Phase diagram of $\mathrm{Ce}{\mathrm{Sb}}_2$ from magnetostriction and magnetization measurements: Evidence for ferrimagnetic and antiferromagnetic states

Cerium diantimonide ($\mathrm{Ce}{\mathrm{Sb}}_2$) is one of a family of rare-earth-based magnetic materials that exhibit metamagnetism, enabling control of the magnetic ground state through an applied magnetic field. At low temperatures, $\mathrm{Ce}{\mathrm{Sb}}_2$ hosts a rich phase diagram with multiple magnetically ordered phases for many of which the order parameter is only poorly understood. In this paper, we report a study of its metamagnetic properties by scanning tunneling microscopy (STM) and magnetization measurements. We use STM measurements to characterize the sample magnetostriction with subpicometer resolution from magnetic field and temperature sweeps. This allows us to directly assess the bulk phase diagram as a function of field and temperature and relate spectroscopic features from tunneling spectroscopy to bulk phases. Our magnetostriction and magnetization measurements indicate that the low-temperature ground state at zero field is ferrimagnetic. Quasiparticle interference mapping shows evidence for a reconstruction of the electronic structure close to the Fermi energy upon entering the magnetically ordered phase.

Figure 1

Magnetic measurements of $\mathrm{Ce}{\mathrm{Sb}}_2$. (a) Phase diagram of $\mathrm{Ce}{\mathrm{Sb}}_2$ following Ref. [11]. Four different magnetic phases have been identified (labelled I–IV) in addition to the paramagnetic phase (PM) at high temperatures $T>15.5\mathrm{K}$. (b) Zero-field cooled (ZFC), field cooled (FC), and field warmed (FW) magnetization measurements all recorded using an applied field of $10\mathrm{mT}$. Three transitions can be observed at 1, 2, and 3. (c) Magnetic isotherm recorded at $T=4.2\mathrm{K}$ for fields up to $3\mathrm{T}$. The three kinks at $0.4\mathrm{T}$ (A), $1\mathrm{T}$ (B), and $2.8\mathrm{T}$ (C) in the isotherm are associated with metamagnetic transitions. (d) Zero-field cooled magnetization curves as a function of temperature and for magnetic fields up to $2.7\mathrm{T}$.

Figure 2

$M\text{−}H$ loops. (a) $M\text{−}H$ loop recorded at $4.2\mathrm{K}$ in the lowest temperature magnetic phase (phase III in Fig. 1). Clear hysteretic behavior can be seen. (b) $M\text{−}H$ loop recorded at $8.2\mathrm{K}$. The sample shows less hysteresis in this phase and a linear magnetic response at low field. (c) $M\text{−}H$ loop recorded in phase I, showing only a paramagnetic response. (d) $M\text{−}H$ loop recorded in the paramagnetic phase at $T=20\mathrm{K}$. The sample shows a smaller coupling to the field than in phase I. Between the measurements of $M\text{−}H$ loops at different temperatures, the sample was warmed up to $20\mathrm{K}$ and then cooled in zero field.

Figure 3

Topographic imaging of $\mathrm{Ce}{\mathrm{Sb}}_2$. (a) Topographic STM image of a step edge showing two different surface terminations T1 and T2 of $\mathrm{Ce}{\mathrm{Sb}}_2 [V=100\mathrm{mV}, I=40\mathrm{pA}, (50\times 30){\mathrm{nm}}^2, T=2.2\mathrm{K}]$. (b) Line profile taken along the red line in (a). (c) Topographic image showing the termination labeled T1 attributed to the Sb layer $[V=100\mathrm{mV}, I=40\mathrm{pA}, (20\times 20){\mathrm{nm}}^2, T=2.2\mathrm{K}]$. (d) Fourier transformation of the topographic image shown in (c). (e) Topographic STM image of the termination labeled T2 attributed to the CeSb layer $[V=100\mathrm{mV}, I=40\mathrm{pA}, (20\times 20){\mathrm{nm}}^2, T=2.2\mathrm{K}]$. (f) Fourier transformation of (e). (g) Average STS spectra recorded on the different terminations taken from a spectroscopic map across the step in (a) ($V_{\mathrm{s}}=100\mathrm{mV}, I_{\mathrm{s}}=100\mathrm{pA}, V_{\mathrm{mod}}=2.5\mathrm{mV}$). (h) Crystal structure of CeSb. The Sb termination (tentatively attributed to T1) is indicated by the blue-dashed line, the blue box on the right shows its surface structure. The CeSb termination (T2) is indicated by the green dashed line with its surface structure in the green box.

