Ultrafast scattering dynamics of coherent phonons in Bi_{1-x}Sb_{x} in the Weyl semimetal phase
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Ultrafast scattering dynamics of coherent phononsin Bi − x Sb x in the Weyl semimetal phase Yuta Komori , Yuta Saito , Paul Fons , , Muneaki Hase Department of Applied Physics, Faculty of Pure and Applied Sciences, University ofTsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan. Device Technology Research Institute, National Institute of Advanced IndustrialScience and Technology, Tsukuba Central 5, 1-1-1 Higashi, Tsukuba 305-8565, Japan. Faculty of Science and Technology, Department of Electronics and ElectricalEngineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa223-8522, Japan.E-mail: [email protected]
Abstract.
We investigate ultrafast phonon dynamics in the Bi − x Sb x alloy systemfor various compositions x using a reflective femtosecond pump-probe technique. Thecoherent optical phonons corresponding to the A g local vibrational modes of Bi-Bi, Bi-Sb, and Sb-Sb are generated and observed in the time domain with a few picosecondsdephasing time. The frequencies of the coherent optical phonons were found to changeas the Sb composition x was varied, and more importantly, the relaxation time of thosephonon modes was dramatically reduced for x values in the range 0.5–0.8. We arguethat the phonon relaxation dynamics are not simply governed by alloy scattering, butare significantly modified by anharmonic phonon-phonon scattering with implied minorcontributions from electron-phonon scattering in a Weyl-semimetal phase. Keywords : coherent phonon, phonon scattering, ultrafast spectroscopy, topologicalmaterials. ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ...
1. INTRODUCTION
In the past decade, topological materials, such as topological insulators (TIs), Diracsemimetals, and Weyl semimetals, have attracted much attention in condensed matterphysics owing to growing interest in the fundamental properties of their surface (bulk)band structure [1, 2, 3, 4, 5], massless quasiparticle dynamics [6], spin dynamics [2, 7],carrier transport [8], and many other new physical effects [9, 10, 11, 12]. Topologicalmaterials, in particular TIs, have been considered also as promising materials forthermoelectric devices, taking advantage of their intriguing properties, such as lowthermal conductivities [13], and large Seebeck coefficients [14]. Bi − x Sb x was firstexperimentally discovered to be a 3D TI in 2008 (Ref. [6]), and has been recentlypredicted to be a Weyl semimetal for some compositions x with a specific atomicarrangement [15]. In addition, structural changes in Bi − x Sb x depending on thecomposition x have been theoretically predicted [16, 17] and are thought to be animportant factor in the stabilization of topological properties in the alloy system.Although among TIs, Bi Se , Bi Te and Sb Te have been extensively investigatedfrom the point of view of ultrafast dynamics [18, 19, 20, 21, 22, 23], an alloy systemlike Bi − x Sb x has rarely been studied because of the difficulty in precisely controllingthe composition x , with which the electronic band structure of the Bi − x Sb x systemchanges from semiconducting to a simple semimetal (SM) with band crossover [24, 25],a TI with a gapless surface state, and even to a Weyl semimetal with a Weyl node at anon zone-center position [26, 27].In practice, the physical information for the Bi − x Sb x system has been limited[28, 29], and only a few experimental studies have been reported on Dirac quasiparticles[2, 3, 6]. Moreover, while the properties of lattice vibrations, phonons, have beenexamined for the Bi − x Sb x system by Raman scattering [30, 31, 32], the presenceof instrumental broadening due to slit and laser line widths [31] as well as seriousbackground from Rayleigh scattering [32] may limit the accuracy of such measurements.These limitations make the indirect measurement of precise phonon lifetimes onpicosecond and femtosecond time scales generally difficult to achieve using conventionalRaman measurements. Our motivation here is the direct measurement of the dephasingdynamics of coherent phonons on sub-picosecond time scales and this approach offersstrong advantages regarding the accuracy of the dephasing time values over thoseindirectly estimated by line widths from conventional Raman measurements.Phonon scattering, such as electron-phonon, anharmonic phonon-phonon, defect-phonon (or alloy) scatterings, are