Experimental Determination of the Topological Phase Diagram in Cerium Monopnictides

Experimental determinations of bulk band topology in the solid states have been so far restricted to only indirect investigation through the probing of surface states predicted by electronic structure calculations. We here present an alternative approach to determine the band topology by means of bulk-sensitive soft x-ray angle-resolved photoemission spectroscopy. We investigate the bulk electronic structures of the series materials, Ce monopnictides (CeP, CeAs, CeSb, and CeBi). By performing a paradigmatic study of the band structures as a function of their spin-orbit coupling, we draw the topological phase diagram and unambiguously reveal the topological phase transition from a trivial to a nontrivial regime in going from CeP to CeBi induced by the band inversion. The underlying mechanism of the phase transition is elucidated in terms of spin-orbit coupling in concert with their semimetallic band structures. Our comprehensive observations provide a new insight into the band topology hidden in the bulk states.

Figure 1

(a) 3D Brillouin zone (BZ), showing the $k_z-k_x$ sheet at $k_y=0$. (b) Calculated bulk Fermi surfaces (FSs). (c1) SX-ARPES result for FS mapping on the $k_z\text{−}{k}_x$ plane [yellow plane in (a)] with different $h\nu$. (c2),(c3) SX-ARPES band maps cut along different $k_x$ crossing $X$ and $W$ [red and blue lines in (c1), respectively]. (d1) VUV-ARPES FS mapping on the $k_z\text{−}{k}_x$ plane by changing $h\nu$. (d2),(d3) VUV-ARPES band maps at different $k_z$ as shown in (d1). The $k_z$ value was obtained with the inner potential $V_0=12\mathrm{eV}$.

Figure 2

(a1)–(a3) SX-ARPES band maps for $\mathrm{Ce}X$’s cut along the $X\text{−}\mathrm{\Gamma }\text{−}X$ line obtained by the $h\nu$ of (CeAs) 590 eV, (CeSb) 530 eV, and (CeBi) 515 eV. (b1)–(b3) Comparison of (red and green) the $p_{3/2}$ and (black) $p_{1/2}$ energy dispersions around $\mathrm{\Gamma }$. The energy dispersions are determined by tracing the peak position of the momentum distribution curves (MDCs). (c) Experimentally determined strength of their SOC. The size was defined as the energy difference for the top of $p_{3/2}$ and $p_{1/2}$ bands at $\mathrm{\Gamma }$ [black arrows in (b1)–(b3)]. The top of the bands is deduced by fitting analysis with quartic function (solid lines). The energy positions of $X1$ and $X2$ bands are obtained by the results presented in Fig. 3.

Figure 3

(a1)–(a3) FS mapping on the $k_x\text{−}{k}_y$ plane at $k_z=0$ for $\mathrm{Ce}X$’s. (b1)–(b3) Enlarged SX-ARPES band maps of Figs. 2 and (inset) their MDCs at $E_F$. The hole (electron) pockets are shown by red (blue) arrows above the MDCs. (c1)–(c3) The energy distribution curves (EDCs) around $X$ within the $E-k_x$ window in (b1)–(b3), displaying (red circles) $X1$ and (black circles) Ce $t_{2g}$ dispersions. The EDC at $X$ is highlighted by the black line. (d1)–(d3) The calculated band structures around $X$ (Supplemental Material, Note 3 [36]). Red and blue represent the pnictogen $p$ and Ce $t_{2g}$ orbital contributions. For the calculation, we used the on-site-energy shift for Ce $t_{2g}$, ${\mathrm{\Delta }}_{t2g}=0.85\mathrm{eV}$ (CeAs), 0.60 eV (CeSb), and 0.54 eV (CeBi). (e) Experimentally determined topological phase diagram with the energy positions of (blue) Ce $t_{2g}$, (red) $X1$, and (green) $X2$ bands (Table S1 [36]). (Inset) The schematics of the band structures around $X$.

Figure 4

(a) (001) surface BZ of CeBi. (b) Schematics of the observed surface dispersions induced by the parity inversion within the bulk projection gap. (c) Calculated (001) surface band structures along $\overline{\mathrm{\Gamma }}\text{−}\overline{\mathit{M}}$ line [blue line in (a)]. For our semi-infinite slab calculations, two CeBi layers are regarded as a unit for the calculation scheme of the surface Green’s function presented in Ref. [38], and so the surface spectral weight is defined for the top two CeBi layers. To fit our SX-ARPES data, $E_F$ in the calculation results is shifted to be 0.10 eV. (d) Enlarged two surface states around $\overline{M}$ point. $s1$ and $s2$ label the surface bands (Note 5 [36]). (e) VUV-ARPES band map cut along $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ with $h\nu =55\mathrm{eV}$. The dashed lines guide the inverted bulk-band dispersions taken from Fig. 3. The signals from $s1$ and $s2$ are indicated by arrows. (f) The MDCs around $\overline{M}$. The observed $s1$ and $s2$ are guided by colored lines.