Thermal conductivity measurements of sub-surface buried substrates by steady-state thermoreflectance
Md Shafkat Bin Hoque, Yee Rui Koh, Kiumars Aryana, Eric Hoglund, Jeffrey L. Braun, David H. Olson, John T. Gaskins, Habib Ahmad, Mirza Mohammad Mahbube Elahi, Jennifer K. Hite, Zayd C. Leseman, W. Alan Doolittle, Patrick E. Hopkins
aa r X i v : . [ phy s i c s . i n s - d e t ] F e b Thermal conductivity measurements of sub-surface buriedsubstrates by steady-state thermoreflectance
Md Shafkat Bin Hoque, Yee Rui Koh, Kiumars Aryana, Eric Hoglund, Jeffrey L. Braun, David H.Olson, John T. Gaskins, Habib Ahmad, Mirza Mohammad Mahbube Elahi, Jennifer K. Hite, Zayd C.Leseman, W. Alan Doolittle, and Patrick E. Hopkins
Md Shafkat Bin Hoque, Yee Rui Koh, Kiumars Aryana, Jeffrey L. Braun, David H. Olson, John T.GaskinsDepartment of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia22904, USAEric HoglundDepartment of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia 22904,USAJennifer K. HiteU.S. Naval Research Laboratory, Washington, D.C. 20375, U.S.A.Mirza Mohammad Mahbube ElahiDepartment of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NewMexico 87131, USAZayd C. LesemanDepartment of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran, East-ern Province 31261, Saudi ArabiaHabib Ahmad, W. Alan DoolittleSchool of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332,United StatesPatrick E. HopkinsDepartment of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia22904, USADepartment of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia 22904,USADepartment of Physics, University of Virginia, Charlottesville, Virginia 22904, USAEmail: [email protected] bstract Measuring the thermal conductivity of sub-surface buried substrates are of significant practical interests. However, this remains challengingwith traditional pump-probe spectroscopies due to their limited thermal penetration depths (TPD). Here, we experimentally and numericallyinvestigate the TPD of recently developed optical pump-probe technique steady-state thermoreflectance (SSTR) and explore its capabilityfor measuring the thermal properties of buried substrates. The conventional definition of the TPD does not truly represent the upper limitof how far beneath the surface SSTR can probe. For estimating the uncertainty of SSTR measurements of a buried substrate a priori,sensitivity calculations provide the best means. Thus, detailed sensitivity calculations are provided to guide future measurements. Due tothe steady-state nature of SSTR, it can measure the thermal conductivity of buried substrates typically inaccessible by traditional pump-probe techniques, exemplified by measuring three control samples. We also discuss the required criteria for SSTR to isolate the thermalproperties of a buried film. Our study establishes SSTR as a suitable technique for thermal characterizations of sub-surface buried substratesin typical device geometries.
Keywords: thermal conductivity, buried substrate, steady-state thermoreflectance, thermal penetra-tion depth
Thin films with thicknesses ranging from nanometer to micrometer length scales have become an inte-gral part of transistors, thermoelectrics, optical coatings, solar cells, and memory devices. Asthe device efficiency and reliability are often dictated by the thermal performance, it is of crucial im-portance to properly characterize the thermal properties of the thin films and substrates. Traditionalnon-contact, optical pump-probe techniques such as time-domain thermoreflectance (TDTR) andfrequency-domain thermoreflectance (FDTR) are widely used to measure the thermal conductivity ofthin films. However, TDTR and FDTR often cannot measure the thermal conductivity of buried sub-strates located beyond 1 µ m due to shorter thermal penetration depths (usually < 1 µ m). The recentlydeveloped optical pump-probe technique steady-state thermoreflectance (SSTR) offers a solution to thisissue as its thermal penetration depth can be much larger than those produced during TDTR and FDTRmeasurements.
Therefore, a detailed study into the thermal penetration depth of SSTR techniqueand its ability to measure the thermal conductivity of sub-surface buried substrates is highly warranted.Using the principle of thermoreflectance, SSTR employs co-axially focused pump and probe beamsfrom continuous wave (CW) lasers to directly measure the thermal conductivity of a material by applyingFourier’s law. Schematics of the SSTR measurement configuration and principle are shown in Figure 1(a)and 1(b), respectively. Using a low modulation frequency ( f ), the pump laser generates a periodic heatflux at the sample surface for an extended period. The low modulation frequency provides enough timefor the system to reach steady-state. The probe beam then measures the resultant steady-state temperaturerise by monitoring the reflectance change with a balanced photodetector and a lock-in amplifier. Alinear relation between the heat flux and temperature rise is established by varying the pump power andmonitoring the reflectance change at each pump power. From this relation, the thermal conductivity ofany material can be determined by using Fourier’s law. The thermal conductivity tensor measured bySSTR is different from that of TDTR or FDTR. For bulk materials, whereas TDTR and FDTR usuallymeasure the cross-plane thermal conductivity, SSTR measures √ k r k z , where, k z and k r are cross-planeand in-plane thermal conductivity, respectively. In alignment with TDTR and FDTR, in SSTR, the thermal penetration depth (TPD) is defined as thedistance normal to the surface at which the temperature drops to the 1/e value of the maximum surfacetemperature (T max ). According to this definition, the 1/e heater (pump) radius represents the upperlimit of the TPD when the modulation frequency is low ( f → However, such a a)Pump T e m pe r a t u r e r i s e Pump power ((cid:181) heat flux) (b)
