Disturbed and scattered, the path of thermal conduction through diamond lattice

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Disturbed and scattered, the path of thermal conduction through diamond lattice Firooz Faili 1*, William Huang 1, Julian Calvo 2, Martin Kuball 2 and Daniel Twitchen 1 1) Element Six Technologies,

1 Disturbed and scattered, the path of thermal conduction through diamond lattice Firooz Faili 1*, William Huang 1, Julian Calvo 2, Martin Kuball 2 and Daniel Twitchen 1 1) Element Six Technologies, 3901 Burton Drive, Santa Clara, CA, 95030, USA 2) University of Bristol, Center for Device Tomography and Reliability, Bristol, UK, Tyndal Avenue 1, BS8 TL1 * firooz.faili@e6.com ABSTRACT With more phonons carrying the energy in the lattice, the phonon density of states in diamond extends to a much higher frequencies than that of any other material. This is related to the Debye temperature of diamond, being the highest of any bulk materials and of having the highest sound velocity of any known bulk materials. However, the thermal conductivity not only depends on the number of phonons and how fast they are, but also on how long they can travel without being disturbed or scattered. The measurement of this length of travel is the Mean Free Path of the phonons, l, which depends on the number of phonons in the lattice through the 3-phonon processes (Normal and Umpklapp), and the imperfections in the lattice (boundaries, grain boundaries, non sp3 bonds, isotopes, impurities, extended defects, dislocations, etc.). Consequently, the real world thermal conductivity of a given piece of diamond will depend on the quality of the lattice, yielding values from 1 W/mºK (ultra-nanocrystalline diamond) to more than 3400 W/mºK for isotopically pure single crystal diamond. KEY WORDS: CVD diamond, thermal management, thermal interface, grain boundaries, defect density, extended defects, impurities NOMENCLATURE k thermal conductivity D thermal diffusivity C p heat capacity (constant pressure) R thermal resistance L length A cross-sectional area C v heat capacity (constant volume) l mean free path sound velocity INTRODUCTION To develop any fundamental model of phonon scattering related to its thermal properties requires a range of samples measured over a wide temperature range which have been characterized for factors such as grain size, point defect density, extended defect density, sp2 fraction and isotopic purity. Extensive measurements have been made over wide range of temperature, developing a baseline of thermal conductivity values for different grades of diamond. This paper also reports on the phonon scattering model using the theory of Callaway-Holland built on an off-the-shelf software platform with input terms for point defect density; extended defect density (e.g. dislocations and stacking faults); grain boundary barrier resistance and grain size, and tested against the measured thermal conductivity and other material properties. The primary objective and novelty of this work is in furthering the understanding that the state of the diamond lattice is the dominating determinant in quality and level of thermal conductance in CVD diamond, and the development of a predictive mathematical model capable of forecasting the responses as a function variation in such states. CHARACTERIZATION (BULK EFFECTS) In order to develop a fundamental understanding of phonon heat transport in polycrystalline diamond, it is essential a) have understanding of what are the governing factors impacting the thermal conductivity and b) to have a carefully measured set of thermal and material data for a range of thermal grades of diamond [1]. With those data sets in hands one can proceed to provide a definitive phonon scattering model that address outstanding questions about in-plane k xy and cross-plane k z thermal conductivity and its relation to grain size and intrinsic grain quality. Grain Size While there are CVD diamond growth chemistries that continue to promote re-nucleation to avoid grain size development, in general for a parameter such as thermal conductivity it is highly desirable to choose conditions consistent with [2]: Exceptional intra grain purity with respect to point and extended defect Exceptionally well inter-grown and low defect density grain boundaries Large grains Figure 1: Showing Nomarski microscopy of polycrystalline diamond surface and method of grain size characterization.

Figure 1 a shows typical cross-section of polycrystalline CVD diamond, with the fine grain nucleation material developing in size and volume with thickness.

