First-principles investigations on elasticity properties of FeSi under high pressure and temperature

Physica B: Condensed Matter 557 (2019) 82–87

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First-principles investigations on elasticity properties of FeSi under high pressure and temperature

T

Shao-Peng Qia, Xiu-Lu Zhanga, Zhen-Wei Niua,∗, Cheng-An Liua, Ling-Cang Caib a b

Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology, 621010 Mianyang, Sichuan, China National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, Chinese Academy of Engineering Physics, Mianyang 621900, China

A R T I C LE I N FO

A B S T R A C T

Keywords: FeSi Elastic constants Sound velocity High pressure and temperature

Taking into account the influence of temperature, we investigated the thermoelasticity properties of FeSi under high pressure. The obtained relationship of pressure and volume agrees well with the experimental data at high pressures. We found the anisotropy factor and sound velocities are significantly affected by pressure whereas they are insignificantly affected by temperature, especially at high pressures. The different propagating directions almost have no effect on the compressional wave velocities at high pressures and different temperatures.

1. Introduction The intermetallic compound FeSi has attracted much attention over decades owing to the significant electrical and magnetic properties. It is currently of considerable interest from the functional applications in thermoelectric conversion and solar cells [1–6]. Among these compounds, the elastic property of FeSi at high pressure and temperature has also attracted much scientific attention due to the remarkable application in earth science [7–10]. Nevertheless, some remarkable properties of the phase transition in FeSi are not understood. Two stable structures in FeSi system, B20 phase with P213 symmetry and B2 phase with CsCl structure, were reported via both experiments and calculations at high pressure and temperature. In earlier experiments [11–13], only B20 phase was measured at pressures lower than 10 GPa. Even if the experimental conditions are elevated to 50 GPa and 1500 K, the phase transition from B20 to B2 remains undiscovered in the study of Knittle and Williams [14]. However, subsequent theoretical studies [15,16] showed that B2 phase will become thermodynamically stable compared with B20 phase when pressure is above 20 GPa. Dobson et al. [17] pointed out the reason why previous experimental studies fail to obtain B2 phase at high pressure and temperature is the slightly enriched silicon of the starting sample. By using Fe0.52Si0.48 sample, they found B2 phase is stable at 24 GPa and at the temperatures above 1950 ± 50 K. In the following experimental studies, Lord et al. [18] and Fischer et al. [19] showed sharp boundaries between the two phases at high pressure and temperature, while Geballe and Jeanloz [20] suggested a wide B2+B20 two-phase field. Fischer et al. [19] and Geballe and Jeanloz [20]

∗

showed no detectable temperature dependence in the process of phase transformation when temperature is above 1000 K. Nevertheless, the two studies gave two different transition pressure boundaries: ∼42 GPa for Fischer et al. [19] and ∼30 GPa for Geballe and Jeanloz [20]. It should also be noted that refs 17 and 18 do show temperature dependence to the boundary. Computer simulations also provided entirely different transition pressures. Caracas and Wentzcovitch [7] found the B20-B2 transition takes place at ∼40 GPa, whereas Wann et al. [10] reported that the phase transition occurs from ∼11 GPa at 300 K to ∼3 GPa at 2000 K. From these experiments and calculations, there exists only B2 phase at the pressure of more than 50 GPa. The elasticity simulation of FeSi under high pressure and temperature will help to understand the unusual phenomena of transition metal silicides in deep, such as the phase transition from B20 phase to B2 phase. Despite a great many theoretical and experimental studies performed on FeSi recently, the reports about the elasticity of FeSi under high pressure and temperature are still relatively lacking. This leads us to take an interest in studying the thermoelasticity and thermodynamic properties of B20 and B2 phases of FeSi under high pressure and temperature using the first-principles calculations, in which the effects of temperature are considered by the Quasi harmonic approximation (QHA) [21]. 2. Calculation methods Most of our calculations are based on density functional theory [22,23] as implemented in the Cambridge Sequential Total Energy Package (CASTEP) [24]. The generalized-gradient approximation

Corresponding author. E-mail address: [email protected] (Z.-W. Niu).

https://doi.org/10.1016/j.physb.2019.01.009 Received 19 September 2018; Received in revised form 2 November 2018; Accepted 7 January 2019 Available online 08 January 2019 0921-4526/ © 2019 Elsevier B.V. All rights reserved.

