Ab initio studies on phase transition, thermoelastic, superconducting andthermodynamic properties of the compressed cubic phase of AlH3Yong-Kai Wei, Ni-Na Ge, Xiang-Rong Chen, Guang-Fu Ji, Ling-Cang Cai, and Zhuo-Wei Gu Citation: Journal of Applied Physics 115, 124904 (2014); doi: 10.1063/1.4869735 View online: http://dx.doi.org/10.1063/1.4869735 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Elastic, superconducting, and thermodynamic properties of the cubic metallic phase of AlH3 via first-principlescalculations J. Appl. Phys. 114, 114905 (2013); 10.1063/1.4821287 Isotope effect in the superconducting high-pressure simple cubic phase of calcium from first principles J. Appl. Phys. 111, 112604 (2012); 10.1063/1.4726161 Ab initio calculation of lattice dynamics and thermodynamic properties of beryllium J. Appl. Phys. 111, 053503 (2012); 10.1063/1.3688344 Ab initio investigations of phonons and superconductivity in the cubic Laves structures LaAl2 and YAl2 J. Appl. Phys. 111, 033514 (2012); 10.1063/1.3681327 First-principles calculations of phase transition, elastic modulus, and superconductivity under pressure forzirconium J. Appl. Phys. 109, 063514 (2011); 10.1063/1.3556753

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Ab initio studies on phase transition, thermoelastic, superconductingand thermodynamic properties of the compressed cubic phase of AlH3

Yong-Kai Wei,1,2 Ni-Na Ge,1,2 Xiang-Rong Chen,1,3,a) Guang-Fu Ji,2,a) Ling-Cang Cai,2

and Zhuo-Wei Gu2

1College of Physical Science and Technology, Sichuan University, Chengdu 610064, China2National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics,Chinese Academy of Engineering Physics, Mianyang 621900, China3International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China

(Received 17 December 2013; accepted 16 March 2014; published online 27 March 2014)

The phase transition, thermoelastic, lattice dynamic, and thermodynamic properties of the cubic

metallic phase AlH3 were obtained within the density-function perturbation theory. The calculated

elastic modulus and phonon dispersion curves under various pressures at 0 K indicate the cubic phase

is both mechanically and dynamically stable above 73 GPa. The superconducting transition

temperature Tc was calculated using the Allen-Dynes modification of the McMillan formula based on

BCS theory. The calculations show that Tc for the cubic phase AlH3 is 8.5 K (l� ¼ 0:1) at the onset

of this phase (73 GPa), while decreases to 5.7 K at 80 GPa and almost disappears at 110 GPa,

consisting with experimental phenomenon that there was no superconducting transition observed

down to 4 K over a wide pressure range 110–164 GPa. It is found that the soft phonon mode for

branch 1, namely, the lowest acoustic mode, plays a crucial role in elevating the total EPC parameter

k of cubic AlH3. And the evolution of Tc with pressure follows the corresponding change of this soft

mode, i.e. this mode is responsible for the disappearance of Tc in experiments. Meanwhile, the

softening of this lowest acoustic mode originates from the electronic momentum transfer from M to

R point. This phenomenon provides an important insight into why drastic changes in the diffraction

pattern were observed in the pressure range of 63–73 GPa in Goncharenko’s experiments.

Specifically, once finite electronic temperature effects are included, we find that dynamical

instabilities can be removed in the phonon dispersion for P � 63 GPa, rendering the metastability of

this phase in the range of 63–73 GPa, and Tc (15.4 K) becomes remarkably high under the lowest

possible pressure (63 GPa) compared with that of under 73 GPa (8.5 K). Our calculations open the

possibility that finite temperature may allow cubic AlH3 to be dynamically stabilized even for

pressures below 73 GPa. It is reasonable to deduced that if special techniques, such as rapid

decompression, quenching, and annealing, are implemented in experiments, higher Tc can be

observed in hydrides or hydrogen-rich compounds under much lower pressure than ever before.VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869735]

I. INTRODUCTION

The current interest in the development of hydrogen-

rich compounds stems from their potential use as possible

high-temperature superconductors1 and hydrogen-storage

materials.2 Various hydrogen-rich compounds, especially for

group IV hydrides3–5 and some three metal-hydride systems

of MH3 form,6 such as ScH3, YH3, and LaH3, have been

elaborately studied based on this purpose. However, since

the enthalpy difference is not the only criteria to estimate

whether these hydrides form or not, some new predicted

high-Tc H2-rich compounds based on the enthalpy difference

relations, such as SiH4, PbH4, and PtH, may exist decompo-

sition under pressure in experiments.5,7,8 As a promising

hydrogen-storage material with the largest hydrogen content

(10.1 wt. %), and a very important material for its application

as an energetic component in rocket propellants, aluminum

hydride, has also been extensively studied both theoretically

and experimentally in recent years.9–13 Compared with the

new predicted hydrogen-rich compounds mentioned above,

aluminum hydride is a known solid compound at both ambi-

ent conditions and high pressures,14 making it become a per-

fect candidate in the process of exploring superconductive

properties of hydrogen-rich compounds. Furthermore, as a

crucial additive to rocket fuels and high explosives, the com-

prehensive analysis of thermodynamic properties of AlH3,

especially under extreme conditions such as high pressures

and temperatures, is also expected indeed.

Aluminum hydride exists at least seven different poly-

morphs15 and many of them have been extensively studied in

past several years.10,16 However, limited studies have been

conducted on the cubic phase AlH3, which was found first

with the symmetry of Pm�3n by Pickard et al.14 using random

structure searching technique. The studies showed that the

cubic phase AlH3 is energetically favorable above 70 GPa,

and contains two formula units per primitive unit cell with

Al atoms forming a body-centered-cubic array while H

atoms forming linear chains along each Cartesian direction.