important issues in the field of thermoelectric andtopological devices, since phonon scattering determines the thermal conductivity [13]and the mobility of Dirac quasiparticles [6]. Thus, exploring the ultrafast dynamics of,e.g., anharmonic phonon-phonon scattering and alloy scattering, on a sub-picosecondtime scale, is a challenging issue required for the development of new topologicaldevices. Such dynamical studies of the phonon scattering processes will offer keyphysical information not only for device applications, but also for the understanding ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... Figure 1. (a) Schematic band structure near the Fermi-level for different values of theSb concentration x , where the alloy changes from a semimetal (SM) to a topologicalinsulator (TI), and then back to a SM phase [1, 24]. The emergence of the Weylsemimetal phase was theoretically predicted for x = 0.5 and 0.83 near the zone center(Γ) (Ref. [15]). (b) XRD patterns obtained from thin film samples with different Sbcompositions x . (c) 2D mapping (256 µ m × µ m) of Bi (left) and Sb (right) atomsusing EPMA for a Bi . Sb . sample, showing the homogeneous distribution of bothelements. of the fundamental physical properties of topological alloy systems.Here we explore the ultrafast scattering dynamics of coherent phonons in theBi − x Sb x alloy system for various compositions x by using a femtosecond pump-probetechnique with ≈
20 fs time resolution. The coherent optical phonons corresponding tothe local vibrations of Bi-Bi, Bi-Sb, and Sb-Sb bonds were detected, and the relaxation ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... x ≈
2. MATERIALS AND METHODS
The samples used in this study were highly orientated polycrystalline Bi − x Sb x alloyfilms prepared by radio frequency magnetron sputtering [33]. Thin film samples weresuccessfully deposited onto a Si (100) substrate with different compositions x = 0, 0.36,0.49, 0.72, 0.77, 0.82 and 1.0 by carefully tuning the sputter power of each target.According to the reported electronic band structures in the literature for the Bi-Sbsystem, the addition of Sb to Bi shifts the T and upper L bands downwards and thelower L band upwards [figure 1(a)], resulting in the alloy changing from a semimetalto a topological insulator [24, 25]. The topological insulating composition range x isbetween 0.07 and 0.22, indicating that the L -point band is inverted with respect tothe electronic band in Bi [6, 34]. For our samples, the x = 0.36 alloy was a semimetalwith a narrow band gap (less than several tens of a meV) at the L -point [4, 25]. Thetwo samples x = 0.49 and 0.82 are close to the compositions theoretically predictedto form in the Weyl semimetal phase [15]. All films were approximately 16-nm-thickand X-ray diffraction (XRD) measurements showed that they were highly orientatedpolycrystalline films as can be seen in figure 1(b) except for the pure Bi sample whichshowed the presence of off-orientation peaks. The average grain size in the out-of-planedirection estimated from Scherrer’s formula [35] was ∼
17 nm, a value nearly equivalentto the film thickness, suggesting that single grains were formed from the substrate/filminterface to the film surface. This trend was observed regardless of the Sb content, andthus the effect of grain boundary scattering on x is expected to be negligible. It shouldbe noted that the (003) and (006) peaks shift toward higher angle with increasing Sbcontent suggesting that the lattice constant, c , decreases upon alloying with Sb whichhas a smaller atomic radius.The composition of the films was evaluated by electron probe micro analysis(EPMA) as shown for the case of x = 0.49 in figure 1(c). To reduce uncertainty in thecomposition, the value of the Sb content, x , was measured at five different positions onthe sample, and averaged. We have also measured the transient reflectivity change signalat different sample positions, confirming that the signal did not vary. Furthermore, thecompositional mapping displayed in figure 1(c) confirms a uniform distribution of bothelements over the substrate surface. To prevent oxidation, samples were capped by a 20-nm-thick ZnS-SiO layer without breaking the vacuum. ZnS-SiO capping layers havelong been used as a dielectric layer in optical disc applications [36] and are completelytransparent to near infrared light making no contribution to the observed signal. ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... !" ! ’ ( ) ( * + , - . / " +.:. =. *6/ !