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Cubic magnets with Dzyaloshinskii–Moriya interaction
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 1602–1603 www.elsevier.com/locate/jmmm
Cubic magnets with Dzyaloshinskii–Moriya interaction S.V. Maleyev Petersburg Nuclear Physics Institute, Leningrad District 188300, Russia Available online 13 November 2006
Abstract Ground-state energy and spin-wave spectrum are studied theoretically using conventional exchange, Dzyaloshinskii–Moriya interaction, anisotropic exchange and cubic anisotropy. The spin-wave spectrum is strongly anisotropic: excitations with momentum q along and perpendicular to the helix wave vector k have linear and quadratic dispersion, respectively, if q5k. It is a result of the umklapp interaction connecting the spin-waves with q and q k. The classical ground-state energy depends on the magnetic field component along the vector k only. Transition to the ferromagnetic state holds at H4H c where g mB H c ¼ Ak2 andpAffiffiffi is the spin-wave stiffness at qbk. For low perpendicular field the helical order is stabilized by the spin-wave gap D. For g mB H ? oD 2 there is Bose condensation of the spin-waves with momenta k and zero. The perpendicular susceptibility and the second harmonic of the spin rotation appear. For larger field the vector k establishes along the field and the condensation disappears. The theory is in agreement with the existing experimental data. r 2006 Elsevier B.V. All rights reserved. PACS: 61.12.Bt Keywords: Spin-wave; Bose condensation; Magnetic field
Cubic magnets MnSi, FeGe, etc. have attracted a lot of attention due to their specific electronic and magnetic properties (see Ref. [1] and references therein). However, their low-T magnetic properties such as the spin-wave spectrum and the magnetic field dependence of the groundstate energy are not well understood up to now. In this paper we give a brief survey of corresponding theoretical studies and their comparison with the existing experimental data published recently in Ref. [2]. In zero field the helical structure of the considered compounds is determined by competition of the conventional exchange, the Dzyaloshinskii–Moriya interaction (DMI) and anisotropic exchange [3,4]. In particular the helix wave vector is given ^ by k ¼ SD0 ½a^ b=A, where A is determined by J q ¼ J 0 Aq2 =S, J q and S are the exchange integral and the unit-cell spin, respectively, D0 is the strength of the DMI at q ¼ 0, a^ and b^ are mutually perpendicular unit vectors in the plain of the spin rotation. We will see below that A is the spin-wave stiffness at qbk. Tel.: +7 812 552 5164; fax: +7 813 713 1963.
E-mail addresses: [email protected], [email protected] (S.V. Maleyev). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.629
The DMI violates the total spin conservation law and in the case of incommensurate helical structure gives rise to the umklapp interaction connecting the spin-waves with q and q k and different energies. As a result the spinwave spectrum becomes strongly anisotropic [2]. In zero magnetic field the excitations with q along k are not affected by the umklapps and we have ðqk Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aqk k2 þ q2k . If q is perpendicular to k the umklapps mix the excitations with q and q k and we obtain two branches. The lower branch pffiffiis ffi gapless. At q? 5k we have ¼ Aq2? =2 and þ ¼ Ak2 2. All three modes are shown in Fig. 1. If the magnetic field H4H c , where gmH c ¼ Ak2 we have ferromagnetic spin configuration and the umklapps vanish. In this case the spin-wave energy has the form ðqÞ ¼ Aq2 þ g mB ðH H c Þ.
(1)
At HoH c we have obtained the gapless spin-wave modes. However, one can show that in the magnetic field perpendicular to the vector k we have 2 ð0Þ ¼ ðgmH ? Þ2 =2o0 and the helical structure becomes unstable. Meanwhile in FeGe and MnSi it remains stable
ARTICLE IN PRESS S.V. Maleyev / Journal of Magnetism and Magnetic Materials 310 (2007) 1602–1603
ε/ AK2
20
10
0
0
2 q/k
4
Fig. 1. Spin-wave dispersion for different directions of the wave vector q. The vector q is along the helix vector k (dashed line), gapless and gapped branches for q ? k (solid and dot-dashed lines, respectively). Three curves do not merge at qbk due to corrections to the asymptotic law ðqÞ ¼ Aq2 [2].
below some critical perpendicular field of order of 0.01 and 0.1 T, respectively [5,6]. This stability is ensured by the spin-wave gap D and we have 2 ðH ? Þ ¼ D2 ðgmH ? Þ2 =2.
