Electronic structures of quasi-one-dimensional cuprate superconductors Ba$_2$CuO$_{3+\delta}$

TL;DR

This study uses first-principles calculations to explore the electronic and magnetic properties of Ba$_2$CuO$_{3+\delta}$, revealing its antiferromagnetic insulating state and the role of CuO chains in potential superconductivity.

Contribution

It provides a detailed theoretical analysis of Ba$_2$CuO$_{3+\delta}$, highlighting the importance of CuO chains and magnetic fluctuations in its superconducting behavior, which differs from traditional cuprates.

Findings

Ba$_2$CuO$_3$ is an antiferromagnetic Mott insulator.

Oxygen vacancies prefer planar sites in CuO chains.

Magnetic frustrations can be induced by moderate charge doping.

Abstract

An intact CuO$_2$ plane is widely believed to be a prerequisite for the high-$T_c$ superconductivity in cuprate superconductors. However, an exception may exist in the superconducting Ba$_2$CuO$_{3+\delta}$ materials where CuO chains play a more important role. From first-principles density functional theory calculations, we have studied the electronic and magnetic structures of Ba$_2$CuO$_{3+\delta}$. The stoichiometric Ba$_2$CuO$_3$ and Ba$_2$CuO$_4$ contain quasi-one-dimensional CuO chains and intact two-dimensional CuO$_2$ planes, respectively. In comparison with the nonmagnetic metal Ba$_2$CuO$_4$, Ba$_2$CuO$_3$ is found to be an antiferromagnetic (AFM) Mott insulator. It possesses a nearest-neighbor intra-chain antiferromagnetic (AFM) coupling and a weak inter-chain interaction, and its lowest unoccupied band and highest occupied band are contributed by Cu 3$d_{b^2-c^2}$-orbital (or $d_{x^2-y^2}$-orbital if we denote the $bc$-plane as the $xy$-plane) and O 2$p$-orbitals, respectively. Total energy calculations indicate that the oxygen vacancies in Ba$_2$CuO$_{3+\delta}$ prefer to reside in the planar sites rather than the apical oxygens in the CuO chains, in agreement with the experimental observation. Furthermore, we find that the magnetic frustrations or spin fluctuations can be effectively induced by moderate charge doping. This suggests that the superconducting pairing in oxygen-enriched Ba$_2$CuO$_{3+\delta}$ or oxygen-deficient Ba$_2$CuO$_{4-\delta}$ is likely to be mainly driven by the AFM fluctuations within CuO chains.

Tables

Table 1. Table 1: Relative energies (in unit of meV/Cu) of several magnetic states with respect to the nonmagnetic (NM) state for undoped Ba 2 CuO 3 .

	Néel	collinear	AFM1	dimer	FM
Ba2CuO3	-180	-22	-180	-116	-22

Table 2. Table 2: Relative energies (in unit of meV/Cu) of the AFM1 state, the AFM dimer state, and the FM state with respect to that of the nonmagnetic state for Ba 2 CuO 3 under different electron ( e 𝑒 e ) and hole ( h ℎ h ) doping concentrations (per Cu atom).

doping	AFM1	dimer	FM
0	-180	-116	-22
0.1$e$	-97	-70	-35
0.15$e$	-61	-58	-47
0.15$h$	-79	-36	+11
0.3$h$	-27	-13	-3

Table 3. Table 3: Relative total energies (in unit of eV) for all 26 structures of Ba 2 CuO 3.25 shown in Figs. 6 and 7 with oxygen vacancies in a 2 × 2 × 1 2 2 1 2\times 2\times 1 supercell which contains 16 Ba atoms, 8 Cu atoms, 26 O atoms, and 6 O vacancies.

	BaO-1	BaO-2	82-1	82-2	82-3	73-1	73-2	64-1	64-2	64-3	64-4	64-5	64-6
$\Delta E$	-1.871	0	-5.168	-4.450	-4.480	-6.350	-6.190	-8.173	-7.319	-7.069	-7.614	-7.380	-7.142
	55-1	55-2	55-3	55-4	64-1*	64-3*	64-4*	64-5*	64-6*	55-1*	55-2*	55-3*	55-4*
$\Delta E$	-7.779	-7.736	-7.779	-7.711	-8.201	-7.042	-7.606	-7.401	-7.206	-7.729	-7.657	-7.704	-7.683

