Lattice-Mediated Magnetic Order Melting in TbMnO3
Edoardo Baldini, Teresa Kubacka, Benjamin P. P. Mallett, Chao Ma, Seyed M. Koohpayeh, Yimei Zhu, Christian Bernhard, Steven Lee Johnson, Fabrizio Carbone
LLattice-mediated magnetic order melting in TbMnO E. Baldini ∗ ,
1, 2, 3
T. Kubacka, B. P. P. Mallett, C. Ma, S. M.Koohpayeh, Y. Zhu, C. Bernhard, S. L. Johnson, and F. Carbone Institute of Physics and Lausanne Center for Ultrafast Science (LACUS),´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland Institute of Chemical Sciences and Engineering and Lausanne Center for Ultrafast Science (LACUS),´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA Institute for Quantum Electronics, Eidgen¨ossische Technische Hochschule (ETH) Z¨urich, CH-8093 Z¨urich, Switzerland Department of Physics, University of Fribourg, Chemin du Mus´ee 3, CH-1700 Fribourg, Switzerland Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, China Institute for Quantum Matter, Department of Physics and Astronomy,Johns Hopkins University, Baltimore, Maryland 21218, USA Department of Condensed Matter Physics, Brookhaven National Laboratory, Upton, New York 11973, USA (Dated: January 16, 2018)Recent ultrafast magnetic-sensitive measurements [Phys. Rev. B , 184429 (2015) and Phys.Rev. B , 184414 (2017)] have revealed a delayed melting of the long-range cycloid spin-order inTbMnO following photoexcitation across the fundamental Mott-Hubbard gap. The microscopicmechanism behind this slow transfer of energy from the photoexcited carriers to the spin degreesof freedom is still elusive and not understood. Here, we address this problem by combining spec-troscopic ellipsometry, ultrafast broadband optical spectroscopy and ab initio calculations. Uponphotoexcitation, we observe the emergence of a complex collective response, which is due to high-energy coherent optical phonons coupled to the out-of-equilibrium charge density. This responseprecedes the magnetic order melting and is interpreted as the fingerprint of the formation of anti-Jahn Teller polarons. We propose that the charge localization in a long-lived self-trapped statehinders the emission of magnons and other spin-flip mechanisms, causing the energy transfer fromthe charge to the spin system to be mediated by the reorganization of the lattice. Furthermore,we provide evidence for the coherent excitation of a phonon mode associated with the ferroelectricphase transition. I. INTRODUCTION
In the last years, increasing attention has been devotedto the study of type-II multiferroics, with the aim ofusing electric fields to switch magnetic order, or mag-netic fields to switch the spontaneous electric polariza-tion [1]. One of the ultimate goals in this research areainvolves the ability to achieve dynamic switching on ul-trafast timescales by means of tailored laser pulses. Tobetter understand the dynamics of magnetic order, sev-eral time-resolved techniques that are directly sensitive tolong-range spin order are currently being explored. Forexample, time-resolved cryo-Lorentz transmission elec-tron microscopy holds huge promise for imaging the dy-namical evolution of magnetic correlations in real spacewith nanometer spatial resolution [2, 3]. This approachhas recently reached a temporal resolution of 700 fs, thusallowing for the observation of fast spin-related phenom-ena occurring below the 50 ps timescale [4]. An alter-native method is represented by time-resolved resonantelastic x-ray scattering (trREXS), which is currently ca-pable of a time resolution of 100 fs in slicing facilities atthird-generation synchrotron sources and in free-electronlasers [5–10]. Within this approach, after the photoexci-tation with an ultrashort laser pulse, a soft x-ray probetuned to a resonance can provide sensitivity to spin, or- bital, and charge order in combination with element se-lectivity.Within the class of type-II multiferroics, one of themost studied systems is TbMnO , which crystallizes in aGdFeO -distorted structure (space group Pbnm ). Withregard to its magnetic properties, many competing ex-change paths exist between nearest neighbour and nextnearest neighbour Mn ions, which lead to frustration ofthe magnetic moments on the Mn sites [11]. This mag-netic frustration is at the origin of the exotic long-rangespin ordering patterns that characterize TbMnO [12–17]. At room temperature this material is paramagnetic.When the temperature is decreased below 100 K, short-range spin correlations start to develop and a magneticand structural fluctuating regime dominates over a widetemperature range. The first magnetic phase transitionis observed at T N1 = 42 K with the establishment of aparaelectric spin-density wave (SDW) state, in which asinusoidal antiferromagnetic spin structure forms alongthe b -axis (Fig. 1(a)). A second magnetic phase transi-tion occurs around T N2 = 27 K, at which the spins on theMn sites order in a spin cycloid in the bc plane (Fig.1(b)). An additional transition related to the 4 f spins ofthe Tb ions is finally observed at 7 K [13, 18]. The cy-cloid spin ordering of the Mn sites below T N2 createsa spontaneous ferroelectric polarization along the c -axis a r X i v : . [ c ond - m a t . s t r- e l ] J a n FIG. 1. Schematic representation of spin order in TbMnO (a) in the paraelectric SDW state (T N2 < T < T N1 ) and (b) inthe spin cycloid phase (T < T N2 ). Here T N1 = 42 K, T N2 =27 K. Mn atoms are reported in blue, Tb atoms in violet andO atoms are omitted at the vertices of the MnO octahedra.Ellipticity in (b) has been neglected. due to the combined effects of the inverse Dzyaloshinskii-Moriya interaction and of the Tb ionic displacement[13, 16, 18, 19].By using trREXS upon resonant THz excitation of anelectromagnon in the cycloid spin-ordered phase, it hasbeen recently demonstrated that a coherent dynamics ofthe magnetic structure can be driven [6]. On the otherhand, optical excitation in the near-infrared/visible rangehas been shown to lead to a delayed melting of the long-range magnetic order on a time scale of ∼
20 ps [7]. Time-resolved measurements of magnetic x-ray scattering havesuggested that the loss of magnetic order and the as-sociated Mn 3 d orbital reconstruction follows a quasi-adiabatic pathway from the cycloidal phase through anintermediate sinusoidal phase, similar to what is seenwhen slowly increasing the temperature [10]. Unlike thesituation seen in thermal equilibirum, the wavevector ofthe spin ordering in the intermediate sinusoidal phaseremains at a value equivalent to the cycloidal orderingwavevector. A direct coupling between magnetic andorbital orders was also found by monitoring the orbitalreconstruction induced by the spin ordering [10]. Al-though these studies shed light on various mechanismsoccurring in TbMnO upon photoexcitation, a completemicroscopic explanation for the delayed demagnetizationprocess is still lacking. Such slow dynamics imply a de-layed transfer of energy from the photoexcited carriers to the spin system, which is at odds with the behaviorobserved in other correlated oxides with long-range spinorder [8, 20]. As such, the dynamic bottleneck may bedue to the lattice or to the spin system itself, or to acombination of both.A promising route to elucidate the energy pathway andthe possible involvement of other degrees of freedom inthe melting process is to perform ultrafast optical spec-troscopy with a combination of broad wavelength rangeand high temporal resolution [21–24]. A broad wave-length range is crucial when large shifts of the opticalspectral weight (SW) are induced by magnetic order-ing phenomena. Photons in the optical range can accessmagnetic and other complex ordering effects, as changesof the symmetry in a system upon ordering are often re-flected in the optical properties of the material. Thismechanism is at play in the insulating perovskite man-ganites and manifests on the d - d intersite transitions ofthe optical spectrum [25]. With a time resolution below50 fs, one can reveal the timescales on which the excitedelectronic states redistribute their excess energy, unveil-ing the fingerprint, if any, of specific collective modes as-sociated with the ultrafast rearrangement of the lattice orspin subsystems. In this work, we combine steady-statespectroscopic ellipsometry (SE), ultrafast broadband op-tical spectroscopy with a time resolution of 45 fs and abinitio calculations to elucidate the hierarchy of phenom-ena following the photoexcitation of TbMnO across thefundamental Mott gap and leading to the delayed melt-ing of the cycloid spin-order. Our results are suggestiveof a scenario in which anti-Jahn-Teller (JT) polarons areformed upon photoexcitation, leading to the emergenceof a fast coherent lattice response. We propose that thecharge localization in a long-lived self-trapped state hin-ders the emission of magnons and other spin-flip mecha-nisms, causing the energy transfer from the charge to thespin system to be mediated by the reorganization of thelattice.The paper is organized as follows: In Sec. II we reportthe steady-state optical properties of the material usingSE and we perform ab initio calculations to assign thefeatures in the optical spectra. Next, in Sec. III, weshow the out-of-equilibrium experiments and describe thespectro-temporal signatures for the melting of the long-range spin order. We conclude with Sec. IV, offeringa complete picture of the ultrafast dynamics followingphotoexcitation of TbMnO across the Mott gap. II. STEADY-STATE OPTICAL PROPERTIESA. Spectroscopic ellipsometry
