psec acoustics andtime-domain thermoreflectance• Optical constants andreflectivity depend onstrain and temperature• Strain echoes giveacoustic properties orfilm thickness• Thermoreflectance givesthermal properties
Schmidt et al., RSI 2008• Heat supplied bymodulated pumppbeam (fundamentalFourier componentat frequency f)• Evolution of surfacetemperaturetime
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 thermal wave• Hankel transform ofsurface temperature• Multiply by transformof Gaussian heatsource and takeinverse transform• Gaussian-weightedsurface temperature8
Iterative solution for layered geometries
Frequency domain solution for 3ω andTDTR are essentially the same3ω• “rectangular” heatsource and temperatureaveraging.• One-dimensional Fouriertransform.• “known” quantities inthe analysis are Jouleheating and dR/dTcalibration.TDTR• Gaussian heat sourceand temperatureaveraging.• Radial symmetric Hankeltransform.• “known” quantity in theanalysis is the heatcapacity per unit area ofthe 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 anddifference over sidebands• 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 softwareauthor: 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”thermal conductivityof the thermallygrown SiO 2 layer(interfaces notmodeled separately)
TDTR: early validation experimentsCostescu et al., PRB (2003) Zhao et al., Materials Today (2005)
Each have advantages and disadvantages3ω• High accuracy, particularly for bulk materialsand low thermal conductivity dielectric films• Accuracy is reduced for semiconducting thinfilms and high thermal conductivity layers– Need electrical insulation: introduces anadditional thermal resistance.– Cannot separate the metal/film interfacethermal conductance from the thermalconductivity• 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 thermalconductance from the thermal 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 thermalconductivity film on high thermal 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 thebiggest issue.– use physics to fix the calibrationBloch-Grüneisen resistivity of a metal
Calibration of Au thermometer line• Materials with largecoefficient of thermalexpansion create aninteresting ti problem– during calibration ofR(T) substrate strainis homogeneous–but during 3ωmeasurement, acstrain field is complexso the determinationof dR/dT is not reallycorrect.
High thermal expansion coefficients• Add terms to account for effect of strain onthe Bloch-Grüneisen resistivity and theresidual 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– thermal 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 andvary 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 thermalconductivitya-SiO 2InGaAs, InGaPInP, GaAsSiKoh and Cahill PRB (2007)
Same data but fit Λ at each frequency fFrequency dependentthermal conductivityof semiconductoralloysKoh and Cahill PRB (2007)
How can thermal 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-timeapproximate” 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 )
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 extrathermal 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.