Figure 1: Schematics of the SSTR measurement (a) configuration and (b) principle. description of the SSTR TPD fails for multilayer material systems (i.e., a thin film on a substrate). In suchsystems, the TPD can change widely based on the ratio of thin film to substrate thermal conductivity, andthe thermal boundary conductance ( G ) between the thin film and substrate. This is further complicatedby the presence of thin metal film transducers at the sample surface, which are often a requirement inoptical pump-probe techniques for optothermal transduction. In this study, we numerically and experimentally analyze the TPD definition of SSTR measurements.We also discuss the implications of metal film transducers, thermal boundary conductances, and role ofmultilayer material systems on the heuristic approximations for the SSTR TPD. Notably, our experimen-tal results indicate that although the traditional TPD definition provides a convenient estimate of howfar beneath the surface SSTR can probe, it does not represent the absolute upper limit of SSTR probingdepth. In specific cases, SSTR can probe beyond both the 1/e temperature drop distance and the heaterradius. Furthermore, the thermal conductivity measurements of buried layers or substrates by SSTR arenot solely dictated by the TPD. The uncertainty associated with such measurements are often governedby the transducer thermal conductivity, the thermal boundary conductances, and the thermal resistancesoffered by different layers of the multilayer material system. Thus when determining whether the ma-terial of interest at some depth under the surface is measurable within acceptable limits of uncertainty,sensitivity calculations provide the best means. Moreover, we show that due to the continuous wavenature of the pump laser source and low modulation frequency, SSTR can measure the thermal conduc-tivities of buried substrates that are typically inaccessible by TDTR and FDTR. This is illustrated bymeasuring the thermal conductivities of buried substrates in three different samples: i) ∼
130 nm amor-phous silicon dioxide (a-SiO ) thin film on a silicon (Si) substrate, ii) ∼ µ m gallium nitride (GaN)thin film on a n-type GaN substrate, and iii) ∼ µ m aluminum nitride (AlN) thin film on a sapphire sub-strate. In addition, it is also established that using large 1/e pump and probe radii, SSTR can measurethe thermal conductivity of highly resistive buried films. Results and discussion
We first review how the TPD of SSTR changes as a function of substrate thermal conductivity in a 2-layer system: metal transducer/substrate. The substrate here represents a bulk isotropic material. TheTPD is calculated by solving the cylindrical heat diffusion equation, detailed descriptions of which aregiven elsewhere. For these calculations, 1/e pump and probe radii ( r o and r , respectively) of 10 µ mare used. The modulation frequency is chosen to be 100 Hz as it represents a realistic value usable inan experiment. We further assume that all the energy is absorbed in an infinitesimal thin layer on thesurface (i.e., surface boundary condition). distance1/e distance T he r m a l pene t r a t i on dep t h ( m ) Substrate thermal conductivity (W m -1 K -1 ) T / T m a x Depth ( m) max (b)
Figure 2: (a) Thermal penetration depth as a function of substrate thermal conductivity for a 2-layer system: metal trans-ducer/substrate. (b) Normalized temperature drop ( △ T/T max ) as a function of depth for a substrate thermal conductivity of1000 W m − K − . The calculations correspond to f = 100 Hz, d = 80 nm, r = r = 10 µ m, k = 100 W m − K − , C V , = C V , = 2 MJ m − K − , and G = 200 MW m − K − . Here, d and C V represent thickness and volumetric heat capacity,respectively. In Figure 2(a), the TPD corresponding to the 1/e temperature drop distance from the surface is pre-sented for two scenarios, with and without the inclusion of a transducer. When no transducer is present,the change in the TPD is very small with respect to the substrate thermal conductivity. The small decreasein the TPD with substrate thermal conductivity reduction can be attributed to the choice of modulationfrequency. For a given pump and probe radii, the lower the substrate thermal conductivity, the longer ittakes for the system to reach steady-state. Thus, as the substrate thermal conductivity decreases, thesystem slightly deviates from the ideal steady-state condition ( f = 0). To keep the TPD constant, themodulation frequency needs to be lowered in accordance with the substrate thermal conductivity reduc-tion. However, for the chosen modulation frequency of 100 Hz, the deviation from the ideal steady-statecondition is quite small for the substrate thermal conductivities considered here and therefore, the systemcan still be reasonably approximated to be in steady-state. The presence of a transducer drastically changes the TPD. When the substrate thermal conductivityis low (< 10 W m − K − ), the TPD with the transducer is higher than the TPD without the transducer.This stems from the radial heat spreading in the transducer and a corresponding increase in the overallheater radius. On the other hand, when the substrate thermal conductivity is high (> 100 W m − − ), the TPD with the transducer sharply deceases. The higher the substrate thermal conductivity iscompared to that of the transducer, the lower the temperature rise is in the substrate for a given amountof heat flux. As a result, in high thermal conductivity substrates, a large temperature drop exists at theinterface between the transducer and substrate. This is exemplified in Figure 2(b), where we present thenormalized temperature drop as a function of depth for a substrate thermal conductivity of 1000 W m − K − . In this example, the temperature decreases by nearly 41% at the transducer/substrate interface,leading to a TPD of 2.48 µ m.We also calculate the distance normal to the surface at which the temperature drops to 1/e value ofmaximum surface temperature and present it in Figure 2(a). It is evident that the 1/e distance (calculatedwith a transducer) is much higher than the 1/e distance for all substrate thermal conductivities. This is tobe expected as the temperature decay increases with depth. We now extend the TPD discussion to a 3-layer system with the following geometry: metal trans-ducer/thin film/substrate. When the thin film and substrate thermal conductivities are nearly equal, theTPD will