The first-principles research on the role of surface in the heavy fermion compound CeRh_2Si_2
Yue-Chao Wang, Yuan-Ji Xu, Yu Liu, Xing-Jie Han, Xie-Gang Zhu, Yi-feng Yang, Yan Bi, Hai-Feng Liu, Hai-Feng Song
TThe first-principles research on the role of surface in the heavy fermion compoundCeRh Si Yue-Chao Wang, Yuan-Ji Xu, Yu Liu, ∗ Xing-Jie Han, Xie-GangZhu, Yi-feng Yang,
2, 4, 5
Yan Bi, Hai-Feng Liu, and Hai-Feng Song † Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Science, Beijing 100190, China Science and Technology on Surface Physics and Chemistry Laboratory, 621908 Jiangyou, Sichuan, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Center for High Pressure Science and Technology Advanced Research, Beijing 100094, China (Dated: February 23, 2021)In the heavy fermion materials, the characteristic energy scales of many exotic strongly correlatedphenomena (Kondo effect, magnetic order, superconductivity, etc.) are at milli-electron-volt order,implying that the heavy fermion materials are surface sensitive. Here, we investigate the electronicstructures for Si- and Ce-terminated surfaces of CeRh Si by first-principles methods. Our researchreveals three notable impacts of surface effects on electronic structures, which are consistent withrecent angle-resolved photoemission spectroscopy (ARPES) experiments. Firstly, the relaxationof surface crystal structures changes the relative position of Fermi level, adjusts the dispersion ofbands and enhances the Kondo resonance. Secondly, the decrease of the hybridization between theCe-4 f and conduction electrons in the surface layer leads to a weaker Kondo resonance peak andthe shift of spin-orbit bands. Thirdly, the variation of crystal electric field around surface Ce atomsaffects the splitting of Kondo resonance peaks, and also pushes down the lower-Hubbard bands ofsurface 4 f electrons. Moreover, we find the characteristic of bulk’s lower-Hubbard bands, which wasoverlooked in previous works. Our investigation suggests that these surface effects are potentiallyimportant and highlighted in the future researches on properties of strongly correlated materials. PACS numbers: 71.27.+a, 73.20.-r, 31.15.E-
I. INTRODUCTION
Cerium-based compounds have many exotic and inter-esting properties, such as heavy fermion behavior, su-perconductivity, magnetic order, which are believed tobe originated from the strongly correlated 4 f electronsand its hybridization with the conduction electrons .Among these compounds, CeRh Si has been extensivelystudied for its strong crystal electric field and anisotropiccrystal structure . The de Haas-van Alphen and neu-tron scattering techniques have been used to reveal thehybridization of f -electrons with conduction electrons ( c -electrons) . More importantly, the layered structureof CeRh Si single crystal can be technically cleaved withdifferent terminated atoms . Recently, different elec-tronic structures of Si- and Ce-terminated CeRh Si werereported by high-precision ARPES and temperature de-pendent UV-APRES . In these experimental results,the strength of Kondo resonance peak, Kondo tempera-ture and other fine structure around Fermi level shownotable differences between the samples with Si- andCe-terminated surfaces, which indicates the hybridiza-tion strength and crystal electric field are affected by thedifferent surface environments. These experimental phe-nomena have shown that the environment of surfaces canbe of great difference and the surface have significant im-pact on the electronic structures.On the theoretical side, many studied focused on the bulk properties and have been done by model Hamilto-nian approaches and first-principles simulations .The equilibrium volume, c/a ratio and bulk modulus ofCeRh Si are obtained from density functional theory(DFT) calculations . The crystal electric field effectand Kondo resonance of bulk 4 f -electrons are studiedby density functional theroy plus dynamical mean-fieldtheory (DFT+DMFT) . All these theoretical workshave discribed some phenomena of CeRh Si properly,such as the anisotropic hybridization interaction, split-ting of Kondo peak and mixed valence Ce atomic con-figuration. However most of these works concentrate onthe bulk of CeRh Si , and the theoretical investigationsfocusing on its surface influence are rare.As ARPES experiments mainly provide informationabout the surface-related electronic structures , andconsidering that the electronic structures of CeRh Si are sensitive to the surface as many characteristic energyscales are ∼ meV, a first-principles investigation with ex-plicit treatment of surface is necessary (such as using aslab model). Although much progress has been made,detailed theoretical studies about the surface effect ofCeRh Si are still lacking.In this work, three aspects of the electronic proper-ties of CeRh Si with