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1.
ACS Appl Mater Interfaces ; 15(40): 47260-47277, 2023 Oct 11.
Artículo en Inglés | MEDLINE | ID: mdl-37751537

RESUMEN

Variants of garnet-type Li7La3Zr2O12 are being intensively studied as separator materials in solid-state battery research. The material-specific transport properties, such as bulk and grain boundary conductivity, are of prime interest and are mostly investigated by impedance spectroscopy. Data evaluation is usually based on the one-dimensional (1D) brick layer model, which assumes a homogeneous microstructure of identical grains. Real samples show microstructural inhomogeneities in grain size and porosity due to the complex behavior of grain growth in garnets that is very sensitive to the sintering protocol. However, the true microstructure is often omitted in impedance data analysis, hindering the interlaboratory reproducibility and comparability of results reported in the literature. Here, we use a combinatorial approach of structural analysis and three-dimensional (3D) transport modeling to explore the effects of microstructure on the derived material-specific properties of garnet-type ceramics. For this purpose, Al-doped Li7La3Zr2O12 pellets with different microstructures are fabricated and electrochemically characterized. A machine learning-assisted image segmentation approach is used for statistical analysis and quantification of the microstructural changes during sintering. A detailed analysis of transport through statistically modeled twin microstructures demonstrates that the transport parameters derived from a 1D brick layer model approach show uncertainties up to 150%, only due to variations in grain size. These uncertainties can be even larger in the presence of porosity. This study helps to better understand the role of the microstructure of polycrystalline electroceramics and its influence on experimental results.

2.
ACS Appl Mater Interfaces ; 14(37): 42757-42769, 2022 Sep 21.
Artículo en Inglés | MEDLINE | ID: mdl-36075055

RESUMEN

A non-ideal contact at the electrode/solid electrolyte interface of a solid-state battery arising due to pores (voids) or inclusions results in a geometric constriction effect that severely deteriorates the electric transport properties of the battery cell. The lack of understanding of this phenomenon hinders the optimization process of novel components, such as reversible and high-rate metal anodes. Deeper insight into the constriction phenomenon is necessary to correctly monitor interface degradation and to accelerate the successful use of metal anodes in solid-state batteries. Here, we use a 3D electric network model to study the fundamentals of the constriction effect. Our findings suggest that dynamic constriction as a non-local effect cannot be captured by conventional 1D equivalent circuit models and that its electric behavior is not ad hoc predictable. It strongly depends on the interplay of the geometry of the interface causing the constriction and the microscopic transport processes in the adjacent phases. In the presence of constriction, the contribution from the non-ideal electrode/solid electrolyte interface to the impedance spectrum may exhibit two signals that cannot be explained when the porous interface is described by a physical-based (effective medium theory) 1D equivalent circuit model. In consequence, the widespread assumption of a single interface contribution to the experimental impedance spectrum may be entirely misleading and can cause serious misinterpretation.

3.
ACS Appl Mater Interfaces ; 14(31): 35545-35554, 2022 Aug 10.
Artículo en Inglés | MEDLINE | ID: mdl-35878322

RESUMEN

In an all-solid-state battery, the electrical contact between its individual components is of key relevance in addition to the electrochemical stability of its interfaces. Impedance spectroscopy is particularly suited for the non-destructive investigation of interfaces and of their stability under load. Establishing a valid correlation between microscopic processes and the macroscopic impedance signal, however, is challenging and prone to errors. Here, we use a 3D electric network model to systematically investigate the effect of various electrode/sample interface morphologies on the impedance spectrum. It is demonstrated that the interface impedance generally results from a charge transfer step and a geometric constriction contribution. The weights of both signals depend strongly on the material parameters as well as on the interface morphology. Dynamic constriction results from a non-ideal local contact, e.g., from pores or voids, which reduce the electrochemical active surface area only in a certain frequency range. Constriction effects dominate the interface behavior for systems with small charge transfer resistance like garnet-type solid electrolytes in contact with a lithium metal electrode. An in-depth analysis of the origin and the characteristics of the constriction phenomenon and their dependence on the interface morphology is conducted. The discussion of the constriction effect provides further insight into the processes at the microscopic level, which are, e.g., relevant in the case of reversible metal anodes.

