Thermal Measurements

CHAPTER 11

Thermal Measurements Coauthor Pierre-Olivier Chapuis Centre for Energy and Thermal Sciences of Lyon (CETHIL), Villeurbanne (Lyon), France

11.1 Really a Hot Topic? Thermal effects are extremely important in many fields of nanoscale science and technology. An important impact on the development of microelectronic components, like microchips, is to be underlined because hot spots generated as by-products of the energy dissipation are a crucial limit for the integration of electronic components. Roughly said, if semiconductor components are heated, then they produce more errors, so it is necessary to reduce energy dissipation or to remove the heat from the device; in practice, both are done. In addition, local overheating can induce a stress field that can lead to mechanical failure. This takes place especially at the contacts between dissimilar materials where the bimaterial effect can open voids. As a result, both during the phase of development and during testing, it is important to know the local temperature distribution and the thermal properties of all the materials. Scanning probe microscopy can do both, enabling local temperature mapping with a resolution of tens of nanometers and allowing us to measure thermal properties of very small objects or thin films [1]. Although the focus of this chapter is on thermal measurements, heated probes are also used for thermochemical and thermal-based nanolithography due to the very fine resolution achieved with such techniques [2]. In these techniques, the probe is heated and locally changes the surface temperature, where a chemical reaction or melting changes the matter state on the surface. The concepts and instrumentation presented in this chapter can also be used for this purpose. The temperature of an object is an important physical property, related to the mean kinetic energy of particles forming the object, and is consequently related to the internal energy of the object. Heat is known to be a measurable energy flowing between bodies having different temperatures, leading ideally toward a thermodynamic equilibrium if the system is let relaxing. Whereas the temperature itself is a state function and can therefore be used as a property describing the state of a system (with other properties like entropy), heat is a variable process that can be defined only when it is related to a change, for example, when it flows from a warmer part of a body to a colder one. Quantitative Data Processing in Scanning Probe Microscopy DOI: 10.1016/B978-0-12-813347-7.00011-X Copyright © 2018 Elsevier Inc. All rights reserved.

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304 Chapter 11 Since temperature is defined statistically, we can expect that there is some limit for the size of a system that can be described by it. If we start with a microscale size object, decreasing its size toward nanoscale, we can expect that there will be some border at which the classical statistical description of temperature will become problematic. Usually, we consider that this border is the mean free path of the energy carrier, as local thermodynamic equilibrium (the condition for the existence of thermodynamic temperature, requiring that the statistics is the same in all directions) is not defined if the object is smaller. The mean free paths in solids are typically electrons in metals and phonons—quanta of lattice vibrations—in nonmetallic materials. The typical mean free path is around 10 nm, but it can be as large as few hundreds of nanometers in some materials. Typical scales of objects employed in SPM are often below this border. Usual thermal parameters defining heat transfer, such as thermal conductivity or thermal diffusivity, also require the temperature to be defined everywhere: their definitions may also become questionable at small scales. Trying to keep the situation easier, working in the classical heat transfer regime (heat diffusion), we can run many interesting experiments. With the instruments known as “scanning thermal microscopes,” two classes of experimental techniques are usually available in commercial instruments: • •

Scanning Thermal Microscopy (SThM) producing images of local temperature or local thermal conductivity (conductance). Local Thermal Analysis (LTA) characterizing the mechanical response to localized heat source, also called Nanoscale or Microscale Thermal Analysis, and aiming providing information similar to Differential Scanning Calorimetry (DSC) at the local scale.

The main theoretical considerations for both classes of methods are very similar. In this chapter, we review the basic tip–sample junction models for heat flow, which are to some extent very similar to what we have discussed in Chapter 10 for electron transport, at least from the point of contact formation. Similarly to other SPM methods, thermal probes and other instrumentation used in SThM and LTA are manufactured to get the best possible spatial resolution and sensitivity to the investigated quantities, not to get the best possible accuracy. There are some well-established conventional techniques, like the 3ω method and the laser flash method for thermal conductivity and diffusivity, and like infrared thermometry for temperature measurements. Spatial resolutions of these methods as collected in Ref. [3] are far below the possibilities of SThM, limited by the sensor size in the range of tens of micrometers (e.g. for thermocouples) or limited by the diffraction limit for optics-based techniques (e.g. for reflectance-based measurements). However, if we need a superior temperature or conductivity resolution, without care for spatial resolution, the above methods can be more precise by several orders of magnitude than SThM-based techniques can provide.

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11.2 Nano- and Microscale Heat Flow Several processes can contribute to the heat flux between two bodies. Here we will discuss here conduction, convection, and radiation heat transfer. As usual in SPM field, all the processes can take part on the energy transfer at once, and often it is hard to estimate their mutual ratio. Here we briefly describe the heat transfer basics and their relation to Scanning Thermal Microscopy and Local Thermal Analysis measurements.

11.2.1 Conduction in Solids In a material, heat can be transferred by vibrations of particles forming the material without physical transfer of the particles itself, for example, by vibrations of the crystalline lattice. A typical example is heat transfer through a rod having ends at different temperatures. Heat flux in a material can be described phenomenologically by Fourier’s law as Q = −κ∇T ,

(11.1)

where Q is the transferred energy (heat), and κ is a parameter called thermal conductivity, representing the ability of the material to transfer heat in the stationary regime. Thermal conductivity, even if here treated as a constant, depends on the temperature and may depend on the direction in material as well. Another important parameter is the heat capacity defined as C=

∂Q ∂T

(11.2)

and thus corresponds to the amount of heat related to a change of temperature of the material. Thermal conductivity and heat capacity can be predicted for a given material if a model describing microscopic phenomena in the material is constructed. As the simplest solution, from kinetic theory of gases, we would get for monoatomic gas C = (3/2)NkB , where N is the number of atoms, kB is the Boltzmann constant, and κ = (Cvλ)/3, where λ is the mean free path. Both thermal conductivity and heat capacity of different materials can be found in tables (thermal conductivities of some SPM-related materials were already mentioned in Chapter 3). Note that since the scale of objects in SPM is very small, the bulk values of the tabulated constants are not always usable for our experiments. For data interpretation, it is therefore important to introduce some physical models of the heat transfer parameters. Treating heat phenomena microscopically, we can distinguish different heat carriers in a solid. For all of them, we can use the traditional diffusive model of heat transfer only in the

