Thermal conductivity and air-mediated losses in periodic porous silicon membranes at high temperatures

Abstract

Heat conduction in silicon can be effectively engineered by means of sub-micrometre porous thin free-standing membranes. Tunable thermal properties make these structures good candidates for integrated heat management units such as waste heat recovery, rectification or efficient heat dissipation. However, possible applications require detailed thermal characterisation at high temperatures which, up to now, has been an experimental challenge. In this work we use the contactless two-laser Raman thermometry to study heat dissipation in periodic porous membranes at high temperatures via lattice conduction and air-mediated losses. We find the reduction of the thermal conductivity and its temperature dependence closely correlated with the structure feature size. On the basis of two-phonon Raman spectra, we attribute this behaviour to diffuse (incoherent) phonon-boundary scattering. Furthermore, we investigate and quantify the heat dissipation via natural air-mediated cooling, which can be tuned by engineering the porosity.Nanostructuring of silicon allows acoustic phonon engineering, but the mechanism of related thermal transport in these structures is not fully understood. Here, the authors study the heat dissipation in silicon membranes with periodic nanoholes and show the importance of incoherent scattering.

Fig. 1

Samples for two-laser Raman thermometry experiment. a Schematic picture of the periodic porous membrane—square lattice of cylindrical holes in the free-standing membrane, where t = 250 nm is the membrane thickness, d is the hole diameter, a is the lattice parameter and n stands for the neck. b Top view of the sample design depicting the temperature scan direction and crystallographic orientation. The porous area is enclosed by two circles with diameters of about 5 and 100 μm. c, d Scanning electron microscope images of sample S2 (a = 250 nm and d = 140 nm). Scale bars in c, d are 20 and 2 μm, respectively

Fig. 2

Two-laser Raman thermometry results. Linear and corresponding logarithmic temperature profiles of a, b pristine 250 nm thick silicon membrane and c, d sample S3 with lattice parameter of a = 200 nm and hole diameter of d = 130 nm. Red circle-line and blue square-line plots indicate experimental data obtained in vacuum and air, respectively. Solid lines in b, d denote theoretical fits using Eq. (2). Error bars in a, b, c, d represent experimental uncertainties of the measured temperature (see Methods Section). e Volume reduction factor ε as a function of porosity calculated by FEM and using analytical expressions. f Measured temperature map of the sample S1 with lattice parameter a = 300 nm and hole diameter d = 135 nm

Fig. 3

Temperature dependence of the thermal conductivity. a Thermal conductivity of porous membranes (S1, S2 and S3) and 250 nm thick membrane as a function of temperature. Solid and dashed lines denote measured and extrapolated κ, respectively. The shaded areas indicate experimental uncertainty derived in Methods Section. b Normalised thermal conductivity of porous membranes as a function of the neck size n for three example temperatures; circles—300 K, triangles—600 K, squares—900 K. The error bars are experimental uncertainties of κ derived in Methods Section. The dashed lines are guides to the eye connecting data points of the same temperature. The solid line is a reference plot indicating n ² dependence. c The exponent β governing the temperature dependence of κ as a function of the neck size n. The dashed line is a guide to the eye and the arrow indicates β of the 250 nm thick membrane. Horizontal and vertical error bars indicate the experimental uncertainties of β and n determined from the data fit using Eq. (2) and from scanning electron microscope images, respectively

Fig. 4

One- and two-phonon Raman spectra. Data obtained for pristine 250 nm membrane and S1 at room temperature in $x_3\left(x_1{x}_1\right){\overline{x}}_3$ scattering geometry. The arrows indicate critical points of the first Brillouin zone of bulk silicon, where TA and TO are transverse acoustic and optical modes, respectively, and LO are longitudinal optical modes

Fig. 5

Air-mediated heat losses. a Relative (red diamonds, left axis) and absolute (blue triangles, right axis) losses caused by air-mediated cooling as a function of surface-to-volume ratio. b Measured κ _exp = κε (red circles) and intrinsic κ (blue triangles) thermal conductivity at 600 K as a function of surface-to-volume ratio. The dashed lines in a, b are guides to the eye. The errors bars are derived using the error propagation of κ derived in Methods section

Fig. 6

Schematics of the two-laser Raman thermometry experiment. The setup is based on the triple-grating Raman spectrometer (T64000, Horiba) and the vacuum temperature controlled microscope stage (THMS350V, Linkam). The non-polarising cube BS and three powermeters are used to determine the absorbed power P ₀ from intensities of the incident, transmitted and reflected laser beams

Fig. 7

The two-laser Raman thermometry calibration. a Temperature as a function of silicon longitudinal optical phonon frequency. The circles indicate experimental data points. The solid line stands for the linear fit with slope of -43.43 ± 0.05 K(cm⁻¹)⁻¹ calculated for data ranging between 300 and 870 K. b Representative Raman spectra obtained in vacuum for S1 at different distances r from the heating spot; black, blue and red circles correspond to r = 10, 20 and 40 μm, respectively. Solid lines indicate Lorentzian function fits of the corresponding experimental data