## 3 Results and Discussion
### Temperature-Dependent Raman Spectroscopy  
The rhombohedral crystal structure of Bi\({}_{2}\)Se\({}_{3}\) belongs to the \(D^{5}_{3d}\)(R\(\bar{3}\)m) space group symmetry. There are five atoms per unit cell in the Bi\({}_{2}\)Se\({}_{3}\). This compound contains sandwich structure of the five layers [Se(1)-Bi-Se(2)-Bi-Se(1)], which are held together by van der Waals force. According to the group theory, there are 12 optical modes in Bi\({}_{2}\)Se\({}_{3}\). However, four modes (2A\({}^{1}_{u}\) and 2E\({}_{u}\)) are infrared active and can not be observed in the Raman spectrum. So, only eight modes were explained [36; 40]. The irreducible representation for the zone-center phonons can be written as [42]:  
\[\Gamma=2\mathrm{E}_{g}+2\mathrm{A}_{1g}+2\mathrm{E}_{u}+2\mathrm{A}_{1u} \tag{1}\]  
It must be noted that the thickness-dependent vibrational frequency of E\({}^{1}_{g}\) mode is mainly due to the weak van der Waals interlayer interaction. Similar with graphene and MoS\({}_{2}\)[43; 44], this mode completely vanishes under single-layer limitation. Here, \(a\), \(b\), and \(c\) is a crystal lattice constant of Bi\({}_{2}\)Se\({}_{3}\), respectively. The Raman tensor of Bi\({}_{2}\)Se\({}_{3}\) are expressed as: [45]  
\[\mathrm{A}_{g}=\left( \begin{array}{ccc}a&0&0\\ 0&a&0\\ 0&0&a\end{array} \right),\mathrm{E}_{g}=\left( \begin{array}{ccc}c&0&0\\ 0&-c&d\\ d&0&0\end{array} \right),\left( \begin{array}{ccc}0&-c&d\\ -c&0&0\\ d&0&0\end{array} \right), \tag{2}\]  
For temperature-dependent Raman scattering, the Raman spectra were collected by varying the temperature in the range of 83-603 K. As shown in Figure 3a, the intensities of Raman peaks increase significantly with decreased temperature. When the temperature increased from 83 to 603 K, A\({}^{1}_{1g}\), E\({}^{2}_{g}\), and A\({}^{2}_{1g}\) modes are observed to follow a monotonous shift to low wavenumber direction. Such as: the frequency of A\({}^{1}_{1g}\) mode moved from 72 cm\({}^{-1}\) to 64 cm\({}^{-1}\) (Figure 3b). The frequency of E\({}^{2}_{g}\) mode moved from 130 cm\({}^{-1}\) to 123 cm\({}^{-1}\) (Figure 3c). The frequency of A\({}^{2}_{1g}\) mode moved from 174 cm\({}^{-1}\) to 163 cm\({}^{-1}\) (Figure 3d). We analyzed and discussed the red-shift of Raman peaks and temperature coefficients in Figures 3 and 4. Generally, the research of temperature-dependent Raman spectroscopy is an efficient way to investigate the thermal expansion, thermal conductivity, and interlayer coupling [46; 47]. In our experiment, all the Raman peaks had an obvious red-shift with the temperature increasing from 80 to 603 K. In addition, the peak shifts had a linear dependence with the temperature, yielding the first-order temperature coefficient (\(\chi\)). It has been proposed that  
Figure 2: **(a) SEM image. (b) Low magnification TEM image. (c) SAED pattern. (d) HRTEM image. (e) Raman spectrum at room temperature of single Bi\({}_{2}\)Se\({}_{3}\) nanoplate. **the temperature-dependent peak positions in Raman spectroscopy may result from anharmonicity that results from the thermal expansion and contraction of the crystal and phonon modes [48]. The temperature-dependent line width and intensity could be attributed to the zone-center optical phonon decaying into an optical phonon and an acoustic phonon [48; 49; 50]. Figure 4a-d plot the peak frequency and FWHM of A\({}_{1g}^{1}\), E\({}_{g}^{2}\), and A\({}_{1g}^{2}\) modes from 83 to 603 K in detail to investigation the temperature effect. The FWHM increased with increase temperature. The peak frequency and FWHM of phonon modes at each temperature can be predicted by theoretical Raman line shape, which can be written as [51]:  
\[I(\omega)=\frac{\chi_{0}\Gamma_{0}\omega\omega_{0}^{2}(\tilde{n}+1)}{(\omega_{ 0}^{2}-\omega^{2})^{2}+\omega^{2}\Gamma_{0}^{2}}, \tag{3}\]  
where \(\tilde{n}=exp[\frac{\hbar\chi}{\kappa_{B}T}]-1\) is the phonon occupation number, \(\omega_{0}\) and \(\Gamma_{0}\) are the peak position and the line width, \(\chi_{0}\) is related with peak intensity, respectively.