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306 Chapter 11 case where the mean free path is significantly smaller than the scale in which we are interested. This is the so-called Fourier regime, which can be described by Eq. (11.1). Otherwise, a regime involving at least partially ballistic transport (where there are no collisions of the energy carriers in the medium in at least one direction) occurs, where the usual thermal conductivity cannot be considered anymore. If heat transfer is analyzed perpendicularly to the small dimension, the so-called Casimir regime of effective heat diffusion is to be considered. If the heat transfer under study is along the small dimension, pure ballistic transport between the material boundaries takes place. In both cases, it is often practical to consider an effective thermal conductivity that depends on the material size (in contrast to the bulk case where thermal conductivity is independent of the material size). When the mean free path is comparable to the material size, an intermediate region where collisions occur, but not as frequently as in the diffusive regime, happens. Material-size-dependent thermal conductivity is also to be considered. Different particles and quasi-particles can carry heat, namely these two: Phonons are collective excitations of atoms in a condensed matter. A phonon itself is a quantum of mechanical vibrations of a given lattice of atoms, representing a long-range mechanical oscillation of the lattice. A lattice can be excited in different manners—adjacent particles can be in the same or opposite phase, which leads to different modes of phonons (acoustic for the same phase and optical for the opposite) having different dispersions. Phonons are scattered at material imperfections (e.g. crystalline lattice defects like vacancies or impurities), can recombine with other phonons, or can interact with other quasi-particles (e.g. with photons in infrared spectroscopy). Phonon scattering is the main reason of the differences in thermal conductivity between many materials since every material can transfer energy via phonons with different efficiency. Phonons are the main heat carriers in semiconductors and electrical insulators, where heat conduction by means of electrons is negligible. The mean free path of phonons is of order 10–100 nm, so for larger objects, we can assume that the heat is flowing diffusely. The values of phonon thermal conductivities in nonmetallic crystalline solids usually range between few W·m−1 ·K−1 and few thousands of W·m−1 ·K−1 . In disordered solids, such as amorphous materials, the situation is much more complex since there is no long-range material ordering originally connected with the concept of phonons. A shorter-range excitations and their mutual interaction can be defined to obtain thermal conductivity using different approaches [4,5]. The values of thermal conductivities of amorphous materials usually range between 0.05 and 3 W·m−1 ·K−1 . Regarding the material heat capacity, the contribution of phonons to this property can be described under the assumption that we have N oscillators per unit volume at frequency ω as [6]   eωkB T ω 2 , (11.3) Cph = 3NkB kB T (eωkB T − 1)2

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where  is the reduced Planck constant. Heat capacity is independent of the mean free path and is therefore independent of size unless very small sizes are reached (few nanometers at room temperature). Electrons in a solid can be bounded to atoms (in nonconductors electrically) or form a free electron gas (electrons above the Fermi energy in metals). The latter can significantly affect the thermal conductivity, so electrons are the main heat carriers in metals. Electrons in a free electron gas can move easily through the material transferring heat. Efficient values of thermal conductivities of order few hundreds of W·m−1 ·K−1 are usually found for metals. The contribution of free electrons to the material heat capacity can be described by the relation [6] Cel =

π 2 NkB2 T , 2F

(11.4)

where N is the number of electrons per unit volume, kB is the Boltzmann constant, F is the Fermi energy, and T is the temperature. A similar expression to that of phonons can be used for the thermal conductivity due to free electrons. Usually, their mean free paths are of order few tens of nanometers at maximum. It can be slightly affected by the crystallinity of thin films, and the thermal conductivity of thin films used in the thermoresistive probes can differ a little bit from the tabulated values.

11.2.2 Convection and Conduction in Fluids Convective heat flux is involved when particles having higher kinetic energy (statistically having higher temperature) are physically transferred in the system, generating a macroscopic mass transfer. A typical example is the movement of water in a kettle when we are warming it up, where warmer water is going up and colder going down due to the different densities. This motion is called advection and is often negligible in SPM since all the scales are usually too small to let matter develop macroscopic mass motion between two bodies of different temperatures. As liquid or gas particles exhibit random motion, there is always also a second mechanism, diffusion, which will be present everywhere if a medium exists between two bodies. If we think of measurements in ambient conditions, there is always the presence of air around the tip–sample contact. This will certainly lead to some diffusion between the probe and sample. Generally, a diffusive and advective heat flux can be described by the equation ∂T = ∇ · D∇T − v · ∇T , ∂t

(11.5)

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308 Chapter 11 where v is the advection velocity, and D is the thermal diffusivity given as D=

κ , ρCp

(11.6)

where κ is thermal conductivity, ρ is the density, and Cp is the heat capacity of the medium (per unit mass). For typical media in SPM, thermal diffusivity ranges from 1.9 × 10− 5 m·s2 for air up to 1.4 × 10−7 m·s2 for water. It is considered that, for usual probe/sample scales in air, the convective heat flow is negligible; however, for significantly heated probes in liquids, there might be some contribution. When a sharp tip is close to the material (or in contact), ballistic heat transport from the tip to the sample through the air becomes significant in comparison to the diffusive heat transfer through air. Indeed, air molecules traveling from the hot tip to the cold sample do not collide often with other molecules. This effect is especially important in the last nanometers before contact or if the tip is placed in partial vacuum. Numerical techniques allow us to compute the transfer (see “Numerical modeling”).

11.2.3 Radiation Every object with a temperature above absolute zero is emitting some radiation. In the ideal case, this process is described by the black body radiation law. Energy radiated by a black body having temperature T has the spectral radiance I (ν) =

1 2hν 3 , hν 3 c e kT − 1

(11.7)

where h is the Planck constant, and k is the Boltzmann constant. At room temperature, the maximum of the spectral density is in the infrared range. Radiated energy is lost by one body and can be absorbed by another body. If we considered that the materials are black bodies, they would absorb all the incoming energy. The difference between the real ability of a material to radiate and the ability of a black body is described by emissivity, which is a function of the material and surface finish. The energy emitted by a body is proportional to I (ν), its surface, and its emissivity. If the materials of two bodies have the same temperature, an equilibrium is preserved, and there is no heat transfer. If the temperatures differ, a heat transfer can be observed. Radiative heat transfer becomes more complicated if we are in the so-called “sub-wavelength regime,” that is, when characteristic scales of objects and their distances become of the order of or smaller than the radiation wavelength (about 10 µm for room temperature). For instance, decreasing the object size usually reduces the thermal radiation emitted more strongly than

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what macroscopic thermal radiation predicts, whereas reducing the distance between the objects dramatically increases the heat transfer (radiation tunneling in the “near field”). Luckily enough, radiation is usually a minor contributor to heat transfer in SPM in comparison to other means, at least while performing measurements in air. It is only important when the experiment is performed in vacuum and when the tip is not in contact with the sample.

11.2.4 Heat Sources in Scanning Probe Microscope Before developing models for scanning thermal microscopy and nanothermal analysis, it is important to note that in other SPM experiments thermal phenomena also may occur. The full range of effects leading to different tip–sample temperatures and heat flow would be as follows: •

•

•

•

Active heating elements are the key components of the thermal techniques described in this chapter. Using Joule heating, they produce heat locally to characterize local thermal properties, melt the surface, and so on. These will be discussed in the Instrumentation section of this chapter in more detail. These are also the key components for nanolithography. Laser feedback itself produces some heating of the cantilever. Even when the cantilever is coated by a reflecting metallic layer (aluminum, gold, etc.), part of this light is absorbed in the cantilever (typically up to 10 percent). The incoming radiation, which is of order of milliwats, can therefore transfer tens of microwatts of heat energy to the cantilever. This is enough to be observed as a temperature change in the scanning thermal microscope (in temperature contrast regime). The estimated temperature rises for standard cantilevers and a 1-mW laser power is around 25 K on a silicon cantilever [7], which is certainly not negligible. Although this heat source has also been used intentionally in scanning thermal microscopy, it is usually not the case. Light coming from other sources than the feedback laser, for example, illumination bulb, other laser sources used either in photoconductivity or in apertureless scanning near field optical microscopy measurements (SNOM), or light coming through probe in aperturebased near field optical microscope. Temperatures up to several hundreds of degrees Celsius have been estimated for some SNOM experiments [8]. Electrical currents flowing through a probe–sample system in conductive AFM measurements.

