Introduction to broadband frequency-domain thermoreflectance (BB-FDTR)



What is broadband frequency-domain thermoreflectance (BB-FDTR)?


The modulation frequency of the pump beam in TDTR/FDTR is an important parameter that affects the sensitivity to various thermal properties and the thermal penetration depth. For investigating the size effect of the thermal conductivity in nanoscale materials, wide-range modulation of the heating frequency is needed to change the thermal penetration depth and measure the thermal conductivity accumulation function kaccum over a broad distribution of mean free paths (MFPs).


(Details are described in "Investigation of size effect in nanoscale thermal transport by BB-FDTR".)


Broadband frequency-domain thermoreflectance (BB-FDTR) is a variation of FDTR that extends the modulation frequency to 200 MHz, which is 10 times higher than the limit for typical FDTR, through heterodyne detection.[1]


BB-FDTR is used for the characterization of various thermal properties of bulk and thin films such as TDTR/FDTR and is more suitable (owing to its higher modulation frequency) for investigating quasi-ballistic thermal transport, which is of increasing importance in nanoscale engineering—particularly for thermoelectric applications.



Overview of BB-FDTR measurement [1]


The experimental setup for BB-FDTR is almost identical to that for FDTR with two CW lasers of different wavelengths. The pump CW laser undergoes intensity modulation at frequency f1 by EOM1 and is focused on the sample by an objective lens. The probe CW laser is focused on the sample by the same objective lens to detect the frequency-modulated thermoreflectance signals at f1.


TypicalExperimentalSetupofBBFDTR

Figure 1. Schematic of the BB-FDTR technique.


EOM2 induces additional intensity modulation in the reflected probe beam at frequency f2, heterodyning the reflected pump and probe beams to create frequency-modulation components at f1 − f2 and f1 + f2 based on the following trigonometric formula:


TrigonometricFormula


The reflected pump beam and high-frequency component f1 + f2 are filtered using a bandpass filter (BPF) and a low-pass filter (LPF), respectively. The amplitude R and phase φ of the probe signal at f1 − f2 are detected using a lock-in amplifier.


The additional modulation frequency f2 is determined such that f1 - f2 is maintained at a constant value of < 100 kHz, which is selected to be close to the upper limit of the frequency range of the lock-in amplifier so that the higher harmonic components are excluded. Under such conditions, the thermoreflectance signal caused by a higher heating frequency can be detected at a far lower frequency, with minimal noise.



Investigation of size effect in nanoscale thermal transport by BB-FDTR


Changing the modulation frequency of the pump beam results in a variation in the thermal penetration depth dp:


ThermalPenetrationDepth


where α represents the thermal diffusivity of the sample, and fmod and ωmod represent the frequency and angular frequency, respectively, of the pump beam.


When dp is comparable to the phonon MFPs, the quasi-ballistic thermal transport effect becomes apparent, and phonons with MFPs longer than dp do not contribute to the apparent thermal conductivity measured by BB-FDTR (TDTR/FDTR).[2] Under such conditions, BB-FDTR can be used to measure the thermal conductivity accumulation function kaccum, which describes the cumulative contributions to the bulk thermal conductivity from phonons with MFPs shorter than dp.[3] By varying the modulation frequency over a wide range, BB-FDTR can be used to measure a phonon MFP spectrum that illustrates the size effect of the thermal conductivity in nanodevices and clarifies the nanoscale thermal transport.


ThermalTransport

Figure 2. (a) Illustration of diffusive and quasi-ballistic transport at low and high modulation frequencies, respectively, of a pump beam. (b) Typical image of the phonon MFP spectrum, which shows the normalized kaccum as a function of the phonon MFP.



Advantages of BB-FDTR


  • In addition to bulk samples, thin films having thicknesses ranging from tens of nanometers to a few micrometers can be measured.
  • By utilizing different laser spot sizes and modulation frequencies, various thermal properties, such as the cross-plane thermal conductivity Kz, in-plane thermal conductivity Kr, interface thermal conductance G, and heat capacity C, can be evaluated.
  • Non-contact measurements work either under regular ambient conditions or through the window of a vacuum chamber.
  • In contrast to TDTR, BB-FDTR (along with FDTR) avoids the complexity of a long mechanical time delay. Additionally, an expensive pulsed laser is not necessary.
  • Frequency selection, which is closely related to the unknowns and consequently difficult to perform before TDTR measurement, can be avoided in BB-FDTR (and FDTR) measurements.
  • The thermal conductivity accumulation function kaccum can be measured over a wider range of MFPs compared with FDTR.


  • References


    [1] K. T. Regner, S. Majumdar, and J. A. Malen,
    “Instrumentation of broadband frequency domain thermoreflectance for measuring thermal conductivity accumulation functions”
    Rev. Sci. Instrum. 84(6), 064901 (2013).

    [2] Y. K. Koh and D. G. Cahill,
    “Frequency dependence of the thermal conductivity of semiconductor alloys”
    Phys. Rev. B 76(7), 075207 (2007).

    [3] C. Dames, G. Chen,
    “Thermal conductivity of nanostructured thermoelectric materials”
    Thermoelectrics Handbook: Macro to Nano, Chapter 42, CRC Press, ed. D. Rowe, (2005).


    Jan. 2022