Full text

I Introduction

The interplay between superconductivity and dimensionality is an interesting issue in condensed matter physics. In the well-known cuprate superconductors Bednorz86 ; Stewart17 , whose superconducting transition temperature is highest at ambient pressure Schilling93 ; ChuCW94 , a common feature is that they all contain quasi-two-dimensional CuO2 planes Dagotto94 ; Leggett06 from which the superconducting pairing emerges while other layers just serve as a charge reservoir. Similarly, superconducting condensation is also believed to take place predominately in quasi-two-dimensional Fe$X$ ($X$=As, Se, …) layers in iron-based superconductors sc1 ; sc2 ; sc3 ; sc4 ; sc5 . In recent years, superconducting transitions have also been discovered in Bechgaard-salts organic superconductors Bechgaard80 ; Bechgaard81 , molybdenum chalcogenides Armici80 ; Potel80 and pnictides RenZA18 , chromium pnictides CaoGH15a ; CaoGH15b ; CaoGH15c ; RenZA18PRM ; RenZA17a ; RenZA17b , bismuth iodide QiYP18 , nickel-bismuth compounds XueQK18 , and other quasi-one-dimensional materials.

More recently, high temperature superconductor Ba2CuO4-δ/Ba2CuO3+δ has been successfully synthesized under high pressure by Jin and coworkers JinCQ18 . There are two possibilities regarding the parent compound of these superconductors. One possibility is that Ba2CuO4 is the parent compound. High-$T_{c}$ superconductivity emerges in Ba2CuO4-δ when some oxygens are removed from Ba2CuO4. Another possibility is that Ba2CuO3 is the parent compound and the superconductivity emerges in Ba2CuO3+δ when more oxygens are doped to it. For the latter, unlike the parent compound of other cuprate superconductors, Ba2CuO3 does not contain CuO2 planes. Instead, it contains only CuO chains and the copper-oxygen bond is compressed along the $c$-axis but stretched along the $b$-axis (the CuO chain direction).

In order to determine the pairing mechanism of electrons, it is crucial to know where the oxygen vacancies reside in the superconducting Ba2CuO4-δ/Ba2CuO3+δ compounds. If the oxygen vacancies mainly occupy the apical sites, the CuO2 planes remain intact as the other cuprate superconductors. Experimental measurements of neutron powder diffraction Shimakawa94 and electron diffraction Zhang95 for similar compounds, however, suggested that the oxygen vacancies reside mainly on the CuO2 planes and the apical sites are fully occupied by oxygen atoms. In this case, there are no perfect CuO2 planes. This implies that intact CuO2 planes are not absolutely necessary in achieving high-$T_{c}$ superconductivity.

In this paper, we address the above questions by calculating the electronic and magnetic structures of Ba2CuO3+δ using first-principles density functional theory. Our study shows that the parent compound is Ba2CuO3, which is an AFM Mott insulator. In Sec. II, we first describe the method, and then discuss the numerical results for both stoichiometric Ba2CuO3/Ba2CuO4 and oxygen doped materials. A summary is given in Sec. III.

II Results

We performed the first-principles density functional theory calculations on the electronic and magnetic structures of undoped or doped Ba2CuO3 or Ba2CuO4 materials using the projector augmented wave (PAW) method PAW with the Vienna ab initio simulation package VASP1 ; VASP2 . The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type was used in the construction of the exchange-correlation functional PBE . The plane-wave basis set with a kinetic energy cutoff of 520 eV and a $12\times 12\times 4$ $k$-point mesh for the Brillouin zone sampling of conventional cell were adopted. The Fermi level was broadened by the Gaussian smearing method with a width of 0.05 eV. Ba2CuO3+δ can be regarded as a compound either by adding oxygen to Ba2CuO3 or by removing oxygen from Ba2CuO4. Various locations of oxygen vacancies in Ba2CuO3+δ (or Ba2CuO4-δ) were investigated with both the virtual crystal approximation (VCA) and the supercell methods. In the latter case, a $2\times 2\times 1$ supercell with different oxygen vacancy distributions was explored, while the lattice constants were fixed to the experimental values, and all internal atomic positions were allowed to relax until the forces on all atoms were smaller than 0.01 eV/Å. The correlation effect among Cu 3$d$ electrons was incorporated using the GGA+U formalism of Dudarev et al. DFTU with a typical effective Hubbard interaction U of 6.5 eV.

II.1 Electronic and magnetic structures of stoichiometric Ba2CuO3 and Ba2CuO4

Figures 1(a) and 1(b) show the crystal structures of Ba2CuO3 and Ba2CuO4, respectively. The former is composed of CuO chains along the $b$-axis, while the latter contains CuO2 planes. In either case, Cu-O layers are separated by Ba-O layers. Cu atoms in both Ba2CuO3 and Ba2CuO4 form a square-like lattice in each $ab$-plane. Ba2CuO4 has the same lattice structure as La2CuO4. However, it is only in Ba2CuO3 that Cu ions have the same nominal valence $+2$ as in La2CuO4.