closely follow those shown in Figure 2(a) with a minor influence from the thermal boundaryconductance between the thin film and substrate. Thus, we consider two extreme cases of this hypothet-ical geometry: an insulating film on a conductive substrate ( k = 10 W m − K − , and k = 100 W m − K − ), and a conductive film on an insulating substrate ( k = 100 W m − K − , and k = 10 W m − K − ).In Figures 3(a) and 3(b), we present the TPD corresponding to the 1/e temperature drop distance asa function of thin film thickness for the first and second case, respectively. It is evident that the TPDwith and without presence of a transducer are nearly identical. This is due to the fact that the thin filmthermal conductivities are 10 and 100 W m − K − , respectively. As shown in Figure 2(a), for thisrange of thermal conductivities, the transducer does not have a significant impact on the TPD. Similar tothe 2-layer system, the 1/e distance is much higher than the 1/e distance in the 3-layer system. FromFigures 3(a) and 3(b), it is also clear that the TPD changes greatly with the film thickness when there is asignificant difference between thin film and substrate thermal conductivities. Interestingly, the influenceof the thin film on the TPD does not subside until the film thickness is approximately 4 times the heaterradius. To understand the rationale behind this, it is necessary to review how the ratio of thin film tosubstrate thermal conductivity influences the heat flow direction.In Figures 3(c) and 3(d), we study the normalized temperature drop as a function of depth for the firstand second case, respectively. When the thin film is insulating and the substrate is conductive, the bulkof the heat flows along the cross-plane direction of the thin film. Due to this, a large temperature gradientexists in the thin film along the cross-plane direction as shown in Figure 3(c). Therefore, in this case,the TPD is much lower than the heater radius unless the thin film thickness is too high or too low. Onthe other hand, when the thin film is conductive and the substrate is insulating, the majority of the heatflows along the in-plane direction of the thin film. Thus, the temperature gradient along the cross-planedirection of the thin film is quite small. As a result, the TPD can be much higher than the heater radiusas evident in Figure 3(d).To provide a more visual representation of this, the temperature profiles of SSTR measurements areshown for a 3 µ m thin film corresponding to the first and second case in Figures 3(e) and 3(f), respec-tively. For the insulating thin film case, temperature decreases greatly along the cross-plane direction ofthe film, whereas for the conductive film case, such temperature decrease is much smaller. However, for .01 0.1 1 10 4011060 T he r m a l pene t r a t i on dep t h ( m ) Film thickness ( m ) distance1/e distance1/e distancewithout transducer (a) T he r m a l pene t r a t i on dep t h ( m ) Film thickness ( m) (b)1/e distance1/e distance1/e distance without transducer T / T m a x Depth ( m)
Heaterradius1/e T max
10 nm1 m3 m10 m20 m (c) T / T m a x Depth ( m) Heater radius 1/e T max
10 nm1 m3 m10 m20 m (d)
Depth ( m) R ad i u s ( m ) Transducer/thin film interfaceThin film/substrate interface (e)1/e distance
Depth ( m) R ad i u s ( m ) Transducer/thin film interfaceThin film/substrate interface1/e distance (f)
Figure 3: [(a) and (b)] Thermal penetration depth as a function of film thickness for a 3-layer system: metal transducer/thinfilm/substrate. [(c) and (d)] Normalized temperature drop as a function of depth for five different thin film thicknesses. [(e)and (f)] Temperature profiles of SSTR measurements for a 3 µ m thin film on a substrate corresponding to an absorbed powerof 5 mW. Figures (a), (c) and (e) represent the case of an insulating film on a conductive substrate ( k = 10 W m − K − and k = 100 W m − K − ), whereas Figures (b), (d) and (f) represent the case of a conductive film on an insulating substrate ( k = 100 W m − K − and k = 10 W m − K − ). The calculations correspond to f = 100 Hz, d = 80 nm, r = r = 10 µ m, k =100 , C V , = C V , = C V , = 2 MJ m − K − , and G = G = 200 MW m − K − . the conductive thin film case, temperature decrease is significant along the in-plane direction. This is inalignment with our previous discussion. .3 Experimental verification of the thermal penetration depth definition Thus far, we have numerically predicted the TPD of 2-layer and 3-layer systems according to the con-ventional definition. We now conduct a series of experiments to check the validity of this conventionalTPD definition for SSTR measurements. Specifically, we address the following questions: i) can SSTRprobe up to the 1/e temperature drop distance defined by the traditional TPD description, and ii) whetherthis 1/e distance or the heater radius represents the absolute upper limit of how deep beneath the surfaceSSTR can probe. V / V ( m V / V ) P (mV) k SiO glass = 1.14 – 0.16 W m -1 K -1 k z-cut quartz = 8.87 – 0.64 W m -1 K -1 k Si = 141 – 10 W m -1 K -1 (a) T h i r d l a y e r t he r m a l c ondu c t i v i t y ( W m - K - ) Second layer thickness ( m)
Siz-cut quartzSiO glass(b) Figure 4: (a) Probe photodetector response, △ V / V ( ∝ temperature rise) as a function of pump photodetector response, △ P ( ∝ pump power) for SSTR fitting of Al coated bulk SiO glass, z-cut quartz and Si. (b) Third layer thermal conductivity asa function of second layer thickness when the three samples are fitted as a 3-layer system: Al transducer/second layer/thirdlayer. For this purpose, the thermal conductivities ( √ k r k z ) of three bulk samples are measured by SSTR:SiO glass, z-cut quartz and Si. Prior to the measurements, the samples are coated with an ∼
80 nmaluminum (Al) film to serve as an optical transducer. A pump radius of ∼ µ m is used for these SSTRmeasurements. The SSTR experimental proportionality constant, γ , is determined from a referencesapphire sample (35 ± − K − ). Details of our SSTR setup and measurement procedure havebeen thoroughly discussed in previous publications. The 1/e temperature drop distance of the SiO glass, z-cut quartz and Si samples are ∼
