Unveiling the Hybridization Process in a Quantum Critical Ferromagnet by Ultrafast Optical Spectroscopy
Y. H. Pei, Y. J. Zhang, Z. X. Wei, Y. X. Chen, K. Hu, Y.-F Yang, H. Q. Yuan, J. Qi
UUnveiling the Hybridization Process in a Quantum Critical Ferromagnet by UltrafastOptical Spectroscopy
Y. H. Pei, Y. J. Zhang, Z. X. Wei, Y. X. Chen, K. Hu, Yi-feng Yang,
3, 4, 5
H. Q. Yuan,
2, 6, 7, ∗ and J. Qi † State Key Laboratory of Electronic Thin Films and Integrated Devices,University of Electronic Science and Technology of China, Chengdu 611731, China Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Science, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Songshan Lake Materials Laboratory, Dongguan 523808, China State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics, Zhejiang University, Hangzhou 310058, China
We report the ultrafast optical pump-probe spectroscopy measurements on the recently discoveredquantum critical ferromagnet CeRh Ge . Our experimental results reveal the two-stage develop-ment of the hybridization between localized f moments and conduction electrons with loweringtemperature, as evidenced by (1) the presence of hybridization fluctuation for temperatures from ∼
85 K ( T ∗ ) to ∼
140 K ( T † ), and (2) the emergence of collective hybridization below the coher-ence temperature, T ∗ , marked by the opening of an indirect gap of 2∆ ≈
12 meV. We also observethree coherent phonon modes being softened anomalously below T ∗ , reflecting directly their cou-pling with the emergent coherent heavy electrons. Our findings establish the universal nature of thehybridization process in different heavy fermion systems. The ferromagnetic (FM) quantum critical point (QCP)is generally believed to be prohibited in a pure FM systemas it is often interrupted by other competing phases orfirst-order phase transitions [1]. The recent discovery of aFM QCP in the stoichiometric Kondo lattice compoundCeRh Ge under pressure has stimulated great intereston its origin and the nature of its associated strange metalphase [2]. One proposal for the existence of a FM QCPis that the local moments may form a triplet resonat-ing valence bound state in the FM state and cause asingular transformation in the patterns of their entan-glement with conduction electrons as the Kondo singletsdevelop at the QCP [3], which leads to an abrupt jumpof the Fermi surface volume. This is similar to the sce-nario of a local QCP in the antiferromagnetic Kondo lat-tice compounds [4]. One may naturally ask if and howdifferent types of inter-site magnetic correlations amonglocalized f moments (even outside the quantum criticalregime) might have an effect on their hybridization pro-cess with conduction electrons and the resulting Fermisurface change. It is thus important to explore the bandevolution of CeRh Ge near the Fermi energy ( E F ) andcompare it with the antiferromagnetic analogues.In this respect, the quantum oscillations and angle-resolved photoemission spectroscopy (ARPES) are themost direct experimental approaches to probe the bandstructure and its evolution. However, ARPES is highlylimited by its energy resolution when applied on theheavy fermion materials [5–7]. So far, it has not beenable to reveal the formation of the indirect hybridizationgap with an order of meV near E F . Fortunately, theultrafast optical pump-probe spectroscopy has recently been demonstrated to provide an alternative route to de-tect the hybridization dynamics over a wide temperaturerange [8]. It can reveal not only the “band bending”probed by the ARPES far above the coherence tempera-ture but also the opening of the indirect hybridizationgap below the coherence temperature, thus proposinga unified picture for spectroscopic and transport mea-surements. Although the measurements can only be per-formed at ambient pressure away from the quantum crit-ical point, one may still expect some useful informationon the microscopic dynamics of the hybridization physics.Indeed, this optical technique provides a unique way toinvestigate the dynamics of excited quasiparticles cou-pled to collective bosonic