Quantum dynamics of hydrogen in iron-based superconductor LaFeAsO0.9D0.1 measured with inelastic neutron spectroscopy

TL;DR

This study investigates hydrogen's quantum behavior in an iron-based superconductor using inelastic neutron scattering, revealing quantum rattling phenomena linked to potential minima and superconducting properties.

Contribution

It provides the first experimental evidence of hydrogen quantum rattling in a superconductor, combining neutron scattering with first-principles calculations.

Findings

Emergence of new excitations below Tc in the 5-15 meV range.

Strong excitations at 14.5 meV and 11.1 meV broaden near 15-20 K.

Quantum rattling of hydrogen linked to multiple potential minima.

Abstract

Inelastic neutron scattering was performed for an iron-based superconductor LaFeAsO0.9D0.1, where most of D (deuterium) replaces oxygen, while a tiny amount goes into interstitial sites. By first-principle calculation, we characterize the interstitial sites for D (and for H slightly mixed) with four equivalent potential minima. Below the superconducting transition temperature Tc = 26 K,new excitations emerge in the range 5-15 meV, while they are absent in the reference system LaFeAsO0.9F0.1. The strong excitations at 14.5 meV and 11.1 meV broaden rapidly around 15 K and 20 K, respectively, where each energy becomes comparable to twice of the superconducting gap. The strong excitations are ascribed to a quantum rattling, or a band motion of hydrogen, which arises only if the number of potential minima is larger than two.

Full text

The discovery of iron-based superconductors has stimulated a growing body of research Kamihara et al. (2008); Paglione and Greene (2010); Stewart (2011); Hosono and Kuroki (2015). For identifying the mechanism of superconductivity, one needs detailed information of the superconducting gap, together with its temperature dependence. In addition to thermodynamic and transport studies, direct measurements of the energy gaps have been tried by various methods such as photoemission Ding et al. (2008); Kondo et al. (2008), tunneling Hanaguri et al. (2010), point-contact Chen et al. (2008); Gonnelli et al. (2009), and infrared Li et al. (2008); Gorshunov et al. (2010) spectroscopies. These methods require high quality of the sample, which is not feasible in some iron-based superconductors made with intricate procedure of synthesis. Therefore, a new direct method is awaited that is also applicable to polycrystalline samples with rough surfaces, magnetic impurities, and other imperfections.

Here, we propose a new method for direct detection of the superconducting gap in the iron-based superconductor via coupling to hydrogen motion at interstitial sites. When two or more interstitial sites of hydrogen are close, the overlap of wave functions causes the splitting of hydrogen energy levels Fukai (2003); Kehr (1978). This quantum phenomenon is known as the tunneling splitting, and has long been studied in conventional superconductors by inelastic neutron scattering (INS)  Fukai (2003); Wipf (1997); Kondo (1986); Kagan and Prokof’ev (1992). The magnitudes of the tunneling splitting in known cases are much smaller than the characteristic value of the superconducting gap. As we shall demonstrate, the hydrogen vibration in the iron-based superconductors is highly anharmonic at the interstitial sites. Accordingly, the generalized tunneling splitting has exceptionally large energy of the order of 10 meV, which is comparable to the superconducting gap. Hence, the hydrogen motion serves as a good probe for superconductivity. Since the local dynamics is not much influenced by the sample quality, the hydrogen probe has a unique advantage over the conventional spectroscopies.

We performed INS measurements for a deuterated LaFeAsO0.9D0.1 with $T_{\rm c}=26$ K Iimura et al. (2012). We found strong intensities in the excitation range of 4–15 meV below $T_{\rm c}$. The source of these excitations is ascribed to hydrogen since the reference material LaFeAsO0.9F0.1, with the same electronic/magnetic phase diagram Kamihara et al. (2008); Iimura et al. (2012), does not show the corresponding intensities. Throughout this paper, we use the term “hydrogen” for both 1H and 2H, and protium (H) or deuterium (D) to distinguish the isotopes explicitly.