4.
Nanoscale ; 13(40): 17116-17124, 2021 Oct 21.
Artículo en Inglés | MEDLINE | ID: mdl-34633010

RESUMEN

Tip-induced optical spectroscopy overcomes the inherent resolution limits of conventional optical techniques enabling studies of sub-nm sized objects due to the tip's near-field antenna action. This statement is true for individual molecules on surfaces or in the gas phase, but does not hold without restrictions for spatially extended samples. The reason is that the perturbations caused by the tip extend into the sample volume. The tip may induce strain, heating or hot-carrier injection locally in the material. These effects add additional degrees of complexity by changing near-field and far-field optical response. The far-field response varies because strain relaxation, heat and carrier diffusion possess areas of influence exceeding the sample area influenced by the short-range near-field effects. Tip-in spectra are not simply enhanced compared to tip-out spectra, they will also vary in spectral appearance, i.e., peak positions, relative peak intensities, and linewidths. Detailed studies of MoS2 samples ranging from a single layer to bulk-like multi-layer MoS2 also reveal that the spectra are sensitive to variations of phonon and band structure with increasing layer number. These variations have a direct impact on the signals detected, but also clearly modify the relative magnitudes of the contributions of the tip-induced effects to the tip-in spectra. In addition, the optical response is affected by the kind of tip and substrate used. Hence, the presented results provide further insight into the underlying microscopic mechanisms of tip-enhanced spectroscopy and demonstrate that 2D materials are an ideal playground for obtaining a fundamental understanding of these spectroscopic techniques.

5.
Adv Mater ; 33(23): e2100518, 2021 Jun.
Artículo en Inglés | MEDLINE | ID: mdl-33951236

RESUMEN

Low-dimensional organic-inorganic perovskites synergize the virtues of two unique classes of materials featuring intriguing possibilities for next-generation optoelectronics: they offer tailorable building blocks for atomically thin, layered materials while providing the enhanced light-harvesting and emitting capabilities of hybrid perovskites. This work goes beyond the paradigm that atomically thin materials require in-plane covalent bonding and reports single layers of the 1D organic-inorganic perovskite [C7 H10 N]3 [BiCl5 ]Cl. Its unique 1D-2D structure enables single layers and the formation of self-trapped excitons, which show white-light emission. The thickness dependence of the exciton self-trapping causes an extremely strong shift of the emission energy. Thus, such 2D perovskites demonstrate that already 1D covalent interactions suffice to realize atomically thin materials and provide access to unique exciton physics. These findings enable a much more general construction principle for tailoring and identifying 2D materials that are no longer limited to covalently bonded 2D sheets.

6.
Phys Chem Chem Phys ; 23(11): 6903-6913, 2021 Mar 21.
Artículo en Inglés | MEDLINE | ID: mdl-33729237

RESUMEN

Square-shaped Ce0.8Gd0.2O2 (GDC) membranes are prepared by microstructuring techniques from (111)-oriented, polycrystalline GDC thin films. The strain state of the membranes is investigated by micro-Raman mapping using polarized excitation light. Using circularly polarized excitation, the maps of the Raman shifts reveal circular contour lines in concordance with the quadratic shape of the membrane and with optical investigations of the residual strain distribution. In contrast, asymmetric contours of the maps of the Raman shifts exhibiting a two-fold symmetry are found when using linearly polarized excitation. The contour plots for a linear polarization perpendicular or parallel to the local curvature are rotated by 90°. This behavior is caused by the polarization dependence of three overlapping Raman modes arising from the splitting of the triply degenerate F2g mode due to strain. The contribution of their Raman intensity to the overall Raman signal depends on the measurement geometry and the polarization of the incoming and scattered light. Varying the polarization of the incoming excitation light results in different averaging of the Raman-active modes contributing to the broad Raman signal observed. These results clearly demonstrate that polarization-dependent Raman measurements have the potential to yield additional insight into the local strain distribution in free-standing oxide membranes.

