THERMAL CONDUCTIVITY MEASUREMENT BY THE 3w METHOD

 

A. Jacquot1, M. Stölzer2, J. Meusel2, O. Boffoué1, B. Lenoir1, A. Dauscher1

 

1Laboratoire de Physique des Matériaux, Ecole des Mines, Parc de Saurupt, 54042 NANCY Cedex, France

jacquot@mines.u-nancy.fr

 

2Martin-Luther-Universität Halle, FB Physik, 06099 HALLE, Germany

stoelzer@physik.uni-halle.de

 

We have developed a 4 wires 3w method based on a digital lock-in amplifier and tested our experimental set-up at 300 K on materials with a large range of thermal conductivity. The bases of the 3w method are reviewed and technical choices are justified. Experimental results are interpreted by a numerical simulation program that is able to take into account a wide range of phenomena neglected in current analytical formulas. The capability of the 3w method to measure both the in-plane and cross-plane thermal conductivity is also discussed.

 

 

Introduction

 

The determination of the thermal conductivity is of great interest because first it contributes to the performance of thermoelectric materials and second it contains information about the microstructure of the material being studied.

In the case of thin films or superlattices, the transport properties can differ considerably from equivalent bulk materials and vary depending on sample preparation. It should be pointed out that the in-plane and cross-plane thermal conductivity (heat flow perpendicular to the film) can differ because of anisotropic film structure (i.e. polycrystalline columnar films, anisotrope materials or superlattice).

Unfortunately, thermal conductivity measurement is difficult on two dimensional structures. Problems arise because of thermal radiation, heat loss in the temperature measuring probes and because the power supply to the sample has to be known with high accuracy.

The use of a thin metal strip as both heater and thermometer and of an AC current in the 3w method overcomes all these difficulties [1].

Although the method has originally been developed for the thermal conductivity measurement of isotropic bulk materials exhibiting low thermal conductivity values, this method was successfully applied to thermal conductivity measurement of thin films deposited on high thermal conducting substrates [2].

 

1. State of the art

 

1.1 Experimental set-up

 

Figure 1: A nickel strip is used as a heater and thermometer. Typically the strip width is 20 µm, the length is 4 mm, the distance between the inner pads is 2 mm and the thickness is 400 nm. The cross section A-A is shown figure 3.

 

A thin metal strip (fig. 1) is deposited on the sample being studied and is used both as a heater and also as a thermometer because of the temperature-dependent electrical resistance of the metal strip.

An AC current with frequency w passes through the strip and heats up the underlying matter. The voltage V(t) is measured during the same time. Because of the temperature dependent electrical resistance of the metal strip, a 3w voltage is superimposed on the initialw voltage [3]:

 

(1)

 

This V3w voltage contains the information about the thermal properties of the underlying matter and allows to measure the temperature oscillation D T2w :

(2)

where I is the current along the strip, R the resistance between the inner pads and a the temperature coefficient of resistivity (a =1/R.dR/dT). The difficulties are (i) to measure the V3w voltage that is typically one thousandth off the primary Vw voltage and (ii) to link the thermal conductivity of the underlying matter to the amplitude D T2w of the temperature oscillation.

 

To measure the 3w voltage, we have first to cancel the w voltage from the V(t) voltage in order to increase the ratio of the 3w to the w voltage.

 

There are two solutions to achieve this aim: use a Wheatstone bridge (2 wires 3w method) [3-4] or use a technology based on differential amplifiers (4 wires 3w method) [1]. Both schemes are presented in figure 2.

In our experiment we have chosen to use a technology based on differential amplifiers. This scheme has some advantages compared to the first one.

Figure 2: Scheme to cancel the primary w voltage. The scheme to the left is based on a Wheatstone bridge and the scheme to the right is based on differential amplifier.

 

Almost all the power supplied by the generator can be put into the sample. The use of a metal strip with four pads allows measurement of the strip resistance with a four-point method. Not only the variation of resistance is obtainable but also the absolute value of the resistance. Therefore the temperature of the strip itself can be known. Moreover as discussed later, to extract the thermal conductivity of the underlying matter a 2D heat conduction is assumed. This hypothesis is more easily satisfied with a 4 wires sample structure because the propagation of heat has to be two-dimensional only between the two inner pads.

Once the w voltage is reduced, the 3w voltage has to be extracted from the remaining signal. This can be achieved with the lock-in amplifier technology.

