Ultrafast dynamics of quasiparticles and coherent acoustic phonons in slightly underdoped (BaK)Fe2As2

We have utilized ultrafast optical spectroscopy to study carrier dynamics in slightly underdoped (BaK)Fe2As2 crystals without magnetic transition. The photoelastic signals due to coherent acoustic phonons have been quantitatively investigated. According to our temperature-dependent results, we found that the relaxation component of superconducting quasiparticles persisted from the superconducting state up to at least 70 K in the normal state. Our findings suggest that the pseudogaplike feature in the normal state is possibly the precursor of superconductivity. We also highlight that the pseudogap feature of K-doped BaFe2As2 is different from that of other iron-based superconductors, including Co-doped or P-doped BaFe2As2.

We have utilized ultrafast optical spectroscopy to study carrier dynamics in slightly underdoped (BaK) Fe 2 As 2 crystals without magnetic transition. The photoelastic signals due to coherent acoustic phonons have been quantitatively investigated. According to our temperature-dependent results, we found that the relaxation component of superconducting quasiparticles persisted from the superconducting state up to at least 70 K in the normal state. Our findings suggest that the pseudogaplike feature in the normal state is possibly the precursor of superconductivity. We also highlight that the pseudogap feature of K-doped BaFe 2 As 2 is different from that of other iron-based superconductors, including Codoped or P-doped BaFe 2 As 2 .
Iron-based high-temperature superconductors have attracted great attention in recent years 1 . Among the more than six families based on their respective compositions of compounds, of which the 1111 (such as SmFeAsO), 122 (such as BaFe 2 As 2 ), 111 (such as LiFeAs) and 11 (such as FeSe) systems have been widely studied. Compounds of the 122 system are the easiest among the iron-based superconductors to synthesize in the form of large single crystals, and most publications have been devoted to 122 compounds 2 . Because superconductor-based devices require high quality crystals/films, the 122 system has also been employed most often among the iron-based compounds to demonstrate devices such as Josephson junctions [3][4][5][6] and superconducting wires/tapes for high current applications [7][8][9][10][11][12][13][14][15][16] . Although the unconventional high-temperature superconductors (including copper-based and iron-based superconductors) have been applied for devices, the fundamental questions such as the forming mechanisms of Cooper pairs remain unclear 17 . K-doped BaFe 2 As 2 was the first compound in the 122 system found to exhibit superconductivity 18 . It should be noted that its parent compound, BaFe 2 As 2, has a structural and magnetic phase transition 19 , but Ba 1−x K x Fe 2 As 2 is only superconducting (SC) for a range of x with maximum T c at x = 0.4 20 . Ba 1−x K x Fe 2 As 2 does not have a structural and magnetic phase transition when x is close to the optimal value. The phase diagram and planar geometry of K-doped BaFe 2 As 2 are similar to those of cuprate superconductors, leading to the intriguing question of whether K-doped BaFe 2 As 2 and cuprate superconductors have the same mechanism of superconductivity.
One of the general features in cuprate superconductors is the presence of a pseudogap above T c 21 . There are generally two energy scales for pseudogaps in cuprate superconductors and the low-energy pseudogap is believed to be the precursor of the SC gap 21 . However, there are contradictory results about the existence of pseudogap in optimally doped (BaK)Fe 2 As 2 (BKFA) from angle-resolved photoemission spectroscopy (ARPES) measurements 22,23 . Although ARPES and scanning tunneling microscopy are capable of directly resolving band structures, they are surface-sensitive techniques. In contrast, ultrafast optical spectroscopy [24][25][26][27][28][29][30] is not surface-sensitive, and is a convenient tool to study the bulk property of materials. By observing the temperature-dependent quasiparticle-relaxation, a pseudogaplike feature has been observed in FeSe 25 , LiFeAs 26 , Ba(Fe,Co) 2 As 2 27 , Ca(Fe,Co) 2 As 2 28 , SmFeAsO 0.8 F 0. 2 29 , and underdoped BKFA 30 . In this article, we have utilized ultrafast optical spectroscopy to study carrier dynamics in slightly underdoped BKFA single crystals without magnetic transition. The signals of coherent acoustic phonons were quantitatively studied. We also found that the quasiparticle relaxation associated with the SC gap has temperature dependence 1 that appears in the SC state and persists at least up to 70 K in the normal state. Our result supports the argument that the pseudogaplike feature in the normal state is possibly the precursor of superconductivity 30,31 . However, in most other iron-based superconductors 26,28,[32][33][34][35][36][37][38] , the pseudogap is not associated with the SC gap and both of them coexist in the SC state. In addition, we suggest that the contradictory reports 22,23 resulted from slightly different concentrations of K near optimally doped BKFA. The pseudogaplike feature could be found in slightly underdoped BKFA but was not found in optimally doped BKFA.

