Effect of f–c hybridization on the γ →α phase transition of cerium studied by lanthanum doping

Yong-Huan Wang(王永歡) Yun Zhang(張云) Yu Liu(劉瑜) Xiao Tan(談笑) Ce Ma(馬策)Yue-Chao Wang(王越超) Qiang Zhang(張強) Deng-Peng Yuan(袁登鵬) Dan Jian(簡單)Jian Wu(吳健) Chao Lai(賴超) Xi-Yang Wang(王西洋) Xue-Bing Luo(羅學(xué)兵) Qiu-Yun Chen(陳秋云)Wei Feng(馮衛(wèi)) Qin Liu(劉琴) Qun-Qing Hao(郝群慶) Yi Liu(劉毅) Shi-Yong Tan(譚世勇)Xie-Gang Zhu(朱燮剛) Hai-Feng Song(宋海峰) and Xin-Chun Lai(賴新春)

1Science and Technology on Surface Physics and Chemistry Laboratory,Jiangyou 621908,China

2Laboratory of Computational Physics,Institute of Applied Physics and Computational Mathematics,Beijing 100088,China

Keywords: structural phase transition,molecular beam epitaxy,ARPES,f-electron system

1. Introduction

Among the f electron systems, i.e., lanthanide and actinide metals,alloys and intermetallic compounds,cerium has its unique importance, in the sense that it is the first element possessing one f electron and serves as the prototypical system in exploring the fascinating properties of f electrons. Historically, cerium metal has been attracting the attention of researchers due to its complex phase transition under moderate pressure and temperature conditions, in which it is believed that its single f electron has been playing important roles. Face-centered cubic(FCC)γ-phase cerium will transform into the same FCCα-phase,with an unconventional volume collapse of up to 16.5%, under a moderate pressure of～0.78 GPa at ambient temperature, or cooled down to low temperature(～141±10 K for bulk materials,[1]or down to around 50 K for single crystalline thin films[2]) at ambient pressure. As for the explanation of the mechanism of this intriguing phase transition,among the various theoretical models, i.e., the promotional model,[3–5]the Mott transition model[6]and the Kondo volume collapse (KVC) model,[7,8]the last one seems to be more adequate. In the KVC model,it is believed that the increase of hybridization between f electrons and conduction electrons drives theγ →αphase transition in Ce metal, and strong f–c hybridization has been revealed in the band structures of single crystalline Ce thin films characterized by high resolution angle resolved photoemission spectroscopy (ARPES) experiments.[2,9,10]However, direct experimental observation of the band dispersions ofα-Ce has not been achieved,due to the incompleteness of theγ →αphase transition and persistent existence ofγphase on the surface of Ce thin films,which makes the identification of f–c hybridization evolution and the role it playing across theγ →αphase transition ambiguous. Therefore,approaches that could manipulate the f–c hybridization and investigate its effects on theγ →αphase transition are demanded.

For the purpose of determining the relationship between phase (atomic arrangement, basically) and electronic properties,pressure is one important variable.[11]It is plausible that only moderate hydrostatic pressure is needed to manipulate theγ →αphase transitions in Ce,however,photoemission is technically non-applicable for hydrostatic pressurized solids.Another commonly adopted variable is alloying doping,which has been usually termed as chemical pressure. The effects on theγ →αphase transition of Ce by doping with some rare earth and actinide elements have been studied by dilatometric and x-ray techniques.[12,13]However,studies on the evolution of the electronic structure are still lacking. Among the various alloying dopants,La is one of a kind,with the reasons as follows: (1) with non-f electron occupancy, La acts as nonmagnetic ion, which could effectively tune the concentration of the localized f electron of Ce;(2)Ce and La alloyed in arbitrary chemical ratio could maintain the FCC structure in thin films; (3) La doping expands the FCC lattice as its concentration increases,from 5.16 ?A for Ce[14]to 5.31 ?A for La,[15]resulting in an effective negative chemical pressure. Here in our work,we have successfully grown Ce1?xLaxthin films by molecular beam epitaxy (MBE), with La concentrationxbeing tuned continuously from zero to 100%. Lattice structure characterization reveals that the thin films were in the FCC structure and the effective negative pressure was realized by La doping. Suppression ofγ →αphase transition by La doping was directly confirmed by electronic transport measurements. High resolution band structures of Ce1?xLaxthin films were obtained by ARPES,confirming that the electronic structure of 4f electron in Ce could be effectively modified by La doping. The evolution of f–c hybridization by La doping was investigated by both a phenomenological periodic Anderson model (PAM) and a modified Anderson impurity model [the Gunnarsson–Sch¨onhammer (GS) model]. We show that the change in f–c hybridization correlates with the suppression ofγ →αphase transition in Ce,which is consistent with the scenario of the KVC model.