Here we present temperature dependent SE measure-ments of a detwinned (010)-oriented single crystal ofTbMnO in the spectral range between 0.50 eV and 6.00 , a () ( - c m - ) (a) (b) -2.5-2.0-1.5-1.0-0.5654321 Energy (eV) , a () ( - c m - ) -2.5-2.0-1.5-1.0-0.5654321 Energy (eV) , c () ( - c m - ) (c) (d) , c () ( - c m - ) FIG. 2. Temperature dependence of the (a,c) real σ and(b,d) imaginary σ parts of the complex optical conductiv-ity of TbMnO ; (a,b) refer to the optical response for lightpolarized along the a -axis, (c,d) for light polarized along the c -axis. The red arrow indicates the pump photoexcitation at1.55 eV in the nonequilibrium experiment. The grey shadedarea represents the spectral region monitored by the probealong the two different axes. eV. Details on the sample preparation and the SE exper-iments are given in the Supplementary Materials (SM).Figures 2(a-d) show the temperature dependence of thereal- ( σ ) and imaginary ( σ ) parts of the optical con-ductivity, measured with light polarized parallel to the a - ( E (cid:107) a) and c -axis ( E (cid:107) c) of the crystal. We observea marked anisotropy between the a - and c -axis response,which confirms the complete detwinning of our TbMnO single crystal. The σ a spectrum is featureless in thenear-infrared region and is dominated by two broad ab-sorption bands, centred around 2.45 eV and 5.15 eV atall measured temperatures. As the temperature is in-creased from 8 K to 300 K, a number of smaller spectralfeatures emerge on top of the optical band at 2.45 eV.In contrast, the σ c ( ω ) spectrum is characterized by astrongly increasing conductivity with frequency, partic-ularly between 2.00 and 3.00 eV, with complex featuresappearing at 3.80 eV and 6.00 eV that evolve with tem-perature. The latter consists of an optical band centeredaround 4.50 eV at 8 K, which progressively loses SWfor increasing temperature and splits into two well dis-tinguished features at 3.90 eV and 5.15 eV above 25 K.The observed anisotropic response is in agreement withthat reported in a previous study on detwinned TbMnO single crystals [26]. From our SE measurements, we calculate theanisotropic reflectivity spectra (R) under equilibriumconditions, since our ultrafast study involves the mea-surements of the transient reflectivity (∆R/R) ofTbMnO . The spectra are shown as a function of temper-ature in Figs. 3(a,b) along the a - and c -axis, respectively.The displayed range is limited to the one covered by thebroadband probe in the nonequilibrium experiment. Thespectrum of R along the a -axis consists of a pronouncedfeature centred around 1.90 eV with a fine structure nearits maximum. A second feature emerges around 2.60 eV,separated from the former by a dip in the reflectivity.Along the c -axis, the reflectivity monotonically increaseswith higher energies and shows a rich structure in the2.40 - 2.80 eV spectral region. Consistent with the SWshifts identified in the previous discussion, we observe anopposite trend of R as a function of temperature for thetwo different axes. While R a undergoes a prominent dropwith increasing temperature, R c increases over the wholespectral range. This aspect enters into the interpretationof the pump-probe data presented in Sec. III. B. Assignment of the optical features
To assign the observed structures, we rely on the re-sults reported in literature on orthorhombic mangan-ites [25–37] and perform ab initio calculations based onDensity-Functional Theory (DFT). First, we focus on themain features characterizing the optical spectrum. Pre-vious work attempted to clarify the origin of the differ-ent absorption bands by identifying possible signaturesof the magnetic ordering on the optical response. Par-ticular attention was devoted to LaMnO , which retainsthe lowest degree of structural distortion from the idealperovskite. Early studies assigned both the low-energyand high-energy bands to p - d charge-transfer (CT) exci-tations [27–29] or associated the low-energy feature withJT orbiton excitations or d - d crystal field transitions[38]. Later on, the validity of these assignments was ques-tioned by the observation of pronounced rearrangementsof the SW between the two bands close to T N = 140K [25, 30, 31]. It was concluded that the broad low-energy band in the ab -plane response of LaMnO origi-nates from intersite Mn d - d CT transitions ( i.e. d i d j → d i d j between neighbouring i and j Mn ions), preserv-ing the electronic spin state. It is therefore a high-spin(HS) transition that is favoured in the antiferromagneticphase. The c -axis low-energy optical response is insteadgoverned by the CT between Hund states of neighbourMn ions with antiparallel spin. Thus, it involves alow-spin (LS) transition. In contrast, the more isotropichigh-energy band was attributed to the manifold of CTtransitions from O 2 p to Mn 3 d levels [35]. Such anassignment has a profound consequence for the interpre-tation of the perovskite manganites electronic structure, R a R c (a) (b) FIG. 3. Temperature dependence of the (a) a -axis (b) c -axisreflectivity of TbMnO . as it implies that these materials are Mott-Hubbard in-sulators and not p - d CT insulators [39]. The HS bandshould be then interpreted as an intersite d - d CT transi-tion across the Mott gap, between the LHB and the UHB.The same conclusion was drawn by calculations based onthe orbitally-degenerate Hubbard model [40]. In addi-tion, a clear manifestation of the d - d CT origin of thelow-energy band comes from the resonant enhancementat 2.00 eV of the B breathing mode in spontaneousRaman scattering, as this mode strongly modulates theintersite d - d CT [41].Here, we confirm the latter assignments by employ-ing ab initio calculations of the electronic structure andthe optical properties of TbMnO . Our calculationswere carried out using DFT via Wien2k code [42]. Theexchange-correlation potential were treated using GGA+ U, where U was set at 3 and 6 eV for Mn 3 d andTb 4 f orbitals, respectively. The computational detailsare reported in the SM. The calculated equilibrium opti-cal responses along the a - and c -axes are shown in Fig.4(a,b) together with the experimental data measured at 8K. For the a -axis response we see a relatively good agree-ment between our calcualtions and the experiment. Forthe c -axis response the agreement is mostly qualitativebut does capture the rough position and magnitude ofthe increase in absorption with increasing energy. Basedon the calculated electronic density of states, the low-energy feature is identified with intersite d - d transitionswhile the higher energy feature is identified with CT fromO 2 p to Mn 3 d levels.These combined experimental-theoretical efforts clar-ify the coarse features of the optical spectrum, but theorigin of the fine structure on the HS band still re-mains largely unexplained. Recently, several authorshave attempted to better understand this fine structurein LaMnO . Initially, it was attributed to Mn d - d crys-tal field transitions split by the JT effect [34]. A completeand detailed study assigned a dual nature ( d - d and p - d )to the fundamental optical gap, since this is also charac-terized by forbidden/weakly-allowed p - d transitions thatact as a precursor to the strong dipole-allowed p - d CT , a () ( - c m - ) , c () ( - c m - ) (a) (b) FIG. 4. Comparison between σ ( ω ) of TbMnO measured at8 K (blue curve) and computed from DFT + U calculations(violet curve): (a) a -axis response; (b) c -axis response. transition at higher energies [36]. Thus, the fine structurereflects these p - d transitions overlapped to the d - d HSabsorption band, giving rise to an intermediate regimebetween the p - d CT and the Mott-Hubbard insulator inthe Zaanen-Sawatzky-Allen scheme [39]. A more recentinterpretation associates the fine structure with quan-tum rotor orbital excitations for the e g electron of Mn ions, disturbed by the lattice anharmonicity [37]. To datethere is no clear consensus on the origin of the spectralfine structure. In contrast to LaMnO , TbMnO retainssuch a fine structure of the d - d HS band even at very lowtemperatures, as evident in the high-resolution spectrumof σ ,a ( ω ) in Fig. S1(a). Moreover, the centre of massof the HS optical band in TbMnO is shifted towardshigher energies than the corresponding one in LaMnO .This effect has been explained by assuming that the JTdistortion increases for smaller R ion sizes, thus enhanc-ing the value of the d - d CT energy [36].
