Biexciton in one-dimensional Mott insulators

Mott insulators sometimes show dramatic changes in their electronic states after photoirradiation, as indicated by photoinduced Mott-insulator-to-metal transition. In the photoexcited states of Mott insulators, electron wavefunctions are more delocalized than in the ground state, and long-range Coulomb interactions play important roles in charge dynamics. However, their effects are difficult to discriminate experimentally. Here, we show that in a one-dimensional Mott insulator, bis(ethylenedithio)tetrathiafulvalene-difluorotetracyanoquinodimethane (ET-F2TCNQ), long-range Coulomb interactions stabilize not only excitons, doublon-holon bound states, but also biexcitons. By measuring terahertz-electric-field-induced reflectivity changes, we demonstrate that odd- and even-parity excitons are split off from a doublon-holon continuum. Further, spectral changes of reflectivity induced by a resonant excitation of the odd-parity exciton reveals that an exciton-biexciton transition appears just below the exciton-transition peak. Theoretical simulations show that long-range Coulomb interactions over four sites are necessary to stabilize the biexciton. Such information is indispensable for understanding the non-equilibrium dynamics of photoexcited Mott insulators.

Mott insulators sometimes show dramatic changes in their electronic states after photoirradiation, as indicated by photoinduced Mott-insulator-to-metal transition. In the photoexcited states of Mott insulators, electron wavefunctions are more delocalized than in the ground state, and long-range Coulomb interactions play important roles in charge dynamics. However, their effects are difficult to discriminate experimentally. Here, we show that in a one-dimensional Mott insulator, bis(ethylenedithio)tetrathiafulvalenedifluorotetracyanoquinodimethane (ET-F2TCNQ), long-range Coulomb interactions stabilize not only excitons, doublon-holon bound states, but also biexcitons. By measuring terahertz-electric-field-induced reflectivity changes, we demonstrate that oddand even-parity excitons are split off from a doublon-holon continuum. Further, spectral changes of reflectivity induced by a resonant excitation of the odd-parity exciton reveals that an exciton-biexciton transition appears just below the exciton-transition peak.
Theoretical simulations show that long-range Coulomb interactions over four sites are necessary to stabilize the biexciton. Such information is indispensable for understanding the non-equilibrium dynamics of photoexcited Mott insulators. the binding energy of the biexciton is 1 /3 accordingly, and the biexciton is stable as well as the exciton. Since the binding energy of 1 /3 originates from 3  This compound is a segregated-stacked charge-transfer (CT) compound consisting of ET (donor) and F2TCNQ (acceptor) columns, as shown in Fig. 1b 37 . An electron is transferred from ET to F2TCNQ. F2TCNQ − molecules are almost isolated, while a finite overlap of wavefunctions with a transfer energy t of 0.2 eV exists between neighbouring ET + molecules along the a axis (Fig. 1b). Because of the large U on ET, the ET columns are in a 1D Mott insulator state. Figure 1c shows the polarized reflectivity spectra. A sharp peak polarized parallel to the a axis (∕∕ ) is observed at 0.7 eV, which corresponds to the Mott gap transition. Such a sharp structure makes us expect an excitonic nature. From the spectral shape, however, we cannot determine whether this peak is attributed to an exciton or an interband transition sharpened owing to the Van-Hove singularity.
To investigate the energy-level structures of the photoexcited states and stabilities of excitons in a 1D Mott insulator of ET-F2TCNQ, we performed terahertz-pulse-pump optical-reflectivity-probe spectroscopy and measured the electric-field-induced changes in the optical reflectivity spectrum, which include information not only about one-photon allowed states but also originally one-photon forbidden states. From analyses of the results, we clarified that the odd-and even-parity excitons are split off from the doublonholon continuum. We next applied pump-probe reflection spectroscopy to ET-F2TCNQ in the near-infrared region with a resonant excitation to the lowest exciton, and investigated the possible bound state of two excitons, that is, a biexciton. We observed the signature of an exciton-biexciton transition in the optical reflectivity spectrum, the spectral shape of which was well reproduced by a theoretical simulation taking into account the Coulomb interactions over up to four sites. The results demonstrate the importance of long-range Coulomb interactions in the dynamics of photoexcited excitons in Mott insulators.
