International Journal of Heat and Mass Transfer 214 (2023) 124376 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt In-plane thermal diffusivity determination using beam-offset frequency-domain thermoreflectance with a one-dimensional optical heat source Kai Xu a, Jiali Guo a, Grazia Raciti b, Alejandro R. Goni a,c, M. Isabel Alonso a, Xavier Borrisé d, Ilaria Zardo b, Mariano Campoy-Quiles a, Juan Sebastián Reparaz a,∗ a Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, Bellaterra 08193, Spain b Physics Department, University of Basel, Klingelbergstrasse 82, Basel CH-4056, Switzerland c ICREA, Passeig Lluís Companys 23, Barcelona 08010, Spain d Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and the Barcelona Institute of Science and Technology (BIST), Building ICN2, Campus UAB, Bellaterra E-08193, Barcelona, Spain a r t i c l e i n f o a b s t r a c t Article history: We present an innovative contactless method suitable to study in-plane thermal transport based on Received 7 March 2023 beam-offset frequency-domain thermoreflectance using a one-dimensional heat source with uniform Revised 16 May 2023 power distribution. Using a one-dimensional heat source provides a number of advantages as compared Accepted 1 June 2023 to point-like heat sources, as typically used in time- and frequency-domain thermoreflectance experi- Available online 22 June 2023 ments, just to name a few: (i) it leads to a slower spatial decay of the temperature field in the direction Keywords: perpendicular to the line-shaped heat source, allowing to probe the temperature field at larger distances Thermal diffusivity tensor from the heater, hence, enhancing the sensitivity to in-plane thermal transport; (ii) the frequency range Contactless method of interest is typically < 100 kHz. This rather low frequency range is convenient regarding the cost of In-plane heat transport the required excitation laser system but, most importantly, it allows the study of materials without the Thermal anisotropy presence of a metallic transducer with almost no influence of the finite optical penetration depth of the Line-shaped 1-dimensional heat source pump and probe beams on the thermal phase lag, which arises from the large thermal penetration depth Patterned transducers imposed by the used frequency range. We also show that for the case of a harmonic thermal excitation source, the phase lag between the thermal excitation and thermal response of the sample exhibits a lin- ear dependence with their spatial offset, where the slope is proportional to the inverse of the thermal diffusivity of the material. We demonstrate the applicability of this method to the cases of: (i) suspended thin films of Si and PDPP4T, (ii) Bi bulk samples, and (iii) Si, glass, and highly-oriented pyrollitic graphite (HOPG) bulk samples with a thin metallic transducer. Finally, we also show that it is possible to study in-plane heat transport on substrates with rather low thermal diffusivity, e.g., glass, even using a metallic transducer. We achieve this by an original approach based on patterning the transducer using focused ion beam, with the key purpose of limiting in-plane heat transport through the thin metallic transducer. © 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 1. Introduction tificial anisotropy” induced by low dimensionality in the case of crystalline (or semi-crystalline) thin films. A large number of ex- The study of the thermal conductivity (or diffusivity) tensor perimental methods to study the out-of-plane components of the (κi j ) in bulk and low dimensional materials has gained consider- thermal conductivity tensor have been developed and successfully able momentum in recent years. For example, in layered materials demonstrated using different methodologies, e.g., based on elec- where the in-plane and out-of-plane components of the thermal trical or optical methods [1–19]. On the other hand, the study conductivity exhibit strong anisotropy, or for the case of the “ar- of in-plane thermal transport is comparatively more challenging due to the lack of sensitivity to this component of most meth- ods, among other reasons. In fact, most experimental approaches ∗ Corresponding author. to study thermal anisotropy are based in achieving experimental E-mail address: jsreparaz@icmab.es (J.S. Reparaz). https://doi.org/10.1016/j.ijheatmasstransfer.2023.124376 0017-9310/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 sensitivity to in-plane thermal anisotropy, which has been demon- their in-plane spatial offset (x), where the slope, ∂ φ ( f )/∂ x, is strated through different experimental configurations by several proportional to the thermal diffusivity of the sample for a given research groups [2,5–8,10,15,20–28]. Electrical methods such as, modulation frequency, f , of the heater. The linear relation between e.g., the recent extension of the 3-Omega method [23,24], where the phase lag and the spatial offset considerably simplifies the data the heater and thermometer are located in different spatial posi- analysis process, i.e., the thermal diffusivity (or thermal conduc- tions on the surface of the sample, have been proven successful to tivity) of the samples is readily obtained through a linear fit of study thermally anisotropic materials, however, these approaches φ (x, f ). Furthermore, the one-dimensional character of the heat result most convenient for electrically insulating samples, other- source sets the frequency range of interest to f < 100 kHz, which wise, the metallic transducer must be electrically insulated from allows the study of materials without the presence of a metallic the sample, a task that in some cases can be challenging due to transducer with almost no influence of the finite optical penetra- the presence of current leakage from the metallic transducer to the tion depth of the pump and probe beams on the thermal phase lag sample. Additionally, electrical methods also require considerable [38]. fabrication efforts since a specific sample must be fabricated for We apply this method to study the in-plane thermal diffusiv- each in-plane direction that aims to be investigated. Alternatively, ity of 2D and 3D materials, which in combination with the 1D several contactless methods typically based on time- or frequency- heat source leads to different heat flow geometries. In particular domain thermoreflectance, have also been