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Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO
Accepted Manuscript Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3 SnO
Enamul Haque, M. Anwar Hossain PII:
S0038-1098(18)30263-1
DOI:
10.1016/j.ssc.2018.09.004
Reference:
SSC 13496
To appear in:
Solid State Communications
Received Date:
11 May 2018
Accepted Date:
07 September 2018
Please cite this article as: Enamul Haque, M. Anwar Hossain, Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO, Solid State Communications (2018), doi: 10.1016/j.ssc.2018.09.004
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Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO Enamul Haque and M. Anwar Hossain* Department of Physics, Mawlana Bhashani Science and Technology University Santosh, Tangail-1902, Bangladesh Email: [email protected], [email protected]
Abstract
We have explored the lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO (Sr3-xBaxSnO, x=0, 1, 2) using density functional theory (DFT) and density functional perturbation theory (DFPT). All the studied alloys show good dynamical stability. These Basubstituted systems show metallic band structure due to the mixing of Sr-4d and Ba-5d with Sn5p orbitals. According to Migdal-Eliashberg theory the calculated electron-phonon coupling constant and logarithmic-averaged phonon frequency for Sr2BaSnO are λ=0.24 and ωln=124.55K, respectively and the corresponding Tc=0.6 K while it is 0.2 K for SrBa2SnO. Our analysis reveals that phonon softening by Ba-substitution is responsible for superconductivity in these alloys.
Keywords: C. Structural stability; D. Lattice thermal conductivity; D. Electronic transport; D. Superconductivity
1. Introduction Lattice dynamics of a solid have a great importance in analyzing different physical properties, such as the thermoelectric figure of merit (ZT) and phase stability of the solid [1,2]. Lattice thermal
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conductivity κl is a key factor appearing in ZT. The materials with low lattice thermal conductivity show good thermoelectric performance. For example, Si has high thermal conductivity showing low performance while SnSe has low thermal conductivity and hence exhibit high figure of merit (ZT) [3–5]. The essential criterion of a crystal is it’s dynamical stability. For example, the growth attempt of MoC in NaCl structure resulted in failure due to the lattice dynamical unstabilty [6]. The structural stability of a solid can be predicted from phonon dispersion relation. For the robust realization of stability of solid, a reliable study of lattice dynamics is required. Thus, the study of lattice dynamical properties of solid has recenly become the subject of intense research. The Sr3SnO (SSO) has some unusual physical properties such as superconductivity [7–9], topological behavior [10,11], ferromagnetic behavior [12–16] and good thermoelectric transport properties [17–19] and thus, gained huge resaerch interest. The SSO is a nearly topological insulator that has been recently found to exhibit topological superconductivity due to Sr deficiency, with a superconducting transition temperature (Tc) ~5 K [7,9,10]. The theoretical and experimental studies confirmed that SSO exhibit topological insulating and semiconducting behavior [9,10,12]. Epitaxial thin films of SSO are grown on a cubic yttria-stabilized zirconia (c-YSZ)/Si (001) is a dilute magnetic semiconductor [12,14]. The study on the transport properties has revealed the enhanced thermoelectric perfotmance in topological insulators [20]. In our recent study, we have found that lattice thermal conductivity is predominant in SSO [19]. Since the alloying can significantly improve the thermoelectric performance by reducing lattice thermal conductivity [21–24] and superconductivity appear in SSO due to Sr-deficiency [7,9], we are ineterested to find the effects of Ba subsititution in SSO on lattice dynamics, transport and superconding properties. The SSO crystallizes in cubic structure with lattice paramter, a=5.1394 Å, and space group 𝑃𝑚3
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𝑚 (#221) [25]. The Wyckoff positions for Sr, Sn and O atoms are 3c (0, 0.5, 0.5), 1a (0, 0 ,0) and 1b (0.5, 0.5, 0.5), respectively [25,26]. In this report, we investigate the lattice dynamics, transport and superconducting properties of Basubstituted Sr3SnO using density functional theory (DFT) [27,28] and density functional perturbation theory (DFPT) [29–31]. We have found that all the studied alloys are dynamically stable and lattice thermal conductivity is significantly reduced in Sr2BaSnO. The Sr2BaSnO shows superconductivity at Tc~0.6K. Small value of electron-phonon coupling and average logerthmic frequency are responsible for such low value of superconducting transition temperature.
