arXiv:1405.3746v1 [cond-mat.supr-con] 15 May 2014
Computational Materials Design of Negative Effective U System for the Realization of High-Tc Superconductors Akitaka Nakanishi∗, Hiroshi Katayama-Yoshida Department of Materials of Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
Abstract We have calculated the effective U of CuAlO2 , AgAlO2 and AuAlO2 using the total energy calculation of different charges states. We found large negative effective U nature for d9 configuration of Cu2+ (d9 ), Ag2+ (d9 ) and Au2+ (d9 ). We propose the general design rule and how to increase the superconducting critical temperature Tc by the attractive electron-electron interactions caused by charge-excitation-induced negative effective U. Keywords: A. Semiconductors; C. Delafossite structure; D. Negative effective U; D. Charge-excitation-induced negative U; E. Density functional theory 1. Introduction For the realization of the superconducting industrial applications, the superconducting critical temperature Tc should be more three times larger than the room temperature (≃ 300K) in order to avoid the superconducting fluctuation. Tc by electron-phonon interaction is limited below 50 K. In order to enhance Tc 10 to 20 times, we need to find the purely electronic attractive interaction caused by the negative effective U system. Up to now, purely electronic negative effective U systems are proposed by Katayama-Yoshida et al. [1] and [2]. The attractive interaction caused by negative effective U is induced by (1) exchange correlation in d4 or d6 electronic configurations ∗
Tel.: +81-6-6850-6504. Fax: +81-6-6850-6407. Email address:
[email protected] (Akitaka Nakanishi)
Preprint submitted to Solid State Communications
May 16, 2014
[1] where d5 configuration is more stable due to the Hund’s rule in the highspin state, or (2) charge excitation in s1 or d9 electronic configurations [2] where the closed shell of s2 and d10 configuration is more stable than the unpaired s1 or d9 configurations. In this letter, we propose the large negative effective U is possible for hole-doped CuAlO2 , AgAlO2 and AuAlO2 in which charge excitation or charge disproportionation in d9 configuration is dominant mechanism for the negative effective U system. For CuAlO2 , AgAlO2 and AuAlO2 , we calculated the total energy of the different charged states, E(N − 1), E(N) and E(N + 1), and estimated the U = E(N + 1) + E(N − 1) − 2E(N). We find that they have large negative effective U which are induced by the charge excitation in Cu2+ (d9 ), Ag2+ (d9 ) and Au2+ (d9 ) electronic configuration. Based on the electron-phonon interaction and the negative effective U, we propose the general design rule to realize the high-Tc superconductor combined ab initio electronic structure calculation and quantum Monte Carlo simulation by using the negative effective U Hubbard model. 2. General Rule of Negative effective U mechanism by the electronic excitations Based on the Periodic Tables of the elements in the compounds, we show the missing oxidation states of the elements in the compounds in the Figure 1, where indicates the missing oxidation states in the existing compounds. In the all of the element as shown in Figure 1, the electronic configurations of the missing oxidation states are only s1 and d9 electronic configurations. It can be naturally concluded that the electronic structure of s1 and d9 electronic configurations are negative effective U system; with U ≡ E(s2 ) + E(s0 ) − 2E(s1 ) < 0 then, we can expect the following charge disproportionation by the energy gaining caused by negative effective U (2s1 → s2 + s0 + |U|), and U ≡ E(d10 ) + E(d8 ) − 2E(d9 ) < 0 then, we can expect the following charge disproportionation (2d9 → d10 + d8 + |U|), where E(sn ) and E(dn ) are the total energies of sn and dn electronic configurations with n-electron systems in the compounds. In the thermal equilibrium condition of the existing compounds in nature, the negative effective U with n-electron system becomes unstable and disproportionate into (n + 1)- and (n − 1)-electron systems, as was shown in schematically in Figure 1 with upword convexity in the total energy as a function of electron occupations. In the insulating compounds, we can stabilize the charge disproportionate state 2
and realize the charge-density-wave (CDW) states, however, in the doped metallic systems, we may stabilize the superconducting states, relative to the CDW states, caused by the attractive electron-electron interactions (negative effective U system). We can propose the Tc -enhancement mechanism in the hole-doped CuBiO2 , CuSbO2 and CuAsO2 , where we can realize Bi4+ , Sb4+ and As4+ which indicates the negative effective U system with missing oxidation states in Figure 1. We may have a possibility of the Tc -enhanced superconductivity in holedoped or electron-doped CuPbO2 and CuSnO2 , where we can realize Pb3+ and Sn3+ in the undoped CuPbO2 (2Pb3+ → Pb2+ + Pb4+ + |U|) and CuSnO2 (2Sn3+ → Sn2+ + Sn4+ + |U|) negative effective U system with missing oxidation states. We also suggest a possible Tc -enhanced superconductivity in electron-doped CuTlO2 , where we can realize Tl2+ , which indicates the negative effective U system with missing oxidation state in Figure 1. We may have another possibility of Tc -enhanced superconductivity caused by the negative effective U system of d9 electronic configurations in the holedoped AuAlO2 , AgAlO2 and CuAlO2 , where Au2+ , Ag2+ and Cu2+ which indicates the negative effective U system in the layered delafossite structure with missing oxidation states as was shown in the case of Au2+ (See Fig. 1). 