In order to understand the choice to make our own digital lock-in amplifier, it is important to understand how a lock-in amplifier works.

Extracting the voltage at the frequency 3w from a periodic V(t) voltage is like finding the 3th Fourier component of a periodic signal [3].

All periodic V(t) functions can be expressed as:

 

(3)

 

With

(4)

N is a whole number (mostly one in theory) and T=2p/.

The complex 3w voltage can be written as:

Then amplitude and phase can be easily calculated:

(5)

To sum up, we can extract V3w and j values, if we are able to build a pure cosine and sine function which is exactly in phase with the generator but with a driving frequency being three times higher. Then we have to do an integration and some simple calculations.

This can be achieved either electronically by using a frequency tripler and an analog lock-in amplifier [1] or in a complete numerical way [3].

Our experimental set-up is based on the scheme figure 2b. Vref and V3w (after amplification by ~ 100) are acquired by a computer equipped with a low noise, 12 bit, 200 kHz data acquisition card (Meilhaus ME260). The V3w signal is still noisy and contains w and 2w parts. The software determines the frequency and an even number of periods of the reference voltage Vref and synthetizes the required cos(3w t) and sin(3w t) using the mathematical sine and cosine function. Then amplitude and phase of V3w are extracted from the signal according (4,5) where the integration is carried out typically over N=10 periods.

1.2. Data analysis

Usually the analysis of the experimental temperature oscillation of the strip deposed on insulating material follows the method developed by Cahill and his co-workers [1-2]:

The complex oscillation amplitude of the strip is given by:

 

(6)

 

where 2b is the width of the strip, Pl the power per unit of length in the strip, |1/q| the wave penetration depth (WPD) and l the material thermal conductivity.

The wave penetration depth is defined by:

 

(7)

 

where r and cp are the density and heat capacity of the material.

Equation (6) is based on the assumptions, that the thickness of the strip can be neglected, the strip has an infinit length (2D heat flow), the size of the sample is semi-infinite and its thermal conductivity is isotropic.

The real part of eq. (6) can be approximated well by:

(8)

 

where h is a constant which is normally close to 0.923 but that is often adjusted to fit the experimental data [5].

Equation (8) is derived under the assumption that the WPD is large compared to the width of the strip. It shows that the thermal conductivity of the material can be determined from the slope of the temperature oscillation amplitude as a function of the logarithm of the frequency.

 

We consider now the case when a thin film with a lower thermal conductivity than the substrate is located between the substrate and the strip. The heater has to be insulated from the film studied if this film is a conductor (see fig. 3) by an additional layer.

The increase of temperature oscillation due to the film can be expressed in terms of a thermal resistance [5]:

 

(9)

 

where e is the film thickness and l film its thermal conductivity.

This equation is derived assuming that the heat flows along the cross-plane axis of the film only. This hypothesis is valid only if the width of the strip is large compared to the thickness of the film. In this case, the cross-plane thermal conductivity of the film is measured. By reducing the width of the strip both the in-plane and cross-plane thermal conductivity can be obtained in principle [4].

 

From the previous discussion it is clear that under certain assumptions the thermal conductivity of a film on a massive substrate can be determined in a straightforward way with analytical formulas. But we have to know if all hypotheses are valid. That is why we have developed a numerical simulation of the 3w experiment. The numerical simulation allows to check if the assumptions required to use anlytical formulas are fulfilled in an experimental case. In addition, it allows to outline interesting features of the 3w method applied to the thermal conductivity measurement of films deposed on high and low thermal conducting substrates.

 

The simulation programme is based on the finit volume method. The half cross-section of a real sample structure (substrate, film studied, insulating layer, heater strip) is devided into 900 cells typically (fig. 3). In each cell, a linear equation is derived which takes into account the flux of heat coming in, coming out, being stored and created. All these equations form a matrix that is solved with an iterative process. Each temperature oscillation is divided into 200 time steps. A steady amplitude of temperature oscillation is typically reached after three temperature oscillations. The boundary conditions are either constant temperature or infinite thermal resistance at the bottom of the sample. Assuming a constant temperature means the sample is mounted on a perfect heat sink whereas an infinite thermal resistance means the flux of heat through this surface is equal to zero.

Figure 3: Computer modelling of the 3w method.