III. ULTRAFAST BROADBAND OPTICALSPECTROSCOPY
In this Section, we present an extensive study of theultrafast dynamics occurring in our detwinned (010)-oriented single crystal of TbMnO . Details on our ex-perimental setup are given in Ref. [43] and in the SM.Briefly, we drive the system out-of-equilibrium using anultrashort 1.55 eV pump pulse, which mimics the exper-imental conditions of previous trREXS studies [7, 10].The pump beam is polarized along the a -axis and itmainly promotes intersite d - d transitions [25] (red ar-row in Fig. 2(a)). The possibility that the excitation of p - d CT transitions could lead to the magnetic dynamicswas ruled out via trREXS experiments by observing thepersistence of the average scattering cross-section inten-sity. Although this argument is not sufficient to excludethat tails of the p - d CT excitation could contribute toother channels probed in the optical range, in the follow-ing we will assume that the main effect of photoexcitation (b)(d)(a)(c) R / R Delay time (ps) E ne r g y ( e V ) -10-505x10 -3 E ne r g y ( e V ) -3 Delay time (ps)
Pump aProbe a Pump aProbe c R / R FIG. 5. (a,b) Colour-coded maps of ∆R/R at 8 K with a -axis pump polarization and (a) a -axis, (c) c -axis probe po-larization. The pump photon energy is 1.55 eV and the ab-sorbed pump fluence is 4.4 mJ/cm . (c,d) Temporal traces atspecific probe photon energies of the respective ∆R/R maps.Each temporal trace results from the integration over 0.10 eVaround the indicated probe photon energy. is to promote intersite d - d transitions. After the interac-tion with the pump pulse, the ultrafast variation of thematerial reflectivity (∆R/R) is monitored over a broadvisible range, extending between 1.72 eV and 2.85 eV,along both the a - and c -axis. This is highlighted by agrey shaded area in Fig. 2(a,c). As described in Sec. II,this spectral region involves the intersite HS and LS d - d transitions along the a - and c -axes, respectively [25]. Asthe optical features depend on the establishment or dis-appearance of the magnetic order in the crystal, they canrepresent suitable observables for extracting valuable in-formation on the optical properties of the magnetic phasewith high time resolution. Indeed, although all-opticalpump-probe methods do not offer a direct measure ofthe magnetic order dynamics, the change in the opticalproperties of the magnetic phase still provides a qualita-tive estimate of the timescale involved in the magneticorder melting [10]. A. Low-temperature ∆ R/R
Figures 5(a,b) display the color maps of ∆R/R as afunction of the probe photon energy and of the time de-lay between pump and probe at 8 K, for a probe po-larization along the a - and c -axes. In both cases, thepump polarization lies along the a -axis and the absorbed pump fluence is 4.4 mJ/cm . Importantly, in this set ofmeasurements, the dynamics are detected up to 14 ps,while sampling with a time step of ∼
26 fs. The a -axis∆R/R consists of a negative signal in correspondence tothe HS intersite d - d transition, which increases its ab-solute weight over time. The spectral shape of ∆R/Ralso acquires more features over time, displaying a se-ries of satellites. Simultaneously, a very sharp featureemerges around 2.60 eV and changes the sign of ∆R/Rto positive. In contrast, the c -axis response undergoes aremarkably different behavior. It is dominated by a pos-itive signal, which increases for longer time delays andshows a maximum close to 2.60 eV.More insight on the temporal dynamics can be gainedby selecting different time traces at representative photonenergies. These are shown in Figs. 5(c,d). We first con-sider the dynamics seen for probe light polarized alongthe crystal a -axis, as shown in Fig. 5(c). Here we seethat for all probe photon energies at short times thereis a sudden decrease in reflectivity which reaches a lo-cal minimum at approximately 190 fs. This time scale issignificantly longer than the 45 fs response time of the ap-paratus [43]. After this prompt decrease, there is a fastpartial recovery of the reflectivity that is characterizedby a time scale of several hundred femtoseconds, a recov-ery that is particularly evident at lower photon energies( e.g. , in which the fast components in the a -axisresponse was not detected [44].Along the c -axis, the temporal traces show a markedlydifferent behavior. The rise time of the signal is ∼ a - and c -axis are respectively shown asviolet curves in Figs. 6(a,b). We find that the simplestmodel that captures the dynamics consists of three ex-ponential functions (with time constants τ , τ and τ m )convolved with a Gaussian accounting for the temporalshape of the pump pulse. The shortest lived compo-nent appears immediately after the pump photoexcita-tion, whereas the other two display a slow rise. More -8-6-4-20 R / R ( - ) R / R ( - ) (a) (b) Pump aProbe a Pump aProbe c
FIG. 6. Temporal traces of the ∆R/R response at 8 K alongthe (a) a - and (b) c -axis, resulting from the integration over0.10 eV around the probe photon energy of 1.80 eV. In bothcases, the pump polarization lies along the a -axis and theabsorbed pump fluence is estimated around 4.4 mJ/cm . Theblack lines represent the results of the fit based on our modelfunction. details about the fitting procedure are reported in theSM.While all three exponential functions are necessary forfitting the a -axis response, only two are needed to re-produce the c -axis dynamics. The results of the fit aredisplayed in Figs. 6(a,b) as black lines superimposed onthe original data. The timescales retrieved along the a -axis are τ = 120 ±
10 fs, τ = 3 ± τ m =9 ± c -axis are τ = 2 ± τ m = 9 ± τ m due to the limited temporal window of ∼
12 ps probed in our experiment. Between the two axes,we notice a strong mismatch in the time constant τ anda close correspondence in the time constant τ m . Con-sidering the a -axis response alone, the emerging pictureof the temporal dynamics reconciles the results of sepa-rate two-color pump-probe experiments performed in thepast on LaMnO and TbMnO [44, 45]. Analysis of ahigh time-resolution experiment on LaMnO attributeda short-lived component similar in time-scale to our τ to electron thermalization, while the intermediate time-scale τ was interpreted as electron-phonon relaxation[45]. Another experiment on TbMnO had significantlyworse time resolution and considered only the slowestcomponent of the time-resolved changes along both axes(corresponding to our τ m ), interpreting this as the timescale for melting of magnetic order [44]. Despite observ-ing also the faster component along the c -axis, the latterexperiment explicitly neglected it due to the difficultyin fitting the data. Using our model function, we findthis component to decay with a time constant of τ = 2ps, which is one order of magnitude larger than expectedfrom a typical electron-electron thermalization timescale.As an alternative way of examining the data, in Fig.7(a,b) we show the spectral evolution of ∆R/R at differ-ent time delays along the a - and c -axes, respectively. In R / R ( - ) R / R ( - ) (a) (b) FIG. 7. Transient spectrum of ∆R/R at different delay timesfor a probe polarization set along (a) the a -axis, (b) the c -axis. The pump photon energy is 1.55 eV and the absorbedpump fluence is 4.4 mJ/cm . Fig. 7(a), the a -axis ∆R/R spectrum for early time de-lays is almost featureless and shows only a broad struc-ture around 2.60 eV. As time evolves, a fine peak-dipstructure clearly emerges in the spectrum, with shoul-ders covering the range from 2.05 eV to 2.50 eV. Thesefeatures coincide with the fine structure superimposed onthe d - d HS absorption band, which can also be observedin the steady-state reflectivity spectrum of Fig. 3(a). At2.60 eV the former broad feature sharpens and increasesits weight until the sign of the response is preserved.Above 2.85 eV the ∆R/R response changes sign and be-comes positive, as evidenced by the gradual redshift ofthe zero-crossing energy at the edge of the probed spec-trum. In Fig. 7(b), the c -axis ∆R/R spectrum for earlytime delays appears as a featureless background, whilefor long time delays some fine structure also arises. Re-markably, these features are not apparent under steady-state conditions (Fig. 3(b)) and cannot be related to aleakage of the a -axis response, since their energies arewell distinct from the ones of the a -axis fine structure.Thus, we conclude that our pump-probe experiment pro-vides a higher contrast to resolve elementary excitationsthat are hidden in the equilibrium spectra. However, atthis stage, no insightful information can be retrieved onthe origin of this fine structure characterizing the opticalspectrum of TbMnO .From this preliminary analysis, we can conclude thatthe photoexcitation of TbMnO along the a -axis with apump photon energy of 1.55 eV leads to an overall re-duction of the reflectivity along the a -axis, accompaniedby an increase of the reflectivity along the c -axis. Thisanisotropic behavior goes beyond the results of previoustwo-color pump-probe measurements performed at differ-ent temperatures [44]. In these experiments, the pumppulse was set at 3.00 eV, thus at the edge between theintersite d - d transitions and the tail of the CT transition,while the probe pulse at 1.50 eV was monitoring the low-est tail of the HS and LS bands in the material. As aconsequence, the probe was not capable of revealing the R / R -12-10-8-6-4-202 R / R ( - ) (a) (b)
