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Abstract
We evaluated the linear thermal expansion coefficient (α) of various laser host crystals with cubic symmetry by first principles calculations, where the relation of calculated α was CaF2 >> Sc2O3 > ZnS > ZnSe ∼ Lu3Al5O12 > Y3Al5O12 > Y2O3 > Lu2O3 at 300 K. The variation of temperature dependent α for these materials in past reports are approximately 15%, and it is comparable to the difference from our calculated α. We also proposed numerical models for thermal expansion from 150 K to 900 K that reproduced our calculated α with the error below 2%.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently the evolution of ultra-high intensity lasers has been pioneering the various cutting edges in the field of basic sciences [1]. When we develop highly intense laser systems, the heat management of laser gain media is very important. Insufficient care for the thermal design of these lasers brings the unstable laser output, and this is caused by thermal effects such as the stress birefringence, the thermal lensing, and the optical path change. These effects are described using several material parameters such as the specific heat [2], the thermal conductivity [3], the temperature coefficient of refractive index [4], and the linear thermal expansion coefficient (α) [4]. In the worst case, laser gain media can fracture by the thermal expansion. Most simply speaking, the power extraction limit from the laser gain medium is defined by the thermal shock parameter RT that can be expressed by [5]
where ν, σmax, and E are the Poisson ratio, the maximum surface stress at which fracture occurs, and the Young’s modulus, respectively. Hence the precise value of α is essential not only for the careful thermal design of lasers but also for the prediction of maximum laser power. Furthermore, the comparative study on α of laser gain media has been strongly desired for the selection of the host crystal.Since thermal expansion is caused by the anharmonicity in the phonon structure, α severely depends on the quality of specimens. Thus, it is necessary for comparative study on α of various laser gain media to use specimens with the same crystal quality under the same experimental condition. However, it is difficult to collect specimens with the same crystal qualities. Moreover, the external stress should be induced into specimens during measurements of α inevitably, especially in measurements under wide temperature range. For example, the reported experimental temperature-dependent α of Y3Al5O12 (YAG) ranges from 6.0 to 7.7 ×10−6 /K around the room temperature [4,6–12], as tabulated in Table 1.
Table 1. Reported linear thermal expansion coefficient of YAG (Unit: 10−6/K)
Because the first principles calculation (FPC) can give completely relaxed structure of crystals, we tried to use FPC for the evaluation of α using samples with the same crystal qualities under the equivalent evaluation condition. In this work, we investigated the comparative study on α of laser host crystals as garnets (Y3Al5O12 and Lu3Al5O12 (LuAG)), bixbyites (Y2O3, Sc2O3, and Lu2O3), sphalerites (ZnS and ZnSe), and fluorite (CaF2) by use of FPC under 1000 K. We also proposed numerical models for α of those crystals as functions of temperature T.
2. Methods
2.1 Geometry optimizations of host crystals
At first, we executed the geometry optimization of primitive cells for host crystals where there is no internal stress. Initial lattice parameters of host crystals were collected from previous literatures [13,14], and they were optimized by Broyden-Fletcher-Goldfarb-Shanno (BFGS) procedure [15] using the variable-cell relaxation program in Quantum Espresso Suite v.6.5 (QES) [16].
Under self-consistent field (SCF) calculation using QES, we applied the projector-augmented wave method, where we selected revised Perdew-Burke-Ernzerhof correlation functions for solids and their surfaces (PBEsol) [17] in PSliblary1.0.0 [18] were selected. k-point sampling was designed to realize higher than 0.2 / Å resolution in the reciprocal lattice according to Monkhorst-Pack method [19]. Although the default solver for self-consistent eigenvalues in QES was the efficient block Davidson method [20], calculations on CaF2 did not converge. Therefore, we used the conjugate gradient method (CG) [20] for solving eigenvalues of CaF2. Other parameters used in BFGS procedures are summarized in Table 2. As in Table 2 convergence thresholds for SCF, the energy, and the force were 1×10−8 Ry, 1×10−5 Ry, and 1×10−4 Ry, respectively. All calculations in this work were carried on a Linux 18.04 LTS workstation with two Intel Xeon Gold 5220 processors and 384 GB memory.
Table 2. Parameters used in BFGS proceduresa
2.2 Frozen phonon calculations
By using SCF calculations of the crystal structures with (or without) displacements of atoms we can estimate the force set for phonon kinetics at 0 K. From this set we can evaluate isovolumic thermal properties of various crystals calculated by the frozen phonon method [21]. We used Phonopy 2.6.1 [22] as a creator of the set of atom-displacements in supercells, and as a solver for frozen phonon method. In this work the frozen phonon calculations were carried out under the reciprocal lattice divided into 51×51×51 mesh.