2. Computational methods The structural optimization was performed after Ba-substitution in 2 × 2 × 2 supercell structure of SSO by using a full potential linearized augmented plane wave method (FP-LAPW) as implemented in WIEN2k [32]. After applying symmetry, the Ba-substitution reduces the symmetry in all the studied alloys, and the space group was found to be P4/mmm (#123). For structural optimization Perdew-Burke-Ernzerhof (PBE) functional [33,34] and 225 k-point for irreducible Brillouin zone (BZ) integration were used. The muffin tin radii: 2.3, 2.5, 2.3, and 2.3 Bohr for Sr, Sn, O, and Ba, respectively and the kinetic energy cutoff (RKmax) 8.0 were chosen for good convergence. The optimized lattice parameters of SSO were taken from our previous study [35] and for x=1, and 2, the lattice parameters were found to be a=5.0971, c=5.66, and a=5.0840, c=4.805 Å, respectively. With these ground state structures, the band structure calculation was performed in WIEN2k using Tran-Blaha modified Becke-Johnson potential (TB-mBJ) [36,37]. Since the calculations of transport coefficients require denser k-mesh, the self-consistent field calculation (SCF) was performed again with 1276 k-point. The transport coefficients were
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calculated by solving semi-classsical Boltzmann transport equation as implemented in BoltzTraP [38]. The BoltzTraP calculates the electron transport coefficients within the constant relaxation time approximation (cRTA). The temperature dependent values of transport coefficients were taken at the Fermi level of 0 K. The phonon-dispersion calculation was performed by using 40 Ry cutoff energy for wavefunctions and 280 Ry for charge density within density functional perturbation theory (DFPT), as implemented in QUANTUM ESPRESSO [39]. For phonon calculation, 888, 886, and 668 k-point; 444, 443 and 334 uniform q-point grid for SSO, Sr2BaSnO, and SrBa2SnO, respectively were used. The k-point used for electron-phonon linwidth calculation was twice as mentioned above. The second-order harmonic and third-order anharmonic IFCs were computed in Vienna ab initio simulation package ( VASP) [40–42] within finite displacment approach. Since PBE-GGA underestimates the lattice thermal condcuctivity [43], the lattice thermal conductivity was calculated using GW method [44–48]. For this, we used PAW-GW [49] pseudopotentials (provided in VASP 5.4.4) and kept other settings same to calculate second-order harmonic and IFCs in VASP. Phonopy code [50] was used to extract second order force-constant and thirdorder_vasp.py to extract third-order IFCs. The lattice thermal conductivity was computed by solving linearized phonon Boltzmann transport equation in ShengBTE [51]. For SSO, we used the results of our previous study [19]. In ShengBTE, 8 × 8 × 8-mesh of the q-point grid was used.
3. Results and discussions 3.1.
Lattice dynamics
Before proceeding to main calculation, the dynamical stability of these alloys were examined through phonon calculations using the optimized structures. Phonon calculations consist of two steps. Firstly, structural relaxation performed in QUANTUM ESPRESSO (QE) and relaxed
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structures were found to be in good agreement with WIEN2k results. Secondly, the phonon dispersions were computed using 2 × 2 × 2 q-point in QE. Phonons are the normal modes of lattice vibrations that are considered to measure the crystal structure stability. The phonon frequency must be the real quantity for a stable crystal structure [50,52]. The imaginary phonon frequency will appear for an unstable crystal structure. The phonon frequencies of SSO are real quantity as illustrated in Fig. 1. Thus, SSO is dynamically stable; it should be because SSO has been experimentally found to be stable [25]. The substitution of Ba leads to generate some new phonon branches in the phonon band structure as shown in Fig. 2. These two alloys have no imaginary phonon frequency and hence these structures are dynamically stable. Therefore, the real phonon frequencies of SSO and Ba-substituted SSO ensure their dynamic stability. Now, we will study the impact of Ba-substitution on phonon transport properties. Since some phonon bands are introduced by Ba, we may expect the significant change in the phonon scattering. Hence, lattice thermal conductivity may be reduced drastically. The calculated lattice thermal conductivity of SSO and its alloys is shown in Fig. 3. We see that the lattice thermal conductivity of one Basubstituted SSO is much reduced and its heat capacity is also reduced, as shown in Fig. 4. It is expected, because phonon scattering is much increased by Ba substitution. The large value of Grüneisen parameter indicates high anharmonicity in the crystal and hence the large phonon scattering. We see that for x=1, the Grüneisen parameter is much larger than that of SSO, indicating large phonon scattering in Sr2BaSnO. But surprisingly the heat capacity is decreased more in SrBa2SnO and Grüneisen parameter becomes positive, much smaller than pure SSO (ignoring sign). Thus, phonon scattering is much reduced and hence, lattice thermal conductivity is larger than pure SSO. Such kind of unexpected result may be due to the phonon transport properties of elemental Ba. Further experimental studies are required to clarify these results. However, we
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conclude that lattice thermal conductivity may not be effectively reduced when a large number of atoms are substituted by others.
