Experimental evidence for Zeeman spin-orbit coupling in layered antiferromagnetic conductors

TL;DR

This paper provides experimental evidence for Zeeman spin-orbit coupling in layered antiferromagnetic conductors, showing that it causes spin degeneracy in Landau levels, confirmed through studies on organic and cuprate superconductors.

Contribution

First experimental demonstration of Zeeman spin-orbit coupling effects in layered antiferromagnetic conductors, confirming its generic presence across different material classes.

Findings

No spin modulation of Shubnikov-de Haas oscillations in the Nél state of $ppa$-(BETS)$_2$FeBr$_4$

Absence of spin modulation in electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_4$

Evidence of Nél order in cuprate superconductor at optimal doping

Abstract

Most of solid-state spin physics arising from spin-orbit coupling, from fundamental phenomena to industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin-orbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet, experimental proof of this phenomenon has been lacking. Here, we demonstrate that the N\'eel state of the layered organic superconductor $\kappa$-(BETS)$_2$FeBr$_4$ shows no spin modulation of the Shubnikov-de Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin degeneracy of Landau levels, a direct manifestation of the Zeeman spin-orbit coupling. Likewise, we show that spin modulation is absent in electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_4$, which evidences the presence of N\'eel order in this cuprate superconductor even at optimal doping. Obtained on two very different materials, our results demonstrate the generic character of the Zeeman spin-orbit coupling.

Equations

$${\mathcal{H}}_{\mathcal{Z}}^{so}=-\frac{{\mu }_B}{2}\left[g_{\Vert }({\mathbf{B}}_{\Vert }\cdot \mathbf{\sigma })+g_{\perp }(\mathbf{k})({\mathbf{B}}_{\perp }\cdot \mathbf{\sigma })\right],$$

$$R_s=\cos \left[\frac{\pi }{\cos \theta }\frac{{\bar{g}}_{\perp }{m}_0}{2m_{\mathrm{e}}}\right],$$

$$A_{\delta }=A_0\frac{m^2}{\sqrt{B}}{R}_{\mathrm{MB}}\frac{\exp (-Km{T}_{\mathrm{D}}/B)}{\sinh (KmT/B)}{R}_s(\theta ),$$

Full text

Introduction

Spin-orbit coupling (SOC) in solids intertwines electron orbital motion with its spin, generating a variety of fundamental effects Winkler ; Dyakonov . Commonly, SOC originates from the Pauli term $\mathcal{H_{P}}=\frac{\hbar}{4m_{e}^{2}}{\bm{\sigma}}\cdot{\bf p}\times{\bm{\nabla}}V({\bf r})$ in the electron Hamiltonian LL-IV ; Kittel , where $\hbar$ is the Planck constant, $m_{e}$ the free electron mass, ${\bf p}$ the electron momentum, ${\bm{\sigma}}$ its spin and $V({\bf r})$ its potential energy depending on the position. Remarkably, Néel order may give rise to SOC of an entirely different nature, via the Zeeman effect RevazPRL2008 ; RevazPRB2009-1 :

[TABLE]

where $\mu_{B}$ is the Bohr magneton, ${\bf B}$ the magnetic field, while $g_{\|}$ and $g_{\perp}$ define the $g$-tensor components with respect to the Néel axis. In a purely transverse field ${\bf B}_{\perp}$, a hidden symmetry of a Néel antiferromagnet protects double degeneracy of Bloch eigenstates at a special set of momenta in the Brillouin zone(BZ) RevazPRL2008 ; RevazPRB2009-1 : at such momenta, $g_{\perp}$ must vanish. The scale of $g_{\perp}$ is set by $g_{\|}$, which renders $g_{\perp}({\bf k})$ substantially momentum dependent, and turns $\mathcal{H}^{so}_{\mathcal{Z}}$ into a veritable SOC BraLuk1989 ; BraLuRa1989 ; RevazPRL2008 ; RevazPRB2009-1 . This coupling was predicted to produce unusual effects, such as spin degeneracy of Landau levels in a purely transverse field ${\bf B}_{\perp}$ KabanovAlexandrov ; RevazPRB2009-2 and spin-flip transitions, induced by an AC electric rather than magnetic field RevazPRB2009-2 . Contrary to the textbook Pauli spin-orbit coupling, this mechanism does not require heavy elements. Being proportional to the applied magnetic field (and thus tunable!), it is bound only by the Néel temperature of the given material. In addition to its fundamental importance as a novel spin-orbit coupling mechanism, this phenomenon opens new possibilities for spin manipulation, much sought after in the current effort Rashba-91 ; Shafiei-13 ; Berg-13 to harness electron spin for future spintronic applications. While the novel SOC mechanism may be relevant to a vast variety of antiferromagnetic (AF) conductors such as chromium, cuprates, iron pnictides, hexaborides, borocarbides, as well as organic and heavy-fermion compounds RevazPRB2009-1 , it has not received an experimental confirmation yet.