2 Figure 1 a shows typical cross-section of polycrystalline CVD diamond, with the fine grain nucleation material developing in size and volume with thickness. The grain structure can easily be observed by either a plasma decoration etch (opens up the grain boundaries due to different etch rates) or using non-destructive optical techniques such as Nomarski microscopy. Based on this test, the analysis shows that for this plasma and substrate chemistry, the grain size is approximately 10% of growth thickness. Heat Capacity One of the most critical factors associated with methods relying on measuring heat diffusivity to calculate the thermal conductivity or thermal boundary resistance for diamond is the understanding of the responses of fundamental properties such as heat capacity and density to factors such as the measurement temperature or the growth process conditions [3, 4]. In essence if one extract thermal data from the correlation: Diamond Thickness Sensitivity of CVD diamond thermal conductivity with respect to the changing thickness has been well established [6]. The plotted data in Figure 3 shows the improvement of thermal conductivity of diamond, using laser flash and FTIR, as more nucleation layer is removed from two different grades of diamond. This trend appears to be reaching a steady value after the removal of 25 µm of nucleation layer for 1500 W/mºK material grade. In case of the 1800 W/mºK grade diamond, the thermal conductivity continues to improve even after removal of 50 µm of material from the nucleation interface. Equation 1: It should be based on relevant, reliable and current values for. It is imperative to recognize that singularly assumed values for used for a wide range of temperature measurement would yield inaccurate results. With regards to a collaborative work with partners in US and in Europe focused on a series of internal round robin measurements for two extreme grades of diamond [5]. The intention was to determine if special allowance was needed for specific grades of diamond, or if a single value could be used whether it was single crystal, heavily doped polycrystalline, high-quality polycrystalline or nanocrystalline diamond. The two extreme cases taken were boron (ca 0.1 atomic percent; B/cm -3 ) doped polycrystalline CVD diamond (BDD), and high purity single crystal (3A) diamond (no grain boundaries and point defect density <1x10 17 cm -3 ). The indications are that a single would work for both extreme cases, and the results holds significance in two regards. Firstly, one can appreciate the significance of variation as a function of temperature (Figure 2). Second, a modest change in measurement temperature would result in a steep (17% change over 20ºK) calculated equivalence from 1250 to 1500 W/mºK thermal conductivity. Figure 3: The impact of interface material on the thermal conductivity of diamond With the exception of laser flash measurement of 1800 W/mºK grade material, the measured negative contribution of the nucleation layer to the thermal conductivity of the material is of the order of 14% to 18%. The substantially higher slope of change for the laser flash measurement of 1800 W/mºK (dashed blue line) is attributed to the lower accuracy of the technique for thinner films. CHARACTERIZATION (LATTICE EFFECTS) Phonon Scattering Model Fundamentals The special diamond lattice properties lead to very advantageous vibrational properties which boosts the thermal conductivity. The phonon density of states in diamond extends to much higher frequencies than in any other material, i.e. there are more phonons carrying the energy in the lattice. This is related to the Debye temperature of diamond, being the highest in any bulk materials and also having the highest sound velocity of any known bulk materials. For reference comparison of diamond and silicon properties are highlighted in Figure 4 and Table 1. Figure 2: Variation of heat capacity with temperature. Figure 4: Density of states of bulk Diamond vs. Silicon.

Properties Diamond Silicon Sound speed (Transversal acoustic branch) Debye Temperature (Transversal acoustic branch) 17 500 m/s 8430 m/s 2138-2922K 586-919 K Table 1: Salient properties of diamond

The thermal conductivity is given by:, where C v is the heat capacity, which depends on number of phonons carrying the heat, and v the sound velocity.

The measurement of this is the Mean Free Path of the phonons, l, which depends on the number of phonons in the lattice through the 3-phonon processes (Normal and Umpklapp), and the imperfections in

Therefore, in the real world the thermal conductivity of a given piece of diamond will depend on the quality of the lattice, being possible to observe values from 1 W/mºK (ultrananocrystalline

