Studies of Electronic Structure across a Quantum Phase Transition in CeRhSb 1−x Sn x
R. Kurleto, J. Goraus, M. Rosmus, A. Ślebarski, P. Starowicz
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Eur. Phys. J. B
This is a post-peer-review, pre-copyedit version of this article.The final version is available online at: https://doi.org/10.1140/epjb/e2019-100157-3.
Studies of Electronic Structure across a Quantum PhaseTransition in CeRhSb − x Sn x R. Kurleto , J. Goraus , M. Rosmus ,A. ´Slebarski , P. Starowicz a,1 Marian Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków,Poland Institute of Physics, University of Silesia, ul. 75 Pułku Piechoty 1A, 41-500 Chorzów, PolandPublished in Eur. Phys. J. B, September 2019
Abstract
We study an electronic structure of CeRhSb − x Sn x system, which displaysquantum critical transition from a Kondo insulator to a non-Fermi liquid at x = . / final state is detected. Spectral intensity at the Fermi edge has a general ten-dency to grow with Sn content. Theoretical calculations of band structure are realizedwith full-potential local-orbital minimum-basis code using scalar relativistic and fullrelativistic approach. The calculations reveal a depletion of density of states at theFermi level for CeRhSb. This gap is shifted above the Fermi energy with increas-ing Sn content and thus a rise of density of states at the Fermi level is reflected inthe calculations. It agrees with metallic properties of compounds with larger x . Thecalculations also yield another important effects of Sn substitution. Band structure isdisplaced in a direction corresponding to hole doping, although with deviations froma rigid band shift scenario. Lifshitz transitions modify a topology of the Fermi surfacea few times and a number of bands crossing the Fermi level increases. Quantum phase transition (QPT) is a matter of particular interest because it is relatedto an instability of a ground state [1–3]. Heavy fermions are a perfect playground forstudying quantum criticality. The transition from one to another type of the groundstate is associated with non-Fermi liquid behavior. Quantum critical point (QCP)which separates metallic phase from so called Kondo insulator (KI) has attractedparticular attention so far [4–6]. KIs are materials with a narrow gap or a pseudo-gap in an electronic structure, which opens due to hybridization between correlatedelectrons and conduction band [6–8].CeRhSb − x Sn x is an example of the system with QCP. Representatives of CeRhSb − x Sn x system crystallize in orthorombic ε -TiNiSi structure (Pnma space group) for x < . a e-mail: [email protected] and in hexagonal Fe P (P62m space group) for x > . . ≤ x ≤ . x = x c = .
13, which is associated with QCP [6]. For x > x c the systembecomes metallic with non-Fermi liquid ground state [9, 14]. Thorough studies ofphysical properties of CeRhSb − x Sn x in the vicinity of the transition have been madeso far [6, 8, 12]. According to them a universal scaling law ρ · χ = const ( ρ - electricalresistivity, χ - magnetic susceptibility) is obeyed in KI phase ( x < .