Here, we present experimental evidence for the spin degeneracy of Landau levels in two very different layered conductors, using Shubnikov–de Haas (SdH) oscillations as a sensitive tool for quantifying the Zeeman effect shoe84 . First, the organic superconductor $\kappa$-(BETS)2FeBr4 (hereafter $\kappa$-BETS) fuji01 is employed for testing the theoretical predictions. The key features making this material a perfect model system for our purposes are (i) a simple quasi-two-dimensional (quasi-2D) Fermi surface and (ii) the possibility of tuning between the AF and paramagnetic (PM) metallic states, both showing SdH oscillations, by a moderate magnetic field fuji01 ; kono05 . We find that, contrary to what happens in the PM state, the angular dependence of the SdH oscillations in the AF state of this compound is not modulated by the Zeeman splitting. We show that such a behavior is a natural consequence of commensurate Néel order giving rise to the Zeeman SOC in the form of Eq. (1).

Having established the presence of the Zeeman SOC in an AF metal, we utilize this effect for probing the electronic state of Nd2-xCexCuO4 (NCCO), a prototypical example of electron-doped high-$T_{c}$ cuprate superconductors armi10 . In these materials, superconductivity coexists with another symmetry-breaking phenomenon manifested in a Fermi-surface reconstruction as detected by angle-resolved photoemission spectroscopy (ARPES) mats07 ; sant11 ; song17 ; he19 and SdH experiments helm09 ; helm10 ; kart11 ; higg18 . The involvement of magnetism in this Fermi-surface reconstruction has been broadly debated luke90 ; moto07 ; mang04a ; saad15 ; yama03 ; kang05 ; daga05 ; yu07 ; dora18 ; das08 ; sach10 ; chak01 ; silv15 ; sach19 ; green20 . Here, we present detailed data on the SdH amplitude in optimally doped NCCO, tracing its variation over more than two orders of magnitude with changing the field orientation. The oscillation behavior is found to be very similar to that in $\kappa$-BETS. Given the crystal symmetry and the position of the relevant Fermi-surface pockets, this result is firm evidence for antiferromagnetism in NCCO. Our finding not only settles the controversy in electron-doped cuprate superconductors—but also clearly demonstrates the generality of the novel SOC mechanism.

Theoretical background — Before presenting the experimental results, we recapitulate the effect of Zeeman splitting on quantum oscillations. The superposition of the oscillations coming from the conduction subbands with opposite spins results in the well-known spin-reduction factor in the oscillation amplitude shoe84 : $R_{s}=\cos\left(\pi\frac{g}{2}\frac{m}{m_{\mathrm{e}}}\right)$; here $m_{\mathrm{e}}$ and $m$ are, respectively, the free electron mass and the effective cyclotron mass of the relevant carriers. We restrict our consideration to the first-harmonic oscillations, which is fully sufficient for the description of our experimental results. In most three-dimensional (3D) metals, the dependence of $R_{s}$ on the field orientation is governed by the anisotropy of the cyclotron mass. At some field orientations $R_{s}$ may vanish, and the oscillation amplitude becomes zero. This spin-zero effect carries information about the renormalization of the product $gm$ relative to its free-electron value $2m_{e}$. For 3D systems, this effect is obviously not universal. For example, in the simplest case of a spherical Fermi surface, $R_{s}$ possesses no angular dependence whatsoever, hence no spin zeros. By contrast, in quasi-2D metals with their vanishingly weak interlayer dispersion such as in layered organic and cuprate conductors, spin zeros are wosn92 ; kova93 ; meye95 ; uji02 ; kart16 ; RamshawNatPhys2011 a robust consequence of the monotonic increase of the cyclotron mass, $m\propto 1/\cos\theta$, with tilting the field by an angle $\theta$ away from the normal to the conducting layers.