(2)
There are two contributions to the square of the gap [2]: cubic anisotropy and the spin-wave interaction considered in the Hartree–Fock approximation. The former contribution depends on the k direction. The latter is a result of the DMI, which violates the total spin conservation law. The Hartree–Fock contribution is positive, whereas the cubic one can have arbitrary sign. So changing their relative contributions for example by pressure one can do D2 to be negative and brings to the first-order transition to the disordered spin-liquid state with strong chiral fluctuations instead of the helical structure. May be it is a reason for the transition observed in Ref. [1]. In the field along the vector k the flat helix transforms to the conical structure and at HXH c we have ferromagnetic spin configuration. It is a pure classical phenomenon. In weak perpendicular field the Zeeman energy contains linear combination of the spinwave operators ak and aþ k as well as the terms mixing them with the zero-momentum excitations. Bose condensation of all these excitations occurs and the field depending part of the ground-state energy for H5H c has the form ( ) 2 g mB H 2 sin C EH ¼ cos2 C , (3) Hc 2½1 ðg mB HÞ2 =ð2D2 Þ where C is the angle between k and the field. The final spin configuration is determined by competition of this energy with contributions of the anisotropic exchange and cubic anisotropy to the ground-state energy [2]. The spin-wave Bose condensation determines the perpendicular spin susceptibility and the second harmonic of
1603
the spin rotation. The critical perpendicular fields for FeGe and MnSi mentioned above were determined from observation of this harmonic in FeGe [5] and MnSi [6]. Up to now detailed experimental work was done in the case of MnSi compound only. We now compare some of the known experimental results obtained at ambient pressure with the predictions of our theory and discuss possibilities of the further experimental studies. The principal parameters are: k ’ 0:035 A˚ 1 , saturated magnetization M ¼ 0:4 mB =a3 ’ 0:016 T ð4pM ¼ 0:20 TÞ, critical field H c ¼ 0:520:6 T [7] and spin-wave stiffness A ’ 52 meV A˚ 2 [8]. From these data we obtain H c ’ Ak2 =ðg mB Þ ’ 0:55 T. This value coincides with experimentally observed critical field. For more precise comparison one must measure all parameters including the demagnetization N cc considered in Ref. [2] using single sample. To the best of my knowledge the EPR in MnSi was studied in Ref. [9] only. Several resonances were observed but only one was studied qualitatively as a function of the magnetic field. Its frequency in zero field is equal to 0.93 T. It corresponds to k mode. Taking into account the magnetic dipolar interaction for corresponding frequency one has k ¼ ½2H 2c þ 4pMH c 1=2 [2]. Using the value of M given above we obtain 0.85 T. The agreement is within the error bars. The above theory explains some experimental findings. However, further experimental studies have to be done. We mention here some of them: (i) Observation of the spinwave anisotropy by neutron scattering. (ii) Investigation of the specific heat in connection with the umklapp suppression by the field. (iii) Study of competition of the spin chirality originated by the DMI and the applied magnetic field using polarized neutrons [10]. (iv) More detailed study of the EPR and direct observation of the gap D. The work was supported by RFBR (Grant nos. SS1671.2003.2, 03-02-17340, 00-15-96814 and 06-02-16702), and Russian State Programs ‘‘Quantum Macrophysics’’, ‘‘Strongly correlated electrons in semiconductors, metals, superconductors and magnetic materials’’, ‘‘Neutron research of solids’’ and Japan-Russian collaboration 050219889-Jp Physics-RFBR.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
C. Pfleiderer, et al., Nature 427 (2004) 227. S.V. Maleyev, Phys. Rev. B 73 (2006) 174402. O. Nakanishi, et al., Solid State Commun. 35 (1980) 995. P. Bak, M. Jensen, J. Phys. C 13 (1980) L881. B. Lebech, et al., J. Phys. Condens. Matter 1 (1989) 6105. A.I. Okorokov, et al., Physica B 356 (2005) 259. K. Koyama, et al., Phys. Rev. B 62 (2000) 986. Y. Ishikawa, et al., Phys. Rev. B 16 (1977) 4956. M. Date, et al., J. Phys. Soc. Japan 42 (1977) 1555. S.V. Maleyev, Phys. Usp. 45 (2002) 569.
