Pierre Argoul
email: argoul@lami.enpc.fr

Silvano Erlicher

On the use of continuous wavelet analysis for modal identification

This paper reviews two different uses of the continuous wavelet transform for modal identification purposes. The properties of the wavelet transform, mainly energetic, allow to emphasize or filter the main information within measured signals and thus facilitate the modal parameter identification especially when mechanical systems exhibit modal coupling and/or relatively strong damping.

Introduction

The concept of wavelets in its present theoretical form was first introduced in the 1970s by Jean Morlet and the team of the Marseille Theoretical Physics Center in France working under the supervision of Alex Grossmann. Grossmann and Morlet [START_REF] Grossman | Decompositions of functions into wavelets of constant shape and related transforms[END_REF] developed then the geometrical formalism of the continuous wavelet transform. Wavelets analysis has since become a popular tool for engineers. It is today used by electrical engineers engaged for processing and analyzing non-stationary signals and it has also found applications in audio and video compression. Less than ten years ago, some researchers proposed the use of wavelet analysis for modal identification purposes. However, use of wavelet transforms in real world mechanical engineering applications remains limited certainly due to widespread ignorance of their properties. This paper presents two ways of using the continuous wavelet transform (CWT) for modal parameter identification according to the type of measured responses : free decay or frequency response functions (FRF). In 1997, Staszewski [START_REF] Staszewski | Identification of damping in MDOF systems using timescale decomposition[END_REF] and Ruzzene [START_REF] Ruzzene | Natural frequencies and dampings identification using wavelet transform: application to real data[END_REF] started to use wavelet analysis with the Morlet wavelet for the processing of free decay responses, but the identification of the mode shapes was not performed. Wavelet transform allows to reach the time variation of instantaneous amplitude and phase of each component within the signal and makes the identification procedure of modal parameters much easier. Compared to other mother wavelets, Morlet wavelet provides better energy localizing and higher frequency resolution but has a disadvantage: frequency-coordinate window shifts along frequency axis with scaling while other wavelet types only expands the window. To remedy it, Lardies et al. [START_REF] Lardies | Modal parameter estimation based on the wavelet transform of output data[END_REF] and Slavic et al. [START_REF] Slavic | Damping identification using a continuous wavelet transform: application to real data[END_REF] preferred the Gabor wavelet. Argoul et al. [START_REF] Argoul | Instantaneous indicators of structural behaviour based on continuous Cauchy wavelet transform[END_REF][START_REF] Argoul | Wavelet analysis of transient signals in civil engineering[END_REF] chose the Cauchy wavelet and Le et al. [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF] established a complete modal identification procedure with improvements for numerical implementation, especially for a correct choice of the time-frequency localization. Edge effects are seen by all the authors and some attempts to reduce their negative influence on modal parameter identification are presented in [START_REF] Slavic | Damping identification using a continuous wavelet transform: application to real data[END_REF], [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF] and [START_REF] Boltezar | Enhancements to the continuous wavelet transform for damping identifications on short signals[END_REF]. More recently, Staszewski [START_REF] Staszewski | Identification of non-linear systems using multi-scale ridges and skeletons of the wavelet transform[END_REF], Bellizzi et al. [START_REF] Bellizzi | Identification of coupled non-linear modes from free vibration using time-frequency representations[END_REF] and Argoul et al. [START_REF] Argoul | Instantaneous indicators of structural behaviour based on continuous Cauchy wavelet transform[END_REF] adapted their identification procedure in order to process free responses of nonlinear systems. In [START_REF] Argoul | Instantaneous indicators of structural behaviour based on continuous Cauchy wavelet transform[END_REF], the authors proposed four instantaneous indicators, the discrepancy of which from linear case facilitates the detection and characterization of the non-linear behaviour of structures. In [START_REF] Bellizzi | Identification of coupled non-linear modes from free vibration using time-frequency representations[END_REF], an identification procedure of the coupled non-linear modes is proposed and tested on different types of non-linear elastic dynamic systems. Typically, CWT is applied to time or spatial signals. Processing frequency signals is unusual, but a first application was proposed in [START_REF] Argoul | Linear dynamical identification : an integral transform seen as a complex wavelet transform[END_REF] and [START_REF] Yin | Transformations intégrales et identification modale[END_REF]. Argoul showed in [START_REF] Argoul | Linear dynamical identification : an integral transform seen as a complex wavelet transform[END_REF] that the weighted integral transform (WIT) previously introduced by Jézéquel et al. [START_REF] Jézéquel | A new integral transform for linear systems identification[END_REF] for modal identification purposes can be expressed by means of a wavelet transform using a complex-valued mother wavelet close to the Cauchy wavelet. The WIT can be applied to either the ratio of the time derivative of the FRF over the FRF or the FRF itself. The representation with CWT provided a better understanding of the amplification effects of the WIT. When the FRF signal is strongly perturbed by noise, its derivative is hard to obtain; thus, Yin et al. [START_REF] Yin | Transformations intégrales et identification modale[END_REF] then proposed a slightly modified integral transform directly applied to the FRF and called the Singularities Analysis Function (SAF) of order n. It allows the influence of the FRF's poles to be emphasized and direct estimation of the eigen frequencies, eigen modes, and modal damping ratios is performed from the study of the extrema of the WIT.

Theoretical background for the continuous wavelet analysis

A wavelet expansion uses translations and dilations of an analyzing function called the mother wavelet ψ ∈ L 1 (R)∩L 2 (R). For a continuous wavelet transform (CWT), the translation and dilation parameters: b and a respectively, vary continuously. In other words, the CWT uses shifted and scaled copies of ψ(x) : ψ b,a (x) = 1 a ψ( x-b a ) whose L 1 (R) norms ( . 1 ) are independent of a. In the following, ψ(x) is assumed to be a smooth function, whose modulus of Fourier transform is peaked at a particular frequency Ω 0 called the "central" frequency. The variable x may represent either time or frequency; when necessary, the time and circular frequency variables will be referred to t and to ω respectively. The CWT of a function u(x) ∈ L 2 (R) can then be defined by the inner product between u(x) and ψ b,a (x)

T ψ [u](b, a) = u, ψ b,a = 1 a +∞ -∞ u(x) ψ x -b a dx (1.1)
where ψ(.) is the complex conjugate of ψ(.). Using Parseval's identity, Eq.

(1.1) becomes

T ψ [u](b, a) = 1 2π u, ψ b,a = 1 2π +∞ -∞ u(Ω) ψ(aΩ) e iΩb dΩ (1.2)
where u(Ω), ψ b,a (Ω) and ψ(Ω) are respectively the Fourier transform (FT) of u(x), ψ b,a (x) and ψ(x); for instance for u(x) : u(Ω) = +∞ -∞ u(x) e -iΩx dx. Moreover, when ψ and u are continuous and piece-wise differentiable, and ψ is square and absolutely integrable, and u is of finite energy, the CWT of u with ψ is linked to the CWT of u with ψ :

T ψ [ u](b, a) = - 1 a T ψ[u](b, a)
From [START_REF] Chui | An introduction to Wavelets[END_REF], it can be seen that the CWT at point (b, Ω =

D(b, Ω = Ω ψ a ) = [b+ax ψ -a∆x ψ , b+ax ψ +a∆x ψ ]×[ Ω ψ a - ∆Ω ψ a , Ω ψ a + ∆Ω ψ a ] (1.3)
where x ψ and ∆x ψ are called centre and radius of ψ, stated in terms of root mean squares :

x ψ = 1 ψ 2 2 +∞ -∞ x |ψ(x)| 2 dx and ∆x ψ = 1 ψ 2 +∞ -∞ (x -x ψ ) 2 |ψ(x)| 2 dx
, and similar definitions hold on for the frequency centre Ω ψ and the radius ∆Ω ψ of ψ. The area of D is constant and equal to four times the uncertainty : 4∆x ψ ∆Ω ψ = 4µ(ψ). The Heisenberg uncertainty principle states that this area has to be greater than 2.