In an AF metal, the $g$-factor may acquire a $\mathbf{k}$ dependence through the Zeeman SOC mechanism. It becomes particularly pronounced in the purely transverse geometry, i.e., for a magnetic field normal to the Néel axis. In this case, $R_{s}$ contains the factor $\bar{g}_{\perp}$ averaged over the cyclotron orbit, see Supplementary Information (hereafter SI) for details. As a result, the spin-reduction factor in a layered AF metal takes the form:

[TABLE]

where $m_{0}\equiv m(\theta=0^{\circ})$. Often, the Fermi surface is centered at a point ${\bf k^{*}}$, where the equality $g_{\perp}({\bf k^{*}})=0$ is protected by symmetry RevazPRB2009-1 – as it is for $\kappa$-BETS (see SI). Such a ${\bf k^{*}}$ belongs to a line node $g_{\perp}({\bf k})=0$ crossing the Fermi surface. Hence, $g_{\perp}({\bf k})$ changes sign along the Fermi surface, and $\bar{g}_{\perp}$ in Eq. (2) vanishes by symmetry of $g_{\perp}({\bf k})$. Consequently, the quantum-oscillation amplitude is predicted to have no spin zeros KabanovAlexandrov . For pockets with Fermi wave vector $k_{F}$ well below the inverse AF coherence length $1/\xi$, $g_{\perp}({\bf k})$ can be described by the leading term of its expansion in ${\bf k}$. For such pockets, the present result was obtained in Refs. KabanovAlexandrov ; RevazPRB2009-2 ; RevazPRL2010 . According to our estimates in the SI, both in $\kappa$-BETS and in optimally doped NCCO ($x=0.15$), the product $k_{F}\xi$ considerably exceeds unity. Yet, the quasi-classical consideration above shows that for $k_{F}\xi>1$ the conclusion remains the same: $\bar{g}_{\perp}=0$, see SI for the explicit theory.

We emphasize that centering of the Fermi surface at a point ${\bf k^{*}}$ with $g_{\perp}({\bf k^{*}})=0$ – such as a high-symmetry point of the magnetic BZ boundary RevazPRB2009-1 – is crucial for a vanishing of $\bar{g}_{\perp}$. Otherwise, Zeeman SOC remains inert, as it does in AF CeIn3, whose $d$ Fermi surface is centered at the $\Gamma$ point (see SI and Refs. Sett01 ; Ebih04 ; Gork06 ), and in quasi-2D EuMnBi2, with its quartet of Dirac cones centered away from the magnetic BZ boundary Mats18 ; Bori19 . With this, we turn to the experiment.

Results

AF Organic Superconductor $\kappa$-(BETS)2FeBr4 — This is a quasi-2D metal with conducting layers of BETS donor molecules, sandwiched between insulating FeBr${}_{4}^{-}$-anion layers fuji01 . The material has a centrosymmetric orthorhombic crystal structure (space group $Pnma$), with the $ac$ plane along the layers. The Fermi surface consists of a weakly warped cylinder and two open sheets, separated from the cylinder by a small gap $\Delta_{0}$ at the BZ boundary, as shown in Fig. 1 fuji01 ; kono05 ; uji01c .

The magnetic properties of the compound are mainly governed by five localized $3d$-electron spins per Fe3+ ion in the insulating layers. Below $T_{\mathrm{N}}\approx 2.5$ K, these $S=5/2$ spins are ordered antiferromagnetically, with the unit cell doubling along the $c$ axis and the staggered magnetization pointing along the $a$ axis fuji01 ; mori02 . Above a critical magnetic field $B_{c}\sim 2-5$ T, dependent on the field orientation, antiferromagnetism gives way to a saturated PM state kono04b .

