A Numerical Fitting Routine for Frequency-domain Thermoreflectance Measurements of Nanoscale Material Systems having Arbitrary Geometries
Ronald J. Warzoha, Adam A. Wilson, Brian F. Donovan, Andrew N. Smith, Nicholas T. Vu, Trent Perry, Longnan Li, Nenad Miljkovic, Elizabeth Getto
AA Numerical Fitting Routine for Frequency-domain ThermoreflectanceMeasurements of Nanoscale Material Systems having Arbitrary Geometries
Ronald J. Warzoha , Adam A. Wilson , Brian F. Donovan , Andrew N. Smith , Nicholas T. Vu , TrentPerry , Longnan Li , Nenad Miljkovic , and Elizabeth Getto Department of Mechanical Engineering, United States Naval Academy, Annapolis, MD 21402, USA Research Scientist, CCDC Army Research Laboratory Adelphi, MD 20783, USA Department of Physics, United States Naval Academy, Annapolis, MD 21402, USA Department of Mechanical Science and Engineering, University of Illinois, Champaign, IL 61821, USA* Corresponding Author, [email protected]
Abstract
In this work, we develop a numerical fitting routine to extract multiple thermal parameters using frequency-domain thermoreflectance (FDTR) for materials having non-standard, non-semi-infinite geometries. Thenumerical fitting routine is predicated on either a 2-D or 3-D finite element analysis that permits the inclusionof non semi-infinite boundary conditions, which can not be considered in the analytical solution to the heatdiffusion equation in the frequency domain. We validate the fitting routine by comparing it to the analyticalsolution to the heat diffusion equation used within the wider literature for FDTR and known values ofthermal conductivity for semi-infinite substrates (SiO , Al O and Si). We then demonstrate its capacityto extract the thermal properties of Si when etched into micropillars that have radii on the order of thepump beam. Experimental measurements of Si micropillars with circular cross-sections are provided and fitusing the numerical fitting routine established as part of this work. Likewise, we show that the analyticalsolution is unsuitable for the extraction of thermal properties when the geometry deviates significantly fromthe standard semi-infinite case. This work is critical for measuring the thermal properties of materials havingarbitrary geometries, including ultra-drawn glass fibers and laser gain media. Introduction
Measurements of thermal transport properties in nanoscale thin-films are conventionally made using opticalpump-probe thermoreflectance techniques, principally due to the non-contact nature with which they areable to interrogate nanoscale thermal transport characteristics for nearly any material type [1]. In contrastto Raman spectroscopy [2] and 3- ω [3] techniques, thermoreflectance-based measurements can separatethe impacts of thermal boundary conductance (G) across interfaces and thermal conductivity ( κ ) withinindividual material layers [4] and have overwhelmingly served as the thermal characterization technique ofchoice for nanoscale material systems over the course of the last decade [1, 5–8]. However, current modelsused to extract the thermal properties of nanoscale materials limit the geometries that can be interrogated tothose which are (1) semi-infinite in the radial direction and (2) have finite or semi-infinite thickness transverseto the direction of the applied heat source [9]. In this work, we establish a finite element-based numericalfitting routine in order to extend the utility of thermoreflectance techniques for use with any planar geometryhaving finite dimensions.The two thermoreflectance systems most used over the course of the last decade include time-domainthermoreflectance (TDTR) [10] and frequency-domain thermoreflectance (FDTR) [11]. Both techniques relyon two separate laser beams to (1) heat the sample surface and (2) probe the reflectivity (i.e. temperature)on the sample surface. The beams that heat and probe the surface are referred to as the “pump” and “probe”beams, repsectively. Typically, a metal transducer (50-150 nm of Au or Al) is deposited on the sample surfacein order to convert the optical energy of the pump beam to thermal energy prior to the sample surface and due to a well-established wavelength-dependent relationship between the reflectivity of the transducer and itssurface temperature (often referred to as the coefficient of thermoreflectance) [12, 13]. The pump beam ismodulated at a single frequency (TDTR) or across a range of frequencies (FDTR) such that we can use alock-in amplifier to detect small changes in the reflectance as heat penetrates into the sample. In TDTR,a pulsed laser source is used and split into two different pump and probe paths. A delay stage is used tophysically delay the arrival of the probe beam relative to the arrival of the pump beam to monitor changes in1/13 a r X i v : . [ phy s i c s . a pp - ph ] S e p eflectance (i.e. temperature) at the sample surface over time. On the other hand, FDTR utilizes two separatecontinuous wave (CW) lasers to establish pump and probe beams, where