Charge-stripe Fluctuations in Nd_{4}Ni_{3}O_{8} as Evidenced by Optical Spectroscopy
Jiahao Hao, Xinwei Fan, Qing Li, Xiaoxiang Zhou, Chengping He, Yaomin Dai, Bing Xu, Xiyu Zhu, Hai-Hu Wen
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Charge-stripe Fluctuations in Nd Ni O as Evidenced by Optical Spectroscopy Jiahao Hao, Xinwei Fan, Qing Li, Xiaoxiang Zhou, ChengpingHe, Yaomin Dai, ∗ Bing Xu, Xiyu Zhu, and Hai-Hu Wen National Laboratory of Solid State Microstructures and Department of Physics,Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Department of Physics and Fribourg Center for Nanomaterials,University of Fribourg, Chemin du Mus´ee 3, CH-1700 Fribourg, Switzerland (Dated: January 12, 2021)We present an investigation into the optical properties of Nd Ni O at different temperaturesfrom 300 down to 5 K over a broad frequency range. The optical conductivity at 5 K is decomposedinto IR-active phonons, a far-infrared band α , a mid-infrared band β , and a high-energy absorptionedge. By comparing the measured optical conductivity to first-principles calculations and the opti-cal response of other nickelates, we find that Nd Ni O features evident charge-stripe fluctuations.While the α band may be related to impurities, the β band and the high-frequency absorption edgecan be attributed to electronic transitions between the gapped Ni- d x − y bands due to fluctuatingcharge stripes and the onset of transitions involving other high-energy bands, respectively. Further-more, an analysis of the temperature-dependent optical spectral weight reveals a T law, which islikely to originate from strong correlation effects. Since the discovery of unconventional superconductiv-ity in copper oxide compounds [1], tremendous effortshave been devoted to the exploration of cuprate-like ma-terials based on different transition-metal ions in thehope of finding new superconducting systems and gain-ing insights into the paring mechanism in cuprates. Dueto the proximity of Ni to Cu in the periodic table, nicke-lates have attracted great attention [2–11]. Anisimov etal. have pointed out that nickelate analogs to the super-conducting cuprates may be realized only if the Ni + (3 d )ions are forced into a planar oxygen coordination, whichmimics the parent compounds of the cuprates, and thendoped with low-spin Ni (3 d ) holes [2]. On the otherhand, Lee and Pickett [3] have compared LaNiO (Ni + ,3 d ) to CaCuO (Cu , 3 d ), and found very differentbehavior, such as significantly reduced 2 p -3 d hybridiza-tion and the existence of La 5 d at the Fermi level inLaNiO , arguing against the analogy between nickelatesand cuprates. Recently, superconductivity with a crit-ical temperature ( T c ) up to 15 K has been reported inhole-doped infinite-layer nickelates Nd − x Sr x NiO [12–14], reigniting the debate on whether nickelates representanalogs of cuprates [15–18].Hole-doped 3 d Ni ions in a planar oxygen coordinationalso exist in the trilayer R Ni O ( R = La, Pr or Nd).These compounds are 1/3 self hole doped with a nom-inal 3 d . filling, which should fall into the overdopedFermi-liquid regime in the phase diagram of hole-dopedcuprates [11]. From an experimental perspective, whilePr Ni O seems to be a metal without any phase tran-sition in the temperature range 2–300 K [6], La Ni O exhibits a semiconductor-insulator transition at about105 K, resulting in a highly insulating ground state [4–7].The formation of long-range antiferromagnetic order hasbeen detected below the transition at 105 K by La nuclear magnetic resonance (NMR) [4, 9] and neutron diffraction [19]; a recent synchrotron X-ray diffractionstudy has revealed that the ground state of La Ni O is a quasi-two-dimensional charge-stripe-ordered insula-tor [5]. Nd Ni O also shows insulating behavior withno signature of a phase transition from room tempera-ture down to 2 K [20]. Theoretically, the ground stateof R Ni O can be metallic with a low-spin (LS) con-figuration or insulating with a high-spin (HS) configu-ration [8, 21], depending on the relative magnitude ofHund’s rule coupling ( J H ) and the crystal field splitting(∆ cf ) between the d x − y and d z − r bands induced bythe absence of apical oxygen ions. Moreover, Botana etal. have pointed out that a low-spin charge stripe (LS-CS) insulating ground state can also be obtained froma combination of charge-order-related structural distor-tions and magnetic order [10].To date, questions regarding the ground state of thetrilayer R Ni O compounds, and to what extent theyare similar to the cuprates are still open to debate [4–7, 9, 19, 22]. We notice that the theoretically proposedLS metallic, HS insulating, and LS-CS insulating statesfor R Ni O are characterized by entirely distinct bandstructures, which consequently give rise to completelydifferent optical responses. Therefore, investigating theoptical properties of R Ni O and comparing them withfirst-principles calculations and the optical response ofother nickelates may provide pivotal information on thenature of its ground state.In this paper, we present an optical study of Nd Ni O at 15 different temperatures between 5 and 300 K in thefrequency range 30–50 000 cm − (3.75 meV–6.25 eV).The optical conductivity of Nd Ni O at 5 K consistsof several different components: IR-active phonons, afar-infrared band α , a mid-infrared band β , and a high-energy absorption edge. A comparison of our experi-mental results to theoretical calculations and the opticalTypeset by REVTEX
100 1000 100000.00.20.40.60.81.0 5 K 50 K 100 K 150 K 200 K 250 K 300 K Nd Ni O Wave number (cm -1 ) R e f l e c t i v i t y ( c m ) T (K)
10 100 1000Photon energy (meV)
Figure 1. Reflectivity of Nd Ni O up to 50 000 cm − atseveral selected temperatures from 300 down to 5 K. Inset:Resistivity of Nd Ni O as a function of temperature. properties of other nickelates manifests that charge-stripefluctuations exist in Nd Ni O . The α band, the β band,and the high-energy absorption edge can respectively beassociated with electronic transitions involving impuritybands, gapped Ni- d x − y states due to fluctuating chargestripes, and bands lying further away from E F . In addi-tion, the temperature-dependent optical spectral weightexhibits a T law even for a cutoff frequency as highas 12 000 cm − ( ∼ Ni O wereobtained by reacting Nd Ni O with CaH [20, 23]. Theresistivity ρ of our sample, as shown in the inset of Fig. 1,rises with decreasing temperature, which is a clear sig-nature of insulating behavior [20]. The near-normal in-cidence reflectivity R ( ω ) of Nd Ni O was measured at15 different temperatures between 5 and 300 K in thefrequency range from 30 to 12 000 cm − using a BrukerVertex 80v Fourier transform spectrometer. An in situ gold overfilling technique [24] was utilized to obtain theabsolute reflectivity. We then used an AvaSpec-2048 × R ( ω ) to 50 000 cm − at room temperature.Figure 1 shows the measured R ( ω ) of Nd Ni O up to50 000 cm − at several representative temperatures from300 down to 5 K. The far-infrared R ( ω ) below about600 cm − is dominated by IR-active phonons (sharp fea-tures). Below the phonon bands, R ( ω ) approaches a con-stant that is much lower than unity, which is the proto-typical optical response of an insulator [25]. This is con-sistent with the transport measurement, as shown in theinset of Fig. 1, which also reveals insulating behavior inthe temperature range of 2–300 K [20]. Above the IR-active phonon bands, R ( ω ) exhibits a broad hump-like
100 1000 100000.00.51.01.52.0 5 K 50 K 100 K 150 K 200 K 250 K 300 K () ( - c m - ) Nd Ni O Wave number (cm -1 ) () ( - c m - ) Wave number (cm -1 )T = 5 K
10 100 1000Photon energy (meV)
