Topological Phase Transition in an Archetypal f-electron Correlated System: Ce

TL;DR

This paper reveals a topological phase transition in elemental Ce associated with the gamma-alpha transition, driven by f-d band inversion and coherent 4f band formation, controllable by pressure.

Contribution

It uncovers a topological phase transition in Ce linked to the gamma-alpha transition, demonstrating a pressure-controlled 'topology switch' in a Kondo system.

Findings

Topological phase transition occurs in Ce during gamma-alpha transition.

The transition is driven by f-d band inversion and 4f band formation.

Pressure around 1 GPa can switch the topological state at room temperature.

Abstract

A typical f-electron Kondo lattice system Ce exhibits the well-known isostructural transition, the so-called gamma-alpha transition, accompanied by an enormous volume collapse. Most interestingly, we have discovered that a topological-phase transition also takes place in elemental Ce, concurrently with the gamma-alpha transition. Based on the dynamical mean-field theory approach combined with density functional theory, we have unravelled that the non-trivial topology in alpha-Ce is driven by the f-d band inversion, which arises from the formation of coherent 4f band around the Fermi level. We captured the formation of the 4f quasi-particle band that is responsible for the Lifshitz transition and the non-trivial Z2 topology establishment across the phase boundary. This discovery provides a concept of 'topology switch' for topological Kondo systems. The 'on' and 'off' switching knob in Ce is versatile in a sense that it is controlled by available pressure (around 1 GPa) at room temperature.

Figures

Full text

††preprint: J. Kim $et~{}al.$

Physics of strongly-correlated $f$-electron materials has been a longstanding subject of special interest due to complex interplay among the underlying interactions, such as strong Coulomb correlation, spin-orbit (SO) coupling, and the hybridization of the localized $f$ and conduction electrons. More intriguing is that the interplay is very sensitive to small changes in the external parameters. Elemental Ce, which has one occupied $f$ electron in its atomic phase, is a prototypical $f$-electron Kondo lattice system exhibiting such sensitivity. Indeed Ce shows a rich phase diagram (see Fig. 1) and many interesting physical properties as a function of temperature ($T$) and pressure ($P$) mag_suscep ; qc1 ; qc2 ; superconduct . The first-order isostructural volume-collapse transition from ${\gamma}$ to ${\alpha}$ phase of face-centered cubic (fcc) Ce is the most representative phenomenon that experiences the sensitivity. However, the driving mechanism of the ${\gamma}$-${\alpha}$ transition is still under debate, between the two well-known models: Mott transition Mott_transition vs. Kondo volume collapse Kondo_Volume_Collapse1 . The current consensus is that there exists at least a significant change in the Kondo hybridization between the localized $4f$ electrons and conducting electrons across the transition Magnetic_form_factor ; DMFT_f-local ; dmft_opt_cond_Ce ; Haule15 . This peculiarity in Ce could facilitate the emergence of nontrivial topology in the ground-state ${\alpha}$-phase of Ce.

In a recent theoretical work on topological Kondo insulator Coleman_KTI , it is shown that the Kondo hybridization in $f$-electron systems can play an important role in the formation of non-trivial topology. Since then, many subsequent studies have been reported to search for non-trivial topological materials, where the Kondo hybridization gap exists, e.g., CeNiSn, CeRu4Sn6, Ce3Bi4Pt3, SmB6, SmS, and YbB12 Coleman_KTI ; CeRu4Sn6 ; Ce3Bi4Pt3 ; Fu13 ; Neupane13 ; SmS ; YbB12 . Despite extensive studies, however, the topological nature of mother elements, Ce, Sm, and Yb, supplying the correlated $f$-electrons to the above Kondo insulator compounds, has not been explored yet. Here we report, based on the dynamical mean-field theory (DMFT) approach combined with density functional theory (DFT) that has been successful to describe electronic structures of Ce and Ce compounds dmft_opt_cond_Ce ; DMFT_f-local ; CeIrIn5 ; CePd3 , that a narrow $f$-band metal ${\alpha}$-Ce has the non-trivial topology of topological-insulator (TI)-type and topological-crystalline-insulator (TCI)-type nature, and the topological phase transition and the Lifshitz electronic transition occur concomitantly with the ${\gamma}$-${\alpha}$ volume collapse transition in Ce.