13) [9]. A singu-lar behavior of magnetic susceptibility was observed for x > x c , which is related to anon-Fermi liquid formation [14]. At a higher value of temperature there is a crossoverfrom a non-Fermi-liquid to a Fermi liquid [6]. Strong valence fluctuations have beenfound in all synthesized representatives of CeRhSb − x Sn x system [12]. Such instabil-ities of Ce valency are believed to play an important role in the pseudogap formationfor x < x c [15].Photoelectron spectroscopy (PS) combined together with theoretical modelingof a band structure is a powerful tool in studying effects of hybridization in heavyelectrons systems. Several studies devoted to the electronic structure of KIs havebeen conducted with application of PS, so far [16–19]. Particularly, the abrupt stepbehavior of the single ion Kondo scale ( T K ) at a critical point has been inferred fromphotoemission data [19].Both, CeRhSb and CeRhSn compounds were studied with application of PS. Incase of CeRhSn [20, 21], an enhancement of signal attributed to 4f / final state,together with a small intensity of 4f feature was interpreted as a symptom of va-lence fluctuations [20, 21]. Valence band (VB) spectra of CeRhSb obtained previ-ously show not only characteristics of a cerium compound with Kondo effect: f , f / and f / features but also a depletion of the spectral weight at ε F due to the hybridiza-tion gap formation below 120 K [22–25].Transition from KI in CeRhSb to a metallic state in CeRhSn via the critical pointwas studied by means of density functional theory (DFT) calculations before [26].It was shown that QPT is manifested by a change of Mulliken occupation of Ce 5dstates as a function of x . Indeed, DFT calculations properly reproduced experimentalvalue of critical Sn concentration ( x c =0.13) and it was testified that atomic disorderis not a necessary factor for the formation of QCP. However, such a disorder leads tothe Griffiths phase, which was evidenced for CeRhSn [13, 14, 20, 27]. The schematicphase diagram of CeRhSb − x Sn x system in T − x plane has been proposed so far [9].The metallization of a system was explained in terms of the model, which takes intoaccount both, itinerant character of Ce 4f electrons and collective Kondo singlet state.A transition from KI to metallic state is related to destruction of such a collectivesinglet state accompanied by delocalization of 4f electrons.In this paper we present studies of the electronic structure of CeRhSb − x Sn x per-formed as a function of hole doping x in a light of QPT. VB spectra collected withapplication of PS are confronted with the results of theoretical modelling. DFT calcu-lations show that the increase of Sn content in CeRhSb − x Sn x results in a growth ofdensity of states at the Fermi energy ( ε F ), which is observed also in the experiment. The theoretical results also yield a sequence of Lifshitz transitions in Fermi surfaceand band structure.
Description of synthesis and characterization of polycrystalline samples of CeRhSb − x Sn x ( x =
0, 0.06, 0.16, 1) was provided elsewhere [6, 13]. The quality of used sampleswas checked with application of x-ray diffraction and energy dispersive x-ray spec-troscopy (EDXS). Diffraction patterns testified that the specimens subjected to thescrutiny consist of single phase. Good homogenity of samples was proven by meansof EDXS. The ultraviolet photoemission spectroscopy (UPS) data have been col-lected with application of a VG Scienta R4000 photoelectron energy analyzer. Mea-surements have been conducted at temperature equal to 12.5 K. Both, He II (40.8 eV)and He I (21.2 eV) radiation have been used. Base pressure in the analysis chamberwas equal to 4 · − mbar. Specimens were cleaned with diamond file in the prepara-tion chamber (base pressure 4 · − mbar) prior to the measurement. For comparisonwe also made measurements on samples which were cleaved in the ultra-high vacuumconditions, but results obtained in this way have got worse quality. Overall resolutionwas not greater than 30 meV.Partial densities of states (DOSes) have been simulated with application of full-potential local-orbital (FPLO) package [28–33]. Calculations in scalar relativisticscheme were performed with the virtual crystal approximation. Irreducible Brillouinzone (BZ) was divided for 343 k-points. Perdew-Wang exchange correlation poten-tial was used. Partial DOSes, band structure along high symmetry directions in thefirst BZ and Fermi surfaces have been resolved also with application of FPLO codein full relativistic approach.Photoelectron spectra have been simulated using calculated partial DOSes and as-suming atomic cross sections for photoionization [34]. Experimental broadening hasbeen introduced by a convolution with the gaussian function of the width of 10 meV. − x Sn x measured in a wide range of binding energy ( ε B ) are pre-sented in Fig. 1 a. Sn 4d / and 4d / core levels have been observed at ε B equalto − .
74 eV and − . x =
0, in He II spec-tra. Ce 5p doublet is visible for each investigated specimen. It consists of a humpedstructure at ε B ≈ −
21 eV, followed by a broad peak at − . ε B ≈ −
10 eV except forthe compound without Sb. Signal in this energy region observed for CeRhSn is com-pletely flat, so we can attribute this structure to Sb derived states. In fact, the positionof the maximum corresponds to the Sb 5s states. ε B ≈ − . − . ε F in He II measurements. The last structure ( ε B ≈ − . x . The peakcorresponding to higher absolute value of ε B is located at about − .