3 Properties Diamond Silicon Sound speed (Transversal acoustic branch) Debye Temperature (Transversal acoustic branch) m/s 8430 m/s K K Table 1: Salient properties of diamond compared with silicon relevant to extreme thermal properties. The thermal conductivity is given by:, where C v is the heat capacity, which depends on number of phonons carrying the heat, and v the sound velocity. However, the thermal conductivity not only depends on how many phonons and how fast they are, also it depends on how long they can travel without being disturbed/scattered. The measurement of this is the Mean Free Path of the phonons, l, which depends on the number of phonons in the lattice through the 3-phonon processes (Normal and Umpklapp), and the imperfections in the lattice (boundaries, grain boundaries, non sp3 bonds, isotopes, impurities, extended defects, dislocations, etc.). Therefore, in the real world the thermal conductivity of a given piece of diamond will depend on the quality of the lattice, being possible to observe values from 1 W/mºK (ultrananocrystalline diamond) to more than 3400 W/mºK for isotopically pure single crystal diamond. Defect and impurities Analysis and measurements TEM and scanning thermal AFM To develop any fundamental model of phonon scattering related to its thermal properties requires a range of samples measured over a wide temperature range which have been characterized for factors such as grain size, point defect density, extended defect density, sp2 fraction and isotopic purity. Many of these parameters will use specific techniques optimized for diamond such as electron paramagnetic resonance (EPR) and secondary ion mass spectroscopy (SIMS) [7, 8]. Using TEM it was possible to delineate the difference in presence of defects amongst various grades of diamond (Figure 5). Figure 5: Bright field TEM analysis reveals dislocations By combining Scanning Thermal Microscopy (SThM), which combines AFM with an electrically conductive tip to allow operation in constant current mode to probe differences in local thermal conductivity and TEM, the relationship between dislocation density and the thermal conductivity was established (Figure 6). Figure 6: Stacking faults/dislocations within a grain can have significant impact on the thermal properties. Table 2 contains a summary of experimentally determined values for disruptive properties such as nitrogen concentration, dislocation density and C-Vacancies as well as grain size for various grades of diamond. Table 2: Table of values for some disruptive properties in diamond grains measured for various grades of CVD diamond. MEASURING THE THERMAL CONDUCTIVITY Two different techniques for measuring the thermal conductivity of CVD diamond samples were used. Laser flash was used for measuring the cross-plane thermal conductivity and heated bar was used for measuring the in-plane thermal conductivity. Laser Thermal Flash Diffusivity Technique A laser-pulse technique (Figure 7) is used as the reference laboratory methodology for cross-plane thermal conductivity measurements [9]. This technique makes use of a short, highenergy, laser pulse of approximately 8 ns in duration. The pulse impinges on one face of the diamond plate. The temperature rise on the opposite face is measured via a fast (20 MHz), far-ir point photo-voltaic detector. The temperature rise as measured by the detector is read into a digital storage oscilloscope from which the thermal diffusivity, and hence, thermal conductivity is deduced. The diamond is mounted in a cryostat to ensure that the temperature of the sample is known precisely. A 5 o K variation in the temperature can cause more than a 3% error in the determined thermal conductivity at 300 o K. This technique directly measures the thermal diffusivity D, which is related to the thermal conductivity k, via the specific heat capacity C p and density ρ of the material by equation (1):

Figure 7: Laser-flash setup and methodology. The laser flash technique measures the thermal diffusivity perpendicular to the plane of the diamond plate.

perpendicular to the plate surface. A carefully designed and constructed steady state heated bar technique was used to measure the in-plane TC.