excitations in quantum sys-tems [9–12], and thus can help us to detect simultane-ously the exotic fermionic and bosonic responses nearand far from the coherent temperature T ∗ , below whichthe heavy electron state is developed.In this work, we report the optical pump-probe mea-surements on CeRh Ge for the first time. Hybridizationprocesses between the localized f moments and conduc-tion electrons are unraveled via the photoexcited quasi-particle dynamics in this material. Specifically, we haveobserved that the quasiparticle relaxation rate ( γ ) in theshort timescale shows a clear reduction as the tempera-ture decreases across T † ≈
140 K. Remarkably, γ exhibitsan apparent fluence-dependence for T < T ∗ (=85 ± T ∗ < T < T † . Such findings enable us to pre-dict in CeRh Ge (1) the occurrence of a narrow indi-rect hybridization gap of about 12 meV in the densityof states (DOS) associated with the formation of co- a r X i v : . [ c ond - m a t . s t r- e l ] F e b herent heavy electrons below T ∗ , and (2) the existenceof precursor hybridization fluctuations between T ∗ and T † . These results are in resemblance of that reportedpreviously for CeCoIn [8] and independent of the low-temperature magnetic properties. By contrast, recentmeasurements of quantum oscillations and calculationsof band structure suggest a localized nature of 4 f elec-trons in CeRh Ge in the ground state [13], suggestingthe necessity of a better theoretical understanding.The ultrafast time-resolved differential reflectivity∆ R ( t ) /R measurements were performed on the singlecrystal CeRh Ge from 5 K to room temperature us-ing a Ti:sapphire femtosecond laser producing a pulsewidth of ∼
35 fs at centre wavelength of 800 nm ( ∼ R ( t ) /R within the first tens of picoseconds can be attributed tothe electron-electron (e-e) and electron-boson scatteringprocesses in strongly correlated and metallic-like systems[14, 15]. The bosons may involve phonon or other bosonicexcitations [16, 17].We first focus on the non-oscillatory part of the signal.As shown in Fig. 1(b), the initial decay below ∼ . ∼
20 K, and leadsto the initial decay hidden inside a tiny peak at very lowtemperatures. Similar signals with a second rise havealso been obtained in many heavy fermion materials andother strongly correlated systems, and could originatefrom the excitations of electronic origins entangled withthe electron-phonon (e-ph) and nonthermal e-e scatter-ings [8, 9, 18]. We can fit the data within ∼ R ( t ) /R signal using a single exponentialformula, ∆ R/R = Ae − γt , to investigate the quasiparti-cle relaxation quantitatively. Here, A and γ are the am-plitude and decay rate, respectively. Figure 1(c) showsthe derived γ as a function of temperature under variouspump fluence. An evident fluence-dependent trend is ob-served below a critical temperature of 85 ± T ∗ , which is very close to the temperature where themagnetic resistivity exhibits a coherence peak (see theSupplemental Material) [2]. This concurrence indicatesthat the quasiparticle relaxation is influenced by the co-herent heavy electron state emerging below T ∗ , the sameas that have been observed in CeCoIn .The fluence-dependent phenomena of the decay rate γ below T ∗ can be explained by the widely used Rothwarf-Taylor (RT) model [19, 20], where the dynamics of quasi-particles ( n ) and bosons ( N ) are well described by thepresence of a narrow energy gap ∆ in DOS. Specifically, (b)(c)
293 K160 K140 K120 K100 K80 K60 K40 K20 K D R / R ( a r b . un i t s ) t (ps) (d)
10 50 2500.10.53.5 T † m J/cm m J/cm m J/cm g ( p s - ) T (K) T * T =26 K D R / R ( · - ) t (ps) m J/cm m J/cm m J/cm (a)