Referring to the conventional frequency analysis of constant-Q filters, the parameter Q defined as the ratio of the frequency centre Ω ψ to the frequency bandwidth (2∆Ω ψ ):

Q = Ω ψ a 2 ∆Ω ψ a = Ω ψ 2∆Ω ψ , (1.4)
is introduced in [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF] to compare different mother wavelets and to characterize the quality of the CWT. Q is independent of a.

The notion of spectral density can be easily extended to the CWT (see [START_REF] Carmona | Practical Time-Frequency Analysis[END_REF] for others properties such as linearity, admissibility and signal reconstruction, etc.). From the Parseval theorem applied to Eq. 1.2, it follows that

+∞ -∞ |T ψ [u](b, a)| 2 db = 1 2π +∞ -∞ | u(k)| 2 ψ(ak) 2 dk
and it leads to the "energy conservation" property of the CWT expressing

that if u ∈ L 2 (R), then T ψ ∈ L 2 (R × R * + , db da a ), and T ψ [u] 2 = C ψ u 2 = C ψ 2 π u 2 .
Finally, after changing the integration variable: a = Ω0 Ω and because u(Ω) is hermitian (u(x) being real valued), one gets

+∞ 0 +∞ -∞ T ψ [u](b, Ω o Ω ) 2 db dΩ Ω = C ψ π +∞ 0 | u(Ω)| 2 dΩ (1.5)
where

C ψ = +∞ -∞ | b ψ(ω)| 2 ω dω < ∞.
This allows to define a local wavelet spectrum:

E u,CW T (Ω, b) = 1 2C ψ Ω T ψ [u](b, Ωo Ω )
2 and a mean wavelet spectrum:

E u,CW T (Ω) = +∞ -∞ E u,CW T (Ω, b
)db; and Eq. 1.5 can be rewritten:

+∞ 0 E u,CW T (Ω) -1 2π | u(Ω)| 2 dΩ = 0.
When the mother wavelet ψ is progressive (i.e. it belongs to the complex Hardy space ψ(Ω) = 0 : for : Ω ≤ 0), the CWT of a real-valued signal u is related to the CWT of its analytical signal Z u (see. [START_REF] Carmona | Practical Time-Frequency Analysis[END_REF])

T ψ [u](b, a) = 1 2 T ψ [Z u ](b, a) (1.6)
From 1.3, the CWT has sharp frequency localization at low frequencies, and sharp time localization at high frequencies. Thanks to a set of mother wavelets depending on one(two) parameter(s), the desired time or frequency localization can be yet obtained by modifying its(their) value(s). In [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF], two complex valued mother wavelets were analyzed : Gabor and Cauchy. Morlet wavelet is a complex sine wave localized with a Gaussian envelope and the Gabor wavelet function is a modified Morlet wavelet with a parameter controlling its shape. Cauchy wavelet ψ n of n order for n ≥ 1, is defined by:

ψn(x) = " i x + i « n+1 = " 1 1 -ix « n+1 = » 1 √ x 2 + 1 -n+1 e i (n+1)Arctg(x) . (1.7)
The main characteristics of ψ n (x) follow: its FT:

ψ n (Ω) = 2πΩ n e -Ω n!
Θ(Ω) where Θ(Ω) is the Heaviside function (ψ n is progressive), its central frequency: Ω 0n = n, its L2 norm: ψ n 2 2 = (2n)! 2 2n (n!) 2 π, its x-and frequency centres: x ψ = 0 and Ω ψ = n + 1 2 , its radius:

∆x ψ = 1 2n-1 , its frequency radius: ∆Ω ψ = √ 2n+1 2
, its uncertainty:

µ ψ = 1 2 1 + 2 2n-1 , its quality factor: Q = n+ 1 2 √
2n+1 and its admissibility factor:

C ψ = 4π 2 1 2 2n (2n-1)! (n!) 2 = 2π n ψ n 2 2
. The use of the Cauchy wavelet is legitimate when Q is less than 5/ √ 2; as Q increases, both wavelets give close results, a little better with Morlet wavelet due to its excellent time-frequency localization, and behave similarly in the both time and frequency domains when Q tends toward infinity. Some numerical and practical aspects for the computation of the CWT are detailed in [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF] where

Ω = Ω ψ a )
, is then proposed to take into account the decreasing properties of ψ and ψ by introducing two real positive coefficients c x and c Ω : . Let us consider a frequency Ω j for which u(Ω) exhibits a peak and introduce a frequency discrepancy dΩ j (for example, the distance between two successive peaks of u(Ω)). From the intersection of the two hyperbolae at the point (b = L 2 , Ω = Ω j ) and by imposing the frequency localization along the straight line: a = Ω0 Ωj , to be included into [Ω j -dΩ j , Ω j + dΩ j ], some upper and lower bounds are found for

D ext (b, Ω = Ω ψ a ) = [b + ax ψ -a c x ∆x ψ , b + ax ψ + ac x ∆x ψ ] × [ Ω ψ a -c Ω ∆Ω ψ a , Ω ψ a + c Ω ∆Ω ψ a ].
Q: c Ω Ωj 2dΩj ≤ Q ≤ L Ωj
2cx and finally, the authors proposed c x = c Ω ≃ 5.

Modal analysis and modal identification with CWT

Experimental identification of structural dynamics models is usually based on the modal analysis approach. One basic assumption underlying modal analysis is that the behaviour of the structure is linear and time invariant during the test. Modal analysis and identification involve the theory of linear time-invariant conservative and non-conservative dynamical systems. In this theory, the normal modes are of fundamental importance because they allow to uncouple the governing equations of motion. Also, they can be used to evaluate the free or forced dynamic responses for arbitrary sets of initial conditions. Modal analysis of a structure is performed by making use of the principle of linear superposition that expresses the system response as a sum of modal responses. For linear MDOF systems with N degrees of freedom, the transfer function H ij (p) of receptance type is defined as the Laplace transform of the displacement at point j to the impulse unit applied at point i and it can be expressed as a sum of simple rational fractions

H ij (p) = N r=1 (A r ) ij p -p r + A r ij p -p r (1.8)
where A r ij and p r are respectively the conjugate of the residues (A r ) ij and of the poles p r of H ij (p). When the system is stable, p r = -ξ r ω r + i ω r where ω r , ω r are the undamped and the damped vibration angular frequencies and ξ r is the damping ratio for the mode r. (A r ) ij can also be expressed from the complex modes χ r by (A r ) ij = (χr ) i (χr ) j γr

where γ r = χ T r Cχ r +2p r χ T r M χ r , M and C being respectively the mass and viscous damping matrices. Moreover, the FRF of receptance type H ij (ω) can be usually related to H ij (p) by:

H ij (ω) = H ij (p = iω), leading to H ij (ω) = N r=1 -i (A r ) ij (ω -ω r ) -iξ r ω r + -i A r ij (ω + ω r ) -iξ r ω r (1.9)
The free responses at point j in terms of displacements u (f ree) j (t) can be expressed according to the modal basis of complex modes: χ jr = η jr + i κ jr (r being the mode number, 1 ≤ r ≤ N )

u (f ree) j (t) = N r=1 A rj (t) cos (Φ rj (t)) (1.10)
where A rj (t) = |χ jr | ρ r e -ξrωr t and Φ rj (t) = ω r tϕ r + arctan κjr ηjr . ρ r and ϕ r are defined from initial displacement and velocity of mode r (cf. [START_REF] Geradin | Théorie des vibrations Application à la dynamique des structures[END_REF]). In the case of proportional viscous damping (Basile assumption), introducing the real eigenvectors Ψ r of mode r for the associated conservative system, and replacing the residues (A r ) ij with (Ψr) i (Ψr) j 2i e ωr mr in Eq. 1.9 lead to

H ij (ω) = N r=1 (Ψ r ) i (Ψ r ) j m r (ω 2 r -ω 2 + 2i ξ r ω r ω) (1.11)
Moreover, in Eq. 1.10, κ jr = 0; thus |χ jr | can be replaced by (Ψ r ) j and the term arctan κjr ηjr can be equal to 0 or π according to the sign of the real part η jr .