Although in the following sections we will discuss mostly intentional heat sources as in scanning thermal microscopy, most of the theoretical and numerical approaches can be also used for the latter cases. www.elsevierdirect.com

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11.2.5 Tip–Sample Heat Transfer Basics There are three key quantities related to the heat transfer and temperature measured using SPM. We might want to know the local temperature distribution on the sample, for example, on an active device like a microchip. We might also be interested in the local thermal conductivity distribution. Finally, we might want to determine some thermochemical response of the sample, like local expansion or surface melting. To be able to perform the three types of experiments, we need to have a probe that is able to both produce heat and measure the temperature. A thermocouple-based tip can be only used for temperature mapping if it is not heated, so an active heating element can be added to the probe if thermal conductivity and thermochemical measurements are to be performed. A thermoresistive (i.e. electrical-resistance based) tip can in principle be used directly for the three purposes. We will now analyze the energy balance of the tip–sample system. Ideally, we would assume that the heat generated in the probe is transferred into the sample and assume that the temperatures of probe and local sample area where the probe resides are the same. As discussed in the next sections, this is a heavy simplification, but we will first use it here to discuss the basic scanning regimes. Another problem for all these cases will therefore be the probe–sample interaction, including the apparent radius of the probe–sample contact and the tip–sample thermal resistance at the contact, also discussed later. Scanning thermal microscopy and local thermal analysis experiments are typically performed in contact mode using the same feedback principle as in the Atomic Force Microscopy. We can therefore start the interaction description with Fig. 11.1, describing the probe and sample in contact and showing all possible heat transfer paths. We can see that the heat transfer can be realized via conduction, convection, and radiation. In the literature, we can find some estimates for the amount of heat transferred by these three components. An example of how the approach curve—probe readout that is proportional to probe temperature dependence on probe–sample distance—can look like shown in Fig. 11.2. Let us assume that we have a probe that can be heated by an electrical current and the temperature of which can be measured from its resistance or by a thermocouple. We can use the following measurement regimes to obtain a quantitative information about the thermal properties of the sample: DC scanning thermal microscopy is based on the measurement of the instant probe temperature, optionally with heating above room temperature using a DC current passing through its resistive element. It can be used for temperature or conductivity estimation: in the temperature mode the self-heating of probe is minimized, and only the probe temperature is measured based on its electrical resistance. In the conductivity mode, we heat the probe electrically, and we observe how it is cooled by the sample. This cooling depends monotonically on the local thermal conductivity (it can be proportional to it in some thermal conductivity range). In

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Figure 11.1: SThM heat transfer mechanisms.

Figure 11.2: An example of the approach curve in DC SThM measurements, measured with the microscale Wollaston wire probe (thanks to E. Rousseau).

the stationary regime, the heat loss of a probe, equal to the heating Qtotal , will probably have many components, for example, Qtotal = Qcant + Qair + Qrad + Qts,cond + Qts,other ,

(11.8)

where Qcant is the conductive heat loss via the cantilever, Qair is the convective heat loss to surrounding air, Qrad is the radiative heat loss (partially to the sample), Qts,cond is the conductive heat transfer to the sample, and Qts,other is the heat transfer to the sample via air and adsorbed water layer. The determination of Qts,other is certainly not easy. First of all, it is important to determine thermal heat loss properties of the probe itself. If the probe is far from sample, then heat is lost by conduction of the probe support and by con-

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312 Chapter 11 vection to surrounding air. By measuring the microscope signal far from sample we can determine this heat loss assuming that it will be the same when the probe is in contact. The measured value of this can then be subtracted from the signal in contact. In this way, we determine the sum Qcant + Qair of components, and if we are not too far from sample, then also most of Qrad . The most important fact is that in this way we remove the component Qcant , which from simulations presented in Ref. [9] is assumed to contain 94% of the free probe heat loss. Note that Qair might be negligibly close to contact since most of the heat dissipated in the air tends to flow toward the highest thermal conductivity zone, that is, the sample. When we come to contact, the heat transfer via Qts,cond and Qts,other should be determined. Conductive heat transfer depends on the tip and sample thermal conductivity, tip–sample contact resistance, and contact radius. To find an analytical equation relating the sample conductivity κs to the signal obtained from SPM is not easy without knowledge of the probe geometry and many assumptions on the probe composition and sample geometry. Such an equation was found by Lefevre et al. [10] for the case of a Wollaston wire probe (see further), under the following assumptions: • • • •

the mean free path of heat carriers is smaller than the size of all the bodies and contact radius, heat losses to the air environment are neglected, mean probe temperature and resistance remain constant (they are actively controlled), the silver part of the Wollaston probe acts as a heat sink.

Then the difference between the Joule power dissipated far from contact (denoted with index o for “out of contact”) and at contact (denoted with index i, “in contact”), when the probe temperature is maintained constant, can be written as U2 − U2 3 G/(G + GP t )

P = i 2 o = , P 4 GGP t /(4bκs (G + GP t )) + 1 Ui

(11.9)

where G is the contact conductance, b is the contact radius, κs is the sample thermal conductivity (what we search for), GP t is the probe (platinum) conductance, and Uo and Ui are the measured voltages (microscope signals) when out of and in contact. The factor 4 assumes that the heat from the tip through the sample through a disc of radius b (conductance is 4bκs ). Many of the factors influencing the probe–sample heat transfer are very hard to be estimated in real measurements; therefore, the P /P dependence is often used for measurement calibration on the basis of known samples without trying to fully understand all the heat transfer paths. In this case, the functional dependence can be written as A

P = − C. P B/κs + 1 www.elsevierdirect.com

(11.10)

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Figure 11.3: A schematic calibration curve for DC SThM measurements using Wollaston wire probe (thanks to A.M. Massoud).