different terminated surfaces areinvestigated by first-principles simulations. Firstly, wefocus on the surface crystal structure relaxation. TheDFT+DMFT simulations of band structures are per- a r X i v : . [ c ond - m a t . s t r- e l ] F e b xyzBulk Si-terminated case Ce-terminated case(a) (b) (c) (d) Ce (S) Ce (B) FIG. 1: The crystal structure of CeRh Si . The left panelshows the cases of bulk (a), Si-terminated slab (b) and Ce-terminated slab (c). The exposed surface in both slab modelsis (100) direction. We point out the surface Ce and bulk Cereferred in our work with label Ce (S) and Ce (B) in both slabs.The right panel (d) is the first Brillouin zone of the bulk (blue)and surface (red). formed on relaxed surface crystal structures of Si- andCe-terminated surfaces. Compared to the unrelaxedcrystal structures, the relaxed crystal structures giveband structures (relative position of Fermi level andbands’ shape) similar to APRES. We show that this iscaused by the different distribution of the electron den-sity due to relaxation. Secondly, we investigate the effectof hybridization on the surface, and we reveal the differ-ence between surface and bulk electronic properties in Si-and Ce-terminated surfaces respectively for the decreas-ing of c - f hybridization strength on the surface. Thirdly,we take the crystal electric field effect into considerationand reproduce the electronic structures from ARPES re-sults successfully, and find the surface electronic informa-tion can cover the main feature from experiment, whichimplies surface states matter in the first-principles simu-lations of CeRh Si .The paper is organized as follows. In Sec.II, we intro-duce the methods and parameters used in this paper. InSec.III, the results of Si- and Ce-terminated slab modelare exhibited, and the analysis on the surface effect on theelectronic properties of CeRh Si is performed. Sec.IVcloses the the paper by a summary of the main findingsof this work and some general remarks. II. METHOD
For crystal structure relaxation, the DFT+ U methodis exploited for both Si- and Ce-terminated cases as a re-laxation of slabs on DFT+DMFT is expensive , andthe validity of this treatment is discussed in our Supple-mental Material. All the exchange-correlation functionalused in the this work is the conventional Perdew-Burke-Ernzerhof (PBE) functional . We consider the spin- orbit coupling during our simulation, while long-rangemagnetic orders are not considered in the simulations.The DFT+ U simulation is performed in the Vienna Abinitio Simulation Package (VASP) code with the projec-tor augmented wave (PAW) method . The f electrons istreated as valence electrons in PAW pseudopotential, anda plane wave energy cut of 350 eV is used. For the sim-ulation of slabs, the k-mesh is set to 25 × ×
1, and theGaussian smearing is used to avoid sample error along k z direction. We have also tested the density of states(DOS) results on a 31 × × × × U and Hunds exchange J ,used for DFT+ U are U =6.0 eV and J =0.7 eV which is aconventional choice . As shown in Fig.1, CeRh Si crystal takes a body-centered tetragonal ThCr Si -typestructure belonging to the D h point group (space groupI4/mmm No. 139). The lattice parameters of CeRh Si bulk is fixed as experimental results . The slab modelused in this work for Si- and Ce-terminated (100) sur-faces are also displayed in Fig.1. The vacuum added onthe slab is 15 angstrom, and the convergency is tested bythe DOS calculation with a 20 angstrom vacuum. Dur-ing the relaxation, the central layer Ce, and the Si-Rhlayers beside it are fixed at the bulk position to simulatethe bulk, and the lattice parameters inside the surface iskept as the experiment ones . The first Brillouin zone ofthe bulk and surface and the high symmetric points usedin this work are shown in Fig.1(d). As shown in Fig.1(a-c), the Ce atoms in the surface and bulk are marked byCe (B) and Ce (S) respectively. If not specified, all the sim-ulations in this work are performed with relaxed struc-tures.The electronic structures simulations in isotropic envi-ronment (including band structures and density of states)are done with a charge fully self-consistent DFT+DMFTcalculation . The DMFT part is solved by theeDMFT software package, and DFT part is performedin WIEN2k software package . The DFT performedin WIEN2k is based on the the full potential linearizedaugmented plane-wave method (LAPW), with R mt K max = 8.0 and muffin-tin radii, R mt , 2.5 a.u. for Ce, 2.4a.u. for Rh and 2.0 a.u. for Si. The multi-orbital An-derson impurity model is solved by the hybridization ex-pansion continuous-time quantum Monte Carlo impuritysolver (CTQMC) . The temperature is T = 50 K andthe Hilbert space of atomic eigenstates is truncated intoelectron occupancy from 0 to 3. The crystal structures,k-points and on-site U and J values are the same as re-laxation part.The electronic structure simulations with crystal elec-tric field effect are perform based on the isotropic sim-ulation. The crystal parameters are calculated from aconstrained-DFT approach introduced by Nov´ak withLAPW method in WIEN2k, and it is used to avoid theself-interaction error in DFT . The electron of 4 f isconstrained to one on Ce atoms, and the radial part lo-cal orbital of 4 f is the same as in DFT+DMFT. Other M M0.00.51.0-0.5-1.0 E n e r gy ( e V ) (a) M M0.00.51.0-0.5-1.0 E n e r gy ( e V ) (b)M M0.00.51.0-0.5-1.0 E n e r gy ( e V ) (c) M M0.00.51.0-0.5-1.0 E n e r gy ( e V ) (d) HIGHLOW relaxed unrelaxed