Modal identification using free decay responses

The processed signals u (f ree) j (t), whose expression is given in Eq. 1.10, can be considered under the assumption of weak damping (ξ r ≪ 1 √ 2 ), as a sum of N modal components: A rj (t) cos(Φ rj (t)) consisting in asymptotic signals. The approximation of asymptotic signal means that the oscillations resulting from the phase term: Φ rj (t) are much faster than the variation coming from the amplitude term: A rj (t); it entails that the analytical signal associated with A rj (t) e iΦrj (t) can be approximated by: A rj (t) e iΦrj (t) . Therefore, Eq. (1.6) and the linearity of the CWT entail that:

T ψ [u (f ree) j ](b, a) = 1 2 N r=1 T ψ [A rj (t) e iΦrj (t)
](b, a). Taking the Taylor expansion of each amplitude term of this sum, it follows that

T ψ [u (f ree) j ](b, a) = 1 2 N r=1 A rj (b) e iΦrj (b) ψ(a ω r ) + R rj (b, a) .
(1.12) When each modal pseudo-pulsation ω r is far enough from the others, this straight line's equation easily provides an approximation of ω r . Eq. 1.12 implies that the CWT of asymptotic signals has a tendency to concentrate near a series of curves in the time-frequency plane called ridges, which are directly linked to the amplitude and phase of each component of the measured signal. In [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF], a complete modal identification procedure for natural frequencies, viscous damping ratios and mode shapes, is given in the case of viscous proportional damping and applied to a numerical case consisting in the free decay responses of a mass-spring-damper system with four degrees of freedom (4-DoF). The procedure is applied here to the free responses of a 4-DoF system whose natural frequencies are: f 1 = 0.0984 Hz, f 2 = 0.1871 Hz, f 3 = 0.2575 Hz and f 4 = 0.3027 Hz and modal damping ratios: ξ 1 = 0.0124, ξ 2 = 0.0235, ξ 3 = 0.0324, ξ 4 = 0.0380. Fig. 1.1 shows the FT and the CWT of the displacement of the second mass for which the four eigenfrequencies are clearly visible. Fig. 1.2 presents for the third mode and for the four masses, the modulus of the CWT and its logarithm that will allow to estimate the value of the damping ratio ξ 3 (cf. [START_REF] Le | Continuous wavelet transform for modal identification using free decay response[END_REF]). The edge effect is delimited by two hyperbolae in the time frequency plane; Q being chosen to 20 (15.3 ≤ Q ≤ 43.1). Identified values are very close to the exact ones and identification errors are negligible inside the domain D ext . This procedure was also applied in [START_REF] Argoul | Wavelet analysis of transient signals in civil engineering[END_REF] to a set of accelerometric responses of modern buildings submitted to non destructive shocks. From the CWT of measured responses, the ridge and the corresponding amplitude and phase terms for the two first modes were extracted. For the first instantaneous frequency ω 1 , the processing revealed a slight increase of ω 1 just after the shock. The origin of this non linear effect was attributed to the non-linear behaviour of the soil-structure interaction. Other preliminary results can be found in [START_REF] Argoul | Instantaneous indicators of structural behaviour based on continuous Cauchy wavelet transform[END_REF] where instantaneous indicators based on CWT are computed from the accelerometric responses of a non-linear beam to an impact force. A Duffing non-linearity effect was then identified thanks to the first and the super-harmonic components.

Modal identification using FRF functions

The first definition of the SAF of order n (n ∈ N) uses the derivative of a transfer function H(p) of the linear mechanical system, where p = a + ib :

H n (b, a) = (-1) n 2 π a n+1 2 d n dp n {H(p)} (1.13)
H n (b, a) can also be expressed as the FT of the impulse response function h(t) filtered by: 2πa :

H n (b, a) = n! a n+1 2 T ψ n [H](b, a) = n! a n+1 2 R H(ω)ψ n ω -b a dω (1.15)
Let us now consider a transfer function restricted to only one term F λ,ω0 (ω) of the sum given in (1.9) and perturbed by a Gaussian white noise n(ω) so that :

H(ω) = F λ,ω0 (ω) + n(ω) = -B i (ω-ω0)-iλ + n(ω).
Let us introduce the coefficient ρ 2 defined as the ratio of the square absolute value of the CWT of the pure signal upon the common variance of the CWT of the noise itself:

ρ 2 = |T ψ [H](b,a)| 2 E{|T ψ [n](b,a)| 2 } = | F λ,ω 0 , ψ b,a | 2 E{| n , ψ b,a | 2 }
, it looks like a signal to noise ratio.

The use of the Cauchy-Schwarz inequality gives that : ρ 2 ≤ F λ,ω0 2 and the maximum of ρ 2 is reached for (b, a) = (ω 0 , λ) when ψ b,a (ω) = K F a,b (ω) = K -B i (ω-b)-ia where K is a constant. In conclusion, ψ(ω) = K B -i ω-i = K B ψ 0 (ω); the analyzing functions ψ(ω) which allow the minimization of the noise effect on the peaks of |T ψ [H](b, a)| are proportional to ψ 0 (s) defined in Eq. 1.7. The above definitions of the SAF of order n (n ∈ N * ) allow to better understand its effect when applied to discrete causal linear mechanical systems. In Eq. 1.13, the successive derivatives of a rational fraction make the degree of the denominator increase and cause an amplification of the pole's effect. So, when Eq. 1.8 is introduced in Eq. 1.13, the SAF of order n becomes The absolute value of H n (b, a) exhibits 2N maxima located symmetrically from the vertical axis in the phase plane. The effect of parameter n is to facilitate the modal identification, especially when the structure has neighboring poles. As soon as n is large enough, the coordinates (b max , a max ) of these 2N maxima allow the estimation of the real and imaginary parts, respectively ℜ {p r } and ℑ {p r }, of each of the 2N conjugate poles of the system :

ℜ {p r } = -ξ r ω r ≈ -a max r and ℑ {p r } = ω r ≈ b max r (1.17)

The residue A r can be then estimated from the complex valued amplitude of the extremum called H maxr As n increases, the estimation of the extrema is better for signals without noise; but for noisy signals, noise effects are also increased. Recently, Yin et al. [START_REF] Yin | Natural frequencies and damping estimation using wavelet transform of a frequency response function[END_REF] proposed the use of analyzing wavelets which are the conjugate of Cauchy wavelets in which n is replaced by a positive real number. Applications both to numerical and experimental data showed the efficiency of this technique using SAF [START_REF] Argoul | Integral transforms and modal identification in[END_REF]. It was applied to the FRFs obtained with H1 estimators, of a test structure designed and build by ONERA for the GARTEUR-SM-AG19 Group. This structure was made of two aluminum sub-structures simulating wings/drum and fuselage/tail. Finally, the results obtained with SAF were very similar to those obtained with the broadband MIMO modal identification techniques provided by the MATLAB toolbox IDRC.

In conclusion, the pre-processing of measured signals (free decay responses or FRFs) with CWT proved to be an efficient tool for modal identification.

a

  ) picks up information about u(x), mostly from the localization domain D(b, Ω = Ω ψ a ) of the CWT, defined as

  , especially when applied to signals with time bounded support. Let us note D u = [0, L]×[0, 2πf N yq ] the validity domain of the discretized signal in the (b, Ω) plane where f N yq is the Nyquist frequency (f N yq = 1 2T with sampling period T ). The edge-effect problem due to the finite length (L) and to the discretization of measured data record and to the nature of the CWT (convolution product) is tackled. An extended domain D ext (b, Ω) ⊃ D(b, Ω)

  Forcing D ext to be included into D u leads the authors to define a region in the (b, Ω) plane where the edge effect can be neglected; this region is delimited by two hyperbolae whose equations are:Ω = 2 b c x Qµ ψ and Ω = 2L-b c x Qµ ψ and two horizontal lines whose equations are: Ω = 0 and Ω =

Fig. 1 . 1 .