10, 9 and 7 µ m, respectively, according to the conventional TPDdefinition presented in Figure 2(a).The SSTR best-fit curves for the thermal conductivities of the three samples are shown in Figure 4(a).The SSTR-measured thermal conductivities of the SiO glass, z-cut quartz and Si are ∼ ± ± ±
10 W m − K − , respectively. The uncertainty of the measured values stem from theuncertainty associated with the γ value (sapphire reference), Al transducer thermal conductivity and thethermal boundary conductance. Details of these parameters are listed in Table 1. The measured thermalconductivities of the three specimen are in agreement with literature. In Figure 4(b), the SiO glass, z-cut quartz and Si samples are approximated as a 3-layer materialsystem: Al transducer/second layer/third layer. Here, the second and third layer represent thin films andburied substrates, respectively. We fit for the thermal conductivity of the third layer assuming that thesecond layer possesses the value presented in Figure 4(a). The thermal boundary conductance between he second and third layer is kept fixed at 1000 MW m − K − . Figure 4(b) shows that with the increaseof second layer thickness, the uncertainty of third layer thermal conductivity increases. When the sec-ond layer thickness is equal to the 1/e distance ( ∼
10, 9 and 7 µ m for SiO glass, z-cut quartz and Si,respectively), the third layer thermal conductivities are ∼ ± ± ± − K − , respectively. Furthermore, when the second layer thickness is 14 µ m, the third layer thermalconductivities are ∼ ± ± ±
61 W m − K − , respectively.It is possible to answer the previously posed questions from Figure 4(b). As shown here, SSTRcan measure the thermal conductivities of layers located at 1/e temperature drop distance although suchmeasurements have relatively high uncertainty. However, it is also evident that SSTR can probe beyondthis conventional 1/e distance and the heater radius. This indicates that immediately beyond the 1/edistance or the heater radius, SSTR measurement sensitivity does not drop to zero. This phenomenoncan be explained by reviewing Figure 2(a) which shows that the temperature does not drop to 1/e valueof maximum surface temperature until the distance is much higher than the 1/e distance or the heaterradius. Thus, even though the traditional TPD definition can be used as a convenient estimate of SSTRprobing depth, neither the 1/e distance nor the heater radius should be taken as an absolute upper limitTable 1 : Parameters used in the SSTR measurements and sensitivity calculationsSamples Layers a Thermal conductivity b Thermal boundary conductance c (W m − K − ) (MW m − K − ) G G Al/SiO glass Al 126 ±
13 150 ±
20 -Al/Quartz Al 108 ±
12 230 ±
30 -Al/Si Al 117 ±
12 180 ±
30 -Al/130 nm SiO /Si Al 180 ±
18 230 ±
50 -SiO ± ± µ m GaN/GaN Al 130 ±
13 240 ±
40 -GaN 184 ±
15 - 150 ± µ m AlN/sapphire Al 190 ±
19 380 ±
80 -AlN 281 ±
26 - 150 ± µ m Si/1 µ m SiO /Si Al 180 ±
18 100 ±
10 -Si 127 ±
11 - 230 ± ea The thicknesses of the layers are measured by picosecond acoustics and transmission elctronmicroscopy (TEM). The uncertainty associated with layer thicknesses are about ∼ b The thermal conductivities of the Al transducers are measured by 4-point probe. The SiO , GaN,AlN and Si thin film thermal conductivities are measured by TDTR. c G is measured by TDTR. G is estimated from related literature. As SSTR has negligiblesensitivity to G , the estimated values do not have an appreciable influence on SSTR measure-ments. d In the Al/SiO glass sample, the SiO substrate is a commercial glass slide, whereas in Al/130nm SiO /Si sample, the SiO thin film is laboratory grade SiO grown via dry oxidation. As aresult, the thermal conductivity of SiO is different between the two samples. e For the Al/2.5 µ m Si/1 µ m SiO /Si sample, the thermal boundary conductances of the Si/SiO ( G ) and SiO /Si ( G ) interfaces are considered to be the same. f how far beneath the surface SSTR can probe. To empirically study how different parameters of multilayer material systems impact the thermal conduc-tivity measurements of buried layers or substrates, we use the same example used in section 2.3. Figure5(a) shows the % uncertainty of the third layer thermal conductivity as a function of second layer thick-ness corresponding to Figure 4(b). The uncertainty of the third layer thermal conductivity is highest forSiO glass, followed by z-cut quartz and Si. This may seem counter intuitive as SiO glass has the high-est 1/e temperature drop distance among the three materials. Thus, one might expect the third layer ofthe SiO glass to have the lowest uncertainty among the samples. To understand this apparent anomaly, itis necessary to review how sensitivity to different parameters influence the SSTR measurements of thirdlayer thermal conductivity. % un c e r t a i n t y o f t h i r d l a y e r t he r m a l c ondu c t i v i t y Second layer thickness ( m)
SiO glassz-cut quartzSi(a) -3 -2 -1 G S x Thickness ( m) k k G k (b) SiO glass -3 -2 -1 G S x Thickness ( m) k k G k (c) z-cut quartz -3 -2 -1 G S x Thickness ( m) k k G k (d) Si Figure 5: (a) % uncertainty of the third layer thermal conductivity as a function of second layer thickness correspoding toFigure 4(b). The sensitivity, S x , as a function of second layer thickness for (d) SiO glass, (e) z-cut quartz, and (f) Si. For allspecimen, k represents √ k r k z . The sensitivity of SSTR measurements to different parameters for SiO glass, z-cut quartz and Si arepresented in Figures 5(b), 5(c) and 5(d), respectively. It is evident from these sensitivity calculations thatwhen the second layer thickness is high, measurements of third layer are greatly impacted by the secondlayer thermal conductivity. However, for SiO glass, there is also significant sensitivity to the transducerthermal conductivity, whereas for z-cut quartz and Si, sensitivity to nearly all other parameters are very mall. Due to the influence of second layer and transducer thermal conductivity, the uncertainty ofSiO glass is highest. For similar reasons, the uncertainty of the third layer thermal conductivity ishigher for z-cut quartz compared to Si when the second layer thickness is high. At