Difference analysis model for the mismatch effect and substrate-induced lattice deformation in atomically thin materials

Atomically thin materials hold mutative electronic transition properties by virtue of the inevitable strains adding to the lattice distribution. A quantitative approach and clear systematical verification have seldom been presented to effectively illustrate the substrate-induced strain along the lattice distribution, although the substrate perturbations for monolayers have been repeatedly mentioned. Here, a model of difference analysis is introduced to reveal the structural strain of monolayers from the substrate, which is derived from the mismatch effect based on the variation of the thermal expansion coefficient between the substrate and monolayers. A coupling coefficient is proposed to quantitatively manifest the substrate-induced strain and mismatch effect. Furthermore, combined with the coupling coefficient, the lattice disorder from the mismatch effect is demonstrated, experimentally using the ${\mathrm{WS}}_2$ monolayer as reference. This study could promote broad investigations of some fundamental issues from preparation to electrical structural performances in atomically thin materials and further practical applications.

Figure 1

(a)–(d) Raman mappings of the four monolayer samples. The insets show optical images from the corresponding mapping regions. The color bars correspond to the intensity of Raman spectra. (e)–(h) The PL spectra and multiple-peak fittings of the four monolayers. The intensity magnitudes of PL are displayed at the top. (i)–(l) The AFM topographies, confirming the monolayer nature of the samples. (m)–(p) The surface potential distributions of the four monolayers. (q) The Gaussian distribution of CPD for every sample. The scale bars of (a), (b), (c), and (d) are 9, 7, 9, and 13 $\mu \mathrm{m}$, respectively.

Figure 2

Temperature-dependent Raman spectra of samples A–D. (a)–(d) Color contour mappings of Raman spectra (250–$450{\mathrm{cm}}^{-1})$ as a function of temperature. The color bars correspond to the Raman intensity, which can be normalized to the Si peak. The Raman modes (e)–(h) for $E_{2g}^1$ and (i)–(l) $A_{1g}$ from temperature-dependent Raman shifts analyzed by experiment (triangles) and theoretical arithmetic (dots). The experimental results are preliminarily fitted with the linear model.

Figure 3

(a) Calculated TEC of ${\mathrm{SiO}}_2$ and monolayer ${\mathrm{WS}}_2$. (b) TEC difference between the monolayer and substrate $\left({\alpha }_{{\mathrm{WS}}_2}-{\alpha }_{{\mathrm{SiO}}_2}\right)$. (c) The evolution of basic mismatch strain $\varepsilon$ with temperature increasing from 77 to 900 K. The inset illustrates the schematic of the thermal contraction of monolayers on the substrate during the cooling process. (d) The effect of substrate-induced strain $\mathrm{\Delta }{\omega }_M\left(T\right)$ when the coupling coefficient $\beta$ changes from 0 to $10{\mathrm{cm}}^{-1}/\%$. (e)–(h) $\mathrm{\Delta }{\omega }_M\left(T\right)+\mathrm{\Delta }{\omega }_{RT}[\mathrm{\Delta }{\omega }_M\left(T\right)$ is the substrate-induced strain effect; $\mathrm{\Delta }{\omega }_{RT}$ is the residual strain effect] from frequency variations of Raman spectra for the $E_{2g}^1$ mode in samples A, B, C, and D. The parameter $\beta$ is the coupling coefficient, which indicates the substrate coupling strain.

Figure 4

Temperature-dependent PL spectra of four monolayers. (a)–(d) Color contour mappings of PL spectra (1.54–2.14 eV) as a function of temperature. The color bars correspond to the intensity of the PL spectra. (e)–(h) Normalized temperature-dependent PL spectra of samples A, B, C, and D, respectively.

Figure 5

Gaussian fitting results for monolayers. (a)–(d) Peak position of the intrinsic transition exciton $X$ and defect-bound exciton emission $L$ of samples A–D. (e)–(h) PL intensity of exciton $X$ and $L$ of samples A–D, respectively. The green dots in (f)–(h) show the intensity ratios of intrinsic to defect-bound excitons ($X$/$L$).