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Magnetic properties of R1−xR′x Mn2(Si,Ge)2 (R, R′ = rare earth) compounds
Journal of Alloys and Compounds 442 (2007) 108–110
Magnetic properties of R1−xRx Mn2(Si,Ge)2 (R, R = rare earth) compounds M. Duraj a,∗ , A. Szytuła b a
Institute of Physics, Cracow University of Technology, Podchor˛az˙ych 1, 30-084 Cracow, Poland b Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland
Received 24 July 2006; received in revised form 25 September 2006; accepted 25 September 2006 Available online 30 January 2007
Abstract Based on a phenomenological mean-field model the (P,T) magnetic phase diagram for the R1−x R x Mn2 (Si,Ge)2 series of compounds were discussed. The exchange parameters derived from the experimental data were used for the theoretical calculation of canting angle 2θ. © 2007 Elsevier B.V. All rights reserved. PACS: 75.30.Kz; 75.50.Ee Keywords: Rare-earth compounds; Intermetallics; Magnetic phase diagram
1. Introduction The magnetic properties under pressure of the intermetalic compounds R1−x R x Mn2 (Si,Ge)2 , (R, R —rare earth) were studied [1–4]. These compounds crystallize in the tetragonal ThCr2 Si2 -type crystal structure which consists of the monoatomic layers along the c-axis with the sequence Th–Si–Cr2 –Si. In these systems an especially interesting Mnsublattice ordering appears. Below the N´eel temperature TN there exists a collinear magnetic structure of the AFl type, arranged of antiferromagnetic collinear (0 0 1) Mn planes. As the temperature decreases, below the transition temperature TCinter , a canted ferromagnetic Fmc structure is established. With decreasing temperature, the canted phase transforms into the antiferromagnetic one accompanied with metamagnetic transition at T2inter . The metamagnetic transition at T1 is accompanied by a change in the rare-earth sublattice from disordered to an ordered state. It is known that the magnetic properties of R1−x R x Mn2 (Si,Ge)2 compounds are sensitive to the atomic distance. The magnetic phase transitions at T2inter from the ferromagnetic to the antiferromagnetic state occurs at the intralayer Mn–Mn distance RaMn–Mn = 0.286 (5) nm (for SmMn2 Ge2 ). The mag-
∗
Corresponding author. Tel.: +48 12 637 06 66; fax: +48 12 637 14 46. E-mail address: [email protected] (M. Duraj).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2006.09.147
netic structure can be easily modified by external or chemical pressure and temperature. 2. Model In order to make an analysis of critical behaviour of non-collinear antiferromagnetic Mn subsystem of R1−x R x Mn2 (Si,Ge)2 compounds, the standard two-sublattice meanfield model was discussed in Ref. [5]. The model presumes that the two equivalent Mn-sublattice magnetic moments are canted with 2θ angle. The full thermodynamic potential calculated per initial volume is the sum of exchange energy Φe , the entropy term Φs , the elastic energy Φd , the pressure term Φp and the Zeeman energy ΦH . The exchange interaction Φe takes the following form: A 2 B 2 a 2 b l + m + (lx + ly2 ) + (m2x + m2y ) 2 2 2 2 Ci D Di C i 2 2 + li4 + m4 + (li mi )2 + l m 4 4 2 2 i i
Φe = Φ0 +
i
i
(1)
i
៝ 2 of the system where the overall magnetic moment m =μ 1 + μ is perpendicular to the antiferromagnetic vector l = μ 1 − μ 2, and mi , li are components of m, l, respectively. The Mn–Mn interaction is assumed to be linearly dependent on relative changes of crystal volume, giving rise to magnetostrictive phenomena.
M. Duraj, A. Szytuła / Journal of Alloys and Compounds 442 (2007) 108–110
Fig. 1. The magnetic phase diagram (P,T) of the SmMn2 Ge2 compound [1]. The relative change of the unit cell V = −3.8 × 10−3 (nm3 /GPa) × P. Pcr = 1.00 ± 0.02 GPa, Vcr ≈ 0.1757 (3) nm3 and Tcr ≈ 330 ± 2.