In the later X-ray diffraction experiments, to provide further

a)Authors to whom correspondence should be addressed. Electronic

addresses: xrchen@scu.edu.cn and cyfjkf@caep.ac.cn

0021-8979/2014/115(12)/124904/11/$30.00 VC 2014 AIP Publishing LLC115, 124904-1

JOURNAL OF APPLIED PHYSICS 115, 124904 (2014)

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experimental evidence for the superconducting properties of

hydrogen-rich compounds, Goncharenko et al.17 proved the

existence of this phase and presented the first experimental

evidence for a pressure-induced hydrogen-rich metallic state

in the pressure range of 100–165 GPa. This high symmetry

structure is the same as that of the superconductor Nb3Sn (ni-

obium stannide),14 which has important applications where

high magnetic fields exist. It is very interesting that H-H dis-

tance (1.54) in this phase is slightly smaller than the shortest

observed H-H distance in any metal hydride.18 Thus, with

the exception of H2 molecule, this high-pressure cubic phase

of AlH3 exhibits the shortest H-H distance ever measured.17

The striking characters mentioned above enhance our great

interest to have a further study on the properties of the cubic

phase AlH3.

In this work, our main aim is to investigate the phase tran-

sitions, thermoelastic, lattice dynamic, superconducting and

thermodynamic properties of the cubic phase AlH3 using den-

sity functional perturbation theory (DFPT). In Sec. II, we

describe our theoretical methods and the computation details

briefly. In Sec. III, we first examine the phase transitions of

AlH3 in the pressure range of 0–200 GPa according to the dif-

ferences of Gibbs free energy for several interested structures.

Then the phonon dispersion curves, Gr€uneisen parameters and

superconducting properties of the cubic phase AlH3 were

determined under high-pressure regime. Based on the obtained

phonon frequencies, the thermodynamic properties of cubic

AlH3 was deduced within the framework of the quasiharmonic

approximation (QHA). The anharmonic effects were also esti-

mated by using a modification term of the Helmholtz free

energy in the studies of the thermodynamic properties of AlH3.

The main results and conclusions are summarized in Sec. IV.

All of our work are implemented on Quantum-Espresso (QE)

package,19 which is a very useful and credible tool based on

the first-principles theory. Using above methods, we have suc-

cessfully investigated the phase transition, elastic, and thermo-

dynamic properties of several materials.20–23

II. COMPUTATIONAL DETAILS

In the framework of QHA, the Helmholtz free energy of

a system is written as

F V; Tð Þ ¼ Estatic Vð Þ þ Fel V; Tð Þ þ Fzp V; Tð Þ þ Fph V; Tð Þ;(1)

where Estatic Vð Þ is the first-principles zero-temperature

energy of a static lattice at volume V and FelðV; TÞ is the

electronic free energy arising from the electronic thermal

excitations at temperature T and volume V. FelðV; TÞ can be

evaluated via the standard methods of finite-temperature

DFT developed by Mermin (the Fermi-Dirac distribution),24

FelðV; TÞ ¼ EelðV; TÞ � TSelðV; TÞ: (2)

The electronic energy due to the electronic excitations is

given by

EelðV; TÞ ¼ð

n eð Þf eð Þede�ðeF

n eð Þede; (3)

where n eð Þ is the electronic density of states (EDOS) at the

energy eigenvalues e, f is the Fermi distribution function,

and eF is the energy at the Fermi level.

The electronic entropy is calculated by

SelðV; TÞ ¼ �2kB

Xfi ln fi þ ð1� fiÞln 1� fið Þ; (4)

where kB is Boltzmann’s constant.

The term Fzp V; Tð Þ in Eq. (1) is the zero-point motion

energy of the lattice given by

Fzp ¼1

2

Xq;j

�hxj q;Vð Þ; (5)

where xj q;Vð Þ is the phonon frequency of the jth mode of

wave vector q in the first Brillouin zone (BZ).

The last term in Eq. (1) is the phonon free energy due to

lattice vibrations, and it can be expressed as

Fph V; Tð Þ ¼ kBTXq;j

ln 1� exp ��hxj q;Vð Þ=kBT� �� �

: (6)

The calculation of phonon dispersion curves of AlH3 has

been performed within DFPT25,26 using local-density approxi-

mation27 with the parametrization of the Perdew and Zunger.28

A nonlinear core correction to the exchange-correlation energy

function was introduced to generate a norm-conserving pseu-

dopotential29 for Al and H with the valence electrons configu-

ration 3s23p1 and 1s1, respectively. To ensure the convergence

of the free energy and phonon frequencies, we made a careful

test on k and q grids, as well as the kinetic energy cutoff and

smearing parameters. For cubic AlH3, dynamical matrices

were computed at 10 wave (q) vectors using an 4� 4� 4 qgrid in the irreducible wedge of Brillouin zone. The plane

wave cutoff for the wave functions was 60 Ry. The

Monkhorst-Pack (MP) meshed were 16� 16� 16 and the

Fermi-Dirac smearing width was 0.02 Ry.

The geometric mean phonon frequency �w is defined by

ln �w ¼ 1

Nqj

Xqj

ln wqj; (7)

where wqj is the phonon frequency of the branch j at the

wave vector q and Nqj is the number of q points in the sum.

For all volumes, the geometric mean phonon frequency wwas converged to 1 cm�1 with respect to the k mesh and ki-

netic energy cutoff was used.

