A first-principles study on the structural, lattice dynamical and thermodynamic properties of beryllium chalcogenides

Physica B 406 (2011) 4666–4670

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A first-principles study on the structural, lattice dynamical and thermodynamic properties of beryllium chalcogenides Xudong Zhang n, Caihong Ying, Guimei Shi, Zhijie Li School of Science, Shenyang University of Technology, Shenyang 110870, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 March 2011 Received in revised form 19 September 2011 Accepted 20 September 2011 Available online 22 September 2011

First-principles calculations, which are based on the plane-wave pseudopotential approach to the density functional theory and the density functional perturbation theory within the local density approximation, have been performed to investigate the structural, lattice dynamical and thermodynamic properties of zinc blende (B3) structure beryllium chalcogenides: BeS, BeSe and BeTe. The results of ground-state parameters and phonon dispersion are compared and contrasted with the experimental and theoretical data of previous literature. The phonon frequencies at the zone-center are analyzed. We also used the phonon density of states and quasiharmonic approximation to calculate and predict some thermodynamic properties such as entropy, heat capacity, internal energy and free energy of the B3 phase beryllium chalcogenides. & 2011 Elsevier B.V. All rights reserved.

Keywords: First-principles calculations Beryllium chalcogenides Lattice dynamics Thermodynamic properties

1. Introduction Because they can offer a wide range of physical properties, alkaline earth chalcogenides (AECs, AX: A ¼Be, Mg, Ca, Sr, Ba; X¼O, S, Se, Te) have attracted special attention both scientific and technological in recent years. AECs can exhibit the properties of the semiconductor with large valence band width and can also exhibit the properties of classical insulators with wide band gap. These properties make them have the promising use for luminescent devices and microelectronics in a wide range. The beryllium chalcogenides belong to the AECs family. BeS, BeSe and BeTe differ from BeO (crystallize in wurtzite structure). BeS, BeSe and BeTe are similar to the boron compounds BN, BP and BAs having the same crystal structure. At ambient conditions, BeS, BeSe and BeTe can crystallize in cubic zinc blende structure. In previous studies, literature exhibits their strong covalent character, because of their short bond length, chemical bonding and large hardness [1–4]. Because these materials are important for blue–green laser diodes and light-emitting diodes, they can also be applied in the area of luminescent devices [5–8]. These compounds are potentially good materials for technological applications. Over the last few years, researches have done a lot of investigations on the physics properties of the beryllium chalcogenides. But because they have very high toxic nature, only a few experimental investigations [8–15] have been performed on the

n

Corresponding author. Tel.: þ86 24 23971846; fax: þ86 24 25496502. E-mail address: [email protected] (X. Zhang).

0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.09.056

beryllium chalcogenides. On the theoretical side, the ab initio calculations have been extensively used to obtain the structural [12,16–23], electronic [21–25], elastic [21,22,25,26], lattice dynamical [4,27–30], thermodynamic [31,32], optical [23,33,34] and their alloy [35–39] properties for beryllium chalcogenides. The study of the thermodynamic properties of materials is important to extend our knowledge on their physical properties. To the best of our knowledge, there are no theoretical or experimental works exploring the Helmholtz free energy DF, the internal energy DE and the entropy S. Thus, the aim of this work is to calculate the phonon dispersions and complete the data of the thermodynamic properties. The paper is organized as follows. In Section 1, we introduced the research progress briefly. In Section 2, we describe the computational method used in this work. We present our results and discussions in Section 3. Finally, a summary of the work is given in Section 4.

2. Computational methods The ABINIT code [40–42], which is based on the plane-wave pseudopotential approach in the framework of the density functional theory (DFT) and the density functional perturbation theory (DFPT), has been applied to calculate the structural, dynamical and thermodynamic properties of zinc blende phase beryllium chalcogenides. The exchange-correlation term has been considered within the local density approximation (LDA). The normconserving local density approximation pseudopotentials of Be, S, Se and Te are generated in the scheme of Troullier and Martins

