Topological properties and the dynamical crossover from mixed-valence to Kondo-lattice behavior in golden phase of SmS
We have investigated temperature-dependent behaviors of electronic structure and resistivity in a mixed-valent golden phase of SmS, based on the dynamical mean-field theory band structure calculations. Upon cooling, the coherent Sm bands are formed to produce the hybridization-induced pseudogap near the Fermi level, and accordingly the topology of Fermi surface is changed to exhibit a Lifshitz-like transition. The surface states emerging in the bulk gap region are found to be not topologically protected states but just typical Rashba spin-polarized states, indicating that SmS is not a topological Kondo semimetal. From the analysis of anomalous resistivity behavior in SmS, we have identified universal energy scales, which characterize the Kondo/mixed-valent semimetallic systems.
pacs:71.27.+a, 71.18.+y, 75.30.Mb
Kondo and mixed-valent physics in strongly-correlated electron systems have been subject of longstanding controversy. It includes various interesting phenomena such as -wave superconductivity and non-Fermi-liquid behavior in heavy-fermion systems Stewart84 (); Coleman01 (); Gegenwart08 (), and recently proposed topological Kondo insulator behavior in a typical mixed-valent insulator SmB Dzero10 (); Takimoto11 (); Dzero12 (); Kang13 (); Xu13 (); Neupane13 (); Kim14 (); Kim13 (); Kim13-2 (); Jonathan13 (); Chmin14 (). The topological Kondo insulator of our present interest has attracted a great deal of recent attention. However, the realization of the topological properties in SmB is still under debate. The immediate question was addressed whether a similar mixed-valent system, SmS, also has the topological properties or not. Indeed, the mixed-valent golden phase of SmS (g-SmS) was reported to be a topological Kondo semimetal Li14 ().
SmS has been studied for last four decades Varma76 (); Villars91 (); Campagna74 (); Coey76 (); Maple71 (); Jayaraman70 (); Deen05 (); Shapiro75 (); Martin79 (); Barla04 (); Imura09 (); Ito02 (); Mizuno08 (); Matsubayashi07 (); Flouquet04 (); Wachter94 (); Antonov02 (); Svane05 (); Lapierre81 (); Konczykowski81 (); Konczykowski85 (); Imura09-2 (), but there remain several issues still unresolved. SmS crystallizes in a face-centered cubic (fcc) structure of rock-salt (NaCl-type) type. At the ambient pressure, SmS has a so-called black phase, which is a semiconductor with indirect and direct band gaps of 90 meV and 0.4 eV, respectively Mizuno08 (). In the black phase of SmS (b-SmS), the valence state of Sm is divalent (2+), and so the system is nonmagnetic. Under high pressure above 6.5 kbar, SmS undergoes a first-order isostructural phase transition from b-SmS to g-SmS, in which Sm ions are mixed-valent with the average valency of 2.6+ 2.8+ Coey76 (); Maple71 (). This isostructural transition is accompanied by the volume collapse as much as 15% Maple71 (), as shown in Fig. 1(b).
The energy gap decreases monotonically with increasing the pressure. It is controversial whether g-SmS has a real gap or a pseudo-gap Matsubayashi07 (); Wachter94 (). The resistivity behavior of g-SmS is quite anomalous in the sense that the overall behavior is Kondo-lattice like but it exhibits a couple of abnormal hump structures Lapierre81 (); Konczykowski81 (); Konczykowski85 (); Imura09 (); Imura09-2 (). Applying the pressure further, g-SmS has a magnetic instability at about 19.5 kbar with the antiferromagnetic order Shapiro75 (); Barla04 (); Imura09 (). Above 19.5 kbar, the resistivity shows a metallic behavior and the valence state of Sm increases toward 3+ Imura09 ().
There have been several reports on the density-functional theory (DFT)-based band structure study of SmS Antonov02 (); Svane05 (); Li14 (). But even the ground-state insulating nature of b-SmS is not properly described by the DFT-based schemes, and so the advanced methods like the dynamical mean-field theory (DMFT) should be employed to investigate the topological properties in g-SmS that has the strongly-correlated 4 electrons. Here we have investigated electronic structures of SmS, based on the DMFT scheme. First, we have shown that the electronic properties of b-SmS are described properly only by the DMFT scheme. Then we have examined the -dependent electronic structure evolution in g-SmS. Upon cooling, the states form the coherent bands with a pseudo-gap feature near the Fermi level (), and accordingly the topology of the Fermi surface (FS) is changed, which is reflected well in the anomalous resistivity behavior in g-SmS. We have demonstrated that the surface states realized in g-SmS are not the topological states, but are just the typical Rashba states.