150 200 250 3002.53.03.54.0 g ( p s - ) T (K) m J/cm FIG. 1. (a) ∆ R ( t ) /R of CeRh Ge as a function of tem-perature. (b) ∆ R ( t ) /R at 26 K under various pump fluence.Strong fluence-dependence behaviour is clearly observed. (c)The decay rate γ as a function of temperature for differentpump fluences. Two critical temperatures ( T ∗ and T † ) areidentified. (d) γ at high temperature regime for a fluence of0.8 µ J/cm . The red line is a fit using the nonequilibriummodel described in the main text. the relaxation of the excited quasiparticles with an en-ergy larger than the gap ( (cid:126) ω > dndt = I + βN − Rn ,dNdt = 12 [ Rn − βN ] − ( N − N T ) τ − γ , (1)where I is the external excitation, n is the total numberof quasiparticles, R is the recombination rate of electron-hole pairs, N is the density of high frequency bosons withthe energy larger than 2∆, β is the probability per unittime for generating the nonequilibrium quasiparticles bysuch bosons, τ − γ is the escaping rate of the high fre-quency bosons, and N T is the thermal-equilibrium bosondensity. As long as R or τ − γ is large enough [20], thequasiparticle relaxation dynamics is dominated by theso-called bimolecular recombination process, which con- DOS (b) E (k) E F k ǀ ǁ (c) (d) EE F E pump boson A ( · - ) T (K) (a) g - ( p s ) T (K)0 20 40 60 800.00.20.4 n T T (K) FIG. 2. (a) Diagram of the collective hybridization betweenlocal f and conduction electrons below T ∗ , resulting in theformation of an indirect gap, 2∆. (b) The DOS near theFermi energy for a Kondo lattice below T ∗ . Nonequilibriumquasparticles are excited with an energy much larger than2∆. These Quasiparticles decay to the gap edge via emissionof high frequency bosons, i.e. phonons or other bosonic exci-tations. Subsequently, the bimolecular recombination domi-nates the decay and the relaxation rates become fluence de-pendent. (c) The density of thermally excited quasiparticles n T as a function of temperature below T ∗ . The inset showsthe temperature dependence of the amplitude A . (d) Decaytime γ − as a function of temperature below T ∗ . Values of A and γ here are for pump fluence of 0.8 µ J/cm . The redcurves are the fitted results using the RT model. tributes the nonlinear n term and is fluence-dependent.This type of process is schematically shown in Figs. 2(a)-(b), and can exactly elucidate the physics behind thefluence-dependent γ in Fig. 1(c).The gap formation in the DOS can be studied by fittingthe γ ( T ) and n ( T ) using the RT model [21–23], γ ( T ) ∝ (cid:20) δζn T + 1 + 2 n T (cid:21) (cid:0) ∆ + αT ∆ (cid:1) ,n T ( T ) = A (0) A ( T ) − ∝ ( T ∆) p e − ∆ /T , (2)where α , ζ and δ are fitting parameters, respectively. n T is the density of quasiparticles thermally excited acrossthe gap and p (0 < p <
1) is a constant determined bythe shape of the gapped DOS [24]. For a typical DOSof the Bardeen-Cooper-Schrieffer (BCS) form, we mayfix p = 0 . ≈
12 meV,reflecting the formation of an indirect hybridization gapbelow T ∗ due to collective hybridization associated withthe emergence of coherent heavy electron states near E F .For temperatures above T ∗ , the fluence-independency clearly indicates the closing of the indirect hybridizationgap. However, we notice that the behaviour of γ ( T )can be further separated into two regimes. For T > T † ( ∼
140 K) , γ firstly shows a saturation behavior and thendecreases slightly as the temperature increases. Such T -dependence cannot be explained by the conventionaltwo-temperature model [25]. Rather, it indicates a non-thermal process, i.e., the relaxation time due to the e-ecollisions can be longer than the e-ph relaxation time( τ e − e > τ e − ph ) [14], or the thermal distribution by e-e scatterings cannot be instantaneously attained, eventhough the excited fermionic quasiparticles may relaxclose to E F . Such process may be described by a nonequi-librium model [14]: 1 /τ e − ph = 3 (cid:126) λ (cid:104) ω (cid:105) / (2 πk B T l ), where λ (cid:104) ω (cid:105) represents the e-ph coupling and T l is the latticetemperature. If we assume γ = 1 /τ e − ph in the high tem-perature regime, i.e. >
230 K, this model can well explainwhy the measured γ becomes smaller as T increases (seeFig. 1(d)), which strongly supports the existence of thenonthermal process within the initial several picoseconds.We further obtain λ (cid:104) ω (cid:105) (cid:39)