FIG. 2: The momentum-resolved spectral functions calcu-lated by DFT+DMFT. (a) and (c) are the results of relaxedcrystal structures of Si- and Ce-terminated cases respectively.(b) and (d) are the results of unrelaxed crystal structures ofSi- and Ce-terminated cases respectively. The green framespoint out the regions with main differences between the bandstructures with and without the relaxation. parameters are kept the same as those mentioned above.
III. RESULTS AND DISCUSSIONA. Surface structure relaxation
The first issue that we want to address is the impor-tance of including the surface crystal structure relaxationin the electronic structures calculations . The concept ofsurface crystal structure relaxation refers to the changeof geometric structure of surface caused by the unbal-anced force performed on the surface layer atoms. Af-ter relaxation, in the Si-terminated case, the surface Silayer moves into the center by 0.204 ˚A, and the outmostCe layer moves into the center by 0.064 ˚A. In the Ce-terminated case, the surface Ce layer moves toward thecenter by 0.186 ˚A. Consequently, the external potentialsare changed by different surface atomic structures andwe will see below that the electronic band structures arefound to be significantly influenced before and after re-laxation.Fig.2 compares the calculated electronic band struc-tures for the relaxed and unrelaxed crystal structuresalong the high-symmetry line of the Brillouin zone.Quantitative differences can be observed for the low-energy spectral functions around the Fermi level. Forthe relaxed Si-terminated surface in Fig.2(a), we observea hole-like, linearly dispersing surface resonant band(square region) with a cusp around the Fermi level at¯Γ point. This cusp is shifted much above the Fermi levelfor the unrelaxed case as shown in Fig.2(b). On the con-trary, the location of the electron-like band (the circle re- gion) around the ¯M point for the relaxed structure movesabove the Fermi level. In both cases, the experimentallyobserved Shockley-type surface states around -0.5 eV at¯M point can be well reproduced , which may originatefrom the dangling bonds of surface Si. For Ce-terminatedsurface as shown in Fig.2(c,d), the most apparent discrep-ancy appears around -0.25 eV at ¯Γ point (diamond re-gion). The crossing point observed in the unrelaxed caseis separated in the relaxed case. The reported rocket-shaped features below -0.5 eV are also shown in our cal-culations for both cases . Furthermore, we find that ourresults after relaxation are consistent with experimentalresults . This implies that relaxation plays an impor-tant role in calculating the electronic band structures ofCeRh Si .Besides the band dispersion around Fermi level, it canalso be observed that the f / band is enhanced with thesurface relaxation. This feature can not be obtained fromARPES results, as ARPES probes the information belowFermi level. Together with this feature, the Kondo peaksand spin-orbit bands are also enhanced sightly. The de-tailed results of DOS and hybridization strength are dis-played in the Supplemental Materials Fig.S5.These differences can be qualitatively understood bythe enhancement of bonding effect between the surfaceand sub-surface layer atoms. Fig.3 displays the electrondensity difference for Si-/Ce-terminated surfaces. In theSi-terminated case, there are more electrons in the rangebetween Rh-Si layer and Ce layer after relaxation, mak-ing the bonding between Ce and Rh-Si layer stronger.The enhancement of bonding effect will make the bandwith bonding characteristic move downwards and withanti-bonding characteristic move upwards in energy. Asimilar but weaker effect can also be observed in Ce-terminated cases between the outmost Ce-layer and Rh-Si sub-layer. All these enhancements of bonding effectare directly related to the shorten of the distance betweenthe surface and the bulk from relaxation. The enhance-ment of f / , Kondo peaks and spin-orbit bands may bepartially attributed to the increasing of hybridization af-ter the relaxation. However, it should be emphasizedthat the hybridization variation caused by surface itselfis much notable than relaxation, which is investigated