Observation of Mermin-Wagner behavior in LaFeO3/SrTiO3 superlattices

Two-dimensional magnetic materials can exhibit new magnetic properties due to the enhanced spin fluctuations that arise in reduced dimension. However, the suppression of the long-range magnetic order in two dimensions due to long-wavelength spin fluctuations, as suggested by the Mermin-Wagner theorem, has been questioned for finite-size laboratory samples. Here we study the magnetic properties of a dimensional crossover in superlattices composed of the antiferromagnetic LaFeO3 and SrTiO3 that, thanks to their large lateral size, allowed examination using a sensitive magnetic probe — muon spin rotation spectroscopy. We show that the iron electronic moments in superlattices with 3 and 2 monolayers of LaFeO3 exhibit a static antiferromagnetic order. In contrast, in the superlattices with single LaFeO3 monolayer, the moments do not order and fluctuate to the lowest measured temperature as expected from the Mermin-Wagner theorem. Our work shows how dimensionality can be used to tune the magnetic properties of ultrathin films.

The properties of magnetic films with thickness in the nanoscale have been a long-standing research topic.The theory of critical behavior predicts that the phase transition temperature should decrease with decreasing film thickness [1], which was observed in several cases [2][3][4][5][6][7].In the 2-dimensional (2D) limit, Mermin and Wagner [8] extended the initial idea of Hohenberg [9] for a superconductor and predicted complete suppression of the long-range magnetic order in models with continuous rotational symmetries (i.e., with the Heisenberg or XY spin Hamiltonian) at finite temperature due to long-wavelength fluctuations.Importantly, this prediction is strictly valid only for the thermodynamic limit, i.e., for samples with laterally infinite sizes.However, since the divergence of the fluctuations in 2D case is only slow (logarithmic in sample size), it was suggested that for any finite-size laboratory samples, the phase order is preserved for superconductivity [10] and even for magnetism [11].
The discovery of magnetic van der Waals materials allowed the investigation of magnetism in samples with thickness down to a single monolayer [12].For example, it was reported that in samples of bulk antiferromagnet NiPS 3 that are two or more monolayers thick, the magnetic order is preserved, whereas it is suppressed in a single monolayer sample [13].Since the Hamiltonian of NiPS 3 has the XY symmetry, this behavior thus follows the prediction of the Mermin-Wagner theorem rather than the suggestions for preserving the long-range order [11].However, due to the small lateral size of the single monolayer NiPS 3 samples obtained by the exfoliation, the antiferromagnetic order was probed relatively indirectly by Raman spectroscopy via coupling of a phonon to a magnon mode [13].
To test the Mermin-Wagner behavior using a magnetic probe, we study the magnetic properties of three to two-dimensional crossover in superlattices composed of antiferromagnetic LaFeO 3 separated by nonmagnetic SrTiO 3 layers.Bulk LaFeO 3 is a prototypical perovskite antiferromagnetic insulator with Heisenberg symmetry of the spin Hamiltonian [14] and with the highest Neel temperature (T N ) of 740 K among ReFeO 3 materials [15], where Re stands for rear earth.It has a high magnetic moment of almost 5 µ B per Fe 3+ ion and the G-type structure of the antiferromagnetic state (where each spin is aligned opposite to the nearest neighbor), thus the antiferromagnetic order is expected to be relatively robust.Thanks to the advancement in deposition technology, it is possible to fabricate heterostructures with sharp interfaces that are composed of perovskite oxides with various order parameters, including magnetism, ferroelectricity, and superconductivity [16,17].Perovskite oxide heterostructures are also promising for applications since they can be used in large-scale samples and devices (see e.g.Refs.[18][19][20][21]).Using pulsed laser deposition, we fabricated superlattices with 1, 2, and 3 monolayers of LaFeO 3 separated by a non-magnetic spacer of 5 monolayers of SrTiO 3 with a large lateral size of 10 × 10 mm 2 that allowed their investigation using a sensitive magnetic probe -low-energy muon spin rotation spectroscopy [22].
To enhance the signal in the muon spin rotation experiment, we prepared superlattices denoted as [(LaFeO 3 ) m /(SrTiO 3 ) 5 ] 10 , where a bilayer with m = 1, 2 or 3 monolayers of LaFeO 3 and five monolayers of SrTiO 3 is repeated 10 times.The scheme of the ideal superlattice structure near the interface with the TiO-terminated SrTiO 3 (001) substrate is shown in Fig. 1(a).Figure 1(b) displays the surface morphology of the m = 2 superlattice measured by an atomic force microscope, which exhibits a flat surface with single unit cell steps copying those of the substrate.The X-ray diffraction spectra, see Fig. 1(c), exhibits zero (SL 0 ), first (SL 1 , SL −1 ) and the second superlattice diffraction peaks (SL 2 , SL −2 ) due to the (LaFeO 3 ) m /(SrTiO 3 ) 5 bilayer, which depict the high structural quality of the superlattices.The thickness of (LaFeO 3 ) m /(SrTiO 3 ) 5 bilayer determined from the first order diffraction peak follows very well the estimates based on the lattice constant of SrTiO 3 and LaFeO 3 , see Supplementary Fig. 1(a).
Investigations of magnetic properties of ultrathin antiferromagnetic layers is a challenging task because of their zero (or very small) average magnetic moment compared to the large total diamagnetic moment of the substrate.To probe the magnetic properties of our superlattices, we have used muon spin rotation spectroscopy, which is sensitive to even very weak local magnetic fields and can distinguish between static and dynamic behavior.We performed the experiments with a low-energy (2 keV) muon beam [22,23], where spinpolarized muons are implanted into the sample only within about 25 nm deep from the surface, see Supplementary Fig. 1(b).Any magnetic field component transverse to the muon spin direction causes its precession with the Lamour frequency ω L = γ µ B, where γ µ = ge/2m µ is the gyromagnetic ratio of the muon and B is the magnitude of the local magnetic field.The time dependence of polarization of the muon spin ensemble (the so-called asymmetry) is measured thanks to the muon decay into a positron preferentially emitted along the muon spin [24].