We have employed the all-electron FLAPW band method implemented in Wien2k Wien2k (). We have checked that both the DFT and the DFT+ (on-site Coulomb ) schemes can not describe the ground state insulating electronic structure of SmS properly (see the supplement Supp ()). Therefore, we have employed the combined DFT and DMFT (DFT+DMFT) approach implemented in Wien2k, which has successfully reproduced many aspects of the strongly-correlated electron systems Kotliar06 (); Haule10 (). We used projectors in the large window of 10 eV, and the on-site Coulomb and exchange energies of = 6.1 eV, = 0.8355 eV were adopted to fit in X-ray photoemission spectroscopy (XPS) data (see Fig. S2 for the DMFT band structure of b-SmS in the supplement) Campagna74 (). To solve the impurity problem, the non-crossing approximation (NCA) is used Jonathan13 (); Kim14 (). To verify the justification of the NCA scheme, we have also used the continuous time quantum Monte Carlo scheme for b-SmS and checked that two schemes give the same result.
The DMFT band structure and DOS of g-SmS ( = 5.6 ) at K are shown in Fig. 2(a). Notable is the flat Sm bands near , which yield sharp Kondo resonance-like peaks in the DOS (see Fig. 3(a)). The -band in the vicinity of is mainly of character (), while bands are located at about 0.17 eV from . These coherent Sm bands hybridize strongly with Sm band to produce the hybridization gap near (see Fig. 2(c)). However, the () band at is above , and so the band structure exhibits the metallic nature having the mixed-valent state of Sm. The band is dispersive with the band width of about 0.03 eV, as in SmB Kang13 (). The DMFT band structure near is analogous to the band structure obtained by the DFT+SOC (SOC: spin-orbit coupling), as shown in the supplement Supp (), but the band width of the former is nearly ten times smaller than that of the latter, which is also similar to the case in SmB Kang13 (); Kim14 ().
Even though g-SmS has metallic nature, the band inversion occurs at , and so it is tempting to anticipate the topologically protected surface states in SmS Li14 (). To check the topological property, we have examined the surface band structure in Fig. 2(b), which was obtained by the model slab calculation based on the DFT bulk band structure Slab (). The surface states, however, have a tiny gap instead of a Dirac cone and the momentum-dependent splitting of spin states, which reveals that they are not topologically protected states but just Rashba spin-polarized surface states. This result is a contrast to that of Li et. al. Li14 (), who argued that g-SmS is to be a topological semimetal. The charge gap protection is essential to have the topological nature. In this respect, La-doped SmS can be a better candidate for a topological Kondo insulator, because SmLaS under pressure was reported to be an excitonic insulator with an energy gap of 1 meV Wachter95 ().
Figure 2(c) shows the -dependent band structures of g-SmS ( = 5.6 ). At K, only the Sm -band is seen to cut . Upon cooling, Sm spectra start to emerge near at K, and bands are observed to be formed below K. At K, the hole bands near ( quartet) forms the coherent band, and the hybridization gap between Sm () and () bands begins to appear at the crossing point near . The coherence temperature is usually defined in metallic Kondo lattices as an onset , at which the hybridization gap appears Choi12 (). Note that is equivalent to that is introduced in the two-fluid model for the Kondo lattice Yang08 (). Hence K is considered to be of g-SmS. In the Kondo lattice systems, can often be identified from the peak position of the resistivity, as will be discussed in Fig. 3(c). With decreasing further, the separation between the upper flat band ( doublet) along and the lower flat band (/ quartet) becomes enhanced, and eventually the former is shifted up above at K. Below K, it is seen that the band becomes completely coherent over the whole Brillouin zone. In this case, the hole band is still above , and so SmS at very low would exhibit metallic nature (see Fig. 3(c)).
