Theoretical investigation of novel electronic, optical, mechanical and thermal properties of metallic hydrogen at 495 GPa
Bo Peng, Ke Xu, Hao Zhang, Hezhu Shao, Gang Ni, Jing Li, Liangcai Wu, Hongliang Lu, Qingyuan Jin, Heyuan Zhu
TTheoretical investigation of novel electronic, optical, mechanical andthermal properties of metallic hydrogen at 495 GPa
Bo Peng , Ke Xu , Hao Zhang ∗ , Hezhu Shao , Gang Ni , Jing Li ,Liangcai Wu , Hongliang Lu , Qingyuan Jin , and Heyuan Zhu Key Laboratory of Micro and Nano Photonic Structures (MOE),Department of Optical Science and Engineering,Fudan University, Shanghai 200433, China Ningbo Institute of Materials Technology and Engineering,Chinese Academy of Sciences,Ningbo 315201, China College of Science, Donghua University,Shanghai 201620, China State Key Laboratory of ASIC and System,Institute of Advanced Nanodevices,School of Microelectronics,Fudan University, Shanghai 200433, China
Atomic metallic hydrogen has been produced in the laboratory at high pressure and lowtemperature, prompting further investigations of its di ff erent properties. However, purelyexperimental approaches are infeasible because of the extreme requirements in producingand preserving the metastable phase. Here we perform a systematic investigation of the elec-tronic, optical, mechanical and thermal properties of I / amd hydrogen at 495 GPa usingfirst-principles calculations. We calculate the electronic structure and dielectric function toverify the metallic behaviour of I / amd hydrogen. The calculated total plasma frequencyfrom both intraband and interband transitions, 33.40 eV, agrees well with the experimen-tal result. The mechanical properties including elastic stability and sound velocity are alsoinvestigated. The mechanical stability of I / amd hydrogen is limited by shear modulusother than bulk modulus, and the high Young’s modulus indicates that I / amd hydrogen isa sti ff material. After investigating the lattice vibrational properties, we study the thermody- a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y namical properties and lattice anharmonicity to understand thermal behaviours in metallichydrogen. Finally, the lattice thermal conductivity of I / amd hydrogen is calculated to be194.72 W / mK and 172.96 W / mK along the x and z directions, respectively. Using metallichydrogen as an example, we demonstrate that first-principles calculations can be a game-changing solution to understand a variety of material properties in extreme conditions. I. INTRODUCTION
In 1935, Wigner and Huntington predicted that, hydrogen molecules would become a metal-lic solid similar to the alkalis at high pressure . Metallic hydrogen is expected to be a wondermaterial: it may be a room-temperature superconductor and have significant applications in en-ergy and rocketry . However, producing metallic hydrogen has been a great challenge. Althougha lot of theoretical investigations have been performed on metallic hydrogen , not until re-cently has atomic metallic hydrogen been synthesized in the laboratory at high pressure and lowtemperature . At a pressure of 495 GPa, hydrogen becomes metallic with a high reflectivity, andthe metallic phase is metastable when the pressure is released. This prompts a systematic study ofits electronic, optical, mechanical and thermal properties for further applications.However, the challenge is to produce and preserve such metastable phase for experimentalinvestigations. Several months after the synthesis, the world’s only metallic hydrogen samplehas disappeared. Even if the reproducibility of the sample is improved, characterizing metallichydrogen under such pressure is extremely di ffi cult due to the limitations of conventional tech-niques. For instance, for hydrogen samples in a diamond anvil cell at extremely high pressure,neutron scattering and X-ray di ff raction experiments are unavailable. Thus it is imperative to de-velop alternative techniques for exhaustive characterization. Theoretical estimation, as a referenceto the experiments, is indeed another option. Recently, computational material techniques havebeen developed to perform an ab initio study on materials in extreme conditions . In addition,state-of-the-art techniques can be used to calculate accurately the ground- and excited-state ,mechanical , and phononic properties , enabling the feasibility of theoretical predictions ina variety of di ff erent properties of metallic hydrogen.Here, we perform an exhaustive study of metallic hydrogen at 495 GPa. Our fully first-principles calculations shed light into its electronic, optical, mechanical and thermal properties.First we focus on ground- and excited-state properties such as electronic band structure, dielec-tric function and plasma frequency. Then, mechanical properties including elastic anisotropy andsound velocity are investigated. After that, we discuss the lattice vibrational properties as wellas lattice anharmonicity. Finally we study the phonon transport properties of metallic hydrogen.Our results provide clear means of characterizing metallic hydrogen via these distinct features,strongly calling for experimental verifications. II. GROUND- AND EXCITED-STATE PROPERTIESA. Crystal structure
