Thermal Conductance of the 2D MoS2/h-BN and graphene/h-BN Interfaces

Two-dimensional (2D) materials and their corresponding van der Waals heterostructures have drawn tremendous interest due to their extraordinary electrical and optoelectronic properties. Insulating 2D hexagonal boron nitride (h-BN) with an atomically smooth surface has been widely used as a passivation layer to improve carrier transport for other 2D materials, especially for Transition Metal Dichalcogenides (TMDCs). However, heat flow at the interface between TMDCs and h-BN, which will play an important role in thermal management of various electronic and optoelectronic devices, is not yet understood. In this paper, for the first time, the interface thermal conductance (G) at the MoS2/h-BN interface is measured by Raman spectroscopy, and the room-temperature value is (17.0 ± 0.4) MW · m−2K−1. For comparison, G between graphene and h-BN is also measured, with a value of (52.2 ± 2.1) MW · m−2K−1. Non-equilibrium Green’s function (NEGF) calculations, from which the phonon transmission spectrum can be obtained, show that the lower G at the MoS2/h-BN interface is due to the weaker cross-plane transmission of phonon modes compared to graphene/h-BN. This study demonstrates that the MoS2/h-BN interface limits cross-plane heat dissipation, and thereby could impact the design and applications of 2D devices while considering critical thermal management.


COMSOL simulation
Considering the non-uniform heat flux across the MoS2 (graphene)/ h-BN and the influence from the contacts between MoS2 (graphene) and the electrodes, we performed a threedimensional COMSOL simulation to correct the thermal interface conductance (G) calculated in equation (1) and (2) in the main text, which is based on the assumption of uniform heat flux. The idea is to iteratively obtain a final value of G that makes the local temperature differences across heterostructure equal to that is measured in our experiments (Figure. S1). The simulation is built based on the real geometries and contact resistances of the samples. The input parameters are listed in Table. S1. Considering the contact resistance, we assigned the total Joule heating power P0 to the channel and the contacts based on the resistance ratio R4p/R2p. That is, P0*( R4p/R2p) is assigned to be generated by the MoS2 or graphene channel, and P0*(1-R4p/R2p) is assigned to be generated by the contacts between metal and MoS2 or graphene. The results of this iterative process are listed in Table S2. The temperature profile of the MoS2/h-BN sample 1 is demonstrated in Figure  S2 as an example.

Bubbles and relationship with G
We find that bubbles are formed and almost unavoidable during the transfer process. To examine the morphology of the bubbles, AFM is carried out on both MoS2/h-BN and graphene/h-BN samples with the height images shown in Figure S3. Hence, we report an 'effective' value of thermal conductance, which includes any thermal conductance through the bubbles. The measured G depends on the interface quality as well as intrinsic thermal properties. To consider the relationship between G and bubble density for a fair comparison of G between different samples, we link the bubble density with G as follows.
a. Bubble density The bubble densities can be characterized with AFM. Typical morphology of the MoS2/h-BN and graphene/h-BN is shown in Fig. S3. Regions are defined as bubbles when their heights are larger than the roughness (root mean square Rq ~1.5 nm) of h-BN. The reason for choosing this cut-off height is that the roughness of sample surfaces comes from two parts: (1) bubbles between the 2D superstrate (either MoS2 or graphene) and the substrate h-BN and (2) PMMA residues on the top of the 2D superstrate from the transfer process (note that the bottom surface of the MoS2/graphene as well as the top surface of the h-BN of the heterostructures do not have any PMMA residue). Based on bearing analysis in the NanoScope Analysis software, which reveals how much of a surface lies above or below a given height, the bubbles take up to 28.7% and 13.9% for MoS2/h-BN and graphene/h-BN interfaces, respectively. It is to be noted that this estimate is conservative, as the AFM roughness includes any additional PMMA residue on top of the superstrate, which does not affect the thermal interface conductance between the superstrate and substrate.

b. Relationship between G and bubble density
With the bubble density defined as Abubble/Atotal, where A denotes 'area', the intrinsic thermal conductance Gintrinsic can be estimated based on the measured Gtotal by the relationship of where Atotal = Aintrinsic + Abubble is the total contacting area, including the good contact area Aintrinsic, and the bubble area Abubble.
We note that if the bubble gives a lower G than an intrinsic G, the reason of lower G in MoS2/h-BN than that in graphene/h-BN could be the bubble density. To clarify this, based on the bubble density of Abubble/Atotal from the AFM analysis above, even if we assume bubbles have a lower G than the intrinsic interfaces, for instance, Gbubble = 0.1*Gintrinsic, Gintrinsic can be estimated as 22.9 MW/m 2 K for MoS2/h-BN, which is still much smaller than 59.7 MW/m 2 K for graphene/h-BN. Actually, as the bubbles take up less than ¼ of the total contact area, the thermal conductance through the bubbles doesn't contribute substantially to the measured Gtotal. In the worst case where Gbubble = 0, we still have GMoS2/h-BN = 23.8 MW/m 2 K much smaller than Ggraphene/h-BN = 60.6 MW/m 2 K, confirming our measurement that intrinsically the MoS2/h-BN interface has a lower G than that of graphene/h-BN.

Electrical contact resistance
To obtain the electrical contact resistances, 2-probe/4-probe measurements were conducted ( Figure S4)

NEGF simulation
We simulate the heat flux between a semi-infinite superstrate of graphene (or MoS2) sheets and a semi-infinite substrate of h-BN sheets. The graphene layers are stacked in the A-B configuration while the h-BN and MoS2 are stacked in the A-A' configuration. In each structure, the interfacial area is 2.49×2.16 nm 2 . In the directions orthogonal to the direction of the heat flux, we impose periodic boundary conditions. The interatomic potentials in the structures are taken from Refs. [10][11][12] After optimizing the structures in GULP, 13 we compute their force constant matrices which we use to calculate the transmission spectrum and thermal conductance of the interface. The thermal conductance is given by the formula 14 where S is the interfacial area, N is the Bose-Einstein distribution at temperature T, and Ξ(ω) is the transmittance at frequency ω.