4 Figure 7: Laser-flash setup and methodology. The laser flash technique measures the thermal diffusivity perpendicular to the plane of the diamond plate. It has been reported that, in some cases, CVD diamond exhibits anisotropy in k owing to its columnar grain structure; the thermal diffusivity parallel to the plane of the plate may be lower than that perpendicular to the plate surface. A carefully designed and constructed steady state heated bar technique was used to measure the in-plane TC. Heated bar technique for measuring in-plane thermal conductivity The in-plane thermal conductivity of the diamond is calculated by using simple Joule heating thermometry [10]. In this technique a resistive heater is attached to the specimen at one end and a heat sink is attached to the opposite end. A differential temperature is measured along the length of the specimen by placing two K-type thermocouples between the heat source and the heat sink. A high precision IV probe is used to measure the power delivered to the specimen. Figure 8 shows the basic schematic of the setup. MODELING THERMAL CONDUCTIVITY Callaway-Holland Model The Callaway method has been extensively used for analysing the thermal conductivity of diamond. However, typically it has been implemented without splitting longitudinal and transversal phonon branches and making use of a big set of free parameters. This model has been implemented ignoring some nonessential integral values and accounting for phonon-phonon scattering as well as scattering caused due to point defects (substitution impurities and interstitial impurities and vacancies) and extended defects (dislocations and grain boundaries). Starting with the basic heat flux equation: 1 1, C V vl V k 3 Equation 2 : vx( E) f r p Figure 9: Thermal flux Where: C V is heat capacity (density of phonons carrying the heat) v is sounds velocity (how fast phonons can carry the heat) and l is mean free path (MFP) of phonons (how long they can travel undisturbed) Attempting to determine phonon distribution through Boltzmann transport equation: Equation 3: is non-trivial. However, one can attempt to apply relaxation time approximation to the following: Figure 8: The basic layout of the measured sample, heater, thermocouples and heat-sink. The most important engineering aspect of the in-plane k measurement tool is in achieving optimum thermal isolation, minimizing heat losses and ensuring temperature measurement accuracy while utilizing long (>30 mm) high free-standing diamond films. Once the primary factors are accurately measured, the calculation is as follows: Thermal resistance is calculated by equation 7: Thermal conductivity is calculated by equation 8: Equation 4: ( ) MATHEMATICAL MODEL This section reports on the phonon scattering model using the theory of Callaway-Holland built on a commercially available software platform and tested against the measured k and other material properties. The mathematical model was built in Mathematica with the following input characteristics (sensitivities) and accounting for the following factors that includes certain diamond lattice defects: Diamond thickness Boundary/Grain boundary scattering Isotope/Impurity impact Impact of extended and point defects Dislocations densities Electron/hole-phonon scattering

The model s primary outputs are: Thermal conductivity vs. temperature Thermal conductivity vs.

model was verified by showing the exceptional between the measured data and the predicted trend (Figure 10).

The accuracy of the model is particularly notable when considering the significant differences between the four various grades of diamond.

5 The model s primary outputs are: Thermal conductivity vs. temperature Thermal conductivity vs. impurity concentration (or defect concentration) Heat capacity as a function of temperature Mean Free Path (MFP) as a function of temperature Model fit versus measured data The capability of the model was verified by showing the exceptional between the measured data and the predicted trend (Figure 10). The discrete measured thermal conductivity data points for different grades of diamond are superimposed on modeled plots. The accuracy of the model is particularly notable when considering the significant differences between the four various grades of diamond. From the optical grade 2000 W/mºK thermal conductivity diamond to the opaque 1000 W/mºK, mechanical grade material. Figure 10: Model and experimental data for different grades of diamond versus temperature Figure 11: Anisotropy in thermal conductivity of diamond as a function of temperature and diamond thickness Lastly, examining the reported data from various researchers, one can see that an explanation through simplified grain boundary scattering would be a good fit for polycrystalline diamond larger than 10 µm grain size [11]. With smaller grain size the effects of excess grain boundary and the contribution of non-diamond inclusions (nucleation face growth) could explain the significant scattering of the data. It would be fair to suggest that below 1 µm grain size, a simple Calloway model will be insufficient to explain the observed scatter and the model should be adjusted for interfacial thermal resistance (Kapitza) contribution [12, 13]. In Figure 12, the data from Angadi et al was superimposed with the results of the predictive model for two different scenarios, focusing on the changes in thermal conductivity for 1500 W/mºK grade CVD diamond. One (in green) is based on Callaway alone, while the second (in red) considers additional Kapitza contribution. Another powerful outcome of the model is to enable the user to investigate the changes in anisotropy of thermal conductivity of CVD polycrystalline diamond. The illustration of that effect is presented in Figure 11, for 1800 W/mºK diamond grade. The model shows that the ratio of cross-plane to in-plane thermal conductivity of CVD diamond has a value of ~4 at 5 ºK which reduces near unity at about 400 ºK. Simply put, the lower the temperature the larger the level of anisotropy. This is primarily due to transport of heat by long MFP phonons which are more sensitive to grain size effects. Conversely, the higher the temperature the higher the isotropy as the short MFP phonons began to dominate the function of transporting the heat and the fact that they are less impacted by the grain size. Figure 12: Impact of grain size on thermal conductivity of CVD diamond While the Callaway model alone, correctly predicts the reducing trend of the thermal conductivity as a function of