13 ps 2.20 eV
FIG. 8. (a) Temperature dependence of the ∆R/R spectrumfor a time delay of 13 ps. (b) Temperature dependence of the∆R/R temporal traces for a probe photon energy of 2.20 eV.The temperatures are indicated in the labels and all curveshave been cut from the maps of Fig. S2. details of the ultrafast electronic response at early timedelays, nor of providing spectrally resolved informationacross the whole spectral region of the intersite d - d ex-citations. In the following, we demonstrate instead thatour approach bridges the gap between the conclusionsdrawn by ultrafast two-color optical spectroscopy and tr-REXS experiments, thus providing a unified picture ofthe nonequilibrium dynamics of TbMnO triggered by anear-infrared pump pulse. B. Temperature dependence
As highlighted in Sec. II, the spectral region of the a -axis HS optical band is indirectly sensitive to the mag-netic correlations in the insulating perovskite manganites[25]. We can then expect that measurements of opticalproperties in this frequency range can give us some infor-mation on the dynamics associated with magnetic orderchanges. A useful quantity that can be extracted fromthe nonequilibrium experiment is the transient complexoptical conductivity ∆ σ = ∆ σ + i ∆ σ . This is esti-mated without the need of a Kramers-Kronig transform,relying on our steady-state SE data of Fig. 2(a) as astarting point and performing a Lorentz analysis of the∆R/R maps at different temperatures [46, 47]. As a re-sult, the determination of the real part ∆ σ gives accessto the temporal evolution of the SW in the visible range.To this aim, we perform a complete temperature de-pendence of ∆R/R. The full set of data is shown in Fig.S2. We observe that the fine structure of the low-energyband manifesting at 8 K is gradually lost as the temper-ature is increased. A strong variation of the intensityof the response also occurs, and the measured changesbecome smaller than the noise level above 100 K. Thefine structure of the low-energy optical band of TbMnO becomes more evident when the long time delay ∆R/Rspectra are directly compared at different temperatures. -20-15-10-50 S W ( c m - ) = 12 ps-30-25-20-15-10-50 121086420 Delay time (ps) 8 K 11 K 15 K 20 K 25 K 30 K 40 K 45 K 50 K 100 K S W ( c m - ) T N2 T N1 (a) (b) FIG. 9. (a) Comparison of the nonequilibrium ∆ SW temporaldynamics at different temperatures, which are indicated in thelabel. At every time delay, the SW is calculated by comput-ing the integral of the corresponding ∆ σ map over the wholeprobed spectral range. The absorbed pump fluence is 4.4mJ/cm . (b) Temperature evolution of the nonequilibriumSW integrated over the whole probe spectrum at 12 ps de-lay time. The blue shaded region highlights the temperaturerange where the material is in the multiferroic spin-cycloidphase, the violet region depicts the region where the SDWphase emerges and the red region represents the paramag-netic phase with short-range spin correlations. The respectivetemperature scales T N2 and T N1 are indicated on top. These spectra are shown in Fig. 8(a) for a time delayof 13 ps and are vertically shifted for clarity. The tem-poral evolution of the system around the probe photonenergy of 2.20 eV is displayed in 8(b) at different tem-peratures. Upon entering the magnetic phase below T N1 ,the dynamics slow down as observed previously [44]. Thisincrease in the τ m time constant for decreasing temper-ature was assigned to the signature of a photoinducedmagnon-assisted hopping along the c -axis of the mate-rial, which leads to an increase in the magnon numberdensity and in turn affects both the a - and c -axis opticalresponse. In the following, we will show that our datasupport instead a scenario in which an ultrafast latticereorganization following the formation of small polaronsis the source of the bottleneck observed in the spin ordermelting of TbMnO .From the temperature dependent ∆R/R data, we cal-culate ∆ σ as a function of probe photon energy andtime delay. The color-coded maps at the different tem-peratures are shown in Fig. S3 and feature a prominentdrop of ∆ σ at large time delays. The determination of∆ σ at all temperatures allows us to follow the temporalevolution of the change in the partial SW (∆ SW ) over theprobed range. The quantity ∆ SW is defined as∆ SW = (cid:90) ω ω ∆ σ dω (1)where ω = 1.72 eV and ω = 2.85 eV. A direct compar-ison among the ∆ SW temporal dynamics at the differenttemperatures is established in Fig. 9(a). Here, our pri-mary interest is to reveal whether the magnetic orderdynamics gives rise to detectable temperature anoma-lies in ∆ SW at long time delays. For this reason, wetrack the value of ∆ SW at 12 ps for the different tem-peratures. The results are presented in Fig. 9(b). Forclarity, we also indicate the T N1 and T N2 temperaturescales by highlighting different temperature regions withdistinct colors. Starting from high temperatures, we ob-serve that ∆ SW decreases its value when approaching thefirst magnetic phase transition at T N1 and decreases evenfurther close to the second magnetic phase transition atT N2 . Two kinks are observed in the proximity of T N1 and T N2 . While the former had been already observedat the single-wavelength ∆R/R level [44], the latter is apreviously undetected feature that can be accessed whenthe whole spectral region of the intersite d - d transitionsis covered by the probe pulse. Although the presence ofthese kinks could be correlated with the magnetic orderdynamics, this interpretation is not unambiguous. Irre-spective of the origin of this SW loss, the carriers lostin the visible range are likely to be redistributed to highenergies, in correspondence to the optical band at 5.15eV [25, 26]. C. Coherent collective response
The possible detection of a dynamical SW transfer as-sociated with the loss of magnetic order in photoexcitedTbMnO provides new insights into the nonequilibriumresponse of the material. However, this observable repre-sents an indirect effect of the order parameter melting onthe optical properties of the system. As such, our mea-surements only suggest that a delayed transfer of ther-mal energy from the excited carriers to the spin system isat play in the material, consistent with previous experi-mental results [7, 10]. Thus, the microscopic mechanismbehind the magnetic order melting remains unclear. Onescenario that has been envisioned behind the delayed en-ergy transfer is related to relaxation of the JT distortion,with the subsequent localization of the carriers in theform of small polarons [7]. The charge localization wouldhinder magnon-assisted hopping and therefore would re-quire that the energy transfer to the spin system to bemediated by changes in the lattice structure.This represents a crucial aspect, since the polaronicbehavior of the charge carriers is widely recognized asone of the peculiarities of the orthorhombic manganites[48–50]. Indeed, doping carriers [51] or photoexciting theintersite d - d CT transitions in perovskite manganites ispredicted to lead to the creation of the so-called anti-JTpolarons [52]. Similar effects are envisioned when the p - d CT transitions are photoexcited [53]. Indeed, as thelattice shows a cooperative JT distortion, the presenceof an extra charge ( i.e. electron or hole) on the Mn ions can produce a strong structural rearrangement to R / R ( - ) FFT A m p li t ude ( a . u . ) (a) (b) A g (5) A g (4) A g (1) A g (3) -6-4-20 43210 Delay time (ps) 21Delay time (ps) FIG. 10. (a) Temporal dynamics and (b) FT of the spectralresponse at 1.90 eV, 2.02 eV and 2.19 eV, averaged over theregion indicated in the label. The pump polarization is setalong the a -axis. The absorbed pump fluence is 4.4 mJ/cm .Inset in (a) Details of the coherent oscillations on the trace at2.19 eV between 0 and 2 ps. The assignment of the coherentmodes is given in (b). locally remove and relax the JT distortion. In this way,the system tries to minimize the energy cost generatedby the existence of additional charge in a collectively JT-distorted crystal. This site acts as a defect that becomesstrongly pinned, since the hopping to other Mn sitesrequires moving along the lattice distortion. In magneti-cally ordered phases, also the spin degree of freedom canbe affected by the charge localization, as the polaron isexpected to produce canting of the spins from their natu-ral directions. Behaving similarly to a defect, the anti-JTpolaron is a prototypical example of a small (Holstein)polaron and, as such, it involves the presence of a lo-cal deformation around the self-trapped carrier [54, 55].Therefore, the local symmetry associated with the dis-placement is expected to retain a totally symmetric char-acter (A g symmetry).To investigate the validity of this scenario, we searchfor the signatures of coherent optical phonon modes withtotally symmetric character that are coupled to the pho-toexcited carriers and signal a rearrangement of the lat-tice structure consistent with the relaxation of the JTdistortion. Here, we go beyond the results of SectionIII.A and monitor the a -axis ∆R/R at 8 K by decreasingthe time step for the detection to ∼