 11 Fig. 1.1. Modulus of FT and of the CWT for the 4DoF system (Q = 18)

Fig. 1 . 2 .

 12 Fig. 1.2. Modulus of FT and of CWT around the 3rd mode for all channels. Logarithm of the modulus of the CWT (L = 250s, T = 0.175s, ct = cΩ = 4.7, Q = 20).

H

  n (b, a) = 2π n! a

n

  = H max n (a max r , b max r ) A r ≈ 2 n (a max r )


Hong Wu

Xiangnan Gong

Yi Peng

Long Zhang

Bin Zhang

Kunling Peng

Jie Liu

Guang Han

Aifeng Wang

Yisheng Chai

Mingquan He

Haoshuang Gu

Emmanuel Guilmeau

Guoyu Wang
email: guoyuw@cigit.ac.cn

Xu Lu
email: luxu@cqu.edu.cn

Xiaoyuan Zhou
email: xiaoyuan2013@cqu.edu.cn

Identification of vibrational mode symmetry and phonon anharmonicity in SbCrSe3 single crystal using Raman spectroscopy

Keywords: SbCrSe3 single crystal, in-situ Raman, angle-resolved polarized Raman spectroscopy, lattice dynamics, thermal transport ^

published or not. The documents may come

Identification of vibrational mode symmetry and phonon

anharmonicity in SbCrSe3 single crystal using Raman spectroscopy

INTRODUCTION

Low-dimensional bulk materials, with quasi one-dimensional (1D) or 2D structure, have been received growing attention owing to the extraordinary electronic and thermal properties, and emerged as highly attractive candidates used for next-generation electronic, optoelectronic and thermoelectric devices [START_REF] Chhowalla | Two-dimensional semiconductors for transistors[END_REF][START_REF] Dresselhaus | New directions for low-dimensional thermoelectric materials[END_REF][START_REF] Geim | Van der waals heterostructures[END_REF][START_REF] Shivananju | The roadmap of graphene-based optical biochemical sensors[END_REF][START_REF] Yuan | Polarization-sensitive broadband photodetector using a black phosphorus vertical p-n junction[END_REF][START_REF] Zhao | Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals[END_REF][START_REF] Zhou | Routes for high-performance thermoelectric materials[END_REF][START_REF] Peng | Ultra-high average figure of merit in synergistic band engineered SnxNa1-xSe0.9S0.1 single crystals[END_REF]. In particular, the low dimensional intrinsic magnetic semiconductors, CrMX3 ternary chromium trichalcogenides (M as a non-transition metal and X = S, Se or Te), have garnered wide interests in magnetism and thermoelectricity fields due to appropriate band gap and intrinsically low " $ [START_REF] Gong | Discovery of intrinsic ferromagnetism in twodimensional van der waals crystals[END_REF][START_REF] Kong | Anisotropic magnetic properties of the ferromagnetic semiconductor CrSbSe3[END_REF][START_REF] Mathew | Density functional study of magnetic, structural and electronic properties of quasi-one-dimensional compounds CrSbX3 (x = S, Se)[END_REF][START_REF] Odink | Synthesis and structure of chromium antimony triselenide (CrSbSe3): A pseudo-one-dimensional ferromagnet[END_REF][START_REF] Tang | Dopant induced impurity bands and carrier concentration control for thermoelectric enhancement in p-type Cr2Ge2Te6[END_REF][START_REF] Yang | Cr2Ge2Te6: High thermoelectric performance from layered structure with high symmetry[END_REF][START_REF] Yang | Intrinsically low thermal conductivity from a quasione-dimensional crystal structure and enhanced electrical conductivity network via pb doping in CrSbSe3[END_REF]. Intriguingly, the crystal structure of these compounds, by virtue of the different arrangement of CrX6 octahedra, can transform from quasi-2D layered structure to 1D needlelike structure. For example, CrMX3 holds a laminar structure when M = Si, Ge, or Sn, with CrX6 octahedra adopting a honeycomb lattice; in contrast, when Sb or Ga appears on M sites, it exhibits a pseudo-1D crystal structure, with CrX6 octahedra forming infinite, edge-sharing and double rutile chains, in which M atoms are linked to the neighboring chains. The relative angle between the double rutile chains is determined by the type of M atoms [START_REF] Volkov | Hrem image analysis up to structure determination of SbCrSe3: A new 1D ferromagnet[END_REF]. Although the crystal structures of these compounds have been fully comprehended, the advancement of investigation on their fundamental lattice vibration properties is still relatively slow.

Raman spectrum, as a fast and nondestructive tool, has been widely employed to characterize the phonon characteristics in the vicinity of Brillouin zone center, the crystal structure, and electron-phonon interactions, especially in many low dimensional materials [START_REF] Balandin | Phononics in low-dimensional materials[END_REF][START_REF] Ghosh | Dimensional crossover of thermal transport in few-layer graphene[END_REF][START_REF] Zhang | Review on the Raman spectroscopy of different types of layered materials[END_REF][START_REF] Wu | Identifying the crystalline orientation of black phosphorus using angle-resolved polarized Raman spectroscopy[END_REF]. As a characteristic example of CrMX3 compounds, SbCrSe3 has an orthorhombic structure with 20 atoms in its primitive cell and is subject to the Pnma space group and D2h point group. Based on the group analysis, 60 vibrational modes at the Γ point in Brillouin zone are expectantly generated, in which eight inequivalent irreducible are included in the D2h point group as following: # = 10! ! + 5) %! + 10) "! + 5) &! + 5! ' + 10) %' + 5) "' + 10) &' . One can see that the symmetries of group theory. Recently, the ARPRS has emerged as a powerful tool to resolve and assign Raman modes symmetries for some important crystals, and to determine the crystalline orientation of films and nanotubes [START_REF] Li | GeP: An unexploited low-symmetry semiconductor with strong in-plane anisotropy[END_REF][START_REF] Li | Probing the domain architecture in 2D α-Mo2C via polarized Raman spectroscopy[END_REF][START_REF] Ribeiro | Unusual angular dependence of the Raman response in black phosphorus[END_REF][START_REF] Zhang | Raman spectroscopy of shear and layer breathing modes in multilayer MoS2[END_REF]. However, to the best of our knowledge, such a technique has never been employed to identify the vibrational symmetries for the experimentally observed Raman modes in the low dimensional CrMX3 semiconductors.

Moreover, SbCrSe3 exhibits intrinsically low " $ [START_REF] Yang | Intrinsically low thermal conductivity from a quasione-dimensional crystal structure and enhanced electrical conductivity network via pb doping in CrSbSe3[END_REF], indicating that it could be a promising thermoelectric material after electrical property optimization [START_REF] Bell | Cooling, heating, generating power, and recovering waste heat with thermoelectric systems[END_REF][START_REF] Wu | Strong lattice anharmonicity securing intrinsically low lattice thermal conductivity and high performance thermoelectric SnSb2Te4 via Se alloying[END_REF][START_REF] Shi | Advanced thermoelectric design: From materials and structures to devices[END_REF][START_REF] Wu | Realizing high thermoelectricity in polycrystalline SnS via manipulating fermi surface anisotropy and phonon dispersion[END_REF][START_REF] Chen | Rationalizing phonon dispersion for lattice thermal conductivity of solids[END_REF][START_REF] Xu | Anisotropic thermoelectric properties of layered compound SnSe2[END_REF].

Nevertheless, the underlying physics for the low thermal conductivity of SbCrSe3, which relies on the analysis of phonon modes, is still obscure until now. By virtue of in-situ Raman spectroscopy analysis, the temperature-dependent lattice vibration characteristics can be precisely attained, which further illustrates the anharmonic interactions between phonons by analyzing the temperature-dependent evolution of Raman peaks and corresponding full width at half maximum (FWHM) [START_REF] Lan | Phonon anharmonicity of rutile SnO2 studied by Raman spectrometry and first principles calculations of the kinematics of phonon-phonon interactions[END_REF][START_REF] Peercy | Pressure and temperature dependences of the Raman-active phonons in SnO2[END_REF][START_REF] Zhang | Extraordinary photoluminescence and strong temperature/angle-dependent Raman responses in few-layer phosphorene[END_REF]. Hence, it is essential to unveil the phonon scattering mechanisms in bulk low dimensional materials like SbCrSe3 using the combination of in-situ Raman experiment and lattice dynamics.