In order to plot the temperature dependences of magnetization and (P,T) magnetic phase diagrams, it is necessary to calculate equilibrium thermodynamic potentials of different phases, and then to find the lowest one. Such numerical calculations were previously proposed in Ref. [5]. 3. Magnetic phase transitions at TNinter and TCinter Basing on the experimentally obtained (P,T) magnetic phase diagrams [1–4], values of the critical temperature Tcr and the critical pressure Pcr (i.e. the Fmc phase disappears and only the antiferromagnetic interaction remain) were obtained. Fig. 1 shows the magnetic (P,T) phase diagram of SmMn2 Ge2 . The unit cell volume, the critical pressure Pcr and the critical temperature Tcr for Sm1−x (Y,Gd)x Mn2 Ge2 and SmMn2 (Six Ge1−x )2 are listed in Table 1. From this results the relative change of the unit cell volume as function of external pressure was estimated to be V = κP, where κ = −3.8 × 10−3 (nm3 /GPa). This dependence allows us to determine the parameters of the model. The intralayer Mn–Mn exchange interaction value Iintra = 245 T/B has been derived from the N´eel temperature of GdMn2 Ge2 [6]. The intralayer interaction value calculated for SmMn2 Ge2 (the N´eel temperature 385 K) was found as equal to Iintra ≈ 143 T/B . For the R1−x R x Mn2 (Si,Ge)2 series of compounds, the magnetic phase transition from the AFmc antiferromagnetic structures to the AFl ones at temperature TNinter Table 1 The critical point parameters of the R1−x R x Mn2 (Si,Ge)2 compounds
SmMn2 Ge2 Sm0.9 Gd0.1 Mn2 Ge2 Sm0.85 Gd0.15 Mn2 Ge2 Sm0.9 Y0.1 Mn2 Ge2 Sm0.85 Y0.15 Mn2 Ge2 SmMn2 (Si0.2 Ge0.8 )2
V (nm3 )
Pcr (GPa)
0.1797 0.1792 0.1791 0.1784 0.1777 0.1767
1.00 0.92 0.79 0.72 0.58 0.20
V is the unit cell volume at room temperature.
± ± ± ± ± ±
0.02 0.02 0.02 0.02 0.02 0.02
Tcr (K) 330 332 330 328 328 318
± ± ± ± ± ±
2 2 2 2 2 2
109
was established under pressure. Furthermore, TNinter can be written as TNinter = TON (1 − βAF V/V ). So we conclude, that the decrease of cell volume stimulates the increment of the interlayer antiferromagnetic Mn–Mn interaction. Based on the pressure variation of the TNinter the value of dTNinter /dP = −βAF κT0 = 18 ± 3 K/GPa (compressibility κ < 0) was found and calculated and the model parameters βAF = 2.67 and T0N = 312 K were estimated. For Sm1−x (Y,Gd)x Mn2 Ge2 and SmMn2 (Six Ge1−x )2 compounds the ferromagnetic ordering disappears when the unit cell volume and the temperature approach Vcr ≈ 0.1757 (3) nm3 and Tcr ≈ 330 ± 2 K, respectively. According to the experimental results, it seems reasonable to obtain interlayer ferromagnetic Mn–Mn exchange interaction Iinter from Tcr ∼ 330 K (see Table 1). The Iinter value of Mn-sublattice is evaluated to be ∼123 T/B . Below TCinter , the canted ferromagnetic Fmc-type structure of the Mn-sublattice was established and the interlayer Mn–Mn exchange interactions increases with V. Using the pressure variation of the critical temperature TCinter we obtained TCinter = T0C (1 + βF V/V ) and dTCinter /dP = βF κT0 = −11 ± 3 K/GPa, so estimated values of βF , T0C and βF /βAF were found as 1.60, 319 K and 0.598, respectively. As reported in Refs. [7,8], the determined value for the antiferromagnetic component μab in (0 0 1) plane is almost independent