Tricritical point of the $f$-electron antiferromagnet $\mathrm{US}{\mathrm{b}}_2$ driven by high magnetic fields

In pulsed magnetic fields up to 65 T and at temperatures below the Néel transition, our magnetization and magnetostriction measurements reveal a field-induced metamagneticlike transition that is suggestive of an antiferromagnetic to ferrimagnetic ordering. Our data also suggest a change in the nature of this metamagneticlike transition from second- to first-order-like near a tricritical point at $T_{\mathrm{tc}}\sim 145\mathrm{K}$ and $H_{\mathrm{c}}\sim 52\mathrm{T}$. At high fields for $H>H_{\mathrm{c}}$ we found a decreased magnetic moment roughly half of the moment determined by neutron powder diffraction. We propose that the decreased moment and lack of saturation at high fields indicate the presence of a field-induced ferrimagnetic state above the tricritical point of the H-T phase diagram for $\mathrm{US}{\mathrm{b}}_2$.

Figure 1

Experimental setup for the fiber Bragg dilatometer showing the orientations of the $\mathrm{US}{\mathrm{b}}_2$ crystals. The samples shown in (a) were stacked along the $c$ axis to make a total height of 1 mm in order to increase the signal. (b) The stack was glued to the fiber optic cable so that the $c$ axis//$H$. (c) Crystal of $\mathrm{US}{\mathrm{b}}_2$ glued with the $a-b$ plane of the crystal attached to the fiber. (d) Complete experimental setup showing the $c$-axis stack (lower left red box), the $a-b$ plane attached along the crystal edge, the position of the thermometer (upper left corner), the magnetic pick-up coil used for magnetic field calibration wrapped around the holder (between the two samples), the gold wires attaching the samples to the thermometer in order to achieve thermal equilibrium.

Figure 2

Thermal contraction as a function of temperature during cooling of $\mathrm{US}{\mathrm{b}}_2$ along the $a$−$b$ plane. A picture of the samples attached to the fiber Bragg grating oriented with $H$ along the $a$−$b$ plane (shown in the top, right inset), and the stack of six crystals attached to the FBG with $H$ along the $c$ axis (shown in the top, left inset). The coefficient of thermal contraction (α, defined as the derivative of $\mathrm{\Delta }L/L$ with respect to the temperature) during cooldown (lower inset) is shown to emphasize the location of the AFM transition at $T_{\mathrm{N}}=202.58\mathrm{K}$, as previously reported in the literature.

Figure 3

Comparison of the magnetostriction as a function of applied magnetic field, for $H//c$ axis and $H$ perpendicular to the $c$ axis at $T=125\mathrm{K}$. The anisotropy of the system is clearly demonstrated by the metamagneticlike transition seen for $H//c$ axis, but not with $H$ perpendicular to $c$ axis. All of the $\mathrm{\Delta }L/L$ data shown in the following figures is for $H//c$ axis. Oscillations for $H//c$ are likely due to vibrations resulting from the small misalignment of magnetic moment with the applied magnetic field.

Figure 4

$\mathrm{\Delta }L/L$ as a function of applied magnetic field, measured with the $H//c$ axis. Measurements at temperatures above and below the AFM transition are shown. For $T>T_{\mathrm{N}}$ there is no evidence of a transition, but for $T<T_{\mathrm{N}}$ there is a change in slope that grows increasingly sharp as the temperature is decreased, with a second-order-like to first-order-like tricritical point at $T_{\mathrm{tc}}\sim 145\mathrm{K}$. There is also a change in the slope of the $\mathrm{\Delta }L/L$ for $H>H_{\mathrm{c}}$ and $T<140\mathrm{K}$. All traces are offset vertically for clarity.