successfully demon- we studied: (i) suspended Si and PDPP4T (poly(3-([2,20:50,20’- strated to address thermal anisotropy [9,11,14,19,29,30]. Three main terthiophen]-5-yl)-2,5-bis(2-decyltetradecyl)-6-(thiophen-2-yl)- experimental strategies were followed: (i) beam-offset experiments 2,5-dihydropyrrolo[3,4-c]pyrrole-1,4-dione)) thin films with dif- based on time- or frequency domain thermoreflectance using a fo- ferent thicknesses (quasi-2D samples; 1D heat flow), (ii) Bi bulk cused Gaussian (0D, isotropic) heat source [8,14,15,29], (ii) collinear sample without metallic transducer (3D sample; 2D heat flow), beam experiments using an elliptically-shaped (2D, anisotropic) and (iii) Si and glass substrates with a 60 nm thick Au transducer heat source, Li et al. [11], Jiang et al. [29] and (iii) collinear beam (3D samples; 2D heat flow). Finally, we generalize the method experiments using a line-shaped (1D, anisotropic) heat source for situations where the use of a metallic transducer is unavoid- [19] (resembling the geometry used in the 3-Omega method but able showing that it is possible to control (and avoid) in-plane in a contactless fashion). Although all these methodologies have heat flow within the transducer by patterning, hence, enhancing been proven successful to determine the complete elements of the the sensitivity to the in-plane thermal diffusivity of the sample thermal conductivity tensor, they present some aspects which have independently of the presence of the metallic transducer. the potential to be improved as we demonstrate through the novel approach developed in this work. For example, the methods devel- 2. Solution of the heat equation for 1D heat sources oped within (i) and (ii) typically suffer from the influence of the shape of the heat source (actually its spatial energy distribution) In this section we provide the solution of the heat diffusion on the acquired data, which originates from the rather small spa- equation (parabolic approximation) for the case of a harmonic 1D tial offsets that can be set between the heat source and the probe. heat source with uniform power distribution along its principal Typically, for 0D heat sources (approximated as focused Gaussian axis, applied to quasi-2D and 3D materials. We note that the key beams) large offsets cannot be achieved due to the rapid spatial advantage of using a 1D heat source is that no heat propaga- decay of the thermal field. On the other hand, the method devel- tion occurs along the direction parallel to the line-shaped heat oped in (iii) avoids the explicit knowledge of the shape of the heat source, hence, reducing the dimensionality of the heat flow geom- source, similarly to the case of the 3-Omega method [16], but at etry and, consequently, of the corresponding heat diffusion equa- the expense of precisely knowing the temperature coefficient of tion. Figure 1 displays a schematics of each of the heat flow ge- reflectance of the surface of the sample (or transducer in most ometries we studied. cases), since the absolute temperature must be known to com- pute the thermal conductivity. Furthermore, several authors have 2.1. 1D heat flow: 1D heat source & 2D sample developed an approach based infrared (IR) detection which mea- sures the phase lag between a spot-shaped harmonic thermal ex- We start by analyzing the case of a 1D heat source with uni- citation and thermal detection using IR imaging. This method, la- form power distribution along its long axis applied to a quasi-2D beled as “slope method”, relays on the linearity between the phase material such as suspended thin membranes. This case is well de- lag and the offset between the excitation spot and the detection scribed by a 1D heat flow geometry, i.e., no heat flow is present position, and has been proven useful to determine the in-plane along the direction parallel to the line-shaped heat source. The dif- component of the thermal diffusivity for millimeter sized samples ferential equation which describes the system response is simply [31–37]. However, it has also been shown that this approach suffer the 1D heat diffusion equation as follows: from different error sources such as convection and IR diffraction ∂ 2T ∂ T QV D − =− (1) coupling to the detector at positions close to the heater region, ∂ x2 ∂ t ρC p which makes it unsuitable for micrometer sized specimens. Here, we demonstrate an original experimental approach with where T = T (x, t ) is the temperature, t and x are the temporal enhanced sensitivity to in-plane heat transport, which is based on and spatial coordinates, respectively, and x is perpendicular to the using a 1D heat source with uniform power distribution along its line-shaped heater, D is the thermal diffusivity, ρ is the density of long axis, but a point-like probe spot. We show that the 1D ge- the material, C p is the specific heat capacity, and QV is the vol- ometry of the heat source leads to a slower spatial decay of the umetric heat generation rate. We follow the solution by Salazar temperature field as compared to 0D heat sources, hence, allow- [39] where Eq. (1) is solved under harmonic excitation condi- ing to probe the temperature field at relatively large spatial offsets tions, p(t ) = p0 [1 + cos(ωt )]/2 = [ p0 (1 + eiωt )/2], where p(t ) is from the heat source. The present approach is based on measuring the time-dependent injected power. In short, the time depen- the phase lag between the thermal excitation and the detection, dent solution for T (x, t ) is obtained by separation of variables as: hence, rendering the thermal diffusivity of the studied samples. In Tac (x, t ) = [θ (x )eiωt ]. Solving the spatial part of Eq. (1) for θ (x ) addition, one of its key advantages is that, for harmonic excitation and multiplying by eiωt renders the complete solution:  p  sources, the phase lag (φ ) between the thermal excitation (line- 0 −qx iωt T (x, t ) =  e e (2) shaped heater) and the detection exhibits a linear relation with 2κ q 2 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 Fig. 1. Schematics of the different heat transport geometries studied. In all cases the heater has a line-shaped geometry, whereas the probe beam is given by a diffraction limited focused Gaussian spot. (a) A quasi-2D sample consisting on a suspended thin film. (b) Bulk sample with no transducer. (c) Bulk sample with a transducer or thin film deposited on its