FIG. 1. Conventional cell of I / amd hydrogen at 495 GPa, and 3D electron localization function (isosur-face = × × We start by structural optimization using the Vienna ab-initio simulation package (VASP) basedon density functional theory (DFT) under the generalized gradient approximation (GGA) ex-pressed by the Perdew-Burke-Ernzerhof (PBE) functional . A plane-wave basis set with kineticenergy cuto ff of 800 eV is employed with 51 × ×
21 (41 × × k -mesh for conventional (prim-itive) cell at a pressure of 495 GPa during structural relaxation, until the energy di ff erences areconverged within 10 − eV, with a Hellman-Feynman force convergence threshold of 10 − eV / Å.The conventional cell of the I / amd phase at 495 GPa is shown in Figure 1. The calculatedlattice parameters are a = b = . c = . . The electron localization function maps of I / amd hydrogenat 495 GPa show a typical metallic behavior: Regions with constant 0.5 (corresponding to theelectron-gas like pair probability) are connected one to the other by channels, forming infinitethree-dimensional networks . In order to confirm the structure of the experimentally reportedmetallic hydrogen, we simulate the X-ray di ff raction patterns of the conventional cell . The results TABLE I. Calculated X-ray di ff raction patterns for the conventional cell of I / amd hydrogen.No. hkl θ ( ◦ ) d (Å) FWHM ( ◦ )1 001 28.6 3.1222 0.0752 002 59.2 1.5611 0.0813 100 79.0 1.2114 0.0914 011 86.1 1.1294 0.0965 003 95.6 1.0407 0.1046 110 107.6 0.9571 0.118 are shown in Table I. B. Electronic structure (a) (b)
XM P Γ N Γ FIG. 2. (a) Electronic structure and (b) imaginary part of dielectric function for I / amd hydrogen at 495GPa. Figure 2(a) shows the electronic band structure of I / amd hydrogen at 495 GPa. The Fermilevel is crossed by five di ff erent bands, confirming the metallic behaviour. The huge dispersion isassociated to the dominating kinetic term in the energies of the electronic states. Di ff erent fromthe free-electron approximation , the band gap open across the Brillouin zone at about 20 eV dueto the interaction of the electrons with the proton lattice.Borinaga et al has pointed out that the electron-electron interactions such as Hartree, exchangeand correlation e ff ects give no significant contribution to the band structure . However, afterphoto-excitation, an electron is excited from valence band into conduction band, leaving behinda positively charged hole. This single-particle excitation cannot be described by non-interactingparticles with infinite lifetime. Due to interactions with other particles, the excited electrons andholes become quasiparticles with finite lifetimes. This acquires a self-energy to account for all theelectron-electron interactions . In addition, the negatively charged quasielectron is attracted bythe quasihole, stabilizing the electron-hole pair and forming a new quasiparticle, an exciton. Apartfrom this attraction, a possible electron-hole exchange is also an important many-body e ff ect.Moreover, the electron-hole pairs are weakened by the screening of the electronic system. There-fore, self-energy, excitonic, and other many-electron interaction e ff ects may strongly influence theexcited-state properties of metallic hydrogen . C. Optical properties
Subsequent to the standard-DFT results, self-consistent GW corrections are undertaken with eight iterations of G . The energy cuto ff for the response function is set to be 300 eV. Atotal of 20 (valence and conduction) bands are used with a k -point sampling of 27 × × calculation with the Tamm-Danco ff approximation . No profound e ff ects on the optical properties are found by includingthe excitonic e ff ects. Thus we present the GW results hereafter. TABLE II. Calculated plasma frequency from intraband transitions ω intrap and interband transitions ω interp and electrical conductivity σ of I / amd hydrogen at 495 GPa.direction ω intrap (eV) ω interp (eV) σ (kS / m) x z The calculated imaginary part of dielectric function of I / amd hydrogen at 495 GPa is presentin Figure 2(b). Here we use current-current correlation function instead of density-density cor-relation function to account for the contribution of Drude terms in metals . The dielectric func-tion diverges at zero frequency because of the free-electron contribution. We also list the plasmafrequency from intraband and interband transitions, as well as the electrical conductivity σ in Ta-ble II. The calculated total plasma frequency along the x direction is 33.40 eV, agreeing well withthe experimental value of 32.5 ± . III. MECHANICAL PROPERTIESA. Mechanical stability