The $a$ and $b$ axes in Ba2CuO4 are equivalent because there are no O vacancies in the CuO2 planes. However, they are not equivalent in Ba2CuO3 [Fig. 1(a)]. To determine the ground states of Ba2CuO3 and Ba2CuO4, we calculated the energies of several representative magnetic states formed by Cu2+ ions and compared with that of the nonmagnetic state. These magnetic ordered states, as partially shown in Fig. 2, include the standard AFM Néel state, the collinear AFM (or stripe) state, the AFM1 state in which Cu spins are antiferromagnetically coupled along the $b$-axis but ferromagnetically coupled along the $a$-axis, the AFM dimer state Gong15 ; Liu16 along the $b$-axis which is ferromagnetically coupled along the $a$-axis, and the ferromagnetic (FM) state.

For Ba2CuO3, the relative energies of the above magnetic states with respect to the nonmagnetic state are given in Table 1. All these magnetic states have lower energies than the nonmagnetic one. Among them, the AFM Néel state is energetically degenerate with the AFM1 state, and the collinear AFM state is energetically degenerate with the FM state. The ground state is found to have either the AFM Néel or the AFM1 order, indicating that there is a strong intra-chain AFM coupling along the $b$-axis but a weak inter-chain coupling along the $a$-axis. The AFM dimer state is next to the ground state in energy. In contrast, for Ba2CuO4, all the above magnetic states are unstable and the ground state is nonmagnetic. From the calculations, we find that the weak inter-chain coupling in Ba2CuO3 is about 0.2 meV, smaller than the corresponding values in Ca2CuO3 (3.6 meV) and in Sr2CuO3 (0.8 meV) Rosner97 ; Graaf00 .

Figure 3 shows the total density of states (DOS) for Ba2CuO3 in the AFM Néel state and Ba2CuO4 in the nonmagnetic state, respectively. In contrast to Ba2CuO3 which is an AFM insulator, Ba2CuO4 is found to be a nonmagnetic metal. The calculated band gap of Ba2CuO3 is 1.2 eV, close to the experimental results for Ca2CuO3 (1.7 eV) and Sr2CuO3 (1.5 eV) Maiti98 . It should be pointed out that the AFM insulating ground state of Ba2CuO3 is obtained only when the on-site Hubbard interaction is included in the calculation. Without this interaction, the ground state is metallic. This suggests that Ba2CuO3 is an AFM Mott insulator Schlappa12 , similar to La2CuO4 Wen06 .

We also calculated the partial density of states (PDOS) for Ba2CuO3 in the AFM Néel state (Fig. 4). For convenience, here we denote the $b$ and $c$ axes as the $x$ and $y$ axes [Fig. 1(a)]. Due to the nominal valence $+2$ of Cu ions in Ba2CuO3, nine electrons fill the 3$d$ orbitals of Cu atom. Among these 3$d$ orbitals, the spin-minority $d_{x^{2}-y^{2}}$ orbital is mostly unoccupied [Fig. 4(a)]. The occupied bands are mainly contributed by O 2$p$ orbitals. This results from the crystal field effect created by four O atoms surrounding a Cu atom in the $bc$ plane. The PDOS peaks of unoccupied Cu $d_{x^{2}-y^{2}}$ and O 2$p_{x}$ states appear at almost the same energy above the band gap. This indicates that there is a strong $p$-$d$ hybridization between these orbitals.

Figures 5(a) and 5(b) show the calculated band structures of Ba2CuO3 along high-symmetry paths of the Brillouin zone in the nonmagnetic and the AFM Néel states, respectively. In the AFM state, the valence band maximum is located at the M point. Since the CuO chain is along $b$ axis and the inter-chain couplings are rather weak, the band dispersions along the $k_{b}$ direction are much larger than those along the other two directions.

In oxygen-enriched Ba2CuO3+δ, the extra oxygens doped to Ba2CuO3 introduce hole doping, which is equivalent to shifting the Fermi energy down to the valence bands. In Figs. 5(c) and 5(d), we show the Fermi surface contours when the Fermi level is 0.05 and 0.1 eV below the top of the valence bands in the AFM Néel state, respectively. Upon small hole doping, a hole pocket emerges around the M point. When more holes are doped into the system, the hole pockets merge together to form flat Fermi surface sheets in the $k_{a}$-$k_{c}$ plane.

II.2 Electronic and magnetic structures of Ba2CuO3+δ

In high-$T_{c}$ cuprates, superconductivity emerges when charge carriers, either holes or electrons, are doped into the parent compounds. It is commonly believed that AFM spin fluctuations play a very important role in pairing electrons Moriya00 ; Wen06 ; Scalapino12 . One can dope holes to Ba2CuO3 either by introducing extra oxygen atoms or by substituting Ba atoms with alkali atoms Ueno08 ; Bollinger11 . The superconducting material Ba2CuO3+δ JinCQ18 ; Yamamoto97 ; Yamamoto00 ; Karimoto03 can be regarded as a compound either by introducing more oxygens to Ba2CuO3 or by removing oxygens from Ba2CuO4.

