Magnetocrystalline anisotropy and spin reorientation transition of
DyMn6Sn6 compound
GUO Guang-hua(郭光华), QIN Jiang(秦 江), ZHANG Hai-bei(张海贝)
School of Physics Science and Technology, Central South University, Changsha 410083, China
Received 5 September 2006; accepted 4 December 2006
Abstract:
The spin-reorientation transition of intermetallic compound DyMn6Sn6 was investigated by applying the molecular field theory. The temperature dependence of easy magnetization direction of compound and the magnetic moment directions of Dy and Mn ions were theoretically calculated and they have good agreement with the experimental data. In the framework of single ion model, the temperature dependence of magnetocrystalline anisotropic constants K1R and K2R of Dy ion were also calculated. The results show that the fourth-order crystal field parameter,
, and the corresponding second-order magnetocrystalline anisotropic constant, K2R, of Dy ion must be taken into account in order to explain the spin-reorientation transition satisfactorily. The competition between K2R and K1R plays a key role in the spin-reorientation transition of DyMn6Sn6.
Key words:
rare earth-transition metal compound; spin-reorientation transition; magnetocrystalline anisotropy;
1 Introduction Recently, the rare earth-transition metal compounds RM6X6 (R=rare-earth elements, M=transition metal and X=Sn and Ge) have attracted much attention due to their abundant magnetic properties[1-15]. RMn6Sn6 with heavy rare earth element crystallizes into HfFe6Ge6-type structure (space group is P6/mmm), which is composed of hexagonal R layers comprising Sn atoms and Mn Kagomé nets, stacked in the sequence -Mn-R-Mn-Mn-R-Mn- along the c-axis[1]. From the magnetic point of view, RMn6Sn6 have two different magnetic subsystems: R- and Mn-subsystems. Neutron diffraction studies and magnetic measurements[1-3] indicate that Mn- and R-subsystems simultaneous transfer from paramagnetic state into ferromagnetic state for compounds with Tb, Dy and Ho below Curie temperature TC (where TC=423, 393 and 376 K for Tb, Dy and Ho, respectively) due to the strong R-Mn exchange interaction. Mn-subsystem and R-subsystem are antiferromagnetically coupled with each other. When temperature drops to TSR, spin reorientation transition occurs in the compounds and the easy magnetization direction transfers from c-plane to c-axis. The spin reorientation temperatures TSR are 330, 300 and 200 K for Tb, Dy and Ho compounds, respectively. For TbMn6Sn6, when temperature is below 250 K, the easy magnetization direction completely transfers to c-axis. While for DyMn6Sn6 and HoMn6Sn6, the directions do not completely transfer to c-axis. The angles of easy direction with c-axis are 45? for Dy at 220 K and 48? for Ho at 100 K and keep constants until 2 K. Many researches[2,4-5] show that the mechanism of spin reorientation transition is different for three compounds. The spin reorientation transition in TbMn6Sn6 comes from the competition of magnetocrystalline anisotropy energies of Tb- and Mn-subsystems. While for DyMn6Sn6 and HoMn6Sn6, the competition between the first- and the second-order magnetocrystalline anisotropy constants K1 and K2 plays a very important role (which means the fourth-order crystalline field parameter of rare earth ions cannot be ignored). In this paper, the mechanism of spin reorientation transition of DyMn6Sn6 compound was quantitatively investigated in the frame work of molecular field theory. The temperature depend- ences of magnetocrystalline anisotropy constants of Dy ion were also calculated based on the single ion model.
2 Theoretical model
DyMn6Sn6 is composed of two magnetic subsystems: Dy- and Mn-subsystems. According to the molecular field theory, the Hamiltonians of Dy and Mn ions have the following forms:
where gJ and gS represent the g-factors of Dy and Mn ions; J and S are the total angular momentum operator of Dy ion and spin momentum operator of Mn ion, respectively; μB is Bohr momentum; 
and describe the crystalline field parameters of Dy3+; 
and 
are Stevens operators; K1t is magnetocrystalline anisotropy constant of Mn ion and θt is the angle of Mn ion magnetic moment with respect to c-axis; H is external magnetic field; the molecular fields acting on the Dy and Mn ions are Hex, Dy and Hex, Mn, which can be written as
λDy-Dy, λDy-Mn and λMn-Mn represent the molecular field constants of Dy-Dy, Dy-Mn and Mn-Mn exchange interaction, respectively. 
and 
are the thermal average of magnetization of Dy- and Mn-subsystems, respectively.
The free energy of system can be given as
where kB is Boltzmann constant; ZDy and ZMn are partition functions of Dy and Mn ions, respectively.
