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Published on November 26, 2015

Author: IzabelaFirkowska

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1. The Origin of High Thermal Conductivity and Ultralow Thermal Expansion in Copper−Graphite Composites Izabela Firkowska, André Boden, Benji Boerner, and Stephanie Reich* Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany *S Supporting Information ABSTRACT: We developed a nanocomposite with highly aligned graphite platelets in a copper matrix. Spark plasma sintering ensured an excellent copper−graphite interface for transmitting heat and stress. The resulting composite has superior thermal conductivity (500 W m−1 K−1 , 140% of copper), which is in excellent agreement with modeling based on the effective medium approximation. The thermal expansion perpendicular to the graphite platelets drops dramatically from ∼20 ppm K−1 for graphite and copper separately to 2 ppm K−1 for the combined structure. We show that this originates from the layered, highly anisotropic structure of graphite combined with residual stress under ambient conditions, that is, strain-engineering of the thermal expansion. Combining excellent thermal conductivity with ultralow thermal expansion results in ideal materials for heat sinks and other devices for thermal management. KEYWORDS: Metal−matrix composites, graphite, graphene, thermal conductivity, thermal expansion, strain, thermal stress Our daily life is governed by highly mobile electronic devices. Their operation generates heat in a confined space making the cooling of semiconductors a key challenge for industry.1−4 Metals, especially copper, are the prime materials for heat sinks because of their excellent thermal conductivity and low cost.5 However, metals are heavy and their thermal expansion is four to eight times larger than the expansion of semiconductors.6,7 This mismatch causes thermal strain under operation, which limits the lifetime of high-power electronic devices.8,9 Engineering thermal expansion has proven extremely challenging because it is ultimately controlled by the vibrations of a crystal that are difficult to manipulate.10 Composite materials combine the properties of two (or more) components. Prime candidates for advanced materials for thermal management (engineering heat flow and expansion, low weight) are metal matrix composites (MMC) with nano- and microsized fillers. Carbon and carbon nanostructures, that is, graphite (nano)platelets,11,12 single- and few-layer gra- phene,13−15 carbon nanotubes,16−18 and nanodiamonds19,20 are excellent candidates to reinforce metals due to their excellent thermal properties.21−39 However, often the addition of highly conductive nanoscale fillers reduces the thermal properties of the matrix, for example, because a large thermal interface resistance (Kapitza) hinders heat flow though the compo- site.23,40 For sp2 carbon nanostructures, the alignment of the fillers proved to be a key factor in enhancing thermal conductivity through MMCs. We recently predicted that thermal conductivities above copper are achievable by adding graphite platelets.21 sp2 carbon was also envisioned as a solution for the challenge of thermal expansion, because its in-plane expansion is negative at room temperature (αgr,1 = −1 ppm K−1 ).12 MMCs of aluminum and graphite platelets with expansion coefficients around 12 ppm K−1 for 50% filler fraction were indeed reported.30 This remains by a factor of 3−6 higher than the expansion of semiconductors.6,7 More troubling, however, is that optimizing both conductivity and expansion along the in- plane direction of an aligned filler results in a heat-spreader configuration that distributes heat over an area. Much more desirable are heat sinks, that is, the direction of maximum thermal conductivity is perpendicular to the direction of tailored thermal expansion. Only such a material will efficiently cool and attach smoothly to a semiconductor device. In this Letter, we present a copper composite with graphite platelets that combines 500 W m−1 K−1 thermal conductivity with 2 ppm K−1 thermal expansion in the heat-sink configuration. The copper-graphite composite called SuCoLEx (Superior Conductivity Low Expansion) is obtained by aligning graphite platelets within a copper matrix through spark plasma sintering (SPS). Most remarkable is the enormous reduction in thermal expansion from 28 ppm K−1 for the c-axis of graphite and 17 ppm K−1 for copper to 2 ppm K−1 in the composite. We show that the temperature-dependent in-plane strain in the graphite platelets embedded in copper results in a negative through-plane expansion, which excellently explains the values Received: April 28, 2015 Revised: June 11, 2015 Published: June 17, 2015 Letter pubs.acs.org/NanoLett © 2015 American Chemical Society 4745 DOI: 10.1021/acs.nanolett.5b01664 Nano Lett. 2015, 15, 4745−4751