surface. or, due to the radial symmetry, we use the radial coordinate, r, in- √  x 2 p0 μω −x/μω π stead of the Cartesian coordinate, x. In other words, the heat flow T (x, t ) = e cos − ωt + (3) geometry is 2D, and the solution for the ideal case of an uniform 4κ μω 4 and infinitely long and narrow line-shaped heater is given by the where p0 is the maximum amplitude of the absorbed power, ω modified zeroth-order Bessel function of the second kind, K0 (qr ) is the angular frequency,κ is the thermal conductivity, D is the [16,40]. Similarly as we have proceeded in the previous section, a thermal diffusivity, μ = 2D/ω is the thermal penetration depth, harmonic thermal excitation source leads to the expression for the  and q = iω/D. Equation (3) results particularly interesting if con- time-dependent temperature oscillations, i.e., after multiplication sidering the spatial dependence of its argument, whose time in- by eiωt : dependent component corresponds to the phase lag measured in T (r, t ) = [K0 (qr )eiωt ] (7) a frequency-domain thermoreflectance (FDTR) experiment. Besides   √ the time dependent component (ωt) which averages to zero after where we recall that q = iω/D, |q| = ω/D, i−1/2 = [1 − i]/ 2, √ each cycle, the phase lag, φ = x/μω , exhibits a linear dependence i1/2 = [1 + i]/ 2, and i1/4 = −i. It is particularly interesting to on the spatial coordinate, x. Experimentally, φ (x, f ) is evidenced √ cases with |qr|  1, and |qr|  1. The case with study the limiting as the spatial offset between the line-shaped heat source (1D) and |qr|  1 (|r|  2μ), leads to the solution obtained for the case of the probe beam (0D) positions increases. Taking the derivative of the 3-Omega and ATT methods [16,19]. Here, the thermal penetra- the phase lag, φ , with respect to the spatial coordinate, x, renders: tion depth (μ) is much larger than the typical values of radial co-   ordinate, r. Note that in the previous example the pump (heater) ∂φ ω π and probe (thermometer) beams are collinear. For |qr| → 0, then = μ−1 ω = = f 1/2 (4) ∂x 2D D K0 (qr ) ∝ −ln(qr ), which is the approximation used to obtain the where ω = 2π f , and f is the excitation frequency in Hz. Taking the thermal conductivity, e.g., through the 3-Omega method [16].On second derivative with respect to f 1/2 of the previous equation re- the other hand, in the limit where |qr|  1, it can be shown [41] √ sults in: that K0 (qr ) ≈ π /2e−qr / qr. Given the nature of our experiments,  we are particularly interested in the limit with |qr|  1, which ∂ 2φ π = (5) originates in the rather large spatial offset (r = x) imposed be- (∂ x )(∂ f 1/2 ) D tween the 1D heat source and the probe spot as shown in Fig. 1. which renders the thermal diffusivity of the specimen. We remark Figure 2 (left) displays the approximations of K0 (qr ) for the limit- that taking the second order derivative of the phase lag, φ , with ing cases with |qr|  1, and |qr|  1. In Fig. 2 (right)  we show √ the respect to x and f 1/2 is a convenient approach since it avoids the error introduced by the approximation K0 (qr ) ≈ π /2e−qr / qr, in explicit dependence on the absolute calibration of the spatial and terms of the ratio between the spatial coordinate (r) and the ther- frequency scales. This is particularly relevant for the spatial coordi- mal penetration depth (μ). We consider this approximation to be nate (x), which is prone to larger determination errors as compared valid for r > μ, since the error introduced is similar to the experi- to the thermal excitation frequency ( f ). mental error (≈ 5%) Hence, the expression for the temperature os- cillations is obtained as follows: 2.2. 2D heat flow: 1D heat source and 3D sample p0 p0 π T (r, t ) = [K0 (qr )eiωt ] ≈  e−qr eiωt (8) In this section, we study the case of a 1D heat source with uni- πκl πκl 2qr form power distribution along its long axis applied to a 3D semi- where l is the length of the line-shaped heat source. The former infinite isotropic material. This case is well described by a 2D heat expression can be rearranged after taking the real part of the ar- flow geometry, since no heat flow is present along the direction gument as follows: parallel to the line-shaped heat source, similarly as in the pre-  r vious section. The differential equation which describes the sys- p0 μω −r/μ π T (r, t ) ≈ √ e cos − ωt + (9) tem response is the 2D heat diffusion equation, which we write 4 2κ l 2π r μω 8 in cylindrical coordinates given the symmetry imposed by the 1D Taking the second order derivative of the argument of the pre- heat source as follows: vious expression (φ = r/μω − ωt + π /8) with respect to r and  1 ∂ ∂T ∂T QV f 1/2 , we obtain: D·r − =− (6)  r ∂r ∂r ∂t ρC p ∂ 2φ π = (10) The solution for T (r, t ) is similar to that originally obtained for (∂ r )(∂ f 1/2 ) D the case of the 3-Omega method, [16] and its optical analogue, which is the same expression that we have obtained for the case Anisotropic Thermoreflectance Thermometry [19] (ATT). Note that of 1D heat flow in the previous section (see Eq. (5)). The origin 3 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376  different√approximations to the modified zeroth order Bessel function of the second kind, K0 (x ). The right panel shows the error introduced Fig. 2. The left panel displays the by the approximation K0 (qr ) ≈ π /2e−qr / qr, which is valid for |qr|  1, which is the limit where the thermal penetration depth is smaller than the spatial coordinate. of such similarity between the 1D and 2D heat flow cases, is the 3. Experimental details exponential limit of K0 (qr ) for |qr|  1. We note that the previous expression is not valid for |qr|  1, since in this case the spatial Figure 3 displays a schematic illustration of the experimen- dependence of the temperature field is T ∝ K0 (qr ) ∝ −ln(qr ). tal setup. The wavelength of the pump laser was set to λpump = 405 nm (Omicron A350), whereas the probe laser wavelength was 2.3. 