Raman spectrum of 1T'-WTe2 under tensile strain: A first-principles prediction

TL;DR

This study uses first-principles simulations to predict how tensile strain affects the Raman spectra of monolayer 1T'-WTe2, revealing anisotropic responses and phonon softening at critical strains.

Contribution

It provides the first theoretical prediction of strain-induced Raman spectral changes and phonon behavior in 1T'-WTe2, highlighting anisotropic effects and critical strain points.

Findings

Phonon branches soften to negative frequencies at specific critical strains.

Raman shifts of A1g, A3g, and A4g modes show anisotropic responses to strain.

A3g mode exhibits a parabolic frequency shift with a transition at about 5% strain.

Abstract

Monolayer WTe2 attracts rapidly growing interests for its large-gap quantum spin Hall effect,which enables promising apllications in flexible logic devices. Due to one-dimensional W-W chains,1T'-WTe2 exhibits unique anisotropic structure and promising properties, which can be modified by simply applying strains. Based on the first-principles simulations, we show that phonon branch undergoes soft down to negative frequency at special q points under different critical strains, i.e., epsilon_a = 11.55 percent along a-axis (with W-W chains) direction, epsilon_b = 7.0 percent along b-axis direction and epsilon_ab = 8.44 percent along biaxial direction. Before each critical strain, the Raman-shift of A1_g, A3_g, and A4_g modes, corresponding to the main peaks in Raman spectra of 1T'-WTe2 , shows anisotropic response to uniaxial strain but most sensitive to biaxial strain. Interestingly, we find that the frequency shift of A3_g mode show parabolic characters of strained 1T'-WTe2, then we split it into two parts and it shows a Raman-shift transition at about 5 percent strains. While for the A1_g and A4_g modes,the frequencies change linearly. Through careful structure and vibration analysis, we try to explain these Raman irregularity in strained 1T'-WTe2.

Tables

Table 1. Table 1: The phonon frequencies of nine Raman-active modes at Γ Γ \Gamma point for 1 T ′ superscript 𝑇 ′ T^{\prime} -WTe 2 from Raman experiments and first-principles calculations. The unit of frequency has been converted to c ​ m − 1 𝑐 superscript 𝑚 1 cm^{-1} for direct comparison with the Raman measurements.

symmetry	Exp.[31]	Exp.[32]	Exp.[33]	Exp.[34]	Exp.[35]	Cal.[31]	Cal.[32]	Cal.[35]	Cal.[45]	Cal.(this work)
$A_g^1$	213.2	217	212	216		222.6	219	221.9	208.9	210.3
$A_g^2$					216.1	220.3		220.5	208.4	208.7
$B_g^1$						185.7		173.7	161.3	158.5
$A_g^3$	161.9	164	165	164	164.3	164.5	168	165.9	156.5	155.1
$A_g^4$	132.8				135.9	137.2	135	136.9	129.9	128.9
$A_g^5$					119.7	118.5	119	119.7	114.4	114.1
$B_g^2$					109.4	108.5		110.4	105.8	105.1
$B_g^3$				86	88.9	93.2	89	92.5	87.2	84.5
$A_g^6$			81		85.7	81.4		84.6	80.0	81.4

Full text

I Introduction

In a robust two-dimensional materials family of transition metal dichalcogenides (TMDs) 1 , monolayer 1$T^{\prime}$-WTe2 has large-gap quantum spin Hall (QSH) insulator 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 , unconventional spin-torque 12 and gate-tunable superconductivity 13 , which attracts rapidly growing interests, and enable promising applications in spintronics, dissipationless transport, and quantum computations. Among 1$T^{\prime}$-TMDs—MX2 with M = (W, Mo) and X = (Te, Se, S)—theoretically predicted to be a new class of QSH insulators 2 , WTe2 is the only one for which the 1$T^{\prime}$ phase is most energetically favoured 2 and can be observed under ambient conditions 1 ; 14 . 1$T^{\prime}$-WTe2 with a distorted orthorhombic crystal structure is distinct anisotropy in contrast to the other monolayer TMDs crystallized in 1$H$ (trigonal prismatic coordination) or 1$T$ (octahedral coordination) structures. In plane, the covalently bonded $W$ atoms form a zigzag $W$-$W$ chain, which makes WTe2 structurally one-dimensional and electronically a semimetal 15 . When cutting 1$T^{\prime}$-WTe2 into nanoribbons perpendicular to the $W$-$W$ chains in our previous study 16 , the electronic band opens a gap, and semimetal transforms to semiconductor. Other theoretical studies also found that only 1% tensile strain along the $W$-$W$ chains can lead to a semimetal to semiconductor or QSH insulator transition 17 ; 18 . Moreover, the Poisson ration, the in-plane stiffness and the absorption spectrum of monolayer 1$T^{\prime}$-WTe2 are strongly dependent on and tunable by tensile strain 17 .