the pump beam is modulated over arange of frequencies. Modulating frequency allows for corresponding changes to the penetration depth of theheat deposited by the pump beam and thus establishes sensitivity to multiple thermal properties and/or thethermal properties of several underlying layers of material in a multi-layer stack [9].Recent advances in thermoreflectance-based techniques include extensions to a steady-state system togain sensitivity to so-called “buried interfaces” [14], the use of a magneto-optical kerr effect (MOKE) to gainsensitivity to interfaces having large thermal conductance [15] and phonon-magnon coupling effects [16] andthe development of a transient thermo-transmission technique to measure the thermal boundary conductanceof nanoparticles suspended in transparent media [17]. Recently, transient grating spectroscopy has also beenused to probe non-diffusive thermal regimes in films that are suspended over micron-sized regions [18–20].However, conventional measurements are still limited to geometries that are semi-infinite, which still limitsthe utility of the technique to material systems that can be fabricated in such a way. For instance, thesecharacterization techniques are blind to the thermal properties of material systems whose thermal propertiesare expected to change with geometry, such as ultra-drawn glass fibers (i.e. fiber-optic components), strainedpolymers and laser gain media. In this work, we integrate a finite element-based numerical simulation(constructed in COMSOL Multiphysics v. 5.5) into the conventional fitting routine for frequency-domainthermoreflectance measurements such that the thermal properties of nanoscale and microscale materialsystems having non-standard geometries can be accurately extracted. We take measurements on standardsubstrates having well-known thermal properties (SiO , Al O and Si) to validate the numerical model againstvalues obtained analytically and to those that exist within the wider literature. We then demonstrate thedifference in the numerical solution to the phase lag (i.e. temperature response) at the sample surface due tochanges in the radial geometry of Si in the form of Si micropillars with varying height. Finally, we obtain thethermal conductivity of Si when in micropillar form using the numerical fitting routine and an experimentalmeasurement made using FDTR. Experiment
In this work, FDTR is used to characterize the thermal properties of bulk substrates and Si micropillars. Pastworks provide a detailed description of the FDTR system used in this effort [21, 22]. Briefly, FDTR measuresthe phase lag at the sample surface relative to the imposed phase applied by the modulated heating event.One useful analogy (despite occurring in the time-domain) is the lag in the temperature response of waterrelative to the temperature of a stove-top’s heater. Because our experiment applies a sinusoidal modulationto the heating event, we can observe the temperature response at the sample surface by measuring the phaselag at the sample surface. A representative schematic of the mechanism used to track temperature on thesample surface is provided in Fig. 1.
Figure 1. ( a ) Time-domain representation of modulated heatsource (blue) and temperature response (green), where the peak-to-peak difference is the phase lag as measured by the lock-inamplifier ( φ LI ) and ( b ) Phase lag vs. measured beam intensity.We note that the temperature difference between the probe beamand the probe reference (i.e. the temperature rise/fall at thetransducer surface in response to the modulated heating event)is very small, requiring the use of a lock-in amplifier). In Fig. 1 (a), the blue curve represents thepower of the pump laser, which is modulatedbetween an upper and lower power (1 and -1,respectively). The corresponding temperatureresponse of the surface is shown in green, whichlags behind the applied temperature and fluctu-ates between and upper and lower temperature(again 1 and -1, respectively). The differencein the phase between the pump and probe isthe phase lag measured by the lock-in ampli-fier, φ LI , and this provides information on thethermal properties of the material when usedin tandem with a well-established multi-layeranalytical model [11].Figure ?? (b) displays the intensity of thepump beam as well as the intensity of the re-flected probe. The intensity and apparent mod-ulation of the probe beam comes from the tem-perature response of the material induced bythe pump. The intensity of the reflected probedepends on the surface reflectivity, which is pro-portional to the change in temperature. The 2/13 igure 2. Schematic of frequency-domain thermoreflectance system built at USNA. Note the following acronyms:PBS (polarizing beam splitter), WP (waveplate) and EOM (electro-optic modulator). probe reference that did not interact with the surface—and therefore remains unmodulated—is shown by thedotted black line. By subtracting the probe signal from the reference and taking into account the