Angular-dependent planar metamagnetism in the hexagonal compounds TbPtIn and TmAgGe

Detailed magnetization measurements $M(T,H,\theta )$, were performed on single crystals of TbPtIn and TmAgGe (both members of the hexagonal ${\mathrm{Fe}}_2\mathrm{P}∕\mathrm{Zr}\mathrm{Ni}\mathrm{Al}$ structure type) for the magnetic field $H$ applied perpendicular to the crystallographic $c$ axis. These data allowed us to identify, for each compound, the easy axes for the magnetization, which coincided with high symmetry directions ([120] for TbPtIn and [110] for TmAgGe). For fixed orientations of the field along each of the two sixfold symmetry axes, a number of magnetically ordered phases is being revealed by $M(H,T)$ measurements below $T_N$. Moreover, $T\simeq 2\mathrm{K}$, ${\phantom{\mid }M\left(H\right)\mid }_{\theta }$ measurements for both compounds (with $H$ applied parallel to the basal plane), as well as $T=20\mathrm{K}$ data for TbPtIn, reveal five metamagnetic transitions with simple angular dependencies $H_{ci,j}\sim 1∕\cos (\theta \pm \phi )$, where $\phi =0°$ or 60°. The high-field magnetization state varies with $\theta$ as $2∕3{\mu }_{\mathrm{sat}}\left(R^{3+}\right)\cos \theta$, and corresponds to a crystal field limited saturated paramagnetic state. Analysis of these data allowed us to model the angular dependence of the locally saturated magnetizations $M_{\mathrm{sat}}$ and critical fields $H_c$ with a three coplanar Ising-like model, in which the magnetic moments are assumed to be parallel to three adjacent easy axes. Furthermore, net distributions of moments were inferred based on the measured data and the proposed model.

Figure 1

Projection of the ZrNiAl-type crystal structure along the hexagonal $c$ axis. $R$: large circles, $M$ (Pt or Ag): medium circles, $X$ (In or Ge): small circles. Light circles: $z=0$ plane, dark circles: $z=1∕2$ plane. [Note: the Pt or Ag atoms (medium circles) appear in both planes.]

Figure 2

Anisotropic inverse susceptibility of TbPtIn (symbols) and the calculated average (line); inset: low-temperature anisotropic susceptibilities.

Figure 3

(a) Low-temperature $d\left({\chi }_{\mathrm{ave}}T\right)∕dT$, with vertical dotted lines marking the peaks positions; (b) specific heat $C_p\left(T\right)$; inset: magnetic entropy $S_m\left(T\right)$; (c) low-temperature resistivity $\rho \left(T\right)$, for current flowing in the basal plane $(i\Vert ab)$.

Figure 4

Anisotropic inverse susceptibility (symbols) and calculated average (line) of ${\mathrm{Tb}}_{0.025}{\mathrm{Y}}_{0.975}\mathrm{Pt}\mathrm{In}$ for $H=1\mathrm{kG}$; upper inset: low-temperature anisotropic susceptibility; lower inset: field-dependent anisotropic magnetization, for $T=2\mathrm{K}$.

Figure 5

(a) Anisotropic field-dependent magnetization (increasing and decreasing field data indicated by arrows) and (b) transverse magnetoresistance, for increasing (symbols) and decreasing (line) field.

Figure 6

$M\left(\theta \right)$ of TbPtIn (full circles) and ${\mathrm{Tb}}_{0.025}{\mathrm{Y}}_{0.975}\mathrm{Pt}\mathrm{In}$ (open circles) for $T=2\mathrm{K}$ and $H=55\mathrm{kG}$, $H\perp c$. The solid line represents the calculated $M_{\max }\cos (\theta -n60°)$, $n$ integer. Note: $\theta =0°$ is defined at the [120] direction.

Figure 7

$M\left(T\right)$ data for various fields for two in-plane orientations of the applied field: (a) $H\Vert \left[120\right]$ for $H=1$, 2, 7.5, 15, 20, 25, 30, 40, 45, 50, and $55\mathrm{kG}$ and (b) $H\Vert \left[110\right]$ for $H=1$, 2.5, 3.5, 5, 7.5, 10, 15, 20, 26, 30, 37.5, 40, 42.5, 45, 50, and $55\mathrm{kG}$. The insets show enlarged $M\left(T\right)\bullet T$ derivatives (dotted lines) for $H=20\mathrm{kG}$, together with the Lorentzian fits of the maxima (solid lines), to examplify how the points represented by full symbols on the $H\text{−}T$ phase diagrams were determined.