The SdH oscillations in the high-field PM state and in the Néel state are markedly different (see Fig. 1b). In the former, two dominant frequencies corresponding to a classical orbit $\alpha$ on the Fermi cylinder and to a large magnetic-breakdown (MB) orbit $\beta$ are found, in agreement with the predicted Fermi surface uji01c ; kono05 . The oscillation amplitude exhibits spin zeros as a function of the field strength and orientation, which is fairly well described by a field-dependent spin-reduction factor $R_{s}(\theta,B)$, with the $g$-factor $g=2.0\pm 0.2$ in the presence of an exchange field $B_{J}\approx-13$ T, imposed by PM Fe3+ ions on the conduction electrons Cepas02 ; kart16 . In the SI, we provide further details of the SdH oscillation studies on $\kappa$-BETS.

Below $B_{c}$, in the AF state, new, slow oscillations at the frequency $F_{\delta}\approx 62$ T emerge, indicating a Fermi-surface reconstruction kono05 . The latter is associated with the folding of the original Fermi surface into the magnetic BZ, and $F_{\delta}$ is attributed to the new orbit $\delta$, see Fig. 1c. This orbit emerges due to the gap $\Delta_{\mathrm{AF}}$ at the Fermi-surface points, separated by the Néel wave vector $(\pi/c,0)$ Peierls.QTS .

Figure 2 shows examples of the field-dependent interlayer resistance of $\kappa$-BETS, recorded at $T=0.42$ K, at different tilt angles $\theta$. The field was rotated in the plane normal to the Néel axis (crystallographic $a$ axis). In excellent agreement with previous reports kono05 ; kono06 , slow oscillations with frequency $F_{\delta}=61.2\mathrm{\,T}/\cos\theta$ are observed below $B_{c}$, see inset in Fig. 2b. Thanks to the high crystal quality, even in this low-field region the oscillations can be traced over a wide angular range $|\theta|\leq 70^{\circ}$.

The angular dependence of the $\delta$-oscillation amplitude $A_{\delta}$ is shown in Fig. 3. The amplitude was determined by fast Fourier transform (FFT) of the zero-mean oscillating magnetoresistance component normalized to the monotonic $B$-dependent background, in the field window between $3.0$ and $4.2$ T, so as to stay below $B_{c}(\theta)$ for all field orientations. The lines in Fig. 3 are fits using the Lifshitz-Kosevich formula for the SdH amplitude shoe84 :

[TABLE]

where $A_{0}$ is a field-independent prefactor, $B=3.5$ T (the midpoint of the FFT window in $1/B$ scale), $m$ the effective cyclotron mass ($m=1.1m_{\mathrm{e}}$ at $\theta=0^{\circ}$ kono05 , growing as $1/\cos\theta$ with tilting the field as in other quasi-2D metals wosn96 ; kart04 ), $K=2\pi^{2}k_{B}/\hbar e$, $T=0.42$ K, $T_{\mathrm{D}}$ the Dingle temperature, and $R_{\mathrm{MB}}$ the MB factor. For $\kappa$-BETS, $R_{\mathrm{MB}}$ takes the form $R_{\mathrm{MB}}=\left[1-\exp\left(-\frac{B_{0}}{B\cos\theta}\right)\right]\left[1-\exp\left(-\frac{B_{\mathrm{AF}}}{B\cos\theta}\right)\right]$, with two characteristic MB fields $B_{0}$ and $B_{\mathrm{AF}}$ associated with the gaps $\Delta_{0}$ and $\Delta_{\mathrm{AF}}$, respectively. The Zeeman splitting effect is encapsulated in the spin factor $R_{s}(\theta)$. In Eq. (1), the geometry of our experiment implies $\mathbf{B}_{\|}=0$, thus in the Néel state $R_{s}(\theta)$ takes the form of Eq. (2).