2D heat flow: 1D heat source and 3D multilayer set to λprobe = 532 nm (Cobolt SAMBA series). The pump laser power was harmonically modulated (intracavity modulation) using The solution for the case of a 3D multilayer sample can be ob- a wave function generator (Rigol 54DG5332) between 10 Hz and tained using the approach developed by Borca-Tasciuc et al. [42] as 100 kHz, whereas the probe laser intensity was approximately con- well as the derivation for line-shaped heat sources in Refs. [18,19]. stant in time. The pump laser power was continuously increased Here, we only provide the expression that must be numerically until a temperature rise of the order of several kelvin was observed solved to obtain the solution for the temperature oscillations. The (between 1 and tens of mW depending on the thermal properties solution is obtained considering that the line-shaped heater has of the sample), whereas the power of the probe laser was kept Gaussian power distribution in direction perpendicular to its long low in order to avoid any additional heating effects (typically of axis, and that the focused probe beam is considered as a Delta the order of tens of μW). A noise eater (Thorlabs NEL01A/M) was function of the spatial coordinate, δ (x − x0 ), where x0 is the abso- used to reduce the intrinsic noise from the probe laser, which was lute position of the line-shaped heater. The previous approximation particularly useful for frequencies below 5 kHz. Both lasers were is valid when the diameter of the focused probe beam is diffraction coupled into the main optical axis using beam splitters (BS) and limited, dprobe < 1 μm. Using the same notation as Borca-Tasciuc, finally focused on the sample through the objective lens (Mitutoyo the frequency-dependent temperature oscillation for different off- MY10X-803). The diameters of the pump and probe beams at the sets, x = x − x0 , between the heat source and the probe spot po- output of the lasers were 1100 μm and 800 μm, respectively, and sitions are: their spatial intensity distribution was Gaussian. p0 ∞ exp[−σ p2 ξ 2 /8] cos(ξ x ) In order to obtain a line-shaped uniform intensity distribution T (x, ω ) = dξ (11) of the pump beam, we have used a custom designed holographic πκ ⊥l 1 0 A1 ( ξ )B1 ( ξ ) diffractive optical element (DOE) purchased from HOLO/OR Ltd, where A1 and B1 are defined as: which was placed in the optical path of the pump beam. The di- κ⊥B j ameter of the pump beam was tuned to meet the specifications A j κ⊥ j B − tanh(B j−1 d j−1 ) (beam input diameter) of the DOE using a variable optical beam j = ( 2, . . . , n ) j−1 A j−1 = , j−1 κ ⊥j B j (12) expander. We note that the DOE is one of the key elements in 1 − A j κ⊥ B j−1 tanh(B j−1 d j−1 ) j−1 this experimental setup, since it is responsible for homogenizing  the intensity of the pump laser, i.e., the incident Gaussian dis- κj i2C j ρ j (ω ) tribution is converted into a line-shaped spot with uniform in- Bj = ξ2 + (13) tensity distribution along the main axis, and with diffraction lim- κ ⊥j κ ⊥j ited Gaussian distribution in the perpendicular direction after be- and, ing focused. Regarding the probe laser, we simply use a diffrac- tion limited focused Gaussian beam as shown in the schemat- An = − tanh(Bn dn )s (14) ics of Fig. 3. The spatial displacement between the heater (line- where n is the number of layers counting from the top surface, shaped pump beam) and the detection (spot-shaped probe beam) i.e., n = 1 at the surface where the pump and probe lasers are fo- was controlled using a deflectable mirror prior to the focusing ob- cused, n = “total number of layers” at the bottom layer, C j and ρ j jective, which was calibrated using a Complementary-Metal-Oxide- are the specific heat and density of each layer, respectively. The pa- Semiconductor (CMOS) camera placed at the focal plane of the ob- rameter s sets the type of boundary condition at the bottom layer jective. The position of the line-shaped pump at the CMOS camera with s = 0 for a semi-infinite substrate. When the substrate thick- was measured for each deflection angle of the deflectable mirror. ness is finite, s = 1, for adiabatic boundary conditions, and s = −1, Figure S3 of the Supplementary Materials Section displays an ex- for the case of isothermal boundary conditions. Finally, the thermal ample of the calibration curve. boundary conductance between two layers, KT BC , is modeled using The back reflection of the pump and probe lasers after focusing the usual assumptions, i.e., a 1 nm thick layer with a small heat on the sample was coupled into the detection arm using a 90:10 capacity (e.g, C p ≈ 1 J Kg−1 C−1 ). BS (R:T), and the laser wavelength was selected by two mechani- 4 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 Fig. 3. Schematic illustration of the experimental setup used to study in-plane heat transport based on a balanced detection scheme. The inset shows an optical photograph of the line-shaped heat source (pump) as well as the spot-shaped probe. A diffractive optical element (DOE) is used to modify the shape of the pump beam and a deflectabe mirror is used to control the offset between the pump and probe lasers. cally controlled notch-filters, and further purified using interfero- It is also relevant to briefly describe the measurement proce- metric filters. As detector we used a large area (5 mm in diam- dure used to determine the thermal phase lag, since this is the eter) AC-coupled balanced detector (Thorlabs PDB210A/M-AC-SP). key experimental quantity which is linked to the thermal diffu- A sample of the probe laser (532 nm) was independently coupled sivity of the studied samples. The phase lag of the thermal signal to the second detector input. The output voltage of the detector is always relative to the thermal excitation. Hence, for each exci- is sensitive to the difference between both inputs provided that tation frequency we always measure the phase lag of the pump the difference signal is time-dependent. This is achieved by placing and probe lasers reflected on the sample, and we compute their high pass filter before the amplification stage inside the detector difference, φsample = φprobe − φpump . We note that for cases where (AC-coupled detector). Hence, the output of the balanced detector the temperature coefficient of reflectance of the top layer is nega- is a frequency modulated voltage arising from the thermal oscilla- tive, dR/dT < 0, it is necessary to subtract from φsample an angle of tions on the surface of the sample. In order to minimize the laser 180◦ . Hence, in order to obtain the phase lag of the thermal signal noise, the optical path of the signal and reference laser were kept (φsample ), first the phase of the reflected pump laser (φpump ) was similar within 1 cm, and the input power on both optical detec- measured by the lock-in amplifier, after