2. observed for SuCoLEx. This opens ways to tailor thermal expansion though strain engineering. To demonstrate the performance of our composite as a heat-sink material, we show that SuCoLEx reduces the thermal strain in silicon by up to a factor of 4 compared to copper and aluminum and outperforms copper in cooling a high-power light-emitting device (LED). Graphite platelets with lateral size of 300 and 5 μm thickness (Figure 1a) were mixed with copper powder by ball milling (see Experimental Methods). The extracted material was consoli- dated by SPS39,41,42 resulting in the metal−matrix composite SuCoLEx that can be shaped, cut, and polished (Figure 1c). This bulk synthesis approach is a big advantage for thermal Figure 1. Structure and density of SuCoLEx. (a) Scanning electron microscopy (SEM) image of a graphite platelet. (b) SEM image of the SuCoLEx cross-section. The arrow highlights the graphite alignment. (c) Density (black square) and expected density (dashed line) of SuCoLEx as well as pictures of the bulk material compared to one cent. (d) Raman intensity of the G peak as a function of the angle between the polarization of the light and in-plane direction of SuCoLEx for 8 (black square), 20 (red circle), 40 (green triangle), and 50 vol % (blue diamond). Figure 2. Thermal properties of SuCoLEx. (a) Thermal conductivity of SuCoLEx in the in-plane kx (blue circle) and through-plane kz (black square) direction obtained from the measured thermal diffusivity, density, and heat capacity (Supporting Information). Dashed lines were calculated with the effective medium approximation and σG = 0.37 alignment. The right axis is the thermal conductivity enhancement (TCE, ratio between composite and matrix thermal conductivity). (b) Through-plane αz (black square) and in-plane coefficient of thermal expansion αx (blue circle) of SuCoLEx. The dotted and dashed lines are the modeling predictions. (c) Sandwich-like structure of copper and graphite. Nano Letters Letter DOI: 10.1021/acs.nanolett.5b01664 Nano Lett. 2015, 15, 4745−4751 4746

3. management applications over techniques that produce MMC films.36−38 The addition of graphite to the copper matrix reduced the density (Figure 1c) and increased the heat capacity (Supporting Information Figure 1S) following the rule of mixture. We characterized the nanofillers and the composite by Raman scattering and electron microscopy. The crystalline structure of the graphite platelets remained intact during composite synthesis as verified by the constant D-line intensity in Raman scattering (Supporting Information Figure 2S). The internal structure of SuCoLEx shows strong alignment of the graphite flakes, Figure 1b. This alignment occurs because of the platelet geometry (small thickness to length ratio) combined with the forces during consolidation. The thin graphite particles orient during SPS, because it is performed under uniaxial pressure, see Methods. The c-axis of the platelets is preferentially oriented along the force direction, which we call the through-plane direction or z-axis in this paper; compare inset of Figure 2a. To quantify the platelet orientation, we measured the Raman intensity of the G peak as a function of the polarization angle of the incoming and scattered light (Figure 1d).21,43 The intensity drop at 90° and 270° confirms the alignment of the platelets, which is strongest for 40 and 50 vol % platelet concentration (further details can be found in Supporting Information). We evaluated the polarization dependent intensity following ref 21 and obtained a standard distribution for the orientation σG = 0.69 for 8 vol %, σG = 0.61 for 20 vol %, and σG = 0.37 for a graphite volume fraction above 20 vol %. The alignment at high filler concentration exceeds the maximum value observed in our previous findings,21 which indicates that graphite alignment can be tuned not only as a function of lateral size but also as a function of filler concentration. The thermal conductivity of SuCoLEx was