Excluding $R_{s}(\theta)$, the other factors in Eq. (3) decrease monotonically with increasing $\theta$. By contrast, $R_{s}(\theta)$ in Eq. (2), generally, has an oscillating angular dependence. For $\bar{g}_{\perp}=g=2.0$ found in the PM state kart16 , Eq. (2) yields two spin zeros, at $\theta\approx 43^{\circ}$ and $64^{\circ}$. Contrary to this, we observe no spin zeros, but rather a monotonic decrease of $A_{\delta}$ by over two orders of magnitude as the field is tilted away from $\theta=0^{\circ}$ to $\pm 70^{\circ}$, i.e., in the entire angular range where we observe the oscillations. The different curves in Fig. 3 are our fits using Eq. (3) with $A_{0}$ and $T_{\mathrm{D}}$ as fit parameters, and different values of the $g$-factor. We used the MB field values $B_{0}=20$ T and $B_{\mathrm{AF}}=5$ T. While the exact values of $B_{0}$ and $B_{\mathrm{AF}}$ are unknown, they have virtually no effect on the fit quality, as we demonstrate in the SI. The best fit is achieved with $g=0$, i.e. with an angle-independent spin factor $R_{\mathrm{s}}=1$. The excellent agreement between the fit and the experimental data confirms the quasi-2D character of the electron conduction, with the $1/\cos\theta$ dependence of the cyclotron mass.

Comparison of the curves in Fig. 3 with the data rules out $\bar{g}_{\perp}>0.2$. Given the experimental error bars, we cannot exclude a nonzero $\bar{g}_{\perp}\lesssim 0.2$, yet even such a small finite value would be in stark contrast with the textbook $g=2.0$, found from the SdH oscillations in the high-field, PM state  kart16 . Below we argue that, in fact, $\bar{g}_{\perp}$ in the Néel state is exactly zero.

Optimally doped NCCO — This material has a body-centered tetragonal crystal structure (space group $I4/mmm$), where (001) conducting CuO2 layers alternate with their insulating (Nd,Ce)O2 counterparts armi10 . Band-structure calculations mass89 ; ande95 predict a holelike cylindrical Fermi surface, centered at the corner of the BZ. However, angle-resolved photoemission spectroscopy (ARPES) armi02 ; mats07 ; sant11 ; song17 reveals a reconstruction of this Fermi surface by a $(\pi/a,\pi/a)$ order. Moreover, magnetic quantum oscillations helm10 ; kart11 ; higg18 show that the Fermi surface remains reconstructed even in the overdoped regime, up to the critical doping $x_{c}$ ($\approx 0.175$ for NCCO), where the superconductivity vanishes helm15 . The origin of this reconstruction remains unclear: while the $(\pi/a,\pi/a)$ periodicity is compatible with the Néel order observed in strongly underdoped NCCO, coexistence of antiferromagnetism and superconductivity in electron-doped cuprates remains controversial. A number of neutron-scattering and muon-spin rotation studies luke90 ; moto07 ; mang04a ; saad15 have detected short-range Néel fluctuations, but no static order within the superconducting doping range. However, other neutron scattering yama03 ; kang05 and magnetotransport daga05 ; yu07 ; dora18 experiments have produced evidence of static or quasi-static AF order in superconducting samples at least up to optimal doping $x_{\mathrm{opt}}$. Alternative mechanisms of the Fermi-surface reconstruction have been proposed, including a $d$-density wave chak01 , a charge-density wave silv15 , or coexistent topological and fluctuating short-range AF orders sach19 .

To shed light on the relevance of antiferromagnetism to the electronic ground state of superconducting NCCO, we have studied the field-orientation dependence of the SdH oscillations of the interlayer resistance in an optimally doped, $x_{\mathrm{opt}}=0.15$, NCCO crystal. The overall magnetoresistance behavior is illustrated in Fig. 4(a). At low fields the sample is superconducting. Immediately above the $\theta$-dependent superconducting critical field the magnetoresistance displays a non-monotonic feature, which has already been reported for optimally doped NCCO in a magnetic field normal to the layers helm09 ; li07a . This anomaly correlates with an anomaly in the Hall resistance and has been associated with magnetic breakdown through the energy gap, created by the $(\pi/a,\pi/a)$-superlattice potential helm15 . With increasing $\theta$, the anomaly shifts to higher fields, consistently with the expected increase of the breakdown gap with tilting the field.