7.
RSC Adv ; 11(48): 29887-29895, 2021 Sep 06.
Artículo en Inglés | MEDLINE | ID: mdl-35480291

RESUMEN

2D materials have been intensively studied for almost two decades and are now exhibiting exceptional properties. Thus, devices that integrate 2D materials offer many novel functionalities that will contribute significantly to the transition into an era beyond 'Moore'. Lithographic methods are key technologies in the context of materials' integration into devices. However, to fully leverage the capabilities of these potential devices, it is vital to keep the integrity of the 2D materials intact and to minimize damage induced by device processing. This requirement is only partially met when employing conventional lithography methods, as they induce structural defects in the delicate materials. We demonstrate that exposing graphene to typical electron doses used in conventional electron beam lithography induces significant defect formation. The defect density is proportional to the electron dose and the structural integrity cannot be fully recovered by thermal annealing. We introduce a novel approach of mild lithography which combines traditional processing methods with a subsequent transfer step of the patterned mask onto the 2D material. We demonstrate that this separation of pattern definition and pattern application allows the lithographic process to be performed without exposing and potentially damaging the 2D material being processed. Finally, as an example relevant in terms of innovative device architectures, we present how the mild lithography approach can be used to achieve ordered arrangements of gold nanoparticles on 2D materials.

8.
Rev Sci Instrum ; 91(1): 013905, 2020 Jan 01.
Artículo en Inglés | MEDLINE | ID: mdl-32012588

RESUMEN

Ion-beam techniques, i.e., ion-beam sputtering for material deposition or ion beam etching for a controlled modification of surfaces, are well established for planar thin-film processing. The primary beam of ions exiting a broadband source typically used exhibits a macroscopic spatial beam profile. In general, the beam profile is considered an undesirable but unavoidable feature of the ion source, as it may introduce inhomogeneities in the thickness of the deposited thin-film or the etched surface. Ways of circumventing these effects are sought, e.g., by using rotating substrates or large ion sources compared to the size of the substrate. Here, we demonstrate that an active control of the spatial beam profile may become advantageous when attempting to achieve homogeneous coatings on nonplanar substrates or to etch nonplanar macroscopic structures, e.g., when coating free-form optics.

9.
Nano Lett ; 20(1): 618-624, 2020 Jan 08.
Artículo en Inglés | MEDLINE | ID: mdl-31829616

RESUMEN

We study the magnetotransport properties of single InAs nanowires grown by selective-area metal-organic vapor-phase epitaxy. The semiconducting InAs nanowires exhibit a large positive ordinary magnetoresistance effect. However, a deviation from the corresponding quadratic behavior is observed for an orientation of the applied magnetic field perpendicular to the nanowire axis. This additional contribution to the magnetoresistance can be explained by diffuse boundary scattering of free carriers in the InAs nanowire and results in a reduction of the charge carrier mobility. As a consequence, angle-dependent magnetotransport measurements reveal a highly anomalous behavior. Numerical simulations have been conducted to further investigate the effect of classical boundary scattering in the nanowires. On the basis of the numerical simulations, an empirical description is derived, which yields excellent agreement with the experimental data and allows one to quantify the contribution of boundary scattering to the magnetoresistance effect.

10.
Opt Express ; 27(15): 20347-20357, 2019 Jul 22.
Artículo en Inglés | MEDLINE | ID: mdl-31510130

RESUMEN

Active use of phase transition phenomena for reversibly tuning the properties of functional materials in devices currently is an attractive research area of materials science. We designed and fabricated two kinds of metasurface modulators for dynamically controlling the wavefront of terahertz (THz) radiation based on the temperature-induced insulator-to-metal phase transition of vanadium dioxide (VO2). The modulators designed are based on the C-shaped slot antenna array. The slot antennas are made of the VO2 films on c-sapphire substrates. The C-shaped slot antennas are active only when the VO2 is in its metallic phase, i.e. at temperatures T > TC ∼68 °C. At T > TC, the first kind acts as a THz multi-focus lens which converges an incident THz plane wave into four focal spots and the second kind as an Airy beam generator. We characterized the function of two THz wavefront modulators over a broad frequency range, i.e. from 0.3 to 1.2 THz. Such thermally switchable THz wavefront metasurface modulators with a capability of dynamically steering THz fields will be of great significance for the future development of THz active devices.