Figure 2. The real part of the optical conductivity σ ( ω ) ofNd Ni O at different temperatures from 300 down to 5 K.The inset shows σ ( ω ) in the far-infrared range at T = 5 K.The blue solid circle at zero frequency denotes the dc conduc-tivity from transport measurements. feature in the frequency range 600–5000 cm − ; a similarfeature has been reported in the charge-stripe-orderedLa − x Sr x NiO [26, 27] and La NiO δ [28].In order to obtain further information about the chargedynamics of Nd Ni O , the real part of the optical con-ductivity σ ( ω ) is determined through a Kramers-Kroniganalysis of R ( ω ). Since R ( ω ) exhibits a tendency to-wards saturation in the far-infrared range, a constantwas used for the low-frequency extrapolation. On thehigh-frequency side, we adopted a constant R ( ω ) up to12.5 eV followed by a free-electron ( ω − ) response. Fig-ure 2 displays σ ( ω ) of Nd Ni O at different tempera-tures from 300 down to 5 K. The inset shows σ ( ω ) at 5 Kin the far-infrared range, and the blue solid circle at zerofrequency denotes the dc conductivity σ dc at 5 K fromtransport measurements. The zero-frequency extrapola-tion of the optical conductivity σ ( ω →
0) agrees quitewell with σ dc , testifying to the self-consistency of our ex-perimental results. The far-infrared σ ( ω ) is dominatedby IR-active phonon modes which are characterized assharp peaks below 600 cm − . The mid-infrared σ ( ω )features a broad absorption band lying in the frequencyrange of 600–5000 cm − , which grows in amplitude asthe temperature is lowered. Above 10 000 cm − , σ ( ω )increases sharply, leading to an absorption edge at about20 000 cm − .The measured σ ( ω ) of Nd Ni O can be fit to a seriesof Lorentzian oscillators, σ ( ω ) = 2 πZ X j γ j ω Ω j ( ω j − ω ) + γ j ω , (1)where Z ≃
377 Ω is the vacuum impedance; ω j , γ j
10 100 1000 100000.00.51.01.52.0 Wave number (cm -1 ) T = 5 K MIR band () ( - c m - ) IR-active phononsFIR band
10 100 1000Photon energy (meV)
Figure 3. The blue solid curve denotes σ ( ω ) of Nd Ni O measured at 5 K. The red dashed line through the data isthe fitting result, which is decomposed into an FIR band α (orange dashed line), an MIR component β (green dashedline), and a high-energy absorption edge (cyan dashed line). and Ω j correspond to the resonance frequency, linewidthand strength of the j th oscillator, respectively. The reddashed line in Fig. 3 represents the fitting result at 5 K,which reproduces the measured σ ( ω ) (blue solid line)quite well. As illustrated in Fig. 3, the fit allows us to de-compose σ ( ω ) of Nd Ni O at 5 K into several differentcomponents: (i) IR-active phonon modes which are notshown in the figure; (ii) a far-infrared (FIR) absorptionband α under the phonon modes (orange dashed line);(iii) a mid-infrared (MIR) absorption band β centered atabout 2500 cm − (green dashed line); (iv) an absorptionedge at about 20 000 cm − (cyan dashed line). In thefollowing, we compare our experimental results to first-principles calculations and the optical response of othernickelates, aiming at understanding the origins of thesecomponents (except for the phonon modes) in σ ( ω ) andthe ground state of Nd Ni O .First of all, it should be noted that a Drude com-ponent is absent in σ ( ω ) of Nd Ni O , agreeing wellwith the insulating behavior revealed by transport mea-surements [20]. Nevertheless, we would like to mentionthat in systems with strong electronic anisotropy, suchas cuprates and iron pnictides, the optical response of apolycrystalline sample is dominated by the weakly con-ducting c axis [29–34], while the Drude response associ-ated with the metallic ab plane is significantly suppressed.Hence, the absence of a Drude component in σ ( ω ) ofthe polycrystalline Nd Ni O points towards two pos-sibilities: either this material is insulating in both the ab plane and the c direction, or it is highly anisotropicwith a metallic ab plane and an insulating c direction.The latter possibility may be ruled out by the insulatingtransport properties of Nd Ni O [20], because for com-pounds that are metallic in the ab plane but insulating d z -r () ( - c m - ) (b) LS-CSHS LS d x -y xx zz ave = (2 xx + zz )/3 (d) 10 100 1000Photon energy (meV)10 100 1000 10000 024 (f) Wave number (cm -1 ) Figure 4. (a), (c) and (e) display the calculated band struc-tures of Nd Ni O for the LS metallic, HS insulating andLS-CS insulating states, respectively. (b), (d) and (f) depictthe calculated optical conductivity spectra based on the bandstructures in (a), (c) and (e), respectively. along the c axis, for example cuprates and iron pnictides, ρ ( T ) of a polycrystalline sample has been found