Figure 4

Temperature-dependent quasiparticle interference. (a) Fourier transformation (symmetrized) of a map layer at a bias voltage of $+6\mathrm{mV}$ recorded in the high temperature paramagnetic phase ($T=20\mathrm{K}$). (b),(c) Linecuts through the FT-STS data shown in (a) along the $x$ and $y$ directions marked in (a). Red arrows highlight the presence of scattering between bands. (d) Fourier transformation (symmetrized) of an STS map layer with the same bias voltage as in (a) recorded in the low-temperature phase ($T=2.2\mathrm{K}$). (e),(f) Linecut through the FT-STS data along the $x$ and $y$ directions marked in (a). Red arrows in $q_x$ data (b),(e) highlight the absence of the scattering that was present at $20\mathrm{K}$. The yellow parabola in (c) is a fit to the $20\mathrm{K}$ data with a maximum at $22.5\mathrm{mV}$. In the $2.2\mathrm{K}$ data (f) the red solid line is a fit to the data with maximum at $14\mathrm{mV}$, the $20\mathrm{K}$ fit is shown as a dashed yellow line. All spectroscopic data shown in [(a)–(f)] were recorded in the same atomic location and with the same tip and identical tunneling conditions ($V_{\mathrm{s}}=100\mathrm{mV}, I_{\mathrm{s}}=1\mathrm{nA}, V_{\mathrm{mod}}=2\mathrm{mV}$).

Figure 5

Temperature and field dependent spectroscopy. (a) Averaged $dI/dV$ spectroscopy recorded with the quasiparticle interference maps discussed in Fig. 4. Spectra were recorded with the same tip and at the same location on the sample surface ($V_{\mathrm{s}}=100\mathrm{mV}, I_{\mathrm{s}}=1\mathrm{nA}, V_{\mathrm{mod}}=2\mathrm{mV}$). (b) The ratio of the spectrum taken at $2.2\mathrm{K}$ thermally broadened to $20\mathrm{K}$ and the spectrum acquired at $20\mathrm{K}$. A gap can be seen around the Fermi level, which has a half width of $\mathrm{\Delta }=6\mathrm{mV}$. (c) Field dependent point spectra recorded with the same tip apex and averaged over multiple points on the sample surface. The field was applied in the sample $ab$ plane ${16}^{\circ }$ from ${\mathbf{q}}_{\delta }$. The sample density of states can be seen to increase in a $10\mathrm{mV}$ window above the Fermi level for fields above $2\mathrm{T}$ indicated by the arrow. (d) Color plot of the tunneling spectra shown in c as a function of applied field.

Figure 6

Magnetostriction measurements in STM. (a) Schematic of the principle of how an STM can be used to measure the magnetostriction of a sample. The STM tip is held at a constant current $I$, and thus a constant height above the surface, over a defect (shown in the inset) while the magnetic field is ramped. To keep the tip in the same lateral position, an atom tracking algorithm is used. The magnetic-field-induced change $\mathrm{\Delta }L$ in the sample $c$ axis leads to a change in the tip position to keep the tunneling current constant, allowing for a measurement of the magnetostriction. (b) Fourier transformation of a topographic image of the CeSb termination (T2). The cyan arrow indicates the direction of the applied magnetic field ${16}^{\circ }$ from the [110] direction. (c) Measurement of magnetostriction of $\mathrm{Ce}{\mathrm{Sb}}_2$ in the low-temperature phase at fixed temperature as a function of applied field. Clear hysteretic behavior can be seen between the forward and backward sweeps. The sweeps are offset from each other by $3\times {10}^{-5}$ and 0 is defined at a field of $-1\mathrm{T}$. Data between 2 and $6\mathrm{K}$ are in the low-temperature phase (phase III). Data between 8 and $11\mathrm{K}$ are in the intermediate temperature phase (phase IV). The $13.8\mathrm{K}$ data is in phase I and the $16.5\mathrm{K}$ data is outside the magnetically ordered phases.