11.
ACS Appl Mater Interfaces ; 9(51): 44756-44765, 2017 Dec 27.
Artículo en Inglés | MEDLINE | ID: mdl-29199813

RESUMEN

The impact of inclusions of carbon nanotubes (CNT) on the thermoelectric properties of nanostructured Bi1-xSbx alloys with an Sb content between 10 and 20% was investigated for varying amounts of CNT. Three series of Bi1-xSbx pellets with 0, 0.3, and 0.5 wt % CNT were synthesized by mechanical alloying followed by uniaxial pressing. The resistivity was investigated in the temperature range from 30 to 500 K, revealing an enlargement of the band gap due to nanostructuring of the Bi1-xSbx alloy, which is even more pronounced for alloys including CNT. This enlargement is attributed to a modification of the interface between the Bi1-xSbx nanoparticles by a graphene-like coating, which is formed during the fabrication process due to the addition of CNT. Measurements of the Seebeck coefficient and the thermal conductivity were also performed to determine the thermoelectric properties. In total, the CNT-containing samples show a significant improvement of the figure of merit up to 250% for the Bi0.88Sb0.12 composition with 0.3 wt % CNT due to the interface modification between the nanoparticles, demonstrating the beneficial effect of CNT on the thermoelectric properties.

12.
Nano Lett ; 15(12): 7822-8, 2015 Dec 09.
Artículo en Inglés | MEDLINE | ID: mdl-26544014

RESUMEN

The transport properties of Ge-doped single GaN nanowires are investigated, which exhibit a weak localization effect as well as universal conductance fluctuations at low temperatures. By analyzing these quantum interference effects, the electron phase coherence length was determined. Its temperature dependence indicates that in the case of highly doped nanowires electron-electron scattering is the dominant dephasing mechanism, while for the slightly doped nanowires dephasing originates from Nyquist-scattering. The change of the dominant scattering mechanism is attributed to a modification of the carrier confinement caused by the Ge-doping. The results demonstrate that the phase coherence length can be tuned by the donor concentration making Ge-doped GaN nanowires an ideal model system for studying the influence of impurities on quantum-interference effects in mesoscopic and nanoscale systems.

13.
Phys Chem Chem Phys ; 17(8): 5932-41, 2015 Feb 28.
Artículo en Inglés | MEDLINE | ID: mdl-25635837

RESUMEN

The photoluminescence (PL) of semiconductor nanoparticles (SNP) is strongly modified when the semiconductor is in the proximity of a metal surface or a metal nanoparticle (MNP). The effect may be due to two different phenomena which are (a) (Förster) resonant energy transfer ((F)RET) between the semiconductor and the metal and (b) the enhanced electric field around metallic structures that arises from surface plasmon oscillations. Here we present experimental evidence for enhancement and quenching of the PL of dilute SNP colloidal solutions depending on the amount of admixed MNP and the position of the MNP plasmon band with respect to the excitation wavelength and the optical bands in the SNP. The average distance between an MNP and its next neighbor MNP is varied between ∼0.1 and 2 µm by varying the MNP concentration, whereas that between MNP and SNP as well as between SNP and SNP is kept at about 0.1 µm. A model function based on the rate equations of the system is developed that yields a satisfactory description of the measured data by considering solely FRET between the particle species. The derived function is an extension of the Stern-Volmer equation, as it not only accounts for the energy transfer from the fluorescent SNPs to the MNPs, but also for the transfer of excitation from MNPs to SNPs and between MNPs. This theory provides a deeper insight into the mechanisms of metal-enhanced fluorescence and fluorescence quenching phenomena.

14.
ACS Appl Mater Interfaces ; 6(15): 12083-92, 2014 Aug 13.
Artículo en Inglés | MEDLINE | ID: mdl-25006701

RESUMEN

Lithium peroxide (Li2O2), the solid and intrinsically electronic insulating discharge product of Li-O2 batteries strongly influences the discharge and charge kinetics. In a series of experiments, we investigated the growth of Li2O2 upon discharge and the corresponding reduction and oxidation processes by varying the depth of discharge. The results indicate that insulating Li2O2 particles with a disc-like shape were formed during the initial discharge stage. Afterward, the nucleation and growth of Li2O2 resulted in the formation of conducting Li2O2 shells. When the discharge voltage dropped below 2.65 V, the Li2O2 discs evolved to toroid-shaped particles and defective superoxide-like phase presumably with high conductivity was formed on the rims of Li2O2 toroids. Both Li2O2 and the superoxide-like phase are unstable in ether-based electrolytes resulting in the degradation of the corresponding cells. Nevertheless, by controlling the growth of Li2O2, the chemical reactivity of the discharge product can be suppressed to improve the reversibility of Li-O2 batteries.