6 grain size, the addition of Kapitza resistance provides a closer match for the reported data scatter in literature. SUMMARY & CONCLUSIONS In reviewing the results, one may draw the following conclusions: The model provides a broad range of coverage in predicting the thermal conductivity of diamond as a function of temperature, grain size and a number of disruptive lattice features very accurately. The model and the prevailing data suggest that thin ultrananocrystalline CVD diamond will follow the same trend and changes in the thermal conductivity as a function of grain size and thickness as the microcrystalline diamond. When considering thick diamond film properties, the ability to control and the understanding of the scattering impacts of the various forms of defects incorporated in CVD diamond lattice is paramount for optimizing CVD diamond thermal performances. Acknowledgments This work was supported by DARPA Contract No: FA C-7517, managed by Dr. Avram Bar Cohen and Dr. John Blevins with support from Dr. Joseph Maurer and Dr. Abirami Sivananthan. References [1]D.J. Twitchen, C.S.J. Pickles, S.E. Coe, et al., Thermal Conductivity Measurements on CVD Diamond, Diamond and Related Materials, 10(3-7), (2001) [2]J. Hartmann, M. Costello, and M. Reichling, Influence of Thermal Barriers on Heat Flow in High Quality Chemical Vapor Deposited Diamond, Physical Review Letter, Volume 80, Number 1, , (January 1998) [3]Sir C.V. Raman, The Heat Capacity of Diamond between 0 and 1000 o K, Proceedings of Indian Academy of Sciences, A46, , (1957) [4]A.C. Victor, Heat Capacity of Diamond at High Temperatures, The Journal of Chemical Physics, Volume 36, Number 7, , (April 1962) [5]Austrian Institute of Technology, Element Six Technologies, Thermophysical Characterization of Element Six Diamond, Commissioned Report (2013) [6]J. E. Graebner, Measurement of Thermal Conductivity and Thermal Diffusivity of CVD Diamond, International Journal of Thermophysics, Volume 19, Number 2, , (1998) [7]S. Felton, B. Cann, R. J. Cruddace, M. E. Newton and D. Fisher, EPR Measurements on the g = 2.00 Region of HPHT N Doped Diamond, A poster presentation at 57 th Diamond Conference, (2006) [8]D. Zhou, F.A. Stevie, L. Chow, et al., Nitrogen incorporation and trace element analysis of nanocrystalline diamond thin films by secondary ion mass spectrometry, Journal of Vacuum Science and Technology A, Volume 17, Number 4, , (August/July 1999) [9]B. Remy, D. Maillet, S. Andre, Laser Flash Diffusivity Measurement of Diamond Films, International Journal of Thermophysics, Volume 19, Number 3, , (May 1998) [10]S. D. Wolter, D. A. Borca-Tasciuc, G. Chen, et al., Thermal Conductivity of Epitaxially Textured Diamond Films Diamond and Related Materials, 12, 61-64, (2003) [11]M. A. Angadi, T. Watanabe, A. Bodapti, et al., Thermal Transport and Grain Boundary Conductance in Ultrananocrystalline Diamond Thin Films, Journal of Applied Physics, 99, , (2006) [12]S.L. Shinde, E.S. Piekos, J.P. Sullivan, et al., Phonon Engineering of Nanostructures, Sandia Report, SAND (2010) [13]J. A. Calvo, M. Kuball, Control of the in-plane Thermal Conductivity of Ultra-thin Nanocrystalline Diamond Films through the Grain and Grain boundary Properties, Acta Materialia, Volume 103, pp. 104 (2016)

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Supporting Information Large-Area, Transfer-Free, Oxide-Assisted Synthesis of Hexagonal Boron Nitride Films and Their Heterostructures with MoS2 and WS2 Sanjay Behura, Phong Nguyen, Songwei Che, Rousan