coefficientof thermoreflectance, the temperature response of the sample can be determined.A schematic of the FDTR system used in this work is shown in Fig. 2. The system contains twoseparate continuous-wave lasers that act as the “pump” (405 nm Coherent OBIS CW laser) and “probe” (532nm Coherent OBIS CW laser) respectively. We note that at these wavelengths, the transducer absorbs asignificant amount of the pump beam and we are highly sensitive to changes in the reflectivity of the samplesurface by the probe beam.The pump beam is first split into two separate paths: (1) through the electro-optic modulator (EOM,Conoptics Model 350-160) and (2) into the balanced photodetector (Thorlabs, PDB450A-AC) that tracksthe reference phase via a 1% beam splitter. The small amount of pump beam that is immediately directedinto the photodetector is used as a reference to subtract any coherent noise within the laser. The portionof the pump beam that is routed through the EOM is modulated using the built-in waveform generator inour lock-in amplifier (Zurich UHFLI). After exiting the EOM, the pump beam is again spit into separate“primary” (downward direction) and “reference” (leftward direction) paths using a polarizing beam splitter.The primary path is steered into an objective lens (Mitutoyo 50x) using a dichroic mirror and focused ontothe sample surface. The reference path is used to track the applied phase of the pump beam at the samplesurface. In order to match the phase at the sample surface, we modulate at the highest frequency usedin our measurement ( ω max = 20 MHz) and match the numerical phase of the pump beam that leaks intothe primary balanced photodetector (shown on the left of the image) while blocking the probe beam. Thisallows us to measure the phase response at the sample surface (measured with the probe beam) and theimposed phase at the sample surface (applied via the pump beam) simultaneously , which is unique to ourimplementation of FDTR. By subtracting the two signals, we obtain φ LI as described previously.Provided with a measurement of φ LI , one can obtain the underlying thermal properties across an interface(i.e. the thermal boundary conductance, G) or within an individual layer of a multilayer stack (e.g. the thermalconductivity, κ ). The entire formulation of the analytical expression used to extract thermal properties isdescribed in detail elsewhere [11]. However, it is useful to describe several features of the analytical solution tothe frequency-domain version of the heat diffusion equation in order to demonstrate its geometric limitations.The analytical solution is constructed within the framework of a multi-layer material stack whose substrateis semi-infinite in the radial direction, and most often semi-infinite in the through-thickness direction. Aschematic of the general multilayer material stack used in the development of the analytical solution isprovided in the figure below.In the time domain, the equation governing heat diffusion is expressed as, C v ∂θ∂t = κ r r ∂∂r (cid:18) r ∂θ∂r (cid:19) + κ z ∂ θ∂z (1)3/13 igure 3. Arbitrary multi-layer material stack with semi-infinite boundary conditions in r- and z-directions. Pump(blue) and probe (green) beams are depicted above the Au transducer.
The above expression accounts for 2-D heat flow in the radial (r) and through-plane (z) directions. A Fouriertransform is applied to obtain the frequency-dependent heat diffusion equation, written as, κ z ∂ θ ( ω, k, z ) ∂z = ( κ r k + C v iω ) θ ( ω, k, z ) (2)where q is defined for a layer of material n and thickness d as q = κ r k + C v iωκ z (3)We can relate the temperature to the heat flux at the top surface (subscript t ) of a slab made of a certainmaterial in the frequency domain with the bottom surface (subscript b ) using, (cid:20) θ n,b f n,b (cid:21) = (cid:20) cosh ( qd ) − κ z q sinh ( qd ) − κ z ∗ q ∗ sinh ( qd ) cosh ( qd ) (cid:21) (cid:20) θ n,t f n,t (cid:21) (4)The temperature and heat flux between the bottom surface of material n are connected to the top of material n + 1 via (cid:20) θ n +1 ,t f n +1 ,t (cid:21) = (cid:20) − G − (cid:21) (cid:20) θ n,b f n,b (cid:21) (5)where G is the thermal boundary conductance between the two layers. The heat flux boundary condition ofthe top, f t , can be found with f t = A π exp (cid:18) − k w o (cid:19) (6)which is the Hankel transform of a Gaussian spot with a power of A and a 1 /e radius of w . If there aremultiple layers, the solution can be found with (cid:20) θ b f b (cid:21) = M n M n − ... M M = (cid:20) A BC D (cid:21) (cid:20) θ t f t (cid:21) (7)where M n is the matrix of the bottom layer. If the bottom layer is treated as adiabatic or semi-infinite, thesurface temperature can be found using: θ t = − DC f t (8)4/13he final frequency response, H ( ω ), is found by taking the inverse Hankel transform of Eqn. 2 and weightingit with a Gaussian spot with a 1 /e radius of w : H ( ω ) = A π (cid:90) ∞ k (cid:18) − DC (cid:19) exp (cid:18) − k ( w + w (cid:19) dk (9) Figure 4.