An effective method to evaluate the energy-level structures of excitons is electroreflectance (ER) spectroscopy [38][39][40][41] , in which reflectivity changes induced by quasistatic electric fields are measured. This enables us to obtain a wide frequency range of the third-order non-linear susceptibility (3) spectrum without special laser systems.
However, the ER spectroscopy cannot be applied to low-resistivity materials, in which an application of high electric fields sometimes gives rise to a dielectric breakdown, destroying the sample owing to excess electric current. In most organic molecular compounds with small gap energies, nonlinear electric transport and current-induced electric-switching phenomena indeed occur. This makes it impossible to adopt the ER method. To overcome this problem, in the present study, we apply terahertz-pump opticalprobe spectroscopy to ET-F2TCNQ (Fig. 2a). Within a terahertz pulse, an electric current hardly flows owing to the short duration of the electric field (1 ps) 42 . In addition, the magnitude of the electric field can be increased without sample damages.
In Fig. 2b where (3) ( ) and ( ) are the third-order nonlinear polarization and the electric field of the probe light, respectively. Such a reflectivity modulation by a terahertz electric field was previously reported in [Ni(chxn)2Br]Br2 (chxn=cyclohexanediamine) 42 .
To obtain detailed information about the energy level structure, we measured the probe-energy dependence of ∆ / at d = 0 ps (open circles in Fig. 2e). By applying the Kramers-Kronig (KK) transformation to the and ∆ ( d = 0 ps)/ spectra, we obtained the 2 and ∆ 2 spectra, as shown by the solid black line in Fig. 2f and solid blue line in Fig. 2g, respectively. Details of the analyses are reported in Supplementary Note 2. The 2 spectrum has a sharp peak at 0.7 eV, and the ∆ 2 spectrum has a plusminus-plus structure around the sharp peak.
First, we analyse the 2 spectrum with the following Lorentzian-type dielectric function: Here, |0⟩ and |1⟩ show the ground state and the one-photon-allowed odd-parity state, respectively, and 〈0| |1〉 is the transition dipole moment between them. The used parameter values are listed in Table I.
We next analysed the ∆ 2 spectrum showing a plus-minus-plus structure (the solid blue line in Fig. 2g). To analyse this spectrum, we assumed that the frequency of the terahertz electric field, THz , is 0. This assumption is justified under the condition that ℏ THz (~4 meV ) is sufficiently lower than an energy difference between any of two energy levels of excited states 38 . ET-F2TCNQ meets this condition, as will be shown later.
Using this assumption, we calculate Im (3) from ∆ 2 with the following equation: The maximum of |Im (3) | was evaluated to be 1  10 -7 esu.
The previous ER spectroscopy of 1D Mott insulators of transition metal compounds revealed that a plus-minus-plus structure in Im (3) spectra can be interpreted by a threelevel model in which the one-photon forbidden state with even parity (|2⟩) is assumed in addition to the ground state |0⟩ and the odd-parity state |1⟩ [38][39][40][41] . In ET-F2TCNQ, small negative signals appear above 0.85 eV, as shown in Fig. 2g, in addition to the plus-minus-plus structure. Such a negative signal can be explained by considering the secondlowest odd-parity state (|3⟩) 41 . In a four-level model consisting of |0⟩ − |3⟩, the (3) spectrum is represented by the following equation 43 : 〈 | | 〉 shows the transition dipole moment between states | ⟩ and | ⟩ .  Table I.
The splitting between |1⟩ and |2⟩ , ℏ( 2 − 1 ) , was small (26 meV), indicating that the two states are nearly degenerate. In addition, 〈1| |2〉 , which is the most important parameter dominating the magnitude of (3) , was very large at 18 Å. In ET-F2TCNQ, the ratio 〈1| |2〉 〈0| |1〉 ⁄ is equal to 13. The enhancement of 〈1| |2〉 is attributable to the fact that the wave functions of the odd-and even-parity states are similar to each other except for their phases, as schematically shown in Fig. 2h, and the spatial overlap of these wave functions becomes very large. These features are characteristic of 1D Mott insulators 38,39 . The observation of a higher odd-parity state |3⟩ demonstrates that the lower two states |1⟩ and |2⟩ are excitionc states.
To obtain evidence of the excitonic effect from the transport property, we measured Optical-pump optical-reflectivity-probe spectroscopy.
To observe a biexciton, we next performed optical-pump optical-reflectivity-probe spectroscopy (  Therefore, this response is attributable to the coherent response, which is observed in the resonant excitation to an exciton in semiconductors 45

