Pressure dependence of the superconducting transition temperature of compressed LaH 10
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Pressure dependence of the superconducting transition temperature of compressed LaH Chongze Wang, Seho Yi, and Jun-Hyung Cho ∗ Department of Physics, Research Institute for Natural Science,and HYU-HPSTAR-CIS High Pressure Research Center, Hanyang University,222 Wangsimni-ro, Seongdong-Ku, Seoul 04763, Republic of Korea (Dated: August 1, 2019)Two recent experiments [M. Somayazulu et al ., Phys. Rev. Lett. , 027001 (2019) and A. P. Drozdov et al ., Nature , 528 (2019)] reported the discovery of superconductivity in the fcc phase of LaH at acritical temperature T c between 250 ∼
260 K under a pressure of about 170 GPa. However, the dependence of T c on pressure showed different patterns: i.e., the former experiment observed a continuous increase of T c upto ∼
275 K on further increase of pressure to 202 GPa, while the latter one observed an abrupt decrease of T c with increasing pressure. Here, based on first-principles calculations, we reveal that for the fcc-LaH phase,softening of the low-frequency optical phonon modes of H atoms dramatically occurs as pressure decreases,giving rise to a significant increase of the electron-phonon coupling (EPC) constant. Meanwhile, the electronicband structure near the Fermi energy is insensitive to change with respect to pressure. These results indicatethat the pressure-dependent phonon softening is unlikely associated with Fermi-surface nesting, but driven byeffective screening with the electronic states near the Fermi energy. It is thus demonstrated that the strongvariation of EPC with respect to pressure plays a dominant role in the decrease of T c with increasing pressure,supporting the measurements of Drozdov et al . PACS numbers:
In 1968, Neil Ashcroft proposed that a metallic solid-hydrogen could exhibit superconductivity (SC) at high tem-peratures [1]. This metallic hydrogen can be referred to as aconventional Bardeen-Cooper-Schrieffer (BCS) superconduc-tor, where SC is driven by a condensate of electron pairs, so-called Cooper pairs, due to electron-phonon interactions [2].Despite such a theoretical proposal of high-temperature SC,the experimental realization of metallic hydrogen has beenvery challenging, because it requires too high pressures over ∼
400 GPa [3–5]. In order to achieve the metallization ofhydrogen at relatively lower pressures attainable in diamondanvil cells [6, 7], an alternative route has been taken by usinghydride materials in which hydrogen atoms can be “chemi-cally precompressed” to reduce the distances between neigh-boring H atoms [8–10]. This route with hydrides has re-cently been demonstrated to be promising for the achieve-ment of room-temperature SC [11–13] that is one of the mostchallenging subjects in modern physics. Nearly simultane-ously, two experimental groups synthesized a lanthanum hy-dride LaH with a clathrate-like structure [see Fig. 1(a)] atmegabar pressures and measured a superconducting transitiontemperature ( T c ) between 250 ∼
260 K at a pressure of ∼ T c is the highest tem-perature so far among experimentally available superconduct-ing materials, thereby opening a new era of high-temperatureSC [16–20].Although the two experiments [14, 15] agreed well withhigh values of T c in the fcc phase of compressed LaH , thedependence of T c on pressure is conflicting with each other.The measurements of Somayazulu et al . [14] showed a con-tinuous increase of T c up to ∼
275 K at 202 GPa, while thoseof Drozdov et al . [15] displayed a dome-like shape of the T c vs. pressure values, which represents an abrupt decrease of T c with increasing pressure over ∼
170 GPa. The latter ex- y xz
Pressure (GPa) B ond l eng t h ( Å ) La H H (a) (b) d H -H d H -H d La-H d La-H FIG. 1: (Color online) (a) Optimized structure of the fcc phase ofcompressed LaH . The H-rich clathrate structure LaH is com-posed of the H cage surrounding a La atom. The two differenttypes of H atoms, i.e., H and H , are drawn with bright and dark cir-cles, respectively. The positive x , y , and z axes point along the [001],[010], and [001] directions, respectively. (b) Calculated H -H , H -H , La-H , and La-H bond lengths of the fcc LaH structure as afunction of pressure. perimental measurements are similar to the theoretical pre-diction of a previous density-functional-theory (DFT) calcu-lation [11], where T c of the