Zero field muon spin rotation
Figure 2 shows results from the muon spin rotation experiment in zero magnetic field.The time dependence of the muon spin polarization of the superlattices with m = 3 and 2, see Figs. 2(a) and 2(b), respectively, exhibit at high temperature a concave Gaussian-like profile and a transition to a faster exponential-like relaxation at lower temperatures.This behavior is consistent with the following qualitative picture: at high temperatures, LaFeO 3 layers are in a paramagnetic state where the iron electronic moments are fluctuating too fast to be followed by muons, and thus the depolarization is mainly due to the nuclear moments [25].With decreasing temperature, the iron electronic moments, that are much larger than the nuclear moments, start ordering, which manifests as a drop of the initial asymmetry and a faster relaxation.In contrast, the asymmetry of the m = 1 superlattice shown in Fig. 2(c) is qualitatively different because even at high temperatures, it exhibits a faster depolarization with a convex profile.Such behavior indicates that even at high temperatures, the iron electronic moments fluctuate relatively slowly, which masks the fields due to the nuclear moments.

The magnetic volume fraction and the Neel temperature
Muon spin rotation spectroscopy offers a way to determine the volume fraction of a magnetically ordered phase using a measurement where a weak external field is applied transverse to the muon spins.In a paramagnetic state, the fluctuation rate of electronic moments is too high to influence the muon spin direction, and thus the muons precess due to the external magnetic field, which is observed as an oscillation of the asymmetry.Figure 3(a) shows these oscillations in the weak transverse field asymmetry of the m = 3 superlattice at 300 K, which is at this temperature in the paramagnetic state.The solid line represents a fit using the exponentially damped cosine function where A 0 is the initial asymmetry, λ TF is the depolarisation rate, B ext is the applied transverse field, and φ relates to the initial muon spin polarisation.
In an ordered magnetic phase, muon spins quickly depolarize because of the large static fields, which leads to the decrease of the oscillation amplitude, as can be seen in the asymmetry of the m = 3 superlattice at 10 K, see Fig. 3(a).This reduction of the oscillation amplitude is a clear sign of the formation of a static magnetic order at low temperatures.The magnitude of this decrease yields the magnetic volume fraction, f mag , which was calculated as where A 0 (T high ) is the mean of the initial weak transverse field asymmetry above 250 K in the expected paramagnetic state.We have determined f mag of our superlattices using measurements in a transverse field of 10 mT applied in a perpendicular direction to the superlattice surface.We corrected f mag for the muonium formation in SrTiO 3 ; for details, see Supplementary Sec.2.2.
The obtained f mag for the m = 3 superlattice, see Fig. 3(b), exhibits an onset near 175 K and increases with lowering the temperature, which is typical for a magnetically ordered state.At 10 K, f mag is above 0.6, which is more than the LaFeO 3 volume fraction, f V,m=3 = 3/8, which depicts that the antiferromagnetic state is well developed with some stray fields reaching into SrTiO 3 layers.The stray fields are likely caused by the small canting of LaFeO 3 moments [15].In the m = 2 superlattice, f mag (T ) exhibits a weak increase below 200 K, a sharp onset below 35 K and reaches above 0.4 at 5 K.This value is again larger than LaFeO 3 volume fraction f V,m=2 = 2/7, demonstrating that even in this superlattice with only two monolayers of LaFeO 3 , the antiferromagnetic state is well developed at 5 K, although with significantly reduced T N to 35 K.In contrast, f mag of the m = 1 superlattice is zero within the experimental error bars down to the lowest measured temperature of 5 K, showing the absence of formation of a static order in the measured temperature range.The qualitative difference between f mag of m = 3 and m = 2 superlattices on the one hand and of the m = 1 superlattice on the other hand again depicts the qualitative difference in their magnetic ground state.
The dependence of T N on m is summarized in Fig. 3(c).Because muons stop in the superlattice at various sites, it is not possible to determine from the data whether the order is ferromagnetic or antiferromagnetic.However, we assume that the observed order is antiferromagnetic since its transition temperature increases with increasing m, and we expect that for large m, the properties should approach those of bulk LaFeO 3 .In our superlattices with m ≤ 3, T N is still much smaller compared to the bulk value of 740 K. To some extent, this reduction can be due to a change of valency of Fe due to proximity to Sr ions at the interface between LaFeO 3 and SrTiO 3 .This effect is the strongest in the m = 1 superlattice where the iron oxide layer is formed only by one LaO and one FeO 2 layer, see Fig. 1(a), and thus Fe ions are surrounded equally by La and Sr ions.Nevertheless, since bulk La 0.5 Sr 0.5 FeO 3 is still antiferromagnetic with T N of about 250 K [33], we conclude that the strong reduction of T N of m = 2 and m = 1 superlattices is predominantly due to the dimensional crossover rather than due to the change of the Fe valency.