such thicknesses,SSTR measurements of z-cut quartz become sensitive to the transducer thermal conductivity. AlthoughSi measurements also become sensitive to the Al/Si interface conductance, the sensitivity of Si to thisthermal boundary conductance is lower compared to the sensitivity of z-cut quartz to the transducerthermal conductivity. As a result, the uncertainty of Si measurements are relatively lower than z-cutquartz at high second layer thicknesses.From the above discussion, it can be concluded that the TPD cannot provide an estimation of theuncertainty associated with SSTR measurements of a buried substrate. Instead, such uncertainty de-pends on how sensitive SSTR measurements are to different parameters such as the transducer thermalconductivity, the thermal boundary conductances, and the thermal resistances of different layers of themultilayer material system. Therefore, sensitivity calculations can provide the best means for estimatingthe uncertainty of a buried layer or substrate thermal conductivity. Figure 6 shows the sensitivity calculations for a 3-layer system: metal transducer/thin film/substrate forthree different heater radii: 2, 20 and 50 µ m. SSTR measurements are most sensitive to the substratethermal conductivity when the heater radius is much larger than the thin film thickness, irregardlessof what the thin film to substrate thermal conductivity ratio is. When the heater radius is small (i.e.,2 µ m), the sensitivity to the in-plane and cross-plane thermal conductivity of the substrate are nearlythe same. However, as the heater radius increases, sensitivity to the in-plane thermal conductivity ofthe substrate keeps decreasing. This occurs because larger heater radius requires longer time to reachsteady-state. Therefore, to increase the sensitivity to the in-plane thermal conductivity of the substrate,the modulation frequency needs to be lowered. From Figure 6, it is evident that by changing the heaterradius, it is possible to measure the thermal conductivity of buried substrates for different thin filmthicknesses. To further demonstrate the ability of SSTR to measure the thermal conductivity ( √ k r k z ) of buried sub-strates, we choose three samples with the following 3-layer geometry: Al transducer/thin film/substrate.The schematics of the three samples are shown in Figures 7(a)-(c). The first sample is a ∼
130 nm a-SiO thin film on Si substrate. This sample represents an insulating film on a conductive substrate. The secondsample is a ∼ µ m unintentional doped (UID) GaN thin film on hydride vapor phase epitaxy (HVPE)n-GaN substrate. This sample represents the case where the thin film and substrate thermal conductivi-ties are nearly equal. The third sample is a ∼ µ m molecular beam epitaxy (MBE) grown AlN thin filmon sapphire substrate. This sample represents a conductive film on an insulating substrate. Traditionalpump-probe techniques such as TDTR and FDTR often can not measure the thermal conductivity ofburied substrates in such samples due to their limited thermal penetration depths under standard oper-ating conditions. Moreover, for the Si substrate measurements, TDTR and FDTR can also suffer fromnon-equilibrium processes, interfacial phonon-scattering, and ballistic phonon transport. In the ∼
130 nm a-SiO thin film on Si sample, we use co-axially focused 1/e pump and probe -3 -2 -1 k = 1k r,3 k z,2 S x k r,2 G G r = 2 m (a)k z,3 k -3 -2 -1 k = 10k z,3 k r,2 G G k z,2 (b)k r,3 k -3 -2 -1 k = 100k z,3 , k r,3 k r,2 G G k z,2 (c)k -3 -2 -1 k r,3 S x k r,2 G G r = 20 m k z,2 (d)k z,3 k -3 -2 -1 k r,3 k r,2 G G k z,2 (e)k z,3 k -3 -2 -1 k r,3 k r,2 G G k z,2 (f)k z,3 k -3 -2 -1 k S x k r,2 G G r = 50 m k z,2 (g)k z,3 Thickness ( m)k r,3 -3 -2 -1 k r,3 k r,2 G G k z,2 (h)k z,3 Thickness ( m)k -3 -2 -1 k r,3 k r,2 G G k z,2 (i)k z,3 Thickness ( m)k Figure 6: (a) Sensitivity, S x as a function of thin film thickness for a 3-layer system: metal transducer/thin film/substrate.Three different heater radii ( r ) are considered here: 2, 20 and 50 µ m. The sensitivity calculations correspond to f = 100 Hz, d = 80 nm, r = r , k = 100 W m − K − , C V , = C V , = C V , = 2 MJ m − K − , G = G =
200 MW m − K − , and k = 10W m − K − . radii of ∼ µ m to measure the thermal conductivity of buried Si substrate. The sensitivity calculationfor this sample is shown in Figure 7(d). As shown here, SSTR measurements of the Si substrate aresensitive to the cross-plane thermal conductivity of the SiO thin film. TDTR is used to measure thecross-plane thermal conductivity of the SiO thin film. Using this SiO value as an input, the SSTR-measured thermal conductivity of the Si substrate is 141 ±
27 W m − K − . Figure 7(d) indicates thatthe sensitivity to SiO cross-plane thermal conductivity is very high when the pump and probe radii are10 µ m. As a result, the uncertainty associated with the Si thermal conductivity is also high, ∼ and correspondinguncertainty of Si measurement can be reduced. To demonstrate this, we repeat the measurement with1/e pump and probe radii of ∼ µ m. The resultant Si thermal conductivity is 140 ±
18 W m − K − .As predicted, the measurement with the 20 µ m spot sizes has a reduced uncertainty of ∼ ∼ µ m GaN thin film on n-GaN substrate sample, we measure the thermal conductivityof the n-GaN substrate by SSTR using ∼
10 and 20 µ m spot sizes. The sensitivity calculation for thissample is presented in Figure 7(e). The sensitivity to the in-plane and cross-plane thermal conductivitiesof the GaN thin film are considerably lower when the spot sizes are 20 µ m compared to the 10 µ mspot sizes. The cross-plane thermal conductivity of the GaN thin film is measured by TDTR. At roomtemperature, the in-plane and cross-plane thermal conductivity of the GaN thin film can be considered
10 20 30 40 5010 -4 -3 -2 -1 S x (r o2 +r ) ( m) k k , cross-planek G , G k , in-plane(d) 130 nm SiO on Si -4 -3 -2 -1 S x (r o2 +r ) ( m) k k , cross-planek G k , in-plane(e) 2.05 m GaN on GaNG -4 -3 -2 -1 S x (r o2 +r ) ( m) k k , cross-planek G k , in-plane(f) 2 m AlN on sapphireG Figure 7: Schematics of the 3-layer samples measured by SSTR: (a) ∼