Here we have added C to the previous equation, which is the term associated with the heat loss by convection when the tip is out of contact. Note that although A, B, C may depend on conductivity, such a dependence is often neglected when fitting the data obtained with reference samples. A schematic calibration curve for P /P is shown in Fig. 11.3. AC scanning thermal microscopy uses an AC signal to drive the heating probe [11,12]. Let us assume that we have an input signal I = I0 + I1 cos(ωt) used for probe heating. Here we consider a thermoresistive probe again, where the electrical resistance Rp = Rp,0 (1 + α T ) depends on the sensor average temperature T via the temperature coefficient of the tip dR α = R1p dTp . As the probe is used for both heating (using Joule power P = Rp I 2 ) and for temperature monitoring and feedback (using output voltage Rp I ), we can observe different components in the output signal. The time dependence of the probe resistance will have the same frequency components as that of the Joule power P but will not necessarily stay in phase: 1 1 P = Rp (I02 + I12 + 2I0 I1 cos(ωt) + I12 cos 2ωt), 2 2 Rp = Rp,0 [1 + αθDC + αθ1ω cos(ωt − φ1ω ) + αθ2ω cos(2ωt − φ2ω )],

(11.11) (11.12)

where θDC,1ω,2ω and φ1ω,2ω are frequency-dependent components of the temperature field in the probe. The final signal (V = Rp I ) will then have the 3ω component 1 V3ω = Rp,0 I1 αθ2ω cos(3ωt − φ2ω ), 2

(11.13)

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Figure 11.4: Applicability of heat transfer modeling techniques for different scales of the system.

which can be measured using a lock-in technique. Both the amplitude and the phase depend on the thermal properties of the sample and are therefore exploitable. Other harmonics can also be used, but we have to keep in mind that the signals are proportional to αθ with α ≈ [10−4 –10−3 ] K−1 . We need to care about the fact that the tip expands periodically in the harmonic regime, following the temperature field inside the tip. However, usually, the AFM feedback is fast enough to compensate and maintain the tip–sample force interaction constant despite the tip shape periodic variation. Local thermal analysis uses a probe that can be heated to some temperature as well. Here, the probe is kept at one position on the surface, ramping its temperature, and the mechanical response of the sample as being melted is observed [11]. Namely for polymers, material mechanical properties undergo large changes in the range of 50–250 °C (e.g. glass transition temperature or melting point), which can be locally measured. The same approach for DC SThM measurements is used to understand the probe–sample heat exchange. The problem is slightly simplified by fact that we stay on a single sample, single position, and watch only some critical points on the dependence of mechanical properties on probe temperature.

11.2.6 Numerical Modeling Approaches Since there are still many phenomena in nanoscale thermal measurements that are not fully understood, we often need to employ some modeling of the probe–sample heat transfer. Even some very simple model, like a resistor network, can be used to significantly improve understanding the processes in SThM image formation and related uncertainties; however, to progress toward quantitative measurements on real samples (featuring various defects like surface roughness), we need to set up a model that is physically closer to the reality. A graphical overview of applicability of different numerical techniques is shown in Fig. 11.4. If we start from the smallest scale, then we can use a Schrödinger equation to determine the time evolution of a system of particles, predicting the evolution of their kinetic energies and

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statistically providing the resulting temperature or heat transfer. Since the ensembles of atoms need to be rather large for statistical concepts of heat to have any meaning, this approach is far too detailed for practical use. Even simplified quantum approaches, like density functional theory, are able to handle only hundreds of atoms at reasonable time scale, which is not sufficient. If we want to run atomistic simulations, then a better choice is classical molecular dynamics, which uses analytical or empiric two-body or multiple-body potentials to handle the interactions between atoms. Here we can easily model systems of millions of atoms, so heat transfer by conduction, convection, and radiation can be simulated. More about molecular dynamics can be found in Chapter 4. The temperature in MD simulations is calculated as [13] T=

N 1   mi vi2 , 3NkB

(11.14)

i=1

which is easy as we know the velocity for every atom in every time step. There are many ways to handle heat flow in MD simulations; however, the following techniques are most frequently used: •

In Equilibrium Molecular Dynamics (EMD), there is no heat flow through the system, and thermal properties are evaluated from correlation of particle velocities, based on simple idea that, in system with low thermal conductivity, each particle motion influences the other particle less than in system with high thermal conductivity. Evaluation is based on Green–Kubo approach, where the lattice thermal conductivity tensor is obtained from particle velocities correlation using the following formula: kij =

M N−m  V t  (N − m) vi (m + n)vj (n), kB T 2 m=1

•

(11.15)

n=1

where t is the time step of the calculation, and V is the volume of the system. EMD is used namely when we want to simulate thermal properties of bulk materials (which still can mean that the materials are nanostructured and we therefore investigate some nanoscale heat phenomena). In Nonequilibrium Molecular Dynamics (NEMD), there is a heat flow introduced to the system, for example, by using a modified thermostat to assign higher temperature to particles in some region and lower temperature in another region. Thermal conductivity can be then calculated from the necessary energy used in these thermostats to preserve the temperature difference or from the temperature distribution across the system. An example of use of NEMD to estimate the effect of contact thermal resistance changes between probe

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Figure 11.5: (A) Atomistic model for heat transfer in very high-resolution SThM imaging, (B) simulated relative probe–sample conductance while scanning over C60 fullerene.

and sample with very high resolution SThM geometry (still far from experimental stateof-the-art) is shown in Fig. 11.5. It is based on a model of a heated silicon probe scanning over a C60 fullerene located on a highly oriented pyrolitic graphite sample. As it can be seen from Fig. 11.5, here the cooler part of the sample was not used (it was substituted by the cooling effect of the fixed substrate), and simulation was based on observing the time dependence of temperature increase of the upper graphite layer due to heated tip contact. Tip was brought to contact in every scan position independently using the same setpoint force, and a virtual scan over the fullerene was performed. In Fig. 11.5B the resulting heat transfer map (apparent thermal conductivity relative to direct probe–graphite contact) is shown. To handle heat flow in the mesoscopic regime, where the full atomistic models are already too detailed and numerically costly, we can use methods dealing with the ballistic heat transfer phenomena. In this case, we neglect any atomistic properties (e.g. interatomic potentials), and we solve the transport almost classically, assuming that it is formed by particles having some distribution function f (r, v), where r denotes the position and v the velocity. The Boltzmann transport equation (BTE) [6] can be used to determine the evolution of the distribution function as follows:   ∂f ∂f + v∇f + a∇f = , (11.16) ∂t ∂t coll www.elsevierdirect.com

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Figure 11.6: A simple kinetic model of the heat transport between probe and sample. (A) Trajectories of individual particles. (B) and (C) Simulated temperature increase distribution on the sample, computed from the locations where particles hit the sample.

where a is the acceleration dv/dt, and the term on the right-hand to colli  side corresponds ∂f f −f 0 sions (scattering). Often, the relaxation time approximation ∂t = − τ , where f 0 is coll the equilibrium distribution function, and τ is the relaxation time, is used to simplify the approach. Several models for prediction of thermal phenomena like thermal conductivity are derived from the BTE, starting by the Fourier law as shown in Eq. (11.1). Various Monte Carlo methods can be considered to solve this complex equation, treating statistically either phonon transport inside material or particle transport in the medium between the probe and sample. For heat transport in air, the direct simulation Monte Carlo method can be used, simulating motion of a set of particles in an environment of the other particles, combining kinetic and statistic approach [14]. To make the calculation faster, there are also some trials on using some simple approximative solution for SThM case [15], based on a single particle motion from hot probe to cold sample through a bath of other particles. An example of such a calculation is given in Fig. 11.6, where a simple kinetic model setup and performance are shown. If only temperature distribution or heat flow in equilibrium has to be simulated and we assume that we are in the diffusive regime, that is, above the mean free path of the heat carriers, then we can solve the Poisson equation similarly to calculation of the electrostatic quantities. The Finite Element Method or a similar continuum-based technique can be used [16,17]. Many details about use of FEM in SPM probe–sample region modeling are already given in Chapter 4, and one particular implementation is further discussed in the next section dealing with roughness artifacts in SThM. An example of the FEM temperature distribution between the probe apex and sample is shown in Fig. 11.7.