Simulation of exciton-biexciton transition
To investigate the biexciton formation more strictly, we theoretically calculate the imaginary part of the dielectric constant 2 in the ground state and in the presence of the lowest exciton using an extended Hubbard model, as follows: Here, is the Coulomb interaction between two electrons distant for j sites, as mentioned above. We assume again that is inversely proportional to a doublon-holon . In Fig. 3f, we show the 2 spectrum in the ground state, which was calculated by the Lanczos method with a system size (site number) of L = 14. The parameter values in the system are set to be = 0.14 eV, = 1.4 eV, and 1 = 0.6 eV to reproduce the peak energy of the 2 spectrum for the odd-parity exciton.
Next, we calculated the 2 spectrum after the resonant excitation to the odd-parity exciton 42 . The temporal shape of the pump pulse is assumed to be Gaussian, as follows: The oscillation frequency OSC is 82 cm -1 , and the relaxation time is 2.0 ps. The value of (= 14) is small, suggesting that the generation mechanism of the oscillation is the displacive excitation of the coherent phonon 48 . The Fourier power spectrum of the experimental time characteristic of ∆ OSC / and the fitting curve are shown in Fig. 4b by open circles and the solid red line, respectively, which agree with each other.
We performed similar analyses of the coherent oscillations in ∆ / signals at various probe energies and plotted the magnitude of the fitting functions ( OSC ) in Fig.  4c (red circles) together with the original spectrum (black line). The data show a clear peak at 0.64 eV, which corresponds well to the peak (0.630 eV) of ∆ 2 assigned to the exciton-to-biexciton transition shown in Fig. 3d. This suggests that the energy and/or the intensity of the exciton-to-biexciton transition is modulated at a frequency of 82 cm -1 , which is observed as the oscillatory structure of the reflectivity changes. The origin of this oscillation is discussed in the next section.

Discussion
First, we comment on the stabilization mechanism of the biexciton in 1D Mott insulators. In the simulation with an extended Hubbard model, we investigated several parameter sets. When we consider only the intersite Coulomb interactions 1 and 2 , no peak is observed just below the lowest exciton transition in ∆ 2 , even if their magnitudes are enhanced. This demonstrates that the long-range Coulomb attractive interaction characterized by 3 plays a significant role in the stabilization of the biexciton. This is consistent with the simplified picture of the energy gain of biexciton formation in Fig. 1a.
Next, we discuss the origin of the coherent oscillation. As seen in the spectrum of the magnitude of the oscillatory components in Fig. 4c, the oscillation is observed around the exciton-biexciton transition at 0.630 eV (Fig. 3d). In addition, the frequency of the oscillation, 82 cm -1 , is a typical frequency of a lattice mode in organic molecular compounds. It is, therefore, reasonable to consider that the oscillation is ascribed to molecular displacements in the lattice relaxation process of the lowest-energy exciton In summary, we successfully measured the spectra of the ultrafast reflectivity changes by a terahertz electric field and by the resonant excitation of the lowest exciton in a 1D Mott insulator of an organic molecular compound, ET-F2TCNQ. By analysing the spectra of reflectivity changes induced by the terahertz electric field, we revealed the energylevel structures of the exciton and continuum states, and evaluated the binding energy of the lowest-energy exciton to be about 160 meV. In addition, from the spectrum of reflectivity changes by the resonant optical excitation to excitons, we demonstrated that the biexciton is stable owing to long-range Coulomb interactions and that its binding energy is about 60 meV, which is almost equal to one third of the exciton binding energy as predicted in a case with a strong electron correlation limit. Such information about biexcitons as well as excitons is significantly important for the understanding of non-equilibrium dynamics in photoexcited 1D Mott insulators.