fcc LaH phase decreases withincreasing pressure in the range between 210 and 300 GPa.Meanwhile, it is expected that, as pressure increases, the lat-tice constants and the H − H bond lengths of fcc LaH de-crease. The resulting stronger covalent bonding of H atomsshould increase the frequencies of H-derived phonons, whichin turn contribute to enhance T c in terms of the standard weak-coupling BCS expression [1, 2]. Therefore, the two differ-ent experimental observations [14, 15] on the pressure depen-dence of T c of fcc LaH raise an open question of whether T c increases or decreases with increasing pressure, together withits microscopic underlying mechanism.Typeset by REVTEXIn this Rapid Communication, using first-principles DFTcalculations, we investigate the bonding, electronic, andphononic properties of fcc LaH as a function of pressureto identify the pressure dependence of T c with its microscopicmechanism. It is found that the Fermi surface and the vanHove singularity (vHs) near the Fermi energy E F , formed bythe holelike and electronlike bands with a strong hybridiza-tion of the La 4 f and H s orbitals [21], is insensitive to changewith respect to pressure. By contrast, the low-frequency opti-cal phonon modes of H atoms shift dramatically toward lowerfrequencies with decreasing pressure, leading to an overlapwith the acoustic phonon modes of La atoms. Such soften-ing of the low-frequency optical phonon modes at lower pres-sures is thus unlikely associated with Fermi-surface nesting,but is due to an enhanced electron-phonon coupling (EPC)with the electronic states at the vHs. These unique features ofphonons and electronic band structure with respect to pressuregive rise to a significant increase of the EPC constant at lowerpressures, resulting in a nearly linear decrease of T c with in-creasing pressure. Therefore, the present results support themeasurements of Drozdov et al . [15] on the T c vs. pressurerelation of fcc LaH . e /Å (a) (b) (c) FIG. 2: (Color online) Calculated total charge densities of fcc LaH at (a) 220 GPa, (b) 260 GPa, and (c) 300 GPa. The charge densitycontour maps are drawn in the (1¯10) plane with a contour spacing of0.1 e / ˚A . We first optimize the structure of the fcc LaH phase as afunction of pressure using the DFT calculations [22]. Basedon previous experimental [14, 15] and theoretical [11–13]studies, we consider the fcc LaH structure having cages of32 H atoms surrounding a La atom [see Fig. 1(a)] [11, 12].It is noted that there are two types of H atoms: i.e., H atomsforming the squares and H atoms forming the hexagons. Theoptimized fcc LaH structures as a function of pressure showthat the lattice constants decrease monotonously with increas-ing pressure (see Fig. S1 of the Supplemental Material [23]).Consequently, as shown in Fig. 1(b), the H -H , H -H , La-H , and La-H bond lengths (denoted as d H − H , d H − H , d La − H , and d La − H , respectively) also decrease monotonouslywith increasing pressure. At 300 GPa, we obtain d H − H =1.145 ˚A and d H − H = 1.064 ˚A, in good agreement with those( d H − H = 1.152 and d H − H = 1.071 ˚A) of a previous DFTcalculation [11]. Figures 2(a), 2(b), and 2(c) show the cal-culated total charge densities of fcc LaH at 220. 260, and 300 GPa, respectively. It is seen that, as pressure increases,the charge densities at the midpoints of the H -H and H -H bonds increase, indicating that their covalent bonding char-acters increase. Such enhanced H-H covalent-bond strengthswith increasing pressure should contribute to increase the fre-quencies of H-derived phonons, as discussed below. It is no-ticeable that there is also a covalent character between Laatoms and H cages [21]: i.e., the electrical charges of Laand H atoms are connected with each other at 260 and 300GPa. As shown in Figs. 2(a), 2(b), and 2(c), the charge densi-ties between La and H atoms decrease with decreasing pres-sure. This reduced covalent strength between La atoms andH cages is well reflected by the increase of the bond lengths d La − H and d La − H with decreasing pressure [see Fig. 1(b)],which will later be shown to be related with softening of thelow-frequency optical phonon modes of H atoms. It is thuslikely that the increases of d La − H and d La − H lead to inducethe dynamical instability of the fcc LaH phase, as observedby experiments [14, 15]. -10-5
5W L (cid:42)
X W K E ne r g y ( e V ) D O S ( s t a t e s / e V ) (a)(b)
300 GPa260 GPa220 GPa (cid:42)
