Time-domain thermoreflectance is a method by which the thermal properties of a material can be measured, most importantly
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
. This method can be applied most notably to
thin film
A thin film is a layer of material ranging from fractions of a nanometer ( monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ...
materials (up to hundreds of nanometers thick), which have properties that vary greatly when compared to the same materials in bulk. The idea behind this technique is that once a material is heated up, the change in the reflectance of the surface can be utilized to derive the thermal properties. The reflectivity is measured with respect to time, and the data received can be matched to a model with coefficients that correspond to thermal properties.
Experiment setup
The technique of this method is based on the monitoring of acoustic waves that are generated with a
pulsed laser Pulsed operation of lasers refers to any laser not classified as continuous wave, so that the optical power appears in pulses of some duration at some repetition rate. Silfvast, William T. (1996). ''Laser Fundamentals'', Cambridge University Press. ...
. Localized heating of a material will create a localized temperature increase, which induces
thermal stress
In mechanics and thermodynamics, thermal stress is mechanical stress created by any change in temperature of a material. These stresses can lead to fracturing or plastic deformation depending on the other variables of heating, which include mat ...
. This stress build in a localized region causes an acoustic strain pulse. At an interface, the pulse will be subjected to a transmittance/reflectance state, and the characteristics of the interface may be monitored with the reflected waves. A probe laser will detect the effects of the reflecting acoustic waves by sensing the
piezo-optic effect.
The amount of strain is related to the optical laser pulse as follows. Take the localized temperature increase due to the laser,
:
where ''R'' is the sample reflectivity, ''Q'' is the optical pulse energy, ''C'' is the specific heat (per unit volume), ''A'' is the optical spot area, ζ is the
optical absorption
In physics, absorption of electromagnetic radiation is how matter (typically electrons bound in atoms) takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber (for example, thermal energy). ...
length, and ''z'' is the distance into the sample (Ref A). This temperature increase results in a strain that can be estimated by multiplying it with the linear
coefficient of thermal expansion
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kineti ...
of the film. Usually, a typical magnitude value of the acoustic pulse will be small, and for long propagation nonlinear effects could become important. But propagation of such short duration pulses will suffer
acoustic attenuation Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity ...
if the temperature is not very low (Ref B). Thus, this method is most efficient with the utilization of surface acoustic waves, and studies on investigation of this method toward lateral structures are being conducted.
To sense the piezo-optic effect of the reflected waves, fast monitoring is required due to the travel time of the
acoustic wave and
heat flow
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
. Acoustic waves travel a few nanometers in a picosecond, where heat flows about a hundred nanometers in a second. Thus, lasers such as titanium sapphire (Ti:Al2O3) laser, with pulse width of ~200 fs, are used to monitor the characteristics of the interface. Other type of lasers include Yb:fiber, Yb:tungstate, Er:fiber, Nd:glass.
Second-harmonic generation
Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy o ...
may be utilized to achieve frequency of double or higher.
The output of the laser is split into pump and probe beams by a half-wave plate followed by a polarizing
beam splitter leading to a cross-polarized pump and probe. The pump beam is modulated on the order of a few megahertz by an acousto-optic or
electro-optic modulator
An electro-optic modulator (EOM) is an optical device in which a signal-controlled element exhibiting an electro-optic effect is used to modulate a beam of light. The modulation may be imposed on the phase, frequency, amplitude, or polarization ...
and focused onto the sample with a lens. The probe is directed into an optical
delay line. The probe beam is then focused with a lens onto the same spot on the sample as the pulse. Both pump and probe have a spot size on the order of 10–50 μm. The reflected probe light is input to a high bandwidth photodetector. The output is fed into a
lock-in amplifier
A lock-in amplifier is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment. Depending on the dynamic reserve of the instrument, signals up to a million times smaller than noise components, p ...
whose reference signal has the same frequency used to modulate the pump. The voltage output from the lock-in will be proportional to ΔR. Recording this signal as the optical delay line is changed provides a measurement of ΔR as a function of optical probe-pulse time delay.