% !& !" ! ? . @ A @ . * + , - . / %>&-" ?B3CD3@E7.*0FG/ H I. . *J1,J1/H I. . *J1,KL/H I. . *KL,KL/*L/ , ’ ( ) ( * + , % . / >&-" .?1AA1@I*E/ +.:. Figure 2. (a)The reflectivity change obtained for Bi − x Sb x alloy films with variouscompositions at room temperature. (b) Fourier transformed (FT) spectra of timedomain data shown in (a). (c) The result of the fit using equation (1) for the case ofBi . Sb . . Time-resolved measurements using a reflection-type pump-probe setup were carried outto observe the coherent phonon signal [37, 38]. The light source was a mode-lockedTi:sapphire laser oscillator with a center wavelength of 830 nm, providing ultrashortpulses of ≤
20 fs duration and operated at a 80 MHz repetition rate. The pump andprobe beams were focused by an off-axis parabolic mirror on the samples to a diameter of ≈ µ m and ≈ µ m, respectively, assuring nearly homogeneous excitation [39, 40, 41].The fluence of the pump beam was fixed at ≈ µ J / cm , while that of the probe wasset to ≈
5% of the pump beam, ≈ µ J / cm . The penetration depth of the laser beamestimated from the absorption coefficients was about 17–20 nm [42], which is largerthan the sample thickness ( ≈
16 nm). Thus, in the present study the optical excitationwas homogeneous over the entire sample thickness and the effects of the penetrationdepth did not play a role on the observed coherent phonon spectra. The delay betweenthe pump and the probe pulses was scanned up to 15 ps by an oscillating retroreflectoroperated at a frequency of 19.5 Hz [43]. The transient reflectivity change (∆
R/R ) wasrecorded as a function of pump-probe time delay. The time zero was determined as theposition of the initial drop, which was confirmed with a Bi semimetal sample [44]. Themeasurements were performed in air at room temperature. ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ...
3. RESULTS AND DISCUSSION
Figure 2(a) shows the isotropic reflectivity changes (∆
R/R ) observed in Bi − x Sb x films with different compositions. The coherent phonon signal, superimposed on thephotoexcited carrier relaxation background, was observed as a function of the timedelay. It is interesting to note that the frequency and oscillation pattern of the coherentphonons differ depending on the composition x , under a constant pump fluence. We alsonote that the amplitude of the coherent phonon decreased when the Sb concentration x was in the range of x = 0.36 - 0.82. The observed large decrease in amplitude maypossibly be due to damping of the coherent lattice vibrations due to atomic disorder,i.e., alloy scattering. The relaxation time of the coherent phonons was also found toshorten when the Sb concentration x was in the range of x = 0.36 - 0.82. In particular,for x = 0.49, the coherent phonon oscillation was strongly damped for the time delaysof ≈ g modes of Bi-Bi (2.93 THzfor x = 0), Bi-Sb (3.63 THz for x = 0.36), Sb-Sb (4.54 THz for x = 1), respectively.The peak frequencies observed in the FT spectra obtained from the time domain dataare reasonably consistent with Raman scattering data [31, 30, 32].In order to estimate the relaxation time of the coherent phonon mode for various Sbconcentrations x , we carried out curve fitting of the time domain data shown in figure2(a) using a linear combination of damped harmonic oscillators and an exponentiallydecaying function. For the semimetal systems like Bi and Sb, the oscillatory componentis well described by a cosine function under the conditions of the displacive excitationof coherent phonons (DECPs) [37]. Thus, the fitting function used was [28, 45]:∆ RR = H ( t ) h X i =1 A i exp( − t/τ i ) cos( ω i t + φ i ) + A c exp( − t/τ c ) i , (1)where A i , τ i , ω i , φ i are the amplitude, relaxation time, frequency, and initial phaseof the coherent phonons, respectively. The subscripts i indicates the three A g modesof Bi-Bi ( i = 1), Bi-Sb ( i = 2), and Sb-Sb ( i = 3). H ( t ) is the Heaviside functionconvoluted with Gaussian to account for the finite time-resolution, while A c and τ c arethe amplitude and relaxation time of the carrier background, respectively. As can beseen in figure 2(c) the quality of the fit was good.Figure 3 shows the phonon scattering rate 1 /τ , the inverse of the relaxationtime, of the coherent phonon oscillations as a function of the Sb concentration ( x ) ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... !" ! ’ ( ) * + * ,- ./ * , ( ! :0 :0 :0 ; B3,-8*)03C././
Figure 3.