Measured data (open black circles) andcorresponding analytical fits for SiO (solid blue line),Al O (solid red line) and Si (solid purple line) sub-strates. Pump and probe diameters are 5.7 µ m and3.7 µ m, respectively. The thermal model for H ( ω ) is then fitted to the lock-inphase data. By changing the parameters of the thermalmodel to fit the lock-in data, the thermal properties canbe determined. The lock-in phase data measured is givenby φ LI = tan − (cid:61) ( H ( ω )) (cid:60) ( H ( ω )) + φ ext (10)where (cid:61) ( H ( ω )) is the out-of-phase signal, (cid:60) ( H ( ω )) is thereference signal, and φ ext is the external phase shift causedby other aspects not caused by changes in reflectively,such as the optical path length, driving electronics, andphotodetectors. Thermal properties are extracted byfitting Eqn. 10 to measurements of the phase lag viaFDTR for SiO , Al O and Si, shown in Fig. 4, below.The thermal properties obtained above are consistent withthose widely reported in the scientific literature [14]. Thethermal boundary conductance for all three samples ishigh relative to other values reported in the literature [14];however, we utilize an ∼ Figure 5.
Sensitivity (S) to the extracted thermalproperties (where each property is represented by thesubscript “x”) for each substrate shown in Fig. 4.
To determine whether a particular thermal property( κ , C v and G) can be extracted across the range of mod-ulation frequencies applied on the sample surface, wecan determine the sensitivity of each parameter to smallperturbations in the measured phase lag.The phase sensitivity to a particular thermal property, x , at a given frequency can be found with S ( ω ) = ∂φ ( ω ) ∂ ln x (11)The sensitivity to the thermal properties shown in Fig. 4is provided in Fig. 5.A two-dimensional (2D) finite element model is de-veloped in order to extract thermal properties from ge-ometries with non semi-infinite boundary conditions. Themost palatable system to demonstrate the utility of thefinite element model is one in which the geometry is con-fined in the radial direction and one that has been widelyfabricated in laboratory environments. Consequently, wechoose to develop the model based on Si micropillar arrays.The arrays we interrogate have geometries that vary inboth the radial and through-plane directions on the orderof single-digit µ m to 10’s of µ m. As these length remainlarger than the mean free path of phonons in Si [23], thegeometric confinement should not result in any change in κ Si . However, it is likely that the phase lag ( φ LI ) doeschange due to the confinement of heat (recall that thephase lag represents the response in the temperature onthe top of the transducer relative to the modulated signalof the applied heating event). 5/13he finite element model is created in COMSOL Multiphysics v. 5.5. We note here that COMSOL isused with its LiveLink module in order to communicate with Matlab. The Si micropillar is modeled usingthe computational domains shown in Fig. 6. We note that we incorporate the Si substrate and the Simicropillar within our model. An 80 nm Au transducer is constructed above the pillar and a finite valuefor thermal boundary conductance is applied at the Au/Si micropillar interface (in our model this remainsa free parameter). The Si micropillar and the Si substrate are “continuous” in the sense that there is noapplied thermal boundary conductance at the interface between domains. This is physically appropriategiven the nature of the fabrication process; the Si micropillars themselves are etched and are never physicallyseparated/reattached during the process.The numerical model itself is meshed using a graded grid in each independent sub-domain shown in Fig.6, including the Au transducer layer, the Si micropillar, the region immediately below the Si micropillar (i.e.from r = 0 to r = r pillar in the Si substrate) and the remainder of the Si substrate. An image of the meshedsub-domains is provided in Fig. 7. Note that a mesh independence study was completed to ensure that thesolution was independent of the number of nodal points in each sub-domain. As shown, the mesh size (i.e. the size of each mesh element) is increased downward (z → ∞ ) and to the right(r → ∞ ) in sub-domains 2, 3 and 4. However, the mesh size decreases in the downward direction (z → ∞ ).This is done in an effort to capture the relevant thermal transport physics at the transducer/pillar interface(i.e. the thermal boundary conductance, G). This is particularly important when (1) we are sensitive to Gand (2) we need to extract κ independent of G (as we do here). Prior to using the numerical simulationdescribed here to fit any data, we use it to fit to the data in Fig. 4. Figure 6.