13 fs.Figure 10(a) shows some representative temporaltraces when the absorbed pump fluence for is 4.4mJ/cm . The probe photon energies at which thesetraces have been selected are indicated in the labels. Re-markably, the initial ultrafast relaxation of the electronicresponse is now clearly resolved, displaying a sharp andwell-defined negative peak. Simultaneously, a coherentbeating among several modes emerges in the time-domainduring the electronic relaxation and persists up to 4 ps(see the inset of Fig. 10(a)). By performing a Fouriertransform (FT) analysis of the residuals from the fit (Fig.10(b)), we identify the presence of four collective modestaking part to the coherent dynamics triggered by theintersite d - d excitation. Their energies correspond to14.5 meV (A g (5)), 47.2 meV (A g (4)), 61.0 meV (A g (1))and 63.4 meV (A g (3)) and are indicative of four of theseven Raman-active A g modes of TbMnO [56–60]. Inparticular, A g (5) corresponds to a soft mode involvingthe displacement of the Tb ion, A g (4) to the rotationof the MnO octahedra, A g (1) to the anti-stretching JTvibrations of the O atoms in the xz plane and A g (3) tothe bending of MnO octahedra. On the other hand, thedoublet structure centered around 35-37 meV in the FTmay arise from the convolution of the peaks associatedwith the A g (2) and A g (7) modes, which are known to bestrongly intermixed in TbMnO [57]. Indeed, this fea-ture is found to persist over the probed spectral range,despite becoming broader as the probe photon energyis tuned above 2.00 eV. For the purpose of our discus-sion, we neglect the presence of this feature and avoidany speculation in the absence of a clear spectroscopicobservable. Based on the observed frequencies and modeassignments, we also remark that no signature of coher-ent excitations of magnetic origin is detected either inthese temporal traces, or in those measured up to ∼ g phononmodes resonate in the proximity to the d - d HS intersiteabsorption band. The same modes are observed when thepump polarization is set along the c -axis of the material,thus promoting LS intersite d - d transitions (Fig. S4).The emergence of these coherent collective modes with awell-defined symmetry allows us to propose an explana-tion of the ultrafast magnetic order dynamics occurringin the spin cycloid phase of TbMnO .In the past, a two-color pump-probe study with a timeresolution of 10 fs revealed the presence of the A g (4) andA g (1) modes in the ultrafast response of LaMnO [45].The A g (5) and A g (3) modes were not detected. In thisexperiment, the pump photon energy was tuned to beresonant with the intersite d - d transition, thus promot-ing the CT between two neighbouring Mn sites andcreating locally Mn -Mn sites. Moreover, a detailedtemperature study of the ∆R/R was performed in orderto track the relevant parameters of the coherent modes,such as the oscillation amplitude and the damping rates.Surprisingly, both A g (4) and A g (1) modes were foundto sharply increase their intensity below T N ; they alsoshowed a pronounced decrease of their damping rateswhen the temperature was reduced toward T N , followedby an enhanced decay below T N . Although this exper-iment did not provide any information on the ultrafastmagnetic order dynamics occurring in the material, itwas concluded that the generation mechanism of the twomodes proceeds via the displacive excitation [61]. Thetrigger mechanism of mode A g (1) (anti-stretching JTmode) was explained by observing that the Mn andMn ions are no longer JT active, leading to a relaxation of the JT distortion that launches the coherent mode.Importantly, as we will discuss later, the disruption ofthe regular, pure Mn arrangements has been addressedfrom the theory perspective [52], leading to the scenarioof anti-JT polaron formation. The excitation of modeA g (4) (out-of-phase rotation of the MnO octahedra)was instead interpreted by invoking the Goodenough-Anderson-Kanamori rules [62–65]. In the excited stateof the manganite, the presence of neighbouring Mn -Mn sites gives rise to two empty e g orbitals. Hence,according to the Goodenough-Anderson-Kanamori rules,the exchange interaction J becomes negative. In a simi-lar way, the presence of neighbouring Mn -Mn sitesleads to two half-filled e g levels, providing again a neg-ative J. The change in sign of J results in the estab-lishment of a force to reduce the Mn-O-Mn semicovalentbond length. This can be achieved by both the relaxationof the JT distortion and by reducing the bond angle togive a straighter bond, which in turn excite the coherentmode. In other words, the renormalization of J undernonequilibrium conditions triggers the coherent latticemotion via the displacive mechanism.As already observed above, modes A g (5) and A g (3)were not observed in the nonequilibrium dynamics ofLaMnO , and thus they represent novel features detectedby our measurement. The experimental parameters (timeresolution, and pump/probe photon energy) used in Ref.[45] were indeed suitable for revealing the presence, ifany, of both modes. This suggests that these coherentmodes are a peculiar feature of the ultrafast dynamics ofTbMnO , which possesses a higher degree of distortionthan LaMnO and displays multiferroicity. Given thecomplex beating among the different modes, extractingthe phase of each individual mode (to distinguish whetherthe temporal behavior is a sine or a cosine function) be-comes a challenge. Hence, to explain the appearance ofmodes A g (5) and A g (3), we rely on additional consider-ations. Concerning mode A g (3), we base our argumentson spontaneous Raman scattering data for the series oforthorhombic RMnO manganites. It was found that thestretching A g (1) and bending A g (3) modes, while uncor-related for R = La, display a pronounced mixing for R= Sm, Eu, Gd, Tb [57]. This effect is enhanced by theproximity between their phonon energies when the unitcell becomes highly distorted. Two phonon modes ofsame symmetries and close energies can be considered ascoupled oscillators, whose frequencies are given by (cid:126) ω , = (cid:126) ω (cid:48) + (cid:126) ω (cid:48)(cid:48) ± (cid:114) ( (cid:126) ω (cid:48) − (cid:126) ω (cid:48)(cid:48) ) V (cid:126) ω (cid:48) and (cid:126) ω (cid:48)(cid:48) are the mode energies without cou-pling and V is the coupling constant. From spontaneousRaman scattering, V can be estimated ∼ g (3) is strictlyconnected to the coherent excitation of mode A g (1), as0 FIG. 11. Schematic illustration of the lattice displacementassociated with the creation of an anti-JT polaron upon pho-toexcitation of the system via an intersite d - d CT transition. the two modes are mutually dependent.The generation mechanism of the coherent A g (5) modeis instead more subtle. As discussed above, this modecorresponds to the vibration of the Tb ion along the c -axis and was identified as a partially softened phononmode associated with the ferroelectric transition [60]. Inthe past, ferroelectric phase transitions have been widelyinvestigated via pump-probe spectroscopy to track thedynamics of coherent soft modes in order-disorder typeperovskites (such as KNbO and SrTiO ) [66–68] and indisplacive type ferroelectrics (such as GeTe) [69]. Whilein the first class of materials the generation mechanismof the soft modes has been associated with ImpulsiveStimulated Raman Scattering (ISRS) [70, 71], in the sec-ond class a displacive excitation has been proposed [61].In our experiment, the mode is found to strongly res-onate only in the low-energy wing of the visible spectrum(around 1.90 eV). In this region, the absorptive part ofthe optical conductivity is weak and featureless, suggest-ing that the displacive character of the exctiation is weak.Consistent with this hypothesis, we observe that the 1.90eV region in which the A g (5) mode resonates correspondsto the pronounced dip found in the dispersive part of theoptical conductivity (Fig. S1(b)). This leads us to con-clude that the ISRS scenario could explain the excitationmechanism of this partially softened mode in TbMnO .The observation of coherent optical phonons in thenonequilibrium dynamics of TbMnO opens intriguingperspectives in the evaluation of the electron-phonon cou-pling for all these modes. Following this consideration,a specific and quantitative estimate of the coupling be-tween the electrons and the coherent Raman-active A g optical phonons via ab initio calculations will provide in-sightful information, albeit projected at the Γ point. Wewill address this aspect in a future publication. IV. CONCLUSIONS