III. RESULTS AND DISCUSSION

A. Phase transition and thermoelastic properties

At ambient pressure, aluminum hydride exists in the

order of six different crystalline forms, and four of them have

been synthesized in experiments. The earlier high-resolution

synchrotron X-ray diffraction studies30 indicated that a-AlH3

has a rhombohedral structural characterization with the space

group R�3c (No. 167), which was found to be stable up to at

least 35 GPa.31 Using a random searching technique, Pickard

et al.14 found a transition from the low-pressure a-phase to an

124904-2 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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insulating layered structure of space group Pnma at 34 GPa,

then a phase of space group Pm�3n becomes favorable at

73 GPa. In the later high-pressure X-ray diffraction experi-

ment implemented in the diamond anvil cell (DAC),

Goncharenko et al.17 characterized two structure phase transi-

tions of AlH3, and found there was no hydrogen dissociates

up to 164 GPa based on the fact that the absence of Raman

modes associated with molecular hydrogen. This experiment

indicated that there exists Pm�3n phase indeed, and found that

drastic changes in the diffraction pattern were observed in the

pressure range of 63–73 GPa. To have a rigorous identification

of the phase transition pressure of AlH3, the enthalpy differ-

ence of different phases mentioned above in the pressure

range of 0–200 GPa at 0 K is presented in Fig. 1. One can find

the phase transition from R�3c to Pnma structure occurs at

about 34 GPa. Then the Pm�3n phase becomes favorable at the

pressure in the order of 73 GPa. The calculations about the

low-pressure phase transition between R�3c phase and Pnmaphase agrees well with the results obtained by Pickard et al.14

and earlier high-resolution synchrotron X-ray diffraction

experiments,30,31 but conflicts with Goncharenko’s experi-

ments17 in the pressure range of 62–73 GPa. The difference

between our calculations and Goncharenko’s experiments

may, in our opinion, originate from the metastability of cubic

AlH3 near the threshold pressure (63 GPa), which we will dis-

cuss in the following parts. On the other hand, it should be

mentioned that the cubic phase exists indeed in the pressure

range of 80–100 GPa based on the Molodets’s shock wave

experiments,11 in which it is found the conductivity of

shocked alane increases in the range up to 60–75 GPa and is

about 30 (X cm)�1. Specially, the conductivity of the cubic

phase AlH3 achieves approximately 500 (X cm)�1 at

80–90 GPa, and they think the conductivity can be interpreted

in frames of the conception of "dielectric catastrophe" taking

into consideration significant differences between the elec-

tronic states of isolated molecule AlH3 and condensed alane.

These findings inspire us to make us a further study on the

superconducting properties about the cubic phase AlH3, as

can be seen later. In Table I, we give our calculated lattice pa-

rameters and atomic positions of the cubic phase AlH3, com-

bined with other theoretical data and experimental results.

To examine the mechanical stability of the cubic phase

AlH3, the elastic properties are investigated in this work.

The three elastic constants c11, c12, and c44 completely

describe the elastic behavior of a cubic crystal. A more con-

venient set for computations are c44 and two linear combina-

tions K and cs. The bulk modulus

K ¼ c11 þ 2c12ð Þ=3 (8)

is the resistance to deformation by a uniform hydrostatic

pressure; the shear constant

cs ¼ c11 � c12ð Þ=2 (9)

is the resistance to shear deformation across the 110ð Þ plane

in the ½1�10� direction, and c44 is the resistance to shear defor-

mation across the 110ð Þ plane in the ½010� direction. The

bulk modulus K was determined from the equation of state,32

using the Vinet equation.33 We obtained the shear moduli by

straining the cubic lattice at fixed volumes using volume

conserving tetragonal and orthorhombic strains for cs and

c44, respectively, and computing the free energy as a func-

tion of strain. cs was obtained by applying the following iso-

choric strain:

2 ¼d 0 0

0 d 0

0 0 1þ dð Þ�2 � 1

0@

1A; (10)

where d is the magnitude of the strain. Then the strain energy

is

F dð Þ ¼ F 0ð Þ þ 6csVd2 þ O d3ð Þ; (11)

where F 0ð Þ is the free energy of the unstrained system and Vis its volume. Similarly, c44 was calculated from the follow-

ing strain:

2¼0 d 0

d 0 0

0 0 d2= 1� d2ð Þ

0@

1A; (12)

FIG. 1. Enthalpy difference versus pressure for various structures of AlH3,

referenced to the Pnma phase. We also calculated some other low pressure

phases10,12,58 referred in other work of AlH3 to have a comparison with our

results. Below 7 GPa, it can be found the low pressure phases of AlH3 are

competitive within the accuracy of typical DFT calculations, for the reason

that this material is known to be unstable near ambient conditions.59

TABLE I. The lattice constants (A), bulk modulus B (GPa), pressure deriva-

tive B0, and atomic positions of cubic AlH3 under different pressures at 0 K.

The atomic positions are Al 2a (0, 0, 0) and 6c (0.25, 0, 0.5).

P (GPa) a (A) B (GPa) B0

Current work 80 3.1685 343.5 2.6148

110 3.0882 423.1 2.5017

Theory (Ref. 14)a 80 3.169

Experiment (Ref. 17)b 110 3.08

aCalculated using CASTEP code with ultrasoft pseudopotentials by Pickard

et al.bExperimental data measured through X-ray diffraction patterns imple-

mented by Goncharenko et al.

124904-3 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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with the corresponding strain energy

F dð Þ ¼ F 0ð Þ þ 2c44Vd2 þ O d4ð Þ: (13)

The quadratic coefficients of strain energy give the elastic

constants. First order terms due to the initial stress (hydro-

static pressure)34 were eliminated by applying isochoric

strains. Then, the elastic constants c11 and c12 were obtained

from cs and K.