16 eV in caseof CeRhSb and remains in the same position for CeRhSb . Sn . , while in caseof CeRhSn and CeRhSb . Sn . it is found at about − .
25 eV. This effect may beconnected with a small contribution from Sn p and d states. The second structurealso displays a weak dependency on x . This maximum occupies roughly the sameposition, equal to − . − . ≈ − . x =
0, 0.06, 0.16 and 1respectively. The values obtained for CeRhSn differ from these obtained for otherstudied compounds, which is not startling, because CeRhSn adopts a different crystalstructure. Hence, it should have a different electronic structure as well.Shape of VB spectra obtained with application of He I radiation (also shownin the inset to Fig. 1 a) corresponds to the data recorded with He II lamp with twomajor peaks observed at similar binding energies. The spectrum is mainly relatedto Rh 4d states, what is supported by theoretical calculations discussed further inthe text. This is also consistent with previous PS measurements and theoretical cal-culations performed for metallic Rh [35, 36]. The maximum for CeRhSb is locatedat − . x = .
06) shifts the peak towards ε F ,next for x = .
16 the maximum is displaced in opposite direction and finally forCeRhSn its binding energy is − .
33 eV. These results would suggest a non-monotonicdependence with respect to x . However, these displacements are very close to exper-imental uncertainty. He-II spectra do not follow well these trends.High resolution measurements of vicinity of ε F are shown in Fig. 1 b. Spectrameasured with application of He II are compared with those obtained with He I. InHe II data one can see a broad maximum at about − . − . x . A blurred peak-like maximum is observed roughly at the same ε B = − . / final state [38]. I n t en s i t y [ a . u .] -25 -20 -15 -10 -5 0 E B [eV] a) I n t en s i t y [ a . u .] -1.2 -0.8 -0.4 0.0 E B [eV] I n t en s i t y [ a . u .] -1.2 -0.8 -0.4 0.0 E B [eV] f b)c) difference:"He II - He I" I n t en s i t y [ a . u .] -6 -5 -4 -3 -2 -1 E B [eV] He I - dashed lineHe II - solid lline CeRhSb CeRhSb Sn CeRhSb Sn CeRhSn
Fig. 1 (Color online) Valence band of the CeRhSb − x Sn x ( x =
0, 0.06, 0.16, 1) system studied by meansof ultraviolet photoemission spectroscopy. All measurements have been performed at T=12.5 K a) Spectraobtained with application of He II radiation in wide energy range. (Inset) Comparison of valence bandspectra of studied compounds obtained with He I and He II radiation. Solid vertical lines denote peakmaxima obtained with fitting the Gaussian function. b) High resolution spectra of valence band near ε F ,obtained with He I and He II radiation. c) Extracted 4f contribution to spectral function for each studiedcompound. The estimated Ce 4f spectral function of CeRhSb − x Sn x does not reveal any en-hancement of the intensity at ε F even for evidently metallic CeRhSn compound. Thisstands in a difference with some previous PES experiments performed on Kondo in-sulators, which often yield spectra which cannot be distinguished from that of weaklyhybridized Kondo metal [18], because of low experimental resolution or presence ofnonmagnetic impurities. Additionally, the lack of enhancement of signal at ε F canbe explained as follows. Firstly, theoretical calculations, which are discussed below,predict that Ce 4f partial DOS concentrates mostly above ε F for all possible concen-trations of Sn. Secondly, if the so called Kondo peak is present in spectral function ofCeRhSn it is located, according to the Friedel sum rule, above ε F [39]. PES probesspectral function within occupied states, making observation of any enhancement ofDOS almost infeasible in both situations.However, we believe that due to careful normalization of the spectra we can dis-cuss how the spectral intensity at ε F changes with substitution of Sn in place of Sb.Namely, one can see that for x = .