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S.-P. Qi et al.

(GGA) proposed by Perdew et al. [25] is used to describe exchangecorrelation potential. Ultrasoft pseudopotentials are generated on the fly. The electronic wave functions are expanded in a plane wave basis set with a cut-off energy of 800 eV. The K-space integrations are performed using 8 × 8 × 8 Monkhorst-Pack meshes [26] in the first Brillouin-zone. The self-consistent convergence of the total energy is 1.0 × 10−6 eV/atom. The maximum ionic Hellmann-Feynman force is within 0.005 eV/Å and the maximum ionic displacement within 1.0 × 10−5 Å. These parameters are carefully tested in our calculations, and it is found that these parameters are sufficient to lead to a wellconverged total energy. In order to investigate thermoelasticity of FeSi, Quasi harmonic approximation [21] is adopted here. In this method, the effect of temperature mainly relates to the change of volume, and considers electron excitation and phonon interactions less. Even so, this method still can be used to describe B20 and B2 phases of FeSi due to the small intrinsic anharmonic effects at the temperature below half of the melting point. The phonon spectra are obtained from finite displacement and linear response methods. The corresponding supercells with 64 and 108 atoms were adopted here for B20 and B2 phases. In present paper, a series of calculations with different volume are used from 58 to 108 Å3 for B20 and 12 to 26 Å3 for B2. At 2000 K, our simulations are above 50% of the melting point, and only reach 50% at ∼100 GPa. This will make the anharmonic contribution of the free energy slightly overestimated [35,36]. Nonetheless, the effect on the elastic constants is not so significant when the temperature is not close to the melting point [37]. Hence, our simulations at 2000K are valid in some extent.

Table 2 The parameters of FeSi from a Third-Order Birch-Murnagham Equation of State. V (Å) B20 structure 89.135 89.015 90.39 90.193 90.217 88.84 B2 structure 20.751 20.596 21.741 21.323 21.369 21.254

The obtained structural parameters for the two phases of FeSi are presented in Table 1. The obtained zero-pressure lattice constants agree with the experimental values, and the difference is less than one percent. As is known, for the conventional density functional theory (DFT) techniques, local-density approximation (LDA) commonly underestimates crystal structure parameters while generalized-gradient approximation (GGA) commonly overestimates them. However, in our calculation, the lattice constant of B20 phase is in excellent agreement with the experimental data but that of the B2 phase is slightly lower than the experimental data. One possible reason is that the levels of the localization of Fe-3d electronic states are different in the two phases, and the localization is more dramatic in B2 phase. Based on the obtained energy-volume (Emin-V) data, we fit Third-Order Birch-Murnagham equations of state (EOS) to obtain bulk modulus K0 and pressure derivative of bulk modulus K0ʹ, which are shown in Table 2. The obtained bulk modulus of B20 phase is larger than the experimental data but close to the previous first-principles calculations [7,16,and 27]. The difference in compressibility could stem from the pseudopotentials employed, or be the result of non-hydrostaticity in the experiments. Also, some of the experiments performed compression to much Table 1 The lattice constants of FeSi under pressure. P (GPa)

aB20

Knittle and Williams

0 12.5 19.5 48.9 0 0 0 0.77 0 0 0

Wood et al. Guyot et al. Lin et al. Whitaker et al. Dobson et al. Ono et al. Sata et al.

aB2

P (GPa)

aB20

aB2

4.465

0

4.467

2.748

4.388 4.344 4.226 4.496 4.488 4.485 4.482

20 40 60 80 100 120 140 160

4.347 4.263 4.195 4.138 4.089 4.046 4.008 3.974

2.678 2.629 2.591 2.559 2.531 2.507 2.485 2.465

2.791 2.773 2.775

Present

K0′

References

196.1 209 172 184.7 160 221.2

5.14 3.5 4 4.75 4 4.22

This work (finite displacement) Knittle and Williams (1995) Guyot et al. (1997) Lin et al. (2003) Wood et al. (1995) Wann et al. (2017)