From the stable equilibrium condition the magnitude of and , as well as their direction with respect to the c-axis can be evaluated. Consequently, the magnetic structure and the temperature and field dependence of magnetization of compounds can be determined.
In our calculation the values of Dy and Mn magnetic moments at 4.2 K are taken to be 10μB and 2.21μB, respectively[2]. By fitting the experimental data in Ref.[2], including the spin reorientation temperature and thermal magnetization curves, the other magnetic parameters are estimated as follows: λMn-Mn=162.3T/μB, λDy-Mn=-5.8T/μB, λDy-Dy=2.1T/μB, =-0.25 K, = 0.003 K, K1t0=-1.5 K/Mn (valued at T=4.2 K).
Fig.1 shows the theoretically calculated temperature dependences of easy magnetization direction of DyMn6Sn6 and the angles of Dy and Mn magnetic moments with respect to the c-axis. When temperature is above 300 K, as seen from Fig.1, the easy magnetization direction lies in the c-plane. The angles of Dy and Mn magnetic moments are all equal to 90?. At T=300 K the spin reorientation transition occurs and the easy direction begins to turn to c-axis. As temperature reaches 220 K the angle between the easy direction and c-axis is 45? and keeps this value down to 2 K. During the process of spin reorientation the Dy and Mn magnetic moments keep antiparallel. The calculated results are in good agreement with neutron diffraction data[2].
Fig.1 Temperature dependence of easy magnetization direction of DyMn6Sn6 (a) and angles of Dy and Mn magnetic moments with respect to c-axis (b) (Solid lines are theoretically calculated curves, dots denote experimental data)
There are many reasons that can lead to the spin reorientation transition. The competition between the magnetocrystalline anisotropy energies of two magnetic subsystems or the competition among the different order magnetic anisotropy constants in the same subsystem may all give rise to the change of easy magnetization direction. The estimated magnetocrystalline anisotropy constant K1t of Mn-subsystem has negative value, which indicates that its easy magnetization direction lies in the c-plane. Neutron diffraction study also confirms that the easy direction of Mn-subsystem is in the c-plane[2]. The crystal field parameter acting on Dy+3 is negative (corresponding 
is positive), which favors the easy magnetization direction of Dy-sublattice along the c-axis. Investigations in Refs.[2, 4] showed the spin reorientation transition of TbMn6Sn6 can be described in the frame work of competition between the magnetocrystalline anisotropy energies of Tb- and Mn-subsystems. For DyMn6Sn6 and HoMn6Sn6, however, the competition between the first-order and the second-order magnetocrystalline anisotropy constants K1 and K2 play an important role in the process of spin reorientation[5-6]. In fact, the estimated is relatively small compared with other rare earth- transitional metal compounds (corresponding equals to 126 K?
, while of R2Fe14B is as large as 983 K?[2]), which indicates that the higher order crystal field parameter cannot be neglected. Our calculations also show that plays a very important role in the spin reorientation transition in DyMn6Sn6. Furthermore, below 220 K, the easy magnetization direction of DyMn6Sn6 is in cone plane. This requests that K1 and K2 must satisfy the condition: 0＜-K1＜2K2. Therefore K2 must be taken into account in order to explain the spin reorientation transition of DyMn6Sn6, which mainly comes from the fourth-order crystal field parameter, , of Dy ion.
In order to explain the spin reorientation transition of DyMn6Sn6 quantitatively, we calculated the temperature dependence of magnetocrystalline anisotropy constants of Mn- and Dy-subsystems in the frame work of single ion model. We consider the magnetocrystalline anisotropy constant of Mn-subsystem follows the Akulov-Zener law: 
. In accordance with single ion model, the magnetocrystalline anisotropy coefficients of rare-earth sublattice can be written as follows:
where J2=J(J-1/2), J4=J2(J-1)(J-3/2).
The temperature dependence of 
is as
where In+1/2 is hyperbolic Bessel function and L is Langevin function.
The magnetocrystalline anisotropy constants K1R and K2R of rare earth subsystem can be calculated from the following formula:
Fig.2 shows the theoretically calculated temperature dependence of K1R and K2R of Dy-subsystem. It can be seen from Fig.2 that K2R keeps positive in the whole temperature range of magnetically ordered state, which indicates that K2R favors the easy magnetization direction of Dy-subsystem directed along the c-axis. In contrast, K1R changes its sign from positive to negative with the decrease of temperature. In high temperature range K1R favors the magnetic moment of Ho+3 along the c-axis. Below 145 K, K1R becomes negative and makes for magnetic moment of Ho+3 perpendicular to the c-axis. The total magnetocrystalline anisotropy constants of DyMn6Sn6 are K1=K1R+6K1t, K2=K2R. In the whole temperature range of magnetically ordered state, K1 is always negative and K2 keeps positive.
