Comment on "On the nature of magnetic stripes in cuprate superconductors," by H. Jacobsen et al., arXiv:1704.08528v2

TL;DR

This paper discusses how magnetic stripe dynamics in La_2CuO_4+y influence orthorhombicity and how stripe incommensuration measurements can estimate oxygen content.

Contribution

It provides insights into the relationship between magnetic stripe behavior and structural properties in cuprate superconductors.

Findings

Magnetic stripe dynamics reduce orthorhombicity.

Stripe incommensuration correlates with oxygen content.

Potential method for estimating oxygen levels from magnetic measurements.

Abstract

Dynamics reduces the orthorhombicity of magnetic stripes in La_2CuO_4+y. The measured stripe incommensuration can be used to determine the oxygen content of the sample.

Tables

Table 1. Table 1: Incommensuration δ j ¯ ¯ subscript 𝛿 𝑗 \overline{\delta_{j}} and δ j 0 ~ ~ superscript subscript 𝛿 𝑗 0 \widetilde{\delta_{j}^{0}} ( j = h , k 𝑗 ℎ 𝑘 j=h,k ) of static and dynamic magnetic density waves, respectively, in L ​ a 2 ​ C ​ u ​ O 4 + y 𝐿 subscript 𝑎 2 𝐶 𝑢 subscript 𝑂 4 𝑦 La_{2}CuO_{4+y} , expressed in orthorhombic coordinates. Error bars are denoted in parentheses.

Neutron scattering	Peak position	MDW mode	${\delta }_h$ (r.l.u.)	${\delta }_k$ (r.l.u.)
Elastic	($\overline{100}$)	static	0.0967(5)	0.1237(3)
Inelastic	($\widetilde{100}$)	dynamic	0.1038(15)	0.1147(8)
Elastic	($\overline{010}$)	static	0.1233(3)	0.0950(5)

Table 2. Table 2: Incommensuration δ j ¯ ¯ subscript 𝛿 𝑗 \overline{\delta_{j}} and δ j 0 ~ ~ superscript subscript 𝛿 𝑗 0 \widetilde{\delta_{j}^{0}} of Table I expressed in terms of their average, δ h ​ k ≡ ( δ h + δ k ) / 2 subscript 𝛿 ℎ 𝑘 subscript 𝛿 ℎ subscript 𝛿 𝑘 2 \delta_{hk}\equiv(\delta_{h}+\delta_{k})/2 , deviation from the average, δ j = δ h ​ k + Δ ​ δ j subscript 𝛿 𝑗 subscript 𝛿 ℎ 𝑘 Δ subscript 𝛿 𝑗 \delta_{j}=\delta_{hk}+\Delta\delta_{j} ( j = h , k 𝑗 ℎ 𝑘 j=h,k ), and orthorhombicity Ω ≡ | Δ ​ δ / δ h ​ k | Ω Δ 𝛿 subscript 𝛿 ℎ 𝑘 \Omega\equiv|\Delta\delta/\delta_{hk}| .

Scattering (Peak)		Average ${\delta }_{hk}$	Deviation $\Delta {\delta }_h$	Deviation $\Delta {\delta }_k$	Orthorhombicity $\Omega$
Elastic ($\overline{100}$)	${\delta }_h$ , ${\delta }_k$ =	0.1102	- 0.0135	+ 0.0135	0.12
Inelastic ($\widetilde{100}$)	${\delta }_h$ , ${\delta }_k$ =	0.1093	- 0.0055	+ 0.0055	0.05
Elastic ($\overline{010}$)	${\delta }_h$ , ${\delta }_k$ =	0.1092	+ 0.0141	- 0.0141	0.13

Equations

$$\widetilde{{\delta }_j}(ϵ\to 0)\equiv \widetilde{{\delta }_j^0}\ne \overline{{\delta }_j}(j=h,k).$$

$${\delta }_{hk}(y)=\frac{1}{4}\sqrt{2y-x_0^N}.$$

Full text

With elastic and inelastic neutron-scattering experiments on $La_{2}CuO_{4+y}$ Jacobsen et al. have provided a valuable contribution to clarify the longstanding puzzle of static vs. dynamic stripes in the pseudogap state of cuprate superconductors.1 The authors’ main finding is a discrepancy between the incommensuration $\widetilde{\delta_{j}}(\epsilon)$ of dynamic magnetic density waves (MDWs), extrapolated to vanishing energy $\epsilon$, and the value $\overline{\delta_{j}}$ from static MDWs,

[TABLE]

Here $h$ and $k$ label the components of unidirectional MDWs, approximately along the $Cu$-$O$ bonds in the $CuO_{2}$ plane, with respect to the orthorhombic $a$ and $b$ axes. The values are listed in Table I, reproduced from Ref. 1 (supplementary material). The purpose of this comment is twofold: (1) To show that a modified display of the data, Table II, provides more insight into similarities and differences of dynamic vs. static MDWs and qualitatively suggests a plausible explanation. (2) To determine the level of super-oxygenation $y$ from the measured (average) incommensuration $\delta_{hk}$.

