Bu tez çalışmasında, ATl3 (A=Ca, Y, La ve Th) ve La3Tl'deki spin-orbit çiftleniminin (spin-orbit coupling- SOC) rolünü yapısal, elektronik, elastik, mekaniksel, fonon ve elektron-fonon etkileşim özelliklerinin teorik olarak araştırılmasıyla incelenmiştir. Hesaplamalarda Yoğunluk Fonksiyonel Teorisi tabanlı Quantum Espresso paket programı kullanılmıştır. Quantum Espresso, Genelleştirilmiş Gradyan Yaklaşımı (GGA) kullanılarak ATl3 (A=Ca, Y, La ve Th) ve La3Tl bileşiklerinin fiziksel özelliklerini, özellikle de süperiletkenlik özelliklerini araştırmak için kullanılmıştır. Yerel Yoğunluk Yaklaşımında (LDA) r noktasındaki yoğunluk kullanılırken, gerçek uzaydaki yoğunluk noktadan noktaya değişmektedir. Gerçek bir malzemeyi incelerken yoğunluk gradyanının da dikkate alınması gerektiği ve GGA'da da bu değişiklik dikkate alındığı için bu çalışmada GGA kullanılmıştır. Spin-orbit çiftleniminin, incelenen bileşiklerin elektronik bant yapıları üzerindeki etkisi, spin-orbit çiftlenimi olmadan bir skaler relativistik hesaplamada var olabilecek yüksek simetri noktalarındaki bazı dejenerasyonların bu çiftlenim dikkate alınarak ortadan kaldırılmasıdır. İncelenen tüm bileşikler mekanik kararlılık standartlarını karşılamaktadır ve bu nedenle kübik AuCu3-tipi kristal yapılarında mekanik olarak kararlıdırlar. İncelenen tüm bileşikler için C11 değeri C44 değerinden daha yüksektir; bu, incelenen bileşiklerin tek yönlü sıkıştırmaya karşı dirençle karşılaştırıldığında saf kayma deformasyonuna karşı daha zayıf bir direnç gösterdiğine işaret etmektedir. Bulk modülünün kayma modülüne oranı(BH/GH)nın 1.75'ten büyük olması, önemli ölçüde sünekliğin olduğu anlamına gelir. YTl3'ün BH/GH oranı burada incelenen diğer bileşiklerden daha büyüktür ve bu da YTl3'ün diğer bileşiklerden daha iyi sünekliğe sahip olduğunun sinyalini vermektedir. La3Tl'deki La ve Tl atomlarının yer değiştirmesi, Fermi seviyesindeki fonon durum yoğunluğu N(EF)'nin değerini 2,1 kat artırmıştır. Ayrıca, bu yer değiştirme La3Tl'deki tüm fonon modlarını, LaTl3'tekinden daha yumuşak kılmıştır. Hem yumuşak fonon modları hem de yüksek N(EF), La3Tl'deki elektron-fonon etkileşimini LaTl3'tekinden çok daha güçlü kılmıştır. Spin-orbit çiftleniminin varlığı, LaTl3 ve ThTl3'ün Tc değerlerini sırasıyla %34 (1.151 K'den 1.542 K'e) ve %65 (0.479 K'den 0.793 K'e) oranında artırmış ve karşılık gelen deneysel değerler olan 1.63 K ve 0.87 K ile iyi bir uyum sağlamıştır. Sonuç olarak, yapılan hesaplamalara spin-orbit çiftleniminin dahil edilmesi CaTl3, YTl3 ve La3Tl'nin Tc değerleri için deneyle uyumu da geliştirir.
In this work, we have examined the role of spin-orbit coupling (SOC) in ATl3 (A = Ca, Y, La and Th) and La3Tl by theoretical investigation of their structural, electronic, elastic, mechanical, phonon and electron-phonon interaction properties. Intermetallic compounds crystallized in the simple cubic AuCu3-type the structure has been widely investigated in the last 50 years due to their various properties changing from magnetism to superconductivity. For example, PrSn3 and NdSn3 have antiferromagnetic ordering with the Néel temperatures of 8.6 and 4.5 K while a heavy fermion behavior has been observed for CeIn3, CePb3, and CePd3. CeSn3 has been categorized as a dense Kondo compound exhibiting valence fluctuations. Above all else, more than 20 superconducting compounds with the simple cubic AuCu3-type structure have been reported in the literature. Among them the ATl3 systems became of interest initially because of their varied superconducting transition temperatures (Tc) that range from 0.87 to 2.04 K. ThTl3, YTl3, LaTl3, and CaTl3 transform to superconductors at 0.87, 1.52, 1.63, and 2.04 K. Furthermore, the replacement La and Tl in LaTl3 increases the Tc value from 1.63 to 8.86 K by more than five times. An interesting question is: why La3Tl have a much higher Tc than LaTl3? To answer this, one has to study their electronic, phonon, and electron–phonon interaction properties in detail. In this respect, it is worth mentioning that spin-orbit coupling must be included in such studies because Tl 6p electrons have been found to exhibit an intermediate spin-orbit coupling (SOC). The electronic and elastic properties of YTl3 and LaTl3 have been investigated by