The static elastic constants as functions of pressure are

presented in Fig. 2 and Table II. It is found that c11 varies

largely under the effect of pressure compared with the varia-

tions of c12 and c44, which is because c12 and c44 are less

sensitive to the pressure as compared with c11. In general,

the mechanical stability conditions in the cubic structures

can be expressed in terms of elastic constants as:~C44 > 0; ~C11 > j ~C12j, ~C11þ 2 ~C12 > 0, where ~Cii ¼ Cii�Pði¼ 1;4Þ; and ~C12 ¼ C12þP. The elastic constants of AlH3

given in Table II all satisfy these stability conditions, indicat-

ing the mechanical stable of cubic AlH3. Here, it has to

mentioned that our results for c44 underestimate overall

compared with the calculations obtained by Kim et al.,35 and

the difference tends to be larger with increasing pressure.

This difference may result from the discrepancy of

energy-distortion relations obtained by two different pack-

ages (QE and ABINIT). The calculations on the phonon

band structures can also explain the existence of this discrep-

ancy, for the reason that the frequency of the phonon mode

at X point is negative for pressure values below 72 GPa and

above 106 GPa obtained by ABINIT coeds, while our calcu-

lations and experiments17 indicate it is still dynamically sta-

ble for cubic AlH3 in the pressure range of 110–165 GPa.

In shock wave experimental measurements, the relation

between the experimental adiabatic constants cSij and the cal-

culated isothermal elastic constants cTij is as follows:36,37

cSij ¼ cT

ij þT

qCVkikj; (14)

where ki ¼P

kakcTik, ak is the linear thermal expansion ten-

sor, CV is the specific heat and q is the density. For cubic

crystals, Eq. (14) can be simplified to

cS11 ¼ cT

11 þ D; (15)

cS12 ¼ cT

12 þ D; (16)

where

D ¼ T aKTð Þ2= qCVð Þ ¼ qCVTc2 ¼ TaKTc: (17)

Here a is the thermal expansion coefficient, c is the

Gr€uneisen parameter, and KT is the isothermal bulk modulus.

The Gr€uneisen parameter and thermal expansion coefficient

are obtained based on the phonon-dispersion relations under

certain pressures, which will be discussed later. The calcula-

tions showed that D increases with temperature but decreases

with pressure. The elastic constants of cubic AlH3 as func-

tions of temperature at various pressures are presented in

Fig. 3. The calculated moduli decrease with temperature in a

quite linear manner at all pressures. To further examine the

temperature dependences at a given pressure, we fit the

FIG. 2. Static elastic constants of cubic AlH3 as a function of pressure.

Inset: The trigonal shear elastic constants C44 versus pressure obtained from

a quadratic fit to the energy relation by Kim et al.35

TABLE II. The static elastic constants for cubic-AlH3. All the elastic con-

stants and pressure units are GPa.

V(A3) Pressure K c44 cs c11 c12

31.81 80 348.1 140.2 268.4 705.9 169.2

30.95 90 375.3 151.3 283.1 752.7 186.6

30.17 100 402.6 162.4 297.5 799.2 204.3

29.45 110 429.4 173.0 311.1 844.2 222.0

28.80 120 455.6 183.4 324.0 887.6 239.6

28.20 130 481.5 193.6 337.1 931.0 256.8

27.64 140 507.7 203.9 350.3 974.8 274.2

27.12 150 533.7 213.9 363.1 1017.8 291.6

26.63 160 559.3 223.7 375.7 1060.3 308.9

26.18 170 584.6 233.3 388.0 1101.9 326.0

25.75 180 609.6 242.8 400.0 1142.9 343.0

25.35 190 631.9 252.5 413.7 1183.4 356.1

24.97 200 659.5 261.5 426.5 1228.1 375.2

FIG. 3. The calculated temperature dependence of the elastic moduli for the

cubic phase AlH3, at different pressures from 80 GPa (lowest curve) to

200 GPa (uppermost curve) with 20 GPa interval.

124904-4 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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calculated moduli as a second-order polynomial, and tabu-

lated the fitting parameters in Table III.

B. Superconducting properties and Gr€uneisenparameters

The calculated phonon dispersion curves of the cubic

phase AlH3 along the ½00n�, ½1=2nn�, ½nnn�, and ½nn0� high

symmetry directions under different pressures at 0 K are illus-

trated in Fig. 4(a). The calculations indicate that the cubic

phase AlH3 is dynamically stable when the pressure is above

73 GPa. The most striking feature of phonon band structures

under pressure is the presence of soft phonon modes, includ-

ing the decrease in the phonon frequency of the lowest acous-

tic branch (branch 1) and the dip in the lowest optical branch

(branch 4) at X point. It is worth noted that the soft phonon

mode for branch 1 decreases gradually as the pressure

increases, while almost disappears when the pressure reaches

in the order of 110 GPa. Many earlier studies have shown that

the observation of the soft phonon modes in phonon disper-

sion curves often implies the existence of electron-phonon

interaction, which can make a great contribution to the super-

conducting transition temperature (Tc) of materials. The

changes of the observed soft phonon mode for branch 1 with

elevating pressure may indicate that the electron-phonon inter-

action is quite strong at the onset of this phase but decreases

gradually with increasing pressure. To carry out a rigorous

verification of the potential superconducting properties of the

cubic phase AlH3 that results from the electron-phonon inter-

action, the Allen-Dynes modification of the McMillan for-

mula38,39 based on the Bardeen-Cooper-Schrieffer (BCS)

model, i.e., TC ¼ hxlogi1:20

exp � 1:04 1þkð Þk�l� 1þ0:62kð Þ

h i, is applied to

obtain the superconducting properties of this material. In

this equation, l� denotes the Coulomb pseudopotential, k¼Ð1

0dwk wð Þ is the dimensionless electron-phonon coupling

(EPC) parameter and k wð Þ ¼ 2a2F xð Þ=w, hxlogi is the pho-

non frequency logarithmic average, while the Eliashberg spec-

tral function a2F xð Þ, which can be also obtained via phonon

dispersion experiments, is approximately related to the pho-

non linewidth cqj, owing to electron-phonon scattering40–42

a2F wð Þ ¼ 1

2pNf

Xqj

cqj

wqjd �hw� �hwqjð Þ; (18)