 

 

2. Results on bulk materials

 

Our experiment has been tested firstly on glass which thermal properties are well known. The size of the sample, the width of the strip as well as the frequency used have been chosen in order to fulfil all mathematical assumptions. The thermal conductivity measured at room temperature agrees with the value given by the manufacturer (1.05 versus 1.1 W.m-1.K).

 

Then we have measured the thermal conductivity of Kapton foil (130 µm thick). The Kapton foil was glued on the sample holder with silver paint. Measurements were performed between 0.25 Hz and 10 kHz. As a result the wave penetration depth vary between 1.3 and 260 µm, assuming a density of 1420 kg.m-3, a heat capacity of 1090 J.kg-1.K-1 and a thermal conductivity of 0.32 W.m-1.K-1. At low frequencies, the wave penetration depth is larger than the thickness of the sample itself.

 

In addition to the experimental curves, figure 4 shows two couples of theoretical curves. The first one assumes a constant temperature and neglects the thickness of the strip. The second one assumes a infinite thermal resistance at the bottom of the sample and take into account the thickness of the strip.

Figure 4: Measurement on Kapton foil (1) experimental oscillation amplitude and phase, (2) amplitude and phase calculated assuming a constant temperature at the bottom of the sample and neglecting the thickness of the strip (3) amplitude and phase calculated assuming an infinite thermal resistance at the bottom of the sample and taking into account the thickness of the strip.

 

For frequencies lower than 4 Hz, the experimental curves follows clearly better the curve calculated assuming a constant temperature at the bottom of the sample. For high frequencies the experimental phase shift is much better fitted when the thickness of the strip is taken into account. The phase shift calculated neglecting the thickness of the strip tends to it theoretical value (p /4) at high frequencies [6-7].

The thermal conductivity of the Kapton foil measured is nearly two times higher than the thermal conductivity given by the manufacturer (0.32 versus 0.18 W/m/K).

Figure 5: Determination of the appropriate thickness of silicon wafers. Oscillation amplitude when assuming (1) a constant temperature at the bottom of the wafer (2) an infinite thermal resistance.

 

Our 3w method shows interesting features to measure low thermal conductivities on very thin samples. It is however very important to know the lower thickness that a sample should have to be considered as semi-infinite. It is practical to estimate this value in term of the WPD.

 

We have simulated the expected amplitude for measurements on a silicon wafer for different thickness and at a frequency of 1 kHz. Two curves are plotted in figure 5 assuming a constant temperature and an infinite thermal resistance at the bottom of the sample.

 

The result is that the sample thickness should be a little bit more than one time the WPD in order to assume the geometry as semi-infinite.

 

 

3. Results on thin films  

3.1. Films deposited on high thermal conducting substrates

 

Measurements on thin films have been done on a 230 µm thick silicon wafer covered with a thermally grown 620 nm thick silica layer (measured by SEM). The nickel strip used was 25 µm wide. Both the cross-plane thermal conductivity of the silica layer and the thermal conductivity of the silicon are obtainable.

Figure 6: Measurement on silicon (1) experimental amplitude obtained on a silicon wafer covered with a 620 nm thick silica layer (2) Simulated amplitude due to the silicon only on the assumption of an infinite resistance at the bottom of the sample.

 

Figure 6 shows two curves: The experimental one obtained on the silicon covered with the silica layer and a simulated one showing the contribution of the substrate to the oscillation amplitude. The two curves are nearly parallel in the whole frequency range. This demonstrates that the modelling by an infinite thermal resistance of the heat flow across the boundary between the silicon wafer and the sample holder is a good approximation (the sample is glued on the sample holder here with silicon grease).

The thermal conductivity of the silicon can be determined by the slope of the experimental curve between 100 and 1000 Hz. We found a thermal conductivity of 149.7 W.m-1.K-1 versus 148 W.m-1.K-1 in reference [8].

Assuming the silica layer acts as a thermal resistance, the cross-plane thermal conductivity of the thermally grown oxide is found to be close to 1.4 W.m-1.K-1 which agrees well with the result in the literature [5].

Figure 7: Influence of the thermal conductivity ratio on the temperature oscillation.

 

It is also of great interest to know the highest limit of the film to substrate thermal conductivity ratio in order to know which couple film/substrate can be studied by this method.