Sample preparations.
Single crystals of ET-F2TCNQ were grown by the method previously reported 37 .

Pump-probe reflectivity measurements.
In the terahertz-pulse-pump optical-reflectivity-probe experiments, the output of Ti:Al2O3 regenerative amplifier (pulse width: 130 fs , photon energy: 1.58 eV , repetition frequency: 1 kHz ) were divided into two beams. One was used for the generation of terahertz-pump pulses through the optical rectification in a LiNbO3 crystal by the pulsefront-tilting method 51,52 . The other was introduced to an optical parametric amplifier All the experiments were performed at 294 K.

Photoconductivity measurements
The photoconductivity (PC) measurement was performed by the method previously reported 40 at 92 K to avoid thermal excitations of carriers. We ascertained that the photocurrents are proportional to the intensity of the excitation light. The excitation spectrum of PC was obtained by using the formula, PC ∝ PC [ p (1 − p )] ⁄ . Here, PC , P , and p are the photo-current signal, the photon density of the excitation light per unit area, and the reflection loss.

Data availability.
The data that support the findings of this study are available from the corresponding author on request.   Table I. Parameters evaluated from the fitting analysis.

Supplementary Note 2. Kramers-Kronig transformation of reflectivity-change spectra
To perform the Kramers-Kronig (KK) transformations of the ∆ / spectra, we extrapolated the ∆ / spectra since the measurement range was limited. In the ∆ / spectra obtained by terahertz-pulse-pump optical-reflectivity-probe spectroscopy, the measurement range was 0.48 eV to 1.08 eV. Below (above) the lower (higher) energy bound of the measured range, we linearly extrapolated the data using the lowest (highest) two measurement points and set ∆ / to be zero after ∆ / reached zero. We also made the liner interpolation in the ∆ / spectra experimentally obtained from 0.48 eV to 1.08 eV since the number of measurement points was insufficient to perform the KK transformation.
In the ∆ / spectra obtained by optical-pump optical-reflectivity-probe spectroscopy, the measurement range was 0.082 eV to 2.17 eV . Below the lower energy bound of the measurement range, we used an ∆ / value of 0.082 eV. Above the higher energy bound, we linearly extrapolated the data using the highest two measurement points and set ∆ / to be zero after ∆ / reached zero. We also linearly interpolated the ∆ / spectra from 0.082 eV to 2.17 eV.

Supplementary Note 3. Excitation photon density dependence of exciton-biexciton transition
To clarify how the dynamics of biexcitons depends on the density of excitons generated by the pump pulse, we measured the excitation photon density ph dependence of reflectivity changes ∆ / at 0.58 eV, which corresponds to the peak

Supplementary Note 5. Analyses of coherent oscillations on photoinduced reflectivity changes
To analyse the coherent oscillations on the time evolutions of ∆ / obtained by the optical-pump optical-reflectivity-probe measurements, we derived the oscillatory components by applying a high-pass Fourier filter with a cut-off frequency of 1.6 THz to the ∆ / data. To exclude the contribution of the coherent response around the time origin, we used the data only for d > 0.2 ps. We next performed fitting analyses of the oscillatory components thus obtained using eq. (7) as described in the main text. In these analyses, we used the common values of decay time = 2.0 ps and oscillation frequency OSC = 82 cm −1 . (S1) The first term represents the coherent response and its magnitude is denoted by . The positive ∆ / signal at 0.58 eV due to the exciton-biexciton transition also gives the information about the decay dynamics of exciton. As mentioned in the main text, the ∆ / signal at 0.58 eV will include the contribution of the negative ∆ / component due to the coherent response around the time origin, so that we analysed only the data for d > 1 ps using the following formula including only the third term in Eq. (S1).