to exhibitprominent metallic behavior [33, 35, 36].To gain more insight into the origins of the differentcomponents in σ ( ω ), we calculated the band structure ofNd Ni O and corresponding σ ( ω ) using density func-tional theory (DFT) implemented in the full-potentiallinearized augmented plane wave code WIEN2k [37–40].The structural information of Nd Ni O was taken fromRef [20]. The Perdew-Burke-Ernzerhof (PBE) general-ized gradient approximation (GGA) [41] was chosen asthe exchange-correlation potential. To avoid the ambigu-ity of the 4 f electrons of Nd around the Fermi energy, weemployed the so-called GGA+ U scheme with an effectiveHubbard U eff = 9 eV on the 4 f electrons of Nd to repelthem away from the Fermi level. We also applied U eff =5 eV on the 3 d electrons of Ni, since the correlation effectbetween 3 d electrons of transition-metal atoms should beprofound. Since σ ( ω ) of a polycrystal consists of contri-butions from both the ab plane and the c direction, wecalculated both the in-plane σ xx ( ω ) = σ yy ( ω ) and out-of-plane σ zz ( ω ) optical conductivities, and then comparethe average σ ave = ( σ xx + σ yy + σ zz ) / σ xx + σ zz ) / σ ( ω ) of Nd Ni O .Figure 4(a) displays the calculated band structure ofthe LS state of Nd Ni O which reveals a metallic phasewith the partially filled Ni- d x − y bands crossing theFermi level, in good agreement with previous calculationsfor La Ni O [4, 21, 42] and Pr Ni O [11, 43]. The cal-culated σ xx and σ zz for the LS state are shown as red andblue dashed lines respectively in Fig. 4(b). A pronouncedDrude component, the optical signature of metallic be-havior, can be seen in σ xx . Although the absence ofDrude response in σ zz implies insulating behavior alongthe c direction, the metallic σ xx would inevitably pro-duce coherent electronic transport in the temperature-dependent ρ ( T ) of the polycrystalline Nd Ni O . Therobust insulating behavior in ρ ( T ), as shown in the insetof Fig. 1 and Ref. [20], suggests that the LS metallic statefails in describing the experimental results of Nd Ni O .Thus, models yielding an insulating ground state shouldbe invoked.We recall that La Ni O is also highly insulating atlow temperatures. In order to account for the insulatingbehavior in La Ni O , Pardo and Pickett consider theNi trilayer as a molecular trimer with in-plane AFM or-der, from which on-site repulsion U gives rise to a Mottinsulating state, i.e. the HS insulating state [8]. Alter-natively, Botana et al. have pointed out that an insu-lating ground state, the LS-CS insulating state, can alsobe obtained from a combination of charge-order-relatedstructural distortions and magnetic order [10]. We applythese two approaches to Nd Ni O and derive σ xx , σ zz ,as well as σ ave from the corresponding electronic struc-ture in each case.The calculated band structure of Nd Ni O for the HSinsulating state, which is similar to the previous calcu-lation for La Ni O [8], is traced out in Fig. 4(c). Alarge (direct) gap of ∼ d z − r states that are depicted as blue solid circles. Figure 4(d)shows the optical conductivity spectra calculated fromthe band structure in Fig. 4(c). A sharp peak arising fromdirect interband transitions between the d z − r bandscan be identified at about 13 000 cm − in σ xx . On thelow-frequency side, σ xx and σ zz vanish below 10 000 and15 000 cm − , respectively. It is immediately obvious thatthe measured σ ( ω ) of Nd Ni O contrasts with the cal-culated σ ave [purple solid line in Fig. 4(d)]: (i) absorptionbands, i.e. α and β , exist in the low-frequency (below ∼