In this study, we report the determination of mode symmetry for each observed Raman peak in SbCrSe3 single crystal. It is found that three-phonon scattering processes dominate the thermal transport for SbCrSe3 throughout the measured temperature range, as suggested by the temperature dependence of Raman mode frequency and linewidth, which is further corroborated by the good consistency between experimental and calculated thermal conductivity. Our findings highlight the utilization of Raman spectrum as a fast, precise, and nondestructive method to identify the vibrational modes symmetry, crystalline orientation and the phonon anharmonicity for low dimensional materials.

EXPERIMENTAL AND THEORETICAL METHODS

Synthesis

SbCrSe3 single crystals were grown by self-flux method. Starting elements (chunks, 99.99%, Sigma-Aldrich) were weighted according to a molar ratio of Cr:Sb:Se = 2:38:60 and loaded into an alumina crucible. The alumina crucible was placed into a quartz tube, and another alumina crucible with silica wool was placed upside down on the prior one. Then, the quartz tube was evacuated to a pressure of 5 ´ 10 -4 Pa and sealed. Quartz tubes were heated up to 1000 K over 12 h, slowly cooled to 900 K and remained at this temperature for 144 h; after that, the molten liquid was separated from the crystals in a centrifuge, with silica wool serving as a filter. Finally, the obtained needle crystals were evacuated, encapsulated in fused quartz tube, and annealed at 780 K for 24 h. Consequently, millimeter-long malleable single crystals of SbCrSe3 were obtained.

Characterizations

Diffraction intensity data of single crystal SbCrSe3 were collected on an Agilent Technologies SuperNova single-crystal X-ray diffraction (SCXRD) at 295 K equipped with graphite-monochromatic Mo Kα radiation (λ = 0.71073 Å). The structures were solved with the SHELXT structure solution program and refined with SHELXL refinement package. XRD pattern was recorded via a PANalytical X-ray diffractometer that was operated at 40 kV and 40 mA using Cu Kα (λ = 1.5418 Å) radiation. Raman spectra measurements were carried out by HORIBA JobinYvon HR Evolution, and 600 and 1800 groove mm -1 holographic grating combining with thermoelectric cooled Synapse charge coupled device were used for the ARPRS and temperature dependent Raman experiments, respectively. Solid-state Nd:YAG laser with the wavelength of 532 nm was used as the excitation source. The thermal conductivity was measured in Physical Property Measurement System (PPMS) DynaCool 9. The high-resolution transmission electron microscopy (HRTEM) and scanning transmission electron microscopy high-angle annular dark field (STEM-HAADF) were employed to characterize microstructure. The corresponding energy-dispersive X-ray spectroscopy (EDX) mapping was acquired on a transmission electron microscope (Talos F200S G2) operating at 200 kV. Electron probe micro-analyzer (EPMA, JXA-8530F Plus, JEOL, Japan) was used to determine the distribution of Sb, Cr and Se in the single crystal with a spatial resolution of 1 µm.

DFT calculations

Under the precondition of density functional theory (DFT) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF], the Vienna abinitio Simulation Package (VASP) code using the projector augmented plane wave (PAW) methods is implemented for all calculations [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmentedwave method[END_REF]. A generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) is defined as the relevant exchange-correlation functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. For the relaxation of the crystal structure, the total energies of electronic self-consistent interactions were numerically converged to 1 ´ 10 -8 eV with a basis set energy cutoff of 600 eV for and the forces on the atoms were 1 ´ 10 -4 eV Å -1 for optimizing the atomic positions. For the Brillouin zone integrations, the k-mesh grid (5 × 11 × 3) centered at G was used for the primitive cell. To calculate phonon dispersion, the finite displacement methods using 2 × 3 × 1 supercell were performed in Phonopy package to investigate the lattice dynamics of SbCrSe3 [START_REF] Togo | First principles phonon calculations in materials science[END_REF]. Furthermore, the " $ was calculated by ShengBTE package using a cutoff of 10 angstrom for the interaction range and 1 × 3 × 1 supercell to obtain third order anharmonic inter atomic force constants (IFCs) [START_REF] Li | Shengbte: A solver of the boltzmann transport equation for phonons[END_REF].

RESULTS AND DISCUSSION

Phase characterization

The structure of our SbCrSe3 single crystal, with the lattice parameters a = 9.1475(6) Å, b = 3.7872(3) Å and c = 13.4264 [START_REF] Mathew | Density functional study of magnetic, structural and electronic properties of quasi-one-dimensional compounds CrSbX3 (x = S, Se)[END_REF] Å obtained by the refinement of SCXRD, is orthorhombic with Pnma space group (No. 62) and D2h point group.

Structural data are displayed in Table S1. Fig. 1a shows the refined crystal structure of SbCrSe3, which is regarded as a quasi-1D structure with double chains of CrSe6 octahedra aligned parallel to the b-axis shown in Fig. 1b. Within the double chain, the Cr cations follow an edge-sharing triangular arrangement and Sb atoms are linked to the adjacent chains, resulting in the pseudo-1D crystal structure. As revealed in Fig. 1c, the bond distances between Cr and Se atoms in these octahedra range from 2.5301 to 2.5949 Å, leading to the slightly distorted octahedron due to the asymmetric chemical bond lengths, and the Sb-Se1 and Sb-Se2 bond distances are 2.6486 and 2.6714 Å, respectively. The synthesized SbCrSe3 single crystal in Fig. 1d Furthermore, to gain in-depth insight into the structural characterization and the relative orientation of the cleaved (00l) crystal plane of SbCrSe3 crystal, the STEM-HAADF imaging, electron diffraction and EDX mapping were utilized. As displayed in Fig. 1g andh, the clear lattice fringe, in conjunction with the distinguishable fast Fourier transformation (FFT) image, implies the single crystalline feature. Interestingly, the Miller index of the diffraction spot and the measured interplanar distance, marked in Fig. 1g andh, respectively, elucidate the orientation of the long and short edges (such as 49, 144, 172 and 238 cm -1 ) and some at 45°, 135°, 225°, and 315° (e.g., 64, 125, 153, and185 cm -1 ). In contrast, in parallel-polarized configuration (Fig. S3b and Fig. 2c), some Raman modes intensities repeat after 180°, leading to 2-lobed shape (e.g., 64, 125, 185 and 254 cm -1 ), which could be employed to estimate the crystalline orientation (along a or b-direction) of the cleaved (00l) in-plane of SbCrSe3 (as discussed later). Some modes have a 90° variation period, exhibiting 4-lobed shape (e.g., 49, 144, 172 and 238 cm -1 ). These results clearly uncover not only the anisotropic optical feature of SbCrSe3 crystal but also the dependence of polarized Raman spectrum on crystalline orientation. =>?@ ?AB@ 0 -?AB@ =>?@ 0 0 0 1 ;

The intensity (I) of the observed Raman vibration modes can be expressed as:

I ∝ FG ⃗ * • 6 33⃗ • 2 3⃗ • 6 33⃗ )% • G ⃗ + F " (2)
where G ⃗ + and G ⃗ * are the unit polarization vectors of scatter and incident laser, respectively. The incident laser with unit polarization vector along ,-axis (G ⃗ * = (0 1 0))

propagates along the /-direction to impinge on the sample. The scattered Raman signal is collected through the selectivity analyzer with unit polarization vector G ⃗ + , =