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Figure 11.7: The temperature distribution in the probe–sample interaction volume computed by Finite Element Method.

11.3 Instrumentation In most cases, thermal measurements need some additional equipment to be performed in standard commercial instruments. The necessary instrumentation is usually sold as an option for the microscope or can be purchased separately from some third-party producers. Since the basic circuits for SThM measurements are very simple, we can also find numerous custombuilt electronics.

11.3.1 Commercially Available Techniques First of all, we need to mention the probes used in thermal measurements. As shown before, we need to measure temperature and, for most of the experiments, also to generate heat locally. This cannot be done using standard or somehow slightly modified AFM cantilevers. Special probes need to be used. Commercially available resistive probes that form majority of the market can be divided into two categories, both shown schematically in Fig. 11.8: •

Wollaston wire probes. These probes, shown in Fig. 11.8A, are formed by a thin platinum wire surrounded by a thick silver coating. At probe apex, the silver is dissolved, and the platinum wire forms a sharp tip. Since the resistance of a thin platinum wire is much higher than that of a platinum–silver coaxial wire, the tip is the most resistive part of whole probe. Joule heating is therefore highest here if we apply a current to the probe,

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Figure 11.8: SThM probes.

•

and similarly, temperature is measured predominantly there if we measure the probe resistance. These probes are extremely robust; however, the spatial resolution is limited by the radius of the platinum wire (micrometric) and apex radius, and can be hardly better than around 100 nm (in air, it is often micrometric). Due to the large area of heat exchange, some averaging takes place, and the Wollaston wire probes are not necessarily sensitive to small roughness details. Microstructured probes. These probes, shown in Fig. 11.8B, are prepared using the same technology as that used for standard AFM tips and cantilevers preparation. The conductive parts are formed by some patterned thin metal films, coating a sharp tip very similar to an AFM tip. As a result, the spatial resolution can be much better than for Wollaston wire probes, clearly below 100 nm, depending on the sample and environmental conditions. The probes can be tuned for the thermal response speed, maximum temperature, cantilever stiffness, and position of heating elements and offer more flexibility than the Wollaston wire probes. However, the flux exchanged between the tip and the sample is much lower for sharp tips than in the case of Wollaston wire probes, and therefore the thermal sensitivity of the microstructured probes is often lower.

Another important kind of probe is a thermocouple probe [18], and there are certainly many other experimental geometries available in the literature. To perform the measurements, we can use various circuits depending on the quantity we want to determine. As we always need to handle small electrical resistive elements, some variants of Wheatston bridge circuits are typically used. Measurement is performed simultaneously to topography data acquisition, monitoring the probe resistance and eventually heating it in DC or AC regime as discussed earlier in this chapter. In Fig. 11.9A a potential layout of the bridge for temperature measurements is shown, and in Fig. 11.9B, C a bridge for DC and AC conductivity measurements is shown. At writing the first edition of the book (2012), AC measurements were not directly supported by most of the manufacturers, but could be easily performed by only adding an AC source and a lock-in detector. This situation had not much changed after five years; however, availability of lock-in

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Figure 11.9: Bridges for different SThM modes [19,20].

amplifiers in present microscope electronics units is much wider, so the custom-built setup is now easier. For local thermal analysis, the sample is not scanned (or it is scanned prior to analysis to select the region of interest). Here the result is evaluated from a maximum of deflection vs. probe temperature curve. An active probe that can be heated to specific temperature is used. Measurement procedure is as follows: 1. 2. 3. 4.

Probe is kept at room temperature and approached to the sample with given contact force. Probe is kept in contact with feedback on to let the mechanical drifts stabilize. Feedback loop is switched off. The temperature of probe is slowly increased, watching the tip deflection signal, which goes up due to sample expansion. 5. When the tip deflection reaches maximum and drops down fast (as the tip start penetrating the surface), temperature ramp is stopped, the probe brought out of the contact, and finally the probe is set back to room temperature. 6. The resulting value is obtained from temperature corresponding to the maximum deflection. From the viewpoint of instrumentation, the LTA need not any other electronics than SThM. However, since the probes for LTA are typically manufactured with focus on other properties than SThM ones (LTA probes should not bend while heated and should be able to withstand higher temperatures), the probes have usually different range of resistance, and not every SThM electronics is compatible with them.

11.3.2 Other Experimental Approaches Under Development There are also many other techniques being developed by different groups, focusing on obtaining much more from nanoscale thermal measurements. All those need some special equipment or at least special software and typically are not part of commercial instruments.

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The most promising technique is probably Scanning Thermal Impedance Tomography, which is related to Electrical Impedance Tomography and Thermal Slice Diffraction tomography [21]. Some more effects related to local thermal properties can be observed using the following techniques [22,11]: • • • • •

In Joule Expansion Microscopy, we monitor local sample morphology changes due to thermal expansion while the probe temperature is periodically changed. Thermoemission Microscopy uses the SNOM approach for measuring local sample infrared radiation using a small fiber probe. Thermal Radiation Scanning Tunneling Microscopy scatters the thermal radiation emitted in the near field by the sample when it is heated [23]. Combination of elastic and photothermal interaction is used in Scanning Near Field Thermoacoustic Microscopy. Differential Scanning Calorimetry can be used to determine dynamic properties of heat capacity that are connected with vibrations at a molecular level.

11.4 Data Interpretation If we measure temperature, the most problematic effects are connected with the temperature drop between the probe and sample [24], related to probe–sample contact resistance. This depends on the tip–sample force, conductivity, roughness, and contamination, and currently there are no many ways to estimate this effect on unknown samples (and mostly it cannot be estimated even on known samples). As an example, we show in Fig. 11.10 SThM temperature measurements on a heated gas sensor. We can see on the temperature map significant artifacts related to roughness (compared to sample morphology plotted in Fig. 11.10A). Moreover, we can see the temperature difference between the metal coated part (forming “fingers” on the sample surface) and the ceramic part. The sample is heated from the bottom, and the temperature field should be homogeneous, the effects we see are a combination of different thermal conductivities and contact resistances and maybe also some other effects like different emissivities leading to different radiation losses. We can even find a temperature for which the apparent temperature of the sample is the same and therefore the surface temperature is the same. This is shown in Fig. 11.11, where for a certain current passing through the sample, we see no contrast on the temperature map. The aim of this example is to show that even in the temperature contrast mode, which is understood as a “simpler” technique, we are not free of various artifacts related to sample composition and topography (note the effect of surface roughness).

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Figure 11.10: (A) Morphology and (B) temperature map measured on heated gas sensor.

Figure 11.11: (left) Temperature contrast related to different emissivities and conductivities of platinum and ceramics parts of the gas sensor substrate and (right) graph of a thermal response of the heated gas sensor obtained using SThM.