inthe following part.Here, it should be pointed out that the surface crys-tal structure relaxation is calculated at DFT+ U level,and the crystal structure is used as an approximation tothe DFT+DMFT results. Because DFT and DFT+ U stand for the limit of weak and strong correlation re-spectively, we assume that the performance of crystalstructure relaxation of DFT+DMFT will be betweenthese two approaches. The validity of this approxima-tion for CeRh Si system is confirmed by the compari-son of electronic structures results from DFT, DFT+ U and DFT+DMFT and the structure relaxation results ofDFT and DFT+ U . The detailed discussion of this partis in the Supplemental Material part II. FIG. 3: The difference of electron density between results of DFT+DMFT calculation and the superposition of spherical atomicelectron density. The electron density differences of typical slice of Si- and Ce- terminated cases with and without relaxationare shown in the figure (a)-(h). The warm color means the increasing of electron density and cold color means the decreasingof electron density. (i)-(l) show the definition of the slice used in (a)-(h). The position and direction of different slices arespecified by violet layers, and atomic positions in (a)-(h) are shown in red frames.
B. Hybridization strength
In this section, we focus on the hybridization strengthof the local f -electron and the conduction electrons in thesurface area. The hybridization strength between surface f - and c -electrons is anticipated to decrease due to thedisappearance of c -electrons on one side of the surface.As a result, the electronic properties of Ce (S) are differ-ent from Ce (B) in CeRh Si . In Fig.3 the electron densitybetween Si-Rh and Si-Si is shown to be much larger thanother areas in the slab, indicating the strong bonding ef-fect between Si-Rh and Si-Si. The electrons of Si-Rh andSi-Si region make the skeleton of c -electron environment.The electrons on Ce surrounded by this c -electron envi-ronment (like Ce (S) in Si-terminated case and Ce (B) ) de-creases compared with isolate atom. It indicates a stronghybridization of electrons on Ce atom (like 4 f electrons)with c -electrons, which improves the energy of Ce atomic states and transfers the atomic electrons to c -electrons.On the contrary, there are more electrons concentratingon Ce (S) in Ce-terminated case with a hemispherical dis-tribution, which implies a weak hybridization and lessvariation from atomic state. The relation between thehybridization strength and electron distribution in realspace have also been observed with spectroscopic imag-ing scanning tunneling microscopy (SI-STM) in heavyfermion materials .The hybridization strength is closely related to theKondo resonance peaks at Fermi level and a sharperKondo peak generally corresponds to a stronger hy-bridization strength. To explicitly study the effect ofhybridization in surface, we show the 4 f -projected DOS(PDOS) for Si- and Ce-terminated cases of CeRh Si inFig.4(a) and Fig.4(b). The bulk Kondo peaks are sharperthan the surface Kondo peaks for both cases. In contrastto the single Kondo peak obtained in the Si-terminatedcase, the surface Kondo peak for the Ce-terminated casesplits into two small Kondo peaks. The distribution ofelectron density around Ce atom is responsible for thedistinct behaviors of the Kondo peaks. In Si-terminatedcase, the Ce (S) still has an environment similar to thebulk. However, for Ce-terminated case, the Ce (S) directlyexposes to the vacuum which makes it closer to an iso-lated atom, thus, the Kondo peak vanishes.The hybridization strength is directly related to theimaginary part of the hybridization function . Thedefinition of hybridization function is∆( ω ) = X k | V k | ω − (cid:15) k + iη , (1)where V k and (cid:15) k represent the hybridization parameterand the dispersion of the conduction electrons, respec-tively. η → + is an infinitesimal positive real number.The imaginary part of hybridization function, which canbe used to characterize the hybridization strength, is ob-tained from the imaginary part of the hybridization func-tion Im∆( ω ) = − π X k | V k | δ ( ω − (cid:15) k ) . (2)Fig.4(c) displays Im∆( ω ) for the 4 f Ce (S) atom and theCe (B) atom in Si-terminated case. The absolute valuesof the peak at ω = 0 for the Ce (B) atom is 38% largercompared to the 4 f Ce (S) atom. For the Ce-terminatedcase shown in Fig.4(d), Im∆( ω ) for the 4 f Ce (S) aroundthe Fermi level is very small and the absolute value isonly 21% compared with Ce in bulk. The results fromIm∆( ω ) are consistent with PDOS, and our results sug-gest that the electronic structures at Fermi level and thehybridization strength are closely related.According to previous researches , the spin-orbitside peak below the Fermi level is located around ∆ so + T K , where T K is related with the hybridization strength.Stronger hybridization strength would shift the spin-orbitside peak towards the Fermi level. As shown in Fig.4(a)and (b), the spin-orbital side peak is around -0.15 eVand -0.4 eV for Si-terminated case and Ce-terminatedcase, respectively, which are in qualitative agreementwith ARPES experiments . Therefore, the positionof the spin-orbit side peak can be used to characterisethe hybridization strength. C. Crystal electric field effect
