Mapping Thermal Expansion Coefficients in Freestanding 2D Materials at the Nanometer Scale

Two-dimensional materials, including graphene, transition metal dichalcogenides and their heterostructures, exhibit great potential for a variety of applications, such as transistors, spintronics, and photovoltaics. While the miniaturization offers remarkable improvements in electrical performance, heat dissipation and thermal mismatch can be a problem in designing electronic devices based on two-dimensional materials. Quantifying the thermal expansion coefficient of 2D materials requires temperature measurements at nanometer scale. Here, we introduce a novel nanometer-scale thermometry approach to measure temperature and quantify the thermal expansion coefficients in 2D materials based on scanning transmission electron microscopy combined with electron energy-loss spectroscopy to determine the energy shift of the plasmon resonance peak of 2D materials as a function of sample temperature. By combining these measurements with first-principles modeling, the thermal expansion coefficients (TECs) of single-layer and freestanding graphene and bulk, as well as monolayer ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, or ${\mathrm{WSe}}_2$, are directly determined and mapped.

Figure 1

(a) Low-loss EELS spectra from a monolayer of ${\mathrm{WSe}}_2$ for temperatures between 373 K and 723 K. The purple lines indicate the plasmon peak centers for each temperature, determined by fitting two Lorentzian curves to the peak. The black line shows the plasmon peak center at a temperature of 373 K for comparison. (b) The plasmon energy for each spectrum from (a) as a function of the temperature. (c)–(e) The energy shifts ($dE/dT$) as a function of the number of layers of graphene, ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, respectively.

Figure 2

(a) HAADF image of a ${\mathrm{MoSe}}_2$ nanoflake. (b) Corresponding temperature map of ${\mathrm{MoSe}}_2$ at a nominal sample temperature of 573 K. (c) The overlayed image of (a) and (b) showing three zones defined by the different thickness (I, II, III). (d) The temperature distribution for each area.

Figure 3

(a) Calculated low-loss EEL spectra for single-layer graphene with different lattice constants. (b) The plasmon energy for each spectrum from (a) as a function of lattice constant $a$. (c) Measured in-plane TECs in thin films and bulk of graphene and TMDs, compared with reference data [40, 43, 47, 48]. The error bars are shown in blue and are calculated using the experimental uncertainty in determining the plasmon energy shift.

Figure 4

(a) HAADF image of ${\mathrm{MoSe}}_2$ at 623 K and the spatially resolved map of the local thermal expansion coefficient in the edge between double-layer (DL) and four-layer (QL) areas. (b) Line profile of thermal expansion coefficient of the interface indicated by the black line in (a). (c) Representative atomic-resolution HAADF image of ${\mathrm{MoS}}_2$ taken at 573 K. (d) Line profile of image contrast across several layers of ${\mathrm{MoS}}_2$ at 573 K.