Strain engineering in different orientations can lead to anisotropic modifications to the structure and properties of monolayer 1$T^{\prime}$-WTe2, which could play important roles in the application of flexible logic devices 19 . Theoretical and experimental studies on graphene 20 ; 21 , monolayer $h$-BN 22 ; 23 and 1$H$-MX2 with M = (W, Mo) and X = (Se, S) 24 ; 25 ; 26 ; 27 ; 28 ; 29 have found that the phonon spectra are significantly affected by external strains, and their response can be probed by Raman spectroscopy 30 . In contrast to these 1$H$ structures with in-plane isotropic, 1$T^{\prime}$-WTe2 under stain would exhibit more interesting anisotropic properties. The Raman spectrum of WTe2, from monolayer, few-layer to bulk as well as alloys 31 ; 32 ; 33 ; 34 ; 35 ; 36 ; 37 ; 38 ; 39 ; 40 , has been studied to detect the crystal structure, lattice vibration, number of layers and in-plane anisotropy, however, the Raman spectrum of strained WTe2 has not yet been studied by either theory or experiment.

Based on the first-principles calculations, we provide a thorough study on the changes of structures, phonon spectra, and Raman-active modes of monolayer 1$T^{\prime}$-WTe2 under uniaxial (parallel or perpendicular to the $W$-$W$ chains, respectively) and equibiaxial tensile strains. Here, we only focus on 1$T^{\prime}$-WTe2 under tension since a 1$T^{\prime}$-1$H$ phase transition will occur under compression 41 , which may be experimentally challenging to achieve without incurring any buckling response. Through our phonon dispersion curve calculations, we find that the acoustic branches of phonon are the most sensitive to tensile strains, and a soft mode with negative frequency at different $q$ point under different critical strain occurs, which indicate the structural instability and signify the ideal strength of 1$T^{\prime}$-WTe2 withstanding larger strain before rupture. Besides, we also find that the Raman-active modes response quite different to external strains, and their corresponding frequency changes are anisotopic. These results indicate that Raman measurements are sufficient to detect the strain magnitudes and structural stability of 1$T^{\prime}$-WTe2.

II Computational method

Our calculations are based on density functional theory (DFT) and density functional perturbation theory (DFPT) in pseudopotential plane-wave formalism, as implemented in the Quantum ESPRESSO package 42 . The atomic positions and lattice constants for 1$T^{\prime}$-WTe2 are optimized by using Perdew-Burke-Ernzerhof (PBE) 43 functional, and the cutoff energy of 40 Ry and 400 Ry ($1Ry\approx 13.6eV$) for the wave functions and the charge density, respectively. A vacuum spacing $\thicksim 13\mathring{A}$ is used to prevent the interaction between the WTe2 monolayers. A set of 24$\times 1$2$\times 1$ $k$-point sampling is used for Brillouin Zone (BZ) integration over electronic states, and a set of 8$\times$4$\times 1$ $q$-point grid is used for the phonon calculations to obtain the dynamic matrices. The self-consistent solution of the Kohn-Sham equations is obtained when the total energy changed by less than 10*-8* Ry and the Hellmann-Feynman force on each atom is less than 10*-6* Ry/bohr ($1bohr\approx 0.529177\mathring{A}$). The lattice constants thus are determined with $a_{0}=3.49\mathring{A}$ and $b_{0}=6.31\mathring{A}$ and agree with the experimental 14 ; 44 and theoretica 13 ; 17 values.

III Results and discussion

The primitive cell of 1$T^{\prime}$-WTe2 is indicated by green rectangle, and its primitive vectors,  $\overrightarrow{a}$ and $\overrightarrow{b}$, are indicted by red arrows as shown in Fig. 1(b). Along $a$-axis direction, the distorted $W$ atoms form 1D zigzag chains indicated by the pink lines. In the $W$-$W$ chains as show in Fig. 1(c), the stress (corresponding to $E_{bb}=110.14N/m$) is lager than one in the $b$ axis as shown in Fig. 1(a), based on our calculation of Young’s modulus matrix E=\left(\begin{array}[c]{cc}78.59&27.68\\ 28.43&110.14\end{array}\right)N/m which are in agreement with the theoretical 18 and experimental 34 values. Due to the stronger bond strength in $W$-$W$ chains, one can expect that 1$T^{\prime}$-WTe2 could withstand lager strain along the $a$-axis direction compared to the $b$-axis direction. These two directions are the subject of our study. A state of uniaxial strain along $a$-axis and $b$-axis is constructed by applying the nominal strain and , respectively, where $a$ and $b$ are the strained lattice constants of 1$T^{\prime}$-WTe2.