130 nm SiO thin film on Si substrate, (b) ∼ µ mGaN thin film on n-GaN substrate, and (c) ∼ µ m AlN thin film on sapphire substrate. Figures (d), (e), and (f) representthe sensitivity calculations as a function of effective radius, q r o + r for the three samples shown in Figures (a), (b), and (c)respectively. to the same. The SSTR-measured thermal conductivity of the GaN substrate is 194 ±
27 W m − K − when the spot sizes are 10 µ m. Using spot sizes of 20 µ m, the thermal conductivity of the GaN substrateis measured with a lower uncertainty to be 185 ±
16 W m − K − .The thermal conductivity of the sapphire substrate is measured by SSTR in the ∼ µ m AlN thinfilm on sapphire sample. The sensitivity calculation for this sample is shown in Figure 7(f). SSTRmeasurement of the sapphire substrate thermal conductivity is most sensitive to the in-plane thermalconductivity of the AlN thin film. The cross-plane thermal conductivity of this AlN thin film is measuredby TDTR. As the anisotropy in the AlN thermal conductivity of is very small at room temperature, thein-plane and cross-plane thermal conductivities of the 2 µ m AlN thin film can be assumed to be thesame. Using SSTR, the thermal conductivity of the sapphire substrate is measured to be 35.1 ± − K − with 1/e pump and probe radii of 10 µ m. Similar to the other two samples, with 20 µ m spotsizes, the sapphire thermal conductivity can be determined with a lower uncertainty, 34.5 ± − K − .In Table 2, we present the measured substrate thermal conductivities for the two spot sizes. Theuncertainty of the measured values incorporate the uncertainty associated with the γ value (sapphire ref-erence), Al transducer and thin film thermal conductivity, thin film thickness, and the thermal boundaryconductances. The values of these parameters are tabulated in Table 1. As shown in Table 2, the SSTR- able 2 : SSTR-measured substrate thermal conductivity ( √ k r k z ) of the samples shown in Figure 7Substrates Thermal conductivity (W m − K − )spot size 10 µ m spot size 20 µ m literatureSi 141 ±
27 140 ±
18 140 GaN 194 ±
27 185 ±
16 195 Sapphire 35.1 ± ± measured substrate thermal conductivities are in excellent agreement with literature. This proves theability of SSTR to accurately measure the thermal conductivities of buried substrates that are typicallyinaccessible by TDTR and FDTR. -4 -3 -2 -1 S x (r o2 +r ) ( m) k k , cross-plane k G , G k , in-plane G Figure 8: Sensitivity calculation as a function of effective radius, q r o + r for the 4-layer sample: 85 nm Al transducer/2.5 µ m Si film/1 µ m SiO layer/Si substrate. We now discuss the required criteria for SSTR to measure the thermal conductivity of a buried filmin a 4-layer system: metal transducer/thin film/buried film/substrate. Measurement of such a buriedfilm is possible when the thermal resistance of this layer is much greater those of the top thin film andsubstrate. This stems from the fact that for SSTR to measure the thermal conductivity of any layer ina multilayered material system, a significant steady-state temperature gradient must exist in that layer,either in cross-plane or in-plane direction. As the top thin film is in contact with the metal transducer, thetemperature gradient of this layer is often large unless the film thickness is very low. On the other hand,since the substrate is a semi-infinite medium, a measurable temperature gradient exists in the substratewhen large pump and probe radii are used. For a buried film, however, unless the thermal resistance s large, the resulting temperature gradient is relatively small compared to those of the thin film andsubstrate. Therefore, although SSTR probes through the buried film and is influenced by the thermalproperties of this layer, the degree of such influence is also relatively small. As a result, SSTR can notisolate the thermal conductivity of a buried film with low thermal resistance.In addition, large pump and probe radii (> 10 µ m) are needed for buried film measurements. Whenthe thermal resistance of the buried layer is much higher than those of thin film and substrate, bulk of theheat flows along the in-plane direction of the top thin film. For a sufficient thermal gradient to exist inthe buried film, large spot sizes are required.To experimentally show this, we have selected a sample that fits this criteria: 85 nm Al transducer/2.5 µ m Si film/1 µ m SiO layer/Si substrate. The sensitivity calculation for this sample is shown in Figure8. As exhibited here, SSTR can measure the thermal conductivity of buried SiO layer when large spotsizes are used. However, such measurements are also sensitive to the in-plane thermal conductivity ofthe top Si film. TDTR is used to measure the cross-plane thermal conductivity of top Si film as shownin Table 1. The in-plane and cross-plane thermal conductivity of the 2.5 µ m Si film can be consideredto the same. Using 1/e pump and probe radii of ∼ µ m, we measure the buried SiO film thermalconductivity to be 1.34 ± − K − . This value is in agreement with literature, showing thecapability of SSTR to measure the thermal conductivity of sub-surface buried layers. Conclusion
We experimentally and numerically investigate the influences of multilayer material systems, thinmetal film transducers, and thermal boundary conductances on the TPD of SSTR technique. The tra-ditional TPD definition of 1/e temperature drop distance from the maximum surface temperature doesnot represent the absolute upper limit of SSTR probing depth. Thus, when estimating whether the ther-mal conductivity of a buried substrate is measurable within acceptable limits of uncertainty, sensitivitycalculations provide the best means. The low modulation freqency of SSTR enables it to measure thethermal conductivity of buried substrates typically inaccessible by TDTR and FDTR, demonstrated bypresenting experimental data on three control samples. In addition, SSTR has the capability to isolatethe thermal properties of a buried film as long as the thermal properties of this layer is much higher thanthose of the top thin film and substrate. This work marks an advancement in experimental metrology byestablishing SSTR as a robust technique for thermal characterizations of subsurface buried substrates.