using the full-potential linear augmented plane wave (FP-LAPW) method within the local density approximation for the exchange-correlation functional and including SOC. This theoretical calculation demonstrates that the electronic states near the Fermi level are mainly contributed by Tl p states with a non-ignorable contribution from Y d (or La d) states. For YTl3, at the Γ point, the degenerate scalar relativistic state at around −1.5 eV splits into two states in the SOC case with a sizeable splitting of 618 meV. The value of this splitting at the X point is about 790 meV, more significant than the corresponding value of 618 meV at the zone center. In the range −8 to −3.4 eV, the DOS features are characterized by an admixture of Tl 6s, Tl 6p, Y 5p, and Y 4d states. Between −3.4 eV and the Fermi energy, Tl 6p partial DOS has almost the identical shape as Y 4d states, which is a sign of significant hybridization between these states and thus of covalent interaction. Therefore, in this energy region, the presence of SOC affects the Tl-like states, which impacts Y-like states because of hybridization. It is worth mentioning that this hybridization does not exist for CaTl3. Consequently, the substitution of Ca by Y in CaTl3 increases the contribution of A atom to the value of N(EF) more than ten times due to the significant existence of its d states at the Fermi energy. Thereby, different from CaTl3, the states at the Fermi level are mainly composed of Tl 6p states (45%) and Y 4d states (40%), as well as a minor contribution from Tl 6s states (12%). In the case of YTl3, switching on SOC enhances the value of N(EF) from 1.846 to 1.906 states eV−1 (by around 3%). This result means that the presence of SOC only slightly affects the electronic bands of YTl3 near the Fermi energy. Finally, in light of the above results, we can conclude that not only Tl 6p states but Y 4d states are expected to be involved in forming a superconducting state for YTl3. For LaTl3 compound, at the Γ point, a spin-orbit splitting of 540 meV below the Fermi level and 115 meV above the Fermi level is evident. These splittings are more apparent at the X point with 755 and 509 meV values. An obvious similarity between LaTl3 and YTl3 is the proximity of the d levels of the La atom to the Fermi energy, which leads to significant d hybridization in the states near the Fermi energy. On the other hand, a distinct difference between LaTl3 and YTl3 is the closeness of the f levels of La to the Fermi energy, which gives rise to small f hybridization (6%) with the states around the Fermi level for LaTl3, which does not exist for YTl3. Consequently, at the Fermi energy, the largest contribution comes from Tl 6p states (47%) and La 5d states (36%). It is worth mentioning that the inclusion of SOC moderately influences the second part of the valence band region for LaTl3. For ThTl3 compound, at the Γ point, spin-orbit splitting of 200 meV below the Fermi level and of 152 meV above the Fermi level appears. These values increase to 644 and 605 meV at the X point, respectively. At first glance, an explicit similarity between ThTl3 and LaTl3 is the proximity of the d and f levels of the Th atom to the Fermi energy, which brings significant d hybridization and small f hybridization in the states near the Fermi energy. Consequently, there is considerable hybridization of Tl 6p, Th 6d, and Th 5f orbitals for ThTl3 in the vicinity of the Fermi level. This hybridization implies that ThTl3 have solid Th−Tl bonds, and thus, this compound is expected to have higher elastic moduli than the remaining compounds. For ThTl3, a noticeable effect of SOC can be observed in the energy range from −3.7 eV to the Fermi energy. In particular, the peak at −1.45 eV for the SOC-free calculation is divided into two peaks with energies of −1.92 and −1.38 eV for the SOC-included calculation. The N(EF) value for ThTl3 amounts to be 2.357 states eV−1, with percentage contribution from Th and Tl atom being roughly 51% and 49%, respectively. Especially, Tl 6p, Th 6d, and Th 5f states contribute to the value of N(EF) within 38%, 30%, and 21%, respectively. These contributions indicate that in addition to Tl 6p and Th 6d states, Th 5f states are expected to play an