Engineering thermal conductance using a two-dimensional phononic crystal

Abstract

Controlling thermal transport has become relevant in recent years. Traditionally, this control has been achieved by tuning the scattering of phonons by including various types of scattering centres in the material (nanoparticles, impurities, etc). Here we take another approach and demonstrate that one can also use coherent band structure effects to control phonon thermal conductance, with the help of periodically nanostructured phononic crystals. We perform the experiments at low temperatures below 1 K, which not only leads to negligible bulk phonon scattering, but also increases the wavelength of the dominant thermal phonons by more than two orders of magnitude compared to room temperature. Thus, phononic crystals with lattice constants ≥1 μm are shown to strongly reduce the thermal conduction. The observed effect is in quantitative agreement with the theoretical calculation presented, which accurately determined the ballistic thermal conductance in a phononic crystal device.

Figure 1. The 2D membrane PnC device.

(a) Schematic representation of a perforated membrane PnC geometry with a square array of circular holes, fabricated by e-beam lithography. The central region has a heater, which emits thermal phonons into the PnC structure. (b) A false colour scanning electron micrograph (s.e.m.) of the central region of the larger period a=2425, nm PnC sample. The blue (Al) and yellow (Cu) lines are the metallic wiring which form the heater and thermometer elements at the centre of the PnC. The shorter period sample has the same wiring locations and dimensions. The full size of the perforated membrane is 100 μm × 100 μm. (c) A scanning electron micrograph of a region of the shorter period sample (black areas, empty space, grey areas SiN membrane), showing the unit cell size 970 nm × 970 nm and the width of the narrowest region ~60 nm. The sidewalls have a slight angle so that the bottom end of the hole is slightly smaller. Scale bar has length 200 nm. (d) A false colour s.e.m. of the heater/thermometer structure, consisting of a Cu normal metal wire (yellow) sandwiched between two normal metal-insulator-superconductor (NIS) tunnel junctions, connected to the measurement circuit by superconducting Al leads (blue).

Figure 2. Band structure calculations using FEM.

(a) A representative unit cell used in the FEM calculations with hole-filling factor f=0.7. A typical mesh structure is also shown. (b) The first BZ of the square PnC lattice in k-space (orange), with the irreducible zone (where calculations are performed) inside the triangle with corners at points Γ, X and M. One typical set of evenly divided k-points is also indicated. Other, uneven divisions were used, as well. (c) The scaled width of the phononic band gap in SiN (empty region in angular frequency ω) as a function of the ratio of the thickness of the membrane d and the lattice constant a. The red crosses show the chosen design values.

Figure 3. Band structure DOS and group velocity.

Dispersion relations (band structure) in the main symmetry directions of the BZ for the SiN (a) full membrane (d=485 nm), (b) square lattice PnC, with f=0.7 and a=970 nm, and (c) square lattice PnC, with f=0.7 and a=2425, nm. Complete band gap is observable at 3.3 GHz for the PnC with a=970 nm. (d) The corresponding densities of states (PnC a=970 nm, red, PnC a=2425, nm, blue, full membrane, black) with 2D (pink dash) and 3D (grey dash) Debye models. (e) Average group velocity (averaged over the whole 2D BZ) for full (black) and the two PnC membranes (PnC a=970 nm, red, PnC a=2425, nm, blue). The PnC DOS and group velocity curves have been smoothed for visual clarity.

Figure 4. Measured emitted phonon power versus temperature.

Grey squares show the data for the full, uncut membrane, red circles (a=970 nm) and blue triangles (a=2425, nm) the data for the two square PnC samples. The theoretical lines are given by Equation (1), using the computed band structures for each case, with (solid lines) or without (dashed lines) a phonon back radiation power from the substrate at bath temperature (60 mK for full membrane and PnC with a=970 nm, 80 mK for PnC with a=2425, nm). All three theory curves were fitted to the data with one common scale parameter. The size of symbols represent measurement errors.

Figure 5. The spectral phonon power.

We compare the spectral powers P(v) of a full membrane (black line) to the PnC devices (a=970 nm, red line, a=2425, nm, blue line), at (a) T=0.1 K, and (b) T=0.3 K. For the a=2425, nm structure, power was calculated up to 35 GHz.

Figure 6. Superconducting tunnel junction thermometer characteristics.

(a) Measured subgap current–voltage characteristics of a typical superconductor-insulator-normal metal-insulator-superconductor (SINIS) thermometer (2R_T=43 kΩ) at different bath temperatures in log-linear scale. Horizontal dashed lines correspond to bias currents 450 pA and 8 pA. (b) Measured voltage-bath temperature response of the SINIS thermometer in (a) with two different values of bias current, 8 pA (blue circles) and 450 pA (red circles) (same values as the lines in (a)). Solid lines were calculated using the single-particle tunneling theory, Equation 2. Bath temperature was measured with a calibrated RuO thermometer.

Figure 7. Sorting of eigensurfaces.

Low energy spectral branches of a 2D PnC with circular hole perforations plotted over the irreducible BZ (IBZ). (a) Unsorted surfaces , which provide incorrect group velocity data at k-values where the surfaces intersect. (b) Sorted surfaces, which correspond to the eigenmodes continuous in k. (c) Relative error in P(T) for a full membrane with d=485 nm for various IBZ grid sizes, when eigensurface sorting is neglected. Eigensurfaces were calculated up to 70 GHz.