1. Magnetic density waves. The cross components of the incommensurations at the symmetry-related elastic peaks in Table I are found so close to be essentially equal, $\overline{\delta_{h}}(100)\simeq\overline{\delta_{k}}(010)\simeq 0.0959(5)$, and vice versa, $\overline{\delta_{k}}(100)\simeq\overline{\delta_{h}}(010)\simeq 0.1235(3)$. In contrast, a clear difference exists between the static and extrapolated dynamic values, $|\overline{\delta_{h}}-\widetilde{\delta_{h}^{0}}|=0.07$ and $|\overline{\delta_{k}}-\widetilde{\delta_{k}^{0}}|=0.09$, taken at the (100) peak, with the static values bracketing the dynamic ones. This is the surprising result of Ref. 1.

Using the average, $\delta_{hk}\equiv(\delta_{h}+\delta_{k})/2$, and the deviation from the average, $\Delta\delta_{j}$   ($j=h,k$), the same data are displayed in Table II, to be viewed as $\delta_{j}=\delta_{hk}+\Delta\delta_{j}$. It now becomes obvious that the static and dynamic averages are essentially equal, $\overline{\delta_{hk}}\simeq\widetilde{\delta_{hk}^{0}}\simeq 0.1095$, suggesting a commonality of static and dynamic MDWs. The average $\delta_{hk}$ would be the tetragonal approximation of the orthorhombic incommensurations. The deviations from $\delta_{hk}$ mark their orthorhombicity, $\Omega\equiv|\Delta\delta/\delta_{hk}|$, being considerably larger for static MDWs than for dynamic ones, $\overline{\Omega}/\widetilde{\Omega^{0}}\simeq 2.5$.

Why have the dynamic incommensurations $\widetilde{\delta_{j}^{0}}$ less orthorhombicity? A static MDW can be regarded a (magnetic) superlattice of lattice constants $D_{j}=1/\overline{\delta_{j}}$. A dynamic MDW, in contrast, can be considered a standing wave of oscillating magnetic dipoles with wavelength components $\widetilde{\lambda_{j}^{0}}=1/\widetilde{\delta_{j}^{0}}$. Generally, the motional aspect of dynamics promotes isotropy (here, in the $CuO_{2}$ plane). The oscillations of the dynamic MDWs then tend to tetragonalize their orthorhombic wavelengths $\widetilde{\lambda_{j}^{0}}$, preventing a relaxation of the magnetic dipoles to a (static) superlattice of larger orthorhombicity. This notion is corroborated by still less orthorhombicity of dynamic MDWs with higher energy $\epsilon$—albeit to a much lesser degree, $\Delta\widetilde{\Omega}/\Delta\epsilon=-0.007$/meV (based on Fig. 3 of Ref. 1).

2. Oxygen content. A frequent problem with oxygen-enriched cuprates is uncertainty about the exact level of super-oxygenation $y$. In many cases samples are characterized by the superconducting transition temperature $T_{c}$ instead of the value of $y$. The sample used in Ref. 1 has $T_{c}\simeq 40$ K, similar to the sample used by Lee et al.1 ; 2 A thermogravitimetric analysis of the latter sample gave an estimate of oxygen enrichment $y=0.12\pm 0.01$. The main quantity of interest is, of course, the hole doping level $p$ (per $Cu$ atom in the $CuO_{2}$ plane) caused by super-oxygenation $y$. No such problem occurs in the much-studied companion lanthanum cuprates $La_{2-x}Ae_{x}CuO_{4}$ ($Ae=Sr,Ba$) that are hole-doped through infravalent cation doping $x$ with a hole doping level $p=x$. In the latter materials MDWs and charge-density waves (CDWs) appear together, called “stripes.”3 Their incommensuration, $\delta(x)\propto\sqrt{x-x_{0}^{N}}$, depends on the cation doping $x$, diminished by the Néel point $x_{0}^{N}=0.02$ (collapse of 3D antiferromagnetism at $T=0$).4

It is tempting to extend the incommensuration formula from the cation-doped to the oxygen-enriched $La_{2}CuO_{4}$ compounds. Assuming that each enriching oxygen atom in $La_{2}CuO_{4+y}$ gives rise to two doped holes, $p=2y$, the formula for MDWs, expressed in terms of oxygen enrichment and orthorhombic coordinates (but in tetragonal approximation) becomes,

[TABLE]

Solving for $y=8\delta_{hk}^{2}+x_{0}^{N}/2$ and using the average incommensuration $\delta_{hk}=0.1095$ r.l.u. gives $y=0.105\pm 0.005$, comparable with the estimate by Lee et al., $y=0.12\pm 0.01$.

ACKNOWLEDGMENTS

I thank Sonja Lindahl Holm and Henrik Jacobsen for literature links and information about the oxygen content of the sample.