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# Ultralow Thermal Conductivity in Disordered, Layered WSe2 Crystals

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Science  19 Jan 2007:
Vol. 315, Issue 5810, pp. 351-353
DOI: 10.1126/science.1136494

## Abstract

The cross-plane thermal conductivity of thin films of WSe2 grown from alternating W and Se layers is as small as 0.05 watts per meter per degree kelvin at room temperature, 30 times smaller than the c-axis thermal conductivity of single-crystal WSe2 and a factor of 6 smaller than the predicted minimum thermal conductivity for this material. We attribute the ultralow thermal conductivity of these disordered, layered crystals to the localization of lattice vibrations induced by the random stacking of two-dimensional crystalline WSe2 sheets. Disordering of the layered structure by ion bombardment increases the thermal conductivity.

Materials with the lowest thermal conductivity are typically found among electrically insulating amorphous solids and glasses. In these materials, heat conduction is adequately predicted by a simple phenomenological model, the minimum thermal conductivity, in which heat conduction is described by a random walk of vibrational energy on the time and length scales of atomic vibrations and inter-atomic spacings (1). More sophisticated theories of heat conduction in disordered materials support this description: A majority of the vibrational modes [termed “diffusons” by Allen and Feldman (2)] carry heat in this manner, and only a small fraction of the vibrational modes propagate as waves or are localized and therefore unable to contribute to heat conduction. Recently, we (3) and others (4) have shown that the minimum thermal conductivity can be circumvented in multilayer thin films of metals and oxides. When the spacing between the interfaces is only a few nanometers, the thermal resistance of the interfaces reduces the thermal conductivity far below the thermal conductivity of the homogeneous amorphous oxide.

Here, we demonstrated by both experiment and computer simulation an alternative route for achieving ultralow thermal conductivity in a dense solid. The thermal conductivity of disordered thin films of the layered crystal WSe2 can be as small as 0.05 W m–1 K–1, a factor of 6 smaller than the predicted minimum thermal conductivity and, to the best of our knowledge, the lowest thermal conductivity ever observed in a fully dense solid. Disruption of the layered structure and the crystallinity of the WSe2 sheets by ion irradiation actually produces a marked increase in the thermal conductivity of the thin film. Thus, the lowest thermal conductivities are not found in the fully amorphous form of WSe2; rather, ultralow thermal conductivity is achieved by controlling both order and disorder, and hence the thermal pathways, in this anisotropic material.

We synthesized WSe2 thin films by the modulated elemental reactants method (5, 6). Sequential bilayers of W and Se were deposited in an ultrahigh vacuum chamber onto unheated Si (100) wafers with a stoichiometry of 1:2 and then annealed for 1 hour at elevated temperatures in N2 atmosphere to form the desired layered structures (6). The microstructure of the films was stable at room temperature. In the WSe2 structure, a hexagonal plane of W atoms is bonded to two Se layers by strong covalent-ionic bonds, and each two-dimensional (2D) WSe2 sheet is bonded to adjacent sheets by weaker van der Waals forces (7, 8). We purchased a single crystal of WSe2 from Nanoscience Instruments to provide a baseline for comparisons. Thermal conductivity was measured by time-domain thermoreflectance (TDTR) (6, 912). In our implementation of TDTR, we determine the thermal conductivity by comparing the time dependence of the ratio of the in-phase Vin and out-of-phase Vout signals from the radio-frequency lock-in amplifier to calculations made with the use of a thermal model (11). The thermal model has several parameters, but the thermal conductivity of the WSe2 sample is the only important unknown.

We used synchrotron x-ray diffraction to characterize the microstructure of a typical WSe2 film (Fig. 1). Data for (0 0 L) reflections (6) showed that the layering of the 2D WSe2 sheets was very precise; the surface normal to each sheet (hexagonal c axis) was well aligned with the surface normal of the substrate, and the spacing between the centers of WSe2 sheets was highly uniform at 0.66 nm. These highly textured films had completely random crystalline orientation in the a-b plane. We examined the crystalline structure of the film by scanning the diffraction intensity through reciprocal space where the (1 0 3) reflection intersected the Ewald sphere. The relatively narrow line width (0.06 in reciprocal lattice units) in the direction parallel to the surface, [h 0 3], gave a lateral coherence length of 23 nm (Fig. 1C). Scans through the intersection of (1 0 L) reflections with the Ewald sphere probed the coherence of the crystal structure along the direction normal to the WSe2 sheets. The large line widths (Fig. 1B) indicated that crystallographic ordering in the stacking of the WSe2 sheets was limited to < 2 nm.

