Weak phonon scattering effect of twin boundaries on thermal transmission

To study the effect of twin boundaries on thermal transmission, thermal conductivities of twinned diamond with different twin thicknesses have been studied by NEMD simulation. Results indicate that twin boundaries show a weak phonon scattering effect on thermal transmission, which is only caused by the additional twin boundaries’ thermal resistance. Moreover, according to phonon kinetic theory, this weak phonon scattering effect of twin boundaries is mainly caused by a slightly reduced average group velocity.

Scientific RepoRts | 6:19575 | DOI: 10.1038/srep19575 1846 W/mK, which is about 87.7~96% of our calculated thermal conductivity for the bulk perfect diamond K b-per (1920 W/mK). It is well known that the effect of twin boundaries on mechanical properties can be comparable with that of conventional grain boundaries 8 , similarly, to explore the effect of twin boundaries on thermal transmission, bulk twinned diamond thermal conductivity has been compared with that of nanocrystalline diamond in our previous work 6 . By comparison, as shown in Fig. 1 (b), when the grain size of nanocrystalline diamond ranges from 3.6 to 11.1 nm, the thermal conductivity of the nanocrystalline diamond K nano increases from 10.2 to 28.3 W/mK 6 , and these values are only about 0.5% to1.5% of bulk perfect diamond thermal conductivity, much smaller than bulk twinned diamond thermal conductivity with similar sizes of twin thickness. Despite significant difference between the distribution of grain boundaries in the twinned and nanocrystalline diamond, since thermal transmission is only affected by grain boundaries perpendicular to heat flux 13 , and anisotropy of structures shows low effect on thermal transmission 6 , this great difference between the bulk twinned and the nanocrystalline diamond thermal conductivity is mainly caused by the different structures of twin boundaries and conventional grain boundaries. Therefore, it can be concluded that although the twin thicknesses in twinned diamond is in the same order of magnitude with the grain sizes of nanocrystalline diamond, the degree of the effects respectively caused by twin boundaries and conventional grain boundaries on heat transmission is substantially different.
Due to tandem properties of thermal transmission 16 , thermal resistance of twinned diamond can be divided into two tandem parts: one is intragranular part, and the other is intergranular part (i. e. twin boundaries), that is where L/K td is the thermal resistance (m 2 K/W) for twinned diamond with a model size of L (meter). L/K intra and R inter are the thermal resistances (m 2 K/W) of intragranular and intergranular parts for this structure, respectively. In Eq. (1), the intergranular thermal resistance R inter is the total thermal resistances of all twin boundaries. In our analysis, twin boundary thermal resistances are considered as independent ones which are placed in series. Thus, R inter can be described as where D is the twin thickness, and R K is the thermal resistance (m 2 K/W) for one twin boundary. By combining Eq.(1) and Eq.(2), we can obtain Because of the coincident radial distribution functions of twinned diamond and perfect diamond (as shown in supplementary Figure S1), it can be deduced that the C-C bond lengths of the twinned and perfect diamond are similar, and further the phonon-phonon scattering process in the intragranular part of twinned diamond is the same as that in the perfect diamond. Therefore, the intragranular thermal conductivity K intra for twinned diamond can be considered as perfect diamond thermal conductivity. Since 1/K td and 1/K per are both parameters linearly related to 1/L, Eq. where R b−K is the twin boundary thermal resistance in bulk twinned diamond. Moreover, a linear relationship in twinned diamond between R K /D and 1/L can be easily deduced as where P is the slope of intergranular thermal resistance (R K /D) for the inverse of model size (1/L), and Q is the intercept of this regression line which is used to describe the linear relationship between R K /D and 1/L. Thus, for a twinned diamond with a certain twin thickness D, it also keeps a linear relationship between R K and 1/L. The twin boundary thermal resistance R K can be calculated by the following equation 17 where Δ T is the temperature drop across the twin boundary, and J is the introduced heat flux.
To calculate twin boundary thermal resistance R K , simulations of Δ T have been conducted on twinned diamond with different model sizes (L), and results for Δ T in twinned diamond with D = 9.9 nm has been shown in Figure S2. It can be seen that with increasing model size, Δ T are decreased. Same trends can be found in twinned diamond with other twin thicknesses (D) as well (such as 3.72 nm and 7.44 nm). By using Eq. (7), calculated results for R K at different model sizes (L) are plotted in Fig. 2. The predicted linear relationship between twin boundary thermal resistance R K and inverse of model size 1/L can also be seen from Fig. 2.