Schematic of numerical model built in COMSOL mul-tiphysics. h pillar represents the pillar height, t Si is the thicknessof the Si substrate (approximated as semi-infinite, but modeled as500 µ m in the computational domain), r pillar is the radius of themicropillar, r Si is the radius of the Si substrate (approximatedas semi-infinite, but modeled as 300 µ m) and w and w arethe pump and probe radii, in µ m. Purple boundaries representthose boundaries that are insulated, blue boundaries representboundaries which are held at a constant initial temperature, T = 300 K and red boundaries are symmetric. The numerical model described in the previoussection is validated by fitting the measured datain Fig. 4 and comparing it to both the resultinganalytically determined thermal properties andthose available within the wider literature. Theonly changes to the models shown in Figs. 2and 7 include: (1) removal of the pillar domainand (2) the transducer is split into two separatedomains above the Si substrate (one from r =0 to r = 2 · (w + w ) and the other from r =2 · (w + w ) to r → ∞ ). This effectively modelsthe semi-infinite nature of the substrates shownin Fig. 4, which are blanket coated with 80 nmAu and 5 nm Ti. Numerical fits to the dataare provided in the Supporting Information andmatch to well within 1% of the analytical fitsprovided in Fig. 4. Consequently, we considerthe model used in this work to be valid forfits to the thermal properties of Si when thegeometry is confined in the radial direction (e.g.Si micropillars). The micro-pillars were fabricated using the fol-lowing procedure (depicted schematically in Fig. 8). AZ5214E photoresist was spun on to the surface ofa double-side polished silicon wafer (with spin rate of 4000 rpm, leading to a photoresist thickness of 1.7 µ m, which was soft baked at 120 ◦ C for 45 s). The patterned features were then written using a HeidelbergVPG200++ laser writer. The laser power was kept low enough for image reversal to successfully occur. Thewafer was then baked at 110 ◦ C for 60 s to cross-link the photoresist on the written features, and the entire6/13afer was flood exposed with a UV lamp with ample dosage to successfully complete the image reversalprocess. The wafer was then developed in 4:1 diluted AZ400K photoresist developer for 30 s. The photoresistthat was not exposed during the laser write, but that was exposed during the flood exposure washed awayin the developer, whereas on the written features, the photoresist remained. The wafer was then cleanedand placed in a deep silicon etch tool (PlasmaTherm DSE), where the silicon was etched away to a depthof 29.5 ± µ m everywhere but where the photoresist remained. Then the wafer was soaked in acetone toremove the photoresist from the written features, and the surface was cleaned in solvent rinse. Finally, ablanket-coating of 5 nm Ti and 80 nm Au was deposited by electron beam evaporation on the surface toserve as the FDTR transducer layer. Figure 7.
Mesh for Si micropillar on Si substrate with Autransducer. Entire domain contains 64,000 elements split equallyamong four distinct sub-domains (numbered 1-4).
SEM images of the micropillars used in thiswork are provided in Fig. 9. We note that whilewe only use a single pillar for characterizationin this proceedings, an array of pillars was fabri-cated for use in future work. As shown, squarepillars were also created but are not reportedon in this work.For this study, pillar C (r pillar = 50 µ m) isused and measured with pump and probe radiiof 16.8 µ m and 12.8 µ m, respectively. The ratioof pillar radius to pump radius (R = r pillar /w )is therefore R ≈
3. The pillar height is alsomeasured by SEM as h ≈ µ m (see SupportingInformation). Finally, electron beam (e-beam)evaporation is used to deposit an 80 nm Autransducer above the surface of the pillars forthermal characterization with FDTR.In this section, we provide quantitative re-sults that suggest the phase lag (i.e. temper-ature) response at the sample surface is sub-stantially different for the confined geometrycase (e.g. when r → ∞ ) across a wide range offrequencies. After we demonstrate this quanti-tative difference, we use our numerical fittingroutine to determine the thermal conductivity of Si when in micropillar form in order to demonstrate theutility of the technique and distinguish it from the analytical solution. Consequently, we show that (1) theanalytical solution is insufficient to capture the phase lag (and therefore the temperature response) at thesample surface for confined geometries and (2) a unique numerical fitting routine can be used to capture thephysics that govern thermal transport in novel geometries, paving the way for the thermal characterization ofmicro- and nanoscale materials with thermal properties that are expected to depend on material geometry ,such as ultradrawn glass fibers and laser gain media. Figure 8.