232.95 246.13 184 225 221.7 229.79

4.89 4.04 4.2 4 4.167 4.36

This work (finite displacement) This work (linear response) Dobson et al. (2003) Ono et al. (2007) Sata et al. (2010) Wann et al. (2017)

higher pressures than others, and this also has a huge effect on the accuracy of K′ and in turn K because they are highly correlated. The obtained value of pressure derivative of bulk modulus is higher than some experimental data [11,14], but consistent with the experimental data of Lin et al. [8,9]. For the B2 phase, the obtained bulk modulus value is clearly higher than the experimental data due to the small equilibrium volume. The pressure derivative of bulk modulus is close to the experimental data and the latest calculation result of Wann et al. [10], although it is lower than the calculation results [7,16,27]. Moreover, the obtained K′ based on linear response is significantly smaller than the value obtained based on finite displacement. The possible reason is the subtle differences in the E-V curves from the two methods. This means the accuracy of energy calculation with different methods has a huge effect on the accuracy of K′ and K. The pressure-volume (P-V) data for the two phases of FeSi are shown in Fig. 1 at 300 K, 1200 K and 2000 K. To estimate the reliability of our calculations, we also use the QHA based on linear response method to calculate the P-V relations about B2 phase. Both the two methods present a similar result. The obtained results of B20 phase are slightly lower than the calculation results of Caracas and Wentzcovitch [7], but agree well with the experimental data. For B2 phase, the results are slightly lower than the experimental data at low pressures, whereas they are close to the experimental data of Sata et al. [28] at high pressures. Compared our calculation with the result of Caracas and Wentzcovitch [7], the prime difference is our lower trend line at high pressures. This could be because the choice of pseudopotentials in calculations. At room temperature and pressure, the obtained relative volume change from B20 phase to B2 phase is 6.8% at 0 GPa, which is slightly more than the experimental data (5.5%) of Ono et al. [29]. Furthermore, the temperature has a small impact on the change of the equilibrium volume at high pressure especially for the B20 phase. At atmosphere pressure, the obtained respective relative volume changes from 300 K to 2000 K are 4.9% for B20 phase and 5.5% for B2 phase. However, the volume changes are under 2% at 100 GPa for the two phases. It means that the temperature has a weak impact on the change of the equilibrium volume at high pressure. This could be because the localization of Fe-3d electronic states at high pressure. Through the relation between density and volume, the densities of B2 phase are yielded at different temperatures under 140 GPa, which are about 9.08 g/cm3, 9.01 g/cm3, 8.94 g/cm3 at 300 K, 1200 K, and 2000 K, respectively. To calculate the elastic constants, we use the stress–strain method where the symmetry-dependent strains are non-volume conserving [30]. From Fig. 2, it is shown that the elastic constants increase with the applied pressure and decrease with the increasing temperature. For the B2 phase, the elastic constants are also directly calculated from linear response for comparison. It can be found that the elastic constants obtained from linear response method agree well with our results

3. Results and discussion

Exp.

K0 (GPa)

83

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95

1400

(a)

(a)

1200

90

3

Volume (Å )

80 75

This B20-300 K This B20-1200 K This B20-2000 K Caracas and Wentzcovitch

70 65 0

20

40

60

80

300 K

1000

Elastic constants (GPa)

Wood et al. (1995) Knittle and Williams (1995) Wood et al. (1996) Lin et al. (2003) Whitaker et al. (2008)

85

C11

600 C12

400

C44

200

100

120

140

0

160

0

40

23

(b)

1400

Caracas and Wentzcovitch

1200

22

3

Volume (Å )

19

Elastic constants (GPa)

This B2-300 K This B2-1200 K This B2-2000 K

20

Ono (2013) Sata et al. (2010)

18 17 16

0

20

40

60

80

100

120

160

120

140

(b) K 300

1000

15 14

80 Pressure (GPa)

Pressure (GPa)

21

2000 K

800

C11

800 600 C12

400

C44

200 0

160

0K 200

0

40

80

120

160

Pressure (GPa)

Pressure (GPa)

Fig. 2. The variations of elastic constants of FeSi in the B20 (a) and the B2 (b) structures. The red open circles are our results obtained directly from linear response. The grey short dash lines are the theory data obtained from Caracas and Wentzcovitch [7]. The orange dash lines are the theory data obtained from Ono [31].