Therefore we have calculated the ratio of film on substrate to the substrate oscillation amplitude as a function of the film to substrate thermal conductivity ratio (fig. 7). Accurate measurement should be possible if the ratio of amplitude oscillation is not lower than 1.5. That means a ratio of thermal conductivities lower than 0.1. This value should ensure that (i) the temperature oscillation is high enough in order to decrease error in it measurement (ii) the contribution of the film to the amplitude of oscillation is far larger than thermal resistance between the layers. It is also obvious that for a substrate thermal conductivity close to that of the film, the thermal properties of the substrate have to be known with high accuracy.

From all the previous discussions, the ability of our 3w experiment to measure the cross-plane thermal conductivity of thin films on high thermal conducting substrates with good accuracy has been demonstrated.

 

 

3.2. Films deposed on low thermal conducting substrates

 

Figure 8: Effect on the heat flow of a film with higher thermal conductivity than the substrate

 

It is also interesting to know the in-plane thermal conductivity of thins films. It is obvious that it is very difficult to perform measurements of this value without a removal of the substrate [9-10]. However, a removal is quite difficult to achieve and sometimes not possible. That is why we study now the capability of the 3w method to measure also the in-plane thermal conductivity of films deposited on low-thermal-conducting substrates. The discussion on high-thermal-conducting substrates has already been done by other researchers [4].

Figure 9: Choice of the best frequency for in-plane thermal measurement.

 

In order to increase the contribution of the film compared to the contribution of the substrate, we have to choose a frequency that is high enough. But if the frequency is too high, the 3w signal will be too low in the experimental case and could not be measured with a good accuracy. Furthermore for high frequencies the WPD is far smaller than the width of the strip so that the heat conduction is nearly one dimensional. In this case the in-plane thermal conductivity of the film can not be obtained.

We have simulated the effect of the variation of the frequency on the oscillation amplitude of a 1 µm thick PbTe layer on Kapton assuming a thermal conductivity of 2 W.m-1.K-1 for the lead telluride and 0.32 W.m-1.K-1 for the Kapton. The width of the strip is 5 µm.

Figure 10: Effect of the strip width on the temperature oscillation amplitude (1) normalised amplitude on a Kapton foil with a single PbTe layer (2) with an additional 200 nm thick SiO2 layer.

 

The PbTe layer reduces the oscillation amplitude compared to Kapton alone (fig. 9). This can be understood by means of the Cahill formula. The heat flows along the film, and it acts as if the width of the heater would increase in Cahill formula (fig. 8). Figure 9 shows the film effect increase until 1 kHz.

Nevertheless it is probably better to use a lower frequency (40 Hz) because the signal-to-noise ratio is better in the in a real measurement and for high frequencies the thickness of the strip as to be take into account.

 

It is also obvious that the effect of the film is larger with a smaller strip width. The effect on the temperature oscillation is shown in figure 10.

Note the very small effect of an additional 200 nm thick insulating silica layer that is necessary in real experiments.

Figure 11: Reduction of the oscillation amplitude VS. thermal conductivity ratio (1) Amplitude without a 200 nm thick insulating layer

(2) Normalised amplitude with a 200 nm thick insulating layer.

 

Figure 11 shows the normalised oscillation amplitude D T for different ratios of thermal conductivities. It can be estimated that the thermal conductivity of the film has to be at least 5 times higher than that of the substrate in order to enable an accurate determination of the in-plane thermalconductivity l Film.

 

It should be underlined that the thermal properties of the substrate have to be known to make accurate thin films thermal conductivity measurements.

 

Conclusion

 

A 3w method has been successfully tested on bulk materials with low (0.32 W.m-1.K-1) and very high thermal conductivity (149 W.m-1.K).

It has been shown by numerical simulation that (i) a sample of thickness somewhat larger than the WPD is enough to avoid errors due to the finite sample size, (ii) the 3w method is able to measure the cross-plane thermal conductivity of a film on a substrate if its thermal conductivity is at least ten time lower than the substrate one, (iii) it should be possible to measure the in-plane thermal conductivity of a films thicker than 1 µm on low thermal conducting substrates if its thermal conductivity is at least 5 times higher than those of the substrate with a 5 µm wide strip.

 

References

 

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[9] F. Völklein and H. Baltes, J. Microelectromechanical systems 1 (1992), 1993.

[10] F. Vöklein, E. Kessler, Phys. Stat. Sol. A, 81 (1984), 585.