Crystalline coherence length effects on the thermal conductivity of MgO thin films
Phonon scattering in crystalline systems can be strongly dictated by a wide array of defects, many of which can be difficult to observe via standard microscopy techniques. We experimentally demonstrate that the phonon thermal conductivity of MgO thin films is proportional to the crystal’s coherence length, a property of a solid that quantifies the length scale associated with crystalline imperfections. Sputter-deposited films were prepared on (100)-oriented silicon and then annealed to vary the crystalline coherence, as characterized using x-ray diffraction line broadening. We find that the measured thermal conductivity of the MgO films varies proportionally with crystalline coherence length, which is ultimately limited by the grain size. The microstructural length scales associated with crystalline defects, such as small-angle tilt boundaries, dictate this crystalline coherence length, and our results demonstrate the role that this length scale has on the phonon thermal conductivity of thin films. Our results suggest that this crystalline coherence length scale provides a measure of the limiting phonon mean free path in crystalline solids, a quantity that is often difficult to measure and observe with more traditional imagining techniques.
Defect-induced deviations in lattice structure can give rise to phonon scattering processes and changes in the phonon thermal conductivity of crystals. Where ample experimental works have studied and validated classical scattering theories regarding phonon–grain boundary and phonon–impurity thermal resistances on bulk- and nanoscales [1, 2, 3, 4, 5, 6, 7, 8], the phonon scattering mechanisms contributing to thermal resistances at finer scale defects, such as small-angle tilt boundaries and dislocations, have been much less frequently studied . This has led to voids in the depth of understanding of the interplay between, and importance of, phonon and lattice defect scattering relative to the interaction of phonons with other static impurities, such as incoherent grain boundaries, mass impurities, and interfaces between dissimilar materials. Progress in this fundamental understanding of the phonon–lattice defect interaction will have major impacts in the design of novel classes of material systems, such as recently discovered systems synthesized by utilizing screw dislocations [10, 11, 12, 13, 14], high figure of merit thermoelectric materials designed with dense dislocations arrays , nanostructures with dislocation dense interfaces that impact the thermal boundary conductance [16, 17, 18], and thermal transport in ferroelectric materials where coherent ferroelastic domain walls affect the phononic resistance [19, 20, 21].
In the present study, we report on measurements of the room temperature thermal conductivity of a series of magnesium oxide (MgO) nanocrystalline thin films in which the crystalline coherence lengths (the characteristic length of crystal devoid of translational symmetry-breaking defects) of the MgO films are varied. Small-angle tilt boundaries defining crystallites of similar dimensions to the measured crystal coherence length were previously identified in identically processed films. These, in addition to grain boundaries and other crystallographic defects, such as dislocations, were attributed to a damping of optical phonons . It is therefore anticipated that these same defects may affect the transport of heat-carrying phonons. We use time domain thermoreflectance (TDTR)  to measure the thermal properties of the MgO thin films at room temperature and atmospheric conditions; by utilizing a combination of both the in-phase and the ratio of the in-phase to out-of-phase components of the TDTR response in tandem, we demonstrate the ability to measure the thermal conductivity of the MgO films at a single modulation frequency, while separating the influence of thermal boundary conductance across both the front and back thin film interfaces from this thin film thermal conductivity measurement. We show that the thermal conductivities of the MgO films increase with an increase in crystalline coherence length, which is correlated with the defects that limit the crystalline coherence. Our work demonstrates the ability to quantify the influence of defects on the phonon thermal conductivity by an average length scale of crystal translational symmetry—the crystalline coherence length. Our results suggest that this crystalline coherence length scale provides a measure of the limiting phonon mean free path in crystalline solids, a quantity that can be determined via standard X-ray diffraction and is often difficult to measure and observe with traditional microcopy techniques.