measured by a light flash method. The transient method determines the thermal diffusivity from the time dependence of the temper- ature increase after a short energy pulse. A special sample holder determines the in-plane and through-plane component of the diffusivity separately. The thermal conductivity is then obtained by multiplying the diffusivity with the density and specific heat of SuCoLEx that were measured on the same samples, see Methods for details. The thermal conductivity of SuCoLEx in the in-plane direction (along the graphite flakes alignment), Figure 2a, reaches 503 W m−1 K−1 at 50 vol %, which is 40% higher than pure copper. It also exceeds the thermal conductivity of any metal (including silver) and common engineering alloys5 as well as metal matrix composites with randomly dispersed carbon fillers.32,33,36,39 The trough- plane conductivity kz is up to ten times smaller than kx making SuCoLEx a highly anisotropic material with directional heat transport. We model the thermal conductivity of SuCoLEx within the effective medium approximation considering filler anisotropy, geometry, and orientation;44 see Supporting Information for details. In graphite platelets, the in-plane thermal conductivity kgr,1 = 1500 W m−1 K−1 is much higher than the through-plane conductivity kgr,3 = 15 W m−1 K−1 .45 Using the experimental parameters for filler alignment at 40 and 50 vol % concentration, platelet geometry, and a graphite-copper thermal interface resistance Rk = 10−9 m2 K W−1 (refs 21 and 46), we obtain excellent agreement with experimental data for the 40 and 50 vol % composites (Figure 2a). The simulations show that alignment is the key factor for the increase in kx compared to copper. The apparent drop in the SuCoLEx performance at low filler fraction is due to the increased disorder in the platelet orientation. In Supporting Information Figure 4S we present EMA calculations for σG = 0.69 and σG = 0.61 as obtained from polarized Raman scattering at low filler fraction. They nicely reproduce the experimental results for graphite concentration of 8 and 20 vol %, respectively. We also note that the copper− graphite interface resistance is small (Rk = 10−9 m2 K W−1 )21,46 compared to other metal−graphite interfaces.47 Nevertheless, Supporting Information Figure S5 shows that even an increase in the Kapitza resistance by 1 order of magnitude has little effect on the thermal conductivity of the composite. The potential of SuCoLEx as a heat sink material is highlighted by assuming perfect alignment of the filler (σG = 0), which results in an expected maximum for the thermal conductivity of SuCoLEx kx = 880 W m−1 K−1 at f = 0.5. The highly anisotropic layered structure of graphite combined with platelet alignment causes intriguing changes in the thermal expansion of the SuCoLEx composite (Figure 2b). The in-plane expansion decreases slightly with increasing platelet concentration to αx = 12 ppm K−1 at f = 0.5. This trend is expected from the negative expansion of graphite (αgr,1 = −1 ppm K−1 ). The through-plane expansion, however, drops dramatically to αz = 1.9 ppm K−1 at highest loading and becomes comparable to the expansion of semiconductors. The through-plane expansion of SuCoLEx is by a factor of 9 smaller than the expansion of copper (αCu = 17 ppm K−1 )6 and by a factor of 15 smaller than the graphite expansion along c (αgr,3 = 28 ppm K−1 ).12 This means that the thermal expansion of the composite differs significantly from the averaged thermal expansion of its two components. To understand this counterintuitive behavior, we model the mechanical and thermal interplay of graphite and copper within elasticity theory. We consider a sandwich-like structure of graphite and copper (Figure 2c). There is excellent transmission of stress along the in-plane copper-graphite interface; this is in line with the small Kapitza resistance.21 The in-plane lattice constant of graphite follows the expansion coefficient of the composite. This builds up an in-plane strain ε11 = ε22 in graphite that varies with temperature dε11/dT = Δαx = αx − αgr,1 = 13 ppm K−1 . We now derive an expression for the resulting expansion of a hexagonal crystal along its c-axis. The strain ε in a system with the temperature T and the external stress σ as independent variables is given by the equation of state48 ε σ= +S m (1) where S is the stiffness. m is the thermal strain under zero external stress (its temperature derivative is the thermal expansion α). The temperature-dependent biaxial stress σ11=σ22=σ in the in-plane direction yields a strain ε11=ε22=(S12+S22)σ. The strain along the c axis is given by elasticity theory ε σ ε ε= = + = −S S S S C C 2 2 2 33 13 13 11 12 11 13 33 11 (2) where Cij are the elastic compliance constants of graphite. We restrict eq 1 to ε33 and insert eq 2 ε ε= − +T C C T m T( ) 2 ( ) ( )33 13 33 11 33 (3) Nano Letters Letter DOI: 10.1021/acs.nanolett.5b01664 Nano Lett. 2015, 15, 4745−4751 4747