3. Calculation Methods The calculations were performed within the density functional theory [3, 4] with a plane-wave pseudopotential method, as implemented in the Quantum-ESPRESSO code [5]. We employed the Perdew-Wang 91 generalized gradient approximation (GGA) exchange-correlation functional [6] and ultra-soft pseudopotentials [7]. For the pseudopotentials, 3d electrons of transition metals were also included in the valence electrons. In reciprocal lattice space integral calculation, we used 8 × 8 × 8 k-point grids in the MonkhorstPack grid [8]. The energy cut-off for wave function was 40 Ry and that for charge density was 320 Ry. These k-point grids and cut-off energies are fine enough to achieve convergence within 10 mRy/atom in the total energy. CuAlO2 , AgAlO2 and AuAlO2 have delafossite structure, which belongs to the space group R¯3m (No.166) and is represented by cell parameters a and c, and internal parameter z. These parameters were optimized by the constant-pressure variable-cell relaxation using the Parrinello-Rahman method [9]. After the relaxation, we calculated the total energy of the different charged states, and estimated the effective U. These structure op3
timization and total energy calculations were performed with total charge of the system (tot_charge: input parameter of Quantum-ESPRESSO) = 0.0, 0.1, 0.2, · · · , 2.0. A compensating jellium background is inserted to keep charge neutrality. 4. Calculation Results and Discussion Figures 2, 3 and 4 show the total energy E(N) of CuAlO2 , AgAlO2 and AuAlO2 . Calculated results show the strong upward convexity as a function of N and large negative effective U nature. U = −4.54 eV (CuAlO2 ), U = −4.88 eV (AgAlO2 ) and U = −4.14 eV (AuAlO2 ). Due to the negative effective U nature of d9 electronic configuration of (CuAlO2 )+ , (AgAlO2 )+ and (AuAlO2 )+ , the charge disproportionation from 2d9 → d8 + d10 + |U| can be realized the insulating band-gap opening or itinerant superconducting state, which may indicate the CDW with insulating states or SDW with metallic ferromagnetism or insulating antiferromagnetic state, or Superconductivity (Bose condensation) with itinerant metallic states. When only first nearest neighbor hopping is considered, the highest valence band of CuAlO2 is represented as follows: ǫ(kx , k√ y , kz ) = 2t1 cos(kz dCu−O )+ 2t2 cos(kx dCu−Cu ) + 4t3 cos(kx dCu−Cu /2)cos(ky dCu−Cu 3/2) + ǫ0 . Here, dA−B represents the distance between A atom and B atom. dCu−O = 1.88˚ A, dCu−Cu = 2.86˚ A. We fitted the highest valence band of CuAlO2 with the tight-binding model and found that t1 = −0.0041 eV, t2 = −0.2841 eV, t3 = −0.1329 eV, and ǫ0 = 8.2875 eV. Figure 5 shows this result. 5. How to Design New Materials and Calculate Tc by Negative Effective U system If we want to use the superconductor at the room temperature (∼ 300 K), we need the superconductors which have high-Tc at least up to the 1, 000 K (at least three-times higher than the room temperatures ∼ 300 K) in order to avoid the superconducting fluctuations in the realistic applications such as electric power supply or energy-related technologies. In order to realize an ultra-high-Tc superconductivity up to Tc ∼ 1, 000 K, we need to design a system which contains attractive interaction between the electrons (negative effective U system) up to U < −10, 000 K. The computational materials design searching for a possible candidate of this system is to find a negative effective U system, which is originated by purely electronic mechanism with 4
the attractive interaction of -(1∼4)eV. In order to find the purely electronic attractive interactions (negative effective U system) and design the high-Tc superconductors, we propose the following three steps for the computational materials design. (1) Step 1: we should find the superconductors with Tc ≥ 50 K caused by a strong electron-phonon interactions based on the pseudo-two-dimensional flat-band. In order to avoid the successive phase transition, we need to design a new doping method, which should stabilize the atomic structures such as doping the holes at the anti-bonding states or doping the electron at the bonding states. Doping a hole to the d10 configuration or doping a electron to the d10 configuration is good candidate for this purposes. In this case, we can estimate the electron-phonon interaction mediated Tc correctly based on the conventional ab initio electronic structure calculations. (2) Step 2: we should design and calculate the negative effective U system which is originated by purely electronic mechanism with the attractive interaction of U = −(1 ∼ 4) eV. Possible candidates are (i) charge-excitationinduced negative effective U system for s1 , d9 and f 13 electronic configurations, which is purely charge-excitation originated, and (ii) exchangecorrelation-induced negative effective U system for d4 ,d6 , f 6 and f 8 electronic configurations, which is purely exchange-correlation originated by electronic mechanism with the attractive interactions of −(1 ∼ 4) eV. We can calculate and find the system of, U = E(s2 ) + E(s0 ) − 2E(s1 ) < 0, U = E(d10 ) + E(d8 ) − 2E(d9 ) < 0, U = E(d5 ) + E(d3 ) − 2E(d4 ) < 0, U = E(d7 ) + E(d5 ) − 2E(d6 ) < 0, U = E(f 7 ) + E(f 5 ) − 2E(f 6 ) < 0, U = E(f 9 ) + E(f 7 ) − 2E(f 8 ) < 0, where E(sn ), E(dn ) and E(f n ) are the total energies of sn , dn and f n electronic configurations with n-electron systems in the compounds. (3) Step 3: we should estimate the Tc of negative effective U system based on the Monte Carlo simulation of the mapped negative negative effective U Hubbard model by using the negative effective U, hopping integral t and chemical potential µ which are calculated from the first principles. We can perform the multi-scale simulation of the superconductivity based on the hybrid calculation method combined the ab initio calculation and phenomenological model mapping on the negative effective U Hubbard model. In this multi-scale simulation, the calculated Tc is quantitatively and we can design a new class of electronic-originated high-Tc superconductors without any adjustable or empirical parameters.