3.2.
Transport properties
The main purpose of this manuscript is to explain the change in phonon related properties due to alloying of SSO. However, we will briefly review the change in electrical conductivity and electronic part of the thermal conductivity. In order to explain electron conduction in any material, it is required to understand electronic structure of the material. Therefore, we will discuss it very shortly. The empty states in Sr and Ba are 4d and 5d, respectively. Thus, due to the energy difference between them, some new energy bands are induced by Ba substitution, as shown in Fig. 5. The stoichiometric SSO has direct band gap 0.22 eV at Γ-point [35]. When we substitute one Ba at Sr site, the energy of the conduction band minima (blue line) is lowered and hence crosses the Fermi level. Thus, the Sr2BaSnO shows metallic band structure. The energy of conduction band minima is lowered more when we substitute two Ba atoms. Then, the energy of valence band maxima is increased. We see that the density of states at the Fermi level is significantly increased in Ba-substituted SSO, which is consistent with the calculated band structure. The mixing of Sr4d and Ba-5d (shown in right panel of Fig. 5, only for x=1) with Sn-5p orbitals causes the increase of band energy and the DOS. Since the density of states at the Fermi level in Ba-substituted SSO increases, the electrical and electronic part of the thermal conductivities also increase, as shown in Fig. 6. The electrical conductivity and electronic thermal conductivity of the studied alloys are much higher than that for SSO and increase with temperature. This implies that all these material show semiconducting nature. However, this is the limitation of the cRTA, the constant relaxation
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time approximation cannot give a correct trend of the conductivity at low temperature [53–55]. Therefore, further studies are required to clarify these results, especially experimental studies.
3.3
Superconducting properties
We have previously studied the superconducting properties of Sr3SnO using rigid muffin-tin approximation (RMTA) [56–58] and found to exhibit superconductivity with Tc~8.38K [59]. But the recent experimental study has not found superconductivity up to 0.15 K [9]. Many studies have been found that the calculated Tc by RMTA deviates from the experimental value [57,58,60]. For example, Pb has measured Tc= 7.2 K [61] while RMTA produced Tc =3.3 K [60] and LaFeAsO has Tc =45 K while RMTA gave 0 K [62]. Therefore, such large deviation of our previous result may be due to the limitations of RMTA method. The PBE functional underestimates the bandgap [36,63], and RMTA method uses the band structure data directly [58]. This may be another reason for inconsistent result of our previous study. Therefore, we use DFPT in this paper to investigate superconducting properties of SSO and alloys for good accuracy. The calculated phonon density of states is shown in Fig. 7(a). We see that phonon DOS is much reduced in Ba-substituted SSO, which is consistent with our phonon band structure calculations. The electron-phonon coupling constant (EPC) within the Migdal-Eliashberg theory can be calculated as
∫
𝜆=2
2
𝛼 𝐹(𝜔) 𝑑𝜔 𝜔
2
where 𝛼 𝐹(𝜔) is given by 1 𝛼 𝐹(𝜔) = 2𝜋𝑁(𝐸𝐹) 2
∑𝛿(𝜔 ‒ 𝜔 𝒒𝑣
𝛾𝒒𝑣
𝒒𝑣)ћ𝜔 𝒒𝑣
The calculated spectral function of SSO and its alloys are shown in Fig. 7(b). We see that the spectral function is zero for SSO indicating no superconductivity in it, which is consistent with the
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2
recent experimental study [9]. The maximum peak of 𝛼 𝐹(𝜔) is at ~190 cm-1 for one Basubstituted SSO and the EPC for this system is 0.24. The corresponding logarithmic-averaged phonon frequency (ωln) is 145.55 K. This peak is slightly lowered and shifted to a higher frequency in two Ba-substituted SSO alloy. The EPC for this system is 0.195 and logarithmic-averaged phonon frequency (ωln) is 145.46 K. We see that the ωln remains constant under the change of Baconcentration. EPC of these two alloys indicates that these are weakly coupled superconductors. With these parameters, superconducting transition temperature can be calculated by Allen-Dynes equation
𝑇𝑐 =
𝜔𝑙𝑛 1.2
𝑒𝑥𝑝
[(
‒ 1.04(1 + 𝜆)
𝜆 1 ‒ 0.62𝜇
∗
) ‒ 𝜇∗
]
where μ* is the Coulomb pseudopotential, for which zero value can be taken conventionally [64]. Using the parameters calculated above, we obtain Tc=0.6 K for Sr2BaSnO while the value is 0.2 K for SrBa2SnO. We conclude that such low temperature superconductivity arises in Ba-substituted SSO due to Ba-induced phonon softening.