Schematic representation of the process flowto prepare the pillar arrayed investigated in this work.
We computationally model a Si pillar with a 20 µ mheight in order to highlight the expected effects thatpillar diameter has on the phase lag at the sample sur-face. The pump and probe diameters we use are 7.55 µ m and 1.55 µ m, respectively, which is consistent withthose used in [24] (and therefore provides secondaryverification of the numerical results). The thermalboundary conductance is fixed at G = 35 MW m − K − (consistent with [24]), κ = 144 W m − K − andC v = 1.65 MJ m − K − (both consistent with valuesreported in [14] for Si). To demonstrate the power ofthe numerical model, we plot the phase lag as a function of modulation frequency in Fig. 10 as r pillar isvaried from r pillar = w to 6 · w .As shown in Fig. 10, the phase lag in the plot on the left is drastically altered as r pillar → w , andbegins to converge on with the analytical solution for a semi-infinite Si substrate (black dashed line) athigh frequencies. This makes physical sense in that, at higher frequencies, the thermal penetration depth is7/13educed [25] and therefore becomes less influenced by the confinement of heat at the pillar boundary. Ata low modulation frequencies (e.g. ω = 4 kHz, as used for the average temperature images in the right ofFig. 10), one can see that the influence of the boundary on the temperature distribution becomes morepronounced as r pillar → w . Figure 9.
Scanning Electron Microscope image of pillararray used for this work. Note: only the three smallestcircular pillars are shown, with diameters of (A) 20 µ m,(B) 40 µ m and (C) 100 µ m. The height of the pillar is also expected to alterthe phase lag at the sample surface relative to the caseof a semi-infinite substrate. Here, the expected phaselag is simulated with heights that range from h pillar =1 µ m to 50 µ m. Figure 11 provides the relative impactof pillar height on phase lag across a range of pillarradii (r pillar = w to 4 · w ).Figure 11 reveals that for comparatively low pillarheights and large pillar radii, the phase lag at thesample surface approaches what we would expect tosee for measurements on bulk substrates. This isevident in the top left plot of Fig. 11, where pillarradii greater than 2 · w sit neatly on the phase lagcurve for semi-infinite Si (green dashed line). On theother hand, as the Si micropillar height becomes morepronounced, the phase lag at the sample surface beginsto deviate drastically from the phase lag in the semi-infinite case. This is consistent with what we wouldexpect physically; that is, as the pillar height becomessmaller, the semi-infinite substrate below it has anincreasing impact on the temperature response at thesample surface. As before, a decreasing pillar radius also results in increasing deviation from the phaselag. Because these deviations are so apparent, we expect to be sensitive to the geometry of the pillar itself,permitting accurate experimental measurements of thermal transport properties via FDTR. Figure 10. ( left ) Expected phase lag at the transducer surface as a function of modulation frequencyfor different values of pillar radius (relative to the pump radius, w ), ( right ) Cross-sectional view of thetemperature rise within the sample at a modulation frequency of ω = 4 kHz (relative to the initial temperaturewith no applied power, T ) with an applied power of A = 50 mW. Here we provide several measurements of the phase lag at the sample surface for Si micropillars that arebetween 28 and 33 µ m tall and have varying radii. For this work, we utilize the one of the circular micropillarsshown in Fig. 9 (pillar C with r pillar = 49.98 µ m and h pillar = 29 µ m as measured via SEM). We note thatwe can vary the pump and probe radii used in our FDTR measurements by changing the objective lensmagnification. This study utilizes a 20x objectives to establish pump and probe radii of 16.8 µ m and 12.8 µ m, respectively. 8/13 igure 11. Phase lag vs. modulation frequency for ( top left ) h pillar = 1 µ m, ( top right ) h pillar = 2.5 µ m, ( bottomleft ) h pillar = 6.3 µ m and ( bottom right ) h pillar = 50 µ m. As previously described, our numerical technique is used to extract the thermal properties of the pillarmaterial below the transducer. The phase lag is fit to measured FDTR data with an applied pump power ofA = 7 mW, which produces a ∆T < κ ≈ · m − · K − , which remains consistent with the wider literature and within the measurement uncertaintyfor the conditions described in this work (U κ = 9.5%) relative to the measurements made on semi-infinitesubstrates.We note here that an attempted fit of the data using the analytical model was not possible (shown in Fig.12 is the expected phase lag when the sample is semi-infinite in order to clearly identify that the micropillargeometry has an impact on the phase lag at the transducer surface). A comparison between the obtainedthermal properties for the semi-infinite geometry and the micropillar geometry are provided in Table 1. Property SI Analytical SI Numerical MP Numerical κ (W m − K − ) 139 ± ± ± v (MJ m − K − ) 1.5 ± ± ± Table 1.