SdH oscillations develop above about $30$ T. Figure 4(b) shows examples of the oscillatory component of the magnetoresistance, normalized to the field-dependent non-oscillatory background resistance $R_{\mathrm{backg}}$, determined by a low-order polynomial fit to the as-measured $R(B)$ dependence. In our conditions, $B\lesssim 65$ T, $T=2.5$ K, the only discernible contribution to the oscillations comes from the hole-like pocket $\alpha$ of the reconstructed Fermi surface helm09 . This pocket is centered at the reduced BZ boundary, as shown in the inset of Fig. 5. While magnetic breakdown creates large cyclotron orbits $\beta$ with the area equal to that of the unreconstructed Fermi surface, even in fields of 60-65 T the fast $\beta$ oscillations are more than two orders of magnitude weaker than the $\alpha$ oscillations kart11 ; helm15 .

The oscillatory signal is plotted in Fig. 4(b) as a function of the out-of-plane field component $B_{\perp}=B\cos\theta$. In these coordinates the oscillation frequency remains constant, indicating that $F(\theta)=F(0^{\circ})/\cos\theta$ and thus confirming the quasi-2D character of the conduction. In the inset we show the respective FFTs plotted against the $\cos\theta$-scaled frequency. They exhibit a peak at $F\cos\theta=294$ T, in line with previous reports. The relatively large width of the FFT peaks is caused by the small number of oscillations in the field window $[45-64]$ T. This restrictive choice is dictated by the requirement that the SdH oscillations be resolved over the whole field window at all tilt angles up to $\theta\approx 72^{\circ}$. In the SI we provide an additional analysis of the amplitude at fixed field values in the SI confirming the FFT results. Furthermore, in Fig. 4(b) one can see that the phase of the oscillations is not inverted, and stays constant in the studied angular range. This is fully in line with the absence of spin zeros, see Eq. (2).

The main panel of Fig. 5 presents the angular dependence of the oscillation amplitude (symbols), in a field rotated in the $(ac)$ plane. The amplitude was determined by FFT of the data taken at $T=2.5$ K in the field window $45$ T $\leq B\leq 64$ T. The lines in the figure are fits using Eq. (3), for different $g$-factors. The fits were performed using the MB factor $R_{\mathrm{MB}}=[1-\exp(-B_{0}/B)]$ kart11 ; helm15 , the reported values for the MB field $B_{0}=12.5$ T, and the effective cyclotron mass $m(\theta=0^{\circ})=1.05m_{0}$ helm15 , while taking into account the $1/\cos\theta$ angular dependence of both $B_{0}$ and $m$. The prefactor $A_{0}$ and Dingle temperature $T_{\mathrm{D}}$ were used as fit parameters, yielding $T_{\mathrm{D}}=(12.6\pm 1)$ K, close to the value found in the earlier experiment helm15 . Note that, contrary to the hole-doped cuprate YBa2Cu3O7-x, where the analysis of earlier experiments SebastianPRL2009 ; RamshawNatPhys2011 was complicated by the bilayer splitting of the Fermi cylinder RamshawNatPhys2011 , the single-layer structure of NCCO poses no such difficulty.

Similar to $\kappa$-BETS, the oscillation amplitude in NCCO decreases by a factor of about $300$, with no sign of spin zeros as the field is tilted from $\theta=0^{\circ}$ to $72.5^{\circ}$. Again, this behavior is incompatible with the textbook value $g=2$, which would have produced two spin zeros in the interval $0^{\circ}\leq\theta\leq 70^{\circ}$, see the green dash-dotted line in Fig. 5. A reduction of the $g$-factor to 1.0 would shift the first spin zero to about $72^{\circ}$, near the edge of our range (blue dotted line in Fig. 5). However, this would simultaneously suppress the amplitude at small $\theta$ by a factor of ten, contrary to our observations. All in all, our data rule out a constant $g>0.2$.

Discussion

In both materials, our data impose on the effective $g$-factor an upper bound of $0.2$. At first sight, one could simply view this as a suppression of the effective $g$ to a small nonzero value. However, below we argue that, in fact, our findings imply $\bar{g}_{\perp}=0$ and point to the importance of the Zeeman SOC in both materials. The quasi-2D character of electron transport is crucial for this conclusion: as mentioned above, in three dimensions, the mere absence of spin zeros imposes no bounds on the $g$-factor.

