It is traditionally challenging to measure the thermal conductivity of nanoscale devices. In this Letter, we demonstrate a simple method for the thermal conductivity measurements of silicon nanowires by using the silicon nanowire under test as the thermometer and heater. The silicon nanowire (SiNW) arrays are patterned out of a silicon-on-insulator (SOI) wafer by standard microfabrication processes. The thermal conductivity of SiNWs with a width from 150nm to 400nm (while the thickness is fixed at 220nm) are measured in the temperature range of 20-200K. At low temperature range, the nanowire thermal conductivity exhibits a strong size dependency since the effective phonon mean-free path is dominated by the nanowire radial size. At high temperature range, the phonon mean-free path is significantly smaller than the nanowire radial size. The nanowire thermal conductivity is strongly temperature correlated and nearly independent of the nanowire size. Density function theory calculations are also performed on the SiNWs and the calculated thermal conductivity of SiNWs are largely consistent with the experimental data, showing that our method is valid for the thermal conductivity measurements of nanoscale devices.

Thermal transport in one-dimensional nanostructures has attracted significant research interest1–3 in the past decade, due to their potential applications in thermoelectrics4 and thermal interface materials.5 The widely used approaches for the nanowire thermal conductivity measurements include the suspended micro-bridge method2,6–8 and the T-type method.9 These methods often involve the fabrication of Pt coils as heaters and thermometers on suspended micropads in addition to the placement of a nanodevice bridging the micropads. It is a formidable task even with the assistance of the most advanced tools. Here we demonstrate a simple method to measure the thermal conductivity of silicon nanowires by using the nanowires as inherent heaters and thermometers. Although this method was previously applied on Pt wires,10 semiconducting nanowires are highly sensitive thermometers in comparison with Pt wires due to the exponential ionization of dopants.11 More importantly, engineering the thermal conductivity of semiconductor devices may help improve the thermoelectric performance of the devices. It is clearly more interesting to measure the thermal conductivity of semiconductor nanodeivces. Our silicon nanowires (SiNWs) are patterned out of a silicon-on-insulator (SOI) wafer with CMOS-compatible fabrication processes. The SiNWs under test are directly used as the heater and thermometer for the thermal conductivity measurements, which greatly simplifies the fabrication and measurement process. A large number of SiNWs can be characterized after relatively simple fabrication steps. For nanodevices synthesized by bottom-up approaches, the method demonstrated here is also applicable as these devices can be assembled into large-scale arrays.12 

The fabrication process of the SiNW devices starts with a silicon-on-insulator (SOI) wafer with the device layer on top of a 3μm thick SiO2 layer. The top device layer is 220nm thick and has an average doping concentration of ∼1 × 1018 cm-3 by ion implantation of boron dopants. Electron beam lithography (EBL) and Al thermal evaporation were first performed to define the NW pattern and the pattern for two wide beams that support the nanowire, forming an “H” shaped structure (Fig. 1). Photolithography and Al thermal evaporation were then performed to define the four large contact pads. The top silicon layer were patterned by reactive ion etch (RIE) using the Al film as the etch mask. The silicon structure is formed after the Al mask is removed in Al etchant. To facilitate Ohmic contacts, the large contact pads were further doped to ∼1 × 1020 cm-3 by ion implantation of boron dopants. The dopants were activated by rapid thermal annealing at 950 °C for 30s. As the next step, 300nm thick Al electrodes (light grey layer in Fig. 1) were deposited on the large contact pads by thermal evaporation following photolithography. Furnace annealing was then performed at 250 °C for 15mins to form ohmic contacts between the Al and Si contact pads. To form a suspended nanowire, the SiO2 underneath the silicon nanowire region was then etched by buffered oxide etchant (BOE), as shown in Fig. 1. In the end, the whole SiNW sample was transferred into isopropanol (IPA) solution and dried off in a critical point drier to release the nanowires without the impact of surface tension. The scanning electron microscopic (SEM) image of a typical suspended SiNW device is shown in Fig. 1. The SiNWs have a fixed length of 20μm and a constant thickness of 220nm. The width of the SiNWs are varied from 150nm to 400nm.

FIG. 1.

False color SEM image of the suspended silicon nanowire with width of 200nm and length of 20μm.

FIG. 1.

False color SEM image of the suspended silicon nanowire with width of 200nm and length of 20μm.

