Crystal electric field and properties of $4f$ magnetic moments at the surface of the rare-earth compound ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$

The crystal electric field (CEF) plays an essential role in defining the magnetic properties of $4f$ materials. It forces the charge density of $4f$ electrons and the related magnetic moment to be oriented along a certain direction in the crystal. The CEF and related magnetic properties were widely studied in the past with focus on bulk of $4f$ materials, while their surfaces have not received much attention. By the example of the antiferromagnetic material ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ and using first-principles calculations and classical $4f$ angle-resolved photoemission (PE) measurements, we show how the CEF and related magnetic properties, linked with the orientation of $4f$ moments, are modified at the surface region. Precisely, we studied the CEF characteristics in individual Tb layers for Tb- and Si-terminated surfaces of ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$. We show how strongly the CEF changes near the surface and how dramatically it influences the orientation of the $4f$ moments relative to the bulk. The instructive message of our study is that a rather valuable information about the CEF-related phenomena can be derived from the temperature dependence of $4f$ PE spectra. The presented methodology including the theoretical approach can be further applied to many other layered and quasi-2D rare-earth-based materials for unveiling their surface magnetic properties.

Figure 1

Schematic view of the tetragonal AFM ordered ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ crystal and its Tb and Si surfaces. The Tb atomic layers are well separated from each other by the tightly-bound Si-Rh-Si trilayer blocks. The $4f$ magnetic moments of Tb are shown by arrows. Reorientation of $4f$ moments at the Tb termination is caused by the changes of CEF at the surface (see discussion in text).

Figure 2

ARPES spectra taken from (a) Tb and (b) Si terminated surfaces of the AFM ordered ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ crystal at $T=21\mathrm{K}$ along the $\overline{\mathrm{M}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ direction of the surface Brillouin zone (SBZ). (c) Normal-emission PE spectra obtained by integration over the angle range of $\pm 5^{\circ }$ from the data shown in [(a),(b)]. (d) Theoretical normal-emission $4f$ PE spectra for different single-$M_J$ ground states calculated using Eq. (7). The numbers in brackets denote atomic layers from which the signal comes.

Figure 3

(a) CEF energy levels calculated for different Tb atomic layers (1st and 5th for the Tb surface, and 4th and 8th for the Si surface.) The CEF states in respective sequences are shifted horizontally and connected with lines for better visual discrimination of degenerate and closely lying states. (b) The deviation of the CEF potential from its maximal value ($V_{\mathrm{CEF}}^{\max }\left(R\right)-V_{\mathrm{CEF}}(R,\theta ,\varphi )$) in real space for parameters determined for the 1st Tb layer (Tb surface) and for the 4th Tb layer (Si surface) along with the electron density distribution (${\rho }_f$) for the states $|0\rangle$ and $|6\rangle$.

Figure 4

Temperature dependencies of the magnetic susceptibility $\chi$ along the [001] and [100] crystallographic directions. Our experimental data are compared to the calculated $\chi$ for the CEF parameters given in Table 1 and molecular field parameters from Table 2.

Figure 5

(a) Experimental ARPES data taken in the $\overline{\mathrm{M}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ direction of the SBZ from the Si-terminated surface of ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ at $T=195\mathrm{K}$ using $h\nu =200\mathrm{eV}$. (b) Difference between the ARPES data measured at 195 K and 21 K. (c) The respective angle-integrated PE spectra and their difference. (d,e) Calculated PE spectra at 21 K integrated over the polar angle ranges of $0\pm 3^{\circ }$ (d) and from 8 to ${15}^{\circ }$ (e). The intensity from the fourth Tb layer is shown separately from the signal of deeper layers denoted as bulk. (f) Calculated angle-integrated PE spectra and their difference. The calculated spectra in [(d),(f)] were obtained with DFT $+$ Si $3p$ CFPs. [(g),(h)] Measured temperature dependencies for the ratio of $4f$ PE intensities in the regions 1 and 2 shown in (c) are compared with the theoretical curves obtained for different CFPs. In the case of $h\nu =110\mathrm{eV}$ (g) the intensities were averaged over the angle range of $0\pm {15}^{\circ }$, while in the case of $h\nu =200\mathrm{eV}$ (h) we used two different angle regions: $0\pm 3^{\circ }$ and from 8 to ${15}^{\circ }$. Two sets of experimental data shown by differently colored circles in (g) and (h) were obtained from two ${\mathrm{TbRh}}_2{\mathrm{Si}}_2$ crystals.

Figure 6

[(a),(b),(c)] Experimental and calculated PED patterns in orthographic projection for several PE peaks shown in the spectrum (e). Dashed circles correspond to the emission angle of ${90}^{\circ }$. (d) $R$-factor dependence on the distance $d$ between the Tb and Si atomic layers on the Tb-terminated surface. (e) Angle-integrated PE spectrum of Tb multiplet. The data were obtained from mostly Tb-terminated surface at $T\approx 30\mathrm{K}$ and $h\nu =110\mathrm{eV}$. In (b) and (c) the calculated patterns were smoothed to simulate experimental angle resolution of $1^{\circ }$.