Fig. 1. Pressure-volume data for B20-type and B2-type FeSi. The grey dash lines are our results obtained from linear response methods.

obtained based on finite displacement method. This means that the estimation for thermoelasticity in our calculation is effective and reliable. The elastic constants of B2 phase are consistent with those of Caracas and Wentzcovitch [7] and Ono [31], while the B20 phase values are obviously smaller than those of Caracas and Wentzcovitch [7]. The anisotropy factor Λ can be achieved from the elastic constants by the following relation: 2C44/(C11-C12). For B20 phase, it decreases from 1.30 at ambient pressure to 0.92 at 160 GPa. However, the temperature has a negligible influence on the anisotropy factor at high pressure. For example, it increases from 0.92 to 0.93 with the temperature rising from 300 K to 2000 K at 160 GPa. Different from B20 phase, the B2 phase shows a converse trend, which is consistent with the conclusion of Caracas and Wentzcovitch [7]. The anisotropy factor increases from 0.55 at ambient pressure to 1.00 at 160 GPa. The impact of temperature on B2 phase can also be ignored at high pressure. At 160 GPa, the anisotropy factor decreases from 1.00 to 0.99 when temperature rises from 300 K to 2000 K. According to the Voigt–Reuss–Hill (VRH) average scheme [32], the bulk and shear moduli are obtained via the elastic constants. The aggregate bulk modulus K and shear modulus G are shown in Fig. 3 at different pressures and temperatures. The obtained bulk modulus values of B20 phase at different temperatures are in good agreement with the experimental data of Whitaker et al. [33,34] up to 8 GPa and 1273 K. However, different from bulk modulus, the shear modulus values become slightly larger than the experimental data of Whitaker et al. [33,34]. In addition, it can be found that the bulk modulus of B20

phase is more susceptible to the temperature in comparison with shear modulus from our calculations. The bulk modulus K of B2 phase is consistent with the previous calculations of Caracas and Wentzcovitch [7] and Ono [31]. The obtained shear modulus G is slightly lower than the previous calculations of Caracas and Wentzcovitch, and the difference becomes larger at high pressures. The possible reason is that the anharmonic effect is underestimated at high pressure, and the softening behavior of crystal structure is not considered enough when taking into account the effect of temperature. Interestingly, the variation of elastic moduli with temperature is found to be uncontinuous and unsmooth over the applied range of pressures. This means the temperature dependence of the bulk and shear moduli is different at different pressures. To better understand the thermoelasticity of FeSi, we then calculate the isotropic averaged aggregate compressional and shear sound velocities (VP and VS), and plot them in Fig. 4 with different pressures and temperatures. The compressional and shear wave velocities can be described with the following expression:

VP =

(K +

4 G )/ ρ , 3

VS =

G/ρ .

(1)

For the two phases, both VP and VS increase monotonically with the increasing pressure, whereas the impact of temperature on them is 84

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S.-P. Qi et al.

900

700 600

12000

KS

500 400 300

(a) VP

10500 Velocity (m/s)

800

Elastic moduli (GPa)

(a)

Caracas and Wentzcovitch (2004) This 300 K This 1200 K This 2000 K Whitaker (2008) Whitaker (2009)

This 300 K This 1200 K This 2000 K Whitaker (2008) Whitaker (2009)

9000 7500 6000

G

VS

200

Caracas and Wentzcovitch (2004)

4500

100 0

40

80

120

0

160

40

(b)

900 Caracas and Wentzcovitch (2004) Ono (2013) This 300 K This 1200 K This 2000 K

600

VP

10500 KS

500 400 G

300

This 0 K This 1200 K This 2000 K

9000 7500 6000

VS

4500

Caracas and Wentzcovitch (2004)

Ono (2013)

200 100

0

0

40

80

160

(b)

12000

Velocity (m/s)

Elastic moduli (GPa)

700

120

Pressure(GPa)

Pressure(GPa)

800

80

120

160

40

80

120

160

Pressure(GPa)

Pressure(GPa)

Fig. 4. The seismic wave velocities of B20 (a) and B2 (b) structures at different pressures and temperatures.