Differentiation between the static and dynamic magnetism
The zero field and the weak transverse field data indicate that there is no magnetic order in the m = 1 superlattice down to 5 K.This could be explained by two scenarios: a static disorder (e.g., due to structural defects) or dynamic fluctuations of the electronic moments.Muon spin rotation spectroscopy offers a way to unequivocally differentiate between static magnetism and dynamically fluctuating fields by measurements in the magnetic field longitudinal to the muon spin direction.In the presence of static magnetism, muons in the longitudinal field with a magnitude much larger than that of the local fields essentially do not precess (so-called decouple from the local fields) and thus do not depolarize in contrast to the zero field measurements.However, if the Fig. 4: Differentiation between the static and dynamic magnetism (a) Time evolution of normalized muon spin polarization, A N LF , of the m = 1 superlattice at 5 K for various applied longitudinal magnetic fields.Error bars represent one standard deviation.The solid lines represent fit using a model given by Eq. ( 4).The significant decrease of asymmetry at high fields is a hallmark of dynamic magnetism.Panels (b) and (c) display the theoretical Gaussian Kubo-Toyabe functions used in the fit for dynamically fluctuating moments, P dyn , and for static disordered moments, P stat , respectively.local fields are fluctuating, they cause a random muon spin-flip (a transition between the Zeeman split energy levels) and cause the muon-spin depolarization even in the longitudinal field, essentially the same as in zero field [24].Time evolutions of muon spin polarization in the m = 1 superlattice at 5 K in several longitudinal fields are shown in Fig. 4(a); data are normalized as detailed in Supplementary Sec.2.3.The asymmetry increases between zero field and 2.5 mT, which is caused by the decoupling of the muon spins from the static nuclear moments of SrTiO 3 [25].However, for higher fields between 2.5 mT to 125 mT, the asymmetry is essentially field independent and exhibits at 8 µs considerable depolarization to about 40% of the initial value.Such a significant depolarization independent of the longitudinal field is a hallmark of fluctuating electronic moments (see, e.g., Ref. [34]).
We have modeled the normalized asymmetry in the longitudinal field, A N LF , as a sum of the theoretical Gaussian Kubo-Toyabe functions for dynamic fluctuations, P dyn [35] and for the static disorder, P stat [25] A where c is the volume fraction of the fluctuating part; for details see Supplementary Sec.2.3.The global fit for all longitudinal fields B ext , see solid lines in Fig. 4(a), yields the volume fraction c = 0.64 ± 0.06 and the distribution of the static disordered moments σ s /γ µ = 0.32 ± 0.08 mT.The functions P dyn displayed in Fig. 4(b) for the obtained parameter values are essentially field independent and vanish at 8 µs.In contrast, P stat , displayed in Fig. 4(c), sensitively depends on the external magnetic field.This difference allows the model to discern between static disorder and dynamically fluctuating moments.The obtained value of σ s /γ µ = 0.32 ± 0.08 mT is typical for nuclear moments [25].
It corresponds to regions in SrTiO 3 where the dipolar fields from the iron moments are smaller compared to the nuclear fields.This area's volume fraction of (1 − c) = 0.36 corresponds to the width of about 2.2 monolayers, presumably located in the middle of SrTiO 3 layers.The fact that we can fit the data with the model yielding such a small value of σ s /γ µ at all external fields is incompatible with the picture of statically disordered iron moments with local fields expected to be in the order of 100-250 mT [36].If iron moments were static, the increase of the longitudinal field between 10 and 125 mT