Figure 2 shows the DMFT band structures and the densities of states (DOSs) of ${\gamma}$- and ${\alpha}$-Ce. In the DMFT calculations, we have used the Coulomb correlation ($U$) and the exchange ($J$) interaction parameters of $U=5.5$ eV and $J=0.68$ eV for the Ce $f$-electrons (refer to the supplement for the computational details) Supp . The $4f$ spectral weights of both phases have three main parts in common: the lower Hubbard band (LHB) at $-2.0\sim-2.5$ eV corresponding to the $4f^{0}$ final state, the upper Hubbard band (UHB) at $2\sim 4$ eV corresponding to the $4f^{2}$ final state, and the Kondo resonances near the Fermi level ($E_{\textrm{F}}$) corresponding to the $4f^{1}$ final states.

The energy positions of LHB and UHB are in good agreement with photoemission spectroscopy (PES) PES0 ; PES1 ; resonant_PES and inverse PES experiments inverse_PES .

One of the most notable features in Fig. 2 is that the spectral weight of the Kondo resonance around $E_{\textrm{F}}$ is much stronger in ${\alpha}$-Ce than in ${\gamma}$-Ce, and exhibits the coherent quasi-particle band feature in ${\alpha}$-Ce, as is consistent with previous PES PES0 ; PES1 ; resonant_PES ; Chen18 ; inverse_PES and theoretical reports Ce-dmft1 ; Ce-dmft2 ; Amadon15 ; Ce-dmft3 . As will be discussed below, these contrasting Kondo-resonance features between the two phases lead to the quite different topological classes: trivial and non-trivial Z2 topologies for ${\gamma}$-Ce and ${\alpha}$-Ce, respectively.

The incoherent and coherent $4f$ spectral weights for ${\gamma}$-Ce and ${\alpha}$-Ce, respectively, are more clearly shown in the amplified DMFT band structures in Fig. 3. It is seen in Fig. 3a that, for ${\gamma}$-Ce, $4f$ electrons are incoherent, and so mainly the $5d$ band crosses $E_{\textrm{F}}$, which agrees well with the optical spectroscopy result opt_spec_Ce . In contrast, for ${\alpha}$-Ce, the coherent 4$f$ quasi-particle band feature is evident near $E_{\textrm{F}}$ in Fig. 3b, which is the origin of the effective-mass enhancement of charge carriers and the change of the charge carrier character from $5d$ to $4f$. The coherent band feature for ${\alpha}$-Ce is corroborated by the fact that the DMFT bands have almost the same dispersion as the renormalized DFT bands rescaled approximately by 1/2 (dotted green-line in Fig. 3b).

The different electronic structures between the two phases are also reflected in the Fermi surfaces (FSs). The shapes of FSs in Fig. 3d are topologically different, suggesting that ${\gamma}$-${\alpha}$ transition

corresponds to the Lifshitz transition (see the supplement) Supp . It is noteworthy in Fig. 3d that, while the DMFT FS of ${\gamma}$-Ce is very close to that obtained from the DFT-opencore (“4f-opencore”) calculation considering the $4f$ electrons as core electrons, the DMFT FS of ${\alpha}$-Ce is quite similar to the DFT FS. These results indicate that, for ${\gamma}$-Ce, the contribution of 4$f$ electrons to the FS is negligible, and, for ${\alpha}$-Ce, the 4$f$ quasi-particle band at $E_{\textrm{F}}$ can be described properly by the DFT band (see Fig. S1 of the supplement) Supp .