For local thermal conductivity measurements, the temperature distribution in the system is rather complex if we use a heated probe. All the components have different conductivities, there are significant heat losses to probe legs and sample, and therefore the analytical description is rather complex [9]. A calibration technique is therefore often used instead of direct calculation using a sample with several materials with known thermal conductivity or using several samples [25]. The benefit of this approach is that the whole setup consisting of probe, resistive bridge, and surrounding electronics can be calibrated. The procedure is as follows: 1. Measure the probe heat flow to several known materials, taking care that all of them have minimum possible surface roughness. The heat flow when the probe is far from the sample [20] must be used to correct the offset in the transferred heat due to conduction of the probe itself and convection to air.

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Figure 11.12: Local Thermal Analysis curves (cantilever deflection signal vs. probe temperature) obtained for polycaprolactone, polyethylene, and polyethylene terephtalate.

2. Plot the dependence of the apparent vs. table conductivity and fit it using some simple formula (see Refs. [25,10]) as explained previously:

Q A = B − C,

T κ

(11.17)

where A, B, C are fitting parameters, and κ are the conductivities. 3. Calculate a correction factor for further measurements on unknown materials. An accuracy of ±10% is reported for thermal conductivity of an unknown material using this method [25]. However, larger thermal conductivities are often not suitable for SThM measurements, and accuracy for conductivities above 10 W/m/K dramatically decreases. Data obtained using Local Thermal Analysis can be calibrated in a similar way using a set of materials with known melting points [26]. Different polymers, like polycarbonate, polystyrene, or polymethyl metacrylate, are typically used since these have reasonable melting points and glass transition temperatures in the range 50–300 °C. The melting point or glass transition temperature of an unknown material can be then determined easily, with a precision hopefully better than 5 K. An example of measurement using LTA for analysis of melting temperature of three different polymers is shown in Fig. 11.12. A key assumption for this method is that the thermal conductivities of the polymers need to stay close: the flux dissipated in the sample can then be considered as almost independent of the sample, and

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324 Chapter 11 therefore the calibration can be performed only as a function of the melting temperature. If this is not the case, then a correction should be applied. Note that, for all the methods, we need to consider the contact radius related to elastic deformation of both probe and sample, as discussed in Chapter 6. These will directly influence the heat transfer, and if we work with a sample consisting of several materials of differing hardnesses, then we need to expect artifacts related to different contact radii.

11.4.1 Artifacts Treatment In conductivity contrast mode, namely for probes with a higher radius (like Wollaston wire probes), we can observe significant topography artifacts related to changes of the real contact area on different parts of the sample. The simplest way to predict the presence of these effects is detecting the slope and local curvature on the sample and mark areas that are above a certain threshold. However, since the tip convolution effect is rather large for large SThM probes and the studied samples are typically exhibiting a finite roughness, this approach could easily mark the whole surface. To treat issues related to topography such as contact area changes and related heat flow properly, we would need to use a numerical model, like the Laplace equation solution using the finite element method. At smaller scales, a suitable model might be based on BTE solution or even on molecular dynamics. The use of any of these approaches is however limited by the fact that we do not know the topography and probe shape since we can observe only its convolution; however, based on the morphological operations like surface reconstruction and blind tip estimation, some approximate information about the probe and sample surface shape can be obtained. Here we show a few examples of increasing complexity how to calculate the influence of topography artifacts on the SThM data [27]. In all the cases the goal is to obtain a virtual SThM image, assuming that the sample material is everywhere the same. The virtual image consists then only of the topography artifacts. The examples are based on a rough microchip surface SThM measurements performed using Wollaston wire probe, as shown in Fig. 11.13. As a very simple technique for estimation of topography artifacts, we can perform local evaluation of sample material volume in the neighborhood of the probe. We can expect that when more sample material is in vicinity of probe apex, heat flow from probe to sample is larger. This effect corresponds to what we would expect and what we actually see on conductivity contrast images of simple structures like steps or particles. Evaluation is performed using the following steps: 1. Using blind tip estimation algorithm [28], the tip shape was determined from topography data. www.elsevierdirect.com

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Figure 11.13: (A) Sample topography and (B) thermal conductivity contrast image obtained on a rough microchip surface.

2. Surface reconstruction was used to get the real surface shape. 3. The measured surface was used to determine the tip contact point. Then the reconstructed surface, which is considered as real, was used for evaluation. Volume in close vicinity to the contact point was summed with weight of 1/r 2 , where r is the distance from where tip touches the surface. This is a key assumption of this model. 4. The resulting data are relative numbers, so the final step was a linear transformation that adjusted the minimum and maximum values to be the same as in measured data. Use of a neural network (NN) is a general approach that can be used to generate any result from any source data, assuming that we have trained the network to do so. It can be therefore also trained to estimate topography artifacts in conductivity contrast SThM on a homogeneous sample if we have enough data to perform the training. A neural network is usually formed by a set of input and output nodes that are interconnected and multiplied by weights (small real numbers). The simplest neural network (with no hidden layer) would thus output only a weighted sum of inputs. The analytical capability of such an oversimplified neural network would be rather poor, therefore one or more sets of hidden nodes (interconnected with input and output nodes) are usually added into the network. The crucial point of use of neural networks is how to set the weights of the nodes. This is usually done in an iterative training process that uses known sets of inputs and outputs and adjusts the weights to optimize the performance of the neural network. Here we present the results of using a simple feedforward NN trained with backpropagation algorithm as discussed in our previous work [29] and referring also to first attempts to this by Price [30]. The network has an input layer consisting of height differences corresponding to the characterization of the closest neighborhood of certain points in the topography image,

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326 Chapter 11 one hidden layer of typically 10–15 neurons and an output layer representing the modeled thermal output value. Neural network works best if trained on situations closely resembling the actual case of use. For this reason, the NN was trained on a different homogeneous part of the sample, measured by SThM, featuring the same material, roughness, and topography features. In this way, we can estimate apparent thermal conductivity signal value from a local neighborhood of the probe apex. The network used in this study had an input layer consisting of 16 height differences corresponding to a characterization of the closest neighborhood of a certain point in the topography image, one hidden layer of typically 10–15 neurons and an output layer representing the modeled thermal output value. The network was trained using several sets of images representing thermally homogeneous materials (with topography artifacts only) to simulate the topographic artifacts in a good way. To process the SThM data, we applied the following steps within the measurement: The set of SThM and topography images was measured on a part of the surface with uniform material (usually on the same sample where the material is homogeneous). The measured surface was chosen to have a roughness with similar statistical properties as the roughness at the area of interest has. 1. The neural network was trained to produce thermal data output from the topography measurement described before. 2. SThM and topography image were measured at the area of interest producing thermal data that are highly influenced by local geometry of tip/sample. 3. The neural network was used to simulate a thermal signal from the topography image of area of interest. The simulated signal was subtracted from the measured SThM signal. It was found that, for randomly rough surfaces, a single image of a homogeneous material is sufficient to train the neural network properly. However, more images were necessary for the training step for patterned structures. As the data are already trained on measured conductivity contrast signal, the result values are directly obtained in the range of the measured signal and are therefore directly comparable. Note that the same probe should be used for acquisition of data for training and for data we are evaluating. Finite Element Method can be used to solve the Poisson equation related to diffusive heat transfer in the system, which is one of popular approaches for SThM modeling in diffusive regime as discussed earlier. FEM can calculate the steady state of a temperature field on a geometry given by measured topography data and a tip shape calculated by Blind Tip Reconstruction (BTR). In the model presented here, we use thermal conductivities of three materials: platinum (the tip), the substrate, and thin separating layer of air. The tip is modeled as a volumetric source of heat power (not isothermal). For each data point of the resulting thermal