would lead to a significant increase in the asymmetry [24].The field-independent asymmetry exhibiting such a considerable depolarization for fields above 2.5 mT can be explained only as a consequence of the fluctuating iron moments.In summary, the muon spin rotation data in zero, transverse and longitudinal fields consistently show that (i) m = 3 and m = 2 superlattices exhibit a long-range antiferromagnetic order with T N of 175 K and 35 K, respectively, (ii) that the magnetic properties of the m = 1 superlattice are qualitatively different with no long-range order down to the lowest measured temperature of 5 K and (iii) that at this temperature, the electronic moments are fluctuating rather than statically disordered.These findings point towards a dimensional magnetic crossover where for the superlattice with a single monolayer of iron oxide, the static antiferromagnetic order is lost due to enhanced magnitude of spin fluctuations, as expected from the Mermin-Wagner theorem.the structure of our superlattices leads to a broad distribution of internal fields, including the stray fields due to the iron spin canting that spread through SrTiO 3 layers.This corresponds to a large distribution of Larmor frequencies inevitably leading to a fast damping of the oscillations.The time evolution of the zero-field asymmetry presented in Fig. 2(a)-(c) was fitted using the stretched exponential function (1) whose exponent β is shown in Fig. 2(d).For completeness, values of the other fitted parameters, i.e., the initial asymmetry, A 0 , and the depolarization rate, λ, are shown in Supplementary Figs.2(a) and 2(b), respectively.The temperature dependence of A 0 exhibits a noticeable decrease below T N , that is, below 175 K for the m = 3 and below 35 K for the m = 2 superlattice.This decrease is expected in an ordered magnetic phase where the muons quickly depolarize due to strong static local fields.In the m = 1 superlattice, A 0 exhibits only a gradual and relatively weak decrease with decreasing temperature without a sharper onset in agreement with the interpretation that there is no static order in this superlattice.Surprisingly, A 0 of the m = 1 superlattice seems to increase from 10 to 5 K.However, this increase is on the level of one standard deviation, and we do not consider it significant enough.
The temperature dependence of λ, shown in Supplementary Fig. 2(b), is roughly constant for m = 3 and m = 2 superlattices with values of λ in the range from 0.16 to 0.19 µs −1 .It exhibits only a small indication of a decrease with decreasing temperature below T N , particularly in the m = 2 superlattice, connected with the formation of the static magnetic order.In contrast, in the m = 1 superlattice, the values of λ are above 100 K about two times smaller compared to m = 3 and m = 2 superlattices.In addition, λ significantly increases with decreasing temperature below 100 K to about 0.14 µs −1 at 5 K compared to 0.075 µs −1 at 300 K.This almost doubling of the depolarization rate with decreasing temperature is a strong indication of a decrease in the electronic fluctuation rate.The latter is a typical signature Supplementary Fig. 3: Normalized asymmetry due to the muonium formation in SrTiO 3 expressed by Eq. ( 5) (blue solid line) and Eq. ( 7) (red dashed line).
of a fluctuating magnetic ground state [4].The qualitative and quantitative difference of the temperature dependence of λ between m = 3 and m = 2 superlattices on the one hand and the m = 1 superlattice on the other hand again depicts the difference in their respective magnetic ground states.Note that in the case of the m = 2 superlattice, the temperature dependence of λ exhibits a clear maximum at T N of about 35 K with a significant decrease below.This behavior is expected for a static magnetic order that sets in at a phase transition temperature [5].

