3. Experimental facilities and measurement methods
3.2 Electric property measurement methods
3.2.2 Self-heating 3ω method
Knowledge of the thermal conductivity of thin films, multilayer thin-film, and wires structures is critical for a wide range of applications in microelectronics, photonics, microelectromechanical systems, and thermoelectric materials [44-47].
Some methods developed for the determination or measurement of thermal conductivity of materials, such as the steady-state technique, the 3ω technique, and the thermal diffusivity measurement. [48] Each of these techniques has its own advantages as well as its inherent limitations, with some techniques more appropriate to specific sample geometry, such as the 3ω technique for individual nanowires. For steady-state method, the thermal conductivity of solids is usually determined by measuring the temperature gradient produced by a steady heat flow in a one-dimensional geometry. Direct measurements of the thermal conductivity, for example, typically require the determination of the heat flux and the temperature drop between two points of the sample. Figure 3.3 shows a typical sample configuration of a bulk system. Unfortunately, these techniques often require large, precisely shaped samples and extreme care to be used successfully. From a practical view, it’s impossible to achieve this setup for thin-film and nanowire system. In order to study thin-film and nanowire system, a technique was developed to measure thermal properties. One important technique is the 3ω method, which take 3rd harmonic signal to measure the thermal and electrical conductivity along longitudinal direction.
The 3ω technique and methods for the measurement of thermal conductivity of nanowires is discussed in detail as follow.
In this method, if the sample is electrically conductive and with a temperature-dependent electric resistance, the specimen itself could serve as a heater as well as a temperature sensor. Feeding an ac electric current of the form I0sinωt
into the specimen creates a temperature fluctuation on it at the frequency 2ω, and accordingly a resistance fluctuation at 2ω. This further leads to a voltage fluctuation at 3ω across the specimen. Systematic investigations of the 3v method were carried out mainly during the 1960’s [49-51] and in the last ten years, [52-57] which made the method practical. However, in the previous studies the heat conduction equation was solved under the approximations either only for the high frequency limit, [49, 50, 57] or only for the low frequency limit. [52, 54, 55] With those approximations one lost either the information on the thermal conductivity or the information on the specific heat of the specimen. By comparison with other methods, the 3ω method yields more information. This method can measure electrical conductivity (σ), thermal conductivity (κ) and specific heat (Cp) at the same time. In following sessions the explicit solution for the 1D heat-conduction equation, which was reported by L. Lu in [58] would discuss.
ΔT
Heat sink
Sample Heater
Figure 3.3 The typical arrangement for thermal conductance measurement.
Usually, the large bulk sample is fixed on a high thermal conductive heat sink. The heater is placed on top of sample to provide a steady heat flow by a power P, which generate a stable temperature gradient cross the sample. The temperature different ΔT then is measured with a set of thermometry, since the thermal conductivity κ can be calculate with the simple formula κ=P/ΔT.
We consider a uniform filament-like specimen in a four-probe configuration as for electrical resistance measurement. A complete construction for this model is shown in Figure 3.4. The two outside probes are used for feeding an electric current, and the two inside ones for measuring the voltage across the specimen. The specimen between the two voltage probes is suspended to allow temperature fluctuation. All probes have to be highly thermal conductive, to heat link the specimen at these points to the substrate. The specimen has to be maintained in a high vacuum and the whole setup is heat shielded to the substrate temperature to minimize the radial heat loss through convection and radiation. In such a configuration and with an ac electrical current of the form I0sin(ωt) passing through the specimen, the heat generation and diffusion along the specimen can be described by a partial differential equation, which the details will descript in next session.
Heat sink L
Δ
0 L
Figure 3.4 the sketch for sagging structure and the temperature distribution of a wire. As shows in the right figure, a sagging wire presents a temperature gradient by injecting a current into wire. For an appropriate applied current, the temperature of end points of the wire contacting with heat sink is keeping the same with heat sink.
3rd harmonic signal:
In a one-dimensional configuration and with a sinusoid electrical current passing through the specimen, the heat generation and diffusion along the specimen can be described by the following partial differential equation and the initial and boundary conditions:
where κ, Cp, R, ρ, L, and S are the thermal conductivity, specific heat, electric resistance, mass density, length, and cross-section of the specimen at the substrate temperature T0 , respectively. Let Δ(x, t) denote the temperature variation from T0, i.e., Δ(x,t)=T(x,t)-T0 Eqs. 3.1 then becomes
They obtain the temperature distribution along the specimen:
( ) ( )
accumulation at the center of the specimen. Δ0 is only κ dependent. The information of CP is included in the fluctuation amplitude of the temperature around the dc accumulation.By solving the partial differential equation, the resistance fluctuation can be expressed as
The production of the total resistance R+δR and the current I0 sinωt, the voltage across the specimen contains a 3ω component V (t). Only taking the n=1 term at
low frequency range, the 3ω component can be express as
The root-mean-square value of voltage across the specimen contains a 3ω component V3ω(t), we have
where the κ and γ are thermal conductivity and thermal time constant, respectively.
By fitting the experimental data to this result, we can get the thermal conductivity and thermal time constant of the specimen. The specific heat can then be calculated as Cp=π2γκ/ρL2.