Nematic transition and highly two-dimensional superconductivity in BaTi$_2$Bi$_2$O revealed by $^{209}$Bi-nuclear magnetic resonance/nuclear quadrupole resonance measurements

TL;DR

This study uses $^{209}$Bi-NMR/NQR to reveal electronic nematicity and highly two-dimensional superconductivity in BaTi$_2$Bi$_2$O, highlighting differences from related compounds and implications for the superconducting phase diagram.

Contribution

First microscopic investigation showing nematic electronic state and two-dimensional superconductivity in BaTi$_2$Bi$_2$O using NMR/NQR measurements.

Findings

Detection of in-plane anisotropic parameter indicating nematic order.

No change in $1/T_1T$ below $T_c$ despite superconducting transition.

Distinct normal and superconducting properties compared to BaTi$_2$Sb$_2$O and BaTi$_2$As$_2$O.

Abstract

In this Rapid Communication, a set of $^{209}$Bi-nuclear magnetic resonance (NMR)/nuclear quadrupole resonance (NQR) measurements has been performed to investigate the physical properties of superconducting (SC) BaTi$_2$Bi$_2$O from a microscopic point of view. The NMR and NQR spectra at 5~K can be reproduced with a non-zero in-plane anisotropic parameter $\eta$, indicating the breaking of the in-plane four-fold symmetry at the Bi site without any magnetic order, i.e., `the electronic nematic state'. In the SC state, the nuclear spin-lattice relaxation rate divided by temperature, $1/T_1T$, does not change even below $T_{\rm c}$, while a clear SC transition was observed with a diamagnetic signal. This observation can be attributed to the strong two-dimensionality in BaTi$_2$Bi$_2$O. Comparing the NMR/NQR results among BaTi$_2$$Pn$$_2$O ($Pn$ = As, Sb, and Bi), it was found that the normal and SC properties of BaTi$_2$Bi$_2$O were considerably different from those of BaTi$_2$Sb$_2$O and BaTi$_2$As$_2$O, which might explain the two-dome structure of $T_{\rm c}$ in this system.

Equations

$${\mathcal{H}}_Q$$

$$\mathcal{H}$$

$$=-\frac{\gamma }{2\pi }h(1+K)\mathbf{I}\cdot \mathbf{H}+{\mathcal{H}}_{\mathrm{Q}},$$

$$\frac{1}{T_{1}T}=\frac{2{\gamma }_N^2{k}_{\mathrm{B}}}{({\gamma }_e^2\hslash {)}^2}\underset{{\omega }_0\to 0}{\lim }\sum _{\mathbf{q}}{A}_{\mathbf{q}}{A}_{-\mathbf{q}}\frac{\chi {"}_{\perp }(\mathbf{q},{\omega }_0)}{{\omega }_0},$$

Full text

An electronic nematic transition, which is characterized by spontaneous rotational symmetry breaking in the electronic system is often observed in strongly coupled superconductorsOkazaki et al. (2011); Kitagawa et al. (2013); Baek et al. (2014); Hosoi et al. (2016); Sato et al. (2017). The nematic phase extends beyond the superconducting (SC) phase in the phase diagram of most superconductors. The relationship between superconductivity and the nematic state, and the origin of the nematic transition, especially in iron-based superconductors, are phenomena of intensive debate in the superconductivity community. However, the relationship and the origin remain elusive. Therefore, to understand these phenomena, it is important to investigate and compare various superconductors that exhibit a nematic transition.