In the presence of crystal electric field, the f / orbitalssplit into Γ , Γ and Γ oritals. In this section, Γ , Γ FIG. 4: Density of states and imaginary part of hybridiza-tion functions in Si- and Ce-terminated case. (a) and (b) arethe DOS of Ce 4 f -electrons in Si- and Ce-terminated casesrespectively. The red solid lines are the 4 f -electron DOS ofCe (S) . The blue solid lines are the 4 f -electron DOS of Ce (B) .The black dash lines are the total 4 f -electron DOS of Ce. Theinset of (b) shows the 4 f -electron DOS of Ce (S) in a smallerscale. (c) and (d) show the imaginary part of hybridizationfunction of Si- and Ce- terminated case respectively on realfrequency. The solid red and blue lines are for f / of Ce (S) and Ce (B) . The black dash and green dot lines are for f / ofCe (S) and Ce (B) . and Γ are defined asΓ = a (cid:12)(cid:12)(cid:12)(cid:12)
52 ; ± (cid:29) − b (cid:12)(cid:12)(cid:12)(cid:12)
52 ; ± (cid:29) , Γ = b (cid:12)(cid:12)(cid:12)(cid:12)
52 ; ± (cid:29) + a (cid:12)(cid:12)(cid:12)(cid:12)
52 ; ± (cid:29) , Γ = (cid:12)(cid:12)(cid:12)(cid:12)
52 ; ± (cid:29) . (3)Here the parameters a = p / b = p / . Fig.5shows the f -bulk and f -surface DOS for Si-terminatedand Ce-terminated cases with crystal electric field. Theexperimental results from the integrated resonance en-hanced ARPES are also given for comparison. The peaksreferring to Γ , Γ and Γ are marked on the figure .Γ and Γ states are merged in our bulk and sur-face results shown in Fig.5(b) in Si-terminated case. Thesimilar mergence is reported in previous theoretical andexperimental works when temperature is higher than 30K . For the surface f electrons, the splitting betweenthe peak of Γ and the merged peak, which is composedof Γ and Γ peaks, within -0.1 ∼ . In contrast, for the bulk f -electrons, the resonance peaks (Γ , Γ and Γ ) havea smaller splitting energy and the deviation of the spin-orbit peak from the experimental results is more signifi-cant.For the high-energy part, contrary to the experimen-tal results that there are two peaks located around -2.0eV (Fig.5(a) mark A1) and -1.5 eV (Fig.5(a) mark B1),only one peak is generated in our calculations for thesurface and bulk cases. The positions of the high-energypeaks for bulk and surface are close to the locations ofthe two peaks observed experimentally, indicating thatthe APRES results are composed of both surface andbulk properties of the material at high-energy.Fig.5(c) and (d) show the results of the Ce-terminatedcase. For the Ce (S) , the splitting of the Kondo peak isnot observed due to the fact that Γ , Γ and Γ orbitalsare near degenerate. It can be understood from the crys-tal electric field parameters obtained from constrained-DFT calculations listed in Table.I. The energy differencesamong Γ , Γ and Γ are quite small. However, onecan readily see from Table.I that this near-degeneracy islifted for Ce (B) . Then, the peak is splitted again in thiscase. The spin-orbit peak for Ce (S) is closer to the ex-perimental results within -0.3 ∼ -0.4 eV. In Fig.5(c), theexperimentally observed Hubbard peak at position A2around -2.0 eV is also obtained in our calculation forCe (S) . The broadening of the calculated Hubbard peakcan be attributed to the fact that the calculated DOScontains information of all k-spacing while the experi-mental DOS is integrated along a certain k-path. Forposition B2 around -1.3 eV, the experimentally observedhump can be attributed to the contributions from Ce (B) .The distinction of crystal electric field in different sur-faces can be quantitatively accounted for by the crystalelectric field parameters listed in Table.I. The absolutevalues and splitting energies between the triplet splitting f / states in bulk and different surfaces are given. Itis clear that the absolute values of Ce (S) atoms for theCe-terminated case is one order less than that of bulk.As the absolute value are referring to the Fermi energyof corresponding slabs, the smaller value of Ce (S) impliesa weaker crystal electric field compared with Ce (B) . It TABLE I: Crystal electric field (CEF) parameters fromconstrained-DFT calculation. The absolute value of Γ isgiven in the second column. The values of Γ and Γ refer-ring to Γ are given in the third and the forth column. Thedefinitions of Γ , Γ and Γ are the same as Eq.