Thermal conductivity of CaF_{2} at high pressure
aa r X i v : . [ c ond - m a t . o t h e r] F e b Thermal conductivity of CaF at high pressure Somayeh Faraji,
1, 2
S. Mehdi Vaez Allaei, and Maximilian Amsler ∗ Department of Physics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran Department of Physics, University of Tehran, P.O. Box: 14395/547, Tehran, Iran Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, CH-3012 Bern, Switzerland (Dated: February 16, 2021)We study the thermal transport properties of three CaF polymorphs up to a pressure of 30 GPausing first-principle calculations and an interatomic potential based on machine learning. The lat-tice thermal conductivity κ is computed by iteratively solving the linearized Boltzmann transportequation (BTE) and by taking into account three-phonon scattering. Overall, κ increases nearlylinearly with pressure, and we show that the recently discovered δ -phase with P ¯62 m symmetry andthe previously known γ -CaF high-pressure phase have significantly lower lattice thermal conduc-tivities than the ambient-thermodynamic cubic fluorite ( F m ¯3 m ) structure. We argue that the lower κ of these two high-pressure phases stems mainly due to a lower contribution of acoustic modes to κ as a result of their small group velocities. We further show that the phonon mean free paths arevery short for the P ¯62 m and P nma structures at high temperatures, and resort to the Cahill-Pohlmodel to assess the lower limit of thermal conductivity in these domains.
I. INTRODUCTION
Calcium fluoride (CaF ) has a variety of technolog-ical applications due to its remarkable optical proper-ties and its high thermal stability . At ambient con-ditions, α -CaF crystallizes in the cubic fluorite struc-ture with F m ¯3 m symmetry. In this structure, CaF exhibits a superwide band gap of 12 eV with excellentlight transmission over a wide spectrum, and a high laserdamage threshold. These properties render CaF anideal candidate for optical windows, main lens substratesin large scale semiconductor micro-lithography systems,and photo-detectors .Cubic CaF undergoes a sequence of structural phasetransitions at increased pressures . Above 8–10 GPa,CaF transforms to the denser orthorhombic cotunnite γ -phase with P nma symmetry, accompanied by an in-creased coordination number of Ca from 8 to 9. X-raydiffraction and Raman spectroscopy have shown that thishigh-pressure phase is stable up to 49 GPa at room tem-perature . As pressure increases further, the stabilityof γ -CaF decreases, and above 72 GPa, a further tran-sition occurs to a hexagonal P /mmc phase .In addition to these experimentally observed low-temperature high-pressure phases, high-temperaturemodifications thereof have been studied predominantlyusing computational models. Using ab initio structuralsearches, Nelson et al. recently proposed a hypotheticalstructure with P ¯62 m symmetry as a high-temperaturepolymorph of γ -CaF , referred to as δ -CaF . Similarto its ambient-pressure counterpart, δ -CaF is predictedto undergo a transition to a superionic phase with bccstructure at temperatures exceeding ≈ polymorphs. At ambient pressure, thelattice thermal conductivity of α -CaF has been studiedboth through experiments and computations. In two sep-arate early experiments in 1957 an 1960, the near room- temperature value of the lattice thermal conductivity wasmeasured to be 5 . and 9 . Wm − K − , respectively.Later, Slack reported a value of 11 .