Results and Discussions
The magnetic susceptibilities of BKFA as a function of temperature were measured, as shown in Fig. 1. The SC temperature has been evaluated as 36 K. We conducted temperature-dependent transient optical reflectivity measurements of BKFA. Detailed experimental conditions can be found in the Methods section. Figure 2 shows the measurements for six representative temperatures. As shown in Fig. 2(f), the 110 K trace shows a fast exponential function with a time constant of < 1 ps, followed by a broad negative hump at ~12 ps. The ultrafast relaxation time within 1 ps should be attributed to thermalization of non-Fermi distribution of electrons 39 .
We ascribe the negative humps at ~12 ps to the effects of propagating coherent longitudinal acoustic phonons along the depth direction. In transparent or semi-transparent media, the propagation of coherent acoustic phonons induces temporal sinusoidal oscillation due to coherent Brillouin scattering 40 . The oscillation period of multiple cycles is experimentally resolvable and is determined by the refractive index, sound velocity, optical wavelength, and incidence angle of the optical probe in the media 40 . However, the feature of transient reflectivity due to propagating phonons in highly absorptive materials, such as the BKFA we studied, is not trivial. In order to understand the acoustic signals, we have used the finite difference time domain method to simulate the temporal evolution of longitudinal strain distributions S z t ( , ) 33 (or longitudinal acoustic phonons) in BKFA 41,42 . The time-resolved optical reflectivity can be represented as 43 (2) z l 33 33 / n and κ are respectively the real part and the imaginary part of the complex refractive index of the sample, z is the depth position from the surface, λ is the optical wavelength in vacuum, and l is the optical absorption length. φ is a constant, which is related to n and κ 43 . ∂ ∂ n S / 33 and κ ∂ ∂S / 33 are the real part and the imaginary part of the photoelastic constant, respectively. The photoelastic constant can also be expressed as . For calculation, we assumed that the initial condition of S z t ( , ) 33 is proportional to exp(− z/l), where l = 27.7 nm at 400 nm 41 . By using the indices of refraction . + . i (2 8 2 3 ) at 800 nm and . + . i (1 9 1 8 ) at 400 nm for BKFA 41 , ∆R t ( ) can be calculated following Eq. (1). Figure 3 shows the rescaled ∆R t ( ) for a few representative phases of photoelastic constants (θ). Note that ∆R t ( ) for different θ in Fig. 3 can be compared since all of the other parameters, including P, are kept the same. The centers of the negative hump vary for different values of θ. Therefore, one could not determine the phonon oscillation period simply from the first dip time. According to our fitting results in Fig. 3, the best fit of θ was 60° ± 5° for BKFA at 110 K. This is similar to the phase of the photoelastic constant of Cr metal thin film 44 . The amplitude of the photoelastic constant P could not be obtained in the present study because the strain under our experimental condition was unknown. Further information such as thermal expansion coefficients and heat capacities should be determined for P.   Fig. 4(a), the normalized traces are roughly the same after 10 ps. It should be noted that slower processes such as acoustic phonons, carrier diffusions, and heat diffusions should not be sensitive to the temperature. However, we found additional relaxation components with time constants on the order of several tens of picoseconds in SC state such as at 14 K and 33 K, shown in Fig. 4(a). We attribute these temperature-dependent components to quasiparticle relaxation due to the SC gap 45 .
In order to quantitatively study the temperature-dependence of the carrier dynamics, the transient reflectivity A(t) can be represented by where u t ( ) is the step function; . Experimentally, we found the normalized traces above 110 K are almost the same. We assumed that  was used for fitting, and A e , τ e , A S , τ S , a, and b are the fitting parameters. A(t) was convolved with the cross-correlation of pump and probe pulses to quantitatively fit the experimentally recorded ∆R t R ( )/ traces below 110 K. According to our temperature-dependent analysis, A e and τ e did not show any systematic trend and will not be discussed. Fig. 5(a,b) show the temperature-dependent τ S and A S . From high temperature to low temperature, A S dramatically increases beginning at T c (36 K) and roughly saturates below 25 K. τ S also has a peak at T c . These features were similar to those of quasiparticle relaxation due to the SC gap in other iron-based superconductors 26,29,30,46 . Although iron-based superconductors have multiple bands with different SC gaps, a temperature-dependent one-gap system based on the Rothwarf-Taylor (RT) model was still adequate to estimate the SC gap size for FeSe 46  . We also used the RT model to interpret and fit our data 45 . The transient temperature near the surface of BKFA exceeds T c after the optical pulses excite the quasiparticles. It thus creates high frequency bosons with energy ω ≥ ∆ 2 and breaks the Cooper pairs. The relaxation time reflects the population of high frequency bosons and the recovery time of the SC state when the heat escapes from the optically probed depth. In brief, τ S is the recovery time of superconductivity after suppression by photocarriers 45,47 .
