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Comparing the Superconductivity of Mgb2 and Sr2ruo4

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Comparing the superconductivity of MgB2 and Sr2RuO4 Etienne Palosa) (PHYS 232: Electronic Materials) The physical properties of two superconducting materials are compared in their normal and superconducting states. Their similarities and differences are reviewed in a systematic manner, beginning with a description of the unit cell. Structural, thermodynamic, electronic and magnetic properties are compared. In the super- conducting state, re compare the superconducting parameters of the materials and the effect on TC induced by physical or chemical changes to the systems. The sim- ilarities and differences between the conventional superconductor MgB2 and the unconventional superconductor Sr2RuO4 are discussed mainly within the context of experimental evidence reported in the literature. I. INTRODUCTION Some of the most important unresolved questions in condensed matter physics are re- lated to superconductivity. While the phenomena was observed over one-hundred years ago and an theory that explains conventional superconductivity has been established for the better half of a century, there is no unified theory of conventional and unconventional su- perconductivity. Or is there? However, through continuous theoretical, experimental and collaborative efforts, much has been learned about the physics of superconducting materi- als. The two main categories of superconductors are (i) conventional: those explained by the Bardeen{Cooper{Schrieffer (BCS) theory, and (ii) unconventional: materials whose su- perconducting mechanisms do not satisfy the conditions in BCS theory. In these materials, the superconducting mechanism is not phonon-mediated. In this work, a comparison of the physical properties of one conventional and one uncon- ventional superconductors is presented. On one hand, we have magnesium diboride, MgB2, a material whose superconductivity was discovered in 20011, decades after it's synthesis and crystal structure was first reported. This discovery represented a milestone in super- conductor physics, as at the time MgB2 had the highest TC for a (i) binary system and (ii) a non-Cuprate superconductor. A wave of studies followed, demonstrating that MgB2 is a BCS superconductor. We compare the properties of MgB2 with an unconventional superconductor, specifically a spin-triplet superconductor Sr2RuO4. With a similar history to magnesium diboride, Sr2RuO4 was synthesized decades earlier (1959) and it's supercon- ductivity wasn't discovered until 19942. Strontium ruthenate also made headlines, as it too, did not contain copper. In this sense, it is the first layered perovskite superconductor without copper. We aim to review and compare the properties of these two materials in a systematic yet pedagogical manner. II. PHYSICAL PROPERTIES IN THE NORMAL STATE A. Structure and Symmetry Magnesium diboride (MgB2) is hexagonal "omega" structure structured and crystallizes in the hexagonal P 6=mmm space group. The Mg atoms alternate or are "sandwiched" between hexagonal layers of B2. Mg is bonded to twelve equivalent B atoms to form a mixture of edge and face-sharing MgB12 cuboctahedra. All Mg{B bond lengths are ' a)Electronic mail: [email protected] 2 2.50 A.˚ B is bonded in a 9-coordinate geometry to six equivalent Mg and three equivalent B atoms. All B{B bond lengths are '1.77 A.˚ The unit cell of magnesium diboride is illustrated in Fig. 1 (a) and (b). Strontium ruthenate, Sr2RuO4, is (La, Ba)CuO4 structured (i.e. perovskite structured) + and crystallizes in the tetragonal I4=mmm space group. Sr2 is bonded in a 9-coordinate { geometry to nine O2 atoms. There are a spread of Sr{O bond distances ranging from ∼ + { 2.45{2.77 A.˚ Ru4 is bonded to six O2 atoms to form corner-sharing RuO6 octahedra. There are four shorter (1.95 A)˚ and two longer (2.10 A)˚ Ru{O bond lengths, and two { in-equivalent O2 sites as is the case with most perovskites. The unit cell of strontium ruthenate is illustrated in Fig. 1 (c) and (d). The crystallographic properties described here can be corroborated by downloading the publicly available crystallographic information .cif files and using a visualization software such as VESTA, or any other, and comparing to the information in the reviews used as main references for this special topic paper3,4. A comparison of their lattice parameters, density and a few mechanical properties are compared in Table I. While the structures themselves may not be responsible for the superconducting prop- erties of these two materials, they have been a motive for further investigation since, after all, the atomic arrangement of a given material is determined by its electronic structure. In the case of MgB2, for its simplicity and in the case of Sr2RuO4, because of it's intriguing relationship between spin parity and it's superconducting state. More on that later. TABLE I. Summary of structural and some mechanical properties of MgB2 and Sr2RuO4 in the normal state. ˚ ˚ ˚ g Compound Lattice Space Group a (A) b (A) c (A) ρ ( cm3 ) K (GPa) G (GPa) MgB2 Hexagonal P 6=mmm [191] 3.083 3.083 3.521 2.64 118 148 Sr2RuO4 Tetragonal I4=mmm [139] 7.017 7.017 7.017 5.75 108 53 FIG. 1. Illustrations of the crystal structure (conventional unit cell) is shown for both MgB2 (top panel) and Sr2RuO4 (lower panel). The unit cell is enclosed by solid black lines. We show the isostructural (a) and top (xy) in (b) of MgB2, and in (c) and (d) respectively for Sr2RuO4. Colors: Mb = orange, B = dark green; Sr = light green, Ru = grey, and O = red. Image prepared with VESTA. 