To know the electronic and magnetic structures of this material, it is important to know accurately where the oxygen vacancies are located. We investigated this problem using both the virtual crystal approximation approach and the supercell approach in the framework of density functional theory.

In the virtual crystal approach, we considered three different kinds of distributions of oxygen vacancies: (1) the vacancies are all located in the Cu-O layers, (2) the vacancies are all located in the Ba-O layers, and (3) the vacancies are equally partitioned in both kinds of layers. For a nominal component Ba2CuO3.2, the relative energies of the second and third cases with respect to the first case are found to be +1.88, and +92.58 eV/cell, respectively. This suggests that the oxygen vacancies prefer to reside in the Cu-O layers.

In the supercell approach, we calculated the energies for 26 possible structures of Ba2CuO3.25 (Figs. 6 and 7) in a $2\times 2\times 1$ supercell with 6 oxygen vacancies. By comparison, we find that the two configurations shown in Figs. 6(a) and 6(b) have the lowest and the second lowest energies, respectively. In both cases, we also find that the Ba-O layer keeps intact, while the oxygen vacancies also prefer to locate in the Cu-O layers rather than occupying the apical oxygen positions. The relative total energies of these 26 structures are listed in Table 3.

The results obtained from both approaches indicate that the oxygen vacancies are located in the Cu-O layers rather than in the Ba-O layers. This is consistent with the experimental data of neutron powder diffraction Shimakawa94 and electron diffraction Zhang95 measurements for Sr2CuO3+δ.

To see how the magnetic order is changed by the doping in Ba2CuO3+δ, we evaluated the energies of several different kinds of magnetic states. Since the inter-chain coupling is very weak, we ignore the difference of different magnetic ordered states along the $a$-axis and assume that the inter-chain coupling along the $a$-axis is ferromagnetic. Table 2 shows how the energies of four typical magnetic states (i.e. nonmagnetic, AFM1, AFM dimer, and FM states) vary with the electron or hole doping. As expected, the energy difference between the lowest-energy (AFM1) state and the second-lowest-energy (dimer) state is reduced with increasing doping. This suggests that the doping tends to induce magnetic frustrations Liu15 and enhance spin fluctuations, favoring the formation of superconducting Cooper pairs.

The above discussion suggests that it is Ba2CuO3 rather than Ba2CuO4 that is the parent compound of superconducting Ba2CuO3+δ materials Yamamoto97 ; Yamamoto00 ; Karimoto03 ; JinCQ18 . This can be more clearly seen by comparison with the other parent compounds of cuprate superconductors, such as La2CuO4. First, like La2CuO4, Ba2CuO3 is an AFM insulator. Second, Cu ions in Ba2CuO3 have the same $+2$ nominal valences as in La2CuO4. Third, the highest occupied and the lowest unoccupied states in both compounds derive from the strongly hybridized O 2$p$ orbitals and Cu 3$d_{x^{2}-y^{2}}$ orbitals (for Ba2CuO3, the $bc$ plane is defined as the $xy$ plane and $d_{x^{2}-y^{2}}$ is just the $d_{b^{2}-c^{2}}$ orbital).

III Summary

We have studied the electronic and magnetic structures of Ba2CuO3+δ with $\delta$ varying from 0 to 1 by first-principles density functional theory calculations. Unlike Ba2CuO4 whose ground state is a non-magnetic metal, the ground state of Ba2CuO3 is found to be an quasi-one-dimensional AFM Mott insulator. The lowest unoccupied and highest occupied states in Ba2CuO3 are mainly contributed by Cu 3$d_{b^{2}-c^{2}}$ orbitals and O 2$p$ orbitals, respectively. By comparison of the total energies of Ba2CuO3+δ with different kinds of oxygen vacancy structures, we find that the oxygen vacancies reside mainly in the Cu-O layers (planar sites) rather than in the Ba-O layers (apical sites), in agreement with the experimental observation. Furthermore, doping of charge carriers to Ba2CuO3 can reduce the energy differences between different low-energy magnetic states and thus enhance the spin fluctuations. This suggests that Ba2CuO3 is the parent compound of oxygen doped Ba2CuO3+δ superconductors, and the AFM fluctuations in CuO chains play an important role in the superconducting pairing of electrons in these materials.

Acknowledgements.

We thank C. Q. Jin for stimulating discussions. This work was supported by the National Key R$\&$D Program of China (Grant No. 2017YFA0302900) and the National Natural Science Foundation of China (Grants No. 11774422 and No. 11774424). Computational resources were provided by the Physical Laboratory of High Performance Computing at Renmin University of China.