In $\kappa$-BETS the interplay between the crystal symmetry and the periodicity of the Néel state RevazPRL2008 ; RevazPRB2009-1 ; RevazPRL2010 guarantees that $g_{\perp}({\bf k})$ vanishes on the entire line $k_{c}=\pi/2c$ and is an odd function of $k_{c}-\pi/2c$, see the inset of Fig. 1c and Fig. S2 in the SI. The $\delta$ orbit is centered on the line $k_{c}=\pi/2c$; hence $\bar{g}_{\perp}$ in Eq. (2) vanishes, implying the absence of spin zeros, in agreement with our data. At the same time, quantum oscillations in the PM phase clearly reveal the Zeeman splitting of Landau levels with $g=2.0$. Therefore, we conclude that $\bar{g}_{\perp}=0$ is an intrinsic property of the Néel state.

In optimally doped NCCO, as already mentioned, the presence of a (quasi)static Néel order has been a subject of debate. However, if indeed present, such an order leads to $g_{\perp}({\bf k})=0$ at the entire magnetic BZ boundary (see SI). For the hole pockets, producing the observed $F_{\alpha}\simeq 300$ T oscillations, $\bar{g}_{\perp}=0$ by symmetry of $g_{\perp}({\bf k})$ (see inset of Fig. 5 and Fig. S3 in SI). Such an interpretation requires that the relevant AF fluctuations have frequencies below the cyclotron frequency in our experiment, $\nu_{c}\sim 10^{12}$ Hz at 50 T.

Finally, we address mechanisms – other than Zeeman SOC of Eq. (1) – which may also lead to the absence of spin zeros. While such mechanisms do exist, we will show that none of them is relevant to the materials of our interest.

When looking for alternative explanations to our experimental findings, let us recall that, generally, the effective $g$-factor may depend on the field orientation. This dependence may happen to compensate that of the quasi-2D cyclotron mass, $m/\cos\theta$, in the expression (2) for the spin-reduction factor $R_{s}$, and render the latter nearly isotropic, with no spin zeros. Obviously, such a compensation requires a strong Ising anisotropy [$g(\theta=0^{\circ})\gg g(\theta=90^{\circ})$] – as found, for instance, in the heavy-fermion compound URu2Si2, with the values $g_{c}=2.65\pm 0.05$ and $g_{ab}=0.0\pm 0.1$ for the field along and normal to the $c$ axis, respectively Alta12 ; Bast-19 . However, this scenario is irrelevant to both materials of our interest: In $\kappa$-BETS, a nearly isotropic $g$-factor, close to the free-electron value 2.0, was revealed by a study of spin zeros in the paramagnetic state kart16 . In NCCO, the conduction electron $g$-factor may acquire anisotropy via an exchange coupling to Nd3+ local moments. However, the low-temperature magnetic susceptibility of Nd3+ in the basal plane is some 5 times larger than along the $c$ axis Hund89 ; Dali93 . Therefore, the coupling to Nd3+ may only increase $g_{ab}$ relative to $g_{c}$, and thereby only enhance the angular dependence of $R_{s}$ rather than cancel it out. Thus, we are lead to rule out a $g$-factor anisotropy of crystal-field origin as a possible reason behind the absence of spin zeros in our experiments.

As follows from Eq. (2), another possible reason for the absence of spin zeros is a strong reduction of the ratio $gm/2m_{\mathrm{e}}$. However, while some renormalization of this ratio in metals is commonplace, its dramatic suppression (let alone nullification) is, in fact, exceptional. Firstly, a vanishing mass would contradict $m/m_{\mathrm{e}}\approx 1$, experimentally found in both materials at hand. On the other hand, a Landau Fermi-liquid renormalization shoe84 $g\rightarrow g/(1+G_{0})$ would require a colossal Fermi-liquid parameter $G_{0}\geq 10$, for which there is no evidence in NCCO, let alone $\kappa$-BETS with its already mentioned $g\approx 2$ in the paramagnetic state kart16 .