06 the signal at ε F increases by 13%, while in caseof x = .
16, spectral intensity at ε F decreases by 16%, with respect to x =
0. Intensity measured at ε F for CeRhSn is greater by about 33% than that for CeRhSb. Never-theless, samples with x = .
16 are likely to be more defected, which may explain thelower signal at ε F . Hence, we cannot conclude, if a reduction of DOS for x = . ε F with doping.3.4 Theoretical calculations - scalar relativistic approachTotal and partial DOS of CeRhSb − x Sn x have been calculated within scalar relativis-tic scheme for 0 ≤ x ≤ .
3. Calculated DOS has got a similar shape for each compo-sition. We present results for CeRhSb . Sn . (Fig. 2 a). VB is mainly composed ofCe 4f and Rh 4d states with admixture of Sn 5p and Ce 5d states. Partial DOSes aresymmetric for both spin directions, as expected, because we have not included ad-ditional correlations in computational scheme. Ce 4f spectrum concentrates mainlyabove ε F . Peak-like structure corresponding to 4f states is centered at about 0.5 eVand it extends roughly from ε F to 0.9 eV. In the close vicinity of ε F one can notice adepletion of Ce 4f DOS (Fig. 2 b). A "v-shape" gap is formed with non-zero DOS atminimum ( ≈ · unit cell)). The calculations show that such a gap is locatedat ε F for CeRhSb and is shifted above ε F with Sn doping. As a consequence, DOSat ε F is very low for a semimetallic CeRhSb and grows with x . Such a gap is alsopresent in Rh 4d and total DOS. It is in line with ”semiconducting” character of thesystem for 0 ≤ x ≤ x c . Such residual states in the gap are consistent with measure-ments of specif heat [40]. It is noteworthy, that Anderson lattice model at half fillingalso predicts a gap in 4f spectral function, as well as in a spectrum of carriers fromconduction band for KIs with momentum independent hybridization [41, 42].The photoelectron spectra were simulated using theoretical DOS (cf. the lowestpanel of Fig. 2 a). They reproduce the shape of the valence band obtained experimen-tally. It appears that the contribution from Rh 4d states dominates the valence bandspectra. Rh 4d states extend mainly from ε B equal to − ε F , located at about − − − − − − ≈ −
10 eV (Fig. 1 a). A leading contribution from Sb/Sn site to DOScomes from 5p states. These states are smeared over whole VB, with increased con-tribution in ε B range from about − − ε F .3.5 Theoretical calculations - full relativistic approachPartial DOS calculated in full relativistic approach for compounds with x =
0, 0.06,and 0.16, in principle has got a similar shape as this obtained within scalar relativistic -100-50050100 -5 -4 -3 -2 -1 0 1 E B [eV] -100-50050100 D O S [ s t a t e s / ( e V • un i t c e ll ) ] -10010-404 I n t en s i t y [ a . u .] Total DOS Ce Rh Sb/Sn CeRhSb Sn a) Experiment
He I He II
Theory
He I He II D O S [ s t a t e s / ( e V • un i t c e ll ) ]
420 -1.2 -0.8 -0.4 0.0 E B [eV] CeRhSb Sn Ce 4f
CeRhSb
Ce 4f b) Fig. 2 (Color online) Results of FPLO calculations with scalar relativistic approach. a) Partial densitiesof states calculated for CeRhSb . Sn . . The lowest panel presents a comparison between experimental(collected at T = . . Sn . . b) Ce 4f partial DOS calculated forCeRhSb . Sn . (upper panel) and for CeRhSb (lower panel). The black arrow indicates the position ofthe minimum in DOS which is discussed in the text. -80-4004080 -6 -4 -2 0 2 E B [eV] -40040 D O S [ s t a t e s / ( e V • un i t c e ll ) ] -15-10-5051015-4-2024 Total DOS Ce
4f 6s 6p 5d Rh
4d 5d 5s 5p
Sb/Sn
4d 5p 5s 5d
CeRhSb a) -80-4004080 -6 -4 -2 0 2 E B [eV] -40040 D O S [ s t a t e s / ( e V • un i t c e ll ) ] -15-10-5051015-4-2024 Total DOS Ce
4f 6s 6p 5d Rh
4d 5d 5s 5p
Sb/Sn
4d 5p 5s 5d
CeRhSn b) ˜ E [ e V ] x D ( E F ) [ s t/ ( e V u . c . ) ] x full relativistic: Rh 4d Ce 4f d)d)e) D O S [ s t/ ( e V u . c . ) ] -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 E B [eV] ˜ E Ce 4f