2. Experimental methods

Ce1?xLaxthin films were grown by MBE.High purity Ce and La metals(99.9%)were used and thoroughly degassed at 1600?C and 1500?C separately before thermal evaporation.La concentrationxwas controlled by finely tuning the flux ratio of Ce and La during the thermal evaporation. Quartz crystal micro-balance was used to calibrate the flux rate of Ce and La down to an accuracy within 0.002 ?A/s. Single crystalline Ce1?xLaxthin films were deposited on conductive graphene/6H-SiC(0001) substrate forin-situband structure characterization, while polycrystalline thin films were grown on insulating Al2O3(11-20)substrate forex-situx-ray diffraction (XRD) and electronic transport measurements. For the photoemission spectroscopy(PES)experiment,the photoelectrons were excited by the He Iα(21.2 eV)resonance line of a commercial helium gas discharge lamp. The light was guided to the analysis chamber by a quartz capillary. In virtue of the efficient three-stage differential pumping system,the pressure in the analysis chamber was lower than 1×10?10mbar during our experiments,satisfying the harsh criteria for probing the physical properties of chemically high-reactive elements. A VG Scienta R4000 energy analyzer was used to collect the photoelectrons. The total energy and momentum resolutions were better than 10 meV and 0.006 ?A?1, respectively. The transport measurements were performed with the standard four-probe technique using a physical property measurement system(PPMS-9). Pure indium(5N)metal was used to make the electrode contacts. The resistivity versus temperature curves were recorded during a cooling and warming cycle, respectively. It should be noted that different samples with different La concentrations were used in the above experiments to cover a wide range of La concentration. Theoretical methods will be discussed in more detail in the following main context.

3. Results

3.1. Lattice structure and transport properties

Polycrystalline Ce1?xLaxsamples with the thickness of～1 μm were used for transport measurement. XRD analysis was conducted just after the samples were taken out of the MBE chamber,as demonstrated in Fig.1(d). The values of 2θhave been calibrated by the(110)and(220)peaks of the Al2O3substrate,and the intensities have been renormalized by the intensity of pure Ce(111)peak for a better view. According to the standard PDF cards, we know the Ce film (red curve in Fig. 1(d)) corresponds toγ-Ce of FCC crystal structure. The strong intensity of the (111) peaks for all the samples except pure La film indicates that the samples were highly oriented in the [111] crystallographic direction. However, thein-situreflected high-energy electron diffraction(RHEED)showed that all the films grown on Al2O3(11-20)substrates were polycrystalline,which means that the films were mainly consisted with crystallites oriented along[111]out of plane and randomly inplane. The case for pure La film was quite different: the(111)peak was much weaker relative to that of the substrate,and the signal to noise ratio was much lower than that of the Ce film,indicating that the crystallites in the La film were totally randomly oriented. The evolution of the lattice parameters could be seen from the continuous shift of the(111)and(222)peaks between the vertical lines in Fig. 1(d), and in more detail in Fig. 1(e), demonstrating the linear expansion of the lattices by La doping. It should be stressed that the lattice parameters of our pure Ce and La polycrystalline thin films, i.e.,5.166±0.002 ?A and 5.307±0.005 ?A, corresponded to their FCC structures,[14,16]respectively,and are consistent with our RHEED experiment results.