The key ingredient that makes the difference has something to do with the degree of the renormalization factor (quasi-particle weight) $Z$, arising from the Coulomb correlation interaction of 4$f$ electrons. The renormalization factor $Z$ is obtained from the slope of the real-part of the self-energy $\Sigma(\omega)$ at $E_{\textrm{F}}$. As shown Fig. S4 of the supplement Supp , we have obtained qualitatively different behaviors of $\Sigma(\omega)$’s between the ${\alpha}$ and ${\gamma}$ phases, which produce quite distinct electronic structures and resulting physical parameters. Indeed, Fig. 3e shows that

$Z$ increases discontinuously across the ${\gamma}$-${\alpha}$ transition. As a result, both the hybridization strength ${\Delta}({\omega})$ (Fig. 3c) and the $f$-$f$ hopping strength, which are to be effectively proportional to $Z$, are enhanced for ${\alpha}$-Ce, which give rise to the enhanced 4$f$ spectral weight and help to form the coherent 4$f$ band around $E_{\textrm{F}}$ Amadon15 .

The evolution of the electronic structure across the ${\gamma}$-${\alpha}$ transition makes the elemental Ce more interesting in a topological sense.

The coherent quasi-particle band in ${\alpha}$-Ce, which, via the hybridization with the conduction band, brings about the hybridization gap in the ${\alpha}$-Ce phase, as indicated by gray-shaded area in Fig. 3b. The energy gaps are clearly seen at every time-reversal invariant momentum (TRIM) points of ${\Gamma}$, $X$, and $L$, while those at $W$ and in-between $L$-$\Gamma$ are barely gapped.

Then, with respect to the hybridization gap, the 5$d$ band of even parity and the 4$f$ band of odd parity are inverted at the TRIM point X. Since the crystal structure is symmetric under the inversion operation, the additional odd parity to the TRIM points yields the non-trivial Z2 topology of ${\alpha}$-Ce, as shown in Fig. 3f.

Figure 3e shows that the non-trivial Z2 topology in ${\alpha}$-Ce is established at the very starting edge (volume 28 Å3 at 293 K) of $\alpha$-phase in the $\gamma$-${\alpha}$ transition. Note that no gaps are present in the $\gamma$ phase, but the gaps at the TRIM points, $\Delta_{\Gamma}$ and $\Delta_{L}$ of about 30 meV ($\Delta_{X}>2$ eV), are suddenly developed in the $\alpha$ phase.

This implies that the topological phase transition would occur concomitantly with the $\gamma$ to $\alpha$ volume collapse transition. The more detailed evolution of the band structures across the $\gamma$-$\alpha$ transition is given along a $P$-$V$ isotherm at 293 K in Fig. S2 of the supplement Supp .

In order to confirm the non-trivial Z2 topological invariance of ${\alpha}$-Ce, we have performed the surface electronic structure calculations for the slab geometry of ${\alpha}$-Ce with (001) surface and explored the existence of topological surface states (TSSs) in Fig. 4a. Note that, as shown in Fig. 1, one $X$ point is projected onto $\bar{\Gamma}$, while two non-equivalent $X$ and $X^{\prime}$ are projected onto $\bar{M}$ of the (001) surface BZ. Indeed, as shown in Fig. 4a, the TSSs and corresponding Dirac points emerge in the indirect gap region at $\bar{\Gamma}$ (“D1”) and $\bar{M}$ (“D2”). Due to the bulk metallic nature of ${\alpha}$-Ce, most part of the Dirac bands at $\bar{M}$ is buried under the bulk-projected bands and so the band connectivity is not clear. Nevertheless, it is evident in Fig. 4a that the surface states along $\bar{\Gamma}$-$\bar{X}$ are the Dirac-cone states, because the lower surface band reaches the violet-colored band (“L”) below the indirect gap, while the upper one reaches the green-colored band (“U”) above the indirect gap. The helical spin texture of the corresponding Dirac cone FS around $\bar{\Gamma}$ and $\bar{M}$ in Fig. 4b and 4c also manifests the spin-momentum locking behavior, reflecting its topological nature.