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Figure 11.14: Microchip surface conductivity contrast signal: (A) measured and evaluated from topography by (B) neighbor volume, (C) neural network, and (D) Finite Element method. Reprinted from Ref. [27].

image, the geometry and the mesh is different because the tip position changes. The calculation is very demanding even for a supercomputer since in each point of the virtual SThM image the mesh needs to be generated and temperature distribution computed. Because FEM calculations are too slow for practical purposes, we have developed a custom solution for SThM artifacts treatment based on Finite Difference Method implemented on a graphics card [31]. Physically, the same model as for FEM is used; however, the use of a graphics card and easy parallelization of the voxel-by-voxel calculations lead to significant speedup of the calculations. Despite this development, a single virtual image calculation takes few hours on a computer equipped by a high-end graphics card. All the methods are compared in Fig. 11.14 with the real image of the SThM artifacts (see Fig. 11.13). The FDM result is not shown because it produces the same solution of the Poisson equation as FEM, only much faster. It can be seen that even the simplest technique can provide reasonable estimate of the areas affected by topography artifacts and can be used for uncertainty estimation purposes.

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11.5 Try It Yourself As discussed before, to perform a quantitative measurement using thermal techniques, the easiest way is a calibration of our instrument based on a set of samples with known thermal properties (e.g. thermal conductivity). Although this calibration depends on the probe geometry (which is not very practical), it is simpler than trying to perform a full calculation of heat transfer in the instrument. To apply this approach successfully, we need to have some idea about thermal data artifacts related to sample topography.

Example 11.1: Delaminated Thin Film in Thermal Conductivity Contrast Mode This shows a measurement of a polymer-like carbon film partly delaminated from the silicon substrate. Data were collected in the conductivity contrast mode, where the temperature of the probe apex is kept constant via a feedback loop. The output is the voltage necessary to keep this probe temperature constant, that is, the darker areas mean that you need less current passing through the probe as the local heat losses to the sample are smaller.

Example 11.2: Gas Sensor Heater in Temperature Contrast Mode The gas sensor was measured in the temperature contrast mode, keeping the probe current minimal to minimize its self-heating and to use it only for probe resistance measurement. The raw data output for two different currents flowing through the gas sensor is provided. See Fig. 11.10 and related text for more details.

What Might Come to the Uncertainty Budget Not all the effects that can be observed in SThM measurements are currently so well described that a universal uncertainty budget could be set up. Therefore establishing traceability is not an easy task, even if we try to do it using some of the simpler approaches (e.g. via reference samples). For temperature, the following items might appear in the temperature budget: •

•

Sensor calibration, for example, by placing the probe in a bath and measuring its output at different temperatures. The uncertainty will be given namely by the bath itself; however, comparing to the other uncertainties, the effect may be quite small. Electronics drift during the measurement and between calibration and measurements: for some bridges, the effect is surprisingly large, and electronics drifts can be directly seen on the thermal images as shifts in the slow scan axis, which is also a way to estimate them.

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•

•

• •

Temperature difference in the sample and probe: if there is a heat flow during measurement (which is not necessary if we use some of the nulling thermometry methods, trying to preserve the probe and sample temperature the same), then there will also be a temperature gradient in the probe. Since the temperature sensor may be located at some distance from the probe apex, this leads to a systematic error, which can be estimated via numerical calculations, at least, setting up some very simple thermal resistor model. Probe–sample contact resistance: this applies when there is a heat flow as in the previous case and can be one of the greatest uncertainty components. To estimate it, we can make multiple approaches with different settings (e.g. contact force) and watch the changes due to probe shape change during scanning. Specimen temperature control: in temperature measurements, we usually want to see the temperature distribution across the sample that has some active heaters. However, also fluctuation of the room temperature and/or sample temperature will affect the result. Air conditioning systems in laboratories and microscope enclosures can be of different quality, providing temperature stability up to mK in the extreme case; however, often these are much worse, and some fraction of degree can be already significantly affecting our measurements. Effect of laser used for microscope feedback, causing additional probe heating: it can be estimated by switching the laser off and checking the thermal electronics response. Probe anisotropy: in some probes, namely Wollaston wire based, there is a strong anisotropy of the interaction volume based on the probe shape and its angle with respect to the sample normal. This may be source of further artifacts (on top of all the topography artifacts).

If we want to determine thermal conductivity, then the above uncertainty components apply as well, except the sensor calibration, which is replaced by calibration of the sample set used for setting up the curve-defining probe response. This will be the greatest uncertainty influence. Moreover, we cannot use any nulling technique since the heat flow is measured, so that changes in probe sample contact and related topography artifacts will be less influenced as well. This can be partly estimated by the numerical approach presented in this chapter. For local thermal analysis, the following additional items need to be considered: •

•

Mechanical drift: this includes both the drift of the sample and whole microscope. It can be evaluated from known long-term behavior of the microscope using some drift estimation procedure, which is simpler if only z drift is evaluated by performing scan with zero range or even simpler directly during the thermomechanical measurement by observing the response prior to starting any temperature changes. Position-sensitive detector nonlinearity: as some of the measurements are done with feedback loop off, we need to rely on assuming the position-sensitive detector linearity, which

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•

• • • •

•

•

•

is valid only to some level of accuracy. Measurement at different settings of the laser with respect to the position sensitive detector center can be done to estimate the potential influence of this effect. Cantilever stiffness and deflection sensitivity: the conversion factor from volts on the detector to probe z motion and conversion factor to get the probe–sample force needs to be known and will have large uncertainty (e.g. stiffness uncertainty can be around 10 percent as discussed in the previous chapters). This is needed to know how accurately we set the contact force and evaluate the other forces during the experiment. Sample hardness at glassy state: this affects the contact area and probe–sample heat transfer, together with contact force. Sample creep: for slow measurements, the sample can undergo relaxation leading to another change in the slope of the temperature dependencies of mechanical properties. Contact force: this affects the contact area, that is, heat transfer, adhesion, and similar parameters. It will change nonlinearly when the force is changed. Sample/system thermal expansion: this can be evaluated from the initial response of the sample at glassy state when we keep probe in contact and ramp temperature. It will be mixed with other drift components and probably depend on the speed of the ramp. It can be also evaluated while doing the temperature ramp on a known sample that is not a polymer, for example, silicon. Glass transition slope: the dependence of elastic properties of polymers on the temperature usually shows some continuous change around the transition temperature (not an infinitely sharp step). Affected sample volume, sample conductivity effect: this could be best estimated by a series of time-dependent FEM calculations employing a correct viscoelastic behavior model, which is not simple. Dynamic effects: with a slow ramp, we estimate the transition temperature at lower temperatures as sample has more time to get heated. This is a large effect in overall curve shape.

Read More From many different SThM reviews that can be found in scientific literature, we pick Refs. [3], [11], [32], and [33].