Close modal

The nanowire devices were located in a cryostat during the measurements. The temperature of the cryostat can be adjusted by the temperature controller (Lakeshore 335) in the range of 10-300K. The resistances of the SiNW devices were calibrated using the four-probe measurement method, the configuration of which is shown in Fig. 1. The results show that Ohmic contacts were formed between the metal and silicon pad. The contact resistance and the resistance of the supporting beam were excluded. The I-V curves of the SiNW under small voltage bias were measured from 20K to 270K, as shown in the inset of Fig. 2a. We have shown in the previous study that under small voltage bias, the joule heating in the nanowire can be neglected. In this case, the temperature of the nanowire will be the same with the cold head of the cryostat (background temperature T0 which is tunable). The resistance of the nanowire increases exponentially as the temperature decreases due to the temperature dependent dopant ionization rate, as shown in Fig. 2a. The nanowire itself can be employed as a thermometer in the later experiments by using the temperature dependence of the nanowire resistance.11 

FIG. 2.

(a) Resistance of the silicon nanowire as a function of temperature. Inset: I-V curves under different temperature. (b) Average temperature rise as a function of heating power at different background temperature.

FIG. 2.

(a) Resistance of the silicon nanowire as a function of temperature. Inset: I-V curves under different temperature. (b) Average temperature rise as a function of heating power at different background temperature.

Close modal

The thermal conductivities of the SiNWs were measured by assuming one-dimensional steady state heat conduction. Since the nanowire is much higher in resistance than the contact region and the two supporting beams, the joule heating at high voltage bias mostly occurs in the nanowire region. Different from the suspended nanowire, the contact pads and the supporting beams are in direct contact with the SiO2 substrate. As a result, the temperature of the contact pads and the supporting beams stays the same with the tunable background temperature T0 (see supplementary material S1). Besides, steady state during the thermal conductivity measurements can be readily achieved since the SiNW as a heater can operate at a frequency as high as 50 kHz.11 At high current injection (also at high voltage bias), the nanowire temperature rises due to the joule heating effect. The average temperature rise can be found from the temperature dependence of the nanowire resistance shown in Fig. 2a.

Since the measurements were conducted in high vacuum, convective thermal losses via residual gases can be neglected. The average device temperature rise ΔT¯ was kept smaller than 6K above the background temperature during all the measurements. Thermal radiation loss is negligible due to the small size and small temperature difference. Therefore, the heating power of the nanowire equals to the product of current I and voltage VNW. The average temperature rise ΔT¯ as a function of heating power at different background temperature T0 is plotted in Fig. 2b, from which the thermal conductivity k of SiNWs can be calculated with the following equation:10 

k=(l12wt)ΔT¯IVNW
(1)

where l, w, t is the length, width and thickness of the nanowire, respectively. The thermal conductivity k calculated from eq. (1) is actually an average value between the background temperature and the maximum temperature since ΔT¯ in eq. (1) is the average temperature rise (see supplementary material S2). Since ΔT¯ is limited to a very small value (<6K here), the thermal conductivity k calculated by eq. (1) can be approximately taken as the thermal conductivity at the background temperature T0.

Using the simple kinetic theory, the thermal conductivity of a material can be understood by the following equation:13 

k=13Cvνslp=13Cvνs2τ
(2)

where Cv is the specific heat per unit volume, νs is the sound velocity, lp is the phonon mean-free path and τ is the phonon relaxation time. Cv increases with increasing temperature T in the low temperature regime while lp will decrease due to phonon-phonon scattering as the temperature rises. This competing dependence of lp and Cv results in a positive temperature dependence of the thermal conductivity at low temperature range and a negative dependence at high temperature range (Fig. 3a). The nanowire thermal conductivity measured here is compared with other published experimental works, as shown in the inset of Fig. 3a. The data measured in this work agree pretty well with the measured thermal conductivity of SiNW with similar geometry in literature at temperature above 100K. At temperature below 100K, The thermal conductivity of the 150*220nm SiNW measured in this work is lower than the intrinsic 115nm diameter SiNW reported in literature. The discrepancy can be attributed to the enhanced phonon-impurity scattering due to the high boron doping concentration of the measured SiNW in this work.

FIG. 3.

(a) Measured and calculated thermal conductivity of silicon nanowires with different width. Inset: thermal conductivity of SiNW measured in this work (square) compared with corresponding data from published experimental work (red curve is intrinsic SiNW with diameter of 115nm7). (b) Temperature dependent thermal conductivity ratio.