69 Wm − K − in1961. Theoretical room-temperature values from sim-ulations have been predicted in the range of 7.0 and8.6 Wm − K − . . To the best of our knowledge, theonly work on the pressure dependence of κ in α -CaF was reported 9 . < κ < . − K − , measured usinga dynamic two-strip method at room temperature in anarrow pressure range of 0 . . .In this work, we study the thermal conductivity of the α , γ , and δ phases of CaF as a function of pressure inthe range of 0-30 GPa. To alleviate the computationalburden of ab initio calculations, we resort to training anefficient machine learning interatomic potential to accel-erate the assessment of the lattice thermal conductivity κ . We show that the value of κ for the γ -CaF and δ -CaF phases are lower than that of the α phase acrossthe whole pressure domain. In particular, the extremelysmall phonon mean-free-paths in these two phases leadsto a potential break-down of the Boltzmann transportequation (BTE). Hence, we assess the validity of theBTE results based on the amorphous limit using theCahill-Pohl model and draw the associated temperature-pressure transition boundary. II. METHODSA. Interatomic Potential
We use CENT, a neural-network-based interatomic po-tential, that takes into account charge transfers to modelthe ionic bonding in CaF . The construction of theCaF CENT potential is discussed in detail elsewhere ,and we employ its implementation in the FLAME pack-age . The particular parametrization of our CENTpotential has been used elsewhere to obtain physicalproperties of CaF and to study surface morphologiesof CaF . B. Density Functional Theory
Structural relaxations and single-point total energycalculations were performed with density functional the-ory (DFT) calculations at selected pressures, 2, 10,and 30 GPa. We used the plane-wave QuantumESPRESSO simulation package in conjunction withthe Perdew-Burke-Ernzerhof (PBE) parametrization ofthe exchange-correlation functional and ultrasoft pseu-dopotentials. The wave functions and electron densitieswere expanded with a plane wave basis set up to a kineticcutoff energy of 45 Ry and 540 Ry, respectively. The Bril-louin zone was sampled using 16 × ×
16, 14 × × × ×
14 Monkhorst-Pack k-points meshes for α -CaF , δ -CaF , and γ -CaF , respectively. The atomicpositions were relaxed until the maximal force acting onthe atoms was less than 1 × − Ry/Bohr.
C. Phonons
The second order interatomic force constants were cal-culated by the finite difference approach using the super-cell method as implemented in the Phonopy package. .Supercells of dimension 4 × ×
4, 2 × ×
2, and 2 × × α -CaF , γ -CaF , and δ -CaF , respectively,leading to cells consisting of 192, 144, and 144 atoms. Afinite difference step size of 0 .
01 ˚A was applied to displacethe atoms. A q -point mesh of 50 × ×
50 was used forthe BZ integration.
D. BTE thermal transport
The thermal transport calculations were carried outby taking into account anharmonic three-phonon inter-actions. Third-order force constants were computed fromfinite differences using the supercell method with thesame sizes used in the calculations of the second-orderforce constants. Atomic displacements were created us-ing the thirdorder.py script included in the ShengBTEdistribution, taking into account up to the 5 th -nearestneighbors to truncate the three-body interactions, whichgives well converged values of κ . The thermal conductiv-ity in the BTE is given by κ = 13 V N q X qγ C qγ v qγ τ qγ (1)where V is the volume of the cell containing N atoms, q refers to the wave vector in the first Brillouin zone, N q is the number of discrete q-points, γ is the mode indexthat refers to different phonon branches, C qγ denotes themode specific heat capacity at constant volume, v is the phonon group velocity ( v qγ = ∇ q ω qγ ), τ is the phononlife time, which is related to the MFP λ = v · τ .The detailed effects of the cutoff-distance on the ther-mal conductivities are shown in section S3 of the support-ing information. The second- and third-order interatomicforce constants were fed into the ShengBTE package tocalculate κ by iteratively solving the linearized phononBoltzmann transport equation for temperatures rangingfrom 100 K to 900 K. Both isotopic and three-phononscattering were considered. The isotopic scattering rateswere calculated by applying the Pearson deviation co-efficients incorporated in ShengBTE. The so-called pro-portionality constant scalebroad , related to the adaptiveGaussian broadening technique, was set to 0 . × × × ×