of the temperature changes up to TCinter and the ferromagnetic component μz parallel to c-axis disappears at TCinter when the temperature decreases. The latter remains in good agreement with the magnetometric measurements. Based on the temperature dependences of magnetization the approximation y2 = (1 − T/TC )(T/TC )2 10(j + 1)2 /3 (j2 + (j + 1)2 ) for T → TCinter (where y = σ/σ 0 is the normalised magnetization) for the R1−x R x Mn2 (Si,Ge)2 compounds were calculated. Fitting by a function of y2 = ξ(1 − T/TC ) type produces ξ ∼ (3.20–3.80). For j = 1/2, the constant ξ ∼ 3.0, which agrees with Weiss–Heisenberg theory. It is worth to note, that values ξ > 3.0 are typical for metallic ferromagnetism. 4. Antiferromagnetic–ferromagnetic phase transitions at T2inter While temperature decreases, the energy of magnetic anisotropy increases and the first-order phase transition from the Fmc state to the AFmc ones appears. The interlayer Mn–Mn exchange interaction changes the sign from negative to positive as the interatomic distance (or volume) exceeds critical value. The minimum condition of thermodynamic potential yields the following equation for temperatures magnetic phase transition T2inter [5]: −[r + n(ηy2 + PκβF − αβF T )] cos 2θ + y2 K0 + n2 η cos 4θ =0
(2)
where r, n, βF , K0 and η = I0 κβF2 are the model parameters. The theoretical value of dT2inter /dP = κ/α = (177–184) K/GPa agrees with experimental data dT2inter /dP = 171 K/GPa.
110
M. Duraj, A. Szytuła / Journal of Alloys and Compounds 442 (2007) 108–110
In order to analyse the influence of anisotropy (K0 ) or pressure (P) and temperature (T) on canting angle changes, from the experimental results (the relative change of volume at T2inter and the temperature dependences of magnetization for R1−x R x Mn2 (Si,Ge)2 compounds) the model parameters were calculated. For SmMn2 Ge2 compound the dependence of the cos 2θ versus unit cell volume (pressure) was presented (Fig. 2). Moreover, for the isostructural compounds, the same dependence of the cos 2θ obtained from neutron diffraction measurements was also established. References
Fig. 2. The calculated and observed variation cos 2θ as function of the unit cell volume (pressure) and the interatomic distance for R1−x R x Mn2 (Si,Ge)2 and RMn2−x Fex Ge2 compounds [9,10]. 2θ is the canting angle of the Mn-sublattice.
According to the model the value of canting angle can be expressed as: cos 2θ0 =
r + nηy2 + n(PκβF − αβF T ) (K0 + n2 η)y2
(3)
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M. Duraj, R. Duraj, A. Szytuła, J. Magn. Magn. Mater. 73 (1988) 240. M. Duraj, R. Duraj, A. Szytuła, J. Magn. Magn. Mater. 79 (1989) 61. R. Duraj, M. Duraj, A. Szytuła, J. Magn. Magn. Mater. 82 (1989) 319. R. Duraj, M. Duraj, A. Szytuła, J. Magn. Magn. Mater. 166 (1997) 207. M. Duraj, A. Szytuła, Phys. Status Solidi (b) 236 (2) (2003) 470. G. Guanghua, R.Z. Levitin, A.Yu. Sokolov, et al., J. Magn. Magn. Mater. 214 (2000) 301. I. Dincer, A. Elmali, Y. Elerman, et al., J. Alloys Compd. 403 (2005) 53. I. Dincer, A. Elmali, Y. Elerman, et al., J. Alloys Compd. 416 (2006) 22. G. Venturini, B. Malaman, E. Ressouche, J. Alloys Compd. 237 (1996) 61. M.N. Norlidach, G. Venturini, B. Malaman, E. Ressouche, J. Alloys Compd. 248 (1997) 112.