(1 0 0) defined as perpendicular configuration and G ⃗ + , = (0 1 0) as parallel configuration. For an absorptive material, the Raman tensor elements are complex numbers, with real and imaginary parts [START_REF] Ribeiro | Unusual angular dependence of the Raman response in black phosphorus[END_REF][START_REF] Loudon | The Raman effect in crystals[END_REF]. Thus, the Raman tensor of ! ! and ) %! (other modes ) "! and ) &! seeing SI Section I) according to the D2h point group can be expressed as [START_REF] Loudon | The Raman effect in crystals[END_REF]:

2 3⃗ H! ! I = : JG *-! 0 0 0 KG *-" 0 0 0 =G *-# ;, 2 3⃗ H) %! I = : 0 LG *-$ 0 LG *-$ 0 0 0 0 0 ; (3)
The Raman scattering intensities of different modes can further be expressed as:

I(! ! , ⊥) ∝ % . • ?AB " 2@ • (|J| " -2|J||K| • =>?O /0 + |K| " ) (4)
IH! ! , ‖I ∝ |J| " • ?AB . @ + |K| " • =>? . @ + 2|J||K| • =>?O /0 • ?AB " @ • =>? " @ (5)

I() %! , ⊥) ∝ L " • =>? " 2@ (6) IH) %! , ‖I ∝ L " • ?AB " 2@ (7)
where and represent parallel and perpendicular polarizations, respectively, and

O /0 = O / -O 0 is the phase difference.
Consequently, the relations between I and @ based on group theory analysis for all Raman active modes of SbCrSe3 are summarized in Table S2. Notably, the calculated Raman scattering intensities of the ) 1! (* = 1-3) modes, independent with the phase difference O, are only determined by the real parts.

Moreover, when the direction of incident laser is perpendicular to the (00l) surface, only

Raman-active ! ! and ) %! modes can be detected.

In parallel polarizer configuration, it is interesting to note that the angular dependence of the ! ! intensity with a periodicity of 180° is different from that of ) %! modes holding 90° periodicity out of the 2@ dependence. In perpendicularpolarized geometry, though both ! ! and ) %! modes have similar 90° variation period, the ! ! modes intensity as a function of 2@ show sine-like feature and the relation of ) %! modes exhibit cosine-like behavior. Thus, the distinguished difference in periodicity or functional relationship (I versus 2@) in these modes is highly beneficial to assign experimental Raman-active modes symmetry. As can be seen from Table S3, the fitting curves according to Eqs. ( 4)-( 7) show excellent consistency with the experiment data. All ) %! modes marked by the red lines in both configuration and ! ! modes (the blue lines) under cross-polarized geometry show expected 4-lobed shape, and significantly ! ! modes under parallel polarizer configuration exhibit 2-lobed shapes. Unambiguously, the symmetry of experimentally observed vibrational modes can be well confirmed, which results from the reliable fitting and complementary results for both configurations. Furthermore, in parallel polarized configuration, the intensities ! ^ of some modes (e.g., ~64 and ~185 cm -1 ) with a periodicity of 180° reach the maximum values at 0° or 180° along b-axis and obtain their minimum at 90° and 270° along aaxis. Such characteristics could be used as a guide to verify the desirable crystalline orientation of small crystal or films [START_REF] Wu | Identifying the crystalline orientation of black phosphorus using angle-resolved polarized Raman spectroscopy[END_REF][START_REF] Li | GeP: An unexploited low-symmetry semiconductor with strong in-plane anisotropy[END_REF].

For a better understanding and further verification of these modes assigned to corresponding vibrational symmetry, the phonon dispersion curves along the high symmetry point directions and the vibrational symmetry of phonons vertically intersected with the phonon bands at the Γ point in Brillouin zone were calculated. As shown in Fig. 3a, the calculated phonon dispersion of SbCrSe3 along the high symmetry path is absent of imagine frequency, which hints that the calculated lattice structure is thermodynamically stable. Due to the large number of atoms [START_REF] Wu | Identifying the crystalline orientation of black phosphorus using angle-resolved polarized Raman spectroscopy[END_REF] in the primitive cell of SbCrSe3, 60 phonon bands comprised of three acoustic branches and 57 optical branches can be obtained, in which the expected 30 Raman active modes with four inequivalent irreducible (10! ! + 5) %! + 10) "! + 5) &! ) are predicted. These active Raman modes are in agreement with group theory analysis, as listed in Table S4.

Comparing with the DFT calculations results, the experimentally determined vibrational modes show excellent consistency. It should be noted that some modes, which are not experimentally observed, are also predicted. Moreover, the visualization of the atomic vibrations of some Raman modes (such as

) %! % ~47 cm -1 , ! ! " ~58 cm -1 and ) %! & ~163 cm -1
) is displayed in Figs. 3b-d

Anharmonic phonon decay and Thermal properties

To further elucidate the mechanisms of phonon scattering in SbCrSe3 crystal, the temperature-dependent Raman spectrum was measured from 78 K to 310 K in the cleaved (00l) plane, as shown in Fig. 4a and Fig. S4. Based on the above discussion, the vibrational symmetry of each experimentally observed Raman modes can be explicitly identified, as marked in Fig. S4. All Raman peaks obviously show a red shift, and the corresponding linewidths become broadening with increasing temperature throughout the measured temperature range (Fig. 4a), which is mainly attributed to anharmonic interactions between phonons [START_REF] Cuscó | Temperature dependence of Raman scattering in ZnO[END_REF][START_REF] Domènech-Amador | Raman scattering study of anharmonic phonon decay in InN[END_REF]. Here, the Raman modes

) %! % (~49 cm -1 ) and ) %! & (~172 cm -1
), with the characteristics of the atomic vibration along baxis, are selected to investigate inherent phonon scattering processes. Furthermore, the temperature variation of peak frequencies and FWHM of these Raman modes are depicted in Fig. 4b andc, respectively. To commendably illustrate the temperature dependent behaviors for peak position and FWHM, the conventional model assuming the decay of an optical phonon into two (representing three phonon process) or/and three phonons (four phonon process) is applied. Therefore, the temperature dependent phonon frequency and linewidth are commonly expressed as [START_REF] Balkanski | Anharmonic effects in light scattering due to optical phonons in silicon[END_REF]:

P = P 2 + ! Q1 + " 3 ℏ& ' () * + , )% R + )[1 + & 3 ℏ& ' -) * + , )% + & (3 ℏ& ' -) * + , )%) ( ] (8)
# = # 2 + U Q1 + " 3 ℏ& ' () * + , )% R + V[1 + & 3 ℏ& ' -) * + , )% + & (3 ℏ& ' -) * + , )%) ( ] ( 9
)
where P 2 is the intrinsic frequency, # 2 the broadening at 0 K, W 6 the Boltzmann constant, ℏ the reduced Planck's constant and the coefficients (A, B, C and D) are the anharmonic constants. summarized in Table 1. Notably, compared with the obtained value of the anharmonic coefficients (A and C) with reference to three phonon process, the coefficients (B and D) involved in four phonon process can be neglected due to their small magnitude, which indicates the three-phonon interactions are dominant. Similar to the conclusion of ) %! % mode, the temperature variation of the peak position and FWHM of ) %! & mode can be well characterized by the three-phonon scattering process, as listed in Table 1. It is therefore concluded that the three-order anharmonicity in lattice potential is more accurate to describe the temperature-dependent evolution of frequency and linewidth for SbCrSe3 crystal. Intriguingly, in contrast to ) %! % mode, the fitting parameters A and C shown in Table 1 for ) %! & mode exhibit relatively large values, which implies the

) %! & mode (~172 cm -1
) contributed by the vibrations of Se atoms has stronger anharmonic phonon-phonon coupling and thus plays a vital role in the reduction of the " $ . In addition, the relaxation time Y for the decay process (the phonon lifetime), which is closely associated with " $ , can be estimated by the simple relation [START_REF] Chen | Phonon anharmonicity in thermoelectric palladium sulfide by Raman spectroscopy[END_REF], Y = % 78G , where = represents the light velocity and G is FWHM of Raman modes. As shown in Fig. 4b andc, the FWHM for both modes broadens with the increasing temperature, suggesting that the phonon lifetime decreases and thus leads to the decrease in " $ with the elevated temperature. At the same time, it is evident that the change in the FWHM of ) %! & mode is significantly greater than that of ) %! % mode, which suggests the phonon lifetime of ) %! & mode drops faster with raising temperature.