essential role in t e constitution of superconducting state for ThTl3. Finally, when the SOC is omitted, the N(EF) value becomes 2.189 states eV−1, which differs by 8% from the corresponding value obtained with SOC. This result suggests that SOC is moderately effective in the electronic bands near the Fermi level. For La3Tl, at the Γ point, a spin-orbit splitting of 830 meV above the Fermi level appears clearly for this superconductor. In addition to this, the maximum splitting at the X point is found to be 417 meV below the Fermi level. The valence DOS consists of two prominent domains separated by a sizeable gap of 2.8 eV: the lower domain between −7.2 and −5.7 eV and the upper field extending from −2.9 eV to the Fermi level. There is only one peak feature in the lower part, characterized by Tl 6s states with slightly smaller contributions from La 6p and La 5d states. Most La d states appear in the upper domain of valence DOS and hybridize slightly with La f and Tl p states. Consequently, in contrast to LaTl3, the most significant contribution to the value of N(EF) for La3Tl comes from La d states within 93%. In fact, for this superconductor, the contribution of La d states to N(EF) is more significant than that of Tl p states by a factor of 13. This result signals that the geometric exchange between La and Tl atoms in the LaTl3 unit cell makes the DOS at the Fermi level of mainly d character in La3Tl in contrast to the p character as observed for LaTl3. This change in the molecular makeup changes the value of N(EF) from 2.337 states eV−1 for LaTl3 to 4.883 states eV−1 for La3Tl (by a factor of 2.1). An exploration of elastic and mechanical properties reveals that YTl3 and LaTl3 compounds are ductile in nature with elastic anisotropy. Abraham and co- workers have also studied the electronic, elastic, and mechanical properties of LaTl3 using the FP-LAPW method. Still, they have used the generalized gradient approximation (GGA) for the exchange-correlation potential rather than the local density approximation. This GGA work confirms the metallic character of this compound, but the GGA values of elastic constants and elastic moduli are significantly lower than their LDA counterpart. This difference must arise from different exchange- correlation approximations used by these theoretical calculations. Besides, the GGA method has been used to investigate electronic, elastic, phonon, and electron–phonon interaction properties of YTl3. This theoretical calculation confirms that this compound is soft (ductile) due to significant metallic bonding. On the other hand, elastic constants and elastic moduli calculated in this GGA calculation are different from their LDA counterpart by 50%. Electron–phonon interaction of this GGA work reveals superconductivity in YTl3 arises from the conventional electron–phonon interaction, which is mainly dominated by the coupling between electrons from Tl 6p states and low-frequency Tl-related phonons. Recently, the elastic, mechanical, and phonon properties of LaTl3 have been examined using the GGA method. The calculated values of elastic constants and elastic moduli for LaTl3 by this GGA calculation are consistent with the corresponding results presented by a previous GGA calculation. However, in this GGA calculation, the width of the phonon spectrum for LaTl3 has been reported to be 38 unbelievably high THz, as both La and Tl have heavy atomic mass. It may be that the phonon frequency values in that paper were mistakenly multiples by a factor of 10. Indeed, other lattice dynamical calculations on LaTl3, performed by Y.-L. Wan and co-workers show that the width of phonon spectrum for LaTl3 is around 3.8 THz, which is certainly more realistic than the corresponding value of 38 THz reported by L. An and J.