Anisotropic in-plane thermal conductivity of black phosphorus nanoribbons at temperatures higher than 100 K

Black phosphorus attracts enormous attention as a promising layered material for electronic, optoelectronic and thermoelectric applications. Here we report large anisotropy in in-plane thermal conductivity of single-crystal black phosphorus nanoribbons along the zigzag and armchair lattice directions at variable temperatures. Thermal conductivity measurements were carried out under the condition of steady-state longitudinal heat flow using suspended-pad micro-devices. We discovered increasing thermal conductivity anisotropy, up to a factor of two, with temperatures above 100 K. A size effect in thermal conductivity was also observed in which thinner nanoribbons show lower thermal conductivity. Analysed with the relaxation time approximation model using phonon dispersions obtained based on density function perturbation theory, the high anisotropy is attributed mainly to direction-dependent phonon dispersion and partially to phonon–phonon scattering. Our results revealing the intrinsic, orientation-dependent thermal conductivity of black phosphorus are useful for designing devices, as well as understanding fundamental physical properties of layered materials.


Supplementary
Surface element analysis using nano-AES Auger spectra collected from the surface of a 15 min-air-exposed BP flake. AES equipped with a field emission electron source enables nanoscale chemical analysis at ultra-high vacuum (below 10 -10 mbar). Barely visible oxygen peak completely disappears after a mild Ar + milling, while the carbon peak which might come from physisorbed organic species still remains, implying that the small amount of oxygen is mainly attributed to physically adsorbed oxygen species at the surface, because removing oxygen from oxidized phosphorus may need more energy. Therefore, it is believed that surface oxidation of the BP sample is negligible. The carbon contamination was also observed to be surface-limited by further sputtering. Figure 6│Electrical conductivity of BP nanoribbon Temperature dependent electrical conductivity (σ, left axis, logarithmic scale) of an AC nanoribbon along the AC direction, and corresponding electronic thermal conductivity (right axis), estimated by the Wiedemann-Franz law with the Sommerfeld value of Lorenz number (L 0 = 2.45 × 10 -8 WΩK -2 ). The σ in the AC direction is known to be ~one order of magnitude higher than that in the ZZ direction. 1,2 Electrical conductivity increases as temperature increases. I-V curves are always linear in the investigated temperature range. No size dependence was observed in the electrical conductivity of the BP nanoribbons.  2 σ) of an AC (t=170 nm), and a ZZ (t=200 nm) nanoribbons. One device for each direction was measured. The ZT (= 2 / ) of these nanoribbons are ~0.0036 (AC) and ~0.0006 (ZZ) at room temperature. This low ZT is obviously due to the relatively high κ (single-crystalline samples) and low σ (undoped samples).  3 For successful growth of high quality BP bulk crystal, the Sn to SnI 4 ratio is the most critical factor. When the weight ratio was kept to be 1.9~2.1, the largest and cleanest BP crystals was synthesized. In a typical growth process, 20 mg of SnI 4 , 40 mg of Sn, and 1 g of red phosphorus was mixed in a silica glass ampoule (15 cm in length and 1.14 cm in diameter) and evacuated to a low pressure (~1×10 -5 Torr). Synthesis was carried out in a three-zone Lindberg furnace using 1-inch diameter quartz tube. To facilitate the growth, the empty side of the ampoule was set to 50-75 °C below the growth temperature (~700 °C). The furnace was set to 700 °C (ramp time ~3 hr) and kept at this temperature for 3 hr. Then, the ampoule was cooled down to 560 °C in 10 hr, followed by natural cooling down to room temperature. During the natural cooling step, dark orange and red fumes, associated with SnI 4 and red phosphorus, were formed at the colder end. Finally, large shiny BP crystals were formed towards the cold end of the ampoule, well separated from Sn-rich Sn-phosphites, redphosphorus, and SnI 4-x deposits. The BP bulk crystals synthesized following this method is comprised of bundle of axially ZZ-oriented crystallites. The crystal structure, composition, and morphology of the crystallites were carefully investigated as described in the main text. Based on our temperature dependent Raman measurements and purity tests performed by Rutherford Backscattering (RBS), this technique yields higher purity crystals with sharp Raman features (FWHM <~5cm -1 ) compared to other commonly used techniques such as white phosphorus to BP conversion at high pressures. 4,5