It is verified again that ) %! & mode plays a more important role in reducing " $ . Table 1 The calculated anharmonic parameters for ) %! % and ) %! & modes of SbCrSe3 according to Eqs. ( 8) and ( 9), and the corresponding fitting line shown in Fig. 4b andc Mode w0 (cm -1 ) A (cm -1 ) To further explore the mechanisms of thermal transport, the thermal conductivity of SbCrSe3 single crystal along b-axis was measured. The calculated value was also given by solving the phonon Boltzmann transport equation (BTE). Due to the SbCrSe3 crystal possessing a high electrical resistivity (r) (Fig. S5), the electronic thermal conductivity (κe) is negligible, which is also found in previously reported polycrystalline samples [START_REF] Yang | Intrinsically low thermal conductivity from a quasione-dimensional crystal structure and enhanced electrical conductivity network via pb doping in CrSbSe3[END_REF]. As displayed in Fig. 5a, with increasing temperature, the experimental thermal conductivity first sharply rises to the peak value of ~15 W m -1

B (cm -1 ) G0 (cm -1 ) C (cm -1 ) D (cm -1 ) ) %! % 51.
K -1 at 20 K and then slowly deceases to ~2 W m -1 K -1 at room temperature, suggesting the obtained SbCrSe3 bulk crystal is a normal crystalline material and has intrinsically low " $ . Unambiguously, the calculated " $ from first principles has reasonably good agreement with the measured results of single crystal in the temperature range from 50 K to 300 K, and is slightly higher than that of the polycrystalline samples at elevated temperatures that could be owed to the absence of grain boundary and van der Waals layer interface for phonon scattering. This result substantiates that three-order anharmonicity in lattice vibration potential can well describe the thermal transport in SbCrSe3, which is consistent with the conclusions obtained by the in-situ Raman experiment. As shown in Fig. 5b, the anharmonic scattering rates (SRs), which are inversely proportional to the phonon lifetime, are remarkably enlarged with the rising temperature, which is responsible for the temperature-dependent evolution of " $ . In the frequency below 200 cm -1 , the anharmonic SRs displays a globally enhanced tendency, implying that the high frequency phonons have relatively shorter lifetime. In addition, the cumulative " $ at 100 K and 300 K with regard to the phonon frequency is plotted in Fig. 5c. The inset of Fig. 5c shows that the contribution from optical branches for " $ is larger than that from acoustic branches, which discloses the importance of optical branches in the thermal transport of SbCrSe3 crystal. As shown in Fig. 5d, the thermal conductivities are dominated by the phonons with short mean free paths (MFPs), and the MFPs with 50% accumulation of " $ are only ∼18 nm and engineering should be effective to reduce " $ [START_REF] Huang | Facile in situ solution synthesis of SnSe/rGO nanocomposites with enhanced thermoelectric performance[END_REF][START_REF] Li | High-performance nanostructured thermoelectric materials[END_REF][START_REF] Shen | Optimizing thermoelectric properties of BiSe through Cu additive enhanced effective mass and phonon scattering[END_REF][START_REF] Zhang | Twin engineering in solution-synthesized nonstoichiometric Cu5FeS4 icosahedral nanoparticles for enhanced thermoelectric performance[END_REF]. The combination of the ARPRS measurement and group theory calculation was used to identify the Raman modes symmetries for cleaved (00l) crystal plane of SbCrSe3, which is further corroborated via DFT calculations. It is interesting to found that some modes (e.g., ! ! " ~64 cm -1 and ! ! # ~185 cm -1 ) using the ARPRS measurement can be employed to verify the crystalline orientation of small crystals or films. In addition, facilitated by the in-situ Raman experiment, it is unveiled that the three phonon interactions dominate the thermal transport of SbCrSe3 crystal. Furthermore, the theoretically calculated " $ exhibits excellent agreement with the measured results, which also confirms that the approximation considering three-order term in lattice vibrational potential is reasonable. This work not only provides fundamental understanding of vibrational modes and lattice anharmonicity in the quasi-1D crystal of SbCrSe3, but also describes a fast and facile method to explore the thermal transport properties of other novel low dimensional materials. Intensity (a.u.)

拉曼光谱定义SbCrSe3单晶的振动模式对称性和声子非谐性 吴宏 1,2, † ,公祥南 3, † ,彭义 4 ,张龙 1 ,张斌 3 ,彭坤玲 1 ,刘杰 3 ,韩广 5 ,王爱峰 1 ,柴一晟 1 ,何 明全 1 ,顾豪爽 4 ,Emmanuel Guilmeau 6 ,王国玉 2,* ,卢旭 1,* ,周小元 1,
Raman shift (cm From bottom to top, the angles change from 0° to 360° with 10° intervals. Intensity (a.u.)

Raman shift (cm Eq. S(1) Eq. S(3) Eq. S(5) Eq. S(7) Eq. ( 5) Eq. ( 4)

) %! 0 0 0 0 Eq. ( 7) Eq. ( 6)

) "! 0 0 Eq. S(6) Eq. S(8) 0 0

) &! Eq. S(2) Eq. S(4) 0 0 0 0

  exhibits an aciculiform with a length of ~10 mm along the b-axis. The inset graph in Fig. 1e depicts the surface morphology along mirror side. The predominant (00l) diffraction peaks indicate that the c-axis is perpendicular to mirror side, which is employed to acquire other data in this work. Moreover, the XRD pattern in Fig. 1e of SbCrSe3 sample (crushed single crystal) is in agreement with previous report (PDF#50-0853), in which no distinct secondary phase is detected.

Figure 1

 1 Figure 1 Structural analyses and characterization of SbCrSe3 single crystal. (a) The crystal structure of SbCrSe3 along the b-axis. (b) CrSe6 double chains along b-axis. (c) Bond lengths from Cr to Se and Sb to Se. (d) The SbCrSe3 crystal, the b-axis, is along the long crystal dimension, which is about 10 mm. (e) XRD patterns for SbCrSe3 comparing with PDF#50-0853 and single-crystal SbCrSe3 shows predominantly (00l) peaks. The inset shows the image for single crystal SbCrSe3 used for powder XRD. (f) The HRTEM image of SbCrSe3 along [001] zone axis. The corresponding selected-area electron diffraction pattern and HRTEM image for (g) and (h), respectively. (i) The STEM-HAADF image and the corresponding EDX mapping demonstrates the chemical homogeneity of the SbCrSe3 crystal.

  . 1g are [020] and[200], respectively, which establishes that the preferred orientation of long edges for cleaving plane of SbCrSe3 crystal is b-axis. In addition, the EDX mapping for the synthetic SbCrSe3 sample in Fig.1iconfirms the uniform distribution of Sb, Cr and Se elements, which shows excellent consistency with the results from the EPMA in the regions as displayed in Fig.S1.Identifying Raman modes symmetriesRaman spectrum not only contains rich information of phonon properties near the Brillouin zone, but also is closely correlated with crystal structures in terms of symmetry. Here, to determine the phonon vibrational symmetries for the Raman modes and relevant anisotropy, the backscattering geometry ARPRS measurement was performed on the cleaved (00l) crystal plane of the SbCrSe3 crystal at room temperature (the schematic of experimental setup is exhibited in Fig.S2). The a-axis and b-axis of the SbCrSe3 crystal were defined as the *( and , ( directions, respectively, as shown in Fig.S2c. Fig. S2a and b separately display the two configurations for the ARPRS experiment, where the green rectangle symbolizes the sample, the red arrow indicates the polarization direction of the incident laser, and the blue one represents the original polarization of Raman backscattered signal from the sample surface corresponding to the analyzer-selected vertical or horizontal direction (denoted as perpendicular (⊥) or parallel (‖) polarization configuration, respectively). Fig. 2a displays the Raman spectra of SbCrSe3 single crystal measured in the cleaved (00l) crystal plane at an angle of 0° using un-, parallel-, and perpendicular-polarized configurations. Fourteen Raman modes can be obviously observed under unpolarized configuration. With regard to the under ⊥ and ‖ polarization configurations, however, the peak intensity of some vibrational modes can be enhanced, reduced or even disappeared periodically, which indicates the existence of polarization-dependent behavior.