Recently, superconductivity in the vicinity of a charge density wave (CDW) phase was discovered in BaTi${}_{2}Pn_{2}$O ($Pn$ = As, Sb, and Bi)Yajima et al. (2012, 2013a, 2013b); Yajima (2017), which possesses a two-dimensional layered structure as shown in Fig. 1(a). BaTi${}_{2}Pn_{2}$O crystallizes into a tetragonal structure of space group $P4/mmm$ (No.123, $D_{4h}^{1}$) with alternately stacked Ti${}_{2}Pn_{2}$O layers and Ba atoms along the $c$ axis. The Ti${}_{2}Pn_{2}$O layers contain a Ti2O square net, which is an anti-configuration to the CuO2 square net. The edge-shared TiO${}_{2}Pn_{4}$ octahedra form a square lattice, and the electronic state of Ti3+ is in the 3$d^{1}$ state, which is regarded as an electron–hole symmetric state of the 3$d^{9}$ state in Cu*2+*Yajima et al. (2012). BaTi2As2O shows an anomaly in the nematic state at $T_{\rm A}$ = 200 K, which is ascribed to a CDW transitionWang et al. (2010).

This anomaly was suppressed by the substitution of Sb for As and $T_{\rm A}$ becomes $\sim$40 K in the end member BaTi2Sb2OKitagawa et al. (2013). In BaTi2Sb2O, an SC transition was also observed at an SC transition temperature of $T_{\rm c}$ = 1.2 K. 121/123Sb-nuclear magnetic resonance(NMR)/nuclear quadrupole resonance(NQR) measurements in BaTi2Sb2O revealed the breaking of in-plane fourfold symmetry at the Sb site below $T_{\rm A}$ without an internal field appearing at the Sb siteKitagawa et al. (2013), which indicated an electronic nematic transition at $T_{\rm A}$. The muon spin rotation ($\mu$SR) measurements also indicated no internal magnetic field below $T_{\rm A}$Nozaki et al. (2013), and a long-range structural phase transition was found in the neutron diffraction measurementsFrandsen et al. (2014). These results can be understood with commensurate nematic CDW orderingNakaoka et al. (2016) or $p$-$d$ bond orderSong et al. below $T_{\rm A}$. In both cases, the nematic state is realized in the CDW phase. Moreover, 121/123Sb-NMR/NQR measurements strongly suggest that SC symmetry is a conventional $s$ wave in BaTi2Sb2OKitagawa et al. (2013), which is in sharp contrast with those in the cuprate and iron-based superconductors.

With the substitution of Bi for Sb, $T_{\rm A}$ was suppressed, $T_{\rm c}$ shows a dome shape with maximum $T_{\rm c}$ = 3.5 K at $x=0.2$, and superconductivity terminates at $x=0.4$ in BaTi2(Sb1-xBix)2OYajima et al. (2013a). Interestingly, superconductivity reappears upon further substitution, and a two-dome structure in $T_{\rm c}$ was observed as shown in Fig. 1(b). In the end member BaTi2Bi2O, an SC transition was observed at 4.6 K without any trace of the CDW/spin density wave (SDW) transition. Since there is a possibility that the SC properties of BaTi2Bi2O are different from those of BaTi2Sb2O, it is important to investigate BaTi2Bi2O to understand the normal and SC properties of the BaTi${}_{2}Pn_{2}$O system.

In this Rapid Communication, 209Bi-NMR/NQR measurements have been performed to investigate the physical properties of BaTi2Bi2O from a microscopic point of view. From the temperature evolution of the NQR spectra, an electronic nematic transition at $\sim$45 K was discovered. In the SC state, the nuclear spin-lattice relaxation rate divided by temperature, $1/T_{1}T$, does not change even below $T_{\rm c}$, while a clear SC transition was observed by an ac susceptibility measurement. This is ascribed to the strong two-dimensionality in BaTi2Bi2O. Although the NQR spectra in both BaTi2Sb2O and BaTi2Bi2O indicate the breaking of fourfold symmetry at the Sb/Bi site below $\sim$40 K, the normal and SC properties of BaTi2Bi2O are considerably different from those of BaTi2Sb2O.