Acknowledgements
The authors would like to acknowledge the financial support from U.S. Office of Naval Researchunder a MURI program (Grant no. N00014-18-1- 5332429). Z. C. Leseman acknowledges supportprovided by the Deanship of Scientific Research at King Fahd University of Petroleum & Minerals forfunding this work through project SR191001.
References [1] Park, J.-S.; Mo, Y.-G.; Jeong, J.-K.; Jeong, J.-H.; Shin, H.-S.; Lee, H.-J. Thin film transistor andorganic light-emitting display device having the thin film transistor. 2008; US Patent App. 12/076,216.
2] Tokunaga, K. Thin film transistor and method of manufacturing thin film transistor. 2010; US PatentApp. 12/557, 212.[3] Venkatasubramanian, R.; Siivola, E.; Colpitts, T.; O’quinn, B. Thin-film thermoelectric deviceswith high room-temperature figures of merit.
Nature , , 597–602.[4] Xi, J.-Q.; Schubert, M. F.; Kim, J. K.; Schubert, E. F.; Chen, M.; Lin, S.-Y.; Liu, W.; Smart, J. A.Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection. Nature photonics , , 176–179.[5] Peumans, P.; Yakimov, A.; Forrest, S. R. Small molecular weight organic thin-film photodetectorsand solar cells. Journal of Applied Physics , , 3693–3723.[6] Yang, Y.; Ma, L.; Wu, J. Organic thin-film memory. Mrs Bulletin , , 833–837.[7] Sung, S. H.; Boudouris, B. W. Systematic control of the nanostructure of semiconducting-ferroelectric polymer composites in thin film memory devices. ACS Macro Letters , , 293–297.[8] Dames, C. Measuring the thermal conductivity of thin films: 3 omega and related electrothermalmethods. Annual Review of Heat Transfer , .[9] Cahill, D. G. Analysis of heat flow in layered structures for time-domain thermoreflectance. Reviewof scientific instruments , , 5119–5122.[10] Schmidt, A. J.; Chen, X.; Chen, G. Pulse accumulation, radial heat conduction, and anisotropicthermal conductivity in pump-probe transient thermoreflectance. Review of Scientific Instruments , , 114902.[11] Feser, J. P.; Liu, J.; Cahill, D. G. Pump-probe measurements of the thermal conductivity tensor formaterials lacking in-plane symmetry. Review of Scientific Instruments , , 104903.[12] Jiang, P.; Qian, X.; Yang, R. Tutorial: Time-domain thermoreflectance (TDTR) for thermal propertycharacterization of bulk and thin film materials. Journal of Applied Physics , , 161103.[13] Schmidt, A. J.; Cheaito, R.; Chiesa, M. A frequency-domain thermoreflectance method for thecharacterization of thermal properties. Review of scientific instruments , , 094901.[14] Zhao, D.; Qian, X.; Gu, X.; Jajja, S. A.; Yang, R. Measurement techniques for thermal conductivityand interfacial thermal conductance of bulk and thin film materials. Journal of Electronic Packaging , .[15] Zhu, J.; Tang, D.; Wang, W.; Liu, J.; Holub, K. W.; Yang, R. Ultrafast thermoreflectance techniquesfor measuring thermal conductivity and interface thermal conductance of thin films. Journal ofApplied Physics , , 094315.[16] Braun, J. L.; Olson, D. H.; Gaskins, J. T.; Hopkins, P. E. A steady-state thermoreflectance methodto measure thermal conductivity. Review of Scientific Instruments , , 024905.[17] Koh, Y. R.; Cheng, Z.; Mamun, A.; Hoque, M. S. B.; Liu, Z.; Bai, T.; Hussain, K.; Liao, M. E.;Li, R.; Gaskins, J. T., et al. Bulk-like intrinsic phonon thermal conductivity of micrometer thickAlN films. ACS Applied Materials & Interfaces , , 29443–29450.[18] Rosei, R.; Lynch, D. W. Thermomodulation spectra of al, au, and cu. Physical Review B , ,3883.
19] Koh, Y. K.; Cahill, D. G. Frequency dependence of the thermal conductivity of semiconductoralloys.