94 meV . Assuming the De-bye frequency to be 7 THz (or 28 meV) based on ourmeasurements discussed below, we can estimate that λ takes a value of ∼ T ∗ and T † , γ ( T ) decreases with lowering tem-perature. Similar T -dependence has been observed inCeCoIn [8], where, in the absence of the indirect hy-bridization gap, precursor hybridization fluctuations in ashort correlation time- or length-scale were proposed toexplain the reduction of decay rate γ below T † [26]. Inspecific, both the e-e and electron-boson scatterings ofthe excited quasiparticles are expected to be suppressedas the fluctuating f moments start to participate in thehybridization with the conduction electrons. The devel-opment of hybridization fluctuations can further causethe renormalization (bending) of the conduction bandseven in the high temperature regime (e.g. >
100 K) [5],a prediction based on our experimental observation tobe examined using ARPES in CeRh Ge . This suggeststhat the f moments are not decoupled from conductionelectrons in the paramagnetic state as one may naivelythink for a small Fermi surface at ambient pressure. Notethat how the hybridization process evolves below 5 K re-mains to be investigated.The above results demonstrate that the hybridizationdynamics in CeRh Ge also exhibits a two-stage process,with the onset of precursor hybridization fluctuations be-low T † and the opening of an indirect hybridization gapbelow T ∗ , in resemblance of that in CeCoIn . The sepa-ration of the two stages is also manifested by the oscilla-tory component superimposed on the ∆ R ( t ) /R signals.The oscillations with terahertz (THz) frequency gener-ally originate from the coherent optical phonons triggeredby displacive excitations or photoexcited Raman process[27, 28].Figures 3(a) and (b) plot the oscillatory components FFT A m p . ( a . u . ) T = 20 K
Frequency (THz) T ( K ) W /2 p W /2 p W /2 p (a)(b)
293 K
140 K90 K D R / R ( · - ) t (ps) FIG. 3. (a) Extracted oscillatory components at severaltypical temperatures, e.g. 5 K, 90 K, 140 K and 293 K. (b)The Fourier transform spectra in the frequency domain forthe extracted oscillations from 260 K down to 5 K. Threecoherent modes are indicated by the dashed lines. extracted from the decay process after subtracting thenonoscillatory background. Three obvious terahertzmodes were observed at all investigated temperatures in-cluding Ω / π ∼ / π ∼ / π ∼ R/R ) osc = (cid:88) j =1 , , A j e − Γ j t sin(Ω j t + φ j ) , (3)where A j , Γ j , Ω j , and φ j ( j = 1 , ,
3) are the ampli-tude, damping rate, frequency, and phase, respectively.Ω j and φ j are related to an underdamped harmonic oscil-lator. Ω j = (cid:113) ω j − Γ j , where ω j is the natural frequency.The temperature-dependent evolution of ω j is shown in w /2 pw /2 p a =2 a =1.2 T † w / p ( T H z ) T (K) T * w /2 p FIG. 4. The derived ω j ( j = 1 , ,
3) as a function of tempera-ture using Eq. (3). The green lines represent the fit using theanharmonic phonon model. The orange and black lines arethe fit taking into consideration the contribution of Kondosinglets with different α , as described in the main text. Fig. 4. We see a sharp downturn below T ∗ for all threephonon modes, instead of a gradual flattening expectedby the anharmonic decay model [29, 30] (see also theSupplemental Material), as indicated by the green linesin Fig. 4. By contrast, the T -dependent ω j for T > T ∗ can be well explained by the anharmonic phonon-phononcoupling (red lines).The anomaly around T ∗ cannot arise from the phonon-magnon coupling effect because the long-range FM or-dering in CeRh Ge appears below a Curie temperatureof 2.5 K. To understand quantitatively the peculiar be-havior of ω j ( T ) for T < T ∗ , we calculated the values of δω j that represent the deviation between the expectedvalues from anharmonic phonon model and the experi-mental ω j / π . Such deviation must be associated withthe appearance of collective hybridization and the con-sequent gap opening in the DOS. In fact, δω j can beconnected to the quasiparticle density ( n T ) via the den-sity of Kondo singlets (cid:104) b i (cid:105) [8], which is proportional to[1 − n T ( T ) /n T ( T ∗ )]. Our best fit using δω j ∝ (cid:104) b i (cid:105) α ,where α is a fitting parameter, yields α = 1 . ± .