We prepared a powder sample of LaFeAsO0.9D0.1 (D-sample), weighing $\sim$30 g, under high pressure Iimura et al. (2012). Hydrogen anions (D-) were substituted for O2- anions Matsuishi et al. (2014); Hiraishi et al. (2014). In addition, owing to a tiny amount of H in deuterated reagents Iimura et al. (2013), some mixture of H at the hydrogen sites is inevitable. Their cross sections $\sigma_{\rm inc}$ of incoherent scattering are $\sigma_{\rm inc}$(H)$=80.3\times 10^{-24}$ cm2 and $\sigma_{\rm inc}$(D)$=2.1\times 10^{-24}$ cm2. The H/D concentration ratio was estimated to be $\sim$0.02 based on the analysis of excitations at 70–130 meV of H and D confined in the O sites. We also prepared LaFeAsO0.9F0.1 (F-sample) with $\sim$20 g under ambient pressure as the reference for the D-sample.

The INS intensities for D- and F-samples were measured on a chopper spectrometer (4SEASONS) Kajimoto et al. (2011) at BL01 in the pulsed-neutron source at the Japan Proton Accelerator Research Complex (J-PARC) in the Materials and Life Science Experimental Facility (MLF). Data were collected at incident neutron energies of $E_{\rm i}$ = 9.3, 17.5, and 44.5 meV Nakamura et al. (2009). The Fermi-chopper frequency was set to 150 Hz, with a resolution $\hbar\Delta\omega/E_{\rm i}$ of $\sim$2% at the energy transfer $\hbar\omega\approx 14$ meV. The time-of-flight data from position-sensitive detectors were converted into the $\bm{Q}$-angle average $\langle S(\bm{Q},\hbar\omega)\rangle_{\rm angle}$ of the dynamical structure factor by the Utsusemi program Inamura et al. (2013). We use the notation $S(Q,\hbar\omega)$ for the angle average.

Figure 1 shows $S(Q,\hbar\omega)$ for D- and F-samples, written as $S_{\rm D}$ and $S_{\rm F}$ respectively, together with their difference $S_{\rm D-F}$. Comparing $S_{\rm D}$ at 6 K below $T_{\rm c}$ (Fig. 1(a)) and at 41 K above $T_{\rm c}$ (Fig. 1(b)), we find three groups of excitations at 6 K, all of which are insensitive to $Q$ but merge into a broad feature at 41 K with a strong diffusive character over the entire energy range. In sharp contrast to $S_{\rm D}$, the results $S_{\rm F}$ for the F-sample exhibit neither the flat excitations at 6 K (Fig. 1(c)) nor the diffuse scattering at 41 K (Fig. 1(d)). Instead, a small temperature-insensitive feature is visible at $\hbar\omega$ $\approx$ 12 meV in Figs. 1(c) and 1(d). This is ascribed to phonon scattering as previously reported for the F-sample Christianson et al. (2008). On the other hand, the magnetic scattering reported at $Q\approx$ 1.1 Å Shamoto et al. (2010); Wakimoto et al. (2010) is not visible at the present scale because of their weak intensities.

Figures 1(e) and 1(f) display the difference map $S_{\rm D-F}\equiv S_{\rm D}-S_{\rm F}$ at 6 K and 41 K, respectively. At 6 K, the three distinct features are assigned as U, M, and L, as indicated in Fig. 1(e). The D- and F-samples have almost the same electric and superconducting properties Iimura et al. (2012). Consequently, the strong excitations in $S_{\rm D-F}$ must originate from hydrogen.

Figures 2(a) and 2(b) display the energy spectra of $S_{\rm D-F}$ at 6 K and 23.5 K, respectively, with $Q$ = 2.15 $\pm$ 0.15 Å*-1*. For each temperature, the scattering peaks are fitted by a Lorentzian spectrum with the linewidth $\hbar\Gamma_{\alpha}(T)$ where $\alpha$ represents either U or M mode. Figure 2(c) plots the temperature dependence of $\hbar\Gamma_{\alpha}(T)$, which shows that both $\hbar\Gamma_{\rm U}$ and $\hbar\Gamma_{\rm M}$ barely depend on temperature for $T<15$ K. A rapid increase is seen however around 20 K, and then both linewidths are weakly temperature-dependent as the temperature approaches $T_{\rm c}\sim 26$ K. In contrast, each peak position $\hbar\omega_{\alpha}$ is nearly constant against temperature, as shown in Fig. 2(d).