Nf is the electronic density of states per atom and spin at the

Fermi level. The linewidth of a phonon mode j at wave vec-

tor q,cqj, arising from electron-phonon interaction is given

by

cqj ¼ 2pwqj

Xkmn

jgjkn;kþqmj

2d eknð Þd ekþqmð Þ; (19)

where the sum is over the BZ, and ekn is the energy of bands

measured with respect to the Fermi level at point k, and

gjkn;kþqm is the electron-phonon matrix element. This equation

has been found to be highly accurate for all known materials

when k < 1:5,39 and it has successfully characterized the

superconducting properties for many metal-hydride com-

pounds indeed.6 The calculations show that the Tc of cubic

AlH3 is 8.5 K at the onset of this phase (73 GPa) when l�

equals 0.1 (the typical value of l� is 0.07–0.13 but l� ¼ 0:1for hydrogen-rich materials),1 while decreases to 5.7 K at

80 GPa and almost disappears under pressure in the order of

110 GPa [Fig. 5(a)]. Our results agree well with the experi-

mental phenomenon that there was no superconducting tran-

sition observed down to 4 K over a wide pressure range

110–164 GPa.17 On the other hand, to make a quantitative

analysis on the effects of the soft phonon modes to the super-

conducting transition temperature Tc, Fig. 5(b) gives the cal-

culated spectral functions a2FðxÞ obtained at 73 GPa and the

corresponding integrated EPC parameter k as a function of

pressure. The integrated k among the acoustic modes below

400 cm�1 constitutes almost 79% of the total EPC parameter.

The lowest optical modes from 450 cm�1 to 750 cm�1 con-

tribute in the order of 7%, while the remaining 14% of the

total EPC parameter is derived from Al-H-Al torsional and

bending vibrations, as well as Al-H stretching vibrations. It

can be deduced that the soft phonon mode for branch 1 at

low frequencies plays a crucial role in elevating the total

EPC parameter k of the cubic phase AlH3 indeed. However,

as the pressure increases, the soft phonon mode for branch 1

gradually disappears, resulting in the decrease of

electron-phonon interaction, as well as Tc.

It is worth mentioned that our zero-temperature calcula-

tions demonstrate that cubic AlH3 is unstable below 73 GPa,

preceded by the onset of phonon softening. However, as it has

been demonstrated previously,43 such instabilities can often

be removed when a finite electronic temperature Te is taken

TABLE III. Temperature dependence of elastic moduli of cubic phase AlH3, cSij ¼ Aþ BT þ CT2, with temperature up to 2000 K. A is given in units of GPa,

Bis given in GPa/K, and C is given in �10�6GPa/K2.

Pressure (GPa)

80 100 120 140 160 180 200

cS11

A 780.6 886.6 986.7 1088.1 1221.4 1280.9 1378.5

B �0.0110 �0.0107 �0.0079 �0.0075 �0.0054 �0.0035 �0.0055

C �7.035 �7.097 �8.055 �8.109 �8.153 �8.181 �8.451

cS12

A 197.2 239.2 279.2 320.3 354.8 400.8 442.6

B �0.0042 �0.0038 �0.0034 �0.0030 �0.0028 �0.0026 �0.0025

C �2.603 �3.093 �3.158 �3.342 �3.523 �3.919 �3.643

cS44

A 158.1 183.3 206.7 230.1 252.5 273.8 295.8

B �0.0028 �0.0026 �0.0020 �0.0018 �0.0018 �0.0008 �0.0011

C �1.572 �1.712 �1.834 �1.838 �1.779 �1.941 �1.967

124904-5 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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into account. It has considerable significance to study this phe-

nomenon on cubic AlH3 for the reason that as pressure

decreases, Tc presents an exponent increase based on our

above calculations. For the cubic phase AlH3, once

finite-temperature effects are included, we find that dynamical

instabilities in the phonon dispersion disappear [Fig. 6(a)]

under pressures above 63 GPa. For lower pressures, imaginary

frequencies reemerge at X point, rendering cubic AlH3

dynamically unstable. In fact, the softening of the lowest

acoustic branch near X point is closely related with the elec-

tronic momentum transfer from M (0, 0.5, 0.5) to R (0.5, 0.5,

0.5), since the Fermi-surface nesting of AlH3 under

high-pressure region is dominant with a significant nesting

vector in the R-M direction,17 whose transferred momentum

is X �(0.5, 0, 0). According to the calculated band structures

of this phase under pressures [Fig. 6(b)], the minimum of the

conduction band lies at R point and the highest point of the

valence bands lies at M. Thus, this momentum transfer corre-

sponds to an electron-phonon coupling, showing a responsibil-

ity for both the structure instability and the change of the soft

phonon mode (branch 1) with pressure in this phase, as well

as Tc. This complex phenomenon provides an important

insight into why drastic changes in the diffraction pattern

were observed in the pressure range of 63–73 GPa in

Goncharenko’s experiments. It can be also deduced that the

change of the soft mode with pressure observed in cubic AlH3

is the consequence of a complicated balance between the

entire phonon spectrum and soft phonon modes under differ-

ent pressures. Our calculations open the possibility that finite

temperature may allow cubic AlH3 to be dynamically stabi-

lized even for pressures below 73 GPa. Interestingly, the Tc

[Fig. 5(a)] at the lowest possible pressure (63 GPa) at which

cubic AlH3 is still dynamically stabilized is remarkably high,

namely, around 15.4 K (l� ¼ 0:07), based on our calculations.