Figure 7

Magnetic phase diagram of $\mathrm{Ce}{\mathrm{Sb}}_2$. (a) Temperature derivative of the thermal contraction $\frac{d\mathrm{\Delta }L\left(T\right)}{dT}$ measured in the STM at different magnetic fields ${\mu }_{0}H$ applied in the $ab$ plane. Between measurements, the sample was warmed up to $20\mathrm{K}$ and then cooled in zero field (for raw data see Appendix, Fig. 12). (b) Temperature derivatives of the magnetic susceptibility times temperature, $\frac{d\left(\chi \right(T\left)T\right)}{dT}$. Curves in (a) and (b) are vertically offset for clarity. (c) Revised phase diagram of $\mathrm{Ce}{\mathrm{Sb}}_2$ as a function of temperature and magnetic field constructed from magnetostriction data shown in (a) and magnetization in (b) and specific heat data reproduced from Ref. [13]. Filled data points were obtained from warming sweeps recorded at constant field. Open data points were recorded from isothermal measurements with increasing field. Ferri correspond to a ferrimagnetic phases, AFM is an antiferromagnetic phase, MPh a magnetic phase with unknown order and Para the high temperature paramagnetic phase.

Figure 8

Left (green): Topographies of the termination T2 identified as the CeSb layer recorded at $20\mathrm{K}$ and $2.2\mathrm{K}$ and their respective Fourier transforms ($V=100\mathrm{mV}, I=40\mathrm{pA}$). Right (blue): Topographies of the termination T1 identified as the Sb layer recorded at $20\mathrm{K}$ ($V=300\mathrm{mV}, I=50\mathrm{pA}$) and $2.2\mathrm{K}$ ($V=300\mathrm{mV}, I=35\mathrm{pA}$) and their respective Fourier transforms. The same surface features are seen indicating that they are not magnetic in origin.

Figure 9

A typical differential conductance spectrum $dI/dV\left(V\right)$ recorded after the tip has collected material from the termination labeled T1 in the main text (blue line). The corresponding bias-dependence of the tunneling current $I\left(V\right)$ is shown in red ($V_{\mathrm{set}}=100\mathrm{mV}, I_{\mathrm{set}}=200\mathrm{pA}, V_{\mathrm{mod}}=1\mathrm{mV}$).

Figure 10

(a) Topographic image of the CeSb termination (T2) recorded at positive bias voltage, $V_{\mathrm{S}}=50\mathrm{mV}$, and (b) at negative bias voltage, $V_{\mathrm{S}}=-50\mathrm{mV} [(10\times 10){\mathrm{nm}}^2, I_{\mathrm{s}}=40\mathrm{pA}]$. (c) The difference between the images shown in (a) and (b), and (d) its Fourier transform.

Figure 11

Magnetostriction data obtained at $2.2\mathrm{K}$ on a ${\mathrm{CeSb}}_2$ sample and the same measurement recorded on a gold sample for reference.

Figure 12

STM magnetostriction curves recorded at constant temperatures throughout the magnetic phase diagram of ${\mathrm{CeSb}}_2$. There are clear differences in the magnetostriction in the different phases. The measurements are in the low temperature ferrimagnetic phase below $6.5\mathrm{K}$, antiferromagnetic phase between $6.5\mathrm{K}$ and $\sim 11\mathrm{K}$ and the magnetically ordered and paramagnetic phases above $11\mathrm{K}$ and $15.6\mathrm{K},$ respectively.

Figure 13

(a) Zero-field cooled SQUID magnetization measurements $M\left(H\right)$ recorded with different applied fields. Derivatives, $dM/dH$, of which are shown in Fig. 3 of the main text. Points associated with phase transitions are highlighted. (b) STM tip position versus temperature curves. Curves are vertically offset for clarity.

Figure 14

(a) Blue curve and axis: Isothermal magnetostriction data measured in the low-temperature ferrimagnetic phase at $4\mathrm{K}$. Orange curve and axis: Corresponding magnetization data. Field is swept in the direction of the arrow. (b) The derivatives of the data shown in (a) with respect to the applied field. (c) Blue curve and axis: Isothermal magnetostriction data measured in the low temperature antiferromagnetic phase at $8\mathrm{K}$. Orange curve and axis: corresponding magnetization data. (d) The derivatives of the data shown in (c) with respect to the applied field.