4. Conclusions We have investigated the lattice dynamics, transport and superconducting properties of Basubstituted Sr3SnO by using density functional theory (DFT) and density functional perturbation theory (DFPT). All the studied alloys are found to be dynamically stable. Lattice thermal conductivity is much reduced in one Ba-substituted SSO due to the increase of phonon scattering but it is much increased in two Ba-substituted SSO. The predominant lattice contribution of Ba may be responsible for the decrease of phonon scattering and hence increase of lattice thermal conductivity. These Ba-substituted alloys show metallic band structure due to the mixing of Sr-4d
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and Ba-5d states with Sn-5p. The electrical and electronic part of the thermal conductivity are significantly increased due to the increase of density of states at the Fermi level in Ba-substituted systems. The calculated electron-phonon coupling constant for Sr2BaSnO (λ=0.24) and the logarithmic- averaged frequency (ωln=145 K) produce a maximum superconducting transition temperature (Tc) of 0.6 K, within standard Migdal-Eliashberg theory (MET). The Ba-induced phonon softening may be responsible for superconductivity in these alloys. We hope that experimentalist will be interested to investigate these properties to clarify our prediction.
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Figures
600
Frequency (cm-1)
500 400 300 200 100 0
R
M
X
Fig.1: Phonon dispersion of Sr3SnO (SSO).
(a)
600
500
Frequency (cm-1)
Frequency (cm-1)
500 400 300 200 100 0
(b)
600
400 300 200 100
X
M
A
X
0
X
M
Fig.2: Phonon dispersion of (a) Sr2BaSnO and (b) SrBa2SnO.
A
X
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4 Sr3SnO Sr2BaSnO
l (W/m K)
3
SrBa2SnO
2
1
0
200
250
300
350
400
450
500
T(K)
Fig. 3: Lattice thermal conductivity of SSO and its alloys by using GW functional.
8
10
(a)
7
0
5
Sr3SnO Sr2BaSnO
4
SrBa2SnO
-10
Sr3SnO Sr2BaSnO
-20
SrBa2SnO
-30
3
-40
2
-50
1 0
(b)
CV (106 J/m3 K)
6
200
250
300
350 T (K)
400
450
500
-60
200
250
300
350
400
450
500
T (K)
Fig. 4: Variation of heat capacity at constant volume and Grüneisen parameter (γ) of SSO and its alloys with temperature.
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1.0 Sr3SnO
0.5
Sr2BaSnO
Energy (eV)
SrBa2SnO 0.0
-0.5
-1.0
'
0 1 2 3 4 5 DOS (states/eV u.c.)
Fig. 5: Band structure of SSO, Sr2BaSnO, and (b) SrBa2SnO at Γ-point and total density of states (DOS). The dash-dot line represents Fermi level. Γ-point stands for SSO (cubic) while 𝛤' for its alloys (tetragonal). For SSO, data have been taken from our previous study [35].
25
350 (b)
(a)
e 1012 W/m s)
(1018 S/m s)
20 15 10 5 0
Sr3SnO
300
Sr2BaSnO SrBa2SnO
250 200 150 100 50
200
250
300
350 T (K)
400
450
500
0
200
250
300
350
400
450
500
T (K)
Fig. 6. Electrical conductivity (σ/τ) and electronic thermal conductivity (κe/τ) of SSO and its alloys, with constant relaxation time approximation (cRTA).
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Phonon DOS
(a)
0.30
Sr3SnO
0.4
Sr2BaSnO
0.3
SrBa2SnO
Sr2BaSnO
0.20
0.2
SrBa2SnO
0.15 0.10
0.1 0.0
Sr3SnO
(b)
0.25 2F()
0.5
0.05 0
100
200 300 400 Frequency (cm-1)
500
600
0.00
0
100
200 300 400 Frequency (cm-1)
Fig. 7: (a) Phonon density of states and (b) Eliashberg spectral function.
500
600
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Research Highlights
Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO have been studied. All the studied alloys show good dynamical stability. Lattice thermal conductivity is much reduced in Sr2BaSnO. Electron-phonon coupling in these alloys are very weak and all the alloys show superconductivity.