This low pressure puts superconducting AlH3 into much

closer experimental reach than comparable hydrogen-rich

superconductors. It should be noted, however, that due to

metastability near the threshold pressure (63 GPa), it might be

necessary in experiments to approach this state via a rapid

decompression (quenching or annealing) technique.

For the calculation of the Gr€uneisen parameters

c ¼ �d ln x=d ln V, we calculate the phonon-dispersion

FIG. 4. (a) Phonon-dispersion curves

of the cubic phase AlH3 under different

pressures. It can be found the soft pho-

non mode for branch 1 gradually

decreases with increasing pressure, and

disappears when the pressure is above

110 GPa. (b) Phonon Gr€uneisen disper-

sion relations of the cubic phase AlH3

under the pressure of 80 GPa. The red

and blue dotted lines represent the

mode Gr€uneisen parameters for branch

1 and branch 4, respectively.

124904-6 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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relations at different pressures and approximate the deriva-

tive by the ratio of the differences, c � �D ln x=D ln V.

Under each pressure, the phonon Gr€uneisen dispersion rela-

tions for cubic AlH3 are obtained by the phonon-dispersion

relations at the volume of this pressure and for a 0:5% com-

pressed crystal. A compression by 1.2% yields similar

results, although some modes show deviations in the

Gr€uneisen parameter of up to 0.15. In Fig. 4(b), we showed

the calculated phonon Gr€uneisen dispersion relations of the

cubic phase AlH3 under the pressure of 80 GPa at 0 K. For

branch 1 mentioned above, the Gr€uneisen parameter is

between 1.21 and 5.98, and for branch 4 it lies between 2.45

and 6.69. Although branch 4 is the tallest Gr€uneisen mode

among all the dispersion curves, the phonon Gr€uneisen mode

for branch 1 changes most drastically in the first

Brillouin-zone (alone G-X-R direction). The reason for this

large difference results from the anisotropic properties of the

crystal. That means branch 1 is responsible for the dynamical

stability of the cubic phase AlH3. In our calculation of pres-

sure dependence of the phonon dispersion curves, as is given

in Fig. 4(a), the branch 1 exists image frequency at X point

below 73 GPa indeed. For the whole phonon Gr€uneisen dis-

persion modes of cubic AlH3 at 80 GPa, the average value of

the integration in the Brillouin-zone is c ¼ 2:172. The calcu-

lated Gr€uneisen parameter in this method is in reasonable

agreement with shock wave experiments implemented by

Molodets et al.,11 which gets c � þ2:38 under 80 GPa for

the cubic phase AlH3 (T unknown). Our result is a little

smaller compared with the experimental data. However, the

difference is reasonable since the shock wave experiments

are implemented under extreme conditions such as high pres-

sure (80–100 GPa) and high temperature (1500–2000 K).

Generally, Gr€uneisen parameter c is related to the anhar-

monic effects in the vibrating lattice and can be used to pre-

dict anharmonic properties of a material, such as the thermal

expansion coefficient and temperature dependence of phonon

frequencies and line-widths. In the following studies on the

thermodynamic properties of cubic AlH3, we will give a

detailed estimation of the anharmonic effects based on the

obtained Gr€uneisen parameters and Debye temperatures

FIG. 5. (a) Pressure dependence of superconducting transition temperature

Tc of the cubic phase AlH3 for different parameter l�. Our calculations is in

agreement with experiments implemented by Goncharenko et al.,17 who

reported that there was no superconducting transition observed down to 4 K

in the pressure range of 110–165 GPa. The solid lines represent the

pressure-dependence of Tc without considering the effects of electronic tem-

perature Te, while the dashed lines correspond to the variety of Tc with pres-

sure when a finite electronic temperature Te is taken into account. (b) The

Eliashberg phonon spectral function a2FðxÞ and the electron-phonon inte-

gral k xð Þ of the cubic phase AlH3 at 73 GPa.

FIG. 6. (a) Phonon dispersion curves at 63 GPa with zero electronic temper-

ature shown as red dashed lines (only the lowest acoustic branches are

shown, one of which is destabilized near X point), and a full set of disper-

sion curves where 0.015 Hartree electronic temperature has been added

(solid lines). The corresponding physical temperature can be orders of mag-

nitude smaller.35 (b) Electronic band structure of cubic AlH3 under the pres-

sure of 63 GPa. The horizontal line shows the location of Fermi level.

124904-7 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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(Sec. III C). The Gr€uneisen parameter c of cubic AlH3

obtained above is substantially larger than that of trigonal

structure a-AlH3 (ca ¼ 0:76) (Ref. 11) obtained by shock

wave experiments. Larger c is normally related to increased

anharmonicity of the particular normal vibration.44 It can

also be a consequence of a soft mode behavior when the sys-

tem is approaching (or departing from) a structural

instability.45,46

In experiments, the obtainment of the mode Gr€uneisen

parameter based on phonon frequency under certain pressure

is usually expressed as47

cq;j ¼ � dlnxq;j=dlnV� �

¼ B=xq;j

� �dxq;j=dp� �

: (20)

Combined with the calculated average value of Gr€uneisen

parameter hci ¼ 2:172 obtained above, and the correspond-

ing dp � 1:754 for a 0:5% compressed crystal, we can get

the bulk modulus is about 342.24 GPa under 80 GPa, which

agrees well with the results determined by equation

of state, indicating the correctness of elastic properties in

Sec. III A.