Fig.1. (a)RHEED patterns of Ce0.50La0.50 thin film taken with incident electron directions along1ˉ10 (b)evolution of the RHEED streaks during the growth as a function of time; (c) in-plane lattice constants of various Ce1?xLax single crystalline thin films; (d) renormalized XRD curves of polycrystalline Ce1?xLax thin films; (e)the evolution of lattice parameters in the FCC structure and the effective negative pressures by La doping; (f)normalized resistivity as a function of temperature.

The pressure to volume (P–V) relationship of pure Ce was almost linear for pressures lower than～0.70 GPa.[17,18]Therefore, in order to investigate the negative pressure effect by La doping,we could linearly extrapolate theP–Vrelationship of pure Ce to negative pressure. Based on our experimental lattice parameters,the effective negative pressures for various La concentrations were calculated and shown in Fig.1(e).

The resistivity of the above Ce1?xLaxsamples was renormalized by their values at 250 K and shown in Fig.1(f). Hysteresis loops were observed in pure Ce and Ce0.967La0.033thin films,which are the characteristic feature of theγ →αphase transition. The area enclosed by the hysteresis loop shrank a lot by 3.30%La doping,and diminished for La concentration at 9.20%and above,which means that theγ →αphase transition was suppressed by a few percentage of La doping, i.e.,less than 10%of La concentration. It should be noted that the lattice expansion forxless than 10%was smaller than～0.2%,which is almost negligible. Thus, the mechanism of the suppression of the phase transition lies somewhere else, where the electronic properties of the f electrons and their interaction with conduction electrons might be playing important roles.

3.2. Photoemission studies

Detailed studies on the electronic structures of pure Ce thin films by ARPES could be found in Refs. [2,9,10]. The ARPES spectra for our as-grown Ce1?xLaxthin films with various La concentrations are shown in Fig. 2, corresponding to the spectra taken alongK–Γ–Kin the surface Brillouin zone(SBZ).Figure 2(a)shows the main features of the band structures of the Ce thin film: three dispersive bands that are originated from non-f electrons (s, p, d), labeled asα,βandγ,respectively;three non-dispersive bands that locate near Fermi level(EF),～250 meV and～2 eV belowEF,which were conventionally attributed to the 4f15/2,4f17/2and 4f0,respectively.The clear presence of the 4f1levels nearEFindicates the strong hybridization between conduction electrons with the 4felectron in cerium,enhancing its spectral intensity nearEF.As for pure La thin film,the ARPES spectrum in Fig.2(h)shows almost identical dispersive bands as observed in Ce, with the non-dispersive f bands missing,which is a consequence of the absence of f-electron in La. With increasing La concentration in Ce1?xLax,the 4f bands faded continuously and the conduction bands remained similar,as shown in Figs.2(b)–2(g).

To investigate thek-dependent f–c hybridization, we introduce the phenomenological periodic Anderson model(PAM), which gives the band dispersion describing the hybridization as[2,19,20]Hereε0is the 4f ground state energy,which is notk-dependent.ε(k) is the valence band dispersion at room temperature,which is parabolic and concave down,[2]as shown by the black curves in Figs. 3(a)–3(c). AndVkis the renormalized hybridization strength. When the La concentrationxis below 61.7%,all band structures around the Fermi level of the Ce–La alloys can be fitted with the PAM,as shown in Figs.3(a)–3(c).Without regard to the f–c hybridization,only conduction band crosses the Fermi level (see the black lines). When the hybridization strengthens,the conduction band bends to another direction with much smaller Fermi velocities(see the red dots)and this is how the heavy electrons emerge. However,we find that the f–c hybridization gap of the Ce–La alloys decreases as the La concentration increases as shown in Fig. 3(h) and the green marker line in Fig. 3(g), indicating the decreased hybridization strength. Furthermore, when the La concentration is 61.7%or more,as shown in Fig.3(d),the valence band structure cannot be fitted with the PAM,namely,the hybridization effect disappears, mainly resulted from the faint f bands.That is to say, the hybridization strength of the Ce–La alloys monotonously decreases as the La concentration increases until its disappearance.