Enamul Haque, M. Anwar Hossain PII:
S0038-1098(18)30263-1
DOI:
10.1016/j.ssc.2018.09.004
Reference:
SSC 13496
To appear in:
Solid State Communications
Received Date:
11 May 2018
Accepted Date:
07 September 2018
Please cite this article as: Enamul Haque, M. Anwar Hossain, Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO, Solid State Communications (2018), doi: 10.1016/j.ssc.2018.09.004
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Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO Enamul Haque and M. Anwar Hossain* Department of Physics, Mawlana Bhashani Science and Technology University Santosh, Tangail-1902, Bangladesh Email: [email protected], [email protected]
Abstract
We have explored the lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO (Sr3-xBaxSnO, x=0, 1, 2) using density functional theory (DFT) and density functional perturbation theory (DFPT). All the studied alloys show good dynamical stability. These Basubstituted systems show metallic band structure due to the mixing of Sr-4d and Ba-5d with Sn5p orbitals. According to Migdal-Eliashberg theory the calculated electron-phonon coupling constant and logarithmic-averaged phonon frequency for Sr2BaSnO are λ=0.24 and ωln=124.55K, respectively and the corresponding Tc=0.6 K while it is 0.2 K for SrBa2SnO. Our analysis reveals that phonon softening by Ba-substitution is responsible for superconductivity in these alloys.
Keywords: C. Structural stability; D. Lattice thermal conductivity; D. Electronic transport; D. Superconductivity
1. Introduction Lattice dynamics of a solid have a great importance in analyzing different physical properties, such as the thermoelectric figure of merit (ZT) and phase stability of the solid [1,2]. Lattice thermal
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conductivity κl is a key factor appearing in ZT. The materials with low lattice thermal conductivity show good thermoelectric performance. For example, Si has high thermal conductivity showing low performance while SnSe has low thermal conductivity and hence exhibit high figure of merit (ZT) [3–5]. The essential criterion of a crystal is it’s dynamical stability. For example, the growth attempt of MoC in NaCl structure resulted in failure due to the lattice dynamical unstabilty [6]. The structural stability of a solid can be predicted from phonon dispersion relation. For the robust realization of stability of solid, a reliable study of lattice dynamics is required. Thus, the study of lattice dynamical properties of solid has recenly become the subject of intense research. The Sr3SnO (SSO) has some unusual physical properties such as superconductivity [7–9], topological behavior [10,11], ferromagnetic behavior [12–16] and good thermoelectric transport properties [17–19] and thus, gained huge resaerch interest. The SSO is a nearly topological insulator that has been recently found to exhibit topological superconductivity due to Sr deficiency, with a superconducting transition temperature (Tc) ~5 K [7,9,10]. The theoretical and experimental studies confirmed that SSO exhibit topological insulating and semiconducting behavior [9,10,12]. Epitaxial thin films of SSO are grown on a cubic yttria-stabilized zirconia (c-YSZ)/Si (001) is a dilute magnetic semiconductor [12,14]. The study on the transport properties has revealed the enhanced thermoelectric perfotmance in topological insulators [20]. In our recent study, we have found that lattice thermal conductivity is predominant in SSO [19]. Since the alloying can significantly improve the thermoelectric performance by reducing lattice thermal conductivity [21–24] and superconductivity appear in SSO due to Sr-deficiency [7,9], we are ineterested to find the effects of Ba subsititution in SSO on lattice dynamics, transport and superconding properties. The SSO crystallizes in cubic structure with lattice paramter, a=5.1394 Å, and space group 𝑃𝑚3
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𝑚 (#221) [25]. The Wyckoff positions for Sr, Sn and O atoms are 3c (0, 0.5, 0.5), 1a (0, 0 ,0) and 1b (0.5, 0.5, 0.5), respectively [25,26]. In this report, we investigate the lattice dynamics, transport and superconducting properties of Basubstituted Sr3SnO using density functional theory (DFT) [27,28] and density functional perturbation theory (DFPT) [29–31]. We have found that all the studied alloys are dynamically stable and lattice thermal conductivity is significantly reduced in Sr2BaSnO. The Sr2BaSnO shows superconductivity at Tc~0.6K. Small value of electron-phonon coupling and average logerthmic frequency are responsible for such low value of superconducting transition temperature.