Weak transverse field muon spin rotation
In the weak transverse field data analysis, we shall consider the formation of muonium (a bound state of a muon and an electron) in SrTiO 3 .Because the muon spin in muonium precesses at a different frequency than a free muon spin, the formation of muonium occurring below about 50 K in SrTiO 3 [6] influences the weak transverse field data of our superlattices.The temperature dependence of the normalized weak transverse field asymmetry for SrTiO 3 at 1.6 keV implanting energy was described by the empiric equation [6] A N (T ) = 0.1 arctan T − 43 4.427 + 0.85 , see Supplementary Fig. 3. Our measurements were performed at the implanting muon energy of 2 keV, which is close enough to use Eq. ( 5) as a starting point in muonium correction.Assuming that the muonium is formed only in SrTiO 3 layers, the depolarization due to the muonium formation is subtracted from for the muonium formation using Eq. ( 5) and Eq. ( 7), respectively.
the data using the following equation where f SrTiO3 is the volume fraction of SrTiO 3 in a given superlattice, and A 0 (T high ) is the asymmetry at high enough temperature.For A 0 (T high ) we have used a mean value above 250 K where the superlattices are in the paramagnetic state and the influence of muonium is negligible.The correction for the muonium formation is significant only below the temperature of the muonium formation of about 50 K, where A N (T ) is significantly smaller than unity, see Supplementary Fig. 3.At higher temperatures, A N (T ) ≈ 1 and the second term on the right-hand side of Eq. ( 6) vanishes.
The magnetic volume fraction of the superlattice, f mag (T ), is calculated as [7] where A 0,c (T high ) is the mean of the initial weak transverse field asymmetry above 250 K in the expected paramagnetic state.Supplementary Fig. 4(a) shows f mag calculated without the muonium correction (using A N (T ) = 1).The shaded regions show temperatures below about 50 K where the muonium formation takes place.Values of f mag corrected for the muonium formation using Eq. ( 5), see Supplementary Fig. 4(b), suddenly decrease below about 50 K for superlattices with m = 2 and m = 1, which leads, for the case of m = 1, even to nonphysical values significantly below zero.Most likely, the steplike correction for muonium formation using Eq. ( 5), which was obtained on SrTiO 3 single crystal, is sharper and centered at a different temperature than what would be appropriate for ultrathin SrTiO 3 layers of our superlattices.
We have therefore adjusted the temperature and width of the transition in Eq. ( 5) where the muonium formation occurs so that f mag is not negative for LaFeO 3 /SrTiO 3 superlattice the superlattice m = 1.This approach yielded A N,mod (T ) = 0.1 arctan T − 20 6 + 0.85 , see Supplementary Fig. 3(b).Corresponding f mag is shown in Supplementary Fig. 4(c) and in the main part of the paper in Fig. 3(b).Note that we did not adjust the multiplication factor of the step-like arctan function in Eq. ( 8) corresponding to the magnitude of the correction.Consequently, the values of f mag at 5 K resulting from the two corrections [cf.Supplementary Figs.4(b) and 4(c)] are almost the same.Similarly, the main conclusions are the same: the magnetic volume fraction in the m = 1 superlattice at 5 K is essentially zero corresponding to the absence of a static order formed in the measured temperature range in contrast to the m = 2 superlattice where it is significantly above zero (above 0.4 and above LaFeO 3 volume fraction of 2/7) and thus the superlattice exhibits a static antiferromagnetic order.