CeRhSb Sn c) Fig. 3 (Color online) Results of FPLO calculations with full relativistic approach. Partial DOS is shown for(a) CeRhSb, (b) CeRhSn. (c) Ce 4f partial DOS calculated for CeRhSb . Sn . . The black arrow indicatesthe position of the minimum in DOS, which is the closest to the Fermi energy. d) Position of the partialCe 4f and Rh 4d DOS minimum as a function of x in CeRhSb − x Sn x . e) Partial Ce 4f (blue open squares)and Rh 4d (red filled squares) densities of states at the Fermi level for CeRhSb − x Sn x . scheme (Fig. 3 a, b). A difference is visible in Ce 4f spectra, the structure above ε F isdifferent than that discussed previously. In case of x =
0, 0.06 and 0.16, the gap-likedepletion of DOS in vicinity of ε F is also observed (Fig. 3 c).The position of the gap in Ce 4f and Rh 4d DOS is a rising function of x (Fig. 3 d).The gap is located at ε F for CeRhSb. It is shifted above ε F as a result of hole doping.The same trend is observed for Ce 4f and Rh 4d states. The influence of the dopingis also visible in DOSes at ε F (Fig. 3 e). The Rh 4d partial DOS at ε F rises almostlinearly with x , while in case of Ce 4f a small maximum at x = ε F are enhanced with respect to x = ε F . Strictly speaking, in Rh 4d and in Sn 5p spectraldensities, there is a significant, gap-like depletion at ε F . However in case of Ce 4f, ε F is located in the rising slope of the peak in DOS. One can conclude that metallicproperties of CeRhSn are mostly related to 4f electrons.Dispersion of the bands in the first BZ, calculated within full relativistic FPLOmethod, is shown in Fig. 4. For all studied compounds, except for CeRhSn, bandshave similar shape. At the first sight it seems that Fermi level is simply shifted in adirection corresponding to lowering electron band filling. However, a more detailedanalysis shows that there are significant deviations from a rigid band shift. Neverthe-less, due to displacing bands below Fermi energy and shifting another bands to theFermi level one can anticipate interesting modifications of the Fermi surface topol-ogy. The observed band shifts suggests that certain filled bands may become holebands and some electron parts of the Fermi surface may disappear with doping. This ˆ C S U ˆ Z U R T CeRhSb Sn b) -1.0-0.50.00.51.0 E [ e V ] ˆ X S U ˆ Z U R T CeRhSb a) ˆ X S U ˆ Z U R T CeRhSb Sn c) ˆ MK ˆ A L H A CeRhSn d) Fig. 4 (Color online) Band structure along high symmetry directions calculated for CeRhSb − x Sn x ( x =
0, 0.06, 0.16, 1) with FPLO method in full relativistic approach. would mean that Lifshitz transitions occur with growing x . Careful analysis of theevolution of FS may confirm that and it is presented in the next section.The bands related mostly to 4f electrons are characterized by a very weak disper-sion and lie just above ε F . In band structure of CeRhSn there are bands located nearlyat ε F related to 4f electrons and characterized by high effective mass. There is oneband along M-K- Γ direction, which is especially flat and lies just above ε F .3.6 Calculated Fermi surfacesTheoretical calculations performed in full relativistic approach have yielded Fermisurfaces (FSes) of representatives of CeRhSb − x Sn x family (Fig. 5). The shape ofthe FS strongly depends on x . The band structure calculated along high symmetrylines (Fig. 4) shows that the number of bands crossing ε F depends on the sample com-position. The same feature is reflected in the FS contours. The shape of FS changeswith x in a discontinuous manner. The variation of the number of separated parts of FSforeshadows the topological changes of FS, which can be regarded as a sequence ofsubsequent Lifshitz transitions. In case of CeRhSb, four hole-like pockets contributeto the FS. They are transformed under doping ( x = .