2. Computational methods The structural optimization was performed after Ba-substitution in 2 × 2 × 2 supercell structure of SSO by using a full potential linearized augmented plane wave method (FP-LAPW) as implemented in WIEN2k [32]. After applying symmetry, the Ba-substitution reduces the symmetry in all the studied alloys, and the space group was found to be P4/mmm (#123). For structural optimization Perdew-Burke-Ernzerhof (PBE) functional [33,34] and 225 k-point for irreducible Brillouin zone (BZ) integration were used. The muffin tin radii: 2.3, 2.5, 2.3, and 2.3 Bohr for Sr, Sn, O, and Ba, respectively and the kinetic energy cutoff (RKmax) 8.0 were chosen for good convergence. The optimized lattice parameters of SSO were taken from our previous study [35] and for x=1, and 2, the lattice parameters were found to be a=5.0971, c=5.66, and a=5.0840, c=4.805 Å, respectively. With these ground state structures, the band structure calculation was performed in WIEN2k using Tran-Blaha modified Becke-Johnson potential (TB-mBJ) [36,37]. Since the calculations of transport coefficients require denser k-mesh, the self-consistent field calculation (SCF) was performed again with 1276 k-point. The transport coefficients were
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calculated by solving semi-classsical Boltzmann transport equation as implemented in BoltzTraP [38]. The BoltzTraP calculates the electron transport coefficients within the constant relaxation time approximation (cRTA). The temperature dependent values of transport coefficients were taken at the Fermi level of 0 K. The phonon-dispersion calculation was performed by using 40 Ry cutoff energy for wavefunctions and 280 Ry for charge density within density functional perturbation theory (DFPT), as implemented in QUANTUM ESPRESSO [39]. For phonon calculation, 888, 886, and 668 k-point; 444, 443 and 334 uniform q-point grid for SSO, Sr2BaSnO, and SrBa2SnO, respectively were used. The k-point used for electron-phonon linwidth calculation was twice as mentioned above. The second-order harmonic and third-order anharmonic IFCs were computed in Vienna ab initio simulation package ( VASP) [40–42] within finite displacment approach. Since PBE-GGA underestimates the lattice thermal condcuctivity [43], the lattice thermal conductivity was calculated using GW method [44–48]. For this, we used PAW-GW [49] pseudopotentials (provided in VASP 5.4.4) and kept other settings same to calculate second-order harmonic and IFCs in VASP. Phonopy code [50] was used to extract second order force-constant and thirdorder_vasp.py to extract third-order IFCs. The lattice thermal conductivity was computed by solving linearized phonon Boltzmann transport equation in ShengBTE [51]. For SSO, we used the results of our previous study [19]. In ShengBTE, 8 × 8 × 8-mesh of the q-point grid was used.
3. Results and discussions 3.1.
Lattice dynamics
Before proceeding to main calculation, the dynamical stability of these alloys were examined through phonon calculations using the optimized structures. Phonon calculations consist of two steps. Firstly, structural relaxation performed in QUANTUM ESPRESSO (QE) and relaxed
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structures were found to be in good agreement with WIEN2k results. Secondly, the phonon dispersions were computed using 2 × 2 × 2 q-point in QE. Phonons are the normal modes of lattice vibrations that are considered to measure the crystal structure stability. The phonon frequency must be the real quantity for a stable crystal structure [50,52]. The imaginary phonon frequency will appear for an unstable crystal structure. The phonon frequencies of SSO are real quantity as illustrated in Fig. 1. Thus, SSO is dynamically stable; it should be because SSO has been experimentally found to be stable [25]. The substitution of Ba leads to generate some new phonon branches in the phonon band structure as shown in Fig. 2. These two alloys have no imaginary phonon frequency and hence these structures are dynamically stable. Therefore, the real phonon frequencies of SSO and Ba-substituted SSO ensure their dynamic stability. Now, we will study the impact of Ba-substitution on phonon transport properties. Since some phonon bands are introduced by Ba, we may expect the significant change in the phonon scattering. Hence, lattice thermal conductivity may be reduced drastically. The calculated lattice thermal conductivity of SSO and its alloys is shown in Fig. 3. We see that the lattice thermal conductivity of one Basubstituted SSO is much reduced and its heat capacity is also reduced, as shown in Fig. 4. It is expected, because phonon scattering is much increased by Ba substitution. The large value of Grüneisen parameter indicates high anharmonicity in the crystal and hence the large phonon scattering. We see that for x=1, the Grüneisen parameter is much larger than that of SSO, indicating large phonon scattering in Sr2BaSnO. But surprisingly the heat capacity is decreased more in SrBa2SnO and Grüneisen parameter becomes positive, much smaller than pure SSO (ignoring sign). Thus, phonon scattering is much reduced and hence, lattice thermal conductivity is larger than pure SSO. Such kind of unexpected result may be due to the phonon transport properties of elemental Ba. Further experimental studies are required to clarify these results. However, we
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conclude that lattice thermal conductivity may not be effectively reduced when a large number of atoms are substituted by others.