In this work we have applied ultrafast optical spec-troscopy over a broad wavelength range to study the se-quence of events leading to spin-order melting in laser-excited multiferroic TbMnO . The interaction betweenthe pump pulse at 1.55 eV and the system leads to theexcitation of an intersite d - d CT transition, correspond-ing to an optical excitation across the fundamental Mott-Hubbard gap. In other words, the excitation locally pro-motes the creation of Mn -Mn sites, leading to thedisruption of the regular Mn arrangement and to therelaxation of the JT distortion. In this scenario, anti-JTsmall polarons are expected to form, producing a localdistortion in the MnO octahedra [52]. A pictorial rep-resentation of the photoexcitation process and the for-mation of anti-JT polarons is given in Fig. 11. Thechange in the lattice structure is reflected in the emer-gence of the coherent A g (1) anti-stretching JT mode andof the A g (3) bending mode, to which the A g (1) is stronglymixed. Simultaneously, the photoexcited charge densitycouples via the exchange interaction to the A g (4) mode,corresponding to the out-of-phase rotation of the MnO octahedra. Also this structural mode may be involvedin the distortion associated with the anti-JT polaron for-mation. We propose that these coherent modes repre-sent the signatures of the creation of anti-JT polarons inTbMnO . The formation of such a relatively long-livedself-trapped charge hinders magnon-assisted hopping andthus requires the energy transfer to the spin system to bemediated by the rearrangement of the lattice structure.Consistent with this idea, the signal associated with themelting of the long-range magnetic order rises within sev-eral ps and, at our absorbed pump fluence, the completemelting of the magnetic order is expected to take placewithin 22 ps [7]. We expect polaronic effects to manifestalso when the initial photoexcitation couples to the p - d CT transition, as the d -orbitals will be again influencedby the presence of an extra charge. Similar experimentsexploring a pump photon energy at 3.10 eV confirm theinsensitivity of the magnetic order response to the detailsof the initial photoexcitation [10, 44].The scenario proposed above reconciles the results re-ported in different experiments, suggesting the crucialinvolvement of the lattice behind the spin-order melt-ing. More importantly, as the coherent optical phononswere previously detected in LaMnO , it is reasonable tobelieve that the same phenomenology is effectively atplay in all undoped orthorhombic manganites. Indeed,the pronounced coherences appear in both LaMnO andTbMnO , which represent the least and the most dis-torted orthorhombic manganites of the RMnO family,respectively. A different microscopic mechanism may beat play in the hexagonal manganites of the RMnO fam-ily, as the hexagonal crystal field splitting lifts the d -1orbital degeneracy in a different fashion [72, 73]. In thiscase, a pump pulse at 1.55 eV promotes an on-site d - d transition of the Mn electron, which leaves the totalcharge unchanged and does not modify the polaronic po-tential [74].In addition, two previously undetected coherentphonon modes are found to participate to the ultrafastevolution of the system after the photoexcitation. Inparticular, the A g (5) mode corresponds to the partiallysoftened phonon associated with the ferroelectric phasetransition. We speculate that this mode is generatedvia the ISRS mechanism, similarly to the behavior ob-served in the order-disorder perovskite ferroelectrics [66–68]. We suggest that temperature dependent studies ofthis partially softened mode may unveil new intriguingaspects on the origin of the magnetoelectric coupling inthis multiferroic material and shed light on the fate ofthe ferroelectric order parameter following the photoex-citation. One interesting open question is whether theferroelectric and the magnetic order parameters decou-ple simultaneously after the interaction with the pumppulse and follow separate temporal evolution in the sys-tem. In this regard, our measurements set the basis forrevealing the dynamics of the ferroelectric polarization inthe spin-cycloid magnetic phase with more sensitive ul-trafast methods, such as time-resolved second harmonicgeneration spectroscopy [75] and microscopy [76].We acknowledge A. Mann and S. Borroni for the tech-nical support. Work at LUMES was supported by theSwiss National Science Foundation through the NCCRMUST. Crystal growth work at the Institute for Quan-tum Matter (IQM) was supported by the U.S. Depart-ment of Energy, Office of Basic Energy Sciences, Divisionof Materials Sciences and Engineering through Grant No.DE-FG02-08ER46544.* [email protected] S1. EXPERIMENTAL METHODSA. Sample preparation
A high quality stoichiometric TbMnO single crystalwas produced by the optical floating zone technique atthe zoning rate of 0.5 mm/h with rotation rate of 15rpm for the growing crystal and 0 rpm for the feed rodunder static argon. The crystal was oriented using Lauebackscattering and cut to expose the (010) face with anapproximately 3 ◦ miscut. The surface of the sample waspolished and afterwards the sample was annealed in airin 650 ◦ C for 110 hours. The dimensions of the crystal areapproximately 2 mm × × a , b and c axes, respectively. The Pbnm orthorhombic conventionis used to describe the crystal axes. B. Spectroscopic ellipsometry
We used spectroscopic ellipsometry to measure thecomplex dielectric function of the sample, covering thespectral range from 0.50 eV to 6.00 eV. The measure-ments were performed using a Woollam VASE ellipsome-ter. The TbMnO single crystal was mounted in a heliumflow cryostat, allowing measurements from room temper-ature down to 10 K. The measurements were performedat < − mbar to prevent measurable ice-condensationonto the sample. Anisotropy corrections were performedusing standard numerical procedures [77]. C. Ultrafast broadband optical spectroscopy
Femtosecond broadband transient reflectivity experi-ments were performed using a set-up described in detailsin Ref. [43]. Briefly, a Ti:Sapphire oscillator, pumpedby a continuous-wave Nd:YVO laser, emitted sub-50 fspulses at 1.55 eV with a repetition rate of 80 MHz. Theoutput of the oscillator seeded a cryo-cooled Ti:Sapphireamplifier, which was pumped by a Q-switched Nd:YAGlaser. This laser system provided ∼
45 fs pulses at 1.55eV with a repetition rate of 3 kHz. One third of theoutput, representing the probe beam, was sent to a mo-torized delay line to set a controlled delay between pumpand probe. The 1.55 eV beam was focused on a 3 mm-thick CaF cell using a combination of a lens with shortfocal distance and an iris to limit the numerical apertureof the incoming beam. The generated continuum coveredthe 1.72 - 2.85 eV spectral range. The probe was subse- quently collimated and focused onto the sample througha pair of parabolic mirrors under an angle of 15 ◦ . The re-maining two thirds of the amplifier output, representingthe pump beam, were directed towards the sample undernormal incidence. Along the pump path, a chopper witha 60 slot plate was inserted, operating at 1.5 kHz andphase-locked to the laser system. Both pump and probewere focused onto the sample with spatial dimensions of120 µ m × µ m for the pump and 23 µ m × µ m forthe probe. The sample was mounted inside a closed cyclecryostat, which provided a temperature-controlled envi-ronment in the range 10 - 340 K. The reflected probewas dispersed by a fiber-coupled 0.3 m spectrograph anddetected on a shot-to-shot basis with a complementarymetal-oxide-semiconductor linear array. S2. COMPUTATIONAL DETAILS
To assign the features appearing in the optical spec-tra, we performed ab initio
Density-Functional Theory(DFT) calculations of TbMnO . The electronic structureand optical properties of TbMnO were carried out viaWien2k code. The exchange-correlation potential weretreated using GGA + U, where U was set at 3 eV and6 eV for Mn 3 d and Tb 4 f orbitals, respectively. Themuffin-tin radii Rmt were 2.2, 1.95, 1.5 a.u. for Tb, Mn,and O atoms, respectively. The maximum angular mo-mentum of the radial wavefunctions was set to 10, andRmtKmax was fixed at 7.0 to determine the basis size. S3. ADDITIONAL RESULTSA. Fit of the ∆ R/R data
To get more quantitative information on the timescalesgoverning the ultrafast dynamics, we perform a fit ofsome representative ∆R/R traces at 8 K. For this pur-pose, we select the temporal traces around a probe pho-ton energy of 1.80 eV, as in this region the short-livedcomponent becomes more apparent. These traces areshown in Figs. 6(a,b) of the main text as violet curvesalong the a - and c -axis, respectively. Among the sev-eral models that can be implemented for capturing thedynamics, the simplest one consists of three exponen-tial functions convolved with a Gaussian accounting forthe temporal shape of the pump pulse. While the short-est lived component appears immediately after the pumpphotoexcitation, the other two components display a slowrise. The function that is used for fitting the data is f ( t ) = u ( t ) (cid:34) f ( t ) + f ( t ) + f m ( t ) (cid:35) = u ( t ) (cid:34) e − t τR ∗ A e − t − tD τ + (cid:16) − e − t τR (cid:17) ∗ (cid:104) A e − t − tD τ + A m e − t − tD τm (cid:105)(cid:35) (3)3where u(t) is a step function that has u = 0 for t < ≥
0, A , A and A m are the amplitudes ofthe three exponential functions; τ R and τ R are the risetimes of the exponential functions; τ , τ and τ m are therelaxation constants of the three exponentials; t D and t D are delay parameters with respect to the zero time.While all three exponential functions are necessary forfitting the a -axis response, only two of them are sufficientto reproduce the c -axis dynamics. The results of the fitare shown in Figs. 6(a,b) as black lines superposed on theoriginal data. The timescales retrieved along the a -axisare τ = 120 ±
10 fs, τ = 3 ± τ m = 9 ± c -axis are τ = 2 ± τ m = 9 ± τ m due to the limited temporal window of ∼
12 ps probed inour experiment.