We next compared the thermal conductivity of annealed WSe2 films to the conductivity of a single crystal of WSe2 and the predicted minimum thermal conductivity (Fig. 2). The thermal conductivity of single-crystal WSe2 was approximately proportional to 1/T (the reciprocal of absolute temperature), as expected for a dielectric or semiconductor in which heat transport is dominated by phonons with mean-free paths limited by anharmonicity. Calculations of the minimum thermal conductivity require knowledge of the number density of atoms and the speed of sound (1). We used picosecond acoustics to measure the longitudinal speed of sound in the cross-plane direction of nominal 360-nm-thick films and found that vL = 1.6 nm ps–1 (13, 14). This measurement is in good agreement with an independent measurement of the same film (vL = 1.7 nm ps–1) with the use of picosecond interferometry (15) and an index of refraction at the laser wavelength of 800 nm of n = 4.13. If we use the average of these values, vL = 1.65 nm ps–1, and a mass density of ρ = 9.2 g cm–3, we obtain an elastic constant C33 = 25 GPa, which is approximately a factor of 2 smaller than C33 for single crystals of NbSe2 and TaSe2 measured by neutron scattering (16) and single-crystal WSe2 measured by picosecond interferometry. The transverse speed of sound vT is not accessible to the standard methods of picosecond acoustics; instead, we estimated vT = 1.15 based on our measurement of vL and the ratio C44/C33 previously measured for NbSe2 and TaSe2 (16).

The lowest thermal conductivity, Λ, measured at 300 K is Λ = 0.048 W m–1 K–1 for a 62-nm-thick WSe2 film, 30 times smaller than the cross-plane thermal conductivity of a single-crystal sample of WSe2 (Fig. 2) and a factor of 6 smaller than the predicted minimum thermal conductivity. This degree of deviation from the predicted minimum thermal conductivity in a homogeneous material is unprecedented (17). Notably, the conductivity of the 62-nm-thick film is smaller than the conductivity of a thinner film (24 nm) or a thicker film (343 nm). The reasons for these differences are not understood in detail, but we speculate that variations in the degree of crystallographic ordering along the thickness of the films are playing an important role.

The data shown in Figs. 1 and 2 lead us to conclude that the ultralow thermal conductivities are produced by random stacking of well-crystallized WSe2 sheets. To test this idea, we used irradiation by energetic heavy ions to disrupt the crystalline order in the thin film samples (Fig. 3). Because our TDTR measurements require knowledge of the thermal conductivity of the substrate, bare silicon substrates were irradiated with the same range of ion fluences and measured by TDTR (6). At the highest ion dose, 3 × 1015 ions cm–2, we observed a factor of 5 increase in the thermal conductivity of the WSe2 film. This increase in thermal conductivity with ion beam damage is also unprecedented. We inferred from these experiments that ion-induced damage introduces disorder that reduces localization of vibrational energy and enhances the transfer of vibrational energy in the material.

To gain further insight and confidence in our experimental results, we performed molecular dynamics (MD) simulations on model structures. For simplicity and computational efficiency, the atomic interactions in our model compound are described by 6-12 Lennard-Jones potentials: $Math$(1) where ϵ is the energy scale and σ is the length scale. Two sets of ϵ and σ parameters were used: For interactions within a single WSe2 sheet, ϵ = 0.91 eV and σ = 2.31 Å, and for the interaction between layers, ϵ = 0.08 eV and σ = 3.4 Å. These parameters achieved a good fit to WSe2 crystal structure and the C11 (200 GPa) and C33 (50 GPa) elastic constants. For computation efficiency, a cutoff of 5.4 Å was used, with both energy and forces shifted such that they were zero at the cutoff (18). The cross-sectional area of the simulation cell is 15.3 × 13.3 Å. Along the (001) direction, two sizes were selected: 160 and 320 Å. Periodic boundary conditions were used for all directions. Newton's equations of motions were solved by the fifth-order predictor corrector algorithm (18) with an MD time step, Δt = 1.8 × 10–15 s.

The simulation cell for the thermal transport measurement is depicted in Fig. 4. To calculate the thermal conductivity, the we first equilibrated the structure at T = 300 K for 100,000 MD time steps. Next, the global thermostat was turned off and thermal energy was added to one WSe2 sheet and removed from a second sheet, which was located at a distance from the first sheet equal to one-half of the size of the simulation cell along the (001) direction (19, 20). Atomic velocities were scaled such that heat was added or subtracted at a constant rate, 10–6 eV per MD time step (21). We monitored the temperature profile by averaging the kinetic energy of atoms in each WSe2 sheet. Because of the small energy barrier for shearing of the WSe2 structure and the small cross-sectional area of the simulation cell, our model structures exhibited thermally excited local shearing events leading to disorder in the stacking of the WSe2 sheets (Fig. 4).

After 5 to 20 million MD steps (depending on the system size), a steady-state temperature distribution was established (Fig. 4). The temperature gradient, and thus the thermal conductivity, of 16- and 32-nm-long simulation cells were essentially the same within the statistical standard deviation of 10%, Λ = 0.06 W m–1 K–1. Given the approximate form of the potentials used in our computational work, the agreement between the measured and calculated thermal conductivities was better than we expected. Nevertheless, the low thermal conductivity of the model structure suggests that the ultralow thermal conductivity in disordered, layered crystals is a general phenomenon and not restricted to WSe2.

Our WSe2 films are poor electrical conductors in the cross-plane direction; however, if semiconductors with similar structural features and good electrical mobility can be identified, disordered layered crystals may offer a promising route to improved materials for thermoelectric energy conversion.

Supporting Online Material

Materials and Methods

Figs. S1 to S5

References

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