Figure 2

 2 Figure 2 The ARPRS of SbCrSe3. (a) Measuring Raman spectra at angle of 0° under

!

  On the basis of group theory calculation, the sample angle dependent intensities of each Raman active mode in different crystal planes of the SbCrSe3 sample are full evaluated, which is fundamentally important to identify the mode symmetries. The following calculations are suitable for the analysis of the cleaved (00l) crystal plane and the results of the remaining crystal faces are included in supplementary information (SI Section I). When the (00l) plane of the crystal in the *-, plane is rotated, there is an angle θ between laboratory coordinates (*,/) and crystal coordinates (*′,′/′). Raman tensors (2 3⃗ ) of a crystal are defined according to its crystal coordinates, and the laboratory coordinates can be converted as (2′ 333⃗ ) by 2( 3333⃗ = 6 33⃗ • 2 3⃗ • 6 33⃗ )%, where 6 33⃗ is defined as the orthogonal transform matrix to link the two coordinates:

  , respectively. The atomic displacement of ) %! % and ) %! & modes is along b-axis (out of (010)) plane, while that of ! ! " mode belongs to in-plane (010) vibration.

Figure 3

 3 Figure 3 Phonon properties of SbCrSe3. (a) The calculated phonon dispersion and density of states for SbCrSe3 in left and right panels, respectively. The atomic vibrations of some Raman modes (b) ) %! % ~47 cm -1 , (c) ! ! " ~58 cm -1 and (d) ) %! & ~163 cm -1 ,

Figure 4

 4 Figure 4 In-situ Raman spectra of SbCrSe3. (a) Raman scattering spectrum of SbCrSe3

Figure 5

 5 Figure 5 Thermal transport properties of SbCrSe3. (a) Lattice thermal conductivity k $

  Figure S1 (a)-(c) The EPMA images corroborates the chemical homogeneity of

FigureFigure S3

 S3 Figure S2 (a) Schematic diagram for parallel configuration. (b) Schematic diagram for perpendicular configuration. (c) The sample under an optical microscope shows *′ (a) -axis and ,′ (b) -axis.

Figure

  Figure S4The temperature dependence of Raman spectra of SbCrSe3 in the varied

Figure

  Figure S5The temperature-dependent r and κe of SbCrSe3 crystal.

  Table S1 Crystal data and structure refinement for SbCrSe3 Largest diff. peak/hole/e Å -3 2.40 / -1.80 Table S2 The angle dependence of Raman peak intensities for each Raman active modes of SbCrSe3 with exposed (100), (010) and (001) plane in cross-and parallel-polarized

	configuration
	Empirical formula Formula weight Mode ^// %22	^%22	SbCrSe3 410.63 2%2 ^//	^2%2	^// 22%	^22%
	Temperature/K ! !		295(2)
	Crystal system		orthorhombic
	Space group		Pnma
	a/Å		9.1475(6)
	b/Å		3.7872(3)
	c/Å		13.4264(11)
	α/°		90
	β/°		90
	γ/°		90
	Vomule/Å 3		465.14(6)
	Z		4
	ρcalcg/cm 3		5.864
	μ/mm -1		31.414
	F(000)		708.0
	Crystal size/mm 3		0.38 ´ 0.12 ´ 0.11
	Radiation/Å		Mo Kα (λ = 0.71073)
	2θ range for data collection/°	7.53 to 57.634
	Index ranges		-4 ≤ h ≤ 12, -4 ≤ k ≤ 4, -5 ≤ l ≤16
	Reflections collected		1222
	Independent reflections		617 [Rint = 0.0224, Rsigma = 0.0375]
	Data/restraints/parameters		617 / 0 / 31
	Goodness-of-fit on F 2		1.122
	Final R indexes [I>=2σ(I)]		R1 = 0.0392, wR2 = 0.1000
	Final R indexes [all data]		R1 = 0.0443, wR2 = 0.1057

Table S4 Experimental

 S4 Raman peak frequencies (in cm -1 ) and vibrational symmetries of SbCrSe3. The predicted values of peak frequency given by DFT calculations are also listed for comparison. The ! !" # and ! $" # are also well predicted

	Exp.	Symmetry	DFT	Symmetry	DFT
	49	! %" %	47	! $" %	49
	58	" " %	55	! !" %	59
	64	" " !	58	! !" !	67
	89	" " $	87	! !" $	80
	92	" " &	/	! $" !	92
	125	" " '	120	! !" &	128
	144	! %" !	131	! $" $	132
	153	" " (	149	! !" '	148
	172	! %" $	163	! $" &	163
	185	" " )	171/176	! !" (	166
	209	" " *	203	! !" )	178
	238	! %" &	215	! !" *	208
	254	" " +	235	! $" '	214
	265	" " %,	251	! !" +	234
				! !" %,

Acknowledgments

This work was financially supported in part by the National Natural Science Foundation of China (11904348, 11604032, 51772035, 51672270, 52071041), the Fundamental Research Funds for the Central Universities (106112016CDJZR308808). The work conducted at the Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences is also supported by Key Research Program of Frontier Sciences, CAS, (QYZDB-SSW-SLH016). We also would like to thank Ms. Guiwen Wang and Dr. Yang Zhou at Analytical and Testing Center of Chongqing University for thermal transport analysis and EPMA measurement, respectively.

Author contributions Wu H, Gong X, Lu X, Wang G and Zhou X designed the experiment and wrote the paper; Wu H, Gong X, Peng Y, Zhang L, Wang A and He M conducted experimental testing and crystal growth; Zhang B and Liu J measured the microstructure characterization; Peng K, Gu H, Han G, Chai Y and Guilmeau E participated in result analyses. All authors contributed to the general discussion.

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary information

Supporting Information

Identification of vibrational mode symmetry and phonon anharmonicity in SbCrSe3 single crystal using Raman spectroscopy

Hong Wu 1,2, † , Xiangnan Gong

Section I. Raman theory

The intensity (I) of the observed Raman vibration modes can be expressed as:

The Raman tensors of ! ! , ) %! , ) "! and ) &! according to the D2h point group can be expressed as:

When the incident laser propagates along the * direction with unit polarization vector along ,-axis (G ⃗ * = (0 1 0)). The orthogonal transform matrix is obtained by: 6 33⃗ = : 1 0 0 0 =>?@ ?AB@ 0 -?AB@ =>?@ ; For the selectivity analyzer with unit polarization vector G ⃗ + , = (0 1 0) defined as parallel configuration, the Raman scattering intensities of different modes can be expressed as:

^H! ! , ‖I ∝ |=| " • ?AB . @ + |K| " • =>? . @ + 2|=||K| • =>?O 08 • ?AB " @ • =>? " @ S(1)

In perpendicular configuration, the selectivity analyzer has unit polarization vector G ⃗ + 9 = (0 0 1). The Raman scattering intensities of different modes can be expressed as:

where

When the incident laser propagates along the , direction with unit polarization vector

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along /-axis (G ⃗ * = (0 0 1)). The orthogonal transform matrix is obtained by: 6 33⃗ = :

=>?@ 0 -?AB@ 0 1 0 ?AB@ 0 =>?@ ; For the selectivity analyzer with unit polarization vector G ⃗ + , = (0 0 1) defined as parallel configuration, the Raman scattering intensities of different modes can be expressed as:

On perpendicular configuration, the selectivity analyzer has unit polarization vector G ⃗ + , = (1 0 0). The Raman scattering intensities of different modes can be expressed as:

where

Table S3 The ARPRS intensities in parallel and perpendicular polarization configurations (the red and blue lines represent ! !" and " " modes, respectively)

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Peak