(3). Forcomparison, the experimental and previous theoretical valuesare also given .CEF parameters (eV) Γ Γ − Γ Γ − Γ Ce (S) in Si-Case 0.496 0.055 0.081Ce (S) in Ce-Case 0.063 0.004 0.009Ce (B) in Bulk 0.691 0.049 0.084Expt. in Si-Case – 0.048 0.062DFT in Ce-bulk – 0.021 0.048 FIG. 5: The DOS from DFT+DMFT simulation with crystalelectric field effect. (a) and (c) show the DOS of Si-terminatedand Ce-terminated cases respectively. (b) and (d) are theirenlarged views around Fermi level. The experimental resultsfrom Patil are also shown for comparison . The peaks ofspin-orbit coupling, Γ , Γ and Γ are pointed out directlyin the figure. A1 and B1 stand for the peaks of lower Hubbardband of surface and bulk in Si-terminated case respectively,and so do A2 and B2 for Ce-terminated case. All the DOSdata are renormalized respected to the highest resonance stateat around Fermi level (at about -0.01 eV) from experimentalresults respectively. is reasonable that the Ce (S) in Ce-terminated case expe-rience the weakest crystal electric field effect in the ab-sence of half Si-Rh crystal environment. The different inthe absolute values of Ce (S) and Ce (B) also have impactson the 4 f electron levels, and contribute to the shift ofHubbard bands of Ce (S) comparing to the Ce (B) . Mean-while, the splitting between states in Ce-terminated caseis also one order less than the other cases which impliesthat the behavior of the Ce (S) in Ce-terminated case issimilar to the isolate atom. This could also explain thenear degeneracy of f / in Ce-terminated case. A slightenhancement (about 6 meV) of CEF splitting on Ce (S) of Si-terminated case comparing with Ce (B) is also beobserved in constrained-DFT results. It may have con-tribution to the larger splitting between peaks of Γ andΓ &Γ of Ce (S) in Si-terminated case. We suppose thisvariation of CEF splitting may originate from the asym-metric charge distribution of Ce (S) in Si-terminated case.We have shown that the main features from resonanceenhanced ARPES results can be well reproduced by oursimulation of surface 4 f electrons. Despite the muchlower height of resonance peak comparing to the bulkaround Fermi level of surface f state as shown in theFig.4, most of the low-energy information obtained byARPES can be traced back to the Ce (S) electron states,especially in Ce-terminated surface. Some features orig-inate from the Ce (B) electron states can be observed inhigh energy range. It also suggests that for the simula-tion of ARPES, the surface effect should be taken intoconsideration properly in other materials like CeRh Si . IV. CONCLUSION
In summary, we investigate the electronic structuresof Si- and Ce-terminated surface of CeRh Si using first-principles approaches. We have revealed three key as-pects of surface effect on the electronic structures ofCeRh Si . Firstly, the relaxation of surface structurechanges the dispersion of band structure, thus it adjuststhe relative position of some high symmetry points withFermi level. The enhancement of Kondo resonance isalso be observed especially of f / . From the chang-ing of electron density after the surface relaxation, webelieve that it is the enhancement of the bonding be-tween layers renormalized the bands. More precise elec-tric structure can be obtained with the relaxation of sur-face structure. Secondly, the hybridization between 4 f and c -electrons decreases from bulk to surfaces Ce atomsobviously, which suppresses the strength of Kondo peaksand shifts the spin-orbit peak position of surface 4 f elec-trons. Thirdly, the crystal electric field of outmost Ceatoms is different from the bulk, especially for Ce (S) inCe-terminated case whose f / orbitals are nearly degen-erate like an isolate atom. By considering the crystalelectric field effect on Ce 4 f electrons, we have well repro-duced the experimental ARPES results. All these simu-lation results strongly suggest that the surface has vitalinfluence on the CeRh Si electronic properties, and afully self-consist structure and electronic simulation onthe DFT+DMFT may be needed for the investigationon other strongly correlated materials.It should also be emphasized that the comparison of experimental and simulation DOS shows that the mostARPES information can be attributed to the surface elec-trons, but the information of bulk electrons also appearwhere the strength of surface DOS is weak. This mayprovide a different point of view to the interpretation ofARPES results. V. ACKNOWLEDGEMENT
We thank Guang-Ming Zhang, Jian-Zhou Zhao,Xing-Yu Gao, Huan Li, Dan Jian, Ming-Feng Tian,Yin Zhong and Fa-Wei Zheng for helpful discussions.The work was supported by the Science ChallengeProject (NO.TZ2018002 and NO.TZ2016001), NationalNature Science Foundation of China (NO.U1930401,NO.12004048 and NO.11974397) and the Foundation ofLCP. We thank the Tianhe platforms at the NationalSupercomputer Center in Tianjin.