The twin boundary thermal resistance in bulk twinned diamond (R b-K ) can be obtained by extrapolating 1/L to 0, and in the range of our computation accuracy, R b-K remains a constant despite different twin thicknesses, as shown in Fig. 2. Its value is about 2 × 10 −13 m 2 K/W, three orders of magnitude lower than the conventional grain boundary thermal resistance in nanocrystalline diamond (1.43 × 10 −10 m 2 K/W) 6 , thus the effect of twin  boundaries on thermal transmission is much weaker than that of the conventional grain boundaries, and this phenomenon has already been verified by Aubry et al. about the grain boundary Kaptiza resistance in silicon 18 .
According to Eq. (3), for bulk twinned diamond with a unit length (1 meter), the intragranular thermal resistance R b-intra (or 1/K b-intra ) can also be calculated by removing the intergranular thermal resistance R b-inter (or R b-K /D) from the bulk twinned diamond thermal resistance R b-td (or 1/K b-td ). The plots of calculated intergranular and intragranular thermal resistance for bulk twinned diamond with different twin thickness D have been shown in Fig. 3. With increasing twin thickness, the intergranular thermal resistances R b-inter are decreased due to the decreasing number of twin boundaries, while the intragranular thermal resistances R b-intra are almost invariable. Moreover, they are all very close to the thermal resistance of the bulk perfect diamond with a unit length 1/K b-per (5.2 × 10 −4 m 2 K/W), which implies that the size effect on intragranular thermal resistances in bulk twinned diamond is practically negligible, and this result further confirms that the intragranular thermal conductivity K b-intra is equal to the bulk perfect diamond thermal conductivity K b-per .
Moreover, Eq. (5) can also be applied in thermal conductivity calculation for other bulk twinned semiconductors. For one kind of bulk twinned structure materials, its thermal conductivity can be easily obtained from its twin thickness D, its corresponding bulk perfect crystal thermal conductivity K b-per , and its twin boundary thermal resistance R b-K in the bulk twinned structure.
For nanocrystalline diamond, both grain sizes and grain boundaries have played a rather significant role in its thermal transmission. But for bulk twinned diamond, sizes of twin thickness almost have no effect on its thermal transmission. In order to provide some insight into thermal transmission mechanisms, the separate contributions from the intergranular thermal resistance R b-inter and intragranular thermal resistance R b-intra to the bulk twinned diamond (1 meter) thermal resistance R b-td have also been analyzed. The former contribution can be represented by the ratio between the intergranular thermal resistance to the bulk twinned diamond thermal resistance R b-inter / R b-td , and the latter contribution can be represented by the ratio between the intragranular thermal resistance to the bulk twinned diamond thermal resistance R b-intra /R b−td . Calculated results can be found in the inserted table of Fig. 3. With the increasing twin thickness, the value of R b-inter /R b-td is decreased, while R b-intra /R b-td is increased. When the twin thickness is increased to 9.92 nm, R b-intra /R b-td is 0.96, and R b-inter /R b-td is only 0.04, in which case the thermal conductivity is very close to the perfect crystal, and the effect of twin boundaries on thermal transmission can be almost ignored. This critical value (9.92 nm) in bulk twinned diamond is much smaller than that in the nanocrystalline diamond 6 (about 10000 nm), and this inconformity in these two structures also suggests a much weaker phonon scattering effect of twin boundaries than that of conventional grain boundaries.