15.
Nanotechnology ; 25(26): 265302, 2014 Jul 04.
Artículo en Inglés | MEDLINE | ID: mdl-24915959

RESUMEN

Sub-100 nm antenna arrays consisting of a star-like ridge or dome-like structures with needles in their centers are prepared in thin gold films on glass substrates using femtosecond laser pulses. The needles can be bent mechanically to be horizontally aligned to the substrate surface. Controlled variation of the pulse energy allows one to obtain nanostructures of different defined morphologies. These arrays of nanostructures are covered with a thin homogeneous layer of rhodamine molecules. Raman spectra using linearly polarized laser light of 632.8 nm are taken with the laser spot centered on individual nanostructures and at positions on the unstructured film. The average Raman enhancement within the laser spot focused onto a nanostructure is two orders of magnitude higher than on the unstructured film. The nanostructures with bent needles exhibit a polarization dependence of the SERS effect, i.e., typically the enhancement is larger by about a factor of two for excitation light polarized parallel to the needle direction than for the perpendicular case. The enhancement factor of the star-like ridge structures with needles is analyzed by the finite-element method, which agrees with the experiment. We show that the variation of the SERS activity of almost similar structures arises from the inherent randomness of the hot spots created in the fabrication process. Nevertheless, these antenna structures may be useful as elements in novel SERS devices as they can be accurately positioned on a device using a cheap fabrication process compatible with microfabrication technology.

16.
Nanoscale ; 6(10): 5099-105, 2014 May 21.
Artículo en Inglés | MEDLINE | ID: mdl-24728009

RESUMEN

Highly homogeneous surface-enhanced Raman scattering (SERS) substrates were produced on the centimeter scale by annealing solution-processed gold nanoparticles into plasmonic nanoislands. The average size and separation of the nanoislands are controlled by tuning the annealing temperature. SERS measurements yield a global enhancement factor as large as 10(7) over an area of 2 × 2 cm(2) for samples annealed at temperatures ranging from 150 to 200 °C. Spectral "mapping" of the SERS signal shows a homogeneous distribution of hotspots with high contrast over the entire substrate. The relative standard deviation of the SERS signal is less than 5.4% over an area of 50 × 50 µm(2). Theoretical simulations show strong dependence of the near-field electromagnetic enhancement on the size and the separation gap of the gold nanoislands. Both average gap size and average nanoisland size increase with an increase in annealing temperature. Intensive plasmonic coupling between the adjacent gold nanoislands leads to broadband resonance for samples annealed at 150 and 200 °C; thus, the laser excitation within the spectrum of plasmon resonance at 633 or 785 nm produced significantly enhanced SERS for 4-mercaptopyridine molecules modified on the gold nanoislands.

17.
Small ; 8(12): 1937-44, 2012 Jun 25.
Artículo en Inglés | MEDLINE | ID: mdl-22473813

RESUMEN

A polarization-independent optical sensor is created by fabricating a concentric gold ring grating with a period of 900 nm on the end facet of an optical fiber. The sensing function of this miniaturized device is realized by sending white light as a probe to the gold rings and collecting the response signal in the back-reflection through the optical fiber. A pronounced peak due to the Rayleigh anomaly of the gold ring grating is observed in the reflection spectrum, the center wavelength of which is sensitive to the change in the environmental refractive index of the fiber end facet. Theoretical analysis not only shows excellent agreement with the experimental results, but also gives insights into the mechanisms of this kind of sensor. Using the center position of the Rayleigh peak as the response signal, a high sensitivity dλ/dn of 900 nm per unity refractive index is realized for this sensor and a resolution of Δn/n ≈ 1% is demonstrated in preliminary experiments. The sensitivity is solely determined by the period of the grating.