19, and 15 × × q -point grids for α -CaF , δ -CaF , and γ -CaF , respectively. E. Cahill-Pohl model
We estimate the amorphous limit of the thermal con-ductivity κ CP using the Cahill-Pohl model , which isan extension of the Einstein model. While Einstein as-sumed that the thermal energy is transported betweenneighboring atoms vibrating with a single frequency, theCahill-Pohl model proposes that the energy is transferredbetween collective vibrations. Therefore, the model in-cludes a range of frequencies, instead of a single frequencyused by Einstein. In this model, the thermal conductivityis expressed as follows (details in Ref.[43]) κ CP = ( π k B n X α v α ( TT Dα ) Z T Dα /T x e x ( e x − dx,x = T Dα T , (2)where n is the density of the atoms in the solid ( m − ), v α is the low-frequency speed of sound (from acousticphonons) for polarization α . T Dα = ~ k B v α (6 nπ ) is thecharacteristic temperature equivalent to the Debye tem-perature for that polarization which corresponds to theactivation of all phonons. x is the reduced phonon en-ergy and the summation runs over the vibrational po-larizations (one longitudinal and two transverse acousticbranches). The v α of each acoustic group velocity was de-termined using harmonic lattice dynamics from Sec. II C. III. RESULTS AND DISCUSSION
We start out by validating the quality and predictivepower of our CENT potential with respect to DFT re-sults based on the three relevant phases α -CaF , γ -CaF ,and δ -CaF . The thermodynamic properties includingthe transition pressures are well reproduced by CENT.As shown in Fig. S1 of the supplementary materials, thephase transition from α -CaF to γ -CaF occurs at 8 GPawhich is close to experimental measurements. Also, theenthalpy differences of the δ -CaF and γ -CaF decreaseswith increasing pressure. In fact, these two high pres-sure phases are energetically very close to each other,i.e., dropping from 12.7 to 2.1 meV/atom in the pres-sure range between 2 and 30 GPa. We then compare thedynamical properties predicted by the CENT potentialwith DFT values at 2 GPa. The phonons arising from theCENT potential as well as the phonon DOS agree wellwith the results from DFT (see Fig. S2 in the supple-mentary materials). Similarly, the lattice thermal con-ductivities from CENT are in excellent agreement withthe DFT predictions, as shown in Fig. S3 in the supple-mentary material.Next, we study the evolution of the thermal conduc-tivity of the three phases as a function of pressure. Sincethe δ -CaF phase exhibits imaginary phonon modes at0 GPa, we focus on the pressure regime between 2 and30 GPa within which all structures are dynamically sta-ble. Fig. 1 plots the components κ x,y,z of the thermalconductivity at selected temperatures, and shows thattheir values increase almost linearly with pressure. Thisincrease in thermal conductivtiy can be rationalized ina first approximation by the decrease in volume V inthe denominator in Eq. (1) as the pressure increases.The room-temperature thermal conductivity of all threephases at different pressures and room temperature arealso summarized in Table I FIG. 1. (color online) The components of the lattice thermalconductivities κ x,y,z of α -CaF (panels (a) to (c)), γ -CaF (panels (d) to (f)), and δ -CaF (panels (g) to (i)), as a func-tion of pressure at temperatures of 300, 600, and 900 K. DFTdata are shown with blue crosses, while CENT results areshown with yellow squares (300 K), orange circles (600 K),and pink dots (900 K). At all pressures, α -CaF has a significantly highervalue of κ than any of the other two phases at a given temperature. Table I also contains the room-temperaturezero-pressure value of the thermal conductivity, κ ambient ,of α -CaF from other theoretical studies in the literature.We obtain κ ambient = 7 . − K − , which is close tothe value of 7 . − K − reported by Plata et al . In comparison with experimental results from Andersson et al , our value of κ ambient is about 2 . − K − lowerthan the experimental measurement of 9 . − K − through a two-strip method. Our values of κ for δ -CaF and γ -CaF showthat, unlike α -CaF , these two phases exhibit slightanisotropies along their three components. At 300 K,the components of κ x,y,z for the γ -CaF and δ -CaF phases at 2 GPa are approximately { . , . , . } and { . , . , . } Wm − K − , respectively, while κ itself are1.8 and 0 .