VI. AUTHOR CONTRIBUTIONS
H.-F. Song and Y. Liu conceived and supervised theproject. Y.-C. Wang, Y.-J. Xu and Y. Liu performedthe numerical simulations. All authors analysed and dis-cussed the results. Y.-C. Wang, Y. Liu, Y.-J. Xu, X.-J.Han and H.-F. Song wrote the manuscript, with contri-butions from all the authors. ∗ Electronic address: liu [email protected] † Electronic address: song [email protected] M. Sigrist and K. Ueda. Phenomenological theoryof unconventional superconductivity.
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2, 4, 5
Yan Bi, Hai-Feng Liu, and Hai-Feng Song † Laboratory of Computational Physics, Institute of AppliedPhysics and Computational Mathematics, Beijing 100088, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Science, Beijing 100190, China Science and Technology on Surface Physics and Chemistry Laboratory, 621908 Jiangyou, Sichuan, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Center for High Pressure Science and Technology Advanced Research, Beijing 100094, China (Dated: February 23, 2021)
Our supplemental information is organized as follows. Sec.I shows the convergence tests results. Sec.IIgives the detailed results of surface crystal structure relaxation and the discussion on the performanceof difference methods. Sec.III shows our prediction results of band structure on other k-path. Sec.IVdisplays the band structure with crystal effect. Sec.V shows the variation of Kondo resonance peaks withsurface relaxation.
I. CONVERGENCE TESTS (a) (b)
FIG. 1: The convergence tests of k-points number and vacuum thickness in DFT simulation of slabmodels. Figure (a) and (b) are DOS (total DOS and PDOS of f -electrons) of convergence test in Si-and Ce-terminated cases respectively. The solid lines represent the parameters adopted in our work(625 k-points and 15 ˚A vacuum), the dash line represent the simulation with denser k-mesh (961k-points and 15 ˚A vacuum), and dot line represent the simulation with thicker vacuum (625 k-pointsand 20 ˚A vacuum).In our work, the 25 × × a r X i v : . [ c ond - m a t . s t r- e l ] F e b are shown in Fig.1. The DOS results from denser k-mesh and thicker vacuum are almost the same, whichindicates that the parameters used in this work are suitable for our purpose. II. COMPARISON BETWEEN DIFFERENT METHODS
Table I lists the variation of atom coordinate after the relaxation with DFT and DFT+ U . The DFTand DFT+ U results are almost the same with each other in Si-terminated case. In Ce-terminated, theproportion of difference referring to the height of the sandwich shape sub-layer Rh-Si-Ce-Si-Rh is 0.3%at surface Ce atom. It may be caused by the atom-like nature of surface Ce in Ce-terminated case, andthis nature can be described more properly by DFT+ U than DFT. The results above confirm us thatfor CeRh Si the on-site correlation correction may not much crucial for crystal structure relaxation.Considering the surface Ce in Ce-terminated case, and the crystal structure is relaxed at DFT+ U level.To investigate the similar crystal relaxation results of CeRh Si in DFT and DFT+ U , the density ofstates (DOS) from DFT, DFT+ U and DFT+DMFT of Si- and Ce-terminated cases are shown in Fig.2.For particular this material, it is clearly shown that in the range far below Fermi level, the results ofdifferent methods are almost the same. The obvious differences appear at the area around Fermi leveland it is mainly attributed to the f electron of Ce. DOS of DFT+DMFT shows a sharp peak at Fermilevel which is due to the f / resonance state, and another peak about 0.4 eV comes from the f / state.For the two f peaks mentioned above are caused by dynamic correlated effect, no similar pattern is shownin the DOS of DFT or DFT+ U .In the results of DFT+ U , a broaden f peak can be also found around Fermi level, but this is differentfrom the resonance state from