It is likely that the rapid increase of $\hbar\Gamma_{\alpha}$ at $T\lesssim T_{\rm c}$ is related to the coupling of the hydrogen motion to electronic excitations. Namely, with increasing temperature toward $T_{\rm c}$, the superconducting gap $\Delta(T)$ decreases so as to satisfy the condition $\hbar\omega_{\alpha}>2\Delta(T)$. This means that $S_{\rm D-F}$ serves as a probe of the superconducting gap. For comparison, we refer to the INS spectra in Nb(OH)x ($x\approx$ 0.002), where an inelastic peak at $\sim$0.2 meV below $T_{\rm c}$ = 9.2 K merge into the broad quasi-elastic feature above $T_{\rm c}$ Magerl et al. (1986). The origin of the peak has been assigned to the tunneling splitting of the interstitial hydrogen in the double well potential. In contrast to the present case, the inelastic peak in Nb(OH)x is much smaller than $2\Delta_{0}\sim 3$ meV in the system. The peak width below $T_{\rm c}$ decreases toward the resolution limit upon cooling Black and Fulde (1979); Magerl et al. (1986). The change in width is interpreted in terms of a Korringa-type relaxation due to the conduction electrons, which becomes active as the superconducting gap decreases Wipf (1997); Kondo (1986); Magerl et al. (1983, 1986).

It appears from Fig. 2(c) that each peak width saturates at both low and high temperatures. We then use the sigmoid function to fit $\hbar\Gamma_{\alpha}(T)$, as presented in Fig. 2(c). Next, with a trial value for the superconducting gap $\Delta_{0}$ at zero temperature, we plot $2\Delta(T)$ according to the temperature dependence in BCS theory. For the value $\Delta_{0}=7.8\pm 0.7$ meV, the midpoint temperatures $T_{\alpha}$ of both U and M modes agree reasonably well with the relation $\hbar\omega_{\alpha}=2\Delta(T_{\alpha})$ (Fig. 2(d)). The estimated gap gives the ratio $2\Delta_{0}/k_{\rm B}T_{\rm c}=7.0$, which characterizes this compound as a strong coupling superconductor. According to the point-contact spectra on LaFeAsO0.9F0.1 Gonnelli et al. (2009), two gaps have been reported; $\Delta_{1}\approx$ 7.9 meV, and $\Delta_{2}\approx$ 2.8 meV, both of which show strong deviation from the BCS-like temperature dependence. On the other hand, another experiment on SmFeAsO0.85F0.15 Chen et al. (2008) with $T_{\rm c}\sim 42$ K has reported a single gap around 7 meV that shows a BCS-like temperature dependence. The present result for $\Delta_{0}$ corresponds to the value $\Delta_{1}$ in Gonnelli et al. (2009), but with BCS-like temperature dependence. Further study is necessary to reconcile these conflicting observations.

The crossing of $\hbar\omega$ and $2\Delta(T)$ is analogous to the optical threshold in the BCS superconductor Mattis and Bardeen (1958). With finite damping of quasi-particles and/or presence of normal component of electrons, the optical conductivity becomes already finite Li et al. (2008); Gorshunov et al. (2010); Herman and Hlubina (2017) for $\hbar\omega<2\Delta_{0}$ in contrast with the ideal BCS case. In such a case, the midpoint of the increasing conductivity as a function of $\hbar\omega$ corresponds to the relation $\hbar\omega=2\Delta_{0}$. The finite damping should also be present in the hydrogen excitations. Hence, the midpoint of $\hbar\Gamma_{\alpha}(T)$ seems to be a reasonable choice for extracting $2\Delta(T)$.

