Helpful Publications:

• Thermal conductivity measurement from 30 to 750 K: the 3ω method - David G. Cahill; Rev. Sci. Instrum. 61, 802 (1990); doi:10.1063/1.1141498
• Erratum: “Thermal conductivity measurement from 30 to 750 K: The 3ω method” [Rev. Sci. Instrum. 61, 802 (1990)] Rev. Sci. Instrum. 73, 3701 (2002); doi:10.1063/1.1505652
• Thermal conductivity of thin films: Measurements and understanding - David G. Cahill; J. Vac. Sci. Technol. A 7, 1259 (1989); doi:10.1116/1.576265
• Extension of the 3ω method to measure the thermal conductivity of thin films without a reference sample - J. Alvarez-Quintanaa and J. Rodríguez-Viejo; Sensors and Actuators A: Physical , V. 142-1 (2008); doi:10.1016/j.sna.2007.01.013
• Reexamining the 3-omega technique for thin film thermal characterization - Tao Tong and Arun Majumdar;Rev. Sci. Instrum. 77, 104902 (2006); doi:10.1063/1.2349601
• Thermal Conductivity Measurements Using the 3-Omega Technique - David de Koninck (2008)
• Analytical solutions of the heat diffusion equation for 3ω method geometry - J-Y Duquesne; J. Appl. Phys. 108, 086104 (2010); doi:10.1063/1.3486441
• Data reduction in 3v method for thin-film thermal conductivity determination - T. Borca-Tasciuc, A. R. Kumar, and G. Chen; Rev. Sci. Instrum. 72 (2001); doi: 10.1063/1.1353189
• Application of the three omega thermal conductivity measurement method to a film on a substrate of finite thickness - Jung Hun Kim, Albert Feldman, and Donald Novotny; J. Appl. Phys. 86 (1999); doi: 10.1063/1.371314
• Thermal conductivity measurements using 1ω and 3ω methods revisited for voltage-driven setups - J. Kimling, S. Martens, and K. Nielsch; Rev. Sci. Instrum. 82, 074903 (2011); http://dx.doi.org/10.1063/1.3606441
• A generalized 3ω method for extraction of thermal conductivity in thin films - Zhao-Xiang Zong, Zhi-Jun Qiu, Shi-Li Zhang, Reinhard Streiter, and Ran Liu; J. Appl. Phys. 109, 063502 (2011); doi: 10.1063/1.3559299
• Nanoscale thermal transport - D.G. Cahill et. al.; J. Appl. Phys. 93, 793 (2003); http://dx.doi.org/10.1063/1.1524305

Email correspondence from Jean-Yves Duquesne of Université Pierre et Marie Curie to River Wiedle (10 May 2011):

Modern lockins are able to detect coherent harmonic signal. So, in contrast to D.Cahill original setup, there is no need to synthetize a
reference triple harmonic 3w.

The temperature of the sample is regulated.

R_0 is measured accuratly: 4-wires measurement.

R_a is adjusted in order to null the output of the differential amp at the fundamental frequency w.

At every frequency, the current is measured (phase and amplitude),
thanks to the well calibrated R_i. Since the 3w signal is proportional
to I^3, it is important to monitor |I| and to keep it constant trough
the whole frequency range.

To link the measured voltage to the modulated temperature, you have to
measure accuratly the transmittance of A1+A3. For that purpose, A2 input
is shortened. The thermal transducer can replaced with a well calibrated
resistor R_r (its value is close to R_0). The frequency is swept. Input
tension on A1 is deduced from the measured current (in the calibrated
R_i) and from R_r value. Output tension (A3) is measured.

Electrical contacts to the transducer are done with conductive epoxy.
With some practice, it is possible to make small contacts, say ~300x300
µm. I avoid ultrasonic bonding since this can perforate the deposited
films.

Metal/semiconductor contacts are usually non linear, from an electrical
point of view. Then, very large harmonic signals can be produced when a
current is flowing in the thermal transducter. So it is advisable to
introduce an insulating layer between the transducer and the SC film of
interest.

Schematic of Duquesne 3w setup

• SR850 Stanford Research Systems Lock-in Amplifer. SR850 Manual
• AM502 Tektronix Differential Amplifer. Tek AM502 Manual
• SRDS345 Stanford Research Systems Arbitrary Waveform Generator (belongs to McIntyre Lab). DS345 Manual

• Instrumentation amplifiers tested: INA121, INA128, INA111. The INA128s have the least phase error of the three. (RW 05/12)
• Phase difference drifts negatively (at high frequency) for the 121s and 128s, but drifts positively for the 111s. (RW 05/12)
• The 'divide by two' attenuation for calibration and 1w voltage cancellation causes low-pass behavior and limits the frequency bandwidth of the circuit. Adjusting the gain between 1 and 2 doesn't seems to make a significant difference. (RW 05/12)
• GPIB protocol must be used for talking to the lock-in (SR830/850) with LabVIEW. VISA won't work properly. (RW 05/12)