4. with the thermal strain under ambient conditions m33. The thermal expansion of graphite along c is the temperature derivative of eq 3 α ε α ν α ν ε= = + Δ − ≈ − − T T d d d d 26 ppm Kx33 33 gr,3 2D 2D 11 0 1 (4) where v2D = −2C13/C33 = −0.83 is the two-dimensional equivalent of Poisson’s ratio and its temperature derivative dν2D/dT = −4.3 × 10−2 K−1 .49 The elastic constants and their temperature derivatives are C13 = 15 GPa, C33 = 36 GPa, dC13/ dT = −0.8 GPa/K, and dC33/dT = −0.05 GPa/K.49 We used a residual in-plane strain ε11 0 ≈ −10−3 after SPS, which was estimated from the effective sintering temperature of a copper matrix (400 °C).50 The z-axis expansion of copper within the sandwiched structure, Figure 2c, is 24 ppm K−1 due to the compressive in- plane strain. A perfectly aligned, laminated structure of 50% copper and graphite (Figure 2c) has a through-plane thermal expansion αz ≈ −1 ppm K−1 . αz of SuCoLEx at high volume fraction is well described by the sandwich structure (Figure 2c) as represented by the dashed-dotted line in Figure 2b. At low filler fraction and poor alignment of the filler, the thermal expansion follows an isotropic model of vanishing internal stress, see dotted line in Figure 2b.30 A temperature-dependent in-plane strain in graphite results in a shrinking through-plane lattice constant. This surprisingly strong change in the thermal expansion is due to the large two- dimensional Poisson ratio of graphite and its strong temper- ature dependence. They originate from the two-dimensional layered structure of graphite and the negative Grüneisen parameters of the out-of-plane modes (Lifshitz membrane effect).51 Similar mechanical properties might occur in other two-dimensional layered materials promising more flexible engineering of thermal expansion and mechanical properties. SuCoLEx, given its low thermal expansion should induce less thermal strain when used as a heat sink for semiconductors compared to metals with their higher expansion coefficient. A piece of (001) Si was glued on heat sinks; the thermal strain under operation was mimicked by changing the temperature of the device and quantified by the Raman spectra (Figure 3a). The phonon frequency of Si on Cu increased from 522.0 cm−1 for stress-free silicon at room temperature to 526.3 cm−1 at 83 K. The majority of the shift originated from the anharmonicity of the vibrational potential,52 but the thermal strain induced by the mismatch in thermal expansion resulted in Δω (83 K) = 1.3 cm−1 (Figure 3a). The strain-induced frequency shift in Si on SuCoLEx, Al, and Cu (Figure 3b) reveals the highest strain at the Al−Si interface Δω (83 K) = 2.0 cm−1 , whereas SuCoLEx generates the smallest frequency shift of only Δω (83 K) = 0.5 cm−1 , 2.5 times smaller than in silicon attached to copper. The cooling and heating cycles are highly reproducible (error bars in Figure 3b) verifying the reversibility and reusability of SuCoLEx. The frequency change of the Si phonon with temperature is given by (Methods) ω ω ω α α ω= + − − ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ⎥T F q C C p T d d 1 ( ) ( )x s 0 2 12 11 0 2 Si 0 (5) where ω0(T) is the phonon frequency of free-standing Si as a function of temperature.52 (q/ω0 2 = −2.31) and (p/ω0 2 = −1.85) are the phonon deformation potentials of Si.53 F = 1/2 for SuCoLEx and F = 1 for Al and Cu. Equation 5 predicts for the Si−SuCoLEx interface a phonon frequency shift by thermal strain −4.1 × 10−3 cm−1 K−1 compared to the experimental result −3.4 × 10−3 cm−1 K−1 (Figure 3b). The slightly smaller experimental values points toward slip at the interface due to the epoxy glue. Slipping was more pronounced for the Si−metal interfaces because of the higher thermal strain (Cu, predicted −1.2 × 10−2 cm−1 K−1 , observed −0.8 × 10−2 cm−1 K−1 ; Al, predicted −1.6 × 10−2 cm−1 K−1 , observed −1.0 × 10−2 cm−1 K−1 ). The experimental peak positions scattered more