The 80-nm-thick MgO film series was prepared on (100)-oriented silicon substrates via 30° off-axis RF magnetron sputter deposition within a Kurt J. Lesker Lab 18 instrument. Prior to loading in the load-locked sputter chamber, the substrates underwent a 7:1 buffered HF etch (pH of 5.5) and de-ionized-H2O rinse to remove the native silicon dioxide surface. The film was sputtered from a single-phase sintered MgO target in 5 mTorr of argon at room temperature with a power density of 3.7 W/cm2. The wafer was subsequently divided, and sections were processed between 200 and 800 °C in 200 °C intervals in air for 1 h; varying the annealing temperature directly correlates to a change in crystalline coherence length . Silicon was chosen as a substrate due to its predicted phase stability and chemical inertness with MgO in this temperature range, suggesting the formation of clean interfaces with no secondary phases during the deposition and annealing process ; furthermore, the high thermal conductivity of silicon ensures maximum sensitivity to the thermal conductivity of the MgO thin films in our TDTR measurements. Phase purity and crystalline coherence lengths were characterized via X-ray diffraction using a Philips X’Pert MPD with Cu Kα radiation in the Bragg–Brentano geometry. As such, the diffraction vector is normal to the sample surface, and the only lattice planes to which we are collecting diffracted X-rays are those that are parallel to the film surface. Therefore, crystallite dimensions to which this technique is sensitive are those normal to the film surface—the same direction as thermal conductivity is measured. Crystalline coherence lengths were calculated using X-ray line broadening and Scherrer’s formula .
MgO film thickness (average of measurements made with XRR and VASE), MgO surface roughness (average of measurements made with XRR, VASE and AFM), and thickness of SiO2 layer between MgO and Si substrate (average of measurements made with XRR and VASE)
Anneal temperature (°C)
MgO film thickness (nm)
84.0 ± 2.8
78.9 ± 4.9
79.3 ± 3.5
83.6 ± 1.7
82.4 ± 1.6
MgO surface roughness (nm)
2.8 ± 1.1
5.8 ± 3.4
4.72 ± 1.8
5.1 ± 2.3
9.3 ± 5.3
SiO2 layer thickness (nm)
3.8 ± 3.8
3.4 ± 3.1
3.5 ± 2.1
6.1 ± 1.2
The thermal conductivities of the MgO films were determined using TDTR by fitting the data to a multi-layer thermal model described in detail in the literature [31, 32, 33, 34]. Briefly, TDTR is a non-contact optical pump–probe technique that uses a short-pulsed laser to both produce and monitor modulated heating events on the surface of a sample. The laser output from a sub-picosecond oscillator is separated into pump and probe paths, in which the relative optical path lengths are adjusted with a mechanical delay stage. The pump path is modulated to create a frequency dependent temperature variation on the surface of the sample, and the in-phase and out-of-phase signals of the probe beam locked into the modulation frequency of the pump are monitored with a lock-in amplifier. Prior to TDTR measurements, the sample surfaces were coated with a thin aluminum film so that the changes in reflectivity of the surfaces were indications of the change in temperature within the optical penetration depth of the aluminum; this change in reflectivity is driven by the thermal properties of the MgO film, the silicon substrate, and the thermal boundary conductances across the Al/MgO and MgO/Si interfaces. We assume literature values for the heat capacities of the aluminum , MgO , and silicon , leaving the unknowns in our thermal model as the thermal boundary conductances across the Al/MgO and MgO/Si interfaces (hK,Al/MgO and hK,MgO/Si, respectively) and the thermal conductivity of the MgO film, κMgO .
Our reported uncertainties in the values reported for thermal conductivities and thermal boundary conductances are determined by considering three different sources of error. First, we calculate the standard deviation among the entire set of measurements for each sample (multiple measurements on each sample). Second, we assume a ~10 % uncertainty in the aluminum transducer film thickness. Finally, we determine a 95 % confidence interval for each measurement. We take the square root of the sum of the squares of each deviation from the mean values resulting from these sources of uncertainties to construct our error bars. We note that largest uncertainties in our reported values lie in the samples with the highest thermal boundary conductances and highest thermal conductivities. This is consistent with the fact that as the thermal conductivity of the MgO thin films increase (or the interface conductances increase), and hence, the corresponding thermal resistances decrease, our TDTR measurements become less sensitive to these thermophysical properties. However, our reported values still lie within a 95 % confidence bound. Along these lines, it is worth noting that the apparent observed frequency dependence in the thermal conductivity measurements of the MgO samples with the maximum coherence length (Fig. 3a) are nearly constant when considering our aforementioned confidence interval, and still only deviate ~20 % about the mean; in other words, this fluctuation in our measured data for thermal conductivity with frequency is not physical, but just an artifact of the sensitivity of TDTR for measuring relatively thermally conductive thin films (i.e., films with relatively low thermal resistance), especially when using lower pump modulation frequencies where the thermal penetration depth is increased and therefore sampling more of the underlying substrate relative to the thin film. However, as the coherence length in the MgO thin film is decreased, and the thermal conductivity is lowered, TDTR measurements are much more robust and sensitive in measuring thermal conductivity, consistent with the relatively minor uncertainty associated with our fits.