which the notch-filters tors was balanced. The output of the AC-coupled balanced detec- were mechanically switched and the phase of the reflected probe tor was amplified by two orders of magnitude using an AC-coupled laser was obtained (φprobe ). This process was sequentially repeated low noise amplifier (LNA) purchased from FEMTO. The purpose of for each excitation frequency. the LNA is to boost the thermal signal to higher levels in order to The possible influence of the heating power and of the dimen- avoid, e.g., coherent pick-up inherent to the lock-in amplifier. The sions of the line-shaped pump and spot-shaped probe on the mea- pump laser was also coupled to the detection system following the sured thermal phase lag (φsample ) was also investigated for the same optical path as the probe laser. Note that the dichroic beam 2 μm thick Si membrane. Figure S1 in Supplementary Materials splitter which combines the pump (405 nm) and probe (532 nm) section displays the results for three excitation powers of 5 mW, lasers allows the transmission of a very small fraction of the pump 10 mW, and 15 mW. No influence of the excitation power is ob- laser. However, for the pump beam detection we do not use a bal- served on the measured phase lag within the experimental error, anced detection scheme and the secondary input of the AC-coupled which mostly arises from: (i) the rather large spatial offsets im- balanced detector is simply mechanically blocked. This approach posed between the pump and the probe lasers, leading to much works well since for the pump laser there is no large DC back- lower temperatures away from the line-shaped heater spatial po- ground as it is the case for the probe beam, where the thermal sition, hence, minimizing the possible influence of the tempera- oscillations are always relative to its DC incident power, and are ture dependence of the thermal diffusivity on the measured phase typically Tac /Tdc < 10−3 . lag (φsample ), and (ii) almost in all cases the maximum temper- 5 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 ature rise at the spot region does not exceed ≈ 10 K. Regarding the solution of the temperature oscillations always leads to a lin- the influence of the finite dimensions of the focused pump and ear dependence of the phase lag with the spatial offset between probe lasers, we have experimentally studied the influence of the the heat source and the probe as shown by Eq. (3). Hence, regard- pump dimensions on the phase lag response. Figure S3 in the Sup- ing the excitation frequency there is no limitation for the lower or plementary Materials section displays exemplary data showing the higher frequency, once more, besides the observation of apprecia- phase lag as function of the spatial offset. We have used a beam ble signal which is only relevant for higher frequencies where the profiler to determine the 1/e2 width of the line-shaped pump ob- AC thermal signal strongly decreases as also predicted by Eq. (3). taining ≈ 5 μm. At a distance larger than the diameter of the line- On the other hand, for 3D samples the solution for the tempera- shaped pump, the phase lag is already observed to behave linearly ture oscillations is given by Eq. (7) which only leads to a linear with the spatial offset between pump and probe. Regarding the in- response regime of φ (r ) when r >> 1/q ≈ μ, i.e., when the typical fluence of the probe spot diameter, we neglect here its finite size spatial offsets are larger than the thermal penetration depth. This since the probe beam was expanded at its output in order to ob- originate from the mathematical limit of K0 (qr ) as shown in Fig. 2. tain a diffraction limited spot, i.e., < 1 μm in diameter. Hence, for In practice, we have experimentally observed that the approxima- practical purposes the probe spot can be considered as point-like, tion is valid for r ≥ μ, which is used to determine the minimum which also simplifies the data analysis process and the correspond- frequency which leads to a linear response of φ (r ). ing equations as shown in the previous section. Furthermore, we In practice, it is always possible to measure a rather broad spa- note that considering that the probe beam has finite dimensions tial and frequency range and later on select the linear response re- will not influence the obtained values for the thermal diffusivity, gion following the arguments described in the previous paragraph, since it only leads to a local average the phase lag, and since the which is particularly relevant for 3D systems. spatial dependence of the phase lag is linear, the resulting φ (r ) will only exhibit a rigid translation which has no effect on the 4. Results and discussion slope, dφ /dr. Finally, we comment on some general guidelines to appropri- We have applied the methodology described in the previous ately select the spatial offset range and frequency range of inter- sections to several samples with different heat flow geometries. est for each studied material, which naturally depends on their For all studied samples we have measured the dependence of the thermal diffusivity (or effective thermal diffusivity for a multilayer phase lag (φ ) between the thermal excitation (pump) and the re- system). Regarding the spatial offset range the only limitation to sponse of the sample (probe) as a function of their spatial offset, consider is the minimum offset which may be prone to finite size and for different excitation frequencies. In particular, we have stud- effects arising from the finite width of the line-shaped heater. Typ- ied 1D heat flow for two suspended Si thin films with large sur- ically, it is sufficient to start the offset scan at a minimum distance face areas (in the cm2 range), and with thicknesses of 2 μm and given by the full width of the line-shaped heater. On the other 100 nm which we have purchased from NORCADA, as well as a hand, there is no limitation for the maximum offset, besides the 100 nm thick suspended PDPP4T polymer thin film fabricated in- observation of measurable signal. Regarding the frequency range house. The case of 2D heat flow was addressed for a bulk Bi sub- selection, for 2D or quasi-2D materials, e.g., suspended thin films, strate, a Si bulk substrate covered with 60 nm Au (transducer), a Fig. 4. (a) Phase lag vs. offset for a 1 μm thick suspended Si thin film for different excitation frequencies between 5 kHz and 20 kHz. The dashed lines are linear fits to the data points. (b) Slopes obtained from fitting φ vs. x for each excitation frequency. (c) Phase lag vs. offset for a 100 nm thick suspended Si thin film for different excitation frequencies between 4 kHz and 14 kHz. The dashed lines are linear fits to the data points. (d) Slopes obtained from fitting φ vs. x for each excitation frequency. 