Figure 8

$M\left(H\right)$ isotherms for two in-plane orientations of the applied field: (a) $H\Vert \left[120\right]$ for $T=3$, 5, 10, 15, 17.5, 20, 25, 30, 35, 37.5, 40, and $45\mathrm{K}$ and (b) $H\Vert \left[110\right]$ for $T=2$, 5, 10, 15, 20, 25, 30, 35, 40, 42.5, 45, and $47.5\mathrm{K}$. Insets show enlarged $M\left(H\right)$ derivatives (dotted lines) for $T=20$ and $30\mathrm{K}$, for the corresponding field directions, together with the Lorentzian fits of the maxima (solid lines), to examplify how the points represented by open symbols on the $H\text{−}T$ phase diagrams were determined.

Figure 9

$H\text{−}T$ phase diagrams for (a) $H\Vert \left[120\right]$ and (c) $H\Vert \left[110\right]$, as determined from the magnetization data in Figs. 7, 8, with an intermediate field orientation phase diagram (see text) shown in (b).

Figure 10

(a) $M\left(T\right)$ data for various fields: $H=1\mathrm{kG}$ and $5–70\mathrm{kG}(\Delta H=5\mathrm{kG})$ for the intermediate field direction (see text). The small arrows indicate the highest-field transitions at the corresponding temperature. (b) $M\left(H\right)$ for $T=10\mathrm{K}$, and the corresponding derivative, as an example of how the open symbols on the upper-most phase line in Fig. 9b have been determined.

Figure 11

$M\left(H\right)$ isotherms at $T=2.0\mathrm{K}$ for (a) $0°⩽\theta ⩽12°$ and (b) $12°⩽\theta ⩽30°(\Delta \theta =1°)$; inset: enlarged high field, $T=1.85\mathrm{K}$ (see text) isotherms. Arrows indicate the direction of increasing $\theta$.

Figure 12

Enlarged $M(H;\theta =12°)$ plot, and the corresponding derivative, illustrating the criteria used in determining the points in Fig. 13: large dot on the $dM∕dH$ plot indicates the critical field value, determined by the maximum of the Lorentzian fit (solid line) of the peak; straight lines on the $M\left(H\right)$ plot are fits to the magnetization plateau (thin), extrapolated down to intersect the maximum-slope line (thick), giving the $M_2$ value (large dot). Note: the quality of the Lorentzian fit is representative of some of the poorer fits.

Figure 13

(a) Measured critical fields $H_{ci,j}$ and (b) locally saturated magnetizations $M_j$ (full symbols) as a function of angle $\theta$ measured from the [120] direction. Open symbols are reflections of the measured data across the $\theta =0°$ direction. Also shown are the calculated angular dependencies (dotted lines).

Figure 14

Schematic representation of the three Ising-like systems model (with three distinct $R$’s in the unit cell, sitting in unique orthorhombic sites). Solid arrows: “up” and dotted arrows: “down” positions of the magnetic moments along the easy axes.

Figure 15

Calculated magnetizations as a function of $\theta$, based on the three coplanar Ising model. Open circles: $S=1$ and crosses: $S=2$ (see text for details).

Figure 16

Polar plot of the critical fields $H_{ci,j}$, with one possible moment configuration shown for each observed metamagnetic state. Open symbols represent reflections of the measured data—full symbols—across the $\theta =0°$ direction (see text).

Figure 17

Anisotropic inverse susceptibility of TmAgGe (symbols) and the calculated polycrystalline average (line); inset: low-temperature anisotropic susceptibilities.

Figure 18

Anisotropic field-dependent magnetization, with the solid lines being the linear fits of the high-field plateaus, extrapolated down to $H=0$ (see text).

Figure 19

$M\left(\theta \right)$ of TmAgGe (full symbols) at $T=2\mathrm{K}$ and $H=70\mathrm{kG}$, $H\perp c$. Solid line: calculated magnetization $M_{\max }\cos (\theta -n60°)$, $n$ integer. Note: $\theta =0°$ is defined at the [110] direction.

Figure 20

(a) $M\left(T\right)$ data for $H=1$, $5\mathrm{kG}$ and $10–70\mathrm{kG}$ $(\Delta H=10\mathrm{kG})$ for $H\Vert \left[120\right]$, with enlarged high-field data in the inset. (b) $MT$ derivatives for $H=50$, 60, and $70\mathrm{kG}$, together with the Lorentzian fits of the maxima (solid lines), illustrating how the vertical line in Fig. 22c was determined for high applied fields.