11.6 Tips and Tricks The possibility of performing quantitative measurements strongly depends on probes and electronics you use for the experiments; however, generally, to convert your SThM data into quantitative data, keep in mind the following:

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Calibrate your setup using a suitable approach for temperature or conductivity contrast, check your bridge wiring, and measure its components if possible. Note that every probe has a different resistance, which can influence the absolute values of temperature. Every probe needs to be handled individually. Never believe thermal conductivity data obtained on rough surfaces; these need more treatment as the probe sample contact varies across the surface. Clean the probe in nanothermal analysis when it could get contaminated, by heating it to higher temperature to remove the contamination. If you are able to use a set of samples with known thermal properties for calibration of the whole setup (including probe, electronics, and data processing), then do it. This applies to both the thermal conductivity measurements and to local thermal analysis.

References [1] S. Gomes, L. David, V. Lysenko, A. Descamps, T. Nychyporuk, M. Raynaud, Application of scanning thermal microscopy for thermal conductivity measurements on meso-porous silicon thin films, J. Phys. D: Appl. Phys. 40 (2007) 6677–6683. [2] R. Garcia, A.W. Knoll, E. Riedo, Advanced scanning probe lithography, Nature Nanotechnology 9 (2014) 577–587. [3] E. Gmelin, R. Fischer, R. Stitzinger, Sub-micrometer thermal physics: an overview on sthm techniques, Thermochimica Acta 310 (1998) 1–17. [4] David G. Cahill, R.O. Pohl, Heat flow and lattice vibrations in glasses, Solid State Communications 70 (10) (1989) 927–930. [5] S. Alexander, O. Entin-Wohlman, R. Orbach, Relaxation and non-radiative decay in disordered systems. II: two fraction inelastic scattering, Phys. Rev. B 34 (1986) 2726–2734. [6] Charles Kittel, Introduction to Solid State Physics, 6th edition, John Wiley & Sons, Inc., New York, 1986. [7] O. Marti, J. Colchero, J. Mlynek, Combined scanning force and friction microscopy of mica, Nanotechnology 1 (1990) 141–144. [8] J.L. Kann, T.D. Milster, F.F. Froehlich, R.W. Ziolkowski, J.B. Judgins, Heating mechanisms in a near-field optical system, Applied Optics 36 (1997) 5951. [9] S. Gomes, N. Trannoy, Ph. Grossel, F. Depasse, C. Bainier, D. Charraut, D.C. scanning thermal microscopy: characterisation and interpretation of the measurement, Int. J. Therm. Sci. 40 (2001) 949–958. [10] S. Lefevre, S. Voltz, J.-B. Saulnier, C. Fuentes, N. Trannoy, Thermal conductivity calibration for hot wire based dc scanning thermal microscopy, Rev. Sci. Instrum. 74 (4) (2003) 2418. [11] H.M. Pollock, A. Hammiche, Micro-thermal analysis: techniques and applications, J. Phys. D: Appl. Phys. 34 (2001) R23–R53. [12] D.G. Cahill, Thermal conductivity measurement from 30 to 750 K: the 3ω method, Rev. Sci. Instrum 61 (1990) 802. [13] S. Maruyama, Molecular dynamics methods in microscale heat transfer, in: Handbook of Heat Exchange Update, Begell House, New York, 2000. [14] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, 1998. [15] P.-O. Chapuis, J.-J. Greffet, K. Joulain, S. Volz, Heat transfer between a nano-tip and a surface, Nanotechnology 17 (2006) 2978–2981. [16] S. Volz, X. Feng, C. Fuentes, P. Guérin, M. Jaouen, Thermal conductivity measurements of thin amorphous silicon films by scanning thermal microscopy, International Journal of Thermophysics 23 (6) (2002) 1645–1657.

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332 Chapter 11 [17] A. Altes, R. Heiderhoff, L.J. Balk, Quantitative dynamic near-field microscopy of thermal conductivity, J. Phys. D: Appl. Phys. 37 (2004) 952–963. [18] A. Majumdar, J.P. Carrejo, J. Lai, Thermal imaging using the atomic force microscope, Appl. Phys. Lett. 62 (1993) 2501. [19] I.W. Rangelow, T. Gotszalk, N. Abedinov, P. Grabiec, K. Edinger, Thermal nano-probe, Microelectronic Engineering 57–58 (2001) 737–748. [20] F. Ruiz, W.D. Sun, F.H. Pollak, Ch. Venkatraman, Determination of thermal conductivity of diamond-like nanocomposite films using a scanning thermal microscope, Appl. Phys. Lett. 73 (13) (1998) 1802. [21] L. Nicolaides, A. Mandelis, Experimental and image-inversion optimization aspects of thermal-wave diffraction tomographic microscopy, Optics Express 7 (13) (2000) 519. [22] B. Cretin, Scanning near-field thermal and thermoacoustic microscopy: performances and limitations, Superlattices and Microstructures 35 (2004) 253–268. [23] Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-Ph. Mulet, Y. Chen, J.-J. Greffet, Thermal radiation scanning tunnelling microscopy, Nature 444 (2006) 740–743. [24] J. Pelzl, S. Chotikaprakhan, D. Dietzel, B.K. Bein, E. Neubauer, M. Chirtoc, The thermal contact problem in nano- and micro-scale photothermal measurements, Eur. Phym. J. Special Topics 153 (2008) 335–342. [25] H. Fischer, Quantitative determination of heat conductivities by scanning thermal microscopy, Thermochimica Acta 425 (2005) 69–74. [26] H. Fischer, Calibration of micro-thermal analysis for the detection of glass transition temperatures and melting points, Journal of Thermal Analysis and Calorimetry 92 (2) (2008) 625–630. [27] J. Martinek, P. Klapetek, A. Charvátová Campbell, Methods for topography artifacts compensation in scanning thermal microscopy, Meas. Sci. Technol. 155 (2015) 55–61. [28] J.S. Villarubia, Algorithms for scanned probe microscope image simulation, surface reconstruction, and tip estimation, J. Res. Natl. Inst. Stand. Technol. 102 (1997) 425. [29] P. Klapetek, I. Ohlídal, J. Buršík, Applications of scanning thermal microscopy in the analysis of the geometry of patterned structures, Surf. Interface Anal. 38 (2007) 383–387. [30] D.M. Price, The Development and Applications of Micro-thermal Analysis and Related Techniques, PhD thesis, Loughborough University, 2002. [31] P. Klapetek, J. Martinek, P. Grolich, M. Valtr, N.J. Kaur, Graphics cards based topography artefacts simulations in Scanning Thermal Microscopy, Int. J. of Heat and Mass Transfer 108 (2017) 841–850. [32] S. Gomes, A. Assy, P.-O. Chapuis, Scanning thermal microscopy: a review, Physica Status Solidi A 212 (3) (2015) 477–494. [33] B. Gotsmann, M.A. Lanz, A. Knoll, U. Dürig, Nanoscale thermal and mechanical interaction studies using heatable probes, in: H. Fuchs (Ed.), Nanoprobes, in: Nanotechnology, vol. 6, Wiley–VCH, 2009.

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