for Si−metal interfaces (Figure 3b) confirming that thermal strains affect the controllability of an interface between a device and its heat sink. From the phonon frequency shift we calculated the average thermal stress at the Si−SuCoLEx interface dσ/dT = 0.8 MPa K−1 , a strong reduction compared to Cu (1.9 MPa K−1 ). The fracture strength of silicon dies is 200−400 MPa depending on the processing conditions.54 Within a range of operating Figure 3. Thermal strain at the Si-heat sink interface. (a) Temperature dependence of the Raman mode in Si on Cu with a schematic of the Si-heat- sink sample. The dashed spectrum was recorded on free-standing Si. (b) Temperature dependence of the Si phonon frequency (black square) and for Si on SuCoLEx (green inverse triangle), Al (gray circle), and Cu (red triangle). Error bars indicate the standard deviation from repeated cycles. The solid curves are fits with eq 5. Nano Letters Letter DOI: 10.1021/acs.nanolett.5b01664 Nano Lett. 2015, 15, 4745−4751 4748

5. temperatures of −50 to +200 °C, SuCoLEx remains below the critical stress level for silicon (max stress at 200 °C is 160 MPa), whereas Cu induces 380 MPa stress. SuCoLEx has enormous potential as a heat sink material; it induces less thermal stress (Figure 3) thereby preventing buckling and delamination and it provides better cooling (Figure 2a) than conventional heat sinks. The latter is further highlighted by cooling two 3 W light-emitting diodes, which were mounted on heat sinks made of 50 vol % SuCoLEx and pure copper (Figure 4a and Supporting Information). Heat flux and diode temperature were monitored with an infrared camera. SuCoLEx outperformed copper; in particular, the hotspot right under the LED was efficiently eliminated by the SuCoLEx heat sink. The heat sink temperature was reduced thanks to the higher kx (Figure 2a). On an Ashby plot (Figure 4b) SuCoLEx disrupts the correlation between thermal conductivity and expansion that is characteristic for metals and ceramics. Rivaling in their thermal properties are only highly thermally conductive semiconduc- tors, that is , diamond and boron nitride. However, they are prohibitively expensive, difficult to process and manufacture in bulk quantities. An important figure of merit for highly mobile systems is the specific thermal conductivity (ratio between conductivity and density).4 It doubled from 450 W cm2 kg−1 K−1 for copper to 950 W cm2 kg−1 K−1 for 50 vol % SuCoLEx exceeding the value of aluminum (850 W cm2 kg−1 K−1 ). In conclusion, our work showed how to engineer thermal expansion and conductivity in composite materials. The combined contribution of residual and thermal strain was used to strongly reduce the through-plane thermal expansion of graphite. A copper composite with highly aligned graphite platelets then expands like a semiconductor material (2 ppm K−1 ). Alignment was also key for increasing the thermal conductivity by microscale fillers (503 W m−1 K−1 ). Metal composites reinforced by two-dimensional fillers are promising candidates for advanced materials in thermal management. ■ METHODS Fabrication of SuCoLEx. Commercial Cu powder (3 μm dendritic, Sigma-Aldrich) and natural flake graphite (lateral size 300 μm, thickness 5 μm, Graphene Supermarket) were mixed for 3 h by planetary ball milling (Fritsch) at 250 rpm. A 250 mL milling jar was filled with 50 (1 cm in diameter) grinding balls made of zirconia. The graphite concentration ranged 8−50 vol %. SuCoLEx discs with 2.5 cm diameter and 0.1−1 cm thickness were obtained from the composite powders by spark plasma sintering in a Dr.SinterLab Jr.211Lx (Fuji Electronic). The SPS temperature was 600 °C with a heating rate 50 K min−1 and 5 min annealing time. The pulsed sintering current was controlled by a thermocouple inserted in a small pinhole in the graphite die and reached values up to 1000 A. A pressure of 40 MPa was applied during SPS in vacuum (pressure <5 Pa). Starting materials and composites were characterized by SEM (Hitachi SU-8030) and Raman spectroscopy. Thermal Diffusivity and Expansion. The thermal diffusivity was measured by the light flash method (NetzschL- FA447 NanoFlash). The in-plane and through-plane diffusivity were determined on the same sample using a masked sample holder.21 The specific heat was obtained by calibrating the flash signal with a graphite reference. The thermal conductivities kx and kz were calculated by multiplying thermal