A sufficient difference of the quantum-oscillation amplitudes and/or cyclotron masses for spin-up and spin-down Fermi surfaces might also lead to the absence of spin zeros. Some heavy-fermion compounds show strong spin polarization in magnetic field, concomitant with a substantial field-induced difference of the cyclotron masses of the two spin-split subbands Harr-98 ; McCo-05 . As a result, for quantum oscillations in such materials, one spin amplitude considerably exceeds the other, and no spin zeros are expected. Note that this physics requires the presence of a very narrow conduction band, in addition to a broad one. In heavy-fermion compounds, such a band arises from the $f$ electrons, but is absent in both materials of our interest.

Another extreme example is given by the single fully polarized band in a ferromagnetic metal, where only one spin orientation is present, and spin zeros are obviously absent. Yet, no sign of ferro- or metamagnetism has been seen in either NCCO or $\kappa$-BETS. Moreover, in $\kappa$-BETS,the spin-zero effect has been observed in the paramagnetic state kart16 , indicating that the quantum-oscillation amplitudes of the two spin-split subbands are comparable. However, for NCCO one may inquire whether spin polarization could render interlayer tunneling amplitudes for spin-up and spin-down different enough to lose spin zeros, especially in view of an extra contribution of Nd3+ spins in the insulating layers to spin polarization. In the SI, we show that this is not the case.

Thus, we are lead to conclude that the absence of spin zeros in the AF $\kappa$-BETS and in optimally doped NCCO is indeed a manifestation of the Zeeman SOC. Our explanation relies only on the symmetry of the Néel state and the location of the carrier pockets, while being insensitive to the mechanism of the antiferromagnetism or to the orbital makeup of the relevant bands.

Methods

Crystals of $\kappa$-(BETS)2FeBr4 were grown electrochemically and prepared for transport measurements as reported previously kart16 . The interlayer ($I||b$) resistance was measured by the standard four-terminal a.c. technique using a low-frequency lock-in amplifier. Magnetoresistance measurements were performed in a superconducting magnet system at fields of up to $14\,$T. The samples were mounted on a holder allowing in-situ rotation of the sample around an axis perpendicular to the external field direction. The orientation of the crystal was defined by a polar angle $\theta$ between the field and the crystallographic $b$ axis (normal to the conducting layers).

Optimally doped single crystals of Nd1.85Ce0.15CuO4, grown by the traveling solvent floating zone method, were prepared for transport measurements as reported previously helm10 . Measurements of the interlayer ($I||c$) resistance were performed on a rotatable platform using a standard four-terminal a.c. technique at frequencies of $30-70\,$kHz in a $70\,$T pulse-magnet system, with a pulse duration of 150 ms, at the Dresden High Magnetic Field Laboratory. The raw data were collected by a fast digitizing oscilloscope and processed afterwards by a digital lock-in procedure helm09 . The orientation of the crystal was defined by a polar angle $\theta$ between the field and the crystallographic $c$ axis (normal to the conducting layers).

acknowledgments

It is our pleasure to thank G. Knebel, H. Sakai, I. Sheikin, and A. Yaresko for illuminating discussions. This work was supported by HLD at HZDR, a member of the European Magnetic Field Laboratory (EMFL). P. D. G. acknowledges State Assignment #0033-2019-0001 for financial support and the Laboratoire de Physique Théorique, Toulouse, for the hospitality during his visit. R. G. acknowledges financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) via Germany’s Excellence Strategy (EXC-2111-390814868). J. W. acknowledges financial support from the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - $ct.qmat$ (EXC 2147, project-id 390858490).

Author Contributions

T.H., F.K., M.K., E.K., W.B., and M.V.K performed the experiments and analyzed the data. R.R., P.D.G., and M.V.K. initiated the exploration and performed the theoretical analysis and interpretation of the experimental results. H.F. and A.E. provided high-quality single crystals. R.R., P.D.G., T.H., W.B., E.K., J.W., R.G., and M.V.K. contributed to the writing of the manuscript.

Data Availability

The authors declare that all essential data supporting the findings of this study are available within the paper and its supplementary information. The complete data set in ASCII format is available from the corresponding authors upon reasonable request.

Additional Information

Supplementary Information: Supplementary Information is available at [URL to be provided by the publisher].

Competing interests: The authors declare no competing interests.