LK WX
Energy (eV)
FIG. 3: (Color online) Comparison of (a) the electronic band struc-tures and (b) DOS of fcc LaH , calculated at 220, 260, and 300 GPa.The energy zero represents E F . In (a), the Brillouin zone of the fccprimitive cell is also drawn. The inset of (b) shows a closeup of theDOS around the vHs near E F . Figure 3(a) shows the comparison of the electronic bandstructures of fcc LaH , calculated at 220, 260, and 300 GPa.Here, we consider three different pressures 220, 260, and 300GPa, because the experimentally observed fcc LaH phasebecomes stable above a critical pressure between 210 and 220GPa, as discussed below. We find that the dispersions of thebands around E F change very little with respect to pressure,indicating a nearly invariance of the Fermi surface. Therefore,as shown in Fig. 3(b), the pressure dependence of the densityof states (DOS) around E F is minor compared to those at theenergy regions away from E F . It is noted that the double-shaped vHs, produced by the presence of the holelike andelectronlike bands around the high-symmetry L points [21],exhibits a slight decrease of the DOS at E F with increasingpressure [see the inset of Fig. 3(b)]. According to our previousDFT calculations [21], the existence of such vHs near E F is ofimportance for the room-temperature SC observed [14, 15]in fcc LaH . Here, the electronic states composing the vHshave a strong hybridization of the La 4 f and H s orbitals [21].These hybridized electronic states near E F not only contributeto produce the covalent character between La and H atoms(as discussed above) but also could effectively screen the low-frequency optical phonon modes of H atoms, the frequenciesof which are significantly lowered as pressure decreases (asdiscussed below). In contrast to the bands near E F , the valenceand conduction bands away from E F shift downwards and up-wards with increasing pressure, respectively [see Figs. 3(a)and 3(b)]. Such shifts of the valence and conduction bandsare likely caused by the band widening due to increased elec-tron hopping at higher pressures. As shown in Fig. 3(b), thecalculated DOS shows that the bottom of the valence bands islowered from − − as afunction of pressure using the density functional perturbationtheory implemented in QUANTUM ESPRESSO [28]. Fig-ures 4(a), 4(b), and 4(c) dispaly the calculated phonon disper-sions at 220, 260, and 300 GPa, respectively, together withthe projected DOS onto La, H , and H atoms. We find thatat 300 GPa, the acoustic phonon modes of La atoms withfrequencies lower than ∼
315 cm − are well separated fromthe optical phonon modes of H atoms with frequencies higherthan ∼
700 cm − [see Fig. 4(c)]. However, as pressure de-creases, the H-derived optical modes shift downwards in theoverall frequency range. Specifically, at 220 GPa, the low-frequency optical modes around the L and X points signif-icantly shift to lower frequencies, thereby overlapping withthe acoustic modes [see Fig. 4(a)]. Since the Fermi surfacechanges very little with respect to pressure, such softening ofthe low-frequency optical modes at lower pressures is likelydue to their effective screening with the electronic states near E F , rather than Fermi-surface nesting. It is noted that, as pres-sure decreases, the increase in the bond lengths of d La − H and d La − H would induce the effective screening of the low-frequency optical modes with the strongly hybridized La 4 f and H s electronic states around the high-symmetry L pointsat E F . Consequently, the EPC strength in the low-frequencyoptical modes around the L point is much enhanced with de- creasing pressure, as illustrated by the circles in Fig. 4(a).Here, the larger the size of circle, the stronger is the EPC.Figures 4(a), 4(b), and 4(c) also include the Eliashbergfunction α F ( ω ) and the integrated EPC constant λ ( ω ) as afunction of phonon frequency. We find that the contributionsto α F ( ω ) and λ ( ω ) arise from all three phonon modes in-cluding the La-derived acoustic, the H -derived optical, andthe H -derived optical modes. Here, since the projectedphonon DOS onto La atoms is well separated from those ofH and H atoms, we can estimate that at 260 (300) GPa,the acoustic phonon modes of La atoms contribute to ∼ λ = λ ( ∞ ), while the opticalphonon modes of H and H contribute to ∼
61 and ∼ ∼