10 000 cm − ) region of the measured σ ( ω ), while σ ave is vanishingly small in the same spectral range; (ii) thesharp peak at 13 000 cm − in the calculated σ ave is notobserved in the measured σ ( ω ). Such considerable dis-crepancies between the calculated σ ave and the measured σ ( ω ) do not favor the HS insulating state in Nd Ni O .Next, we consider the LS-CS insulating state. Fig-ure 4(e) shows the calculated band structure of Nd Ni O for the LS-CS insulating state, which also resembles theprevious result in La Ni O [10]. A much smaller gapis formed between the Ni d x − y bands (red solid cir-cles), which is completely different from the HS insulatingstate. The calculated σ xx and σ zz are shown in Fig. 4(f)as red and blue dashed lines, respectively. While σ xx diminishes with decreasing frequency and disappears at ∼ − , σ zz exhibits a broad absorption peak at ∼ − with a tail extending down to ∼ − .The peak and the low-energy tail in σ zz are predom-inantly contributed by interband electronic transitionsacross the gap between the Ni d x − y bands. Intrigu-ingly, the calculated σ ave qualitatively reproduces themeasured σ ( ω ) in the MIR and high-frequency range,hinting that the ground state of Nd Ni O might be (oris in close proximity to) an LS-CS insulator as previouslyproposed for La Ni O [10]. Within this framework, the β band may be ascribed to electronic transitions betweenthe gapped d x − y bands, and the high-frequency absorp-tion edge is associated with the onset of electronic tran-sitions involving bands lying further away from E F .It is noteworthy that the optical response of Nd Ni O bears a remarkable resemblance to that of charge-stripe-ordered La − x Sr x NiO [26, 27, 44, 45] andLa NiO δ [28]. An MIR peak akin to the β band inNd Ni O has also been observed in σ ( ω ) of these ma-terials [26–28, 44, 45]. While early studies have assignedthe MIR peak to the formation of small polarons [27, 44–46], Homes et al. [28] have taken into account the factthat the holes doped into the NiO planes form a stripeorder which has been established by neutron and X-rayscattering studies [47, 48], and consequently ascribed theMIR peak to transitions from filled valence states toempty mid-gap states associated with the charge stripes.Furthermore, as the MIR peak persists to temperaturesfar above the charge-stripe-ordering transition temper-ature T CO [28, 44], recent observations of short-range-ordered or dynamic charge stripes on the same temper-ature scale have naturally related it to the presence ofcharge-stripe fluctuations [49–51]. Given the striking re-semblance between the optical response of Nd Ni O andthat of La NiO δ and La − x Sr x NiO , it is plausible toassociate the β band in Nd Ni O with the emergenceof charge stripes or charge-stripe fluctuations. The for-mation of charge stripes below the transition at 105 Kin La Ni O has been confirmed by the synchrotron X-ray diffraction [5] whereas no phase transition has beendetected in Nd Ni O from room temperature down to2 K [20]. In this case, the β band in Nd Ni O is mostlikely to signify charge-stripe fluctuations, i.e. short-range-ordered or dynamic charge stripes.The presence of charge-stripe fluctuations or even astatic charge-stripe order in Nd Ni O is not surprising,because it is structurally and electronically identical toLa Ni O which has been found to order in charge stripesbelow the transition at 105 K [5]. Although the rela-tively smaller atomic radius of Nd may suppress the staticlong-range order in Nd Ni O , charge-stripe correlationsshould not be significantly affected by such a small dif-ference. In addition, similar to the case of La Ni O [10],the calculated LS-CS state has a lower free energy thanthe HS molecular insulating state, representing a favor-able ground state for Nd Ni O . T (10 K ) W ( T ) / W ( K ) Nd Ni O -1 -1 -1 -1 -1 c Figure 5. Spectral weight up to different cutoff frequencies asa function of T . The dashed lines are linear fits. We further notice that neither the HS nor the LS-CSinsulating state can account for the α band in the mea-sured σ ( ω ) of Nd Ni O . A similar FIR absorption bandhas been observed in doped semiconductors Si:P and as-signed to optical transitions between impurity states andthe conduction band [52]; Some bismuth-based topolog-ical insulators, e.g. Bi Te Se and Bi − x Ca x Se , alsoexhibit such an impurity-related absorption band cen-tered at about 200 cm − [53, 54]. By analogy with Si:Pand bismuth-based topological insulators, the α band inNd Ni O may be attributed to transitions involving in-gap impurity states.Finally, we examine the evolution of σ ( ω ) with