Magnetic properties of R1−xRx Mn2(Si,Ge)2 (R, R = rare earth) compounds M. Duraj a,∗ , A. Szytuła b a
Institute of Physics, Cracow University of Technology, Podchor˛az˙ych 1, 30-084 Cracow, Poland b Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, Poland
Received 24 July 2006; received in revised form 25 September 2006; accepted 25 September 2006 Available online 30 January 2007
Abstract Based on a phenomenological mean-field model the (P,T) magnetic phase diagram for the R1−x R x Mn2 (Si,Ge)2 series of compounds were discussed. The exchange parameters derived from the experimental data were used for the theoretical calculation of canting angle 2θ. © 2007 Elsevier B.V. All rights reserved. PACS: 75.30.Kz; 75.50.Ee Keywords: Rare-earth compounds; Intermetallics; Magnetic phase diagram
1. Introduction The magnetic properties under pressure of the intermetalic compounds R1−x R x Mn2 (Si,Ge)2 , (R, R —rare earth) were studied [1–4]. These compounds crystallize in the tetragonal ThCr2 Si2 -type crystal structure which consists of the monoatomic layers along the c-axis with the sequence Th–Si–Cr2 –Si. In these systems an especially interesting Mnsublattice ordering appears. Below the N´eel temperature TN there exists a collinear magnetic structure of the AFl type, arranged of antiferromagnetic collinear (0 0 1) Mn planes. As the temperature decreases, below the transition temperature TCinter , a canted ferromagnetic Fmc structure is established. With decreasing temperature, the canted phase transforms into the antiferromagnetic one accompanied with metamagnetic transition at T2inter . The metamagnetic transition at T1 is accompanied by a change in the rare-earth sublattice from disordered to an ordered state. It is known that the magnetic properties of R1−x R x Mn2 (Si,Ge)2 compounds are sensitive to the atomic distance. The magnetic phase transitions at T2inter from the ferromagnetic to the antiferromagnetic state occurs at the intralayer Mn–Mn distance RaMn–Mn = 0.286 (5) nm (for SmMn2 Ge2 ). The mag-
∗
Corresponding author. Tel.: +48 12 637 06 66; fax: +48 12 637 14 46. E-mail address: [email protected] (M. Duraj).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2006.09.147
netic structure can be easily modified by external or chemical pressure and temperature. 2. Model In order to make an analysis of critical behaviour of non-collinear antiferromagnetic Mn subsystem of R1−x R x Mn2 (Si,Ge)2 compounds, the standard two-sublattice meanfield model was discussed in Ref. [5]. The model presumes that the two equivalent Mn-sublattice magnetic moments are canted with 2θ angle. The full thermodynamic potential calculated per initial volume is the sum of exchange energy Φe , the entropy term Φs , the elastic energy Φd , the pressure term Φp and the Zeeman energy ΦH . The exchange interaction Φe takes the following form: A 2 B 2 a 2 b l + m + (lx + ly2 ) + (m2x + m2y ) 2 2 2 2 Ci D Di C i 2 2 + li4 + m4 + (li mi )2 + l m 4 4 2 2 i i
Φe = Φ0 +
i
i
(1)
i
៝ 2 of the system where the overall magnetic moment m =μ 1 + μ is perpendicular to the antiferromagnetic vector l = μ 1 − μ 2, and mi , li are components of m, l, respectively. The Mn–Mn interaction is assumed to be linearly dependent on relative changes of crystal volume, giving rise to magnetostrictive phenomena.
M. Duraj, A. Szytuła / Journal of Alloys and Compounds 442 (2007) 108–110
Fig. 1. The magnetic phase diagram (P,T) of the SmMn2 Ge2 compound [1]. The relative change of the unit cell V = −3.8 × 10−3 (nm3 /GPa) × P. Pcr = 1.00 ± 0.02 GPa, Vcr ≈ 0.1757 (3) nm3 and Tcr ≈ 330 ± 2.