Figure 2(d) presents the -dependent FS evolution in g-SmS. Also presented is the DFT-FS, for comparison, which has -centered electron ellipses and -centered hole pockets. At K, the DMFT-FS arises solely from -band, which produces the -centered ellipses. Upon cooling, the ellipses become reduced more and more, and so almost disappear at K. Their spectral weights, however, become enhanced due to the contribution from the hybridized band so as to have maximum intensity at K. On the other hand, -centered hole pockets originating from () band begin to appear at K, and are clearly manifested below K. Therefore, the topology of the FS is to be changed twice, at K and K. The former is due to the emergence of the coherent band formation at , while the latter is due to the separation of the upper flat band ( doublet) from .
This behavior is reminiscent of the Lifshitz transition that is a typical continuous quantum phase transition characterized by the topological change of the FS Lifshitz60 (). The feature in SmS is quite interesting because a Lifshitz-like transition occurs with the variation of . Such a Lifshitz-like transition in g-SmS could be explored by ARPES and de Hass-van Alphen experiments. The change of the FS topology is well reflected in the -dependent resistivity behavior in Fig. 3(c).
Upon cooling, the DOS peak at meV becomes sharper and sharper, as shown in Fig. 3(a), resembling the -dependent evolution of Kondo resonance. Noteworthy is the pseudo-gap feature near , which arises from the - band hybridization below K, which is close to K. It is also seen that, upon cooling, the DOS at increases first and then decreases below K Supp (). The DOS near eV, which corresponds to the spin-orbit split side-band, behaves similarly to the DOS upon cooling.
The number of -electrons () in g-SmS versus is presented in Fig. 3(b). Upon cooling, increases monotonically from (trivalent state) at high to (mixed-valence state) at low . The increasing trend of upon cooling has also been observed in SmB Jonathan13 (); Mizumaki09 (). It is worthwhile to observe that curve has an inflection point at K, which indicates that the effective valence-transition (VT) occurs at K.
Figure 3(c) shows the resistivity versus of g-SmS evaluated in the DMFT scheme Choi12 (). The overall behavior is that the resistivity increases upon cooling, but decreases below K, as in metallic Kondo lattice systems. However, the detailed -dependent behavior is quite anomalous. The calculated resistivity has a broad maximum at K, and starts to increase again at K to produce hump and dip structure at K and K, respectively. Upon further cooling, it exhibits another maximum at K. Quite similar behavior is indeed observed in the measured resistivity of g-SmS Lapierre81 (); Konczykowski81 (); Konczykowski85 (); Imura09-2 (), as shown in Fig. 3(c), even though the feature at K is not so obvious.
The anomalous behavior of the resistivity is closely correlated with the -dependent evolution of electronic structure in Fig. 2(c)-(d). Namely, the broad maximum at K is considered to arise from the dynamical valence-fluctuation (VF), as manifested by the effective VT in Fig. 3(b) and the incoherent Sm spectra in Fig. 2(c) at K. Intriguingly, g-SmS has a minimum volume at K Iwasa05 (), which signifies the close connection of a broad maximum in the resistivity with the mixed-valent nature of SmS. Hence we assign K as . Interestingly, SmB also exhibits a similar broad maximum feature in the resistivity Kebede96 () and a minimum volume Trounov93 () near K. The resistivity increases below K, at which the coherent -band starts to emerge in Fig. 2(c). The resistivity hump at K indicates that is around 26 K. In conventional metallic Kondo lattices, the resistivity decreases monotonically to zero below . The behavior near was explained in the two-fluid model by a crossover from the Kondo spin-liquid (KSL) to Kondo Fermi-liquid (KFL) Yang08 (). We use a term of Kondo-liquid (KL) in Fig. 3(c) to comprise both KSL and KFL phases.
Thus g-SmS exhibits a crossover from mixed-valence to Kondo lattice behavior upon cooling Coleman07 (); Kumar11 (). The charge-fluctuation that starts to be effective at K becomes almost frozen at K, as shown in Fig. 3(b). Then, the spin-fluctuation becomes dominating near K, so as to activate the Kondo screening. In fact, the effective hybridization obtained in the DMFT becomes the largest near K Supp ().