X. Zhang et al. / Physica B 406 (2011) 4666–4670

[43], which come from the ABINIT web site. Troullier and Martins pseudopotential describes the interaction between the nuclei core electrons and valence electrons. Per primitive unit cell contains two atoms in the ab initio calculations. To obtain the ground-state parameters, the structures of B3 phase BeS, BeSe and BeTe are firstly optimized using a Broyden–Fletcher–Goldfarb–Shanno (BFGS) procedure [44]. An 8  8  8 Monkhorst–Pack [45] grid is chosen for k-grid samplings in the Brillouin zone. The plane-wave kinetic energy cutoff of 30 hartree is set to guarantee the total energy errors within 0.0001 hartree in all calculations. Convergence tests prove that the Brillouin zone sampling and the kinetic energy cutoff are reliable to guarantee excellent convergence. Phonon frequencies are calculated using the DFPT linear-response method [46,47]. The thermodynamic properties are calculated using the phonon density of states and quasiharmonic approximation.

Table 1 ˚ bulk modulus (B0 in GPa) and its first pressure Calculated lattice constant (a in A), derivative (B00 ) for beryllium chalcogenides are compared to other theoretical calculations and experiments.

BeS Present study Other calculation

Exp.

3. Results and discussions 3.1. Structural parameters The equilibrium lattice parameters a0, bulk modulus B0 and their pressure derivatives of bulk modulus B00 were calculated by means of fitting the data of energy vs. volume to the Vinet equation of state (EOS). We summarized our results and the experimental and other theoretical values in Table 1. Our calculated results of lattice parameters for BeS, BeSe and BeTe are ˚ respectively. These results accord 4.815, 5.126 and 5.561 A, well with the experimental ones of 4.870, 5.137 and 5.617 A˚ [12,13]. The small underestimation in the equilibrium lattice parameter is a common feature with LDA calculations. In Table 1, the results for the bulk modulus and their pressure derivatives are also in good agreement with the other works. The calculated bulk modulus decreases from BeS to BeTe, suggesting that the compressibility increases from BeS to BeTe. We note that the cation atoms are the same in the three compounds, the size of anion atoms is different. The different size of the anion atoms could be responsible for the lattice constant increasing from BeS to BeTe. 3.2. Dynamical properties The calculated phonon dispersion curves and the total density of phonon states (PHDOS) along the principal symmetry direction of the Brillouin zone (BZ) are displayed in Figs. 1–3. The calculated phonon frequencies at the high-symmetry points G, X and L for BeS, BeSe and BeTe are listed in Table 2. They are compared to the experimental data and previous theoretical results. The transverse optical (TO) and longitudinal optical (LO) phonon modes’ frequencies at the zone center (G) agree well with the Raman measurement. TO (G) modes correspond to 502 (501) and 461 (461) for BeSe and BeTe, respectively. LO (G) modes correspond to 564 (579) and 502 (493) for BeSe and BeTe, respectively. The phonon frequencies are also in agreement with theoretical results [4,24]. The shapes of the phonon dispersion spectra of BeX (X¼S, Se and Te) compounds are similar to each other. The transverse optical (TO) phonon modes of BeX show flatness along the G-L directions and a relatively sharp peak in the PHDOS. The transverse optical (TO) phonon modes of BeX show flatness along the W-L directions. Along the G-L symmetry direction, the LO branch shows a downward, flat and upward dispersion for BeS, BeSe and BeTe, respectively, which induces a weak peak in the PHDOS. The LA branches for BeTe show strongly flat, giving a pronounced peak in the corresponding PHDOS. The LO and TO branches are clearly separated in all materials. The separation

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BeSe Present study Other calculation

Exp. BeTe Present study Other calculation

Exp.

˚ Lattice constant (A)

B0 (GPa)

B00

4.815 4.811a 4.800b 4.878b 4.745c 4.839d 4.887e 4.762f 4.773g 4.81h 4.82i 4.745j 4.870k