To evaluate the mechanical stability of I / amd hydrogen at 495 GPa, we calculate the elas-tic tensor coe ffi cients of the primitive cell including ionic relaxations using the finite di ff erencesmethod with a Γ -centered 34 × × k -mesh. The elastic coe ffi cients with the contributionsfor distortions with rigid ions, the contributions from the ionic relaxations and including both, arepresent in Table III. TABLE III. Calculated elastic coe ffi cients C i j for the primitive cell of I / amd hydrogen. C (GPa) C (GPa) C (GPa) C (GPa) C (GPa) C (GPa)Rigid ions 1974.22 1249.60 111.33 18.54 -317.16 598.73Ionic relaxations -13.74 0.0 -45.57 0.0 13.74 0.0Total 1960.48 1249.60 65.76 18.54 -303.42 598.73 I / amd hydrogen is tetragonal crystal system with a = b (cid:44) c . According to Born-Huang’slattice dynamical theory , the mechanical stability conditions for tetragonal phase are given by C > , C > , C > , C > , ( C − C ) > , ( C + C − C ) > , C + C ) + C + C >
0. The calculated elastic constants of metallic hydrogen satisfy the correspondingBorn stability criteria, indicating the I / amd phase is mechanically stable. B. Bulk modulus and shear modulus
The bulk modulus B describes the material’s response to uniform hydrostatic pressure. Fortetragonal structure, the Voigt and Reuss methods are used to evaluate B B V = C + C ) + C + C , (1) B R = ( C + C ) C − C C + C + C − C . (2)The shear modulus G the material’s response to shear stress, and can be given by G V = C − C + C − C + C + C , (3) G R = B V / [( C + C ) C − C ] + / ( C − C ) + / C + / C . (4)The Hill method calculations are as follows: B H =
12 ( B V + B R ) , (5) G H =
12 ( G V + G R ) . (6)The calculated bulk modulus and shear modulus are shown in Table IV. TABLE IV. Calculated bulk modulus B , shear modulus G , B / G ratio, Young’s modulus E , Poisson’s ratio,and anisotropy index A for the primitive cell of I / amd hydrogen. B V (GPa) B R (GPa) B H (GPa) G V (GPa) G R (GPa) G H (GPa) B H / G H E (GPa) ν A U A B A G The bulk modulus of metallic hydrogen is even higher than diamond (443 GPa) and cubic C N (496 GPa). This result can be easily explained by the extreme high synthesis pressure. The shearmodulus, which represents the resistance to shear deformation against external forces, is muchlower than the bulk modulus. This implies that the shear modulus limits the mechanical stabilityof hydrogen.The B / G ratio measures the malleability of materials. A high value represents ductility, while alow value is associated with brittleness. The critical value separating ductile and brittle materialsis approximately 1.75 . Our calculated B / G ratio is 4.15, revealing that metallic hydrogen isductile. C. Young’s modulus and Poisson’s ratio
The Young’s modulus can be evatulated from B H and G H E = B H G H B H + G H , (7)and the Poisson’s ratio is ν = B H − G H B H + G H ) . (8)The calculated Young’s modulus and Poisson’s ratios are present in Table IV.Young’s modulus is a mechanical property of linear elastic solid materials, and measures thesti ff ness of a solid material. A material with a higher Young’s modulus is sti ff er, which needs moreforce to deform compared to a soft material. The high Young’s modulus indicates that metallichydrogen is a sti ff material.The Poisson’s ratio describes the response in the direction orthogonal to uniaxial strain. Pois-son’s ratio close to 0 indicates very little lateral expansion when compressed, while a Poisson’sratio of exactly 0.5 represents a perfectly incompressible material deformed elastically at smallstrains. For I / amd hydrogen, a Poisson’s ratio of 0.39 suggests it is less compressible. D. Elastic anisotropy
In most single crystals, the elastic response is usually anisotropic. Elastic anisotropy exhibits adi ff erent bonding character in di ff erent directions and is important in diverse applications such asphase transformations, dislocation dynamics. The universal anisotropy index, which represents auniversal measure to quantify the single crystal elastic anisotropy, is defined as A U = G V G R + B V B R − , (9) A U is identically zero for locally isotropic single crystals. The departure of A U from zero definesthe extent of single crystal anisotropy. Another way implies the estimation in compressibility andshear A B = B V − B R B V + B R , (10) A G = G V − G R G V + G R . (11)A value of zero denotes elastic isotropy and a value of 100% represents the largest anisotropy. Thecalculated anisotropy indexes are obtained in Table IV.According to universal anisotropy index A U , the average elastic anisotropy is more than twotimes higher than Li . Comparing A B and A G , there is much more anisotropy in shear than incompressibility. E. Sound velocity and Debye temperature