C. Thermodynamic properties

The P� V equation of state (EOS) isotherms of cubic

AlH3 are determined by fitting the calculated Helmholtz free

energy F at certain T, given by Eq. (1), to Vinet EOS33 men-

tioned above. Based on the obtained EOS, the volumetric

thermal expansion coefficient is defined as

aV ¼1

V

@V

@T

� P

; (21)

while the linear thermal expansion coefficients aa for cubic

crystal can be given as47,48

aa Tð Þ ¼ 1

3B0V

Xq;j

cq;jcVq;j Tð Þ; (22)

where cVq;j Tð Þ is the contribution to CV of the j, q phonon

mode at a certain T. Fig. 7 shows the aa of cubic AlH3 as a

function of T under different pressures. At a given pressure,

aa increases exponentially at low temperatures and gradually

approaches a linear increase at high temperatures. As the

pressure increases, the increase of aa with temperature

becomes smaller, especially at high temperatures. However,

aa decreases drastically with the increase of pressure at a

given temperature. When the temperature is above 1000 K,

the linear thermal expansion coefficients aa of 180 GPa is

just a little larger than that of 200 GPa, and the curves of

160, 180, and 200 GPa seem to be parallel to each others,

which means that the temperature dependence of aa is very

small at high pressure.

Diagonalizing the dynamical matrices at a large number

of wave vectors, we can obtained the phonon density of

states g xð Þ and can evaluate thermodynamic function, e.g.,

heat capacity CV ¼ �T @2Fvib=@T2� �

V , as a function of

temperature49

CV Tð Þ ¼ kB

ðxmax

0

�hxkBT

� 2 exp �hxkBT

�

exp �hxkBT � 1 �h i2

g xð Þdx: (23)

In practice, the integration over the phonon frequency,

required to evaluate the above thermodynamic quantities, is

transformed into the summation over the phonon eigen-

states.50 This is because these thermodynamic quantities

depend on q and j only through the frequency x ¼ xq;j. Due

to anharmonicity, the heat capacity at constant pressure, Cp,

is different from the heat capacity at constant volume, CV .

The former, which is what experiments determine directly, is

proportional to T at high temperature, while the latter goes to

a constant which is given by the classical equipartition law:

CV � 3NkB, where N is the number of atoms in the system.

The relation between CP and CV is51

CP � CV ¼ a2V Tð ÞBVT; (24)

where V is the unit cell volume, aV Tð Þ the volumetric ther-

mal expansion and B the bulk modulus. The calculated heat

FIG. 7. Linear thermal expansion coefficient aa of cubic AlH3 as a function

of temperature, at different pressures.

FIG. 8. Calculated temperature dependence of heat capacity of cubic AlH3

at constant pressure (CP-dashed lines) and at constant volume (CV-solid

lines) under different pressures. Inset: Calculated CP for a�AlH3 in QHA

method compared with experimental results.16,60,61

124904-8 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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capacity at constant P, CP, compared with the heat capacity

at constant volume, CV , as a function of T is shown in Fig. 8,

for different pressures. Since the scarcity of experimental

data for cubic AlH3, to examine the correctness of the calcu-

lation using QHA theory, we also compute the heat capacity

at constant CP for a-AlH3 and compare it with experimental

data. It can be seen the results agree well with experiments

in low pressure range. As for the P dependence of CP and

CV , we note that at very low temperatures (below 100 K) CP

and CV is almost insensitive to P, within the considered pres-

sure range. Above this temperature, CP and CV is found to

decrease with increasing P, respectively. However at high

temperatures, CP increases monotonously with the tempera-

ture, while CV approaches about 96 J K�1 mol�1.

It is known that phonon frequencies depend on V in the

QHA calculations, while intrinsic anharmonic effects aris-

ing from phonon-phonon interaction are neglected. These

effects become more important at elevated T, and, hence,

QHA becomes increasingly less adequate at high tempera-

tures. However, anharmonic effects decrease with increas-

ing pressure.52 Indeed, the harmonic terms in the expansion

of the Helmholtz free energy in terms of volume V is

expected to become increasingly predominant with increas-

ing pressure. Thus, the QHA generally works well over a

wide temperature range at elevated pressure. To estimate

the anharmonic free energy FA, we followed the approach

of Wallace53 who showed that the anharmonic part of free

energy can be written in the form of an expansion in terms

of powers of T,

FA ¼ A2T2 þ A0 þ A�2T�2 þ : (25)

Then Eq. (1) can be written as

F V; Tð Þ ¼ Estatic Vð Þ þ Fzp V; Tð Þ þ Fph V; Tð Þ þ FA: (26)

Here, the last term FA denotes the anharmonic free energy.

Experiments for very different crystals showed that

there is an empirical relation between the average Gr€uneisen

parameter hci and A2, which is given per atom by

A2 ¼3kB

HH10:0078hci � 0:0154ð Þ: (27)

Here, HH1 is the high-temperature harmonic Debye temper-

ature defined by

HH1 ¼�h

kB

5

3hx2i

� 1 2 ;=

(28)

where hx2i is the average squared harmonic phonon fre-

quency. In the Debye theory, the average Gr€uneisen parame-

ter hci can be also calculated by

hci ¼ � d ln HH1d ln V

: (29)

For the cubic phase AlH3, the harmonic Debye temperature

HH1 and the average Gr€uneisen parameter hci obtained by

Debye methods are list in Table IV. We can see the value of

average Gr€uneisen parameter based on Eq. (29) under

80 GPa is also in good agreement with the calculations

directly from the phonon-dispersion relations and shock

wave experiments (see above).