103 99.8a 102b 93b 116c 103d 92e 94.2f 102g 93h 113i 116j 105k

3.668 3.64a 3.713b 3.524b

5.126 5.085b 5.179b 5.037c 5.137d 5.178e

3.546 3.500b 3.500b

5.13h 5.037j 84(5.182)l 5.137m

90 84b 75b 98c 82d 75e 77f 80h 99j 92(82)l 92m

5.561 5.557b 5.531c 5.638d 5.663e 5.556f 5.58h 5.531j 5.617m

68 64b 70c 60d 58e 55f 60h 71j 67m

3.995 3.914b

3.700e 3.20f 3.70g 3.34h 3.99i 3.22j 3.500k

4.02e 3.88f 3.56h 3.106j 4.000m

6.210e 3.56f 3.72h 3.377j 4.000m

a

Ref. [32]. Ref. [34]. Ref. [17]. d Ref. [18]. e Ref. [23]. f Ref. [48]. g Ref. [3]. h Ref. [4]. i Ref. [49]. j Ref. [1]. k Ref. [13]. l Ref. [50]. m Ref. [12]. b c

between the optical and acoustic regions increases across BeTe, BeSe and BeS. For the LO phonon modes, the top of the bands are not located at the zone center, but at the X point. The LO–TO splitting at the zone-center is 94 cm  1, 62 cm  1 and 32 cm  1 for BeS, BeSe and BeTe, respectively. These quantities are in good agreement with the previous work [24]. It can be seen from Figs. 1–3 that the mass difference between the cation Be and the anions S, Se, Te significantly affects the shapes of the dispersion curves and the corresponding PHDOS. So it is clear that the TO branches and LO branches have relatively flat dispersion. You can see that a clear gap appears between acoustic and optical branches for each compound. No imaginary phonon frequency is observed in the whole BZ as suggested by the phonon dispersion curves and PHDOS in Figs. 1–3, indicating the dynamical stability of the B3 phase beryllium chalcogenides.

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700

energy DE, entropy S and constant-volume specific heat Cv, at zero pressure of BeX by means of using the phonon density of states and quasiharmonic approximation [51]. Expressions used for the Helmholtz free energy DF, internal energy DE, entropy S and constant-volume specific heat Cv are given as follows [52]:  Z omax  _o g ðoÞdo DF ¼ 3nNkB T ln 2 sinh ð1Þ 2kB T 0

LO

600

Frequency (1/cm)

TO 500 400

LA

300

DE ¼ 3nN

200

S ¼ 3nNkB

TA

_ 2

Cv ¼ 3nNkB

Γ

K X

L

Γ

X W

L

PHDOS

Fig. 1. Calculated phonon dispersion and phonon density of states (PHDOS) of BeS.

700 LO

Frequency (1/cm)

600 500 TO

400 300

LA

200 100 0

TA Γ

K X

Γ

X

L

W

PHDOS

L

Fig. 2. Calculated phonon dispersion and phonon density of states (PHDOS) of BeSe.

550 500

LO

450 Frequency (1/cm)

400

TO

350 300 250 200 LA

150 100 50 0

TA Γ

K X

Γ

L

X

W

L

PHDOS

Fig. 3. Calculated phonon dispersion and phonon density of states (PHDOS) of BeTe.

3.3. Thermodynamic properties After obtaining the phonon spectrum over the entire BZ, we calculated and obtained the Helmholtz free energy DF, internal

