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Comparison of the 3ω method and time-domain thermoreflectance

Time-**domain** **the**rmoreflectance

psec acoustics **and****time**-**domain** **the**rmoreflectance• Optical constants **and**reflectivity depend onstrain **and** temperature• Strain echoes giveacoustic properties orfilm thickness• Thermoreflectance gives**the**rmal properties

Schmidt et al., RSI 2008• Heat supplied bymodulated pumppbeam (fundamentalFourier componentat frequency f)• Evolution **of** surfacetemperature**time**

Schmidt et al., RSI 2008• Instantaneoustemperatures measuredby **time**-delayed probe• Probe signal asmeasured by rf lock-inamplifier

Analytical solution to 3D heat flowin an infinite half-space, Cahill, RSI (2004)• spherical **the**rmal wave• Hankel transform **of**surface temperature• Multiply by transform**of** Gaussian heatsource **and** takeinverse transform• Gaussian-weightedsurface temperature8

Iterative solution for layered geometries

Frequency **domain** solution for **3ω** **and**TDTR are essentially **the** same**3ω**• “rectangular” heatsource **and** temperatureaveraging.• One-dimensional Fouriertransform.• “known” quantities in**the** analysis are Jouleheating **and** dR/dTcalibration.TDTR• Gaussian heat source**and** temperatureaveraging.• Radial symmetric Hankeltransform.• “known” quantity in **the**analysis is **the** heatcapacity per unit area **of****the** metal filmtransducer.

TDTR signal analysis for **the** lock-insignal as a function **of** delay **time** t• In-phase **and** out-**of**-phase signals by series **of** sum **and**difference over sideb**and**s• out-**of**-phase signal is dominated by **the** m=0 term• out **of** phase signal is dominated by **the** m=0 term(frequency response at modulation frequency f)

Windows s**of**twareauthor: Catalin Chiritescu,users.mrl.uiuc.edu/cahill/tcdata/tdtr_m.zip

Time-**domain** Thermoreflectance (TDTR)data for TiN/SiO 2 /SiTiNSiO 2SiCostescu et al., PRB (2003)• reflectivity **of** a metaldepends ontemperature• one free parameter:**the** “effective”**the**rmal conductivity**of** **the** **the**rmallygrown SiO 2 layer(interfaces notmodeled separately)

TDTR: early validation experimentsCostescu et al., PRB (2003) Zhao et al., Materials Today (2005)

Each have advantages **and** disadvantages**3ω**• High accuracy, particularly for bulk materials**and** low **the**rmal conductivity dielectric films• Accuracy is reduced for semiconducting thinfilms **and** high **the**rmal conductivity layers– Need electrical insulation: introduces anadditional **the**rmal resistance.– Cannot separate **the** metal/film interface**the**rmal conductance from **the** **the**rmalconductivity• Wide temperature range (30 < T< 1000 K)– But very high temperatures are not usuallyaccessible for semiconductors

Each have advantages **and** disadvantagesTDTR• Accuracy is typically limited to several percentdue to uncertainties in **the** many experimentalparameters– Metal film thickness– Heat capacity **of** **the** sample if film is thick• But many experimental advantages– No need for electrical insulation– Can separate **the** metal/film interface **the**rmalconductance from **the** **the**rmal conductivity– High spatial resolution– Only need optical access: high pressures,high magnetic fields, high temperatures

Digression: what limits **the** accuracy **of** **3ω** data?• 1990’s: approximations made for low **the**rmalconductivity film on high **the**rmal conductivitysubstrate **and** film thickness

Digression: what limits **the** accuracy **of** **3ω** data?• Contributions from **the** heater line.– Not explicitly included in **the** heat fluxboundary conditions **of** **the** solutions– Heat capacity matters at very highfrequencies, see, for example, Tong et al.RSI (2006).– Lateral heat flow in heater line wasconsidered recently by Gurrum et al., JAP(2008).

Digression: what limits **the** accuracy **of** **3ω** data?• In my experience, **the** dR/dT calibration is **the**biggest issue.– use physics to fix **the** calibrationBloch-Grüneisen resistivity **of** a metal

Calibration **of** Au **the**rmometer line• Materials with largecoefficient **of** **the**rmalexpansion create aninteresting ti problem– during calibration **of**R(T) substrate strainis homogeneous–but during **3ω**measurement, acstrain field is complexso **the** determination**of** dR/dT is not reallycorrect.

High **the**rmal expansion coefficients• Add terms to account for effect **of** strain on**the** Bloch-Grüneisen resistivity **and** **the**residual resistivity.• CTE **of** PMMA is ≈50 ppm/K• CTE **of** PbTe is ≈20 ppm/K

Highest precision measurements at Illinoisusing **3ω**: polymer nanocomposites• PMMA mixed with 60 nm γ-Al 2 O 3 nanoparticlesnanocompositePMMAPutnam et al., JAP (2003)

Something not possible with **3ω**: TDTR datafor isotopically pure Si epitaxial layer on Si• Two free fitting parameters– **the**rmal conductivity, 165 W/m-K– Al/Si interface conductance, 185 MW/m 2 -KCahill et al., PRB (2004)23

Thermal conductivity map **of** a human tooth100 μm3210www.enchantedlearning.com/ΛC/C0 0 (W m -1 K -1 )2.015 1.51.00.50.0dentinenamel-400 -200 0 200Distance from **the** DEJ (μm)

High throughput data using diffusion couples

Thermoreflectance raw data at t=100 ps• fix delay **time** **and**vary modulationfrequency f.• Change in V in doesn’tdepend on f. V outmostly depends on(fΛC) -1/2• semiconductor alloysshow deviation fromfit using a singlevalue **of** **the** **the**rmalconductivitya-SiO 2InGaAs, InGaPInP, GaAsSiKoh **and** Cahill PRB (2007)

Same data but fit Λ at each frequency fFrequency dependent**the**rmal conductivity**of** semiconductoralloysKoh **and** Cahill PRB (2007)

How can **the**rmal conductivity be frequencydependent at only a few MHz?• 2πfτ d do notcontribute to **the** heat transport in thisexperiment.T l if th “ i l l ti ti• True only if **the** “single-relaxation-**time**approximate” fails strongly. For singlerelaxation **time** τ, l

For non-equilibrium, add effusivityinstead **of** conductivity• Consider a "two-fluid" model with• Equilibrium,Λ 1 ≈ Λ 2C 1 >> C 2(ΛC) 1/2 = [(Λ + 1/21 Λ 2 )(C 1 +C 2 )]• Out-**of**-equilibrium,(ΛC) 1/2 = (Λ 1 C 1 ) 1/2 + (Λ 2 C 2 ) 1/2≈ (Λ 1/21 C 1 )

f

Summary **and** Conclusions• Usually, **3ω** has higher accuracy because Jouleheating **and** dR/dT calibration are electricalmeasurements **and** geometry is precisely known.• For semiconducting thin films, because **of** extra**the**rmal resistance **of** electrical isolation layers,accuracy **of** TDTR is comparable.• TDTR has tremendous advantages in experimentalconvenience—once **the** high initial cost **and** set-uphas been overcome.