The scattering rate of the three different coherent A g phonons as afunction of the Sb concentration x . The thick lines indicate the theoretical scatteringrate calculated using the alloy scattering model described in the main text, whereΓ = π/ obtained by a fit using equation (1). Interestingly, the phonon scattering rate is largestfor Sb concentrations of about x = 0.7. The scattering rate 1/ τ of the coherentA g phonon in the Bi − x Sb x alloy is given by the sum of the intrinsic anharmonicphonon-phonon scattering 1 /τ Anharmonic , the alloy scattering 1 /τ Alloy , and the electron-phonon scattering 1 /τ e − p . The 1 /τ Anharmonic term is given by the Klemens formula γ [1 + n ( ω LA , TA ) + n ( ω A )] [46], where γ is the effective anharmonic constant, n ( ω )is the phonon distribution function, and ω LA , TA and ω A are the frequencies of thelongitudinal acoustic (LA) or transverse acoustic (TA) and the A g phonons, respectively[44]. Note that the anharmonic phonon–phonon scattering term is assumed to be aconstant because the anharmonic decay channel depends mainly on the distribution oflower lying TA and LA phonons and thus on the lattice temperature. For the generalcase with a constant crystal structure, changes in 1 /τ Anharmonic would be expected tobe negligibly small at a constant lattice temperature. We have estimated the latticetemperature rise for Bi ( x = 0) upon excitation using the two-temperature model (TTM)[47, 48], which predicts a temperature rise of ≈
50 K [23]. Although the lattice specificheat will slightly increase when the composition ( x ) becomes larger (e.g., by 20% as x varies up to 0.12 [49]), we know from TTM calculations that for larger lattice specificheat values, the lattice temperature rise will decrease. Therefore, we conclude thatthe lattice temperature is nearly constant or the temperature change is small for thedifferent compositions. In the latter sections, however, we will show 1 /τ Anharmonic varieswith the composition x . ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... !" ! ’ ( ) * + * ,- ./ * , ( %$$&$$"$$ B C F "!&"!G"!""!A"! H ( I7 ( / : J K L M %$$&$$"$$ B C Figure 4.
The pump fluence dependence of the relaxation rate (a) and frequency (b)obtained for the A g phonon mode (Sb-Sb) for various Sb concentration x . The solidlines represent a fit with a linear function described in the main text. To examine possible contributions from electron-phonon scattering 1 /τ e − p , thepump fluence dependence of the relaxation rate (the inverse of the dephasing time)and the frequency are presented as a function of the pump fluence F in figure 4. Atpresent, reliable data for the variation of the carrier density over the entire composition x in the Bi − x Sb x system is not available [27], but the photogenerated carrier density isof the same magnitude, as figure 3 was obtained under a constant fluence. As electron-phonon scattering would occur near the Fermi-level (E f ) for the semimetal or Weylsemimetal phase [50], we can examine the effect of the carrier density on the electron-phonon scattering near the E f by the analysis shown in figure 4. Here the frequencyand relaxation rate represent the real and imaginary parts of the phonon self-energy,respectively, depending on the photo-excited carrier density n c [51]. Assuming n c ∝ F ,where F is the pump fluence, we fit the frequency and relaxation rate to ω = ω + AF and 1 /τ = 1 /τ + BF , respectively, to obtain, for x =0.49, ω = 4.35 THz, A = – 0.024THz · cm /mJ and 1 /τ = 0.757 ps − , B = 0.097 ps − cm /mJ, respectively. Both therelaxation rate and frequency exhibited a linear and small change (10 %) as the pumpfluence was varied from 86 to 855 µ J / cm , as can be seen in the fitting parameters(A and B). The contribution from 1 /τ e − p is, however, considered to be insufficient toexplain the observed larger change (50 %) in the relaxation rate (figure 3). As discussedabove, alloy scattering is expected to play a significant role in Bi − x Sb x alloys [52], andtherefore we examined an