0



ocoth

 _o gðoÞdo 2kB T

  Z omax  _o _o _o coth  ln 2sinh gðoÞdo 2kB T 2kB T 2kB T 0

100 0

Z omax

   Z omax  _o 2 _o 2 gðoÞdo csch 2kB T 2kB T 0

ð2Þ

ð3Þ

ð4Þ

where kB is the Boltzmann constant, n is the number of atoms per unit cell, N is the number of unit cells, omax is the largest phonon frequency, o is the phonon frequency and gðoÞ is the normalized Ro phonon density of states with 0 max gðoÞdo ¼ 1. Figs. 4–7 display the calculated DE, DF, S and Cv within the scope of temperature from 0 to 1000 K. In Fig. 4, we can see when the temperature increases the internal energies approach each other gradually and increase almost linearly with temperature. At the same temperature, the values of the internal energies increase with the decreasing anion mass (Te-Se-S). Fig. 5 shows the variations of phonon free energy with temperature for the BeX compounds. The phonon free energy curves are similar and move down with increasing temperature. At the same temperature, the values of phonon free energy also increase with decreasing anion mass (Te-Se-S). Fig. 6 displays the temperature dependence of the heat capacities of the BeX compounds. We notice that the heat capacities follow the Debye model and approach the Petit and Dulong limit (Cv  50 J/mol-c K  1) at high temperatures for BeS, BeSe and BeTe. At low temperature, the curvature of Cv increases with increasing anion mass (S-Se-Te). The shape of the computed Cv curve seems to be typical for III–V compounds [31,32,53–55]. This trend supports our predictive results. Fig. 7 shows the variations of entropy with temperature for BeS, BeSe and BeTe. Obviously, the entropy exhibits a similar tendency for all BeX compounds. We can see that the entropy increases when the size of the anion mass increases (S-Se-Te). In an ensemble where the sample volume and temperature are independent variables, the relevant potential is the free energy, F ¼E TS. So, the entropy is present as a TnS product to allow comparison with the internal energy and free energy. Above 400 K, the value of TnS increases faster than the internal energy as the temperature increases, resulting in the value of the free energy for each compound to change from positive to negative in different temperatures. From Figs. 4–7, one can see the variations of E, F, S and Cv for the beryllium chalcogenides compounds as a function of temperature and the atomic mass of S, Se, Te. Because the mass difference of S, Se, Te, the bond length of Be–S, Be–Se and Be–Te are different. As we know, the bond length strongly affects some physical properties, such as thermodynamics, elastic constant and hardness and so on [56]. So the trends of DE, DF, S and Cv are reasonable.

4. Summary In this work, we have calculated the ground state structure, dynamical and thermodynamic properties of beryllium chalcogenides in the zinc blende structure by means of using the DFT and DFPT within LDA. The ground state parameters are in good

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Table 2 Calculated phonon frequencies (in cm  1) at high symmetry points G, X and L of BeS, BeSe and BeTe, compared to experimental and theoretical results. Phonon mode

BeS

BeSe

BeTe

GLO GTO

654,652a, 647b 560 562a, 562b 647, 654a, 652b 361,357a, 364b 503, 505a, 507b 229, 226a, 237b 602,607a, 607b 359,361a, 362b 540, 545a, 543b 162, 158a, 161b

564, 558a, 556b, 579c, 579d 502, 497a, 498b, 501c, 501d 601, 585a, 601b, 496c 214, 212a, 218b, 252c 453, 452a, 451b, 528c 137, 137a, 139b, 101c 557, 555a, 556b, 538c 213, 214a, 216b, 177c 488, 485a, 485b, 517c 99, 97a, 99b, 72c

493,494a, 495b, 502c, 502e 461, 463a, 468b, 461c, 461e 518, 535a, 540b, 468c 149, 156a, 159b, 140c 408, 416a, 425b, 477c 93, 98a, 97b, 80c 481, 498a, 504b, 481c 145, 155a, 158b, 74c 443, 449a, 455b, 473c 68, 68a, 69b, 67c

XLO XLA XTO XTA LLO LLA LTO LTA a

Ref. [24]. Ref. [4]. c Ref. [25]. d Ref. [26]. e Ref. [27].

55000

60

50000

55

45000

50

40000

45

35000

40

30000 25000 20000

BeS BeSe BeTe

15000 10000

CV (J/mol-c.K)

ΔΕ (J/mol-c)

b

30 25

BeS BeSe BeTe

20 15 10

5000 0

35

5 0

100 200 300 400 500 600 700 800 900 1000 Temperature (K)

0

0

Fig. 4. Calculated internal energy DE of BeS, BeSe and BeTe.

100 200 300 400 500 600 700 800 900 1000 Temperature (K)

Fig. 6. Calculated the constant-volume specific heat Cv of BeS, BeSe and BeTe.

20000

BeS BeSe BeTe

10000

BeS BeSe BeTe

100

-10000 -20000

S (J/mol-c.K)

ΔF (J/mol-c)

0

120

-30000 -40000 -50000

80 60 40

-60000 -70000

20 0

100 200 300 400 500 600 700 800 900 1000 Temperature (K)

Fig. 5. Calculated phonon free energy DF of BeS, BeSe and BeTe.

0

0

100 200 300 400 500 600 700 800 900 1000 Temperature (K) Fig. 7. Calculated the entropy S of BeS, BeSe and BeTe.

agreement with the available theoretical and experimental results. We calculated phonon frequency and described the main features of the phonon dispersion curves and compared with other literature. We calculated and presented the thermodynamic quantities such as the internal energy, free energy, specific heat and entropy.

Acknowledgments This work has been supported by Education Department Foundation of Liaoning Province of China Grant nos. (201064145 and

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