In order to plot the temperature dependences of magnetization and (P,T) magnetic phase diagrams, it is necessary to calculate equilibrium thermodynamic potentials of different phases, and then to find the lowest one. Such numerical calculations were previously proposed in Ref. [5]. 3. Magnetic phase transitions at TNinter and TCinter Basing on the experimentally obtained (P,T) magnetic phase diagrams [1–4], values of the critical temperature Tcr and the critical pressure Pcr (i.e. the Fmc phase disappears and only the antiferromagnetic interaction remain) were obtained. Fig. 1 shows the magnetic (P,T) phase diagram of SmMn2 Ge2 . The unit cell volume, the critical pressure Pcr and the critical temperature Tcr for Sm1−x (Y,Gd)x Mn2 Ge2 and SmMn2 (Six Ge1−x )2 are listed in Table 1. From this results the relative change of the unit cell volume as function of external pressure was estimated to be V = κP, where κ = −3.8 × 10−3 (nm3 /GPa). This dependence allows us to determine the parameters of the model. The intralayer Mn–Mn exchange interaction value Iintra = 245 T/B has been derived from the N´eel temperature of GdMn2 Ge2 [6]. The intralayer interaction value calculated for SmMn2 Ge2 (the N´eel temperature 385 K) was found as equal to Iintra ≈ 143 T/B . For the R1−x R x Mn2 (Si,Ge)2 series of compounds, the magnetic phase transition from the AFmc antiferromagnetic structures to the AFl ones at temperature TNinter Table 1 The critical point parameters of the R1−x R x Mn2 (Si,Ge)2 compounds
SmMn2 Ge2 Sm0.9 Gd0.1 Mn2 Ge2 Sm0.85 Gd0.15 Mn2 Ge2 Sm0.9 Y0.1 Mn2 Ge2 Sm0.85 Y0.15 Mn2 Ge2 SmMn2 (Si0.2 Ge0.8 )2
V (nm3 )
Pcr (GPa)
0.1797 0.1792 0.1791 0.1784 0.1777 0.1767
1.00 0.92 0.79 0.72 0.58 0.20
V is the unit cell volume at room temperature.
± ± ± ± ± ±
0.02 0.02 0.02 0.02 0.02 0.02
Tcr (K) 330 332 330 328 328 318
± ± ± ± ± ±
2 2 2 2 2 2
109
was established under pressure. Furthermore, TNinter can be written as TNinter = TON (1 − βAF V/V ). So we conclude, that the decrease of cell volume stimulates the increment of the interlayer antiferromagnetic Mn–Mn interaction. Based on the pressure variation of the TNinter the value of dTNinter /dP = −βAF κT0 = 18 ± 3 K/GPa (compressibility κ < 0) was found and calculated and the model parameters βAF = 2.67 and T0N = 312 K were estimated. For Sm1−x (Y,Gd)x Mn2 Ge2 and SmMn2 (Six Ge1−x )2 compounds the ferromagnetic ordering disappears when the unit cell volume and the temperature approach Vcr ≈ 0.1757 (3) nm3 and Tcr ≈ 330 ± 2 K, respectively. According to the experimental results, it seems reasonable to obtain interlayer ferromagnetic Mn–Mn exchange interaction Iinter from Tcr ∼ 330 K (see Table 1). The Iinter value of Mn-sublattice is evaluated to be ∼123 T/B . Below TCinter , the canted ferromagnetic Fmc-type structure of the Mn-sublattice was established and the interlayer Mn–Mn exchange interactions increases with V. Using the pressure variation of the critical temperature TCinter we obtained TCinter = T0C (1 + βF V/V ) and dTCinter /dP = βF κT0 = −11 ± 3 K/GPa, so estimated values of βF , T0C and βF /βAF were found as 1.60, 319 K and 0.598, respectively. As reported in Refs. [7,8], the determined value for the antiferromagnetic component μab in (0 0 1) plane is almost independent