Journal of Magnetism and Magnetic Materials 310 (2007) 1602–1603 www.elsevier.com/locate/jmmm
Cubic magnets with Dzyaloshinskii–Moriya interaction S.V. Maleyev Petersburg Nuclear Physics Institute, Leningrad District 188300, Russia Available online 13 November 2006
Abstract Ground-state energy and spin-wave spectrum are studied theoretically using conventional exchange, Dzyaloshinskii–Moriya interaction, anisotropic exchange and cubic anisotropy. The spin-wave spectrum is strongly anisotropic: excitations with momentum q along and perpendicular to the helix wave vector k have linear and quadratic dispersion, respectively, if q5k. It is a result of the umklapp interaction connecting the spin-waves with q and q k. The classical ground-state energy depends on the magnetic field component along the vector k only. Transition to the ferromagnetic state holds at H4H c where g mB H c ¼ Ak2 andpAffiffiffi is the spin-wave stiffness at qbk. For low perpendicular field the helical order is stabilized by the spin-wave gap D. For g mB H ? oD 2 there is Bose condensation of the spin-waves with momenta k and zero. The perpendicular susceptibility and the second harmonic of the spin rotation appear. For larger field the vector k establishes along the field and the condensation disappears. The theory is in agreement with the existing experimental data. r 2006 Elsevier B.V. All rights reserved. PACS: 61.12.Bt Keywords: Spin-wave; Bose condensation; Magnetic field
Cubic magnets MnSi, FeGe, etc. have attracted a lot of attention due to their specific electronic and magnetic properties (see Ref. [1] and references therein). However, their low-T magnetic properties such as the spin-wave spectrum and the magnetic field dependence of the groundstate energy are not well understood up to now. In this paper we give a brief survey of corresponding theoretical studies and their comparison with the existing experimental data published recently in Ref. [2]. In zero field the helical structure of the considered compounds is determined by competition of the conventional exchange, the Dzyaloshinskii–Moriya interaction (DMI) and anisotropic exchange [3,4]. In particular the helix wave vector is given ^ by k ¼ SD0 ½a^ b=A, where A is determined by J q ¼ J 0 Aq2 =S, J q and S are the exchange integral and the unit-cell spin, respectively, D0 is the strength of the DMI at q ¼ 0, a^ and b^ are mutually perpendicular unit vectors in the plain of the spin rotation. We will see below that A is the spin-wave stiffness at qbk. Tel.: +7 812 552 5164; fax: +7 813 713 1963.
E-mail addresses: [email protected], [email protected] (S.V. Maleyev). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.629
The DMI violates the total spin conservation law and in the case of incommensurate helical structure gives rise to the umklapp interaction connecting the spin-waves with q and q k and different energies. As a result the spinwave spectrum becomes strongly anisotropic [2]. In zero magnetic field the excitations with q along k are not affected by the umklapps and we have ðqk Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aqk k2 þ q2k . If q is perpendicular to k the umklapps mix the excitations with q and q k and we obtain two branches. The lower branch pffiffiis ffi gapless. At q? 5k we have ¼ Aq2? =2 and þ ¼ Ak2 2. All three modes are shown in Fig. 1. If the magnetic field H4H c , where gmH c ¼ Ak2 we have ferromagnetic spin configuration and the umklapps vanish. In this case the spin-wave energy has the form ðqÞ ¼ Aq2 þ g mB ðH H c Þ.
(1)
At HoH c we have obtained the gapless spin-wave modes. However, one can show that in the magnetic field perpendicular to the vector k we have 2 ð0Þ ¼ ðgmH ? Þ2 =2o0 and the helical structure becomes unstable. Meanwhile in FeGe and MnSi it remains stable
ARTICLE IN PRESS S.V. Maleyev / Journal of Magnetism and Magnetic Materials 310 (2007) 1602–1603
ε/ AK2
20
10
0
0
2 q/k
4
Fig. 1. Spin-wave dispersion for different directions of the wave vector q. The vector q is along the helix vector k (dashed line), gapless and gapped branches for q ? k (solid and dot-dashed lines, respectively). Three curves do not merge at qbk due to corrections to the asymptotic law ðqÞ ¼ Aq2 [2].
below some critical perpendicular field of order of 0.01 and 0.1 T, respectively [5,6]. This stability is ensured by the spin-wave gap D and we have 2 ðH ? Þ ¼ D2 ðgmH ? Þ2 =2.