Fig.2. (a)–(h)ARPES spectra within large energy scale(from ?3.0 eV to 0.5 eV)for Ce1?xLax alloy thin films along K–Γ–K in the surface Brillouin zone for various La concentrations.

Fig. 3. (a)–(d) Detailed band dispersions of Ce, Ce0.833La0.167, Ce0.638La0.362 and Ce0.383La0.617 thin films along K–Γ–K, in which the f–c hybridization bands are fitted by PAM model. The red dotted lines and green lines are the fitted results,while the black lines are the conduction band dispersions without f–c hybridization. All the PAM fitting results from(a)to(c)are also summarized in(h). (e)Normalized EDCs within±0.04 ?A around the Γ point of the thin films in(a)–(d),together with that of pure La thin film(ARPES spectra not shown here).(f)Normalized EDCs of(a)–(d)in(e)subtracted that of pure La thin films. The spectra are vertically shifted for a better view. (g)Peak areas of 4f15/2 (open squares)and 4f17/2 (open triangles)fitted from(f),together with the fitted indirect hybridization gaps ?g (open circles).

Additionally,we also analyzed the evolution of the f band intensities with La doping, which was done by investigating the energy distribution curves(EDCs)of the spectra.The EDC peak intensities of both 4f15/2and 4f17/2bands gradually decrease as the La concentration increases as shown in Figs.3(e)and 3(f). This phenomenon is quite similar to the f band evolutions of the Ce film as a function of temperature and mainly results from the localization of f electrons with increasing temperature.We can also find that the 4f15/2and 4f17/2peak areas of the EDCs decrease with increasing La concentration as shown in Fig. 3(g). When the f electron concentration is extremely low [see EDC of Ce0.383La0.617in Figs. 3(e) and 3(f)], there still have some 4f15/2intensities around the Fermi level. However, the f–c hybridization does not happen and the band dispersion cannot be fitted with the PAM as shown in Fig. 3(d).With the increasing f electron concentration, the intensity of f band strengthens and the f–c hybridization also emerges as shown in Figs.3(a)–3(c).

3.3. Theoretical consideration

whereV(ε)describes the hopping between the f level and the conduction states,and is related to the coupling strength?by?=πmaxε[|V(ε)|2].Nf1andNf2are the degeneracy of thej=7/2 andj=5/2 levels, i.e., 8 and 6, respectively. ?Eis the lowering of the energy when the f electron impurity is introduced,and could be calculated numerically.εfand ?εfare the position and the spin–orbit splitting of the f level,respectively.The definition of ?gcould be found in the appendix of the literature.[21]To numerically calculateρv(ε),an infinitesimal 0+was introduced(z=ε ?i0+)in the integration procedure,and the imaginary part of the integration corresponds to the valence photoemission spectra.

As for|V(ε)|2, instead of using the over-simplified semi-elliptical form,[21]we have adopted a double-peaked Lorentzian line-shape to account for the coupling between the f level and different conduction bands. As is well known,the matrix element effect would induce different photoemission spectra for various photon energies, which was not explicitly included in the GS model. However, it should be noted that the normal emission spectra of our Ce1?xLaxsamples resemble the resonance photoemission results of Ce, especially for pure Ce thin films.[2]Therefore, we could approximately simulate our experimental spectra as a combination of the total spectral weight of f-electron(regardless of the matrix element effect),and the contribution from the conduction bands,which is related to the linear combination of the Lorentzian peaks in|V(ε)|2.Additionally,the background is simulated by a quadratic curve, which corresponds to the systematic noise and the second order photoelectrons. During the fitting procedure, the spin–orbit splitting ?εfwas fixed at 280 meV.[24,25]The starting values of?andεfwere 0.125 eV and?1.2 eV,respectively. A Fermi–Dirac distribution (FDD) of 80 K was also introduced, which corresponds to the experimental temperature.All the fitting parameters were constrained in reasonable ranges in order to guarantee that the results have physical meaning.