Longitudinal field muon spin rotation
Supplementary Fig. 5 shows the asymmetry of the m = 1 superlattice measured at 5 K for several longitudinal fields.The data are analyzed with the model Here A 0 (B ext ) is a background asymmetry which, in principle, depends on magnetic field B ext .In LEµSR, muons are focused onto the sample by the external magnetic field, and thus different magnetic fields give rise to a different background.A is the normalization constant that is field independent.The depolarization due to the sample is modeled as a weighted average of the theoretical Gaussian Kubo-Toyabe functions for the static disorder, P stat [8], and the dynamic fluctuation, P dyn [9], where ν is the fluctuation rate.The depolarization rate σ appearing in Eqs. ( 10)-( 12) is defined as σ = γ µ < ∆B 2 > where < ∆B 2 > is the second that is due to the comparably large field of 2.5 mT used in the measurements.
To determine this value with better precision, one would need to measure with significantly smaller fields.The errors of other values shown in Tab. 1 are reasonably low (about 10%), which demonstrates that the global fit is well conditioned.Particularly, the fit allowed us to determine the constants A and A 0 (B ext ) with reasonable precision.For the sake of simplicity, we display in Fig. 4 the data as normalized asymmetry 3 Run logs

Fig. 2 :
Fig. 2: Zero field muon spin rotation.Time evolution of the zero-field muon spin polarisation of [(LaFeO 3 ) m /(SrTiO 3 ) 5 ] 10 superlattices with (a) m = 3, (b) m = 2, and (c) m = 1.Error bars represent one standard deviation, and solid lines represent fit by the stretched exponential function (1).(d) Exponent β of the stretched exponential as a function of temperature.Lines represent a guide to the eye.Colored regions show expected values of β for different types of magnetic states.

Fig. 3 :
Fig. 3: The magnetic volume fraction and the Neel temperature a) Time evolution of the muon spin polarization in the weak transverse field of 10 mT in the m = 3 superlattice at 300 K and 10 K shown with a fit (solid lines) using Eq.(2).(b) Magnetic volume fraction, f mag , of [(LaFeO 3 ) m /(SrTiO 3 ) 5 ] 10 superlattices evaluated from the measurement in the weak transverse field.Horizontal dashed lines represent the volume fraction of LaFeO 3 in the superlattices.(c) The Neel temperature with respect to m determined from panel (b).

Supplementary Fig. 2 :
(a) Initial asymmetry, A 0 , and (b) depolarization rate, λ, obtained from the fit of zero field data using Eq.(1).The highlighted areas mark Neel temperature of m = 2 and m = 3 superlattices.

. 4 :
Magnetic volume fractions obtained from the weak transverse field measurement.Panel (a) shows values obtained without the correction for muonium formation, and panels (b) and (c) show those corrected

Table 2 :
Low energy µSR run log for the m = 1 superlattice measured in zero, weak transverse, and longitudinal fields.LaFeO 3 /SrTiO 3 superlattice

Table 3 :
Low energy µSR run log for the m = 2 superlattice measured in zero and weak transverse fields.Springer Nature 2021 L A T E X template LaFeO 3 /SrTiO 3 superlattice 23

Table 4 :
Low energy µSR run log for the m = 3 superlattice measured in zero and weak transverse fields.