The sound velocity is determined by the symmetry of the crystal and the propagation direction.The tetragonal symmetry dictates that pure transverse and longitudinal modes can only exist for0
TABLE V. Calculated longitudinal sound velocity v l , transverse sound velocity v t for the primitive cell of I / amd hydrogen for [100], [001] and [110] directions.[100] (km / s) [001] (km / s) [110] (km / s)[100] v l [001] v t [010] v t [001] v l [100] v t [010] v t [110] v l [001] v t [1¯10] v t [100], [001] and [110] directions. In all other directions the propagating waves are either quasi-transverse or quasi-longitudinal. In the principal directions the acoustic velocities can be simplywritten as For [100] direction:[100] v l = (cid:112) C /ρ, [001] v t = (cid:112) C /ρ, [010] v t = (cid:112) C /ρ ; (12)For [001] direction: [001] v l = (cid:112) C /ρ, [100] v t = [010] v t = (cid:112) C /ρ ; (13)For [110] direction:[110] v l = (cid:112) ( C + C + C ) /ρ, [001] v t = (cid:112) C /ρ, [1¯10] v t = (cid:112) ( C − C ) / ρ. (14)The calculated sound velocities are presented in Table V. TABLE VI. Calculated average longitudinal sound velocity v l , transverse sound velocity v t , average soundvelocity v s and Debye temperature θ D for the primitive cell of I / amd hydrogen. v l (km / s) v t (km / s) v s (km / s) θ D (K)26.42 11.28 12.75 3626.64 The Debye temperature θ D can be calculated as follows θ D = hk B (cid:18) N π V (cid:19) / v s , (15)where h is the Planck constant, k B is the Boltzmann constant, N is the number of atoms in the cell, V is the volume of the unit cell, and v s is the average sound velocity given by v s = (cid:20)
13 ( 1 v l + v t ) (cid:21) − / , (16)1The average sound velocity in crystals can be determined from B and G , v l = (cid:115) B + G ρ , (17) v t = (cid:115) G ρ . (18)The calculated sound velocities and Debye temperature are shown in Table VI.The speed of sound can be used to measure the speed of phonons propagating through thelattice. The ultrahigh longitudinal sound velocity of metallic hydrogen indicates high phononvelocity. The Debye temperature measures the temperature above which all modes begin to beexcited . Thus a high θ D indicates weak three-phonon scattering and hence a high lattice thermalconductivity. IV. PHONON PROPERTIESA. Dynamical stability
The Born-Huang mechanical stability criteria provide a necessary condition for the dynamicalstability, but not a su ffi cient one . Therefore we need to examine the dynamical stability of I / amd hydrogen at 495 GPa by calculating the phonon dispersion of the primitive cell.The harmonic interatomic force constants are obtained using density functional perturbationtheory (DFPT) within a supercell approach . A 7 × × × × q -mesh is used.The phonon dispersion and thermodynamical properties are calculated from the interatomic forceconstants using the PHONOPY code .As shown in Figure 3(a), no imaginary frequencies exist in the whole Brillouin zone, indicatingdynamical stability at 0 K. The calculated Raman active modes are 1200.24 cm − ( E g ) and 2692.70cm − ( B g ) respectively, which can be used as a reference for characterization of I / amd hydrogenat 495 GPa. The calculated phonon dispersion agrees well with previous result . B. Thermodynamical properties
For light elements such as hydrogen, phonons play an important role in determining the thermo-dynamical properties of crystals both at 0 K and at finite temperatures . Using phonon frequencies2 Γ X P N Γ M S (a) (b)(c) (d) FIG. 3. (a) Phonon dispersion of I / amd hydrogen at 495 GPa. (b) Helmholtz free energy, (c) entropy,and (d) heat capacity for I / amd hydrogen at 495 GPa as a function of temperature. in the Brillouin zone, we further examine the thermodynamical properties of metallic hydrogen bycalculating the Helmholtz free energy F , F = E tot + (cid:88) q j (cid:126) ω q j + k B T (cid:88) q j ln[1 − exp( − (cid:126) ω q j / k B T )] , (19)where E tot is the total energy of the crystal, (cid:126) is the reduced Planck constant, ω q j is the phononfrequency of the j -th branch with wave vector q , T is the temperature, and the summation termis the Helmholtz free energy for phonons . The first summation term is a temperature-freeterm corresponding