Figure 5

(a) Amplitude of the jump in $\mathrm{\Delta }L/L$ at $H_{\mathrm{c}}$ as a function of temperature. The red stripe at $T=145\mathrm{K}$ is the tricritical temperature as predicted by our calculations. This data show that there is a discontinuous jump in $\mathrm{\Delta }L/L$ for $T\le 150\mathrm{K}$ and that the amplitude of the jump at $H_{\mathrm{c}}$ increases with decreasing temperature. (b) Plot of the slope of the $\mathrm{\Delta }L/L$ above the transition ($H>H_{\mathrm{c}}$) as a function of applied magnetic field. The slope of the $\mathrm{\Delta }L/L$ at high temperatures increases initially with decreasing temperature and then decreases. For $T\le 140\mathrm{K}$ the slope is nearly zero, within error, which we propose as evidence of a ferrimagnetic state. Error bars for both plots are based on the cumulative variation due to transition width and noise in the data before and after the transition.

Figure 6

Magnetization as a function of applied magnetic field for $H//c$ in units of ${\mu }_{\mathrm{B}}$ per uranium atom for selected temperatures. For $T>T_{\mathrm{N}}$ there is a linear trend of the magnetization but for $T<T_{\mathrm{N}}$ there is a metamagneticlike transition. The increase of the magnetization at $H_{\mathrm{c}}$ is roughly $1{\mu }_{\mathrm{B}}/\mathrm{U}$, which is nearly a factor of 2 less than the previously measured zero-field moment of $1.88{\mu }_{\mathrm{B}}/\mathrm{U}$ [33, 51]. High-field data was calibrated using a Quantum Design Vibrating Sample Magnetometer up to 15 T.

Figure 7

Plot of magnetization as a function of applied magnetic field for various temperatures. All of the traces were fit with a straight line from zero up to $H^{*}$, where the slope of the trace deviates from linearity. All of the traces were then normalized so that the maximum value of the magnetization was equal to one. The upper inset of the figure shows the derivative of the magnetization as a function of magnetic field $dM/dH$. The change from second to first order can be seen as the lower field side of the derivative becomes discontinuous for $T<150\mathrm{K}$.

Figure 8

(a) Width of the second-order-like transition ($H^{*}$) in the magnetization as a function of temperature. The width of the $H^{*}$ transition is clearly connected to the tricritical point at 145 K, as the width of the transition goes to zero by 130 K. (b) Plot of the width of the metamagneticlike transition $H_{\mathrm{c}}$ as a function of temperature. The $H_{\mathrm{c}}$ transition width is not as strongly correlated as $H^{*}$ is to the order of the transition, nonetheless the trend of the data shows a decrease of the transition width as the temperature approaches $T\sim T_{\mathrm{tc}}$. The error bars represent the variation allowed by different straight-line fits to the data.

Figure 9

Magnetic field and temperature phase diagram of $\mathrm{US}{\mathrm{b}}_2$ ($T_{\mathrm{N}}=202.3\mathrm{K}$), compiled from both magnetostriction and magnetization data. The predicted tricritical point is shown at $T_{\mathrm{tc}}=145\mathrm{K}$, with the red dashed line intersecting the phase boundary at $H=52\mathrm{T}$. The phase boundary is drawn with a solid black line for $T>145\mathrm{K}$ to indicate that the metamagneticlike transition is second-order-like, whereas it is drawn as a dashed line for $T<145\mathrm{K}$ where the metamagneticlike transition is first-order-like. The intensity plot above the phase boundary shows the slope of the $\mathrm{\Delta }L/L$ versus magnetic field above the metamagneticlike transition ($H>H_{\mathrm{c}}$).

Figure 10

Magnetostriction as a function of applied magnetic field for three temperatures showing how $H_{\mathrm{c}}$, transition width, and the change in strain at the transition were measured. $H_{\mathrm{c}}$ is defined as the intersection point of two straight lines, one on the nearly vertical part of the trace and one on the part of the trace before going nearly vertical. Error bars represent the maximum and minimum value that could have been chosen by incorporating the curvature of the trace before the transition and the noise in the trace above the transition into the straight-line fits to the data.

Figure 11

Magnetization as a function of applied magnetic field for two temperatures showing how $H_{\mathrm{c}}, H^{*}$, and the related error bars were chosen. $H_{\mathrm{c}}$ is defined as the intersection point of two straight lines, one on the nearly vertical part of the trace and one on the part of the trace before going nearly vertical. The error is taken as the points where the trace deviates from the straight line. $H^{*}$ is taken as the point where there is a clear deviation of the magnetization curve from the straight-line fit.