3 B. Electronic Structure The structural and electronic properties of the materials under exam provide constraints on the superconducting properties. There electronic structure and of these systems has been investigated thoroughly, by means of different theoretical methods such as the Tight-Binding formalism, Density Functional Theory, Dynamical Mean Field Theory and others. Due to the volume of such studies, I will the reader to (in my opinion) representative references5,6. In Fig 2, we present the band structures of the materials side-to-side, along with their orbital decomposed density of states computed within the General Gradient Approximation (GGA) in Density Functional Theory. Here, we briefly discuss the band structure of the materials FIG. 2. Electronic band structure and spin-polarized density of states (DOS) of (a) MgB2 and (b) Sr2RuO4 computed by Kohn-Sham Density Functional Theory. Graphics publically avaible in the Materials Project database. in what is mostly a qualitative manner. This is to not lose generality and focus on technical details which may be specific to certain methodologies, yielding distinct quantitative results (e.g. LDA vs GGA vs DMFT). The band structure of magnesium diboride was reported well before it's superconduc- tivity was dsicovered, but the first "post-enlightenment" comprehensive first-principles in- vestigation of electronic structure and phonon coupling in MgB2 was reported by Mazin 6 and Antropov in 2001 , and refer to them for our discussion. The band structure of MgB2 resembles that of graphite, with three bonding σ bands which correspond to in-plane sp2 (spxpy) hybridization in the Boron honey-comb layer, and two bands, π; π∗ formed by hy- bridized pz orbitals. The main difference between MgB2 and graphite is that two σ bands are only partially filled. The band structure of Sr2RuO4 was first described by Tamio Oguchi shortly after the 5 discovery of it as a superconductor . He showed that Sr2RuO4 is a metal, with 12 energy bands that lye below -2 eV that correspond to the bonding states of the Ru d and O p orbitals, as well as O p non-bonding. Additionally, there are three bands that cross the Fermi energy, EF . These bands posses the antibonding character of the Ru dxz; dyz and dxy orbitals O p; π orbitals. Ultimately, the take away here is that, like in the cuprates, there exists strong dp hybridization due planar network structure. However, while these band structure calculations provide insight into the orbital nature of the material, and provides similarities and dissimilarities with related cuprates, id sheds no real insight into the superconducting mechanism of the system. Further theoretical modeling, including the proposal of effective Hamiltonians have been developed based on Oguchi's work. C. Vibrational Structure Understanding the vibrational properties of materials can provide insight into it's dynam- ical stability, thermal properties and the mechanism of superconductivity in the materials we are currently interested in. In Figure 3 we show the phonon dispersion (phonon band structure) in (a) and the phonon density of states in (b) for MgB2, and in the same order 4 (c) and (d) for Sr2RuO4. FIG. 3. Phonon dispersion bands (a) and phonon density of states (phDOS) (b) are shown for 7{10 MgB2, and in (c), (d) for Sr2RuO4 computed. Image adapted from Refs . III. PHYSICAL PROPERTIES IN THE SUPERCONDUCTING STATE A. Mechanisms of superconductivity To begin the discussion of their superconducting state properties, we will make references again to their electronic structure. Let us begin with the case of MgB2. There is experimental evidence that showcases the mechanism of superconductivity in 10 MgB2, beginning with the observation of a large different in TC between Mg B2 and 11 Mg B2. This difference, ∆TC ∼ 1K is known as the "isotope effect”. While we omit further details regarding the isotope effect coefficient (α), we highlight that in the case of MgB2 such a large isotope effect is a strong indicator that this system is a phonon- mediated superconductor. Particularly, it is the B phonon modes that are dominant for superconductivity, as there is strong coupling between the conduction electrons and the optical E2g optical phonon, where neighboring B-atoms move in opposite directions within the plane. A figure explicitly showing the isotope effect can be found in Ref4. This would classify MgB2 as a typical BCS superconductor, except for an added complexity of the boron phonon mode, which is anharmonic and the multi-band superconductivity of the material.
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