The experimental valence spectra were extracted from Fig.2 by integrating the EDCs within±0.50 ?A?1around theΓpoint,and were further smoothed by the multicomponent function as mentioned in Ref.[2]. The resultant smoothed spectra are demonstrated as solid lines in Fig.4(a). The dotted lines in Fig.4(a)are the fitted valence spectra by the GS model. The overall fittings are quite satisfactory for La concentration below 53.75%,where the location,relative intensities and shape of the 4f0ionization peak, the Kondo peak nearEFand its 4f17/2spin–orbit splitting duplicate are well reproduced. As the La concentration increases further, the spectral intensity of the normal conduction band feature[indicated by the vertical arrows in Fig.4(a)]gradually prevails that of the 4f0peak,and the fitting results deviate from the experimental spectra below?1.5 eV. As mentioned above, the overall fitted spectrum is composed by the background and contributions from felectron and conduction bands,which could be extracted separately during our fitting procedure. To have a better view of the evolutions from various contributions, we took pure Ce and La concentrationx=24.96% as examples and extracted the background,f-electron and conduction bands,as shown in Fig. 4(b). We could see that the backgrounds (dashed black curves in Fig.4(b))show similar behavior, while the spectral contributions from f-electron and conduction bands demonstrate obvious variations. For the f-electron spectra (dashed red curves in Fig. 4(b)), the intensity ratio of the 4f0/4f1forx=24.96%is bigger than that of pure Ce, which means that the f-electron occupancy is higher forx= 24.96%. Additionally, the increase of the spectral weight of the conduction bands (dashed green curves in Fig. 4(b)) reveals the increasing contributions from La. The f electron occupancynfcould be calculated by integrating the square of the module of the wave functions of the states when the f electron impurity was introduced in the ground state.[21]The calculated f electron occupancy for pure Ce is approximate 0.86, and increases to～0.90 forx=24.06%, as shown in Fig. 4(c). The increase ofnfindicates that the f electrons in Ce1?xLaxbecome more localized,or in other word,the coupling strength of the f level with the conduction states weakens, which is consistent with the previous discussed PAM fitting results of the f–c hybridization. However,the relationship betweennfandxchanges asxgoes beyond 24.96%,i.e.,nfdecreases monotonically asxincreases,as shown in Fig.4(c). This could be attributed to the fact that the valence spectra demonstrate more and more La features asxincreases,and Ce atoms act as dopant instead of host, in which cases the GS model seems to be not adequate.Similar behaviors were also observed for the coupling strength?,as shown by the red marker line in Fig.4(c),despite that?decreased with La concentration up to 53.75%.

As mentioned above, the central idea of the GS model originated from the impurity Anderson model, and the coupling of f-level with conduction states resulted in the 3-peak feature (4f0, 4f15/2and 4f17/2) in the valence spectra of many Ce related materials.For the validity of this model,the screening of the local momentum of f electrons by the conduction electrons should be moderate, and the lattice structure of Ce should be intact to ensure the formation of f bands. La doping,on one hand, would effectively increase the ratio of conduction electrons to f electrons, i.e., strengthening the screening of f electron, and on the other hand, would gradually destroy the original Ce lattice, both of which could drive the system from strong f–c coupling to weak coupling,and eventually to no coupling. Additionally,according to the previous XRD results, La doping gradually expands the FCC lattice, exerting effective negative pressure on the lattice, which would also weaken the hybridization between f and conduction electrons,despite the fact that its effect is negligible for a few percentage of La doping. Therefore, we could safely conclude that the suppression ofγ →αphase transition is mainly due to the weakening of f–c hybridization, and on the contrary, the strengthening of f–c hybridization could drive theγ →αphase transition upon the condition of cooling or positive pressure.