To obtain the phonon and Raman spectra of strained 1$T^{\prime}$-WTe2, we firstly calculate the phonon dispersion curves of intrinsic one, and then analyse the characteristics of Raman-active modes based on the group theory. As shown in Fig. 2, there exist 18 phonon branches (3 acoustic and 12 optical branches) with 6 atoms (2 tungsten and 4 tellurium) in the primitive cell of 1$T^{\prime}$-WTe2. Because monolayer WTe2 belongs to the No. 11 space group $P21/m$ and the point group $C_{2h}^{2}$, the irreducible representation of the normal modes at $\Gamma$ point is $6A_{g}+3A_{u}+3B_{g}+6B_{u}$, in which only 9 modes are Raman-active among 18 phonon modes based on the symmetry analysis of calculated atomic displacements. These nine Rman-active modes as shown in Fig. 2 are $6A_{g}+3B_{g}$, whose frequencies are shown in Table I with other theoretical 31 ; 32 ; 35 ; 45 and experimental counterparts 31 ; 32 ; 33 ; 34 ; 35 . Typical Raman spectrum of 1$T^{\prime}$-WTe2 exhibits three prominent Raman peaks at $\thicksim 210cm^{-1}$, $160cm^{-1}$, and $130cm^{-1}$ denoted as $A_{g}^{1}$, $A_{g}^{3}$ and $A_{g}^{4}$, respectively. Later, we will focus our work exclusively on these three Raman-active modes.

Owing to strongly anisotropic mechanical properties of 1$T^{\prime}$-WTe2, we expect that the three Raman-active modes exhibit diverse characteristics for WTe2 under different-direction tensions. Therefore, we then calculate the phonon dispersion relations of strained monolayer WTe2 along $a$-axis, $b$-axis as well as $ab$-biaxis direction, respectively. Interestingly, as red curves shown in Fig. 3, the phonon branches become soft and their frequencies become negative at special $q$ points under different critical strains. For example, when the tensile strain along $a$-axis direction approaches 11.55% ($\varepsilon_{a}=11.55\%$), one of the acoustic modes of 1$T^{\prime}$-WTe2 becomes imaginary near $S$ point indicating structural instability, which consists with Torum’s ($\thicksim 11\%$) 17 and Xiang’s ($\thicksim 12.5\%$) 18 phonon spectra. In particularly, applying strains along b-axis direction, Torun et al 17 found that the strain-stress curve reaches a maximum at the critical strain of 15% (but without supporting phonon dispersion curves), and Xiang et al 18 found that a phonon branch becomes unstable near the $\Gamma$ point (not pronounced) at the critical strain $\varepsilon_{b}=12\%$. However, different from these two works, our study of tensile strain along $b$-axis direction shows that the ideal strength of monolayer 1$T^{\prime}$-WTe2 is 7% with the soft mode occurring at $Y$ point exactly, which is smaller than that of WTe2 along $a$-axis direction with $\varepsilon_{a}=11.55\%$. Such anisotropic phonons might be attributed to the 1D structure formed by the $W$-$W$ chains. Meanwhile, we also calculate the phonon dispersion curves of 1$T^{\prime}$-WTe2 under uniform biaxial strain, as shown in Fig. 3(c), a soft mode occurs near $X$ point at critical strain $\varepsilon_{ab}=8.44\%$ indicating a possible phase transition with the 1$T^{\prime}$-WTe2 fracture.