65 and ∼ λ , respectively. Therefore, the contri-butions of the H -derived optical modes to λ are larger thanthose of the H -derived optical and the La-derived acousticmodes. As shown in Figs. 4(a), 4(b), and 4(c), λ decreaseswith increasing pressure as 4.24, 2.35, and 1.86 at 220, 260,and 300 GPa, respectively. Meanwhile, the logarithmic av-erage of phonon frequencies, ω log , increases with increasingpressure as 467, 797, and 932 cm − at 220, 260, and 300 GPa,respectively [see Fig. 4(d)]. These opposite variations of λ and ω log with respect to pressure are combined to slightly in-fluence the change of T c for the two pressure ranges between220-260 GPa and 260-300 GPa. Here, λ ( ω log ) contributes todecrease (increase) T c with increasing pressure [32]. By nu-merically solving the Eliashberg equations [33] with the typi-cal Coulomb pseudopotential parameter of µ ∗ = 0.13 [11, 12],we find that T c decreases with increasing pressure as 265, 250,and 233 K at 220, 260, and 300 GPa, respectively [see Figs.4(d) and S2 of the Supplemental Material [23]]. Thus, we cansay that λ plays a more dominant role in determining the pres-sure dependence of T c , rather than ω log . It is noticeable thatat 210 GPa, the negative phonon frequencies appear along the L − W line [see Fig. 4(e)], indicating that the fcc LaH phasebecomes unstable. Thus, our results of the T c vs. pressurerelation in fcc LaH show that T c decreases by ∼
16 K forevery 40-GPa increase above the critical pressure P c between210 and 220 GPa [see Fig. 4(d)]. This linear slope of T c asa function of pressure agrees well with a previous DFT re-sult [11], but it is larger than the experimental measurementsof Drozdov et al . [15] where T c decreases with a slope of6 ± P c ≈
170 GPa [15, 34] is muchlower than the present and previous [11] theoretical ones be-tween 210 and 220 GPa. This difference of P c between ex-periment and theory was explained by the anharmonic effectson phonons [34, 35]. Thus, in order to better reproduce theobserved [14, 15] pressure dependence of T c in fcc LaH10,further theoretical investigations with including anharmonicor nonadiabatic effects [36–38] are demanded in the future.In summary, our first-principles calculations have demon-strated that T c of the compressed fcc LaH phase decreaseswith increasing pressure. It is revealed that, as pressure de-creases, the low-frequency optical phonon modes of H atomsare dramatically softened due to their effective screening with W L (cid:42)
X W K05001000150020002500 F r equen cy ( c m - ) W L (cid:42)
X W KW L (cid:42)
X W K
DOS (states/cm -1 ) (a)(c) (e)(d) (b) (cid:302) F (cid:11)(cid:550)(cid:12)(cid:15)(cid:3)(cid:540)(cid:11)(cid:550)(cid:12) W L (cid:42)
X W K
DOS (states/cm -1 ) (cid:302) F (cid:11)(cid:550)(cid:12)(cid:15)(cid:3)(cid:540)(cid:11)(cid:550)(cid:12) F r equen cy ( c m - ) F r equen cy ( c m - ) (cid:302) F (cid:11)(cid:550)(cid:12)(cid:15)(cid:3)(cid:540)(cid:11)(cid:550)(cid:12) (cid:302) F (cid:11)(cid:550)(cid:12)(cid:540)(cid:11)(cid:550)(cid:12) T c ( K ) P (GPa) (cid:90) l og ( c m - ) FIG. 4: (Color online) Calculated phonon spectrum, phonon DOS projected onto La, H , and H atoms, Eliashberg function α F ( ω ) , andintegrated EPC constant λ ( ω ) of fcc LaH at (a) 220 GPa, (b) 260 GPa, and (c) 300 GPa. Here, the size of circles on the phonon dispersionis proportional to the EPC strength. (d) Calculated T c (displayed with circles) and ω log (squares) of fcc LaH as a function of pressure.For comparison, the experimentally observed T c of Drozdov et al . [15] between ∼
170 and ∼
220 GPa are given in (d), where the scale ofexperimental pressure is given in parentheses. (e) Calculated phonon spectrum of fcc LaH at 210 GPa. the electronic states near E F , which are characterized by astrong hybridization of the La 4 f and H s orbitals. Inter-estingly, the electronic band structure around E F changes verylittle with respect to pressure. Therefore, such unique fea-tures of phonons and electronic band structure with respect topressure result in a large increase of the EPC constant withdecreasing pressure. Our findings provided a microscopic un-derstanding of why T c of fcc LaH decreases with increasingpressure, supporting the recent experimental measurements ofDrozdov et al . [15]. The present explanation for the pressuredependence of T c of fcc LaH based on the strong variationof EPC with respect to pressure is rather generic and hence,it could be more broadly applicable to other high- T c hydrideswith structural instability. Indeed, compressed H S having T c = 203 K at ∼