02) into one tori-like structureextended along Γ -X direction. For x = .
03 FS can be regarded as two deformedconcentric spheres. The increase in x leads to increase of the overall volume of FS.For x = .
05 FS contour reaches the border of the BZ, resulting in opening of FS.Moreover, some new parts of FS, attributed to band 1 emerge. In case of x = .
6, FSconsists of four leaf-like structures near the X-U-S BZ wall (band 1) and a barrel-likeshape along Γ -X direction with two plate-like structures on both sides of the X point(band 2). Subsequent increase in x makes leaf like structures merge into a pair ofbow-tie shaped structures. The barrel-like shape becomes bigger, while the bottoms(plate-like structures) disappear. Afterwards, the additional closed part of FS appearsaround the Γ point, for x = .
09. The volume of this structure increases with x , and eventually it touches and merges with the bow tie-like structure near to X-U-S wallof BZ. In parallel, the volume of the barrel like band 2 increases.The presented simulations indicate that Sn substitution in CeRhSb − x Sn x systemmodifies FS topology six times for x ranging from 0 to 0.2. This is interpreted asa sequence of Lifshitz transitions. FS volume is also considerably increased. It isdifficult to assign a particular Lifshitz transition to QPT because the location of ε F is given with some uncertainty in DFT calculations. Hence, Sn concentrations incalculations may correspond to a shifted value of x in a real system. Nevertheless, ageneral increase of DOS at ε F , FS volume and a number of band crossing ε F certainlyhelps to understand the modifications of electronic structure, which are related toQPT. We have investigated electronic structure of CeRhSb − x Sn x family, in which QPToccurs. VB spectra obtained with UPS agree qualitatively with both scalar relativis-tic and full relativistic FPLO calculations. They are dominated by Rh 4d states. Thespectra collected at 12.5 K, do not reveal 4 f electron peak in vicinity of ε F . However,the feature related to 4f / final state is clearly visible in Ce 4f spectral functionextracted for each of the studied compounds. The UPS results exhibit a tendency ofCe 4f derived DOS at ε F to rise with Sn content. Calculated DOS yields a semimetal-lic character of CeRhSb with a gap-like minimum in DOS at ε F . This gap is shiftedabove ε F with Sn doping leading to metallic properties. Full relativistic calculationsshow that due to the effect of hole doping realized with Sn substitution a few mod-ifications of FS topology takes place. These are interpreted as Lifshitz transitions.The calculations also yield that FS volume generally grows and a number of bandcrossings at ε F increases as a function of x . These results visualize the evolution ofthe electronic structure which occurs in QPT in CeRhSb − x Sn x system. Fig. 5 (Color online) a) The first Brillouin zone of CeRhSb − x Sn x . b) The Fermi surface of selectedrepresentatives of CeRhSb − x Sn x system.2 Acknowledgements
This work has been supported by the National Science Centre, Poland within theGrant no. 2016/23/N/ST3/02012. Support of the Polish Ministry of Science and Higher Education underthe grant 7150/E-338/M/2018 is acknowledged.
Author contribution statement
A. ´S. and P. S. came up with the presented idea. A. ´S. synthesized and character-ized the used samples. J. G. performed theoretical calculations. Photoelectron spec-troscopy was performed by R. K., M. R. and P. S. R. K. performed data analysis andcomparison with theoretical results. The project was realized under supervision ofP. S. All authors discussed the obtained results and contributed to the article.
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