Fig.4. (a)Experimental(solid lines)and fitted(dotted lines)valence photoemission spectra of Ce1?xLax thin films;(b)detailed fitting results for pure Ce and La concentration x=24.96%, as shown in the upper and lower sections. The total fitted spectra, background, contributions from f-electron and conduction bands are demonstrated by solid blue,dashed black,red and green curves,respectively.(c)Calculated f electron occupancy nf and fitted coupling strength ?for various La concentration.

4. Discussion and conclusion

La doping in Ce has two major influences: diluting the f electron concentration and expanding the lattice parameters. Our researches have demonstrated that the former would greatly weaken the f–c hybridization effect,and consequently suppress theγ →αphase transition in Ce upon cooling,or in other words, stabilize theγphase. This indicates that greater pressure should be expected for theγ →αphase transition to take place for La doped Ce samples at room temperature,which was indeed the case from the experimental results on bulk samples.[13]Additionally,many rare earth alloying additions on Ce, such as Sc, Pr, Dy and Lu, shrank the lattice by exerting positive chemical pressure and increased the phase transition pressure at room temperature.[12,13,26]One special alloying addition is the actinide element Pu,which also shrank the lattice from 0 up to～25%concentration,whereas decreasing the transition pressure. This means that Ce is prone to phase transition upon Pu doping. It is well known that Ce and Pu are counterparts to each other,[27,28]in the sense that: on one hand, they both undergo huge volume collapse, 17% for Ce and 25%for Pu,across theirγ →αandδ →αphase transitions,respectively;and on the other hand,the 4f-electron of Ce and 5f-electron of Pu both have dual itinerant and localized behaviors, demonstrating strong many-body effects and strong interaction with their conduction electrons. We conjecture that Pu doping would strengthen the f–c hybridization,

whereas other alloying additions would not. Another interesting alloying addition is Th, which also shrank the FCC lattice of Ce,however,the pressure needed for theγ →αphase transition at room temperature only decreased slightly, with a value of～3.6% at 20% of Th doping, compared with the value of～71.5%for 20%of Pu doping.[12,13]What is more,theγ →αphase transition persistently existed upon cooling for a large range of Th doping from 0 to 60%. Despite the atomic ground state configuration 6d27s2,Th metal in the FCC structure has a non-integer 5f-electron count due to the 5f–6d overlapping,[29]or in other word,f–c hybridization. Alloying the two elements most probably did not change the overall valence states and f-electron occupancy too much,thus Th doping had much smaller influence on the phase transition of Ce,compared with the other alloying elements.

To summarize,we have experimentally and theoretically investigated the effects of alloying addition on Ce by La. Lattice structure and electronic transport experiments have revealed linear expansion of the FCC lattice and suppression of theγ →αphase transition. Detailed ARPES measurements and theoretically model calculations have demonstrated the weakening of f–c hybridization and the important role it played in the suppression of theγ →αstructural phase transition. Our results provoke further studies on the electronic origins of the phase transitions and phase stabilities of rare earth and actinide metals and alloys.

Acknowledgements

Project supported by the National Key Research and Development Program of China (Grant Nos. 2021YFA1601100 and 2017YFA0303104), the SPC-Lab Research Fund (Grant No. WDZC201901), the Science Challenge Project (Grant Nos. TZ2016004 and TZ2018002), the National Natural Science Foundation of China(Grant Nos.U1630248,11774320,and 11904334), Special Funds of Institute of Materials(Grant No.TP02201904),and the Development Funds(Grant No.JZX7Y201901SY00900107).