5
6. Conclusions We proposed general rule of purely electronic mechanism for the negative effective U systems; Exchange correlation-induced negative effective U for d4 or d6 electronic configuration, which is caused by the stability of Hund’s rule of d5 electronic configuration, and charge-excitation-induced negative effective U for s1 , p1 , p5 , d9 electronic configuration, which is caused by the stability of closed-shells in s2 , p6 , and d10 electronic configuration. Combined with the first principles calculation of Tc by electron-phonon interaction, the purely electronic mechanisms for the negative effective U system, and multi-scale simulation of Tc and phase diagram by Monte Carlo simulation of negative U Hubbard model, we propose the Computational Materials Design Methodology for the realization of high-Tc superconductors. We applied the total energy calculation to evaluate the effective U of CuAlO2 (U = −4.54 eV), AgAlO2 (U = −4.88 eV) and AuAlO2 (U = −4.14 eV), and found large negative effective U for this system. The negative effective U nature of this system is pure electronic and consistent with the charge excitation-induced negative effective U system. Acknowledgment The authors acknowledge the financial support from the Global Center of Excellence (COE) program ”Core Research and Engineering of Advanced Materials - Interdisciplinary Education Center for Materials Science”, the Ministry of Education, Culture, Sports, Science and Technology, Japan, and a Grant-in-Aid for Scientific Research on Innovative Areas ”Materials Design through Computics: Correlation and Non-Equilibrium Dynamics”. We also thank to the financial support from the Advanced Low Carbon Technology Research and Development Program, the Japan Science and Technology Agency for the financial support. References [1] H. Katayama-Yoshida, A. Zunger, Phys. Rev. Lett. 55 (1985) 1618. [2] H. Katayama-Yoshida, K. Kusakabe, H. Kizaki, A. Nakanishi, Appl. Phys. Express 1 (2008) 081703. [3] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. 6
Figure 1: Missing oxidation states are shown by in the Periodic Tables (Periodic Table of Elements, WILEY-VCH, ISBN 978-3-527-31856-8). The existing oxidation numbers are shown in this figure. The missing oxidation states in the existing compounds are the s1 and d9 electronic configurations, which indicate the negative effective U systems. If the n-electron system shows the negative effective U , the n-electron system becomes unstable and disproportionate into the (n + 1)- and (n − 1)-electron systems, as was shown in schematically with up-word convexity in the total energy as a function of electron occupations.
7
Figure 2: Total energy of CuAlO2 as a function of number of doped holes. Calculated effective U is −4.54 eV.
Figure 3: Total energy of AgAlO2 as a function of number of doped holes. Calculated effective U is −4.88 eV.
8
Figure 4: Total energy of AuAlO2 as a function of number of doped holes. Calculated effective U is −4.14 eV.
[4] W. Kohn, L. J. Sham, Phys. Rev. 140 (1965) A1133. [5] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. Fabris, G. Fratesi, S. de Gironcoli, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, R. M. Wentzcovitch, J. Phys.: Condens. Matter 21 (2009) 395502. [6] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [7] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [8] H. J. Monkhorst, J. D. Pack, Phys. Rev. B 13 (1976) 5188. [9] M. Parrinello, A. Rahman, Phys. Rev. Lett. 45 (1980) 1196.
9
Figure 5: The highest valence band of CuAlO2 and tight binding fitting. Using the calculated effective U in CuAlO2 and fitted tight-binding band structure based on first principles electronic structure calculations, we can make mapping on the negative effective U Hubbard model. We can do Quantum Monte Carlo calculation and determined the phase diagram and Tc quantitatively by multi-scale simulation without experimentally observed or empirical parameters.
10