(2)
There are two contributions to the square of the gap [2]: cubic anisotropy and the spin-wave interaction considered in the Hartree–Fock approximation. The former contribution depends on the k direction. The latter is a result of the DMI, which violates the total spin conservation law. The Hartree–Fock contribution is positive, whereas the cubic one can have arbitrary sign. So changing their relative contributions for example by pressure one can do D2 to be negative and brings to the first-order transition to the disordered spin-liquid state with strong chiral fluctuations instead of the helical structure. May be it is a reason for the transition observed in Ref. [1]. In the field along the vector k the flat helix transforms to the conical structure and at HXH c we have ferromagnetic spin configuration. It is a pure classical phenomenon. In weak perpendicular field the Zeeman energy contains linear combination of the spinwave operators ak and aþ k as well as the terms mixing them with the zero-momentum excitations. Bose condensation of all these excitations occurs and the field depending part of the ground-state energy for H5H c has the form ( ) 2 g mB H 2 sin C EH ¼ cos2 C , (3) Hc 2½1 ðg mB HÞ2 =ð2D2 Þ where C is the angle between k and the field. The final spin configuration is determined by competition of this energy with contributions of the anisotropic exchange and cubic anisotropy to the ground-state energy [2]. The spin-wave Bose condensation determines the perpendicular spin susceptibility and the second harmonic of
1603
the spin rotation. The critical perpendicular fields for FeGe and MnSi mentioned above were determined from observation of this harmonic in FeGe [5] and MnSi [6]. Up to now detailed experimental work was done in the case of MnSi compound only. We now compare some of the known experimental results obtained at ambient pressure with the predictions of our theory and discuss possibilities of the further experimental studies. The principal parameters are: k ’ 0:035 A˚ 1 , saturated magnetization M ¼ 0:4 mB =a3 ’ 0:016 T ð4pM ¼ 0:20 TÞ, critical field H c ¼ 0:520:6 T [7] and spin-wave stiffness A ’ 52 meV A˚ 2 [8]. From these data we obtain H c ’ Ak2 =ðg mB Þ ’ 0:55 T. This value coincides with experimentally observed critical field. For more precise comparison one must measure all parameters including the demagnetization N cc considered in Ref. [2] using single sample. To the best of my knowledge the EPR in MnSi was studied in Ref. [9] only. Several resonances were observed but only one was studied qualitatively as a function of the magnetic field. Its frequency in zero field is equal to 0.93 T. It corresponds to k mode. Taking into account the magnetic dipolar interaction for corresponding frequency one has k ¼ ½2H 2c þ 4pMH c 1=2 [2]. Using the value of M given above we obtain 0.85 T. The agreement is within the error bars. The above theory explains some experimental findings. However, further experimental studies have to be done. We mention here some of them: (i) Observation of the spinwave anisotropy by neutron scattering. (ii) Investigation of the specific heat in connection with the umklapp suppression by the field. (iii) Study of competition of the spin chirality originated by the DMI and the applied magnetic field using polarized neutrons [10]. (iv) More detailed study of the EPR and direct observation of the gap D. The work was supported by RFBR (Grant nos. SS1671.2003.2, 03-02-17340, 00-15-96814 and 06-02-16702), and Russian State Programs ‘‘Quantum Macrophysics’’, ‘‘Strongly correlated electrons in semiconductors, metals, superconductors and magnetic materials’’, ‘‘Neutron research of solids’’ and Japan-Russian collaboration 050219889-Jp Physics-RFBR.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
C. Pfleiderer, et al., Nature 427 (2004) 227. S.V. Maleyev, Phys. Rev. B 73 (2006) 174402. O. Nakanishi, et al., Solid State Commun. 35 (1980) 995. P. Bak, M. Jensen, J. Phys. C 13 (1980) L881. B. Lebech, et al., J. Phys. Condens. Matter 1 (1989) 6105. A.I. Okorokov, et al., Physica B 356 (2005) 259. K. Koyama, et al., Phys. Rev. B 62 (2000) 986. Y. Ishikawa, et al., Phys. Rev. B 16 (1977) 4956. M. Date, et al., J. Phys. Soc. Japan 42 (1977) 1555. S.V. Maleyev, Phys. Usp. 45 (2002) 569.