Asunto(s)
Metales/química , Miniaturización/métodos , Nanopartículas/química , Fibras Ópticas , Simulación por Computador , Diseño de Equipo , Oro/química , Luz , Microscopía Electrónica de Rastreo/métodos , Modelos Estadísticos , Modelos Teóricos , Óptica y Fotónica , Refractometría
18.
Chemistry ; 14(19): 5935-40, 2008.
Artículo en Inglés | MEDLINE | ID: mdl-18491349

RESUMEN

The first synthesis of a chiral periodic mesoporous organosilica (PMO) carrying benzylic ether bridging groups is reported. By hydrolysis and condensation of the new designed chiral organosilica precursor 1,4-bis(triethoxysilyl)-2-(1-methoxyethyl)benzene (BTEMEB) in the presence of the non-ionic oligomeric surfactant Brij 76 as supramolecular structure-directing agent under acidic conditions, an ordered mesoporous chiral benzylic ether-bridged hybrid material with a high specific surface area was obtained. The chiral PMO precursor was synthesized in a four-step reaction from 1,4-dibromobenzene as the starting compound. The evidence for the presence of the chiral units in the organosilica precursor as well as inside the PMO material is provided by optical activity measurements.

19.
J Am Chem Soc ; 126(3): 797-807, 2004 Jan 28.
Artículo en Inglés | MEDLINE | ID: mdl-14733554

RESUMEN

Arrays of highly ordered Zn(1-x)MnxS quantum wires with x ranging from 0.01 to 0.3 and with lateral dimensions of 3, 6, and 9 nm were synthesized within mesoporous SiO2 host structures of the MCM-41 and SBA-15 type. The hexagonal symmetry of these arrays (space group p6m) and the high degree of order was confirmed by X-ray diffraction and transmission electron microscopy (TEM) studies. Physisorption measurements show the progressive filling of the pores of the SiO2 host structures, while TEM and Raman studies reveal the wire-like character of the incorporated Zn(1-x)MnxS nanostructures. X-ray absorption near-edge structure, extended X-ray absorption fine structure, photoluminescence excitation (PLE), and electron paramagnetic resonance studies confirm the good crystalline quality of the incorporated Zn(1-x)MnxS guest species and, in particular, that the Mn2+ ions are randomly distributed and are situated on tetrahedrally coordinated cation sites of the Zn(1-x)MnxS wires for all x up to 0.3. The amount of Mn2+ ions loosely bound to the surface of the Zn(1-x)MnxS nanowires is less than 4% of the total Mn content even for the 3 nm nanostructures up to the highest Mn content of x = 0.3. The effects of the reduction of the lateral dimensions on electronic properties of the diluted magnetic semiconductor were studied by PLE spectroscopy. Due to the quantum confinement of the excitons in the wires an increase of the direct band gap with decreasing particle size is observed.

20.
Chemistry ; 8(1): 185-94, 2002 Jan 04.
Artículo en Inglés | MEDLINE | ID: mdl-11822450

RESUMEN

We present a novel way of synthesising highly ordered arrays of hollow Cd(1-x)Mn(x)S quantum wires with lateral dimensions of 3-4 nm separated by 1-2 nm SiO2 barriers by forming Cd(1-x)Mn(x)S (0 < or = x < or = 1) semiconductors inside the pore system of mesoporous MCM-41 SiO2 host structures. X-ray diffraction and transmission electron microscopy (TEM) studies reveal the hexagonal symmetry of these arrays (space group p6m) and confirm the high degree of order. Physisorption measurements show the filling of the pores of the MCM-41 SiO2. The X-ray absorption near-edge structure (XANES), extended X-ray absorption fine structure (EXAFS), electron paramagentic resonance (EPR), and Raman studies confirm the good crystalline quality of the incorporated (Cd,Mn)S guest. The effects of reducing the lateral dimensions on the magnetic and electronic properties of the diluted magnetic semiconductor were studied by photoluminescence (PL) and PL excitation spectroscopy and by SQUID and EPR measurements in the temperature range 2-400 K. Due to the quantum confinement of the excitons in the wires, an increase of about 200 meV in the direct band gap was observed. In addition, the p-d hybridisation-related bowing of the band gap as a function of Mn concentration in the wires is much stronger than in the bulk. This effect is related to the increase in the band gap due to quantum confinement, which shifts the p-like valence band edge closer to the 3d-related states of Mn in the valence band. Thus, the p-d hybridisation and the strength of the band gap bowing are increased. Compared to bulk (II,Mn)VI compounds, antiferromagnetic coupling between the magnetic moments of the Mn2+ ions is weaker. For the samples with high Mn concentrations (x > 0.8) this leads to a suppression of the phase transition of the Mn system from paramagnetic to antiferromagnetic. This effect can be explained by the fact that the lateral dimensions of the wires are smaller than the magnetic length scale of the antiferromagnetic ordering.

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