of the temperature changes up to TCinter and the ferromagnetic component μz parallel to c-axis disappears at TCinter when the temperature decreases. The latter remains in good agreement with the magnetometric measurements. Based on the temperature dependences of magnetization the approximation y2 = (1 − T/TC )(T/TC )2 10(j + 1)2 /3 (j2 + (j + 1)2 ) for T → TCinter (where y = σ/σ 0 is the normalised magnetization) for the R1−x R x Mn2 (Si,Ge)2 compounds were calculated. Fitting by a function of y2 = ξ(1 − T/TC ) type produces ξ ∼ (3.20–3.80). For j = 1/2, the constant ξ ∼ 3.0, which agrees with Weiss–Heisenberg theory. It is worth to note, that values ξ > 3.0 are typical for metallic ferromagnetism. 4. Antiferromagnetic–ferromagnetic phase transitions at T2inter While temperature decreases, the energy of magnetic anisotropy increases and the first-order phase transition from the Fmc state to the AFmc ones appears. The interlayer Mn–Mn exchange interaction changes the sign from negative to positive as the interatomic distance (or volume) exceeds critical value. The minimum condition of thermodynamic potential yields the following equation for temperatures magnetic phase transition T2inter [5]: −[r + n(ηy2 + PκβF − αβF T )] cos 2θ + y2 K0 + n2 η cos 4θ =0
(2)
where r, n, βF , K0 and η = I0 κβF2 are the model parameters. The theoretical value of dT2inter /dP = κ/α = (177–184) K/GPa agrees with experimental data dT2inter /dP = 171 K/GPa.
110
M. Duraj, A. Szytuła / Journal of Alloys and Compounds 442 (2007) 108–110
In order to analyse the influence of anisotropy (K0 ) or pressure (P) and temperature (T) on canting angle changes, from the experimental results (the relative change of volume at T2inter and the temperature dependences of magnetization for R1−x R x Mn2 (Si,Ge)2 compounds) the model parameters were calculated. For SmMn2 Ge2 compound the dependence of the cos 2θ versus unit cell volume (pressure) was presented (Fig. 2). Moreover, for the isostructural compounds, the same dependence of the cos 2θ obtained from neutron diffraction measurements was also established. References
Fig. 2. The calculated and observed variation cos 2θ as function of the unit cell volume (pressure) and the interatomic distance for R1−x R x Mn2 (Si,Ge)2 and RMn2−x Fex Ge2 compounds [9,10]. 2θ is the canting angle of the Mn-sublattice.
According to the model the value of canting angle can be expressed as: cos 2θ0 =
r + nηy2 + n(PκβF − αβF T ) (K0 + n2 η)y2
(3)
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M. Duraj, R. Duraj, A. Szytuła, J. Magn. Magn. Mater. 73 (1988) 240. M. Duraj, R. Duraj, A. Szytuła, J. Magn. Magn. Mater. 79 (1989) 61. R. Duraj, M. Duraj, A. Szytuła, J. Magn. Magn. Mater. 82 (1989) 319. R. Duraj, M. Duraj, A. Szytuła, J. Magn. Magn. Mater. 166 (1997) 207. M. Duraj, A. Szytuła, Phys. Status Solidi (b) 236 (2) (2003) 470. G. Guanghua, R.Z. Levitin, A.Yu. Sokolov, et al., J. Magn. Magn. Mater. 214 (2000) 301. I. Dincer, A. Elmali, Y. Elerman, et al., J. Alloys Compd. 403 (2005) 53. I. Dincer, A. Elmali, Y. Elerman, et al., J. Alloys Compd. 416 (2006) 22. G. Venturini, B. Malaman, E. Ressouche, J. Alloys Compd. 237 (1996) 61. M.N. Norlidach, G. Venturini, B. Malaman, E. Ressouche, J. Alloys Compd. 248 (1997) 112.