3.2.
Transport properties
The main purpose of this manuscript is to explain the change in phonon related properties due to alloying of SSO. However, we will briefly review the change in electrical conductivity and electronic part of the thermal conductivity. In order to explain electron conduction in any material, it is required to understand electronic structure of the material. Therefore, we will discuss it very shortly. The empty states in Sr and Ba are 4d and 5d, respectively. Thus, due to the energy difference between them, some new energy bands are induced by Ba substitution, as shown in Fig. 5. The stoichiometric SSO has direct band gap 0.22 eV at Γ-point [35]. When we substitute one Ba at Sr site, the energy of the conduction band minima (blue line) is lowered and hence crosses the Fermi level. Thus, the Sr2BaSnO shows metallic band structure. The energy of conduction band minima is lowered more when we substitute two Ba atoms. Then, the energy of valence band maxima is increased. We see that the density of states at the Fermi level is significantly increased in Ba-substituted SSO, which is consistent with the calculated band structure. The mixing of Sr4d and Ba-5d (shown in right panel of Fig. 5, only for x=1) with Sn-5p orbitals causes the increase of band energy and the DOS. Since the density of states at the Fermi level in Ba-substituted SSO increases, the electrical and electronic part of the thermal conductivities also increase, as shown in Fig. 6. The electrical conductivity and electronic thermal conductivity of the studied alloys are much higher than that for SSO and increase with temperature. This implies that all these material show semiconducting nature. However, this is the limitation of the cRTA, the constant relaxation
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time approximation cannot give a correct trend of the conductivity at low temperature [53–55]. Therefore, further studies are required to clarify these results, especially experimental studies.
3.3
Superconducting properties
We have previously studied the superconducting properties of Sr3SnO using rigid muffin-tin approximation (RMTA) [56–58] and found to exhibit superconductivity with Tc~8.38K [59]. But the recent experimental study has not found superconductivity up to 0.15 K [9]. Many studies have been found that the calculated Tc by RMTA deviates from the experimental value [57,58,60]. For example, Pb has measured Tc= 7.2 K [61] while RMTA produced Tc =3.3 K [60] and LaFeAsO has Tc =45 K while RMTA gave 0 K [62]. Therefore, such large deviation of our previous result may be due to the limitations of RMTA method. The PBE functional underestimates the bandgap [36,63], and RMTA method uses the band structure data directly [58]. This may be another reason for inconsistent result of our previous study. Therefore, we use DFPT in this paper to investigate superconducting properties of SSO and alloys for good accuracy. The calculated phonon density of states is shown in Fig. 7(a). We see that phonon DOS is much reduced in Ba-substituted SSO, which is consistent with our phonon band structure calculations. The electron-phonon coupling constant (EPC) within the Migdal-Eliashberg theory can be calculated as
∫
𝜆=2
2
𝛼 𝐹(𝜔) 𝑑𝜔 𝜔
2
where 𝛼 𝐹(𝜔) is given by 1 𝛼 𝐹(𝜔) = 2𝜋𝑁(𝐸𝐹) 2
∑𝛿(𝜔 ‒ 𝜔 𝒒𝑣
𝛾𝒒𝑣
𝒒𝑣)ћ𝜔 𝒒𝑣
The calculated spectral function of SSO and its alloys are shown in Fig. 7(b). We see that the spectral function is zero for SSO indicating no superconductivity in it, which is consistent with the
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2
recent experimental study [9]. The maximum peak of 𝛼 𝐹(𝜔) is at ~190 cm-1 for one Basubstituted SSO and the EPC for this system is 0.24. The corresponding logarithmic-averaged phonon frequency (ωln) is 145.55 K. This peak is slightly lowered and shifted to a higher frequency in two Ba-substituted SSO alloy. The EPC for this system is 0.195 and logarithmic-averaged phonon frequency (ωln) is 145.46 K. We see that the ωln remains constant under the change of Baconcentration. EPC of these two alloys indicates that these are weakly coupled superconductors. With these parameters, superconducting transition temperature can be calculated by Allen-Dynes equation
𝑇𝑐 =
𝜔𝑙𝑛 1.2
𝑒𝑥𝑝
[(
‒ 1.04(1 + 𝜆)
𝜆 1 ‒ 0.62𝜇
∗
) ‒ 𝜇∗
]
where μ* is the Coulomb pseudopotential, for which zero value can be taken conventionally [64]. Using the parameters calculated above, we obtain Tc=0.6 K for Sr2BaSnO while the value is 0.2 K for SrBa2SnO. We conclude that such low temperature superconductivity arises in Ba-substituted SSO due to Ba-induced phonon softening.