Figure 21

$M\left(H\right)$ isotherms for $T=1.85$, $2–4\mathrm{K}$ $(\Delta T=0.25\mathrm{K})$, 4.2 and $4.6\mathrm{K}$ for (a) $H\Vert \left[120\right]$ and (b) $H\Vert \left[110\right]$; insets show the enlarged data around the metamagnetic transitions. (Arrows indicate direction of increasing $T$.)

Figure 22

$H\text{−}T$ phase diagrams for TmAgGe with (a) $H\Vert \left[110\right]$ and (c) $H\Vert \left[120\right]$, as determined from magnetization data in Figs. 20, 21. (b) Intermediate-position phase diagram (see text); the error bar shown in the inset of (b) represents the range of values for the high-field line, as determined from Fig. 23.

Figure 23

$M\left(H\right)$ for $T=1.85\mathrm{K}$ (symbols), and the corresponding derivative (line). The inset shows an enlargement around the high field transition, to exemplify the two criteria used for determining this critical field (see text). Note: the error bar shown in the inset (determined from the position of the peaks in the derivative) gives a caliper of the uncertainty in determining this critical field value.

Figure 24

$M\left(H\right)$ isotherms $(T=2.0\mathrm{K})$ for (a) $0°⩽\theta ⩽10°$, $\Delta \theta =2°$ (enlarged $M_1$ state shown in the inset) and (b) $10°⩽\theta ⩽30°$, $\Delta \theta =2°$. The inset in (b) shows the extrapolation of the two higher metamagnetic states down to lower fields, such that the intersection with the maximum-slope line gives the magnetization values $M_3$ and $M_{\mathrm{CL}\text{−}\mathrm{SPM}}$: solid dots (see text). Arrows indicate increasing $\theta$.

Figure 25

(a) Measured critical fields $H_{ci,j}$ and (b) locally saturated magnetizations $M_j$ (full symbols), as a function of angle $\theta$ measured from the easy axis. Open symbols are reflections across the $\theta =0°$ direction (see text). Also shown are the calculated angular dependencies of $H_{ci,j}$ and $M_j$ (dotted lines). The error bars shown in the low part of (b) give the range of values that we can infer for $M_1$ from the data shown in Fig. 24.

Figure 26

Polar plot of the critical fields $H_c$, with one of the possible moment configurations shown for each observed metamagnetic state.

Figure 27

$M\left(H\right)$ isotherms at $T=20\mathrm{K}$ for (a) $0°⩽\theta ⩽12°$ and (b) $12°⩽\theta ⩽30°$ $(\Delta \theta =1°)$. Inset: enlarged $M_4$ state. Arrows indicate the direction of increasing $\theta$.

Figure 28

(a) Measured critical fields $H_{ci,j}$ and (b) locally saturated magnetizations $M_j$ (full symbols), as a function of angle $\theta$, for $T=20\mathrm{K}$. Open symbols are reflections across the $\theta =0°$ direction. Also shown are the calculated angular dependencies of $H_{ci,j}$ and $M_j$ (dotted lines).

Figure 29

Polar plot of the critical fields $H_{ci,j}$, with one possible moment configuration shown for each observed metamagnetic state (except for $M_1$, where the moment configuration is uncertain—see text). Open symbols represent reflections of the measured data (full symbols) across the $\theta =0°$ direction.

Figure 30

Plausible models for extremely anisotropic, planar compounds (see text) with (a), (b) tetragonal and (c)–(e) hexagonal unit cells. The numbers are used to identify the different magnetic moments in the unit cell. (a) Four position clock model describing tetragonal systems with one $R$ in tetragonal point symmetry. (b) Double coplanar Ising-like model: for orthorhombic point symmetry in tetragonal unit cell, two magnetic moments would be necessary (Ising-like systems, 90° away from each other in the basal plane). (c) Six position clock model: one magnetic moment with six possible orientations (arrows along six high symmetry orientations in the basal plane). (d) Triple coplanar Ising-like model: three $R$ ions in unique orthorhombic point symmetry are needed in a hexagonal compound (three Ising-like systems, 60° away from each other in the basal plane). (e) A double coplanar three position clock model can describe hexagonal systems with two magnetic moments in unique trigonal point symmetry position, with three possible orientations (120° away from each other in the basal plane) for each. In all cases, the corresponding CL-SPM states (described in text) are represented by full arrows (applied field is assumed to be vertically up).