The double Dirac points, which are supposed to be at $\bar{M}$ due to the projection of two non-equivalent X TRIM points (Fig. 1), are to be separated due to the hybridization between the bands of the double Dirac cones.

On the (001) surface Brillouin-Zone (BZ) of ${\alpha}$-Ce, there are two mirror symmetric lines, $\bar{\Gamma}$-$\bar{X}$ and $\bar{\Gamma}$-$\bar{M}$, as shown in Fig. 1, which could play a key role in realizing the TCI-type nature. It is thus obvious that the band crossing along $\bar{X}$-$\bar{M}$ that is not a mirror symmetric line would be gapped, but that along $\bar{M}$-$\bar{\Gamma}$ needs further consideration. However, the surface states along $\bar{M}$-$\bar{\Gamma}$ are completely buried under the bulk projected bands, and so it is not easy to identify the specific TCI-type band feature in Fig. 4a. In view of the surface states inside the dotted-black square and those along $\bar{X}$-$\bar{M}$ designated by red arrows in Fig. 4a, we just conjecture that the band crossing along $\bar{M}$-$\bar{\Gamma}$ would be gapped to have Rashba-type surface states, as reported for the golden phase of SmS ($g$-SmS) that is expected to have the same topological symmetry as ${\alpha}$-Ce CJKang15 . In fact, ${\alpha}$-Ce is found to have the same mirror Chern numbers as $g$-SmS CJKang2 , as shown in Fig. S5 of the supplement Supp .

We have also examined the TSSs for the (110) and (111) surfaces of ${\alpha}$-Ce. For the (110) surface, single and double Dirac points are expected to be located at $\bar{\Gamma}$ and $\bar{X}$, respectively, as shown in Fig. 1. For the (111) surface, only the single Dirac point is expected at $\bar{M}$, as shown in Fig. S6 of the supplement Supp . As shown in Fig. 4d and Fig. S6, however, neither (110) nor (111) surface states show a clear TI-type or TCI-type signature in the hybridization gap region, because, here too, most of the surface states near $E_{\textrm{F}}$ are buried under the bulk-projected bands. In this circumstance, for the (110) surface, one apparent TCI signature is seen at $\bar{X}$ near 240 meV in Fig. 4e, which demonstrated the gapped and protected Dirac points along $\bar{X}$-$\bar{S}$ (red arrow) and $\bar{X}$-$\bar{\Gamma}$ (black arrow), respectively. This suggests that the near-$E_{\textrm{F}}$ TSSs buried under the bulk-projected bands in Fig. 4f would also have the TCI-type band nature.

Our finding highlights that a typical narrow $f$-band metal ${\alpha}$-Ce is a topological Kondo system of TI- and TCI-type nature, and the “on” and “off” topology switch can be operative by using a $P$-tuning or $T$-tuning knob, accompanied by the first-order volume-collapse and Lifshitz transitions. So Ce would be an excellent test-bed for investigating the topological phase transition in $f$-electron Kondo lattice systems. It is thus highly desirable to explore the topological surface states in ${\alpha}$-Ce, preferentially for its (001) surface, by using high resolution angle-resolved PES measurement.

Acknowledgments$-$ We would like to thank J.D. Denlinger, J.-S. Kang, and J.H. Shim for helpful discussions. This work was supported by the NRF (Grant No. 2017R1A2B4005175, Grant No. 2018R1A6A3A01013431, Grant No. 2016R1D1A1B02008461), Max-Plank POSTECH/KOREA Research Initiative (No. 2016K1A4A4A01922028), the POSTECH BSRI Grant, and the KISTI supercomputing center (Grant No. KSC-2017-C3-0057).