We proceed to consider the microscopic origin of the newly observed excitations. Firstly, to investigate stable positions of the interstitial hydrogen, we performed first-principles calculations using the Vienna ab initio simulation package, which is combined with a generalized gradient approximation using the Perdew–Burke–Ernzerhof exchange-correlation function Perdew et al. (1996). The cut-off energy for the plane wave is set to 400 eV, and the $k$-point $3\times 3\times 3$ mesh is sampled in the first Brillouin zone. Interstitial hydrogens are placed in periodically repeated $2\times 2\times 1$ supercell corresponding to the interstitial hydrogen number 1/8 per unit cell. The interstitial hydrogen frequently induces displacements of surrounding atoms from their equilibrium positions, and forms a self-trapped state Sundell and Wahnström (2004); Sundell et al. (2007); Iwazaki et al. (2010). However, it has been discussed that thermal fluctuations recover the equivalent level for all interstitial sites, which enables the tunneling Flynn and Stoneham (1970); Kagan and Klinger (1974). There is no consistent microscopic theory on how quantum tunneling occurs at low enough temperatures where thermal fluctuations are negligible. In view of this situation, the calculation was mainly performed using fixed positions of atoms. In the Supplemental Material, we discuss how the relaxation of surrounding atoms influences the result. To simulate the system with hydrogen implanted, we place a static point charge at location $\bm{r}$ in the supercell together with a neutralizing electron. Then, we compute the total energy $E(\bm{r})$ per supercell, which is minimized at the optimal position $\bm{r}$ of the positive charge. We define the effective potential for a positive charge: $\Delta E(\bm{r})=E(\bm{r})-E_{0}-\mu$ where $E_{0}$ is the corresponding energy without the positive charge, and $\mu$ is the chemical potential. As a result, the dynamics of protons or deuterons are neglected in the present simulation.

According to structure analysis, LaFeAsO0.9D0.1 has a ZrCuSiAs-type structure with alternating stacks of conducting FeAs4 and insulating (O,D)La4 layers (Fig. 3(a)). Figure 3(b) illustrates the contour map of $\Delta E(\bm{r})$ on the $xy$-plane at $z$ = 0.425. There appear four potential minima in $\Delta E(\bm{r})$ at $\bm{r}=(0.11,0.11,0.425)$ and its crystallographically equivalent positions related by the fourfold rotational symmetry, labeled Hint hereafter. The minima are located within the As5La void with Hint–As distance of 2.15 Å. Bader charge analysis gives the charge of hydrogen at Hint to be 1.33, which implies the interstitial hydrogen lies in hydride Tang et al. (2009).

Using the effective potential $\Delta E(\bm{r})$, we now consider the dynamics of the interstitial hydrogen. Figure 4(a) illustrates the line profile through the minima A–B–C–D–A, as indicated in Fig. 3(b). This corresponds to the pathway of hydrogen migration. The profile features a periodic potential with tunneling distance $d$ = 1.13(1) Å and potential barrier $V_{\rm b}$ = 102 meV at the saddle point. In the harmonic approximation for sinusoidal potential, the angular frequency $\omega_{0}$ is given by $\omega_{0}^{2}$ = $2\pi^{2}V_{\rm b}/(md^{2})$, where $m$ is the isotope mass. Next, we obtain $\hbar\omega_{\rm 0,H}$ = 81 meV for H and $\hbar\omega_{\rm 0,D}$ = 57 meV for D. The $V_{\rm b}$ is comparable to the 80 meV estimated for Nb(OH)x with $d$ = 1.17 Å and $\hbar\omega_{0}$ = 107 meV Magerl et al. (1983).

The cross section $\sigma_{\rm inc}$ of incoherent scattering is much larger in H than D, with the ratio $\sigma_{\rm inc}({\rm H})/\sigma_{\rm inc}({\rm D})\approx 40$. Considering the concentration ratio $n_{\rm H}/n_{\rm D}\approx 0.02$, the scattering intensities from the two isotopes should be comparable. Hence, we assume that the features U and M come from the interstitial H and D, respectively. The ratio $\omega_{\rm U}/\omega_{\rm M}\approx 1.3$ is rather close to the square root of the atomic mass ratio $\sqrt{m_{\rm D}/m_{\rm H}}=\sqrt{2}$. This might be regarded as the isotope effect for harmonic phonons. In fact, for H or D substituting oxygen, the Supplemental Material demonstrates the isotope effect with reasonable agreement between the calculation and the experimental observation in the range of 70–130 meV. However, since $\hbar\omega_{\rm U}$ and $\hbar\omega_{\rm M}$ are much smaller ($<20$ meV), it is difficult to ascribe U and M to ordinary vibrations of hydrogen.