16, asshown in Fig. 4. Surprisingly, this value of α is nearly thesame as that obtained in CeCoIn within the experimen-tal errors, while the mean-field theory predicted α = 2.This indicates that a new and generic theory is requiredin order to explain the anomalous phonon softening below T ∗ in two quite different systems. In essence, this theoryshould take into account variation of the electron-phononcoupling induced by the DOS change near E F in the pres-ence of collective hybridization. Clearly, the observed fre-quency softening further proves that the phonon renor-malization can be a useful probe of the coherent heavyelectron states.Altogether, our results reveal the detailed hybridiza-tion process in the FM Kondo lattice compoundCeRh Ge over a wide temperature range. The reduc-tion of the relaxation rate below T † ∼
140 K suggeststhe possibility of “band bending” already in this hightemperature region, which could be detected in futureARPES experiments. Below the coherence temperature T ∗ ∼
85 K, we unveil an indirect band gap of ∼
12 meV,which plays the role of a protector of the coherent heavyelectron state. The associated anomalous softenning inthe frequencies of coherent optical phonons provides abenchmark for further theories. The observed two-stagehybridization process is in close resemblance of that inCeCoIn . The distinction in the magnetic correlationsappear to have no significant influence at least in themeasured temperature range (above 5 K) and at ambientpressure. This seems to suggest a universal mechanismfor the onset of heavy electron coherence independent ofthe details of the inter-site coupling among localized f moments. We should also note that our results in thenormal state are in contrast with the observation of asmall Fermi surface in the ferromagnetic ground state[13]. This distinction was not anticipated in previoustheories. It remains open how the hybridization processchanges upon entering the magnetic state. More elabo-rated studies are needed to clarify this point, and particu-larly its influence on the properties of quantum criticality[31]. Note:
During the submission of our manuscript, werealize an ARPES work on CeRh Ge by Y. Wu et al.[32], providing evidence for anisotropic hybridization be-tween f - and conduction electrons in the high tempera-ture regime.This work was supported by the National NaturalScience Foundation of China (Grants No. 11974070,No. 11974306, No. 12034017, No. 11734006, No.11774401, and No. 11974306), the Frontier ScienceProject of Dongguan (2019622101004), the National KeyR&D Program of China (Grants No. 2017YFA0303100,No. 2018YFA0307400, and No. 2016YFA0300202), theScience Challenge Project of China (No. TZ2016004),the Chinese Academy of Sciences Interdisciplinary Inno-vation Team, the Strategic Priority Research Programof CAS (XDB33010100), and the Key R&D Program ofZhejiang Province, China (2021C01002). ∗ [email protected] † [email protected][1] M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirk-patrick, Rev. Mod. Phys. , 039901 (2016).[2] B. Shen et al. , Nature , 51 (2020).[3] Piers Coleman, Yashar Komijani, and Elio J. K¨onig,
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