Polycrystalline samples of BaTi2Bi2O were synthesized via the conventional solid-state reactionYajima et al. (2013a). Stoichiometric amounts of BaO, Ti, and Bi were mixed and pelletized. The pellet was then wrapped in a Ta foil and sealed in a quartz tube. The reaction temperature was 850oC. To prevent sample degradation by air and/or moisture, the polycrystalline samples were mixed with Araldite adhesive. The mixture was then solidified with random crystal orientation. All the procedures were performed in a glove box filled with Ar. The SC transition at 4.6 K was confirmed by a dc magnetization measurement with a commercial superconducting quantum interference device (SQUID) magnetometer (Quantum Design, MPMS3), and the ac susceptibility measurement was performed using an NMR coil after the mixing. A spin-echo technique was used for the NMR/NQR measurements. The 209Bi-NMR spectra (nuclear spin $I$ = 9/2, nuclear gyromagnetic ratio ${}^{209}\gamma/2\pi=6.842$ MHz/T, and natural abundance 100% ) were obtained as a function of magnetic field in a fixed frequency $f$ = 68.41 MHz ($\sim 10$ T). The 209Bi nuclear spin-lattice relaxation rate $1/T_{1}$ was determined by fitting the time variation of the spin-echo intensity after the saturation of the nuclear magnetization to a theoretical function for $I$ = 9/2.

The 209Bi-NQR measurements indicate an electronic nematic state at low temperatures. Figure 2 shows the 209Bi-NQR spectra that were obtained by the frequency-swept method at 5 K. When $I\geq 1$, a nucleus has an electric quadrupole moment $eQ$ as well as a magnetic dipole moment; thus, the degeneracy of nuclear energy levels is lifted even at zero magnetic field due to the interaction between $eQ$ and the electric field gradient (EFG). This interaction is described as

[TABLE]

where $h$ is the Planck’s constant, $\nu_{zz}$ is the quadrupole frequency along the principal axis ($c$-axis) of the EFG and is defined as $\nu_{zz}\equiv 3e^{2}qQ/2I(2I-1)$ with $eq=V_{zz}$, and $\eta$ is an asymmetry parameter of the EFG expressed as $(V_{xx}-V_{yy})/V_{zz}$ with $V_{\alpha\alpha}$, which is the second derivative of the electric potential $V$ and the EFG along the $\alpha$ direction ($\alpha=x,y,z,$). When 209Bi is in the EFG, the degenerate ten nuclear-spin states are split into five energy levels, yielding four (or more) resonance frequencies as shown in Fig. 2. The NQR parameters $\nu_{zz}=3.63$ MHz and $\eta=0.5$ were obtained by comparing the observed 209Bi-NQR spectra and calculated resonance frequencies obtained from the diagonalization of Eq.(1). A nonzero $\eta$ implies the breaking of fourfold symmetry at the Bi site, which has a $C_{4}$ symmetry in the tetragonal structure (also called the “electronic nematic state”). It was observed that the low-intensity peak at the highest frequency corresponded to the $\nu_{1}$ + $\nu_{2}$, which is caused by the formation of hybrid states due to nonzero $\eta$Karube et al. (2011).

The nonzero $\eta$ was also confirmed by the field-swept NMR spectrum as shown in Fig. 3. For the NMR measurements, although the nuclear energy levels were already split by the electric quadrupole interaction, magnetic fields were still applied to lift the degeneracy of the spin degrees of freedom. The total effective Hamiltonian could then be expressed as

[TABLE]