Meanwhile, K of the dip structure coincides with , at which the upper flat band ( doublet) along is detached from . Thus the resistivity up-turn occurs due to the apparent pseudo-gap (PG) feature, and so we assign K as . Actually, this kind of hump and dip structure in the resistivity has been observed in several Ce compounds of Kondo-insulator type, such as CeNiSn and CeRhSb Takabatake90 (); Malik91 (). The third resistivity drop at K occurs due to the complete formation of the coherent band over the whole Brillouin zone. Below K, the imaginary part of self-energy almost vanishes Supp (), whereby the crossover from the KL to the Landau Fermi-liquid (LFL) takes place Yang08 (). So we assign K as . Therefore, SmS undergoes several crossover transitions upon cooling, from VF liquid (VFL), KL, PG to LFL, as shown in Fig. 3(c). The energy scales , , , and , which are identified for SmS, are considered to be universal to characterize Kondo/mixed-valent semimetallic systems.
Finally, it should be pointed out that the resistivity behavior in SmS is quite different from that in SmB that is a candidate of topological Kondo insulators. g-SmS is not a Kondo insulator but close to a Kondo semimetal, so that the conventional metallic resistivity behavior is expected to be realized at very low .
In conclusion, in g-SmS, the coherent Sm bands are formed upon cooling to produce the hybridization-induced pseudogap feature near , which is accompanied by a Lifshitz-like topological transition in the FS. g-SmS is found to be not a topological Kondo semimetal in view of that the in-gap surface states are not topological states but are typical spin-polarized Rashba states. From the analysis of -dependent resistivity of g-SmS, we have identified characteristic multiple energy scales, which are expected to govern the physics of Kondo/mixed-valent semimetallic systems universally.
Helpful discussions with Junwon Kim, J. H. Shim, B. H. Kim, Ki-Seok Kim, J. -S. Kang, J. D. Denlinger, J. W. Allen, and K. Sun are greatly appreciated. This work was supported by the NRF (No. 2009-0079947 and No. 2011-0025237), POSTECH BK21Plus Physics Division and BSRI grant, and the KISTI supercomputing center (No. KSC-2013-C3-010).
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Topological properties and the dynamical crossover from mixed-valence to Kondo-lattice behavior in golden phase of SmS
Hong Chul Choi Kyoo Kim
B. I. Min
Appendix A A black phase of SmS (b-SmS)
a.1 DFT band structure
For the DFT calculations, we have employed the all-electron FLAPW band method implemented in Wien2k Wien2k (). We used 17 17 17 -point mesh in the full Brillouin zone. The muffin-tin radii were set to 2.50 and 2.30 a.u. for Sm and S, respectively, and the product of and the maximum reciprocal lattice vector was chosen as . We used the maximum value of 10 for the waves inside the atomic spheres and the largest reciprocal lattice vector of 12 in the charge Fourier expansion.
Figure S1 shows the band structures and densities of states (DOSs) of a black phase of SmS (b-SmS) obtained by the DFT schemes. Figure S1(a) shows symmetry decomposed band structure of b-SmS in the GGA + SOC scheme (SOC: spin-orbit coupling). Since Sm 4-electrons feel much larger SOC than the cubic crystal field, the SOC bases incorporating the cubic crystal field should be utilized Kang13 (); Pappalardo61 (). The cubic crystal field splits states at into lower doublet and higher quartet, as in SmB Kang13 (). Sm 4 bands are dominant near the Fermi level (), and and states are located mainly below and above , respectively. The splitting between and states is about 0.75 eV, which is related to the strength of the SOC of 4 electrons in Sm ion. In fact, most of states are to be shifted up in energy due to the strong correlation effect of 4-electrons.
Sm characters are also shown in order to visualize the hybridization between the doublet and band. Since the lobes of and orbitals of Sm electrons are along and away from the anion sulfur ions, respectively, the energy of band is to be lower than that of band. This feature in SmS is contrary to that in SmB, for which the band is lower in energy than the band because the lobes of orbitals are toward anion boron ions Kang13 (). Therefore, in SmB, the - hybridization occurs between the doublet and band, while, in SmS, it occurs between the doublet and band. At , the band is located below bands, and so the order of parities is changed. The valence state of Sm in the GGA + SOC scheme is estimated to be 2.56+, which is far from the semiconducting black phase of the valence state of Sm, 2+. Furthermore, the band gap is not obtained which is contrary to the experiment.