Comparison of thermal properties for semi-infinite (SI) samples using analytical and numericalfitting routines to micropillar (MP) numerical fitting routine. G is not shown due to lack of sufficientsensitivity.Table 1 suggests that the formulation described here appropriately captures the thermal propertiesof materials whose geometries are confined, and are quite different from those solutions which use thestandard analytical fitting routine. Consequently, current analytical formulations [11] for resolving thethermal conductivity of materials having non semi-infinite geometries are insufficient to resolve their thermalproperties.To provide further evidence that the fitting routine is able to capture the effects of the adiabatic boundaryon the phase response at the upper surface of the pillar, we use our fitting routine to determine the thermalconductivity of Si micropillars having different diameters. 9/13 igure 12.
Numerical fit to the thermal properties of the Si micropillar and comparison to the expected phase lagfor a completely semi-infinite geometry.
Figure 13.
Phase lag vs. frequency for micropillars with d pillar = 100 µ m (blue) and 200 µ m (red) and pump beamdiameters of w = 21.2 µ m (circles) and 35.7 µ m (diamonds). Solid and dashed lines represent the numerical fits toour data for the 21.2 µ m and 35.7 µ m diameter pump beams, respectively. We also utilize different pump sizes than those used in Fig. 12 (in this case, w = 21.2 µ m, and w = 35.7 µ m). We first compare the impact of the relative size of the pump beams on the phase lag for micropillardiameters of 100 µ mand 200 µ m) in order to highlight the sensitivity of the phase lag to the presence ofconfined boundaries, as shown in Fig. 13.In Fig. 13, the discrepancy between the phase lag for each of the two pump diameters is more pronouncedfor the pump beam with the larger diameter. This suggests that as the size of the pump beam is increased10/13 igure 14. Phase lag vs. frequency for micropillars of varying size with w = 35.7 µ m. relative to the size of the pillar, the boundaries start to play a larger role on the temperature at the surfaceof the transducer.We observe the same phenomenon when we hold the pump diameter constant and reduce the pillarcross-section, as shown in Fig. 14. In Fig. 14, the phase lag is shown to be dependent on the ratio of thepump diameter to the diameter of the pillar. As the pump beam becomes more confined by the geometry ofthe pillar, its impact on the phase lag in the low frequency regime is more pronounced. We note that allfits resulted in a thermal conductivity of κ = 145 W/m · K, which is consistent with that reported in thewider literature. This fitting routine is therefore critical when characterizing materials that are expectedto have thermal properties that are dependent on finite geometries, such as ultra-drawn glass and polymerfibers [26, 27] and laser gain media.
CONCLUSIONS
In this work, we develop a numerical model that is capable of fitting the phase lag of FDTR measurementson samples having non semi-infinite geometries. We show that, for a known geometry, we can extractaccurate values of thermal conductivity, volumetric heat capacity and the thermal boundary conductanceacross the transducer/material interface for a non-typical geometry (in this case, a Si micropillar). Theimpact of the substrate is included to incorporate the impact of thermal spreading at low pump beammodulation frequencies, which we clearly demonstrate in a numerical parametric analysis. Given this newfitting paradigm, we expect thermal scientists to be able to characterize materials having geometry-dependentthermal properties, to include ultra-drawn glass fibers and laser gain media.
ACKNOWLEDGEMENTS
RJW and BFD would like to acknowledge primary financial support from Mr. Peter Morrison and the Officeof Naval Research (N001420WX00381). We also thank Dr. Mark Spector and ONR for equipment support(N0001420WX01170). 11/13 eferences
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