We assumed a temperature-dependent gap ∆ 2 (T), which is proportional to ∆ 2 (T) BCS following BCS temperature dependence.   where δ and α are fitting parameters following the criteria described in ref. 30. The fitting curves were arbitrarily scaled due to the proportionality. The gap size ∆ = ± (0) 12 meV 1 meV was obtained from the fitting results based on the BCS temperature-dependent one-gap system. The red lines in Fig. 5(a,b) show the fitting curves with ∆ = (0) 12 meV. The temperature-dependent A S below T c shows good agreement with the RT model, while the temperature-dependent τ S does not agree well. However, τ S still shows the characteristic trend of the RT model, the "U" shape below T c . From low temperature to high temperature, τ S first decreases and then dramatically increases when the temperature approaches T c 30 . It should be noted that this is a simplified model. Actually, the transient optical reflectivity should be governed by the contribution of quasiparticle relaxation in all bands due to the multiband feature of iron-based superconductors. According to ARPES measurement results of optimally doped BKFA, Δ is ~12 meV at the inner/outer electron pockets (γ/δ bands) and the inner hole pocket (α band) while Δ is ~6 meV at the outer hole pocket (β band) 22,23,48,49 . Our obtained ∆ = ± (0) 12 1 meV agrees well with that of α, γ, δ bands in BKFA. The uncertainty of the fitting results with the simplified one-gap model might be attributed to different SC gaps of multi-bands in BKFA.
From Fig. 5(b), we have noticed that A S does not vanish above T c , which is similar to the phenomenon observed in underdoped BKFA with magnetic transition 30 . The SC-related component of relaxation persists up to at least 70 K in our slightly underdoped BKFA without magnetic transition. Note that the noise level in Fig. 5(b) was obtained by a strict definition of peak-to-peak noise fluctuations in ∆R t R ( )/ , which means the onset temperature could be even higher. It should be noted that the error bars for τ S are larger above T c due to the relatively small amplitude A S . However, the relaxation component A S (after consideration of the error bars and noise level) indeed persist from below T c to above T c . This phenomenon can also be clearly observed without quantitative analysis. The normalized traces in Fig. 4  (with τ S of several tens of ps) is relatively strong and overwhelms some features of A t ( ) Ph (as shown in Fig. 3 that the signal decreases prior to the minimum at ~12 ps). As shown in Fig. 4(b), these can be clearly observed in the normal state, 49 K trace, which still has the relaxation component τ − A e S t/ S . According to our quantitative analysis in Fig. 5(b), the relaxation component due to the SC gap persists at least up to 70 K.

(b) reflect the percentage of different contributions from
This phenomenon was similar to that found in underdoped BKFA with magnetic transition 30 , and was attributed to the pseudogap (or probably partial gaps) in the normal state. The main difference is that our slightly underdoped samples do not have magnetic transition. In addition, the pseudogap of slightly underdoped BKFA (without magnetic transition) was also found by optical conductivity measurement 31 . Kwon et al. found that the gap feature in the range between 50 cm −1 and 150 cm −1 (corresponding between 6 meV and 18 meV) continuously existed from the SC state to the normal state up to 100 K 31 . Our experimental results and previous reports 30,31 support the proposition that the pseudogap in BKFA should be the precursor of the SC gap because the same SC-related relaxation component in the SC state persists up to the normal state.