where $K$ is the Knight shift and $\bm{H}$ is an external field. As the NMR spectra are varied against the angle between the principal axis of the EFG and magnetic field direction, the sum of the spectrum for all the corresponding angles is observed in the case of the powder samples. The NMR spectrum at 5 K was consistently reproduced by the NQR parameters determined through the NQR measurements at 5 K. In contrast, the NMR spectrum at 50 K can be fitted by the simulation with $\nu_{zz}=3.50$ MHz and $\eta=0.0$. The small difference between experimental and simulated values might have originated from the impurity phase and/or the degree of orientation. It should be noted that the double-horn-shaped satellite signals are the characteristic feature of the $\eta=0.0$. This reflects the preservation of the fourfold symmetry of the crystal structure. To estimate the transition temperature, the temperature evolution of the NQR spectra was measured. Figure 4 shows the temperature dependence of $\eta$ deduced from the first and second largest NQR peaks ($\sim$10 and 14 MHz). The figure indicates that $\eta$ assumes a nonzero value below $T_{\rm anom}~{}\sim$45 K. As such, a nematic phase transition was not reported in previous experimentsYajima et al. (2013b). The most plausible reason behind this is that the anomalous transition may have been so small that it could be detected only by a highly sensitive probe in an electric environment, such as the NQR measurement. A similar nematic transition was observed at $\sim$40 K in BaTi2Sb2O. To reveal the origin of the nematic transitions and the mechanism by which the nematic phases cover the two SC domes in BaTi2(Sb1-xBix)2O, it is important to first understand the relationship between superconductivity and the electronic nematic state.

Figure 5 shows the temperature dependence of $1/T_{1}T$ at $\sim$10 T. $1/T_{1}T$ was measured at the center peak of the Bi-NMR spectrum as indicated by arrows in Fig. 3. $1/T_{1}T$ slightly increases upon cooling and no measurable anomaly was observed around $T_{\rm anom}$. To investigate the SC properties, the temperature dependence of $1/T_{1}T$ at the NQR signal (14.2 MHz peak) as shown in the inset of Fig. 5 was also measured. While a clear SC transition was observed at 4.6 K by the ac susceptibility measurement, $1/T_{1}T$ did not change even below $T_{\rm c}$.

First, we discuss the unaltered behavior of $1/T_{1}T$ in the SC state. In conventional superconductors, $1/T_{1}T$ decreases at temperatures well below $T_{\rm c}$ due to the reduction of the quasi-particle density of states around the Fermi energy $E_{\rm F}$ by opening the SC energy gap. There are two possible reasons why $1/T_{1}T$ did not decrease in the SC state. One possibility is that spin diffusion due to low dimensionality prevented the reduction of $1/T_{1}T$Julien (2008). A similar absence of a clear reduction of $1/T_{1}T$ was observed in (La0.87Ca0.13)FePONakai et al. (2008) and certain cuprate superconductorsKambe et al. (1994); Julien (2003). This brings us to the other possibility that the coupling between the Ti2O layer and Bi atom was so weak that the effect of the SC transition could not be detected even by 209Bi-NQR measurements. Such a scenario was discovered at the Cu(1) site in YBa2Cu3O7-δ and YBa2Cu4O8+δMali et al. (1987); Zimmermann et al. (1989). $1/T_{1}$’s at the first and second largest NQR peaks (10.35 and 14.2 MHz) are not different, although the frequency dependency is expected in the spin diffusion scenarioJulien (2008). In both cases, low dimensionality is important for such an anomalous behavior to occur; therefore, it can be concluded that the NMR results indicate that BaTi2Bi2O is a two-dimensional superconductor, and $T_{\rm c}$ in this system is enhanced by low dimensionalityYanase et al. (2003). Such two dimensionality was not predicted by earlier band calculationsSuetin and Ivanovskii (2013); Nakano et al. (2016), which imply three-dimensional Fermi surfaces. It is plausible that this discrepancy is related to the electronic nematic transition. The electronic state at low temperatures differs from that at room temperature due to the nematic transition at $\sim$45 K.