IV. SUMMARY AND CONCLUSIONS

In summary, we have presented a detailed ab initio study

of the phase transition, thermoelastic, lattice dynamic and

thermodynamic properties of the cubic phase AlH3 in quasi-

harmonic approximation (QHA) theory. The superconducting

transition temperature TC under different pressures is also cal-

culated using the Allen-Dynes modification of the McMillan

formula based on the obtained phonon dispersion curves. The

phase transition from R�3c to Pnma phase occurs at about

34 GPa, while the Pm�3n phase becomes favorable when the

pressure is above 73 GPa. The elasticity of the cubic phase

AlH3 is calculated from first principles in the pressure range

of 73–200 GPa and for temperatures up to 2000 K. The super-

conducting transition temperature TC is 8.5 K at the onset of

this phase when l� equals 0.1, while decreases to 5.7 K at

80 GPa and almost disappears in the order of 110 GPa. The

results consist well with the experimental phenomenon that

there was no superconducting transition observed down to 4 K

over a wide pressure range 110–164 GPa. Through the calcu-

lated total EPC parameter k, it is found the branch 1 of the

soft phonon modes (branch 1 and branch 4) observed in pho-

non band structures makes a crucial contribution to the super-

conducting transition temperature TC at the onset of this cubic

phase. And the change of this soft phonon mode with pressure

can explain experimental phenomenon clearly that why there

was no superconducting transition observed down to 4 K for

TABLE IV. Calculated harmonic Debye temperature HH1 (K), average Gr€uneisen parameter hci at different temperatures, in the considered pressure range

(73200 GPa) of this study.

Pressure (GPa)

T (K) 80 100 120 140 160 180 200

HH1 0 1843.4 1965.1 2070.1 2169.2 2257.6 2341.1 2421.1

1000 1824.3 1947.9 2055.7 2155.9 2246.1 2330.5 2411.0

2000 1786.9 1912.8 2024.9 2126.8 2219.8 2305.9 2387.0

hci 0 2.1734 2.1314 2.0318 1.9770 1.9317 1.8902 1.8524

1000 2.1847 2.1054 2.0412 1.9846 1.9373 1.8958 1.8581

2000 2.2112 2.1262 2.0582 2.0016 1.9506 1.9072 1.8694

124904-9 Wei et al. J. Appl. Phys. 115, 124904 (2014)

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132.174.255.116 On: Fri, 19 Dec 2014 06:07:35

pressures above 110 GPa. On the other hand, the electronic

momentum transfer from M to R point results in the softening

of the lowest acoustic branch at X point. The calculations also

indicate that the cubic phase AlH3 becomes dynamical stabi-

lized again in the pressure range 63–73 GPa when a finite

electronic temperature is considered. This provides an impor-

tant insight into why drastic changes in the diffraction pattern

were observed under 63 GPa in Goncharenko’s experiments.

According to the experimental phase transition studies imple-

mented on many other hydrides at room temperature, such as

ErH3, GdH3, HoH3, LuH3, SmH3, etc.,54–56 the similar phe-

nomenon of structure (that after transition) hysteresis exists

indeed below the transition point during the decrease of pres-

sure. It is reasonable to deduced that higher TC (15.4 K) of the

cubic phase AlH3 can be realized under much lower pressure

(63 GPa) if special techniques (e.g., rapid decompression,

quenching, or annealing) are implemented in experiments.

Although, as a covalent hydride, the polarizability (Ref. 57) of

AlH3 is much less than that of many other H2-rich com-

pounds, which may result in lower superconducting transition

temperature, our studies on AlH3 will provide a comparison in

exploring other high-TC hydrogen-rich compounds. Under

each pressure, the phonon Gr€uneisen dispersion relations for

cubic AlH3 are obtained by the phonon-dispersion relations at

the volume of this pressure and for a 0:5% compressed crys-

tal. The results of Gr€uneisen parameter c calculated by this

methods (c ¼ 2:172) is consonant with shock wave experi-

ments (c � þ2:38), as well as the results obtained in Debye

theory (c ¼ 2:1734) referred in Sec. III C. The calculations

also indicate that the soft phonon mode for branch 1 is respon-

sible for the dynamical stability of the cubic phase AlH3, since

the Gr€uneisen parameter of this mode changes most drasti-

cally in the first Brillouin-zone. The linear thermal expansion

coefficients aa, the heat capacity at constant pressure, Cp, as

well as the heat capacity at constant volume, CV , are investi-

gated under different pressures in the temperature range of

0–2000 K. Through the calculations based on the empirical

relation observed in experiments, it can be found the anhar-

monic effects of the cubic phase AlH3 under pressure is very

small. Considering the promising implication of

hydrogen-rich compounds, and the scarcity of systematic ex-

perimental studies, especially for superconducting properties,

on these materials, we hope our studies on this metal hydride

material will provide a useful comparison in both theoretical

and experimental researches of H2-rich compounds.

ACKNOWLEDGMENTS

The authors would like to thank the support by the

National Natural Science Foundation of China under Grant

Nos. 11174214 and 11204192, the National Key Laboratory

Fund for Shock Wave and Detonation Physics Research of the

China Academy of Engineering Physics under Grant No. 2012-

Zhuan-08, the Science and Technology Development

Foundation of China Academy of Engineering Physics under

Grant Nos. 2012A0201007 and 2013B0101002, the Defense

Industrial Technology Development Program of China under

Grant No. B1520110002, and the National Basic Research

Program of China under Grant Nos. 2010CB731600 and

2011CB808201. We also acknowledge the support for the com-

putational resources by the State Key Laboratory of Polymer

Materials Engineering of China in Sichuan University.

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