B. Temperature dependence
To monitor how the spectral signature of the a -axis HSband varies in the nonequilibrium experiment across thetwo magnetic phase transitions, we perform a completetemperature dependence of ∆R/R. The color-coded mapsof ∆R/R are shown in Fig. S2, in which the differenttemperature values are also indicated. We observe thatthe fine structure of the low-energy band manifesting at8 K is gradually lost as the temperature is increased. Astrong variation of the intensity of the response also oc-curs and the signal declines to the sensitivity range of oursetup around the temperature of 100 K. The fine struc-ture of the low-energy optical band of TbMnO becomesmore evident when the long time delay ∆R/R spectraare directly compared at different temperatures.A useful quantity that has to be extracted from thenonequilibrium experiment is the transient complex op-tical conductivity ∆ σ = ∆ σ + i ∆ σ . This can be cal-culated without the need of a Kramers-Kronig transformby relying on our steady-state SE data of Fig. 2(a) as astarting point and performing a Lorentz analysis of the∆R/R maps at the different temperatures. As a con-sequence, the determination of the real part ∆ σ givesaccess to the temporal evolution of the spectral weight(SW) in the visible range. In Fig. S3 we show the cal-culated ∆ σ at all temperatures, as a function of probephoton energy and time delay. At low temperatures, aprominent drop dominates the whole spectral range espe-cially at large time delays. As the temperature increases,the absolute strength of the response becomes smallerand declines to the order of our sensitivity close to 100K. The determination of ∆ σ at all temperatures allowsfollowing the temporal evolution of the change in SW(∆SW) over the probed range. The dynamics of ∆SWare overlapped to all color-coded maps of ∆ σ in Fig. S3. C. Pump polarization along the c-axis
To test whether the coherent modes modulating the a -axis reflectivity can be excited also under other pumppolarization conditions, we perform a separate experi-ment at 8 K where the pump beam is polarized alongthe c -axis. In these conditions, the pump photon energyat 1.55 eV can promote intersite d - d transitions alongthe c -axis, as it is resonant with the tail of the c -axislow-spin d - d absorption feature shown in Fig. 2(c). Thecolor-coded map of the a -axis ∆R/R is displayed in Fig.S4(a) as a function of probe photon energy and time delaybetween pump and probe. Despite the weak c -axis ab-sorption, we retrieve a sizable ∆R/R signal, retaining thesame spectral shape of the response in Figs. 5(a,b) and10(a). Some representative temporal traces selected fromthe map are shown in Fig. S4(b) and demonstrate theemergence and persistence of the coherent response alsofor a pump polarization along the c -axis. The Fouriertransform analysis of the residuals from a fit of the inco-herent response (not shown) confirms the presence of allpreviously listed phonon modes.4 [1] S.-W. Cheong and M. Mostovoy, Nat. Mat. , 13 (2007).[2] H. S. Park, J. S. Baskin, and A. H. Zewail, Nano Lett. , 3796 (2010).[3] J. Rajeswari, P. Huang, G. F. Mancini, Y. Murooka,T. Latychevskaia, D. McGrouther, M. Cantoni, E. Bal-dini, J. S. White, A. Magrez, et al. , Proc. Natl. Acad.Sci, , 14212 (2015).[4] N. R. da Silva, M. M¨oller, A. Feist, H. Ulrichs, C. Ropers,and S. Sch¨afer, arXiv preprint arXiv:1710.03307 (2017).[5] J. Fink, E. Schierle, E. Weschke, and J. Geck, Reportson Progress in Physics , 056502 (2013).[6] T. Kubacka, J. A. Johnson, M. C. Hoffmann, C. Vicario,S. De Jong, P. Beaud, S. Gr¨ubel, S.-W. Huang, L. Huber,L. Patthey, et al. , Science , 1333 (2014).[7] J. A. Johnson, T. Kubacka, M. C. Hoffmann, C. Vi-cario, S. de Jong, P. Beaud, S. Gr¨ubel, S.-W. Huang,L. Huber, Y. W. Windsor, E. M. Bothschafter, L. Ret-tig, M. Ramakrishnan, A. Alberca, L. Patthey, Y.-D.Chuang, J. J. Turner, G. L. Dakovski, W.-S. Lee, M. P.Minitti, W. Schlotter, R. G. Moore, C. P. Hauri, S. M.Koohpayeh, V. Scagnoli, G. Ingold, S. L. Johnson, andU. Staub, Phys. Rev. B , 184429 (2015).[8] S. L. Johnson, R. A. De Souza, U. Staub, P. Beaud,E. M¨ohr-Vorobeva, G. Ingold, A. Caviezel, V. Scagnoli,W. F. Schlotter, J. J. Turner, et al. , Phys. Rev. Lett. , 037203 (2012).[9] M. Langner, S. Roy, S. Mishra, J. Lee, X. Shi, M. Hos-sain, Y.-D. Chuang, S. Seki, Y. Tokura, S. Kevan, et al. ,in CLEO: QELS Fundamental Science (Optical Societyof America, 2015) pp. FW3B–5.[10] E. M. Bothschafter, E. Abreu, L. Rettig, T. Kubacka,S. Parchenko, M. Porer, C. Dornes, Y. W. Windsor,M. Ramakrishnan, A. Alberca, S. Manz, J. Saari, S. M.Koohpayeh, M. Fiebig, T. Forrest, P. Werner, S. S. Dhesi,S. L. Johnson, and U. Staub, Phys. Rev. B , 184414(2017).[11] T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. T.Takahashi, K. Ishizaka, and Y. Tokura, Phys. Rev. B , 060403 (2003).[12] S. Quezel, F. Tcheou, J. Rossat-Mignod, G. Quezel, andE. Roudaut, Physica B+ C , 916 (1977).[13] M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm,J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong, O. P.Vajk, and J. W. Lynn, Phys. Rev. Lett. , 087206(2005).[14] D. Senff, P. Link, K. Hradil, A. Hiess, L. P. Regnault,Y. Sidis, N. Aliouane, D. N. Argyriou, and M. Braden,Phys. Rev. Lett. , 137206 (2007).[15] S. B. Wilkins, T. R. Forrest, T. A. W. Beale, S. R. Bland,H. C. Walker, D. Mannix, F. Yakhou, D. Prabhakaran,A. T. Boothroyd, J. P. Hill, et al. , Phys. Rev. Lett. ,207602 (2009).[16] H. C. Walker, F. Fabrizi, L. Paolasini, F. de Bergevin,D. Prabhakaran, A. T. Boothroyd, and D. F. McMorrow,Phys. Rev. B , 214415 (2013).[17] S. W. Lovesey, V. Scagnoli, M. Garganourakis, S. M.Koohpayeh, C. Detlefs, and U. Staub, J. Phys. Condens.Mat. , 362202 (2013).[18] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,and Y. Tokura, Nature , 55 (2003). [19] T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, andY. Tokura, Phys. Rev. Lett. , 257201 (2004).[20] M. F¨orst, R. I. Tobey, S. Wall, H. Bromberger,V. Khanna, A. L. Cavalieri, Y.-D. Chuang, W. S. Lee,R. Moore, W. F. Schlotter, J. J. Turner, O. Krupin,M. Trigo, H. Zheng, J. F. Mitchell, S. S. Dhesi, J. P.Hill, and A. Cavalleri, Phys. Rev. B , 241104 (2011).[21] A. Mann, E. Baldini, A. Tramontana, E. Pomjakushina,K. Conder, C. Arrell, F. Van Mourik, J. Lorenzana, andF. Carbone, Phys. Rev. B , 035147 (2015).[22] A. Mann, E. Baldini, A. Odeh, A. Magrez, H. Berger,and F. Carbone, Phys. Rev. B , 115122 (2016).[23] E. Baldini, A. Mann, L. Benfatto, E. Cappelluti, A. Aco-cella, V. M. Silkin, S. V. Eremeev, A. B. Kuzmenko,S. Borroni, T. Tan, X. X. Xi, F. Zerbetto, R. Merlin,and F. Carbone, Phys. Rev. Lett. , 097002 (2017).[24] S. Borroni, E. Baldini, V. M. Katukuri, A. Mann, K. Par-linski, D. Legut, C. Arrell, F. van Mourik, J. Teyssier,A. Kozlowski, P. Piekarz, O. V. Yazyev, A. M. Ole´s,J. Lorenzana, and F. Carbone, Phys. Rev. B , 104308(2017).[25] N. N. Kovaleva, A. V. Boris, C. Bernhard, A. Ku-lakov, A. Pimenov, A. M. Balbashov, G. Khaliullin, andB. Keimer, Phys. Rev. Lett. , 147204 (2004).[26] M. Bastjan, S. G. Singer, G. Neuber, S. Eller,N. Aliouane, D. N. Argyriou, S. L. Cooper, andM. R¨ubhausen, Phys. Rev. B , 193105 (2008).[27] Y. Okimoto, T. Katsufuji, T. Ishikawa, T. Arima, andY. Tokura, Phys. Rev. B , 4206 (1997).[28] K. Takenaka, K. Iida, Y. Sawaki, S. Sugai, Y. Moritomo,and A. Nakamura, J. Phys. Soc. Jpn. , 1828 (1999).[29] K. Tobe, T. Kimura, Y. Okimoto, and Y. Tokura, Phys.Rev. B , 184421 (2001).