150 GPa [16] was also observed to exhibit a de-crease of T c with increasing pressure. Acknowledgement.
This work was supported by the NationalResearch Foundation of Korea (NRF) grant funded by theKorean Government (Grants No. 2019R1A2C1002975, No.2016K1A4A3914691, and No. 2015M3D1A1070609). Thecalculations were performed by the KISTI SupercomputingCenter through the Strategic Support Program (Program No.KSC-2018-CRE-0063) for the supercomputing applicationresearch. ∗ Corresponding author: [email protected] [1] N. W. Ashcroft, Phys. Rev. Lett. , 1748 (1968).[2] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. ,
162 (1957).[3] J. McMinis, R. C. Clay, D. Lee, and M. A. Morales, Phys. Rev.Lett. , 105305 (2015).[4] J. M. McMahon, M. A. Morales, C. Pierleoni, and D. M. Ceper-ley, Rev. Mod. Phys. , 1607 (2012).[5] R. P. Dias and I. F. Silvera, Science , 715 (2017).[6] W. A. Bassett, High Press. Res. , 163 (2009).[7] H. K. Mao, X. J. Chen, Y. Ding, B. Li, and L. Wang, Rev. Mod.Phys. , 015007 (2018).[8] N. W. Ashcroft, Phys. Rev. Lett. , 187002 (2004).[9] J. Feng, W. Grochala, T. Jaro´n, R. Hoffmann, A. Bergara, andN. W. Ashcroft, Phys. Rev. Lett. , 017006 (2006).[10] M. I. Eremets, I. A. Trojan, S. A. Medvedev, J. S. Tse, and Y.Yao, Science , 1506 (2008).[11] H. Liu, I. I. Naumov, R. Hoffmann, N. W. Ashcroft, and R. J.Hemley, Proc. Natl. Acad. Sci. U.S.A , 6990 (2017).[12] F. Peng, Y. Sun, C. J. Pickard, R. J. Needs, Q. Wu, and Y. Ma,Phys. Rev. Lett. , 107001 (2017).[13] I. A. Kruglov, D. V. Semenok, R. Szcze¸´aniak, M. M. D. Esfa-hani, A. G. Kvashnin, and A. R. Oganov, arXiv:1810.01113.[14] M. Somayazulu, M. Ahart, A. K. Mishra, Z. M. Geballe, M.Baldini, Y. Meng, V. V. Struzhkin, and R. J. Hemley, Phys. Rev.Lett. , 027001 (2019).[15] A. P. Drozdov, P. P. Kong, V. S. Minkov, S. P. Besedin, M.A. Kuzovnikov, S. Mozaffari, L. Balicas, F. F. Balakirev, D.E. Graf, V. B. Prakapenka, E. Greenberg, D. A. Knyazev, M.Tkacz, and M. I. Eremets, Nature (London) , 528 (2019).[16] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, andS. I. Shylin, Nature (London) , 73 (2015).[17] T. Bi, N. Zarifi, T. Terpstra, and E. Zurek, arXiv:1806.00163v1(2018).[18] D. V. Semenok, I. A. Kruglov, A. G. Kvashnin, and A. R.Oganov, arXiv:1806.00865v2 (2018).[19] E. Zurek and T. Bi, J. Chem. Phys. , 050901 (2019).[20] J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta, R. Arita, andM. Eremets, arXiv:1905.06693v1 (2019).[21] L. Liu, C. Wang, S. Yi, K. W. Kim, J. Kim, and J. H. Cho, Phys.Rev. B , 140501(R) (2019).[22] Our DFT calculations were performed using the Vienna ab ini-tio simulation package (VASP) with the projector-augmentedwave (PAW) method [24–26]. For the exchange-correlationenergy, we employed the generalized-gradient approximationfunctional of Perdew-Burke-Ernzerhof (PBE) [27]. A plane-wave basis was taken with a kinetic energy cutoff of 500 eV.The k -space integration was done with 24 × × k pointsfor the structure optimization and 72 × × k points for theDOS calculation. All atoms were allowed to relax along thecalculated forces until all the residual force components wereless than 0.001 eV/ ˚A. The lattice dynamics and EPC calcula-tions were carried out by using the QUANTUM ESPRESSO(QE) package [28] with an optimized norm-conserved Vander-bilt pseudopotential (ONCV) [29] and a plane-wave cutoff of1224 eV. Here, we used the 6 × × q points and 24 × × k points for the computation of phonon frequencies. For the cal-culation of EPC, we used the software EPW [30, 31] with the24 × × q points and 72 × × k points. These employed q and