to the zero point energy of phonons; and the second summation term is atemperature-dependent term referring to the thermally induced occupation of the phonon modes.The calculated zero point energy of I / amd hydrogen at 495 GPa is 0.295 eV / atom, which isthe Helmholtz free energies of phonons at 0 K. Temperature is also an important thermodynamicvariable for determining the stability of materials. The Helmholtz free energies F as a functionof temperature are shown in Figure 3(b). At higher temperature, the phonon modes are occupied3according to Bose-Einstein statistics, and the free energy further decreases.From the Helmholtz free energy, the other thermodynamical behavior can be deduced . Theentropy is S = − ∂ F ∂ T . (20)The calculated entropies for the four structures as a function of temperature are shown in Fig. 3(c).The di ff erence between the entropies of di ff erent phases can be used to determine the relativestability .The isometric heat capacity can be calculated as C V = (cid:88) q j k B (cid:32) (cid:126) ω q j k B T (cid:33) exp( (cid:126) ω q j / k B T )[exp( (cid:126) ω q j / k B T ) − . (21)The calculated heat capacities per atom are shown in Figure 3(d), which approach their Dulong-Petit classical limit (2 × / K mol ) at high temperatures. C. Lattice anharmonicity
Accurate simulation of anharmonic properties such as thermomechanics and thermal expan-sion is important for understanding thermal behaviours in solids and their realistic applications.The anharmonic properties can be derived from the volume dependences of phonon dispersion us-ing the quasiharmonic approximation method , and temperature is assumed to indirectly a ff ectvibrational properties via thermal expansion. Using the same supercell and q -mesh in previousphonon calculations, we calculate the phonon spectra of ten volumes, and the thermal expansionand thermomechanics are derived by fitting the free energy-volume relationship.Gibbs free energy G at given temperature T and pressure p is obtained from Helmholtz freeenergy F via finding a minimum value by changing volume V , G ( T , p ) = Min V [ F ( T ; V ) + pV ] , (22)where F is the sum of electronic internal energy and phonon Helmholtz free energy. The calculated G is depicted in Figure 4(a). The minimum value is obtained by the fits to third-order Birch-Murnaghan equation of states.The calculated isothermal bulk modulus is present in Figure 4(b). An anticipated thermal soft-ening is observed in the temperature dependence of B T . From 0 to 1000 K, the softening of the4 (a) (b)(c) (d) FIG. 4. (a) Gibbs free energy, (b) isothermal bulk modulus, (c) Gr¨uneisen parameter, and (d) volume thermalexpansion coe ffi cient of I / amd hydrogen at 495 GPa as a function of temperature. isothermal bulk modulus for metallic hydrogen is 157.98 GPa, which is much larger than those inC and Al O .The Gr¨uneisen parameter γ describes the e ff ect of thermal expansion on vibrational properties,and provides information on the anharmonic interactions. A larger Gr¨uuneisen parameter indicatesstronger anharmonic vibrations. As shown in Figure 4(c), the calculated γ at room temperature is0.84.The volume thermal expansion coe ffi cient β can be calculated from the Gr¨uneisen parameter β = γ C V B T V . (23)Another approach is to obtain β from the calculated equilibrium volumes V at temperature T β = V ( T ) ∂ V ( T ) ∂ T . (24)As shown in Figure 4(d), the β curves from these methods are in excellent agreement with eachother. It should be noticed that it is challenging to measure β accurately in experiment because5thermal expansion varies with the crystalline orientation and there may be some metastable phasesof metallic hydrogen at relative high temperatures. D. Phonon transport
The lattice thermal conductivity κ can be calculated using the self-consistent iterative approach as a sum of contribution of all the phonon modes λ , κ αα = V (cid:88) q j C q j τ q j v α q j , (25)where C q j is the heat capacity per mode, τ q j is the mode relaxation time, and v α q j is the groupvelocity along α direction, respectively. The lattice dynamical properties C q j and v α q j in Eq. (25)can be obtained by the phonon dispersion relation with harmonic interatomic force constants asinput, while the τ q j provides information about the anharmonic interactions and can be obtainedusing the anharmonic interatomic force constants. The anharmonic interatomic force constants arecalculated using a supercell-based, finite-di ff erence method , and a 7 × × × × q -mesh. We include the interactions with the 40th nearest-neighbour atoms. A discretiza-tiona of the Brillouin zone into a Γ -centered regular grid of 32 × × q points is introduced withscale parameter for broadening chosen as 0.1. When the phonon q -points mesh increases from28 × ×