4. Conclusions We have investigated the lattice dynamics, transport and superconducting properties of Basubstituted Sr3SnO by using density functional theory (DFT) and density functional perturbation theory (DFPT). All the studied alloys are found to be dynamically stable. Lattice thermal conductivity is much reduced in one Ba-substituted SSO due to the increase of phonon scattering but it is much increased in two Ba-substituted SSO. The predominant lattice contribution of Ba may be responsible for the decrease of phonon scattering and hence increase of lattice thermal conductivity. These Ba-substituted alloys show metallic band structure due to the mixing of Sr-4d
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and Ba-5d states with Sn-5p. The electrical and electronic part of the thermal conductivity are significantly increased due to the increase of density of states at the Fermi level in Ba-substituted systems. The calculated electron-phonon coupling constant for Sr2BaSnO (λ=0.24) and the logarithmic- averaged frequency (ωln=145 K) produce a maximum superconducting transition temperature (Tc) of 0.6 K, within standard Migdal-Eliashberg theory (MET). The Ba-induced phonon softening may be responsible for superconductivity in these alloys. We hope that experimentalist will be interested to investigate these properties to clarify our prediction.
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Figures
600
Frequency (cm-1)
500 400 300 200 100 0
R
M
X
Fig.1: Phonon dispersion of Sr3SnO (SSO).
(a)
600
500
Frequency (cm-1)
Frequency (cm-1)
500 400 300 200 100 0
(b)
600
400 300 200 100
X
M
A
X
0
X
M
Fig.2: Phonon dispersion of (a) Sr2BaSnO and (b) SrBa2SnO.
A
X
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4 Sr3SnO Sr2BaSnO
l (W/m K)
3
SrBa2SnO
2
1
0
200
250
300
350
400
450
500
T(K)
Fig. 3: Lattice thermal conductivity of SSO and its alloys by using GW functional.
8
10
(a)
7
0
5
Sr3SnO Sr2BaSnO
4
SrBa2SnO
-10
Sr3SnO Sr2BaSnO
-20
SrBa2SnO
-30
3
-40
2
-50
1 0
(b)
CV (106 J/m3 K)
6
200
250
300
350 T (K)
400
450
500
-60
200
250
300
350
400
450
500
T (K)
Fig. 4: Variation of heat capacity at constant volume and Grüneisen parameter (γ) of SSO and its alloys with temperature.
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1.0 Sr3SnO
0.5
Sr2BaSnO
Energy (eV)
SrBa2SnO 0.0
-0.5
-1.0
'
0 1 2 3 4 5 DOS (states/eV u.c.)
Fig. 5: Band structure of SSO, Sr2BaSnO, and (b) SrBa2SnO at Γ-point and total density of states (DOS). The dash-dot line represents Fermi level. Γ-point stands for SSO (cubic) while 𝛤' for its alloys (tetragonal). For SSO, data have been taken from our previous study [35].
25
350 (b)
(a)
e 1012 W/m s)
(1018 S/m s)
20 15 10 5 0
Sr3SnO
300
Sr2BaSnO SrBa2SnO
250 200 150 100 50
200
250
300
350 T (K)
400
450
500
0
200
250
300
350
400
450
500
T (K)
Fig. 6. Electrical conductivity (σ/τ) and electronic thermal conductivity (κe/τ) of SSO and its alloys, with constant relaxation time approximation (cRTA).
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Phonon DOS
(a)
0.30
Sr3SnO
0.4
Sr2BaSnO
0.3
SrBa2SnO
Sr2BaSnO
0.20
0.2
SrBa2SnO
0.15 0.10
0.1 0.0
Sr3SnO
(b)
0.25 2F()
0.5
0.05 0
100
200 300 400 Frequency (cm-1)
500
600
0.00
0
100
200 300 400 Frequency (cm-1)
Fig. 7: (a) Phonon density of states and (b) Eliashberg spectral function.
500
600
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Research Highlights
Lattice dynamics, transport and superconducting properties of Ba-substituted Sr3SnO have been studied. All the studied alloys show good dynamical stability. Lattice thermal conductivity is much reduced in Sr2BaSnO. Electron-phonon coupling in these alloys are very weak and all the alloys show superconductivity.