[30] M. A. Quijada, J. R. Simpson, L. Vasiliu-Doloc, J. W.Lynn, H. D. Drew, Y. M. Mukovskii, and S. G. Karaba-shev, Phys. Rev. B , 224426 (2001).[31] N. N. Kovaleva, A. M. Ole´s, A. M. Balbashov, A. Maljuk,D. N. Argyriou, G. Khaliullin, and B. Keimer, Phys.Rev. B , 235130 (2010).[32] A. Nucara, F. M. Granozio, M. Radovic, F. M. Vitucci,P. Maselli, R. Fittipaldi, A. Vecchione, and P. Calvani,Eur. Phys. J. B , 435 (2011).[33] M. W. Kim, S. J. Moon, J. H. Jung, J. Yu, S. Parashar,P. Murugavel, J. H. Lee, and T. W. Noh, Phys. Rev.Lett. , 247205 (2006).[34] J. F. Lawler, J. G. Lunney, and J. M. D. Coey, Appl.Phys. Lett. , 3017 (1994).[35] A. Rusydi, R. Rauer, G. Neuber, M. Bastjan, I. Mahns,S. M¨uller, P. Saichu, B. Schulz, S. G. Singer, A. I. Licht-enstein, D. Qi, X. Gao, X. Yu, A. T. S. Wee, G. Stry-ganyuk, K. D¨orr, G. A. Sawatzky, S. L. Cooper, andM. R¨ubhausen, Phys. Rev. B , 125110 (2008).[36] A. S. Moskvin, A. A. Makhnev, L. V. Nomerovannaya,N. N. Loshkareva, and A. M. Balbashov, Phys. Rev. B , 035106 (2010).[37] N. N. Kovaleva, K. I. Kugel, Z. Potek, O. E. Kusmart-seva, N. S. Goryachev, Z. Bryknar, E. I. Demikhov, V. A.Trepakov, A. Dejneka, F. V. Kusmartsev, and A. M.Stoneham, J. Exp. Theor. Phys. , 890 (2016).[38] P. B. Allen and V. Perebeinos, Phys. Rev. Lett. , 4828(1999).[39] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev.Lett. , 418 (1985). [40] J. S. Lee, M. W. Kim, and T. W. Noh, New J. Phys. ,147 (2005).[41] R. Kr¨uger, B. Schulz, S. Naler, R. Rauer, D. Budelmann,J. B¨ackstr¨om, K. H. Kim, S. W. Cheong, V. Perebeinos,and M. R¨ubhausen, Phys. Rev. Lett. , 097203 (2004).[42] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, andJ. Luitz, An Augmented Plane Wave + Local OrbitalsProgram for Calculating Crystal Properties (2001).[43] E. Baldini, A. Mann, S. Borroni, C. Arrell, F. vanMourik, and F. Carbone, Struct. Dyn. , 064301 (2016).[44] I. P. Handayani, R. I. Tobey, J. Janusonis, D. A.Mazurenko, N. Mufti, A. A. Nugroho, M. O. Tjia,T. T. M. Palstra, and P. H. M. van Loosdrecht, J. Phys-Condens. Mat. , 116007 (2013).[45] S. Wall, D. Prabhakaran, A. T. Boothroyd, and A. Cav-alleri, Phys. Rev. Lett. , 097402 (2009).[46] E. Baldini, A. Mann, B. P. Mallett, C. Arrell,F. Van Mourik, T. Wolf, D. Mihailovic, J. L. Tallon,C. Bernhard, J. Lorenzana, and F. Carbone, Phys. Rev.B , 024501 (2017).[47] E. Baldini, L. Chiodo, A. Dominguez, M. Palummo,S. Moser, M. Yazdi-Rizi, G. Aub¨ock, B. P. P. Mallett,H. Berger, A. Magrez, C. Bernhard, M. Grioni, A. Ru-bio, and M. Chergui, Nat. Comm. (2017).[48] Y. Yamada, O. Hino, S. Nohdo, R. Kanao, T. Inami, andS. Katano, Phys. Rev. Lett. , 904 (1996).[49] S. Shimomura, N. Wakabayashi, H. Kuwahara, andY. Tokura, Phys. Rev. Lett. , 4389 (1999).[50] A. Daoud-Aladine, J. Rodriguez-Carvajal, L. Pinsard-Gaudart, M. T. Fernandez-Diaz, and A. Revcolevschi,Phys. Rev. Lett. , 097205 (2002).[51] S. Mildner, J. Hoffmann, P. E. Bl¨ochl, S. Techert, andC. Jooss, Phys. Rev. B , 035145 (2015).[52] P. B. Allen and V. Perebeinos, Phys. Rev. B , 10747(1999).[53] T. Mertelj, D. Kuˇsˇcer, M. Kosec, and D. Mihailovic,Phys. Rev. B , 15102 (2000).[54] T. Holstein, Annals of Physics , 325 (1959).[55] T. Holstein, Annals of Physics , 343 (1959).[56] L. Mart´ın-Carr´on, A. de Andr´es, M. J. Mart´ınez-Lope,M. T. Casais, and J. A. Alonso, Phys. Rev. B , 174303(2002).[57] M. N. Iliev, M. V. Abrashev, J. Laverdiere, S. Jandl,M. M. Gospodinov, Y.-Q. Wang, and Y.-Y. Sun, Phys.Rev. B , 064302 (2006). [58] J. Laverdiere, S. Jandl, A. A. Mukhin, V. Y. Ivanov,V. G. Ivanov, and M. N. Iliev, Phys. Rev. B , 214301(2006).[59] P. Kumar, S. Saha, D. V. S. Muthu, J. R. Sahu, A. K.Sood, and C. N. R. Rao, J. Phys-Condens. Mat. ,115403 (2010).[60] P. Rovillain, J. Liu, M. Cazayous, Y. Gallais, M.-A. Me-asson, H. Sakata, and A. Sacuto, Phys. Rev. B ,014437 (2012).[61] H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, G. Dres-selhaus, and M. S. Dresselhaus, Phys. Rev. B , 768(1992).[62] P. W. Anderson, Phys. Rev. , 350 (1950).[63] J. B. Goodenough, Phys. Rev. , 564 (1955).[64] J. B. Goodenough, J. Phys. Chem. Solids. , 287 (1958).[65] J. Kanamori, J. Phys. Chem. Solids. , 87 (1959).[66] T. P. Dougherty, G. P. Wiederrecht, K. A. Nelson, M. H.Garrett, H. P. Jensen, and C. Warde, Science , 770(1992).[67] T. P. Dougherty, G. P. Wiederrecht, K. A. Nelson, M. H.Garrett, H. P. Jenssen, and C. Warde, Phys. Rev. B ,8996 (1994).[68] T. Kohmoto, K. Tada, T. Moriyasu, and Y. Fukuda,Phys. Rev. B , 064303 (2006).[69] M. Hase, M. Kitajima, S. Nakashima, and K. Mizoguchi,Appl. Phys. Lett. , 4921 (2003).[70] R. Merlin, Sol. State Comm. , 207 (1997).[71] T. E. Stevens, J. Kuhl, and R. Merlin, Phys. Rev. B ,144304 (2002).[72] R. C. Rai, J. Cao, J. L. Musfeldt, S. B. Kim, S.-W.Cheong, and X. Wei, Phys. Rev. B , 184414 (2007).[73] A. B. Souchkov, J. R. Simpson, M. Quijada, H. Ishibashi,N. Hur, J. S. Ahn, S. W. Cheong, A. J. Millis, and H. D.Drew, Phys. Rev. Lett. , 027203 (2003).[74] P. Bowlan, S. A. Trugman, J. Bowlan, J.-X. Zhu, N. J.Hur, A. J. Taylor, D. A. Yarotski, and R. P. Prasanku-mar, Phys. Rev. B , 100404 (2016).[75] R. Mankowsky, A. von Hoegen, M. F¨orst, and A. Cav-alleri, Phys. Rev. Lett. , 197601 (2017).[76] M. Matsubara, S. Manz, M. Mochizuki, T. Kubacka,A. Iyama, N. Aliouane, T. Kimura, S. L. Johnson,D. Meier, and M. Fiebig, Science , 1112 (2015).[77] D. E. Aspnes, JOSA , 1275 (1980). , a () ( - c m - ) (a) (b) , a () ( - c m - ) -1600-1500-1400-1300-12003.23.02.82.62.42.22.01.81.6 Energy (eV)12001000800600400200 3.23.02.82.62.42.22.01.81.6 Energy (eV) FIG. S1. (a) Real σ a ( ω ) and (b) imaginary σ a ( ω ) parts of the a -axis complex optical conductivity of TbMnO , measured at8 K via SE.FIG. S2. Temperature dependence of ∆R/R as a function of probe photon energy and time delay between pump and probe.The temperatures are indicated in the labels and the absorbed pump fluence is 4.4 mJ/cm . -30-25-20-15-10-50121086420 Delay time (ps) E ne r g y ( e V ) -30-20-100 T = 100 K -30-25-20-15-10-50 E ne r g y ( e V ) -30-20-100 T = 50 K -30-25-20-15-10-50 E ne r g y ( e V ) -30-20-100 T = 25 K -30-25-20-15-10-50121086420 Delay time (ps) E ne r g y ( e V ) -30-20-100 T = 30 K -30-25-20-15-10-50 S W ( / c m ) E ne r g y ( e V ) -30-20-100 T = 8 K -30-25-20-15-10-50121086420 Delay time (ps) E ne r g y ( e V ) -30-20-100 T = 11 K -30-25-20-15-10-50121086420 Delay time (ps) E ne r g y ( e V ) -30-20-100 T = 45 K -30-25-20-15-10-50 E ne r g y ( e V ) -30-20-100 T = 40 K S W ( / c m ) S W ( / c m ) S W ( / c m ) S W ( / c m ) S W ( / c m ) S W ( / c m ) S W ( / c m ) -30-25-20-15-10-50 -30-20-100 T = 15 K -30-25-20-15-10-50121086420 Delay time (ps) E ne r g y ( e V ) -30-20-100 T = 20 K E ne r g y ( e V ) S W ( / c m ) S W ( / c m ) FIG. S3. Temperature dependence of the transient optical conductivity ∆ σ as a function of probe photon energy and timedelay between pump and probe. Every map also shows the temporal evolution of the nonequilibrium SW (∆ SW ), which iscalculated by computing the integral of the corresponding ∆ σ map over the whole probed range. The temperatures areindicated in the labels and the absorbed pump fluence is 4.4 mJ/cm . E ne r g y ( e V ) -6-4-20 R / R ( - ) (a) (b) Pump cProbe a -4-202x10 -3 FIG. S4. (a) Color-coded maps of ∆R/R at 8 K with c -axis pump polarization and a -axis probe polarization. The pumpphoton energy is 1.55 eV and the absorbed pump fluence is 4.4 mJ/cm2