• Finite substrate effects come into play at low frequency. This limits the range of the 'linear region' for the traditional 3\omega method. High thermal conductivity/diffusivity and small thickness of the substrate make these effects show up at higher frequency. These effects can be modeled with the Borca-Tasuic model by setting either an isothermal (Tac = 0) or adiabatic (heat transfer = 0) at the back of the substrate. We want the isothermal condition as this will be more accurate since heat can be lost through other mechanisms (contacts, wires, conduction to air, etc). For a truly isothermal condition, Tx levels off at a finite value at low frequency and Ty converges to 0. This is because the time it takes for heat to transfer across the substrate is small compared to the period of the thermal wave (the substrate can be treated as a thermal resistance layer). To achieve an isothermal condition, it is important that the substrate be heatsunk to a material of high thermal conductivity (Cu works well). Simply setting the substrate on a copper block doesn't do the trick - placing a layer of silver paint between a sanded copper block and the substrate works well. In practice, this does not create a truly isothermal boundary condition - the 'linear region' for the copper block can be seen at low frequency.
• The lock-in amplifier has a setting for AC/DC coupling at the inputs. At low frequency (<10 Hz or so), AC coupling creates large phase and amplitude errors in the measured signal. However, any DC offset on the signal will create an additional signal at the reference frequency that needs to be filtered out (larger time constants might be required).
• For external voltage sources, a TTL reference is required at low frequency. The SINE reference is AC coupled due to the need for an accurate zero crossing. Using SINE reference at low freq will result in phase errors.

• Instrumentation amplifiers have great amplitude and phase accuracy at low frequency, but their short bandwidth causes problems at high frequency. Even the best IAs start to have problems with this around 100 kHz or below. CMR also starts to decrease quickly above 100 Hz. Errors are compounded when multiple amplifiers are hooked together in series. High speed differential amplifiers might be better suited for high frequency measurements.
• It becomes difficult to cancel the 1w signal at high frequency. This causes amplitude and phase errors in the measured 3w signal due to the finite dynamic reserve of the lock-in. This problem does not seem to be related directly to the amplifiers as it has been reduced using the same amplifiers in a breadboarded circuit. Instead, the problem seems to stem from the individual sample (resistance, contancts?). Problems often start at frequencies as low as 100 Hz.
• Most models show that the in- and out-of-phase voltages should converge to zero amplitude with a phase of -45 deg at high frequency. Instead, this phase is seen to decrease farther than -45 deg. The Borca correction term for finite heater thickness shows that the heater thickness will cause the phase to level off at -90 deg instead. Preliminary comparisons show that this is likely what we are seeing.

• The LIA can be phase-locked to the reference signal at any frequency. The amount of phase drift over a wide frequency range is negligible. However, phase locking must be properly accomplished. Simply hitting 'auto-phase' is not always enough. This feature shifts the internal reference by the calculated amount required to maximize X and set Y=0. If the signal has not stabilized this calculated shift will not be accurate. It often takes multiple attempts to properly phase-lock the LIA.
• The LabVIEW automation does a decent job of phase-locking the system. However, it is not perfect. It is always good practice to make sure that the computer is doing what you expect.
• In the 3w method, we want to measure the relative phase between the 1w current and the 3w voltage response. Thus, we want to phase-lock the LIA to the 1w signal. It is important to note that a phase shift of x for a 1w signal is actually a phase shift of 3x for a 3w signal. Therefore, the phase of the internal reference needs to be multiplied by 3 after phase-locking to the 1w signal to properly lock to the 3w signal. This issue appears when using external voltage/current sources. When using the LIA internal oscillator as the source, this problem does not show up as the reference is generated directly from the source (3*0 = 0).
• The overall 180 deg phase shift still shows up. This may be an additional problem related to the above problem.

• Cu, Ag, Au, Al, Ni, and Pt are all metals that can potentially be used as the microheater. Cu, Ag, Au, and Pt do not like to stick to most surfaces without a thin adhesion layer of either Cr or Ti. Al and Ni will stick to a clean surface if the substrate is heated during deposition. Annealing after deposition may work as well, but oxidation of the heater is a concern.
• Contact pads act as heat sinks! This causes a non-uniform temperature profile across the heater. A few papers have shown that this is not as big of a problem for AC signals above 10 Hz. However, using a heater design that moves the voltage leads away from the contact pads can help to get rid of this problem. Comsol modeling has shown that the voltage leads do not significantly alter the temperature profile of the heater.
• The width of the heater needs to be extremely uniform for accurate measurements. For film measurements, it needs to be measured.

• Most solders (indium, Sn/Pb) react with silver, making it difficult to impossible to make good contacts with this method. This may work better with aluminum contacts.
• Pogo pins seem to work well as they make good (mechanical) and repeatable contact and do little damage to the pads.

The system was set up in 2011 by River Wiedle in collaboration with Mark Warner. Matt Oostman worked on the project Fall 2011→ Nico Schmidt Spring 2012 →