-W. Yang. Although phonon properties of YTl3 and LaTl3 have been studied theoretically, to the best of our knowledge, no theoretical studies exist in the literature regarding phonon properties of CaTl3, ThTl3, and La3Tl. As many physical properties, such as electrical and thermal resistivity, thermal expansion, and superconductivity, are determined by phonons and their interactions with electrons, accurate calculations have to be performed for ATl3 (A = Ca, Y, La, and Th) compounds. Furthermore, these four compounds form a suitable ground for investigating the effect of SOC on their structural, electronic, elastic, mechanical, phonon, and electron–phonon interaction properties since Tl 6p electrons exhibit an intermediate SOC. In addition, the results obtained for these four compounds can be utilized to evince the evolution of SOC impacts with the change of A elements in them. Above all, electronic, phonon, and electron–phonon interactions results obtained for LaTl3 must be compared with the corresponding results for La3Tl in detail to understand the physics behind a relatively much higher Tc of the latter compound. In addition to their electronic properties, elastic and mechanical properties of superconductors must be investigated in three respects: i) scientifically through the electron–phonon coupling; ii) technologically through processing conditions and product design parameters; and iii) economically through the durability and reliability of the manufactured products. In the light of the above discussion, we have attempted ab initio pseudopotential calculations to understand the effect of SOC on the electronic and phonon structures and the effect of the electron–phonon coupling in contributing the bulk superconductivity in ATl3 compounds (A = Ca, Y, La, and Th) via comparison with their isostructural compound La3Tl. We have calculated the second-order elastic constants of CaTl3, YTl3, LaTl3, ThTl3, and La3Tl by utilizing the influential stress–strain method. Then, the values of polycrystalline bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (σ) for all the studied compounds are derived from these elastic constants with the help of the Voigt–Reuss–Hill (VHR) approach. We have conducted lattice dynamic calculations on the studied compounds using the density functional perturbation theory within the linear response approach. All the studied compounds meet the standards of mechanical stability, and therefore, they are mechanically stable in their cubic AuCu3-type crystal structure. The value of C11 is greater than the value of C44 for all the studied compounds, signaling that these compounds present a weaker resistance to pure shear deformation compared to resistance to unidirectional compression. C44 is an important physical parameter that describes the hardness and also represents the shear capacity of the crystal. The C44 value of ThTl3 is significantly larger than that of other studied compounds, revealing that this compound ought to exhibit higher hardness and a larger shear capacity than the other compounds studied here. The elastic moduli of ThTl3 are the largest; thus, the chemical bond's bonding strength between ThTl3 atoms is the strongest. The deformation resistance, governed by the shear modulus, for the studied compounds follows the order: ThTl3>LaTl3>YTl3>La3Tl>CaTl3. The stiffness, governed by Young's modulus, of the studied compounds is also ThTl3>LaTl3>YTl3>La3Tl>CaTl3. The larger ratio of the bulk modulus to shear modulus (BH/GH >1.75) means there is significant ductility. The BH/GH ratio of YTl3 is larger than the remaining compounds, signaling that it has better ductility than others. A similar observation has been made by comparing Cauchy pressure values for the studied compounds. Besides, Poisson's ratio can also define atomic bonding force, the atomic bonding force of the material is the central force when this ratio falls in the range of 0.25-0.5. Consequently, the atom bonding force in all the studied compounds is central according to their high Poisson's ratio values Furthermore, phonon dispersion curves and phonon density of states (DOS) for LaTl3 and La3Tl are compared between each other to establish the role of their phonon properties in their significant Tc difference. Finally, electron–phonon interaction calculations for all these compounds have been made by combining the linear response approach and the Migdal–Eliashberg approach. The results have been utilized to analyze the impact of SOC on the superconducting parameters of all these compounds. The inclusion of SOC also improves the agreement with the experiment for Tc values of CaTl3, YTl3 and La3Tl.
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