It is worthy to note that Chia et al. also investigated the optimally doped BKFA and did not observe a significant pseudogap feature above T c 30 . We argue that the BKFA sample of our present study should be slightly underdoped since its T c = 36 K is slightly lower than the T c = 37 K of BKFA reported in ref. 30. We also found a similar situation for ARPES measurements in that a pseudogap was found in BKFA with T c = 35 K 22 but was not clearly found in BKFA with T c = 37 K 23 . A pseudogap was also clearly observed with optical conductivity measurement in slightly underdoped BKFA without magnetic transition 31 . We thus suggest that the contradictory reports 22,23,30,31 actually resulted from the slightly different concentrations of K near optimally doped BKFA. Our results and previous reports 22,31 indicate that a pseudogap still exists in slightly underdoped BKFA without magnetic transition. It is interesting that no strong evidence of a pseudogap has been found in optimally doped 30 or overdoped BKFA, although a shadow gap was predicted recently 50 . Nevertheless, a pseudogap was found in the overdoped regime of other 122 system compounds 28,32-34 . In the 1111 35,36 , 111 26,37,38 , and cuprate 51-53 systems, it was reported that a pseudogap can coexist with the SC gap. The infrared pseudogap in Co-doped and P-doped BaFe 2 As 2 (122 systems) was also reported to be unrelated to superconductivity 33 . However, our findings and previous results 30,31 lead us to suggest that the pseudogap in K-doped BaFe 2 As 2 is the precursor of superconductivity. We suggest that partial gaps along certain momentum might be opened in the normal state and they evolve to complete SC gap below T c . Note that the term "pseudogap" was widely used in the literature to specify a gap or a partial gap of which the physical origin is unknown. They may have different physical origins. Our experimental evidence indicates that the origin of the pseudogap in BKFA is different from that of other iron-based superconductors with coexisting pseudogap and SC gap in the SC state 26,28,[32][33][34][35][36][37][38] . We would like to highlight that K atoms are doped out of the FeAs planes while Co and P are doped in the FeAs planes. It was suggested that in-plane doping or out-of plane doping would affect how SC gap forms 54 . A similar situation may occur in the formation of the pseudogap, leading to different features between (BaK)Fe 2 As 2 (out-of-plane doping systems) and other in-plane systems such as BaFe 2 (CoAs) 2 and BaFe 2 (AsP) 2 .

Conclusions
We have utilized ultrafast optical spectroscopy to investigate carrier dynamics in slightly underdoped BKFA without magnetic transition. The signals of acoustic phonons were quantitatively studied. We have also analyzed the quasiparticle relaxation due to the SC gap. Although the doping concentration of BKFA does not have a magnetic transition and is nearly optimally doped, we still found pseudogap feature at least up to 70 K in normal state. Our experimental results support the proposition that the pseudogap in BKFA should be the precursor of the SC gap. We have highlighted the unique pseudogap feature in K-doped BaFe 2 As 2 , which is different from that in other iron-based superconductors.

Methods
Experimental Setup. The sample was cleaved to reveal a shining surface, and mounted on the holder of the cryostat in an Ar-filled glove box. After the cryostat was moved out from the glove box, the pressure of the chamber was immediately lowered to below 10 −4 mtorr to avoid oxidation. Typical pump-probe (400 nm-800 nm) measurements were conducted. The polarization of the pump beam is orthogonal to that of the probe beam. But, the polarization was not intentionally aligned to the crystal axis of the sample. The repetition rate was 8 MHz and the optical pump fluence was 5~10 μ J/cm 2 . The pump was modulated at ~1 MHz with an acousto-optical modulator (AOM). The full width at half maximum (FWHM) of the temporal cross-correlation of the pump and probe pulse was ~400 fs, which determined the temporal resolution. Time constants close to the temporal resolution could still be accurately determined after the deconvolution process. A color filter was placed in front of the photodetector to eliminate pump light leakage. We recorded the reflectivity variation of the probe pulse as a function of time delay.
Under our experimental condition, the traces were in the linear region above 50 K. The normalized traces for optical pump fluence of both 5 μ J/cm 2 and 10 μ J/cm 2 did not show significant difference. At the lowest temperature cooled with liquid He, the normalized traces were different for optical pump fluence of 5 μ J/cm 2 and 10 μ J/cm 2 . The steady-state laser heating still occurred under our experimental condition especially at low temperatures. The real temperature of the sample increased with increasing optical pump fluence. The temperature was calibrated as described in the next section. However, it should be noted that fluence-dependent measurements cannot be used to verify the linearity in the SC state of BKFA. It was reported that the decay rate of the quasiparticles due to the SC gap varies with optical pump fluence 55 . In our temperature-dependent measurements, the optical pump fluence was 5 μ J/cm 2 below 50 K, which was reasonably low compared with previous experiments on iron-based superconductors 25,26,29,30,46,55 . For enhancement of the signal to noise ratio, the optical pump fluence was 10 μ J/cm 2 for measurements above 50 K because they are in the linear region. The traces above 50 K were rescaled for comparison. Temperature Calibration. The temperature of the sample was 10 K calibrated from the temperature measured by the thermometer. According to Fig. 2(b) in ref. 55, the features of normalized traces in SC and non-SC states under the same pumping fluence can be easily distinguished. We could thus obtain the temperature difference between the real temperature T c of the sample and the temperature measured by the thermometer.