Next, we compare the NMR/NQR results in BaTi2Bi2O with those in BaTi2Sb2O and BaTi2As2O. In general, $1/T_{1}T$ can be described as

[TABLE]

where $\gamma_{N}$ ($\gamma_{e}$) is the nuclear (electronic) gyromagnetic ratio, $k_{\rm B}$ is the Boltzmann’s constant, $\hslash$ is the reduced Planck’s constant, $A_{\bm{q}}$ is the Fourier transform of the hyperfine coupling, and $\chi^{\prime\prime}_{\perp}(\bm{q},\omega)$ is the transverse component of the imaginary part of the dynamical susceptibility. Then, the value of $1/T_{1}T$ in different nuclei may be compared after the normalization by $\gamma_{N}^{2}$. Figure 6 shows the temperature dependence of $1/(T_{1}T\gamma_{N}^{2})$ in BaTi2Bi2O, BaTi2Sb2O, and BaTi2As2OSong et al. . While the NQR/NMR spectra in those compounds indicate the breaking of the fourfold symmetry at the $Pn$ site, the temperature dependences of $1/T_{1}T$ are considerably different from each other. In BaTi2Sb2O and BaTi2As2O, $1/T_{1}T$ was clearly enhanced toward $T_{\rm A}$. The ratio of $1/T_{1}$ between the two isotopes of the Sb nuclei is close to the ratio of the square of the nuclear gyromagnetic ratio $\gamma_{\rm N}$, suggesting that this enhancement originates from the magnetic natureKitagawa et al. (2013). In addition, a reduction of the constant value of $1/T_{1}T$ below $T_{\rm A}$ was observed, indicating a decrease in the density of states due to the nematic transition. The value of $1/(T_{1}T\gamma_{N}^{2})$ in BaTi2As2O is one order of magnitude smaller than those in BaTi2Sb2O and BaTi2Bi2O, possibly due to the small hyperfine coupling constant. In contrast, $1/T_{1}T$ shows no measurable anomaly around $T_{\rm anom}$ in BaTi2Bi2O. While the system does exhibit strong two dimensionality, the value of $1/T_{1}T$ in BaTi2Bi2O is larger than those in BaTi2Sb2O and BaTi2As2O. This observation can be attributed to the existence of strong coupling between the atoms of Bi and Ba and the lack thereof between the Bi and Ti2O layers. From this, it can hence be inferred that the BaBi2 layer in BaTi2Bi2O plays the role of a block layer whose electronic state is different from that of the Ti2O plane.

Furthermore, a clear coherence peak immediately below $T_{\rm c}$ and an exponential decay at low temperatures as evidence of an $s$-wave superconductivity were observed in BaTi2Sb2O, while no anomaly was observed in BaTi2Bi2O. These differences may have arisen from the difference of the dimensionality between two compounds. Since BaTi2Sb2O is three dimensional, the anomaly in the nematic transition and superconductivity can be detected by 121/123Sb-NMR/NQR. However, such an anomaly was not observed by 209Bi-NMR/NQR in BaTi2Bi2O because of its strong two dimensionality. It is apparent that the SC Ti2O layer is sandwiched by the nonsuperconducting BaBi2 block layer with the different electronic state; thus, BaTi2Bi2O is regarded as a two-dimensional superconductor. Therefore, it is important that the validation of two-dimensional superconductivity is performed using 47/49Ti or 17O NMR measurements.

In conclusion, the 209Bi-NMR/NQR measurements were performed to investigate the physical properties of superconducting BaTi2Bi2O from a microscopic point of view. The temperature evolution of the NQR spectra indicates an electronic nematic order at $T_{\rm anom}\sim$45 K. Comparing the NMR/NQR results among BaTi2$Pn$2O ($Pn$ = As, Sb, and Bi), it was found that the normal and SC properties of BaTi2Bi2O were somewhat different from those of BaTi2Sb2O and BaTi2As2O. The observed two-dome structure in $T_{\rm c}$ on BaTi2(Sb1-xBix)2O may have originated from these differences.

Acknowledgments

The authors acknowledge S. Yonezawa, Y. Maeno, and Y. Matsuda for fruitful discussions. This work was partially supported by the Kyoto University LTM Center and Grant-in-Aids for Scientific Research (KAKENHI) (Grants No. JP15H05882, No. JP15H05884, No. JP15K21732, No. JP15H05745, No. JP15K17698, and No. JP17K14339).