Since the frozen phonon calculation was only for isovolumic conditions, we applied the quasi-harmonic approximation by the phonopy-qha program in Phonopy where isovolumic thermal properties were converted to isopiestic properties by using Vinet equation of state [23]. To include the influence of the expansion/contraction of unit cells, isovolumic thermal properties for the crystals with relaxed unit-cells that have lattice constants modulated within ±3% from the optimized geometry were calculated by the frozen phonon method.
SCF calculations for modulated crystal structures were processed by QES. Used pseudopotentials, cut-off energies, convergence thresholds, and 2 nm−1 resolution of k-point sampling were the same as in Sect. 2.1. Other parameters used in phonon calculations are summarized in Table 3. The difference between α by FPC using the quasi-harmonic approximation with Phonopy and experimentally measured α is generally within 3% [24].
Table 3. Parameters used in phonon calculationsa
3. Results
Figure 1 shows α of YAG evaluated by FPC and reported by the various experimental methods. The discrepancy between the calculated α of YAG and reported value was comparable to the variation of reported value, and their temperature dependences were coincided to each other. The color of markers in Fig. 1 indicates a measurement method, where green, blue, and red markers obtained by the powder the push-rod method [4], the interferometry [6], and X-ray methos [9], respectively.
The comparisons between the calculated α by FPC of other host crystals and previously reported values [4,6–12,25–45] are summarized in Fig. 2 (YAG and LuAG), Fig. 3 (Y2O3, Sc2O3, and Lu2O3), Fig. 4 (ZnS and ZnSe), and Fig. 5 (CaF2). Our calculation results were close to experimental values within 15% of differences except CaF2, and these differences were comparable to the variation of reported experimental values. Although in higher temperature discrepancy between calculations and experimental values became higher in Figs. 1–5, temperature dependences of calculated α corresponded to the major parts of reported experimental values. Figure 6 summarizes calculated α of host crystals in this work.
Fig. 2. Comparison between calculated α (lines) and previously reported α (markers) of garnet crystals: (a) YAG, (b) LuAG.
Fig. 3. Comparison between calculated α (lines) and previously reported α (markers) of bixbyite crystals: (a) Y2O3, (b) Sc2O3, and (c) Lu2O3.
Fig. 4. Comparison between calculated α (lines) and previously reported α (markers) of sphalerite crystals: (a) ZnS, (b) ZnSe.
4. Discussions
4.1 Influence of evaluation methods upon experimentally measured α
In powder X-ray method, the expansion coefficient is estimated from the change in the lattice constant. Thus, it is difficult to find the small lattice constant change by small thermal expansion. It indicates that α measured by X-ray diffraction can contain large error under the cryogenic temperature region, and more accurate under higher temperature. On the contrary, the interferometry should be more accurate under lower temperature. This method and the push-rod method require attaching additional parts (push-rods or mirrors) to the specimen, and it is difficult to eliminate external stresses that prevent the relaxation of lattice structures of the specimens. It means that α measured by the interferometry or the push-rod method can become smaller when large strain induced by larger thermal expansion coefficient under higher temperature.
In Fig. 1, our calculation result was close to the experimental value by the powder X-ray method around room temperature, and close to the value by the interferometry under cryogenic condition. This situation is consistent to above discussion about the difference between X-ray method and interferometry. Thus, we concluded our first principles calculation can give thermal expansion coefficient satisfactory, which does not contradict previously reported experimental value. It indicates that our FPC calculation can be applicable to the comparable studies of α, even though there were still possibilities of certain discrepancies between our calculation and true value. This discrepancy might be caused by the difference between the real crystal structure and the crystal structure described by perfect unit cells.
4.2 Comparison in α of host crystals with ideal crystal qualities
As stated in the previous paragraph we calculated α of crystals with perfect unit-cell structure where there were neither crystal defects nor impurities under our FPC. It means α calculated by FPC is for crystals with ideal crystal qualities. Table 4 shows α by FPC at several temperatures to compare α of host crystals with ideal crystal qualities. CaF2 showed several times higher α than other host crystals above 50 K. Lu2O3 showed the lowest α over 150 K. The comparisons of varying in α around 300 K were as follows: Sc2O3 > ZnS > ZnSe ∼ LuAG > YAG > Y2O3 > Lu2O3.
Table 4. α by FPC of various laser host crystals (Unit: 10−6 /K)
4.3 Difference between calculated α and experimental value
In our calculation, we used primitive cells with perfect crystals that have no crystal defects. On the contrary, there are crystal defects in real crystals due to non-zero entropy at finite temperature or other synthesizing problems. For example, garnets synthesized by Czochralski methods contain many dislocations and impurities from crucibles. The suppression of the crystal structures relaxation can cause the difference between calculated α and experimental value. Because more dislocations generated under higher temperature, discrepancies between FPC and experimental value can become larger.