tem-perature. It is noticeable in Fig. 2 that σ ( ω ) in theFIR and MIR region grows as the temperature is low-ered. This is distinct from materials with charge stripesor charge-stripe fluctuations, such as La − x Sr x NiO andLa NiO δ , in which a suppression of the low-frequency σ ( ω ) alongside an enhancement of σ ( ω ) in the MIRpeak region has been observed [28, 50]. Hence, the un-usual temperature dependence of the optical response inNd Ni O is likely to be dominated by a different effect.The detailed temperature dependence of σ ( ω ) can betracked by inspecting the optical spectral weight W de-fined as W = Z ω c σ ( ω ) dω, (2)where ω c represents a cutoff frequency. The symbols inFig. 5 denote W as a function of T for different cutofffrequencies, and the dashed lines through the symbolsare linear fits. W increases upon cooling even for ω c = 12 000 cm − ( ∼ ∼ W varies linearly with T for all ω c ’s we have cho- sen. Such behavior has been widely observed and exten-sively studied in cuprates [55–59]. In a normal metal, e.g.gold [56], W follows T -linear dependence for ω c < ω p ,but is temperature independent for ω c ≥ ω p . Here, ω p isthe plasma frequency which is proportional to the carrierdensity. However, in cuprates [55–59], W exhibits con-spicuous T -linear dependence even for ω c ≥ ω p , whichis in sharp contrast to a conventional metal. Dynam-ical mean-field theory calculations based on a stronglycorrelated Hubbard model have demonstrated that the T law of W in cuprates arises from strong correlationeffects [57]. The T -linear W in Nd Ni O is peculiar,since it is an insulator without any observable free-carrierresponse. Nevertheless, we notice that as the tempera-ture is lowered, a net spectral weight gain in the FIR-MIRrange also occurs in the slightly doped but still insulat-ing cuprates [60, 61], even though detailed temperaturedependence of the optical spectral weight in these mate-rials has not yet been reported. Considering these facts,the T -linear dependence of W in Nd Ni O may be gov-erned by the same mechanism as the one in cuprates, i.e.strong correlation effects. Therefore, strong electroniccorrelation may also play an important role in Nd Ni O .To summarize, we experimentally obtained the opti-cal conductivity of Nd Ni O at 15 different tempera-tures between 5 and 300 K in the frequency range 30–50 000 cm − (3.75 meV–6.25 eV). Data fitting based onthe Lorentz model allows us to decompose the opticalconductivity at 5 K into IR-active phonons, an FIR α band, an MIR β band, as well as a high-energy absorptionedge. A comparison of the measured optical conductiv-ity to theoretical calculations and the optical response ofother nickelates suggests that Nd Ni O is an insulatorwith charge-stripe fluctuations. The α band is likely tobe associated with impurities; the β band and the high-frequency absorption edge can be ascribed to electronictransitions across the charge-stripe-fluctuation-inducedgap between the Ni- d x − y states and the onset of tran-sitions from other high-energy bands, respectively. Theoptical spectral weight varies linearly with T even for acutoff frequency as high as 12 000 cm − ( ∼ T law, which is also widely observed in cuprates, is adirect consequence of strong correlation effects.We thank Ricardo Lobo, Jiawei Mei, Yilin Wang, andShunli Yu for illuminating discussions. We gratefully ac-knowledge financial support from the National Key R&DProgram of China (Grant No. 2016YFA0300401), theNational Natural Science Foundation of China (GrantsNo. A2008/11874206, E0209/52072170, 12061131001),the Strategic Priority Research Program of ChineseAcademy of Sciences (Grant No. XDB25000000), theFundamental Research Funds for the Central Univer-sities with Grant No. 020414380095, and Jiangsushuangchuang program.Jiahao Hao, Xinwei Fan and Qing Li contributedequally to this work. ∗ [email protected][1] J. G. Bednorz and K. A. M¨uller, Z. Phys. B , 189(1986).[2] V. I. Anisimov, D. Bukhvalov, and T. M. Rice, Phys.Rev. B , 7901 (1999).[3] K.-W. Lee and W. E. Pickett, Phys. Rev. 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