alloy scattering model for the phonon scattering rate in figure3. The scattering rate due to the alloy scattering can be expressed using the phonon ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... ω [53, 54] as,1 τ Alloy ( ω ) = π V Γ Alloy ω D ( ω ) , (2)where V is the volume per atom, Γ Alloy is an alloy constant, and D ( ω ) is the vibrationaldensity of states per unit volume. Here, the total vibrational density of states is givenby the sum over phonon branch b , D ( ω ) = X b Z d~q (2 π ) δ [ ω − ω b ( ~q )] , (3)where ~q is the wavevector. The scattering-rate constant is given by three differentcontributions, Γ Alloy ( x ) = Γ Mass ( x ) + Γ Iso ( x ) + Γ Strain ( x ) , (4)where the first contribution is due to the mass difference, the second due the isotopeeffect, and third strain effects. Here we focus on the mass difference, which plays thedominant role in scattering in the alloy Bi − x Sb x [52]. The mass difference constant isgiven by, Γ Mass = X i f i (cid:18) − M i M (cid:19) , (5)where f i is the mass ratio of atoms with mass M i , and M is the average mass given by M = P i f i M i . The Γ Mass ( x ) term depends on the atomic concentration x , and thereforeis given by Γ Mass ( x ) = x (1 − x ) ( M Sb − M Bi ) [ xM Sb + (1 − x ) M Bi ] = x (1 − x ) (121 . u − . u ) [ x · . u + (1 − x ) · . u ] , (6)where the atomic mass of Bi is 208.98 u , and that of Sb is 121.76 u , with u being givenin unified atomic mass units. Thus, the 1 /τ Alloy ( ω ) term shown in equation (2) can bewritten as, 1 τ Alloy = π V ω D ( ω ) · x (1 − x ) × (121 . u − . u ) [ x · . u + (1 − x ) · . u ] . (7)In addition, here, we apply the Einstein model for the D ( ω ) term shown in equation (3)due to the nearly flat dispersion for the optical phonon mode [55], and thus we assumethat all the phonon modes for the optical phonon branch have the same frequency ω .According to the Einstein model, equation (3) can be reduced to, D ( ω ) = N δ ( ω − ω ) = const ., (8) ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... Figure 5.
The phonon dispersion in Bi − x Sb x at x = 0 .
5. Adapted from Ref. [17].The solid arrows represent possible anharmonic decay paths from the A g phononsof the Bi-Bi ( ∼ ∼ g phonon of the Sb-Sb ( ∼ where N is the number of unit cells. Thus, D ( ω ) is independent of the Sb concentration x . Based on the above considerations, 1 /τ Alloy can be finally written as,1 τ Alloy = Γ ω · x (1 − x ) × (121 . u − . u ) [ x · . u + (1 − x ) · . u ] , (9)where Γ = πV D ( ω ) / V and D ( ω ) constants were not available because of the uncertainly in the crystal structures ofthe Bi − x Sb x system, we held V D ( ω ) = 1 for simplicity. In addition, each model curvehas a background, assuming 1 /τ Anharmonic and 1 /τ e − p do not depend on the composition x as a first trial, i.e., 0.55 ps − for the Sb-Sb mode, 0.3 ps − for the Bi-Bi mode, and0.4 ps − for the Bi-Sb mode. The scattering rate thus obtained based on equation (9),as a function of the Sb concentration x and the frequency ω of each optical phononmode, is plotted in figure 3. As shown in figure 3, the theoretical scattering rate basedon equation (9) reproduces the experimental data well, excluding the Sb-Sb vibrationalmode around x = 0 . x = 0 . ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... Figure 6. (a) The relaxation rate of the carrier response as a function of thecomposition. (b) Schematic band structure near the Fermi level for semimetal Bi(Sb), i.e, x =0 or 1. The photoexcited electrons (e − ) relax via electron-phononintraband scattering, followed by interband recombination. (c) Schematic bandstructure near the Fermi-level for Weyl semimetal Bi . Sb . (Adapted from Ref. [15]).The photoexcited electrons (e − ) relax via electron-phonon intraband scattering, butinterband recombination is prohibited. when the phonon frequency decreases, as seen in Si − x Ge x alloy systems [56]. In thepresent study, around x = 0 . /τ Anharmonic term should exist for the two modes (Bi-Bi and Bi-Sb vibrations) around x = 0 .
