Engineering the thermal conductivity along an individual silicon nanowire by selective helium ion irradiation

The ability to engineer the thermal conductivity of materials allows us to control the flow of heat and derive novel functionalities such as thermal rectification, thermal switching and thermal cloaking. While this could be achieved by making use of composites and metamaterials at bulk length-scales, engineering the thermal conductivity at micro- and nano-scale dimensions is considerably more challenging. In this work, we show that the local thermal conductivity along a single Si nanowire can be tuned to a desired value (between crystalline and amorphous limits) with high spatial resolution through selective helium ion irradiation with a well-controlled dose. The underlying mechanism is understood through molecular dynamics simulations and quantitative phonon-defect scattering rate analysis, where the behaviour of thermal conductivity with dose is attributed to the accumulation and agglomeration of scattering centres at lower doses. Beyond a threshold dose, a crystalline-amorphous transition was observed.

the left two figures are the high resolution TEM of selected areas that are marked by a certain color square along the silicon nanowire, from which, the nanowire is seen to remain singlecrystalline rather than core/shell crystalline/amorphous-like structure after final annealing, which is corresponding to the thermal conductivity measurement of intrinsic silicon nanowire. The unavoidable intrinsic oxidation layer of the measured nanowires is ~2.5nm, which is similar to or even slightly smaller than that found in electroless-etching (EE) silicon

Supplementary Table 1 | Irradiation conditions for helium ion damaged Si nanowires.
All the irradiations were carried out under helium ion energy of 30-36.2keV and current of 0.2-0.7pA. The irradiated length (L 1 ) and distance (L 2 ) were different for each nanowire, as well as the irradiated dose.

Supplementary Note 1 | Damage created by helium ions
In The simulation shows that the damage created by helium ions (the red and green dots in Supplementary Fig. 2 (a)) is discrete rather than continuous. This is because of the light helium ions, which create point and point-like defects rather than large damage clusters, the latter typically resulting from heavy implanting ions 4 . Moreover, the defect profile is relatively spatially uniform (Supplementary Fig. 2 (b)), unlike the case for thick Si substrates, in which a well defined peak exists at the end of range. This is because of the small thickness of Si film and high ion beam energy. Lastly, most of the helium ions penetrate the thin film, leaving only 2% of the helium ions remaining inside the sample. Since each helium ion creates 33 vacancies on average, the number of residual helium atoms is much smaller than the number of damaged lattice sites.
From Supplementary Fig. 2 (b), the silicon vacancies induced by each helium ion are linear with its travelling routes and this relation can be approximated as where (θ) is the vacancy created per ion at unit length, (θ) is each helium ion actual travelling length and θ is the angle as shown in Fig. 2 (C).
The total vacancies for each ion are the integral of Equation 1, which is where C is a constant. Considering the case when (θ) is 0, there would be no vacancies generated, thus this formula is reduces to From Supplementary Fig. 2(C), the actual travelling length of each ion is And the element for integration in the horizontal direction, (θ)is then, Thus, assuming that 10% of the Si vacancies created by helium ions can survive, the number of vacancies remaining is a function of dose. Combining the equations 1-5, we obtain

Supplementary Note 2 | Calculation of the phonon -point defect scattering rate
According to kinetic theory of phonon gas, thermal conductivity (κ) can be calculated as where C, v, and τ are volumetric specific heat, group velocity, and phonon lifetime for the phonon mode with wavevector q and polarization p; N is the number of q-points.
The group velocity component perpendicular to the axis is calculated using the following formula for the phonon mode (q, p), where , , and , , are the x and y component of group velocity of phonon mode (q, p), which is computed using the following formula, The phonon lifetime is estimated from different phonon scattering mechanisms by using Matthiessen's rule, namely, where , , , and are phonon scattering rates (inverse of phonon lifetime) due to anharmonic phonon-phonon, phonon-isotope, phonon-boundary, and phonon-defect scattering, respectively. is widely assumed to be proportional to ω 2 . The isotope scattering rate is of the form, = , in which C is analytically calculated to be 1.32x10 -45 s 3 for silicon 5,6 . The boundary scattering rate, , is estimated by where is the group velocity perpendicular to the transport direction 7 , and d is the diameter of the Si nanowire (d=160 nm); while m represents the smoothness of surface of silicon nanowire, which is equal to 1 for silicon nanowire with a perfectly smooth surface, and 0 for a completely roughened surface with a fully diffusive boundary. As for the phonon-defect scattering rate, it follows the relationship = . Therefore, the total phonon lifetime can be modelled as where A and D are fitting parameters. In our fitting procedures, the phonon dispersion is computed from first principle density functional theory as implemented in Quantum We firstly fit the parameter A from the measured thermal conductivity of undamaged Si nanowire. The best fitting gives rise to 5.39×10 -17 s for A, which is close to widely used value for bulk silicon, i.e., 3.28×10 -17 s 6 . Afterwards, the parameter D is fitted for each helium ion dose. We do two different fits-one with and one without the boundary scattering term. The results are shown in Figure 3(b) in the main text.

Supplementary Note 3 | Non-equilibrium molecular dynamics (NEMD) simulations for effect of point defects
In the simulation, Si nanowires with various cross-sections of 3×3, 6×6 and 9×9 unit cells  Fig. 6). The Si nanowire with vacancies is modeled by randomly removing some Si atoms (up to 20%) and then removing the single-bonded atom pairs ( Supplementary Fig. 7). For a given vacancy concentration, the final results are averaged over eight different realizations.