Supplementary Note 2│ Device structure and measurements of thermal conductivity
For the thermal conductance (K S ) and electrical conductance (G) measurements, suspended-pad microdevices (Supplementary Fig. 2a) were used. Two suspended SiN x pads, each supported by six SiN x arms, are bridged by a nanoribbon, through which heat will flow from the hot pad (Pad 1) to the cool pad (Pad 2). A Pt micro-heater/thermometer (R Pt ~ 2 kΩ) was patterned on each pad to heat up or to sense the temperature of each pad. The arms are also covered with Pt (R arm ~ 1 kΩ each) for electrical reading. The global temperature (T g ) is controlled by an external electrical heater and a cryogenic cooler which are connected to the sample holder. K S and G of nanoribbons were measured simultaneously inside a vacuum chamber (<10 -6 Torr). G was measured by using the four-probe method, and K was measured following the steps described below.
When a DC current (I h ) is applied to the micro-heater on Pad 1 (Supplementary Fig. 2b), the heat Q generated by the joule heating raises the temperature of Pad 1 by ΔT h . The heat will transfer to the substrate through the arms (P 2 ), and to Pad 2 through the sample (P 1 ), then to the substrate through the arms of Pad 2 (P 3 ). In steady state, one can write the following equations: where K and K arm is the thermal conductance of the sample and the arm, respectively, and n is the number of arms. Therefore, , where can be obtained by = h 2 × ( Pt + arm ), and ΔT h and ΔT s can be obtained by measuring the resistance of the Pt micro-heater/thermometer on each pad because the temperature coefficient of resistance of Pt is calibrated. An AC current (amplitude < 500 nA) is applied to measure the resistance change of the micro-heater/thermometer. In order to calculate the temperature of each pad, we use the resistance slope (ΔR/ΔT, where ΔT = 10 K) at each global temperature as shown in Supplementary Fig. 3. All the K measured in this study has < 5% error. 6 Finally, the total thermal conductivity κ is obtained considering the geometric factor as κ = K×l/A, where A is the cross-sectional area, and l is the length of the nanoribbon.
Thermal conductivity measurement of BP above 350 K is not able using this method, because decomposition of BP begins at around 400 K at the low pressure (< 10 6 Torr) of the chamber.

Supplementary Note 3│ Anisotropy in density of states of phonons, and fitting parameters
The temperature dependent κ is calculated based on the relaxation-time approximation model, Eq.(1) of the main text. When counting the phonon states for specific heat, we neglected the effect of anisotropy in density of states at the edge of Brillouin zone, because these phonons' contribution (with large wavevectors) to the total κ is relatively small, due to their small near the Brillouin zone boundary. This assumption must be limited to the high temperature regime where most phonon states are excited. We checked it by calculating the phonons' contribution in one third of Brillouin zone (2/3 q max < q < q max ) in the ZZ direction. (The phonon frequency range is relatively small in this wavevector regime, even though the wavevector range is large.) These phonons' contribution to the total is ~10% at room temperature.
From the fitting to the κ-T data in Fig. 3b-c (in main text), we obtained the parameters B 2 (related to the phonon-phonon scattering) and A i (related to the impurity scattering). The fitting parameters are obtained by fitting the ZZ nanoribbon with thickness 310 nm and AC nanoribbon with thickness 270 nm. After that, we applied the fitting parameters to thinner nanoribbons with the thickness 170 nm (with width 540 nm and 590 nm for ZZ and AC, respectively) to check consistency.
Difference in boundary scattering is less than 5% for the 170 nm-thick ZZ and AC nanoribbons. (The boundary of 220 nm and 210 nm are used for ZZ and AC nanoribbons respectively.) For the impurity fitting parameter A i , they are close to each other in ZZ and AC directions. Since the impurity scattering are not orientation-dependent, we keep the A i the same in the AC direction after we obtain it from the ZZ direction. In addition, the anisotropy is strong in the high temperature regime, where the impurity scattering is not the major mechanism of the phonon scattering. For B 2 as the key phonon-phonon scattering parameter, it is empirically close to ~/3 at low T. However, Slack showed that 2 ≈ when / ≈ 1, and 2 > when / > 1; i.e.,