93 Wm − K − for γ -CaF and δ -CaF , respec-tively. The very low thermal conductivity of the δ -phaseat low pressures can be primarily attributed to the softacoustic phonon mode along K–Γ in the first Brillouinzone. (see section S4 and Fig. S5 for in the SI).There are several factors leading to the deceased κ of γ -CaF and δ -CaF compared to the cubic structure.Eq. (1) contains the product of heat capacity, phonongroup velocity, and phonon mean free path, the effectsof which we can study individually. We first investigatethe heat capacities per unit volume at selected pressureand temperatures, and show its evolution in Fig. S4 of theSI. The heat capacity rapidly increases with temperature T , and is proportional to T at low T , whereas it tendsto a constant value at a high temperature, following theDulong-Petit law. In the case of α -CaF , the obtainedvalue for C v at zero pressure and at temperature 300 K is65 .
57 J/m/K, which is comparable with the experimen-tal value of 67.11 J/m/K. The obtained values of C v at temperature 300 K and pressure 2 GPa for α -CaF , δ -CaF , and γ -CaF are 64 .
92, 65 .
10, and 64 .
64 J/m/K,respectively. The value of C v decreases with increasingpressure (see insets in Fig. S4 of the SI) at given tem-perature, and at 30 GPa reaches 58 .
29, 59 .
77, and 59 . α -CaF , δ -CaF , and γ -CaF , respectively.Overall, the difference in C v among the three phases isminute (within less than 3 %) and cannot account for thestrong deviations of κ .We now turn our attention to the group velocities v g ofthe acoustic phonon modes, which are in general respon-sible for a large fraction of the thermal transport. Fig. 2shows v g of the longitudinal and transverse acoustic(LA and TA) branches of γ -CaF and δ -CaF , plottedon top of the values of α -CaF . Note that the δ -phaseexhibits a particularly soft acoustic branch with a low v g along K–Γ (see Fig. S5 in the SI). Overall, α -CaF haslarger group velocities than either γ -CaF or δ -CaF .To quantify the difference in the group velocities, weconsider the mean values of the LA and the two TAmodes, ¯ v LA g , ¯ v TA g , and ¯ v TA g . The ratios of these averagevelocities of α -CaF with respect to the γ and δ -phases is { ¯ v LA g , ¯ v TA g , ¯ v TA g } α / { ¯ v LA g , ¯ v TA g , ¯ v TA g } γ = { . , . , . } ,and { ¯ v LA g , ¯ v TA g , ¯ v TA g } α / { ¯ v LA g , ¯ v TA g , ¯ v TA g } δ = TABLE I. The components of the lattice thermal conductivity κ x,y,z using CENT in units of Wm − K − at 300 K and atselected pressures, together with available values from the literature. Results from DFT calculations are given in paranthesis.Phase Components of κ α -CaF κ x = κ y = κ z , 8.6 , 7.0 ± . δ -CaF κ x = κ y κ z κ x γ -CaF κ y κ z { . , . , . } . Hence, the group velocities of α -CaF is almost twice as high as the corresponding valuesin γ -CaF and δ -CaF . FIG. 2. (color online) Group velocities of the longitudinaland transversal acoustic modes as functions of frequency forthe γ -CaF ((a) through (c)) and δ -CaF ((d) through (f)) incomparison with α -CaF at 300 K and 2 GPa. Further, the contributions of acoustic modes to thethermal transport is influenced by their