Fig. 3. The variations of bulk and shear moduli of FeSi in the B20 (a) and the B2 (b) structures at different pressures and temperatures.

temperature has a more significant impact on B2 phase compared with B20 phase at 0 GPa, and the curve has an entirely different picture. The maximum difference is about 0.86 km/s in the direction of ∼45° for the surface plane and about 0.93 km/s in the direction of ∼35° for the diagonal plane. Anyway, the maximum difference is within 0.3 km/s at 160 GPa and the wave velocities are almost unaffected by the direction at different temperatures, which mean the effect of temperature at high pressures can be neglected.

small, especially at high pressure. Similar to the case of elastic muduli, the obtained VP of B20 phase agrees well with the experimental data of Whitaker et al. [33,34], while VS is slightly larger than the experimental data of Whitaker et al. [33,34]. The reason is that the compressional and shear wave velocities are directly influenced by the elastic muduli. In terms of B2 phase, the obtained VS at 80 GPa is slightly larger than that of Ono [31] but the VP values quite accord with each other. One reason is that in cubic crystal structure VP is mainly relevant to the normal stress instead of shear stress, which is more likely to be affected by pressure and temperature. In order to gain a further understanding of the temperature impact, the compressional wave velocities along different propagating directions are investigated and the results are plotted in Fig. 5. It can be found that the impact of temperature is significant at 0 GPa but can be negligible at high pressure. For B20 phase, the compressional wave velocities decrease rapidly with the increasing temperature at 0 GPa and the velocities along the diagonal plane are always larger than those along the surface plane at different temperatures. The shapes of the curves are upward salient arcs, and the fastest velocity is located in the direction of ∼45° for the surface plane and ∼35° for the diagonal plane. However, there is an opposite trend at 160 GPa. The change of the velocities are not more than 0.2 km/s when temperature rises from 0 K to 2200 K, and the shapes of the curves become downward salient arcs. The direction of the fastest velocity at 0 GPa has become the slowest at 160 GPa for both the surface and diagonal planes. For B2 phase, the situation is completely different, especially at 160 GPa. The

4. Conclusions In summary, we have investigated the effect of temperature on the elasticity properties for B20 and B2 phases of FeSi under high pressure via the first-principles calculations combined with the Quasiharmonic approximation. Our results show that the temperature has a little impact on the elasticity of FeSi at high pressures, especially for B2 phase. For example, the anisotropy factor of B2 phase decreases by just 0.01 from 300 K to 2000 K at 160 GPa. The temperature also has an insignificant effect on the compressional wave velocities of FeSi. Acknowledgement The authors would like to thank the supports by the National Natural Science Foundation of China (Grant Nos. 41574076, 41704088, and 11604273), the Basic Research of Technology Program of China (Grant No. JSHS2014404B002), the Natural Science Foundation of Sichuan province (Grant No. 15TD0013). 85

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(a)

longitudinal wave velocities (km/s)

11.4 11.2 11.0

160 GPa

10.8

the surface plane

the diagonal plane

300 K 1200 K 2000 K

300 K 1200 K 2000 K

10.6 8.0 7.5 7.0 6.5

0 GPa

0

10

20

30

40

50

60

70

80

90

Angle to original-axis (deg.)

longitudinal wave velocities (km/s)

12.4 12.3 (b) 12.2

160 GPa

12.1

the surface plane 300 K 1200 K 2200 K

12.0 8.5

the diagonal plane 300 K 1200 K 2000 K

8.0 7.5

0 GPa

7.0 6.5

0

10

20

30

40

50

60

70

80

90

Angle to original-axis (deg.) Fig. 5. The seismic wave velocities of B20 (a) and B2 (b) along different propagating direction of the crystal structure.

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