k points were found to give converged values of λ (seeFig. S3 of the Supplemental Material [23]). We note that thelattice constants optimized using a PAW+PBE pseudopotentialin VASP and an ONCV+PBE pseudopotential in QE changeonly by less than 0.1% [see Fig. S1(a) of the Supplemental Ma-terial [23]]. In addition, as shown in Fig. S1(b), the electronicband structures obtained using the two different pseudopoten-tials are nearly the same with each other, especially near the Fermi energy. Therefore, we believe that both atomic relaxationand phonon/EPC calculations hardly change depending on thetwo pseudopotentials.[23] See Supplemental Material athttp://link.aps.org/supplemental/xxxxx for the calculatedlattice constants as a function of pressure, the superconduct-ing energy gap of fcc LaH at different pressures, and theconvergence test of λ with respect to k -point and q -pointmeshes.[24] G. Kresse and J. Hafner, Phys. Rev. B , 13115 (1993).[25] G. Kresse and J. Furthm¨uller, Comput. Mater. Sci. , 15 (1996).[26] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[27] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. ,3865 (1996); , 1396 (1997).[28] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo et. al. , J. Phys.: Condens. Matter , 395502 (2009).[29] M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete,D. R. Hamann, X. Gonze, and G. M. Rignanese, Comput. Phys.Comm. , 39 (2018).[30] F. Giustino, M. L. Cohen, and S. G. Louie, Rev. B ,165108(R) (2007).[31] S. Ponc´e, E. R. Margine, C. Verdi, and F. Giustino, Comput.Phys. Commun. , 116 (2016).[32] Using the Allen-Dynes equation [P. B. Allen and R. C. Dynes,Phys. Rev. B , 905 (1975)] with the typical Coulomb pseu-dopotential parameter of µ ∗ = 0.13 [11, 12], we estimate T c as243, 222, and 204 K at 220, 260, and 300 GPa, respectively,showing a nearly linear decrease of T c with increasing pressure.[33] G. M. Eliashberg, ZhETF , 966 (1960) [Sov. Phys. JETP ,696 (1960)].[34] Z. M. Geballe, H. Liu, A. K.Mishra, M. Ahart, M. Somayazulu,Y. Meng, M. Baldini, and R. J. Hemley, Angew. Chem., Int. Ed. , 688 (2018).[35] H. Liu, I. I. Naumov, Z. M. Geballe, M. Somayazulu, J. S. Tse,and R. J. Hemley, Phys. Rev. B , 100102(R) (2018).[36] I. Errea, M. Calandra, C. J. Pickard, J. Nelson, R. J. Needs, Y.Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, Phys. Rev. Lett. ,157004 (2015).[37] I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J. Needs,Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, Nature (London) , 81 (2016).[38] A. P. Durajski, Sci. Rep. , 38570 (2016). Supplemental Material for ”Pressure dependence of the supercon-ducting transition temperature of compressed LaH ”
1. Comparison of the lattice constants and electronic bands of fcc LaH10, obtained using a PAW+PBE pseudopotentialin VASP and an ONCV+PBE pseudopotential in QE.
FIG. S1: (a) Optimized lattice constants a = b = c of fcc LaH using a PAW+PBE pseudopotential in VASP (grey circles) and an ONCV+PBEpseudopotential in QE (blue triangles). The calculated values (in ˚A) at 220, 260, and 300 GPa are also given in (a). (b) Calculated electronicband structures of fcc LaH at 300 GPa, obtained using the two pseudoptentials.
2. Superconducting energy gap of fcc LaH at different pressures. FIG. S2: Superconducting energy gap ∆ as a function of temperature T , obtained at 220, 260, and 300 GPa. Here, we used the typicalCoulomb pseudopotential parameter of µ ∗ = 0.13.
3. Convergence test of λ with respect to k -point and q -point meshes. FIG. S3: Calculated λ of fcc LaH as functions of k points and qq