6 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 glass bulk substrate covered with 60 nm Au (transducer), and a the corresponding data for the 100 nm thick suspended Si thin highly-oriented pyrollitic graphite (HOPG) bulk sample with 60 nm film. The data exhibits a similar dependence of φ on x and f 1/2 , Au (transducer). as compared to the 2 μm thick suspended Si thin film. How- ever, due to the lower thermal diffusivity of the 100 nm thick 4.1. Quasi-2D samples without transducer suspended thin film, we used a slightly lower x and f ranges. We note that for this sample the phase lag data points exhibit Figure 4 (a) displays the phase lag (φ ) versus the spatial off- larger experimental noise, which partly arises from the compara- set (x) between the pump and probe beams for a suspended tively lower signal (absolute temperature rise, T ) for similar x Si thin film with a thickness of 2 μm, and for different thermal and frequency ranges, and which originates from its lower ther- excitation frequencies in the range between ≈5 kHz and 20 kHz. mal diffusivity. In addition, this sample is more sensitive to addi- All suspended thin films where measured in vacuum at a pres- tional noise arising from mechanical vibrations introduced by the sure lower than 10−7 mbar. In all cases, the data were fitted us- environment during the measurement process. Using Eq. (5) we ing a linear relation, as predicted by Eq. (3). The collected data for have obtained the thermal diffusivity of the 100 nm thick sam- each frequency consists of multiple data points acquired at differ- ple as, D = (4.50 ± 0.23 ) × 10−5 m2 /s i.e., 50% of the Si bulk value ent offsets, x, which are uniformly distributed in the range from and also in good agreement with previous determinations [43]. The ≈30 μm to 145 μm. As shown in Fig. 4(a), the magnitude of φ in- reduced values of the thermal diffusivity obtained for both sus- creases with increasing frequency and x. Figure 4(b) displays the pended Si thin films arise from their different thickness, which slopes, ∂ φ /∂ x, as obtained from the linear fits of φ versus x limits the maximum phonon mean free path of the phonons which in Fig. 4(a), at each excitation frequency. In all cases, the experi- can transport heat, a well known effect which was already dis- mental errors are within the size of the symbols. In good agree- cussed by Chávez-Angel [43] and references therein. ment with the prediction of Eq. (5), the slope, ∂ φ /∂ x, exhibits Figure 5 displays the phase lag vs. spatial offsets for a sus- a linear relation with f 1/2 . Hence, using Eq. (5) we obtained the pended PDPP4T thin film with a thickness of 100 nm. The fre- thermal diffusivity for the 2 μm thick suspended Si thin film as quency range chosen was reduced with respect to the case of both D = 7.15 × 10−5 m2 /s, i.e., 82% of the Si bulk value and in good Si suspended thin films due to the lower thermal diffusivity of agreement with previous determinations [43]. Figure 4(c) displays PDPP4T, which leads to a comparatively smaller in-plane thermal Fig. 5. (a) Phase lag vs. offset for a 100 nm thick PDPP4T suspended thin film for different excitation frequencies between 17 Hz and 150 Hz. The dashed lines are linear fits to the data points. (b) Slopes obtained from fitting φ vs. x for each Fig. 6. (a) Phase lag vs. offset for a Bi single crystal, and for different excitation excitation frequency. frequencies. (b) Slopes obtained fitting φ vs. x for each excitation frequency. 7 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 penetration depth. In general, the response of this sample is quali- 4.3. Bulk samples with transducer tatively similar to the case of both suspended Si thin films, hence, we have applied the same methodology to analyze the experi- We have also studied the influence of using a Au metallic trans- mental data. We have obtained an in-plane thermal diffusivity of ducer on the phase lag response. The use of a metallic transducer D = (5.13 ± 0.26 ) × 10−7 m2 /s, which is a typical value for semi- in thermoreflectance experiments is usually desirable for three crystalline polymer thin films. main reasons: (i) the extinction coefficient of a metal for visible incident light is typically rather small (<100 nm), hence, allowing 4.2. Bulk sample without transducer to approximate the heat source as superficial; (ii) no direct exci- tation of the electronic system of the material under investigation We have also studied the case of 2D heat flow as discussed in occurs, thus, avoiding heat transport through excitons, which could Section 2.2. For this purpose, we used single crystal Bi(0 0 01) sub- be relevant for some materials with indirect optical bandgap; and strate purchased from Mateck GmbH. The pump and probe lasers (iii) the temperature coefficient of reflectance of Au is relatively focused directly onto the surface of the sample, i.e., without using large, R/R0 ≈ 2 × 10−4 , for 532 nm probe wavelength, which nat- a metallic transducer as for the suspended thin films in the pre- urally results in enhanced thermal signal. In particular, we address vious section. Figure 6(a) displays the phase lag as a function of to what extent the present method can be applied to multilayered the spatial offset, and for different excitation frequencies. Similarly systems. Typically, the presence of a metallic transducer can have as for the case of the suspended thin films studied in the previ- a strong impact on the measured phase lag, which originates from ous section, the dependence of φ on x and f 1/2 is also linear as in-plane heat conduction in the transducer layer itself, and which predicted by Eq. (10). However, unlike the case of the suspended Si is particularly relevant when the thermal conductivity of the trans- thin films, the heat flow geometry for a bulk sample is 2D, i.e., heat ducer (κtr ) is much larger than the thermal conductivity of the propagates along both directions perpendicular to the line-shaped substrate (κs ). We have