We note that the surfaces of the MgO film are changing among the samples processed at different temperatures; as previously mentioned, with increasing MgO processing temperature, the MgO surface becomes more rough and the SiO2 layer between the MgO and silicon becomes thicker. However, we measure an increasing thermal conductivity with increased temperature, which would imply that the increased surface roughness and increase in SiO2 thickness, which would add thermal resistance to the system, play only a minor role in our thermal conductivity measurements of MgO compared to the changing crystalline coherence length. This also gives further support to our data representing the intrinsic thermal conductivity of the MgO, and our ability to separate the resistances at the MgO interfaces from our reported values of κ.
We determine ∂ω/∂(a3) from the experimentally measured transverse optical frequencies, which depend on unit cell volume , and scale the Debye temperature for each film assuming θD,film = θD,lit(afilm/alit). Through this analysis, we observe opposite trends when comparing our experimental results to the predicted variation of thermal conductivity due to lattice spacing. This implies that the change in lattice parameter among the sample series is not responsible for the variation in thermal conductivity; rather, that defects responsible for the crystalline coherence are the driving force impacting phonon scattering.
This demonstrates the ability to quantify the influence of defects on the phonon thermal conductivity by the crystalline coherence length of the crystal. This has the advantage of offering limiting length scales for phonon transport in crystalline systems in which imperfections are difficult to characterize and/or model. For example, using molecular dynamics simulations, Ni et al.  showed that localized strain fields, varying atomic spacings, and modifications to the intrinsic anharmonic phonon–phonon interaction strength near defects, such as dislocation cores, must be accounted for to properly model the phonon–lattice defect dynamics in the thermal conductivity. Unlike phonon–grain boundary and phonon–mass impurity-limited thermal transport [1, 2, 3, 4, 5, 6, 7, 8], modeling these processes is not easily or accurately feasible with simplified kinetic theory-type approaches, and therefore, predictions of changes in thermal conductivity due to these imperfections can be daunting. However, our work suggests that the characterization of a crystalline coherence length gives insight to qualitatively compare changes in thermal conductivity of similar materials with different degrees of crystalline imperfections.
As a final note, our experimental measurements in Fig. 3b show a relatively negligible dependence of thermal boundary conductance with crystalline coherence length of the MgO. We have previously observed that interfacial imperfections can lead to changes in thermal boundary conductance [16, 43]. Given that we do not observe any substantial structural changes at the surfaces of the MgO films, we would not expect any changes in thermal boundary conductance, which is consistent with our measurements of hK at each interface. Furthermore, it is interesting to note that the Al/MgO thermal boundary conductance is consistently lower than the MgO/Si thermal boundary conductance regardless of the MgO crystalline coherence length. While more work must be done that specifically focuses on the role of interface defects, our results highlight the potential impact of our previously discussed TDTR analysis to extract the thermal boundary conductance across thin films interfaces and, using this, to assess the role of changes in atomic-scale defects at material interfaces on changes (or lack thereof) in thermal boundary conductance.
In conclusion, we have investigated the effects of crystallinity changes on the thermal conductivity of MgO thin films. We find a systematic increase in thermal conductivity with increasing coherence length. Our thermal model, while sufficient for many other material systems and phonon scattering processes, fails to account for this crystallinity effect. This is consistent with previous studies and implies that much more complex modeling is necessary to understand the effects of dislocations on phonon scattering.
The authors would like to thank J. T. Gaskins for electron beam evaporation of the aluminum transducers. The authors acknowledge the use of the Analytical Instrument Facility (AIF) at North Carolina State University, which is supported by NSF contracts DMR 1337694 and DMR 1108071. This work was supported by the Laboratory Directed Research and Development (LDRD) program at Sandia National Laboratories, the Office of Naval Research (N00014-15-12769), and the National Science Foundation (EECS-1509362). Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04–94AL85000.
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