Modeling materials
The surface temperature of a single layer
The
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
solution for a semi-infinite solid which is heated by a point source with angular frequency
can be expressed by the following equation.
where
(1)
(Λ: thermal conductivity of the solid, D: thermal diffusivity of the solid, r: radial coordinate)
In a typical time-domain thermoreflectance experiment, the co-aligned laser beams have cylindrical symmetry, therefore the
Hankel Transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
can be used to simplify the computation of the convolution of equation (1) with the distributions of the laser intensities.
(The Hankel transform is an
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
equivalent to a two-dimensional
Fourier transform with a radially symmetric integral kernel)
Here g(r) is radially symmetric and by the definition of Hankel transform using Eq. (1),
(2)
Since the pump and probe beams used here have
Gaussian distribution, the
radius of the pump and probe beam are
and
respectively.
The surface is heated by the pump laser beam with the intensity
, i.e.
(3)
where
is the amplitude of the heat absorbed by the sample at frequency
.
Then the Hankel transform of
is
. (4)
Then the distributions of temperature oscillations at the surface
is the inverse Hankel transforms of the product
and
, i.e.
(5)
The surface temperatures are measured due to the change in the reflectivity
with the temperature
, i.e.
,
while this change is measured by the changes in the reflected intensity of a probe laser beam.
The probe laser beam measures a weighted average of the temperature
, i.e.
(6a)
This last integral (6a) can be simplified to an integral over
:
(6b)
The surface temperature of a layered structure
In the similar way, frequency domain solution for the
surface temperature of a layered structure can be acquired. Instead of Eq. (2), Eq. (7) will be used for a layered structure.
(7)
(Λn: thermal conductivity of nth layer, Dn: thermal diffusivity of nth layer, Ln: thickness of nth layer)
Using Eqs. (6) and (7), we can calculate the changes of temperature of a layered structure.
Modeling of data acquired in time-domain thermoreflectance
The acquired data from time-domain thermoreflectance experiments are required to be compared with the model.
(8)
(9)
(10)
(Q: quality factor of the resonant circuit)
This calculated Vf/V0 would be compared with the measured one.
Application
Through this process of time-domain thermoreflectance, the thermal properties of many materials can be obtained. Common test setups include having multiple metal blocks connected together in a diffusion multiple, where once subjected to high temperatures various compounds can be created as a result of the diffusion of two adjacent metal blocks. An example would be a Ni-Cr-Pd-Pt-Rh-Ru diffusion multiple which would have diffusion zones of Ni-Cr, Ni-Pd, Ni-Pt and so on. In this way, many different materials can be tested at the same time. Lowest thermal conductivity for a thin film of solid, fully dense material (i.e. not porous) was also recently reported with measurements using this method.
[Catalin Chiritescu, David G. Cahill, Ngoc Nguyen, David Johnson, Arun Bodapati, Pawel Keblinski, and Paul Zschack, "Ultralow Thermal Conductivity in Disordered, Layered WSe2 Crystals" ''Science'' 315, 351-353 (2007) {{doi, 10.1126/science.1136494]
Once this test sample is obtained, time-domain thermoreflectance measurements can take place, with laser pulses of very short duration for both the pump and the probe lasers (<1 ps). The thermoreflected signal is then measured by a photodiode which is connected to a RF lock-in amplifier. The signals that come out of the amplifier consist of an in phase and out of phase component, and the ratio of these allow thermal conductivity data to be measured for a specific delay time.
The data received from this process can then be compared to a thermal model, and the thermal conductivity and thermal conductance can then be derived. It is found that these two parameters can be derived independently based on the delay times, with short delay times (0.1 - .5 ns) resulting in the thermal conductivity and longer delay times (> 2ns) resulting in the thermal conductance.
There is much room for error involved due to phase errors in the RF amplifier in addition to noise from the lasers. Typically, however, accuracy can be found to be within 8%.
References
Thermodynamics