diffusivity, specific heat, and bulk density (measured by Archimedes’ principle). Thermal expansion was studied on a Dilatometer L75XH1000 (Linseis). The measurements were conducted between 20°−150 °C with constant heating rates of 1 and 2 K min−1 . The in-plane and through-plane expansion were measured on the same sample with 5 × 5 × 5 mm dimension. Graphite Alignment. Polarized Raman spectroscopy was carried out on a fractured cross-section of SuCoLEx (excitation wavelength 532 nm and power 1 mW). The light was focused by a 10× objective; the spectra were recorded on a Horiba T64000 triple monochromator. The polarizations of the incoming and scattered light were parallel to each other. The angle between the polarization direction and the sample normal was rotated with a λ/2 wave plate in front of the microscope objective. Data evaluation is described in the Supporting Information. Phonon Frequencies under Stress. The experiments were performed on 100 μm thick piece of silicon (5 × 5 mm) attached to SuCoLEx, Cu, and Al blocks (10 × 10 × 5 mm) with epoxy glue (UHU Plus endfest 300). To avoid residual stress in the silicon substrate the glue was cured overnight at room temperature. The Raman spectra were obtained with a micro-Raman spectrometer in backscattering geometry (532 nm excitation). The measurements were carried out under nitrogen atmosphere at temperatures 83−298 K using a cooling/heating stage (THMS600 Linkam Scientific). The samples were cooled to 83 K and subsequently heated to 298 K with a rate of 10 K min−1 . The spectra were taken every 5 K. The spectral resolution is 0.05 cm−1 , which corresponds to 12 MPa stress. The heat sink exerts an in-plane stress on the Si resulting in an in-plane strain. The strain induced by SuCoLEx is approximately uniaxial (ε11 = 0, ε22=ε), because αz is close to the Si thermal expansion. For the metal heat sinks the strain is biaxial (ε11 = ε22 = ε). The in-plane strain induces a strain along the z axis ε33 = −C12/C11(ε11 + ε22). The frequency shift of a Si phonon polarized along the z axis under strain is55 (Supporting Information) Figure 4. (a) LEDs on SuCoLEx (right) and Cu (left) heat sinks (top image) and temperature distribution on the LEDs and heat sinks under operation (bottom image). (b) Ashby plot of SuCoLEx and key engineering materials. The green area marks the preferred region for heat sink materials (the darker the better). Nano Letters Letter DOI: 10.1021/acs.nanolett.5b01664 Nano Lett. 2015, 15, 4745−4751 4749

6. ω ω ω ω ω ω ε ω ε ε ω ω ε Δ = − = + + = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ p q F q C C p 1 2 ( ) s 0 s 0 0 0 2 33 0 2 11 22 0 2 12 11 0 2 (6) The temperature derivative of eq 6 was used for eq 5 assuming no temperature dependence of the elastic constants and phonon deformation potentials. To calculate the thermal stress we used the relation ε11 + ε22 = (S12 + S11)(σ11 + σ22). ■ ASSOCIATED CONTENT *S Supporting Information Details on the crystalline quality of the graphite flakes; additional information on the measurements of the thermal transport properties; details on the orientation dependent Raman intensity calculation as well as effective medium approximation; details on the phonon frequency shift in Si; and additional information on the LED cooling setup. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b01664. ■ AUTHOR INFORMATION Corresponding Author *E-mail: stephanie.reich@fu-berlin.de. Tel.: +49 30 838 56162. Fax: +49 30 838 56081. Author Contributions The project was conceived by I.F., A.B., and S.R. I.F. and A.B. designed the experiments. B.B. prepared the samples. A.B. measured and analyzed the thermal properties of SuCoLEx supported by B.B. A.B. modeled the thermal conductivity by EMA and quantified the graphite alignment. I.F. performed SEM characterization and analyzed Raman strain data. S.R. developed the elasticity theory and contributed to the strain analysis. I.F., A.B., and S.R. wrote the manuscript; all the authors contributed to the scientific discussion and revised the manuscript Notes The authors declare no competing financial interest. ■ ACKNOWLEDGMENTS We thank H. Grötzebauch for the thermographic camera imaging, C. Thomsen and J. Maultzsch for providing equipment for the strain measurements, P. Kusch for assistance in the Raman experiments, and M. Gegg for the helpful discussions concerning the thermal expansion. 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