28 to 32 × ×
32, the di ff erence of κ between the two grids is less than 1%, verifying theconvergence of our calculations.Figure 5(a) presents the phonon group velocities calculated from the phonon spectrum withinthe whole Brillouin zone. The group velocity shows strong anisotropy along the x and z directions.The phonon group velocities in long-wavelength limit for the acoustic modes agree well with thesound velocities calculated from elastic coe ffi cients.The phonon scattering depends on two factors: the number of channels available for a phononto get scattered, and the strength of each scattering channel. The former factor is determined bywhether there exist three phonon groups that can satisfy both energy and quasi-momentum con-servations. The latter factor depends on the anharmonicity of a phonon mode, which is describedby the mode Gr ¨uneisen parameter. The mode Gr¨uneisen parameters provide information aboutanharmonic interactions, and can be calculated directly from the anharmonic interatomic forceconstants. Figure 5(b) shows the mode Gr ¨uneisen parameters. Strong anisotropy is found in theGr¨uneisen parameters at low frequencies. To provide more physical insight, we investigate the6 (a) (b)(c) (d) FIG. 5. (a) Phonon group velocity, (b) mode Gr¨uneisen parameters, (c) relaxation time of I / amd hydrogenat 495 GPa at 300 K, and (d) lattice thermal conductivity of I / amd hydrogen at 495 GPa as a function oftemperature. phonon relaxation time of each phonon mode at 300 K in Figure 5(c). The relaxation times arehighly anisotropic at low frequencies as well.The thermal conductivity for I / amd hydrogen at 495 GPa at 300 K are higher than 170 W / mKalong the x and z directions, as listed in Table VII. We extract the contributions of di ff erent phononbranches to κ along the x and z directions, as shown in Table VII. Here the di ff erent modes aresimply distinguished by frequency . For phonon transport along x direction, the contribution ofTA phonons is largest among all phonon modes. For phonon transport along z direction, TA phonons contribute the most to κ . For both directions, the acoustic phonons dominate the heattransport.The intrinsic lattice thermal conductivity κ of I / amd hydrogen at 495 GPa is present in Fig-ure 5(d). The κ is weakly anisotropic ( κ xx = κ yy (cid:44) κ zz ). The anisotropy in thermal transport canbe attributed to the anisotropic natures of phonon dispersion of acoustic modes, which results in7 TABLE VII. κ at 300 K and contribution of di ff erent phonon modes branches (TA , TA , LA, and all opticalphonons) towards the κ of I / amd hydrogen at 495 GPa.direction κ (W / mK) TA (%) TA (%) LA (%) Optical (%) x z the orientation-dependent group velocities and Gr¨uneisen parameters as mentioned above. Theorientation-dependence of thermal transport properties provides a way to determine the optimizedtransport directions for potential applications. V. CONCLUSION
State-of-the-art computational material techniques enable a systematic investigation of di ff er-ent properties of metallic hydrogen. The calculated electronic structure and dielectric functionconfirm the metallic behaviour of I / amd hydrogen at 495 GPa. The calculated total plasmafrequency from intraband and interband transitions of 33.40 eV agrees well with the experimentalvalue. The phase is mechanical stable, and the stability is limited by shear modulus other than bulkmodulus. The high Young’s modulus indicates that I / amd hydrogen is a sti ff material, while thePoisson’s ratio suggests it is less compressible. The elastic anisotropy, sound velocity and Debyetemperature are also investigated in detail. The dynamical stability of I / amd hydrogen is con-firmed by phonon dispersion. The thermodynamical properties, as well as lattice anharmonicityare also calculated to understand thermal behaviours in metallic hydrogen. The lattice thermalconductivity of I / amd hydrogen is 194.72 W / mK and 172.96 W / mK along the x and z direc-tions, respectively, where acoustic phonons dominate heat transport. Our results show the powerof first-principles calculations in predicting a variety of material properties in extreme conditionswhere purely experimental approaches are impractical, and pave way for potential applications ofmetallic hydrogen at extremely high pressure and high temperature. ACKNOWLEDGEMENT
This work is supported by the National Natural Science Foundation of China under Grants No.11374063 and 11404348, and the National Basic Research Program of China (973 Program) under8Grant No. 2013CBA01505.
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