We now check the possibility of whether $\omega_{\rm U}$ and $\omega_{\rm M}$ may come from the tunneling motion of hydrogen. The magnitude of the tunneling splitting is sensitive to distance and mass of isotopes Fukai (2003); Wipf (1997); Wipf and Neumaier (1984). For example, the tunneling splittings are reported as 0.2 meV (H) and 0.02 meV (D) for Nb Magerl et al. (1986); Wipf and Neumaier (1984), while 6.3 meV (H) and 1.6 meV (D) for $\alpha$-Mn Kolesnikov et al. (1999). The latter values are referred to as giant tunneling splittings, and are caused by the very short tunneling distance of 0.68 Å. The mass of the isotopes affects the tunneling splittings $\omega_{\rm H,D}$ more strongly than the case for phonons, as seen in $\omega_{\rm H}/\omega_{\rm D}\approx 10$ for Nb Magerl et al. (1986); Wipf and Neumaier (1984) and $\omega_{\rm H}/\omega_{\rm D}\approx$ 4 for $\alpha$-Mn Kolesnikov et al. (1999). Thus, the present value $\omega_{\rm U}/\omega_{\rm M}\approx 1.3$ is not consistent with the identification of tunneling splitting.

We therefore propose that the excitations at $\hbar\omega_{\rm U}$ and $\hbar\omega_{\rm M}$ are associated with a new type of extremely anharmonic phonons, which can be appropriately called quantum rattling. In this process, the potential barrier is so low that the ordinary excited state, corresponding to the energy $\frac{3}{2}\hbar\omega_{0}$ in the harmonic case, is actually extended along the pathway connecting the four potential minima, as illustrated in Fig. 4(a). In four potential minima system, the eigenstates are characterized by the wave numbers $k=0,\pm\pi/(2d)$, and $\pm\pi/d$, as shown in Fig. 4(b). In addition, the eigenstates are related to the superposition of localized phonons, which have energies $(n+1/2)\hbar\omega_{0}$ with $n=0,1,\ldots$ Figure 4(c) illustrates the wave functions $\Psi_{\rm g}$ stemming from $n=0$, and $\Psi_{\rm e}$ stemming from $n=1$. The exact solution of the Schrödinger equation is obtained in terms of the Mathieu functions. If we assign $\hbar\omega_{\rm M}=11.1$ meV as the deuterium excitation from $\Psi_{\rm g}(k=0)$ to $\Psi_{\rm e}(k=\pi/d)$, and assume the same potential barrier $V_{\rm b}$ for D and H, all other excitation energies are fixed theoretically. Hence, the corresponding protium excitation is expected at 16.6 meV, which compares favorably with the experimental value $\hbar\omega_{\rm U}$ = 14.5 meV. While, we have to take the value of $V_{\rm b}\approx 8$ meV, which is much smaller than the value 102 meV obtained from the first-principle calculation. We note that quantum fluctuations, which are neglected in this approximation, may lower the effective barrier substantially, as discussed in the literature Sundell et al. (2007). Further study is required on the influence of fluctuation and self-trapping effects on the dynamics of hydrogen.

In conclusion, we found new excitations of hydrogen in the iron-based superconductor LaFeAsO0.9D0.1 using inelastic neutron scattering. We have ascribed the excitations to a novel quantum rattling, or a band motion of hydrogen, which arises only if the interstitial site has a number of potential minima larger than two. The new excitations of hydrogen broaden rapidly when twice the superconducting gap matches the energy of the hydrogen excitation. Assuming the BCS-like temperature dependence of $\Delta(T)$, we estimate the superconducting gap $\Delta_{0}=7.8$ meV which agrees with another probe result.

It is evident that the isotropic superconducting gap should be the simplest case for the present probe. Thus, the ion-based superconductors are much more suitable for the analysis than cuprates with nodes in the gap. Moreover, the matching of the superconducting gap with the hydrogen excitation is just fortunate in the iron-based superconductors with appropriate interstitial sites. The hydrogen spectroscopy has a unique feature to scan along the temperature axis with fixed energy, while most other spectroscopies scan along the energy axis with fixed temperature. With further refinement of both measurement and analysis, the hydrogen probe should provide more useful information for clarifying the superconducting gap. Concomitantly, the novel quantum motion of hydrogen reported in this paper deserves more detailed study in its own right.