Physical Review B , , 075207.[20] Braun, J. L.; Hopkins, P. E. Upper limit to the thermal penetration depth during modulated heatingof multilayer thin films with pulsed and continuous wave lasers: A numerical study. Journal ofApplied Physics , , 175107.[21] Braun, J. L.; Szwejkowski, C. J.; Giri, A.; Hopkins, P. E. On the steady-state temperature rise duringlaser heating of multilayer thin films in optical pump–probe techniques. Journal of Heat Transfer , .[22] Olson, D. H.; Braun, J. L.; Hopkins, P. E. Spatially resolved thermoreflectance techniques for ther-mal conductivity measurements from the nanoscale to the mesoscale. Journal of Applied Physics , , 150901.[23] Wilson, R.; Apgar, B. A.; Martin, L. W.; Cahill, D. G. Thermoreflectance of metal transducers foroptical pump-probe studies of thermal properties. Optics express , , 28829–28838.[24] Wang, Y.; Park, J. Y.; Koh, Y. K.; Cahill, D. G. Thermoreflectance of metal transducers for time-domain thermoreflectance. Journal of Applied Physics , , 043507.[25] Rost, C. M.; Braun, J.; Ferri, K.; Backman, L.; Giri, A.; Opila, E. J.; Maria, J.-P.; Hopkins, P. E.Hafnium nitride films for thermoreflectance transducers at high temperatures: Potential based onheating from laser absorption. Applied Physics Letters , , 151902.[26] Wang, L.; Cheaito, R.; Braun, J.; Giri, A.; Hopkins, P. Thermal conductivity measurements of non-metals via combined time-and frequency-domain thermoreflectance without a metal film transducer. Review of Scientific Instruments , , 094902.[27] Radue, E. L.; Tomko, J. A.; Giri, A.; Braun, J. L.; Zhou, X.; Prezhdo, O. V.; Runnerstrom, E. L.;Maria, J.-P.; Hopkins, P. E. Hot Electron Thermoreflectance Coefficient of Gold during Electron–Phonon Nonequilibrium. Acs Photonics , , 4880–4887.[28] Qin, M.; Gild, J.; Hu, C.; Wang, H.; Hoque, S. B.; Braun, J. L.; Harrington, T. J.; Hopkins, P. E.;Vecchio, K. S.; Luo, J. Dual-Phase High-Entropy Ultrahigh Temperature Ceramics. Journal of theEuropean Ceramic Society , , 5037 – 5050.[29] Jang, E.; Banerjee, P.; Huang, J.; Holley, R.; Gaskins, J. T.; Hoque, M. S. B.; Hopkins, P. E.,et al. Thermoelectric Performance Enhancement of Naturally Occurring Bi and Chitosan CompositeFilms Using Energy Efficient Method. Electronics , , 532.[30] Fulkerson, W.; Moore, J.; Williams, R.; Graves, R.; McElroy, D. Thermal conductivity, electricalresistivity, and seebeck coefficient of silicon from 100 to 1300 K. Physical Review , , 765.[31] Kremer, R.; Graf, K.; Cardona, M.; Devyatykh, G.; Gusev, A.; Gibin, A.; Inyushkin, A.;Taldenkov, A.; Pohl, H.-J. Thermal conductivity of isotopically enriched 28Si: revisited. Solid statecommunications , , 499–503.[32] Wilson, R.; Cahill, D. G. Anisotropic failure of Fourier theory in time-domain thermoreflectanceexperiments. Nature communications , , 1–11.[33] Ziade, E.; Yang, J.; Brummer, G.; Nothern, D.; Moustakas, T.; Schmidt, A. J. Thermal transportthrough GaN–SiC interfaces from 300 to 600 K. Applied Physics Letters , , 091605.
34] Braun, J. L.; Baker, C. H.; Giri, A.; Elahi, M.; Artyushkova, K.; Beechem, T. E.; Norris, P. M.;Leseman, Z. C.; Gaskins, J. T.; Hopkins, P. E. Size effects on the thermal conductivity of amorphoussilicon thin films.
Physical Review B , , 140201.[35] Tareq, M. S.; Zainuddin, S.; Woodside, E.; Syed, F. Investigation of the flexural and thermomechan-ical properties of nanoclay/graphene reinforced carbon fiber epoxy composites. Journal of Materi-als Research , , 3678–3687.[36] Cheng, Z.; Mu, F.; Yates, L.; Suga, T.; Graham, S. Interfacial Thermal Conductance across Room-Temperature-Bonded GaN/Diamond Interfaces for GaN-on-Diamond Devices. ACS Applied Mate-rials & Interfaces , , 8376–8384.[37] Giri, A.; Hopkins, P. E. A review of experimental and computational advances in thermal boundaryconductance and nanoscale thermal transport across solid interfaces. Advanced Functional Materi-als , , 1903857.[38] Minnich, A. J.; Johnson, J. A.; Schmidt, A. J.; Esfarjani, K.; Dresselhaus, M. S.; Nelson, K. A.;Chen, G. Thermal conductivity spectroscopy technique to measure phonon mean free paths. Physi-cal review letters , , 095901.[39] Regner, K. T.; Sellan, D. P.; Su, Z.; Amon, C. H.; McGaughey, A. J.; Malen, J. A. Broadbandphonon mean free path contributions to thermal conductivity measured using frequency domainthermoreflectance. Nature communications , , 1–7.[40] Wilson, R.; Cahill, D. G. Limits to Fourier theory in high thermal conductivity single crystals. Applied Physics Letters , , 203112.[41] Florescu, D.; Asnin, V.; Pollak, F. H.; Molnar, R.; Wood, C. High spatial resolution thermal conduc-tivity and Raman spectroscopy investigation of hydride vapor phase epitaxy grown n-GaN/sapphire(0001): Doping dependence. Journal of Applied Physics , , 3295–3300.[42] Cahill, D. G.; Lee, S.-M.; Selinder, T. I. Thermal conductivity of κ -Al 2 O 3 and α -Al 2 O 3wear-resistant coatings. Journal of applied physics , , 5783–5786.[43] Lindsay, L.; Broido, D.; Reinecke, T. Thermal conductivity and large isotope effect in GaN fromfirst principles. Physical review letters , , 095901.[44] Lindsay, L.; Broido, D.; Reinecke, T. Ab initio thermal transport in compound semiconductors. Physical Review B , , 165201.[45] Dong, Y.; Cao, B.-Y.; Guo, Z.-Y. Ballistic–diffusive phonon transport and size induced anisotropyof thermal conductivity of silicon nanofilms. Physica E: Low-dimensional Systems and Nanostruc-tures , , 1–6.[46] Cahill, D. G. Thermal conductivity measurement from 30 to 750 K: the 3 ω method. Review ofscientific instruments , , 802–808.[47] Giri, A.; Chen, A. Z.; Mattoni, A.; Aryana, K.; Zhang, D.; Hu, X.; Lee, S.-H.; Choi, J. J.; Hop-kins, P. E. Ultralow thermal conductivity of two-dimensional metal halide perovskites. Nano letters , , 3331–3337., 3331–3337.