Fig.2 Theoretically calculated temperature dependence of K1R and K2R
Our theoretical calculations (Fig.1(b)) and neutron diffraction data[2] all show that the magnetic moments of Dy and Mn ions are always antiparallel in the whole magnetically ordered state. In this case, the free energy correlated to the spin reorientation transition only includes the magnetic anisotropic energy:
where θ is the angle of the easy magnetization direction of DyMn6Sn6 with respect to the c-axis. The directions of magnetic moments of Dy and Mn ions and the easy magnetization direction of compounds can be decided from equilibrium condition. There are three states: 1) θ1=π/2 corresponding the c-plane easy magnetization direction; 2) θ2=0 indicating the c-axis easy magnetization direction; 3) sin2θ3=-K1/2K2 and sin2θ3≤1 representing the easy magnetization cone. The free energies corresponding the three solutions are E1=K1+K2, E2=0 and E3=
/4K2, respectively. By comparing the values of the three energies we can decide which state is the stablest one. Fig.3 displays the temperature dependences of E1, E2 and E3. As shown in Fig.3, in high temperature range E1 is the smallest one, which indicates that the easy magnetization direction is in c-plane. When temperature is below TSR of 300 K, E3 becomes the smallest one, showing the spin-reorientation transition occurs and the easy magnetization direction begins to transfer to the c-axis (when T＞TSR, sin2θ3= -K1/(2K2)＞1, therefore θ3 is not the solution). At T=2 K, sin2θ3=0.513, θ3=45?. In the whole temperature range of magnetically ordered state E2 is not the smallest energy. This means that the easy magnetization direction does not completely transfer to the c-axis. The calculated θ3 and spin reorientation temperature TSR are in good agreement with neutron diffraction results[2].
Fig.3 Temperature dependences of free energies E1, E2 and E3
4 Conclusions
1) The magnetic parameters of DyMn6Sn6 were estimated in the framework of molecular field model by fitting the different kinds of experimental data. The temperature dependence of magnetocrystalline anisotropy constants K1R and K2R of Dy ion was calculated. It is shown that the K2R keeps positive in the whole magnetically ordered temperature range, which indicates that K2R favors the magnetic moment of Dy ion directed along the c-axis. On the contrary, K1R changes its sign from positive to negative at T=145 K. Above 145 K, K1R is positive and makes for the magnetic moment of Dy ion directed along the c-axis, and below this temperature it favors the moment direction of Dy ion lying in the c-plane. The fourth order crystal field makes great contribution to the magnetocrystalline anisotropy energy. The total magnetocrystalline anisotropy constant K1 is always negative and K2 keeps positive in the whole temperature range of magnetically ordered state.
2) The spin reorientation of TbMn6Sn6 comes from the competition of magnetocrystalline anisotropies of Tb- and Mn-subsystems. For DyMn6Sn6, however, in order to describe the spin reorientation transition satisfactorily, the fourth-order crystal field parameter (corresponding to the second-order magneto- crystalline anisotropy constant K2R) must be taken into account, which plays the key role in the low temperature range. In compound DyMn6Sn6 the magnetocrystalline anisotropy energies of Dy- and Mn-subsystems also compete with each other, but the competition between K2R with K1R and K1t are the main cause of spin reorientation in compound DyMn6Sn6.
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Foundation item: Project(60571043) supported by the National Natural Science Foundation of China; Project(03JJY4044) supported by the Natural Science Foundation of Hunan Province, China
Corresponding author: GUO Guang-hua; Tel: +86-731-8836443; E-mail: guogh@mail.csu.edu.cn
Abstract: The spin-reorientation transition of intermetallic compound DyMn6Sn6 was investigated by applying the molecular field theory. The temperature dependence of easy magnetization direction of compound and the magnetic moment directions of Dy and Mn ions were theoretically calculated and they have good agreement with the experimental data. In the framework of single ion model, the temperature dependence of magnetocrystalline anisotropic constants K1R and K2R of Dy ion were also calculated. The results show that the fourth-order crystal field parameter,, and the corresponding second-order magnetocrystalline anisotropic constant, K2R, of Dy ion must be taken into account in order to explain the spin-reorientation transition satisfactorily. The competition between K2R and K1R plays a key role in the spin-reorientation transition of DyMn6Sn6.
