] R.R. Birss, Symmetry and Magnetism, North-Holland, Amsterdam, 1966. [5] M.F. Collins, O.A. Petrenko, Can. J. Phys. 75 (1997) 605. [6] M. Fiebig, D. FroK hlich, H.-J. Thiele, Phys. Rev. B 54 (1996) R12681.
Determination of magnetic symmetry by optical second-harmonic generation M. Fiebig *, D. FroK hlich , St. Leute , Th. Lottermoser , V.V. Pavlov , R.V. Pisarev Institut fu( r Physik, Universita( t Dortmund, 44221 Dortmund, Germany Iowe Physical Technical Institute, St. Petersburg 194021, Russia
Abstract Optical second-harmonic (SH) spectroscopy is introduced as a powerful supplement for the determination of complex magnetic structures. Experimental e!orts are simpli"ed and new degrees of freedom are opened. Thereby, some principal or technical restrictions of neutron or magnetic X-ray di!raction experiments are overcome. As an example, various antiferromagnetic compounds of the Mn> and Cr> ions are discussed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic symmetry; Second-harmonic generation; Magneto-optics; Antiferromagnetism
The determination of symmetry is a prerequisite for any detailed evaluation of phononic, electronic, and magnetic properties of matter. For magnetic crystals, the structure analysis is usually accomplished by applying neutron di!raction experiments. In practice, however, these experiments are subject to several restrictions [1,2], and auxiliary techniques are required for a complete determination of the crystallographic point and space groups. Here, nonlinear optical spectroscopy is introduced as an important supplement to neutron experiments for the study of magnetically ordered materials. Recent experiments have shown [3] that in magnetic materials new types of nonlinear susceptibilities may exist: The second-harmonic polarization PS has to be written as a sum of two contributions of crystallographic or magnetic origin, respectively; PS" ( (i)# (c))ESES, G GHI GHI H I
* Corresponding author. Tel: #39-231-755-5160; Fax: #49231-755-3674. E-mail address: "[email protected] (M. Fiebig).
where ES and ES are the electric "eld components of the H I fundamental light. The tensor (i) does not couple to GHI the magnetic ordering. The tensor (c) depends on the GHI magnetic order parameter and vanishes at the ordering temperature. Its components can be identi"ed experimentally by measuring the intensity of the SH signal I(2)JP (2) for various directions and polarizations G of the incident and emitted light. When the non-zero components of (c) are known, the magnetic structure GHI can be derived [4]. Two examples will be discussed to demonstrate the advantages of our technique. Hexagonal Mn> compounds: RMnO with R"Sc, Y, Ho, Er, Tm, Yb, Lu is hexagonal. Below 70}130 K, frustrated antiferromagnetic ordering of the Mn> spins occurs, which may lead to a multitude of magnetic structures and highly complex phase diagrams [2,5]. Restricting ourselves to planar triangular arrangements of the Mn> spins, six di!erent magnetic structures exist, which are related to an alignment of spins along the x axes (S x), the y axes (S y) or some in-between axes (S ) with parallel ( model) or antiparallel ( model) coupling between adjacent layers at z"0 and z"1/2 in the magnetic unit cell. The nonzero components ijk of the magnetic tensor (c) are di!erent for the six correGHI sponding space groups, which thus can be distinguished by studying magnetic second-harmonic generation
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 7 3 2 - 0
962
M. Fiebig et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 961}962
Fig. 1. MSHG spectra of YMnO (a), ErMnO (b), and HoMnO at 6K (c) and 50K (d). Closed and open circles denote the MSHG intensities for the yyy and xxx polarization con"guration, respectively.
(MSHG) [2]. Fig. 1 shows the MSHG spectra of YMnO , ErMnO , and HoMnO which were measured with the setup described in Ref. [2]. In YMnO , a strong signal was observed in the yyy, but not in the xxx polarization mode, whereas in ErMnO MSHG was observed for the xxx, but not for the yyy con"guration. HoMnO behaves like YMnO at temperatures below 42K. Between ¹ "42K, where a spin reorientation 0 occurs, and the NeH el temperature, it behaves like ErMnO . The corresponding magnetic point symmetries are therefore 6mm with Sx ( type) for YMnO and
) and 6mm with S y ( type) HoMnO (¹(¹ for 0
ErMnO and HoMnO (¹'¹ ). This assignment was 0 not possible before since neutron di!raction experiments could not discriminate between and structures [1]. Note that the spectra in Fig. 1 are not determined by the ion R, but rather by the magnetic symmetry of the compound. Obviously, spectroscopy can be regarded as an additional degree of freedom which may further support the symmetry analysis. Cr O in the spin-yop phase: Cr O is trigonal and antiferromagnetically ordered below ¹ "307.6 K. In , a magnetic "eld B "5.8 T, the spins #op from the 1$ trigonal z-axis into the xy plane, in which they may be oriented along the x, the y, or some in-between axes, each case corresponding to a di!erent magnetic symmetry. Since there are three sets of x and y axes for the trigonal crystal, six possible orientational domains can be realized for each of the three di!erent magnetic symmetries. The analysis of selection rules shows that with MSHG all the 18 resulting magnetic structures can be distinguished [6]. Fig. 2 shows the SH spectra of Cr O for B"0 and for B'B . The polarization dependence 1$ of the spectra for B"0 is due to the interference of crystalline and magnetic contributions to PS (Eq. (1)). For B'B , the splitting is absent which, according to 1$ the MSHG selection rules, can only be expected if the Cr> spins are aligned along the y axes of the crystal, corresponding to a 2/m point symmetry [6]. The image
Fig. 2. SH spectra of Cr O at 1.8 K in zero magnetic "eld (a) and in the spin-#op state at B"6.6 T (b). Closed and open circles denote right and left circular polarized incident light, respectively. A spin-#opped Cr O sample is shown in (c). All six possible orientational antiferromagnetic domains (l> \ ) could be identi"ed by SH polarization analysis.
of a sample in the spin-#op state shows that by further polarization analysis all six orientational domains for the S y case could be identi"ed. In preceding publications, domains were always neglected which lead to contradictory results. Spatial resolution is therefore yet another degree of freedom which is accessible to the MSHG experiments and in the present case indispensable for a correct symmetry analysis. Future experiments will be dedicated to nonlinear magneto-optical investigations of centrosymmetric or opaque samples, which is more involved since three-photon techniques and a re#ection geometry have to be applied. This work was supported by the Deutsche Forschungsgemeinschaft and the Alexander-von-Humboldt-Stiftung. References [1] G.E. Bacon, Neutron Di!raction, Clarendon Press, Oxford, 1975. [2] M. Fiebig, D. FroK hlich, St. Leute, Th. Lottermoser, K. Kohn, V.V. Pavlov, R.V. Pisarev, Phys. Rev. Lett. 84 (2000) 5620. [3] K.H. Bennemann, (Ed.), Nonlinear Optics in Metals, (Clarendon Press, Oxford, 1998). [4] R.R. Birss, Symmetry and Magnetism, North-Holland, Amsterdam, 1966. [5] M.F. Collins, O.A. Petrenko, Can. J. Phys. 75 (1997) 605. [6] M. Fiebig, D. FroK hlich, H.-J. Thiele, Phys. Rev. B 54 (1996) R12681.