Pis’ma v ZhETF, vol. 109, iss. 10, pp. 677 - 678

May 25

Comparative study on interatomic force constants and elastic

properties of zinc-blende AlN, AlP and AlAs

H. Wang¹⁾, Q. Tan, X. Zeng

Department of Physics and Electronic Information Engineering, Xiangnan University,

423000 Chenzhou, The People’s Republic of China

Submitted 25 February 2019

Resubmitted 25 February 2019

Accepted 21

March 2019

DOI: 10.1134/S0370274X19100060

Almost all epitaxial AlN, AlP and AlAs films are

Oppenheimer (BO) total energy surface of the system

expected to contain some residual strain. A knowl-

(electrons plus clamped ions) where C^ion and C^elec are

edge of their elastic constants and strain deforma-

the ionic contribution and electronic contribution to the

tion potentials is indispensable. Although there are

force constants, respectively for other variables, see [8].

many theoretical [1-3] and experimental [4, 5] results,

For this reason, IFC offer a convenient way of storing

there is a lack of systematic research on their elastic

the information contained in the dynamical matrix at

constants and behaviors under hydrostatic and uniax-

any q into a few independent parameters. Therefore, it

ial stress. The purpose of this study is to study the

is necessary to fully describe the ion motion in DFPT.

above properties of these compounds by pseudopotential

Fourier analysis is used to calculate and tabulate a set of

method, and the linear response by density functional

force constant matrices on uniform meshes in reciprocal

theory (DFT) and density functional perturbation the-

space. Real space force constants can be easily obtained:

ory (DFPT). The norm-conserving non-local Troulliers-

1∑

Martins pseudopotentials [6] is employed. The Kohn-

C_kα,k′_β(R) =

e^iqR

C_kα,k′_β(q),

(2)

Sham orbits are expanded in plane waves basis set. The

Troullier and Martins programs generate soft core pseu-

Here N is the number of unit cells in the crystal. Af-

ter calculating the real space constants by this method,

dopotential. The theoretical calculation is performed by

using the local density approximation of the exchange-

the reciprocal-space dynamic matrix can be obtained

by inverse Fourier transform at any q point of Brillouin

correlation Hamiltonian as implemented in abinit pack-

age [7].

region.

In this calculation, Al

(3s²3p¹), N

(2s²2p³),

The dynamical matrices have been calculated on a

P (3s²3p³), and As (4s²4p³) shells are used as valence

(8 × 8 × 8) reciprocal space face centered cubic (FCC)

band elections. The ground state equilibrium volumes

grid. Fourier deconvolution on its mesh yields real-space

of zinc-blende AlN, AlP and AlAs are determined

interatomic force constants up to the ninth neighbor

by calculating the total energy of each primitive unit

shell. This process is equivalent to calculate real space

force constant using the FCC supercell, which linear size

cell as a function of V. The calculated energy volume

data are fitted with Monahan’s equation of state. The

is four times that of the primitive zinc-blende cell, so it

contains 128 atoms.

obtained structural parameters are

4.352Å,

5.442Å

and 5.613Å respectively.

As seen from Fig. 1, in general, the decay of local

The interactomic force constants (IFC) describing

interaction for cation-cation, cation-anion, and anion-

the atomic interactions in a crystalline solid are defined

anion is faster than the total interaction decay. In the

in real space as [8]:

fourth neighbor, the local part of each species pair tends

to zero, while the total part of cation-cation, cation-

∂²E

C_kα,k′_β(a, b) =

anion and anion-anion interaction of AlAs tends to zero

∂τ^akα∂τ^b

k^′β

only in the eighth neighbor, and the rest in the ninth

= C^ionkα,k′_β(a, b) + C^eleckα,k′_β(a, b).

(1)

neighbor has not yet tended to zero.

The elastic constants of solids provide interesting in-

Here, τ^akα is the displacement vector of k-th atom in

formation about their mechanical and dynamic proper-

the a-th primitive cell along α axis. E is the Born-

ties and provide a link between the dynamics and me-

¹⁾e-mail: whycs@163.com

chanical behavior of crystals. The calculated elastic con-

Письма в ЖЭТФ том 109 вып. 9 - 10

2019

677

678

H. Wang, Q. Tan, X. Zeng

ativity between anions and cations decreases, and the

binding force between ions and ions decreases, which

result in elastic constants of the compounds decrease

in turn. Moreover, the lattice constants of AlN are ob-

viously smaller than those of AlP and AlAs, but the

lattice constants of AlP and AlAs are almost the same,

furthermore, the difference of internal nuclear charge

is not taken into account in the selection of P and As

pseudopotentials, and the distribution of valence charge

density is not very different, so the elastic constants of

AlN are obviously higher than those of AlP and AlAs,

and the latter two values are not very different.

In order to better illustrate the elastic behavior of

these crystals, we schematically drew the young’s modu-

lus surface of these materials. Although the zinc-blende

structure has high symmetry, the elastic behavior of the

whole crystal cannot be represented by a single surface.

Following formula [9], Young’s modulus of three com-

pound semiconductors (101) is calculated.

(

)

-2

(l²¹l²² + l²²l²³ + l²³l²¹).

C₁₁

C₁₂

C₄₄

(3)

Here, l₁, l₂ and l₃ are cosines of azimuth. Young’s

moduli are not isotropic in the cubic system. The varia-

tion with direction depends on (l²¹l²²+l²²l²³+l²¹l²³), and the

quantity is zero in 〈100〉 direction but has a maximum

value 1/3 in the 〈111〉 direction. The sagittal diameter

is directly proportional to the surface of Young’s mod-

ulus, and it is a circular cube in the central depression

of each surface.

Full text of the paper is published in JETP Letters

journal. DOI: 10.1134/S0021364019100035

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Письма в ЖЭТФ том 109 вып. 9 - 10

2019