interaction withthe optical modes, i.e., the amount of heat that is scat-tered through optical phonons. In general, phases withlarger, complex structures tend to have larger contribu-tions from optical scattering, with stronger coupling be-tween acoustic and optical modes. Fig. 3 shows the frac-tion of acoustic modes contributing to the total thermalconductivity, r κ = κ acoustic /κ total for the α , γ , and δ phase. At any pressure and temperature, α -CaF ex-hibits the largest value of r κ . Both γ -CaF and δ -CaF show strong contributions of optical phonon scattering,in particular for δ -CaF at low lower pressures. Again,this behavior can be attributed to the soft-mode in oneof the acoustic branches of δ -CaF at 2 GPa, which be-comes less pronounced with increasing pressure as shownin Fig. S5 in the SI.We also compare the phonon MFP in Fig. 4 at 2 and30 GPa at a temperature of 300 K. Overall, the MFPs ofthe α -phase are longer than either of the high-pressure Fm−3m Pnma P−62m k ac ou s ti c / k t o t a l ( i n % ) a −CaF g −CaF d −CaF T=300K, P=2 GPa72% 51% 37%T=900K, P=2 GPa68% 47% 33%T=300K, P=30 GPa80% 52% 57%T=900K, P=30 GPa75% 47% 52%
FIG. 3. The fraction of of acoustic modes κ acoustic contribut-ing to the total thermal conductivity κ total at pressures of 2and 30 GPa and temperatures of 300 and 900 K for the threerelevant CaF phases. phases. In fact, the MFP of a significant fraction ofmodes are shorter than the average inter-atomic distanceof ≈ . γ and δ -phases, leading to an inac-curate description of thermal transport within the BTEby dramatically underestimating the value of κ .To address this issue, we assess the limitations of theBTE by comparing its results to the Cahill-Pohl model,which provides an estimate of the lower bound in theamorphous limit, κ CP . Fig. 5 shows the values of κ CP asa function of temperature and pressures for the α , γ , and δ -CaF . We observe two very clear trends: (a) κ CP in-creases with temperature at a given pressure, plateauingout above ≈
600 K (see top row in Fig. 5), and (b) κ CP increases steadily with pressure at constant temperature(see bottom row in Fig. 5). The values of κ CP are partic-ularly high for δ -CaF , which indicates that an especiallylarge error can be expected in the BTE model.To assess the limits of the BTE, we map out the bound-ary in T and p where κ BTE drops below the amorphouslimit, κ CP . Fig 6 plots the κ BTE and κ CP at selectedtemperatures as a function of pressure. For α -CaF , thethermal conductivities predicted through the BTE are re-liable, as their values remain above κ CP for all pressuresand temperatures considered here. However, the BTE ✶(cid:0)✲✁✶(cid:0)✵✶(cid:0)✁✶(cid:0)✷✶(cid:0)✸ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0)☎ ✆✝✞ ✟✞✠▼✡☛☞✌✍ a ✎✏✞✑✒ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0)✟✭✠ g ✎✏✞✑✒ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0)✟✓✠ d ✎✏✞✑✒✶(cid:0)✲✁✶(cid:0)✵✶(cid:0)✁✶(cid:0)✷✶(cid:0)✸ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0) ✺(cid:0)(cid:0)✔✕ ✆✝✞ ✟✖✠▼✡☛☞✌✍ ❋✗✘✙✚✘✛✜✢✣✜✤✦✶✮ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0) ✺(cid:0)(cid:0)✟✧✠❋✗✘✙✚✘✛✜✢✣✜✤✦✶✮ (cid:0) ✶(cid:0)(cid:0) ✥(cid:0)(cid:0) ✂(cid:0)(cid:0) ✄(cid:0)(cid:0) ✺(cid:0)(cid:0)✟★✠❋✗✘✙✚✘✛✜✢✣✜✤✦✶✮ FIG. 4. (color online) Phonon mean free path (MFP) of allphonon modes as a function of frequency at pressures of 2 GPa(first row) and 30 GPa (second row) of α -CaF (panels (a) and(b)), γ -CaF (panels (c) and (d)), and δ -CaF (panels (e) and(f)).FIG. 5. (color online) The amorphous limit of the thermalconductivity κ CP based on the Cahill-Pohl model of (a) α -CaF , (b) γ -CaF , and (c) δ -CaF based on CENT calcula-tions as a function of temperature at selected pressures (firstrow) and as a function of pressure for temperatures 300, 600,and 900 K (second row). DFT results at 2, 10, 30 GPa areshown with blue symbols. ✵✶(cid:0)(cid:0)✷✸✁ ✭✂✄ a ✲☎✂✆✝ k ✥✞✟✠✡ ❑✠✡ ✮ ❚ ☛ ✸✵✵ ☞❇❚✌☎❈ ✵✷✽✶(cid:0) ✭✍✄ a ✲☎✂✆✝❚ ☛ ✎✵✵ ☞✵✷✽✶(cid:0) ✭✏✄ g ✲☎✂✆✝ k ✥✞✟✠✡ ❑✠✡ ✮ ✵(cid:0)✷ ✭✑✄ g ✲☎✂✆✝✵✷✽ ✵ ✺ ✶✵ ✶✺ (cid:0)✵ (cid:0)✺ ✸✵✭✒✄ d ✲☎✂✆✝ k ✥✞✟✠✡ ❑✠✡ ✮ ✵(cid:0)✷ ✵ ✺ ✶✵ ✶✺ (cid:0)✵ (cid:0)✺ ✸✵✭✓✄ d ✲☎✂✆✝(cid:0)✵✵✷✵✵✁✵✵✽✵✵✶✵✵✵ ✵ (cid:0) ✷ ✁ ✽ ✶✵ ✶(cid:0) ✶✷ ✶✁ ✶✽ (cid:0)✵ (cid:0)(cid:0)✭✔✄✕✖✟✗✖✘✙✚✛✘✖✥❑✮ ❈P✒✜✜✢P✒ ✭✣❈✂✄ g ✲☎✂✆✝ d ✲☎✂✆✝ FIG. 6. (color online) The thermal conductivity κ as a func-tion of pressure for α -CaF ((a) and (b) ), γ -CaF ((c) and(d)), and δ -CaF ((e) and (f)) Cahill-Pohl model (green tri-angles) at 300 and 900 K together with the results of BTE(red circles) at the same conditions. The values obtainedfrom DFT results are shown with blue symbols. The calcu-lated transition boundary where the κ CP ¿ κ BTE for γ -CaF and δ -CaF are shown in panel (g), where the shaded regionsindicate the T − p -range where BTE is reliable. breaks down for δ -CaF and γ -CaF , especially at lowtemperatures and low pressures. The transition bound-ary where κ BTE crosses κ CP in T and p is mapped out inFig. 6(g), showing that BTE only yields reliable resultswithin the regime of high pressure and low temperatures. IV. CONCLUSIONS
In summary, we studied the effect of pressure and tem-peratures on the thermal transport properties of threecrystalline CaF phases, using DFT and a machine-learning based interatomic potential. Our results showthat the two high-pressure phases, δ -CaF and γ -CaF ,exhibit significantly lower thermal conductivities κ thanthe cubic α -phase. We argue that the source of thislarge difference in κ stems from lower group velocitiesof the acoustic modes, and the larger contributions ofphonon scattering events involving the optical modes inthe δ and γ -phase which additionally impedes the trans-port of heat. A careful analysis of the phonon scatteringshows that the MFPs (and the associated phonon life-times) are extremely short for the δ and γ -phases, lead-ing to the low values of κ . In fact, for high temperaturesand at low pressures the MFPs are so short that theydrop below the mean atomic bond lengths, and we ex-pect that the thermal conductivity will eventually con-verge to the amorphous limit which we estimate usingthe Cahill-Pohl model. Despite these limitations, our re-sults show that the high-pressure phases exhibit arounda factor of 5 times lower thermal conductivity than the ambient ground state. V. ACKNOWLEDGMENTS
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