5, whilethe 1 /τ Anharmonic term for the Sb-Sb vibrational mode should decrease.To test for the above possible scenario based on anharmonic phonon-phonon decaychannels in Bi − x Sb x around x = 0 .
5, we extracted the phonon dispersion from theliterature as schematically shown in figure 5 (Ref. [17]). Although there are somediscrepancies between the frequencies of the optical phonon modes at the Γ point infigure 5 and our observations [figure 2(b)], due to different lattice temperature anda possible mismatch in the lattice structure, there is a large energy gap between theacoustic and optical phonon branches for 1.8 < ω < g mode (4.0 THz) is forbidden (red dashed arrows) while theother two optical phonon modes of the Bi-Bi and Bi-Sb A g modes can decay into the twounderlying acoustic phonons under the condition that both the phonon energy and themomentum are conserved (red solid arrows). Thus, phonon dispersion considerationscan support the idea that the 1 /τ Anharmonic term significantly decreases for the A g modesof Sb-Sb around x = 0 . ltrafast scattering dynamics of coherent phonons in Bi − x Sb x in the Weyl ... /τ c ) of the carrier dynamics extracted by the fit in figure 2 as a function ofthe Sb content x . The carrier relaxation rate decreases for x = 0 .
49, indicating thatelectron-phonon scattering becomes weak near x = 0.5. This result suggests a closerelationship between the Weyl semimetal phase ( x = 0.5) and the fact that electron-phonon scattering near Weyl points, which are close to the Γ point, can become weak[50, 57] [see also figure 6(b) and (c)]. On the other hand, the contribution from electron-phonon scattering to the phonon dephasing rate is expected to be smaller than thatfrom anharmonic phonon scattering, as discussed in figure 4, and therefore, the phononrelaxation dynamics are not simply governed by alloy scattering, but are significantlymodified by anharmonic phonon-phonon scattering with implied minor contributionsfrom electron-phonon scattering in a Weyl-semimetal phase. Although exploring suchnew quasiparticle scattering dynamics will require more experimental and theoreticalstudy, the fact that a reduction in the scattering rate was observed near the Weylsemimetal phase ( x = 0.5) suggests a possible contribution from the quasiparticlescattering process described above.
4. CONCLUSIONS
In conclusion, we have investigated the dynamics of coherent optical phonons inBi − x Sb x for various Sb compositions x by using a femtosecond pump-probe technique.The coherent optical phonons corresponding to the A g modes of Bi-Bi, Bi-Sb, and Sb-Sb local vibrations were generated and observed in the time domain. The frequenciesof the coherent optical phonons were found to change with the Sb composition x , andmore importantly, the relaxation times of these phonons were strongly attenuated for x values in the range 0.5–0.8. We argue that the phonon relaxation dynamics arenot simply governed by alloy scattering, e.g., scattering due to mass differences, butare significantly modified by alloy-induced anharmonic phonon-phonon decay channelswith minor contributions from electron-phonon scattering in the Weyl semimetal phase,which is expected to appear at x = 0.5. Acknowledgments
This work was supported by CREST, JST (Grant Number. JPMJCR1875), and JSPSKAKENHI (Grant Numbers. 17H02908 and 19H02619), Japan. We acknowledge Dr.R. Mondal for helping data analysis and Ms. R. Kondou for sample preparation.
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