DFT+DMFT calculation. The broaden peak of DFT+ U is caused by ahalf-filled f orbital caused by the non-spin polarized condition used in + U calculation. An orbital with0.5 electron occupation in + U approach will lead to nearly zero correction to the orbital, and the other f orbitals without electron will be moved up by U /2 in energy to form the upper Hubbard band. Anotherevidence is that there is a very flat lower Hubbard band around -2.0 eV in DMFT simulation, whilein DFT+ U simulation the corresponding f states concentrate at the broaden peak at Fermi level. ForDFT simulation results, almost all the f states are concentrated in the range 0.0 eV to 1.0 eV, becauseof the lack of strongly correlated effect. As the differences of DOS from these three methods are notsignificant below Fermi level, it indicates that the prediction of total energy concerned properties (likecrystal structure) may be not sensitive to the choice of method. III. SPECTRAL FUNCTIONS ON ANOTHER K-PATH
In Fig.3 (left), the spectral function of bulk on a conventional k-path is shown. It is used to confirm ourDFT+DMFT simulation is consistent with other works on CeRh Si bulk . The Si- and Ce-terminatedcases’ spectral functions on another k-path are shown in Fig. (middle) and Fig. (right) for reference. IV. SPECTRAL FUNCTIONS WITH CRYSTAL EFFECT
Fig.4 displays the spectral function at Fermi level in Si-terminated case the bulk to show the effect ofcrystal effect. The spectral function results of Si-terminated is not so clear as the DOS in main text. Itis because the electron states of outmost Ce and central Ce is mixed, but some characters can still beobserved like the splitting of resonance peak at around -0.05 eV. For comparison a spectral function fromTABLE I: The variation of atom position in Si- and Ce-terminated slabs calculated with DFT andDFT+ U . The reference position is the corresponding atoms coordinates in bulk. The proportion ofchanging referring to the height of a sub-layer (Rh-Si-Ce-Si-Rh, 5.09 ˚A) is shown in the parentheses. Slab Si-terminated case Ce-terminated caseMethod DFT DFT+ U DFT DFT+ U Layer-0 (Ce) (˚A) – – 0.206 (4.0%) 0.186 (3.7%)Layer-1 (Si) (˚A) 0.202 (4.0%) 0.204 (4.0%) -0.055 (-1.1%) -0.042 (-0.8%)Layer-2 (Rh) (˚A) 0.088 (1.7%) 0.087 (1.7%) 0.006 (0.1%) 0.006 (0.1%)Layer-3 (Si) (˚A) 0.060 (1.2%) 0.064 (1.2%) 0.094 (1.8%) 0.083 (1.6%)Layer-4 (Ce) (˚A) 0.066 (1.3%) 0.064 (1.2%) 0.000 (0.0%) 0.001 (0.0%)Note: a. The value in the table is the difference of position between the relaxed atoms and unrelaxed atoms with thecentral layer fixed at the coordinate origin. The positive value means the atoms move to the center after therelaxation. b. Layer-0 to Layer-4 represent the atomic layers from the outer to the inner. Layer-0 refers to the Ce terminatedsurface layer and there is no data for Si-terminated case. The element in every layer is given in the parentheses. ,, ,, ,, ,, ,, ,,
FIG. 2: Density of states of f -electron (red) and total electron (black) from DFT (dot line), DFT+ U (dash line) and DFT+DMFT (solid line) in Si- (a) and Ce- (b) terminated cases.FIG. 3: The band spectral functions of bulk (left), Si-terminated case (middle) and Ce-terminated case(right) from DFT+DMFT calculations. The k-path here are along some common high symmetrypoints. Relaxed structures are used for Si- and Ce-terminated case.FIG. 4: The band spectral functions of Si-terminated case (left) and bulk (right) from DFT+DMFTcalculations with crystal field effect.bulk calculation is also shown in Fig.4. The characteristic f-state resonance bands in bulk also appear inthe slab calculation, and it indicates that a simple simulation of spectral function simulation with slab orbulk may not reflect the experimental results properly, for the weight of bulk and surface states shouldnot be the same. V. HYBRIDIZATION VARIATION WITH RELAXATION