Since phonons are the primary heat carriers in semiconductors 19 , the weak phonon scattering effect of twin boundaries on thermal transmission has also been analyzed by phonon kinetic theory 4 , which describes the thermal conductivity as where C is the constant volume specific heat capacity, v is the average phonon group velocity, l is the phonon mean free path, and τ is the characteristic relaxation time associated with the phonon scattering process. For comparison, phonon transmissions in both twinned diamond (D = 0.62 nm) and the perfect diamond are analyzed by using phonon kinetic theory. In this work, the heat capacities at 300 K are calculated (refer to Supplement discussion 2) for these two structures. As shown in Figure S3, D-value between perfect diamond heat capacity and twinned diamond heat capacity is only −0.02 J/K·mol, and calculated heat capacities of the two structures are both about 6.3 J/K·mol. The average phonon group velocity 20 can be calculated by the following formula: is the gradient between the frequencies and k points of both the acoustic and optical branches in the phonon spectra (Fig. 4).The calculated average phonon group velocities of the twinned diamond v td and the perfect diamond v per are about 4721 m/s and 4953 m/s, respectively. The ratio between v td 2 and v per 2 is about 90%. It can be seen that twin boundaries can slightly reduce the average phonon group velocity. Because of a proportional relationship between the relaxation rate (τ −1 ) and the square of Grüneisen parameter in the phonon-phonon scattering process, as described by the relaxation time (τ) in twinned and perfect diamond can been deduced from the Grüneisen parameter (γ) 21 , which, for each mode i and frequency ω, is defined as the logarithmic derivative of the phonon angular frequency ω with respect to the volume V of the crystal. In this paper, the mode-and frequency-averaged Grüneisen parameters (γ) for the twinned and perfect diamond have been calculated from the first-principles phonon calculations at three different volumes 22 , and their values are 0.79 and 0.78, respectively. Further, according to Eq. (10), it can be deduced that the relaxation time of the twinned diamond is about 98% of that for the perfect diamond. It, therefore, can be concluded that the weak phonon scattering effect of twin boundaries on thermal transmission is caused by both the slightly reduced average phonon group velocity and relaxation time, and the main contribution is arising from the former. According to Lu's conclusion of lower electrical resistivity of twin boundaries than that of the conventional grain boundaries 8 , our results about weak phonon scattering effect of twin boundaries can be further proven, due to the similar transport properties of phonons and electrons 23 . While in Li's study about twin boundaries' effect on thermal transmission of nanotwinned ferroic films, they compared the thermal conductivity of a limited size nanotwinned ferrioc film with that of the perfect ferrioc film, a great reduction of thermal conductivity was thus obtained. They contributed this reduction to the strong phonon scattering of twin boundaries. In fact, in their work, this significant effect of twin boundaries on thermal transmission was caused by the coupling action of twin boundaries scattering and boundary scattering. In summary, NEMD simulations have been performed to calculate the thermal conductivity of bulk twinned diamond with different twin thicknesses (0.62~9.92 nm) in order to study the effect of twin boundaries on thermal transmission. Our calculated results indicate that although the bulk twinned diamond thermal conductivity is reduced due to the existence of twin boundaries, twin boundaries show a weak phonon scattering effect on thermal transmission. With the increase of twin thickness, the bulk twinned diamond thermal conductivity is increased. The reduction of the thermal conductivity is only caused by the additional intergranular (twin boundaries') thermal resistance, which is much smaller than that of conventional grain boundaries (three orders of magnitude lower than the conventional grain boundaries' thermal resistance). The intragranular thermal resistance in bulk twinned diamond is the same as that of the bulk perfect diamond. According to phonon kinetic theory, this weak phonon scattering effect of twin boundaries is caused by the slightly reduced average phonon group velocity and relaxation time, and the former makes a primary contribution.

Methods
In order to study the phonon scattering effect of twin boundaries on thermal transmission, a series of cuboids' [111]-oriented twinned diamond and perfect diamond models have been built. The cross section of these models is 1.75 × 1.25 nm along [110] and [112] directions, respectively, and the model length along [111] direction ranges from 30 to 150 nm. The twinned diamond contains a series of parallel Σ ( ) 3 111 twin boundaries, and Figure S4 (a) is the atomic arrangements of this boundary. A schematic diagram of twinned diamond is shown in Figure S4   and the twin thickness D ranges from 0.62 to 9.92 nm. In this work, thermal conductivity calculation is performed by NEMD simulations 24,25 (refer to Supplement discussion 3) implemented in LAMMPS code 26 , and atomic configurations are visualized by using OVITO package 27 . In our simulated scheme, C-C bonding interactions are described by Tersoff potential 17 , and periodic boundary conditions are imposed in all three directions. The pressure set up is an atmospheric pressure and temperature used is 300 K. The atomic structures of the twinned and perfect diamond are first optimized in an isothermal-isobaric (NPT) ensemble by using the gradient-based minimization method implemented to minimize the stress for 2× 10 5 steps with a time step of 0.1 fs. After that, a heat flux is imposed on the relaxed structures for 5× 10 6 steps in a micro-canonical (NVE) ensemble to allow the systems' temperature distribution reaching a steady state. Temperature profiles are obtained by averaging temperatures of simulated atoms in divided slabs 28 every 1× 10 6 steps. Finally, thermal conductivity can be calculated from the heat flux and temperature gradient by following the Fourier's heat conduction law 29 .