FPC In Ref. [35] ZnS and ZnSe were evaluated with the same experimental setup of the push-rod method, and the same Irtran samples as a standard [46]. The Irtran materials are the microcrystalline, hot-pressed compacts developed by the Eastman Kodak Company of Rochester, New York with Irtran-2 being ZnS and Irtran-4 being ZnSe. Table 5 shows the specification of Irtran samples in Ref. [35]. Here, the ZnS sample (Irtran-2) had quite lower melting point compared to 5N grade [47], while little difference between melting point of the ZnSe sample (Irtran-4) and 5N grade. It indicates that ZnS sample in Ref. [35] contained a plenty of impurities and crystal defects induced by impurities. While α of ZnSe well reproduced experimental values as shown in Fig. 4(b), α of ZnS was 12% higher than experimental values as shown in Fig. 4(a). This did not contradict to the situation where crystal defects suppress the relaxation of the internal stress induced by temperature elevation of samples. The negative α in zinc blend has been studied in many works [30,48].
Table 5. Specification of samples in Ref. [35]
In the case of CaF2, there was the anomalous large difference of 22% between α by FPC and experimental values. This was also consistent to the crystal quality of CaF2 which should contains inevitable Frenkel defects induced during manufacturing [49]. Since syntheses of garnet hosts and bixbyite hosts include processes caried at more than 2000 K, other samples of garnet hosts and bixbyite hosts should contain many crystal defects, which also did not contradict to 10% higher α by FPC than experimental values. In addition, evaluation of α by powder X-ray techniques required powder samples which could include crystal defects during pulverization. Therefore, there was a possibility that defects induced during crystal growth prevented the perfect relaxation and brought the difference of α between FPC and experimental values. This could be one of reasons why reported experimental α variated.
Another reason for the wide variation of experimental α might be too rapid temperature elevation speed to relax crystal structures. For example, samples were heated with the rate of 4 K/min in Ref. [4]. We will study the influence of heating speed on measured α in future.
4.4 Coverage of calculation procedures of α in this work
In this work we evaluated α of laser host crystals with cubic symmetry, because the phonopy-qha program in Phonopy 2.6.1 does not support the quasi-harmonic approximation for uni-axial and bi-axial crystals. However, it is possible to calculate α of crystals with non-cubic symmetry by the evaluation of the minimum free energy of modulated super-cells manually using the Vinet equation of state. Calculation on α of uni-axial crystals and bi-axial crystals can cost more than at least seven times and 49 times longer calculation time than crystals with cubic symmetry, respectively.
4.5 Numerical models of for α of host crystals with ideal crystal qualities
Thermal expansion coefficient is proportional to the specific heat and the anharmonicity in lattice vibrations [2]. Since specific heat depends on temperature, thermal expansion coefficient also depends on temperature. From the viewpoint of the convenience for designing high power lasers, α by FPC should be expressed by the function of T. Since the mounting into structural or thermal analysis simulators must be easy, it is necessary this function should be expressed by simple polynomials. Therefore, we tried to apply numerical models to αFPC which was given by [7]
αFPC can be well reproduced by Eq. (2) using parameters in Table 6, and errors between αFPC for temperature range from 150 K to 500 K and Eq. (2) were below 2%.Table 6. Parameters for Eq. (2)
For higher temperature range from 220 K to 750 K, we used the numerical model by [50]
which was often employed in the field of geophysics or mineralogy [50]. Using parameters in Table 7, Eq. (3) reproduces αFPC within 2% errors. We also show the scaling factor which meant as eye guide for the ratio between αFPC and experimental data in Tables 6,7. These scaling factors can be used as reference value for α of real crystals with a certain crystal quality.Table 7. Parameters for (Eq. (3))
5. Conclusion
α of YAG, LuAG, Y2O3, Sc2O3, Lu2O3, ZnS, ZnSe, and CaF2 was evaluated by FPC, and at 300 K they were 7.26, 7.52, 7.18, 7.95, 6.95, 7.63, 7.55, and 23.5×10−6 /K, respectively. Temperature dependences of calculated α corresponded to the major parts of reported experimental values and are close to experimental values within 15% of differences except CaF2. We also proposed numerical models for α as functions of T, and these numerical models were easy to mount into structural or thermal analysis simulators. We expect our evaluation results will help designing of high-power lasers in future.
Funding
JST-Mirai Program (JPMJMI17A1).
Acknowledgments
Portions of this work were presented at the Advanced Solid State Lasers (ASSL), in OPTICA Laser Congress 2021, AW1A.7.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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