FIG. 3.

(a) Measured and calculated thermal conductivity of silicon nanowires with different width. Inset: thermal conductivity of SiNW measured in this work (square) compared with corresponding data from published experimental work (red curve is intrinsic SiNW with diameter of 115nm7). (b) Temperature dependent thermal conductivity ratio.

Close modal

To further validate our measurements, the thermal conductivity of silicon nanowires are also calculated by the first-principles numerical simulations to validate our measurement results. We have employed the anharmonic lattice dynamics method combining with the phonon Boltzmann transport equations.14 Considering that the nanowire is boron-doped, the phonon-impurity scattering is obtained by fitting the scattering term from existing thin film data.15 The details of simulation are presented in the supplementary material S3. The simulated thermal conductivity of silicon nanowires are well consistent with the experimental data at low temperature range, as shown in Fig. 3a. At temperature region 100-200K, the deviation is slightly higher possibly due to the lowered sensitivity of the SiNW thermometer compared to the low temperature region. The SiNW in this work is boron doped, which is almost fully ionized at temperature above 200K. Therefore it has a low sensitivity at high temperature range. To make the SiNW a high sensitivity thermometer and measure the thermal conductivity at temperature above 200K, deep energy level impurity such as nitrogen can be applied, which is partially ionized at room temperature.11 

To experimentally show the size dependency of lp in our nanowires, we normalize the thermal conductivity of the 400 nm-wide nanowire respective to the ones that are 150 nm and 200 nm wide, respectively. This normalization will remove the complication of the specific heat and group velocity, allowing the phonon mean-free path to play the only role in the thermal conductivity ratio. As shown in Fig. 3b, both ratios rapidly drop and saturate to the unity as the temperature rises. To understand this phenomena, we need to take a look at how the phonon-phonon scattering and boundary scattering affect the effective mean-free path. For quasi-1D nanowires, the effective phonon mean-free path is highly dependent on the nanowire width and thickness in addition to the bulk phonon mean-free path. According to the Matthiessen’s rule:16 

1τ=1τbulk+1τboundary
(3)

where τboundary is the phonon-boundary scattering relaxation time and τbulk is the bulk phonon relaxation time. The Casimir limit gives17 

1τboundary=vsdc
(4)

where dc=2wt/π is the effective diameter of the SiNW.18 w is the SiNW width and t is the SiNW thickness.

At low temperature, the bulk phonon mean-free path lp,bulk is on an order of micrometers, much larger than the nanowire thickness and width. In this case, the effective mean-free path is dominantly limited by the SiNW size (Fig. 3b). This is why the thermal conductivity ratio k400nm/k150nm approaches to 2.7 and the ratio k400nm/k200nm to 2. It is known that the bulk phonon mean-free path by phonon-phonon scattering is 1/T dependent on temperature.19 As the temperature rises, the bulk phonon mean-free path drops quickly to values comparable to or even smaller than the nanowire size. As a result, the effective mean-free path of the SiNWs becomes increasingly dependent on temperature and less on nanowire dimension. This explains that the thermal conductivity ratio of the nanowires exponentially drops as the temperature rises and saturates to the unity (Fig. 3a and b).

In conclusion, the thermal conductivities of SiNWs were measured by the one-dimensional steady state heat method in which the SiNW itself serves as both a heater and a thermometer. The thermal conductivities of SiNWs with different width were measured in the temperature range of 20K-200K. A strong size dependent effect was observed which is mainly attributed to phonon boundary scattering. The thermal conductivity of silicon nanowires were also calculated with the first-principles method. The results were largely consistent with the experimental data, showing that the method demonstrated in this work is valid for the thermal conductivity measurements of nanoscale devices. In comparison with the traditional micropad method, this approach significantly simplified the thermal conductivity measurements for nanoscale devices. But its accuracy depends on the temperature sensitivity of the underlying devices.

See supplementary material for thermoelectric simulations by ANASYS.

This work was financially supported by the National Science Foundation of China (61376001) and the Major Research Plan, Science and Technology Commission of Shanghai Municipality (16JC1400405). The nanowire devices were fabricated at the Center for Advanced Electronic Materials and Devices (AEMD), Shanghai Jiao Tong University.

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Supplementary Material