chosen as case study a bulk crystalline heater. In this case, the spatial dependence of the temperature field Si substrate as well as a bulk glass substrate, hence, addressing is given by Eq. (9). In fact, the phase lag and the corresponding the cases with κtr ≈ κs (Si), and κtr  κs (glass). For the case with slopes, ∂ φ /∂ x, shown in Fig. 6 are prone to larger experimental κtr  κs , the signal is fully dominated by the substrate, as can be error as compared to the case of the suspended Si thin films (1D shown solving numerically Eq. (11). heat flow), which is mainly due to the faster spatial decay of sig- Figure 7 (a) displays the phase lag data for a Si substrate cov- nal as predicted by Eq. (9). In addition, the different temperature ered with a 60 nm thick Au transducer, which was deposited using coefficients of reflectivity (at the probe wavelength of 532 nm) of thermal evaporation with a base pressure better than 10−6 mbar, Bi as compared to Si also account for the different signal levels and at a deposition rate of 1 Å/s. The phase lag curves were col- observed in these systems. The thermal diffusivity of the Bi sam- lected at excitation frequencies between 7 kHz and 13 kHz, and ple was obtained fitting the slopes (∂ φ /∂ x) shown in Fig. 6(b), spatial offsets between 40 μm and 90 μm. The data exhibits a obtaining D = 9.89 × 10−6 m2 /s, which is in good agreement with similar trend as observed for the case of bulk Bi, i.e., even in previously reported values [44]. the presence of the Au transducer the phase lag depends lin- Fig. 7. (a) Phase lag vs. offset for a Si substrate with a 60 nm thick Au transducer for different excitation frequencies between 7 kHz and 14 kHz. The dashed lines are linear fits to the data points. (b) Slopes obtained from fitting φ vs. x for each excitation frequency. (c) Phase lag vs. offset for a glass substrate with a 60 nm thick patterned Au transducer for different excitation frequencies between 17 Hz and 150 Hz. The dashed lines are linear fits to the data points. (d) Slopes obtained from fitting φ vs. x for each excitation frequency for the as-deposited Au transducer as well as for the patterned transducer. The inset displays and image of the pattern used to avoid in-plane heat transport within the Au transducer. 8 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 early on the spatial offset. Figure 7(b) displays the slopes as ex- tracted from the linear fits of the phase lag curves in Fig. 7(a) as a function of f 1/2 . Once more, the relation between ∂ φ /∂ x and f 1/2 is linear. We applied Eq. (10) to compute the thermal diffusivity of the sample obtaining D = (8.55 ± 0.43 ) × 10−5 m2 /s, which is in good agreement with reported values for Si substrates [45–47]. It is interesting to note that even in the presence of the Au transducer, it is still possible to directly obtain the thermal diffu- sivity of the substrate from the double derivative of the phase lag, ∂ 2 φ /[∂ x∂ f 1/2 ], as previously done for the case of the suspended thin films and for the Bi substrate without transducer. We have confirmed this approximation solving numerically Eq. (11) for the studied geometry, and observing that when the thermal conductiv- ity of the transducer is similar or lower to the thermal conductivity of the underlying substrate, the response is mostly dominated by the substrate and, hence, the present method can be applied with- out significant loss of accuracy, which is estimated in < 2% for the present case. We also discuss the case of a glass substrate covered with a 60 nm thick Au transducer deposited through thermal evapo- ration in similar conditions as for the case of the Si substrate. The thermal conductivity of the Au transducer was determined by measuring its electrical conductivity through the Van der Pauw method, and using the Wiedemann–Franz law obtaining, κtr = 246 Wm−1 K−1 . This case is substantially different from the pre- vious since the thermal conductivity of the Au transducer is more than two orders of magnitude larger than the thermal conductiv- ity of the glass sample (κGlass ≈ 1). In fact, it is already expected that the response will be dominated by a combination of both, the glass sample and the Au transducer. In order to further study this effect, we have patterned the Au transducer using focused ion beam with the purpose of substantially reducing its in-plane ther- mal conductivity. Hence, we provide experimental data for the case of a glass substrate with a continuous Au transducer, κ// ≈ κ⊥ , as well as for the case with a patterned Au transducer, κ// ≈ 0. The patterned Au transducer was fabricated in a 150 μm × 150 μm area, and was obtained patterning narrow (≈ 15 nm) vertical and Fig. 8. (a) Phase lag vs. offset for a HOPG bulk sample with a 60 nm thick Au trans- horizontal channels on the as-deposited Au transducer. The depth ducer for different excitation frequencies between 14 kHz and 50 kHz. The dashed lines are linear fits to the data points. (b) Slopes obtained from fitting φ vs. x for of these channels was carefully calibrated in order not to dam- each excitation frequency. age the surface of the sample, hence, the residual thickness of the transducer at the bottom of each channel is < 3 nm. The horizontal and vertical spacing between each channel was 5 μm. The inset of Fig. 7(d) displays an optical image of the pattern. Figure 7(c) dis- of the multilayered system, the response φ (x ) is still linear for plays the phase lag data for the case of the glass substrate with each excitation frequency. However, the second order derivative, the patterned Au transducer in the frequency range from 17 Hz to ∂ 2 φ /∂ x∂ f 1/2 , provides the thermal diffusivity of the multilay- 150 Hz. Note that we have selected a rather low frequency  range ered systems, which depends on the used experimental conditions in order to enhance the thermal penetration depth, μ = D/(π f ), such as the offset and excitation frequency ranges. This result is arising from the low thermal diffusivity of the glass substrate and, simply a consequence of the expansion of K0 (qr ) for qr  1. In hence, to increase the relative sensitivity of the thermal properties this case, it is still possible to obtain the thermal diffusivity of the of the glass substrate with respect to the Au transducer. Figure 7(d) sample through numerical simulations reflecting the experimental displays the slopes, ∂ φ /∂ x, as a function of f 1/2 for both glass conditions. samples, i.e., with and without patterned transducer. Fitting the Finally, in Fig. 8 we show the phase lag data for a HOPG bulk data corresponding to the sample with patterned substrate ren- sample purchased from SPI Supplies. This sample was studied us- ders a thermal diffusivity D = (7.73 ± 0.39 ) × 10−7 m2 /s, which ing a 60 nm thick Au transducer in order to enhance the ac- is in good agreement with previously reported values for simi- quired thermal signal. We have also studied this sample in the lar substrates [48]. In the case of the sample with non-patterned absence of the metallic transducer, however, almost not signal transducer a direct fit to the data points in Fig. 7(d) renders D = could be obtained due to its low temperature coefficient of re- (2.55 ± 0.13 ) × 10−6 m2 /s, which is obviously larger than the typi- flectance for the used pump and probe wavelengths. We note cal values observed for glass substrates. As we have previously an- that the presence of the metallic transducer has a negligible ef- ticipated, the origin of this larger value is heat conduction through fect on the response, similarly as for the case of the Si substrate. the Au transducer. In other words, when in-plane heat transport In fact, the HOPG substrate exhibits a rather large in-plane ther- within the transducer is not negligible, the present method ren- mal diffusivity and, hence, it is insensitive to large extent to the ders the effective thermal conductivity of the system. effect of the transducer. We have estimated a deviation below We note that even for the cases where the transducer has a 1% numerically solving Eq. (11). Figure 8(b) displays the slopes, non-negligible contribution to the in-plane thermal conductance ∂ φ /∂ x, as a function of f 1/2 for the HOPG sample with trans- 9 K. Xu, J. Guo, G. Raciti et al. International Journal of Heat and Mass Transfer 214 (2023) 124376 ducer. Fitting the experimental data renders a thermal diffusivity CRediT authorship contribution statement D = (9.64 ± 0.48 ) × 10−4 m2 /s, which is a typical value for the in- plane thermal diffusivity this system [14]. Interestingly, the out- Kai Xu: Data curation, Formal analysis, Investigation, Methodol- of-plane thermal diffusivity of this sample is typically ≈ 150 times ogy, Software, Validation, Visualization, Writing – review & editing. smaller than its in-plane counterpart, which evidences the poten- Jiali Guo: Formal analysis, Investigation, Resources. Grazia Rac- tial of the present approach to study in-plane heat transport for iti: Formal analysis, Investigation, Resources. Alejandro R. Goni: samples with large thermal anisotropy ratios (in-plane vs. out-of- Funding acquisition, Resources. M. Isabel Alonso: Funding acqui- plane). sition, Resources. Xavier Borrisé: Resources. Ilaria Zardo: Concep- tualization, Funding acquisition, Methodology, Resources. Mariano Campoy-Quiles: Conceptualization, Funding acquisition, Methodol- 5. Conclusions ogy, Resources. Juan Sebastián Reparaz: Conceptualization, Fund- ing acquisition, Methodology, Project administration, Resources, We have developed a new method based on beam-offset Software, Supervision, Writing – review & editing. frequency-domain thermoreflectance (BO-FDTR) suitable to study thermal transport with enhanced sensitivity to in-plane heat flow. Data availability The present method substantially differs from previous approaches since it uses a one-dimensional (1D) heat source combined with All data needed to evaluate the conclusions in the paper are controlled spatial offsets between the heater (pump, line-shaped) present in the paper. Additional data related to this paper and the thermometer (probe, dot-shaped). The key advantage of may be requested from the authors. using a 1D heat source with respect to focused Gaussian heat sources (0D) relies on the spatial dependence of the temperature field. Using a 1D heat source allows to probe the thermal field at Acknowledgements larger distances from the heat source, providing enhanced sensi- tivity to in-plane thermal transport. In addition, the spatial depen- We acknowledge financial support from the Spanish Minis- dence of the thermal response at distances larger than the typical terio de Economía, Industria y Competitividad for its support dimensions of the heat source is insensitive to the specific shape through grant CEX2019-0 0 0917-S (FUNFUTURE) in the frame- of the heat source. Regarding the data analysis procedure, this ap- work of the Spanish Severo Ochoa Centre of Excellence program, proach presents large advantages with respect to other methods and grants PID2020-119777GB-I00 (THERM2MAIN), and PDC2021- since it does not require any computational efforts, i.e., the thermal 121814-I00 (COVEQ). This work was financially supported by diffusivity of the specimen is readily extracted from linear fits to the European Commission through the Marie Sklodowska-Curie the data points. In addition, the use of a line-shaped heater defines project HORATES (GA-955837). We also acknowledge grant 2021- the frequency region of interest to frequencies < 100 kHz, which SGR-00444 (NANOPTO) from AGAUR. K.X. acknowledges a fellow- provides substantial advantages regarding the price of the excita- ship (CSC201806950 0 06) from China Scholarship Council and the tion source. Most importantly, such low frequency range allows to Ph.D. programme in Materials Science from Universitat Autónoma study samples without the use of a metallic transducer minimizing de Barcelona in which he was enrolled. G.R and I.Z. acknowl- the influence of the finite optical penetration depth in the samples edge financial support from Eucor, The European Campus (Marie due to the large thermal penetration depth provided by the low Sklodowska-Curie QUSTEC grant agreement no. 847471) and the frequency range. Swiss National Science Foundation (Project Grant no. CRSII5 We have applied the present method to study 2D and 3D sam- 189924 ). ples without the use of a metallic transducer, showing that the thermal diffusivity can be accurately extracted for both sample geometries. Furthermore, we show that the method is still accu- Supplementary material rate for samples with a thin metallic transducer, provided that in- plane heat transport within the transducer is negligible. 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