Finally, we analyse the response of Raman-active modes ($A_{g}^{1}$, $A_{g}^{3}$ and $A_{g}^{4}$) on the three direction strains before 1$T^{\prime}$-WTe2 rupture. Due to $A_{g}$ modes containing double-rotation symmetry-axis operation $C_{2}(z)$ based on the character table for point group $C_{2h}$, they belong to tangential shear modes with opposite direction vibrations of $W$-$W$ or $Te$-$T$e pairs. Moreover, the vibration displacements of $A_{g}$ modes (see Fig. 2) are in the $bc$ plane of 1$T^{\prime}$-WTe2, perpendicular to the $a$-axis direction, therefore, one can expect that the effects of $a$-axis strain on the $A_{g}$ modes would be weaker than that of $b$-axis strain. As a result, $A_{g}^{4}$ mode stays relatively unchanged when 1$T^{\prime}$-WTe2 under uniaxial strain along $a$-axis direction, with small slope value of $-0.29cm^{-1}/\%$, as the red line shown in Fig. 4(c). In order to make the frequency/strain ratio of three modes comparable, we put them in one figure with the same scale of the frequency and strain graduations, respectively. And in Fig. 4, we do find that the frequency shifts in three modes with $b$-axis strain are all much larger than that with $a$-axis strain, indicating a clear Raman modes anisotropy. This anisotropic Raman-strain response can facilitate the determination of the cystallographic orientation in 1$T^{\prime}$-WTe2. Moreover, when applying biaxial strain to 1$T^{\prime}$-WTe2, the frequencies of three Rman modes also red-shift, but exhibit most sensitive to biaxial strain compared to uniaxial strains, with corresponding largest slope values. To explain this, we plot Figure 5. In Fig.5, one can find that, although the bond angle of $W$-$W$ chain is almost unchanged when 1$T^{\prime}$-WTe2 under equibiaxial strain, the $W$-$W$ bond length, the vertical distances in two pairs of $Te$ atoms ($c_{1}$ and $c_{2}$, respectively) are all dramaticly changed compared to the unixal strains. This structure modification under biaxial strain, in turn, leads to a considerable change in the Raman-active modes.

For $A_{g}^{1}$ and $A_{g}^{4}$ modes, their frequencies decrease almost linearly with applied strains as shown in Fig. 4(a) and (c), respectively, therefore, the strain in 1$T^{\prime}$-WTe2 can be quantified by $A_{g}^{1}$ or $A_{g}^{4}$ mode in Raman spectra. However, for $A_{g}^{3}$ mode, the frequency shift undergoes a sharp turn at about 5% for both uniaxial and biaxial strains, as shown in Fig. 4(b), for example, the calculated biaxial-strain coefficient for $A_{g}^{3}$ mode is down from $-2.22cm^{-1}/\%$ to $-6.03cm^{-1}/\%$. This redshift transition might be result from that the lattice symmetry is maintained before 5% strains and then broken after 5% strains applied. Because the $A_{g}^{3}$ mode consists of both in-plane and out-of-plane motions as shown in Fig. 2, with regard to the relative motions in pairs of $W$ atoms (in-plane) and pairs of $Te$ atoms (out-of-plane), the parallelogram without $W$-$W$ chains could maintain its shape, but the parallelogram with $W$-$W$ chains might be flattened or prolongated. Moreover, the amplitudes of $Te$ atoms are much larger than that of $W$ atoms, especially external strains strengthen the motions of $Te$ atoms but weaken the vibrations of $W$ atoms, which would prompt the symmetry broken at last. So, for this particular $A_{g}^{3}$ transition to external strains and $A_{g}^{3}$ as the most significant peak with the largest intensity in Raman spectra, we hope this phenomena can be tested by future Raman experiments of strained 1$T^{\prime}$-WTe2.

IV Conclusions

In conclusion, by using DFT and DFPT simulations, we have systematically obtained the Raman spectrum of monolayer 1$T^{\prime}$-WTe2 under tensile strains. Our results demonstrate that $A_{g}^{1}$, $A_{g}^{3}$ and $A_{g}^{4}$ modes, corresponding to the main peaks in Raman spectra of 1$T^{\prime}$-WTe2, are more sensitive to uniaxial strain along $b$-axis direction than that along $a$-axis (containing $W$-$W$ chains) direction, exhibiting anisotropic Raman-strain response. In particular, biaxial strain induces the largest red-shift of the Raman modes compared to the uniaxial strains. Through careful structure configuration analysis, we reveal that the irregularity comes from the dramatic change of the $W$-$W$ bond length as well as the vertical distances in two pairs of Te atoms under biaxial strain. More interesting, $A_{g}^{3}$ mode undergoes a Raman-shift transition at about 5% for both uniaxial and biaxial strains, different from the $A_{g}^{1}$ and $A_{g}^{4}$ modes whose frequencies show linear dependence on tensile strains. The underlying symmetry reason is revealed to be that the competition and cooperation of the relative motions between $W$-atom pairs and $Te$-atom pairs modify the shape of 1$T^{\prime}$-WTe2. The revealed Raman modes responses to tensile strain are useful for tracing or detecting operations on 1$T^{\prime}$-WTe2 based flexible devices.

Acknowledgements.

This work was supported by the National Key R&D Program of China under Grant Nos. 2017YFB0403602 and 2016YFB0400603, the National Natural Science Foundation of China under Grant No. 61605014, and the China PFCAEP under Grant No. YZJJLX2016010.