Figure S1(b) presents the symmetry-projected partial DOSs (PDOSs) of b-SmS in the GGA + SOC scheme. Sm DOS is projected into the relativistic double group bases and Sm and PDOSs are also provided. It is seen that the DOS at is not zero, implying that the system is not an insulator. Rather it is semimetallic, as shown in Fig. S1(a).
Figure S1(c) and (d) show electronic structures of b-SmS in the GGA + SOC + scheme (: Coulomb correlation of Sm electrons). When eV, it gives an insulating phase with the band gap of 87 meV, which is quite comparable to the experimental band gap of 0.15 eV Varma76 (). Once the semiconducting phase of SmS is obtained, the parity inversion does not happen, which results in the trivial topological number. The valence state of Sm is estimated to be 2.23+, which is close to 2+. However, the band symmetries at high symmetry -points are different from those in the GGA + SOC scheme, which turns out to be wrong in view of the DMFT results (see Fig. 2(c) in the main text).
Figure S1 (e) and (f) show symmetry decomposed band structure and PDOS of b-SmS in the GGA + SOC with 10 times enhanced SOC of Sm -electron (10-SOC scheme). Due to the artificially large SOC strength, the bands are shifted up to the higher energy side. In contrast to the GGA + SOC + scheme, the band symmetries at high symmetry -point in the 10-SOC scheme are consistent with those of the GGA + SOC scheme. Sm band hybridizes with doublet, resulting in the hybridization gap near . The valence state of Sm is estimated to be 2.27+.
Thus the DFT schemes can not describe the ground state electronic structure of b-SmS properly. Also, according to the angle-resolved photoemission spectroscopy (ARPES) experiment Ito02 (), Sm bands are very flat and distributed broadly from 1 to 4 eV below , which is different from the DFT results. This is due to the strong correlation effect in electrons, which cannot be captured in conventional DFT calculations.
a.2 DMFT band structure
Figure S2 shows the band structure and DOS of b-SmS ( = 5.9 ) obtained by the charge self-consistent DFT + DMFT scheme at K. It is seen that the insulating electronic structure is correctly described by the DMFT scheme. The obtained band gap and the valence configuration of Sm are 0.46 eV and 2.06+, respectively. The band structure in Fig. S2 is very similar to that obtained by the periodic Anderson model calculation Lehner98 (), and consistent with ARPES data Ito02 (). The partial DOS in Fig. S2b shows 4 multiplets, which are in good agreement with XPS data Campagna74 (). Since the Sm -band is above the Sm -band, b-SmS is a topologically trivial insulator.
Appendix B A golden phase of SmS (g-SmS)
b.1 Band structure in the DFT + SOC with 10-SOC scheme
Figure S3 provides electronic structures of g-SmS ( = 5.6 ) in the GGA + SOC with 10-SOC scheme. The electronic structures in Fig. S3 are quite similar to those of g-SmS described by the DMFT at low (see Fig. 2(c) in the main text). But, the number of -electron () in the DFT + SOC with 10-SOC scheme is 5.64 (Sm), while in the DMFT scheme is 5.27 (Sm).
b.2 Temperature-dependent DMFT physical parameters
Figure S4 presents the temperature ()-dependent physical properties of g-SmS ( = 5.6 ) described by the DMFT. Figure S4(a) and (b) show the renormalization factor and effective mass versus . Upon cooling, of () increases (decreases), resulting in decreasing (increasing) of (). Relaxation times of both and diverge upon cooling, as shown in Fig. S4(c), suggesting the crossover to the Landau Fermi liquid behavior at low .
Figure S4(d) provides the -dependent behavior of the hybridization function at (), which is closely related to the Kondo interaction strength. Hence the largest value near K suggests that Kondo screening behavior becomes strongly enhanced at this temperature range. On the other hand, the total DOS at in Fig. S4(e) increases first and then decreases below K. Note that the total DOS at is saturated at low-, indicating the full lattice coherence below K.
Fig. S4(f) shows the resistivity versus obtained by the Drude model. The resistivity drop for is clearly shown below K, which is also captured in the DMFT result (see Fig. 3(c) in the main text). It implies that the crossover from the Kondo liquid to the Landau Fermi liquid occurs below K.
The images for moving pictures containing -dependent band structures and Fermi surfaces (T-band-movie1 and T-FS-movie2) are provided for the additional Supplementary Information.
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