Nanomachining-enabled strain manipulation of magnetic anisotropy in the free-standing GaMnAs nanostructures

Strain perturbs atomic ordering in solids, with far-reaching consequences from an increased carrier mobility to localization in Si, stabilization of electric dipoles and nanomechanical transistor action in oxides, to the manipulation of spins without applying magnetic fields in n-GaAs. In GaMnAs, a carrier-mediated ferromagnetic semiconductor, relativistic spin-orbit interactions – highly strain-dependent magnetic interactions – play a crucial role in determining the magnetic anisotropy (MA) and anisotropic magnetoresistance (AMR). Strain modifies the MA and AMR in a nanomachined GaMnAs structure as measured by the anomalous Hall effect (AHE) and the planar Hall effect (PHE). Here, we report an MA modification by strain relaxation in an isolated GaMnAs Hall bar structure and by applying a range of local strains via fabricating asymmetrically mechanically buckled GaMnAs micro-Hall bar structures. In the AHE and PHE measurements, we observe a reduction in the in-plane MA and an enhancement in the out-of-plane MA as the compressive strain due to the lattice mismatch relaxes in the suspended structure. The functionality of such mechanical manipulation, as well as the two-level mechanical state and the corresponding AHE responses, is demonstrated by a fully scalable binary mechanical memory element in a GaMnAs single Hall cross structure.


experiment and Discussion
Freely suspended strain-relaxation structure. The freely suspended GaMnAs Hall bar structure designed for strain relaxation is grown by low-temperature molecular beam epitaxy (LT-MBE): GaMnAs (100 nm)/GaAs (80 nm)/AlGaAs (2000 nm) is grown on a SI-GaAs (001) substrate with ~6% Mn concentration in the GaMnAs epilayer and ~75% Al concentration in the AlGaAs sacrificial layer. The details of the LT-MBE growth of the GaMnAs epilayer and characterization of the epilayer have been described elsewhere 25,26 . A 60-µm-long and 6-µm-wide freely suspended GaMnAs Hall bar structure with a reference structure (identical shape and dimension, but not suspended) is patterned in a single specimen by e-beam lithography, as shown in Fig. 1(a). Then, the Hall bar structures are defined by citric acid/hydrogen peroxide etching with a volume ratio of 7:2 (etch rate: ~200 nm/min), isolating the Hall bar from the remaining GaMnAs layer. Then, metal electrodes (Au 100 nm/Cr 80 nm) are deposited by standard e-beam lithography and e-beam deposition. The thickness of the metallic layer is chosen so that the deposited metal covers the step edge at the end of the GaMnAs structure from the GaMnAs etch. Selective etching of the AlGaAs layer using 10% HF is utilized to suspend the Hall bar structure, while the reference structure is protected from etching by e-beam resist. After the definition of the structure, the critical point drying method is applied to prevent a possible structural collapse. Figure 1(b) shows an SEM image of the realized freely suspended Hall bar structure, which is completely suspended from the substrate. The GaMnAs layer is isolated from the substrate and clamped by soft metal electrodes, without any GaMnAs beneath, solely supporting the Hall bar structure.
The quality of the freely suspended Hall bar structure is demonstrated by both optical and electrical measurements. As shown in Fig. 1(c), the suspended Hall bar is examined by a non-contact optical profilometer (NANO View-E1000) for any possible physical deformation during the fabrication process. The realized suspended Hall bar structure demonstrates little deformation of less than 100 nm along 60 µm. For further verification, the temperature dependence of the resistivity is plotted for both the free-standing Hall bar structure and the reference structure ( Fig. 1(d)). All magnetotransport measurements are performed in a closed-cycle magnetocryostat (IceOxford DRYICE4 TL) using the standard AC lock-in technique (I = 0.1 µA at 17 Hz and 27 Hz). Both structures clearly exhibit typical behaviours of ferromagnetic semiconductors and identical temperature-dependence anomalies (resistance peak) corresponding to T C ~ 65 K, indicating that the fabrication process causes no significant degradation and does not alter the macroscopic electrical characteristics of the GaMnAs epilayer.

Magnetotransport measurements in the freely suspended strain-relaxation structure.
Magnetotransport measurements are performed to investigate the anomalous Hall effect. The measurements are performed with an applied magnetic field perpendicular to the free-standing GaMnAs Hall bar structure (H < ± 9 T) using the AC lock-in measurement technique (I = 0.1 µA at 17 Hz and 27 Hz). Schematic of the magnetotransport measurement is illustrated in Fig. 2(a). Figure 2(b) plots the longitudinal anisotropic magnetoresistance of both the freely suspended and the reference as the applied magnetic field is swept at T = 4 K. Although both the suspended GaMnAs and the reference exhibit similar resistance changes with a high field (H > ±1 T), the resistivity drop for a low field shows marked differences: the suspended GaMnAs structure exhibits a smaller flat magnetoresistivity region between jumps than that of the reference. The corresponding transverse Hall responses are plotted in Fig. 2(c). While the anomalous Hall effect response from the reference shows a typical hard axis response, in which the resistivity increases sharply as the magnetization reversal occurs, the response from the suspended structure exhibits a more inclined shape as a reduced hard axis response. Both the AMR and AHE responses suggest that the easy magnetization direction in the suspended structure is more out-of-plane-like, while the reference exhibits typical in-plane-like behaviour, as reported elsewhere 27 . The AMR and AHE responses are further discussed and quantified later, along with the planar Hall measurements.
Further investigation of the strain-relaxation-based magnetic anisotropy changes is conducted by measuring the angle-dependent longitudinal resistivity at T = 4 K with an applied field of 1500 Oe, as shown in Fig. 2(d). The longitudinal resistivity in the reference increases rapidly as the angle approaches 180°, where the magnetization reversal occurs, and decreases after the magnetization reversal. In contrast, the suspended structure exhibits a smooth transition in the longitudinal resistivity, which can be attributed to the modification of the magnetic anisotropy from the in-plane direction towards the out-of-plane direction. A similar strain-based magnetic anisotropy modification has been reported in metallic magnetic thin films on a flexible substrate 28 .
The modification of the magnetic anisotropy by strain relaxation is further examined by a planar Hall response (PHR) measurement with an applied magnetic field within the plane of the Hall bar structure, as shown in Fig. 3(a). Figure 3(b) plots the angle-dependent transverse resistance at T = 15 K with an applied magnetic field of 500 Oe. While the reference clearly shows switching behaviour attributed to a two-jump sequence of magnetization, ), the suspended structure shows a rather semi-sinusoidal shape, and the Hall resistance jumping behaviour disappears. Figure 3(c) depicts the transverse resistance as a function of the applied field angle (R xy vs. ϕ) at a temperature of 15 K with applied magnetic fields of 200 Oe, 500 Oe, 1500 Oe and 12000 Oe. The resistance jumping behaviour in the reference GaMnAs structure appears as a sharp transition in the plot, whereas the free-standing GaMnAs does not show this sharp transition. Figure 3(d) shows the longitudinal resistance as a function of the applied field angle (R xx vs. ϕ) at T = 4 K, 15 K, 30 K, 60 K, and 90 K with an applied magnetic field of 1.2 T.
In the reference sample, the longitudinal resistance change appears along 90° and 270°, which corresponds to the < > 110 direction in GaMnAs, and disappears with increasing temperature. This response exhibits the in-plane cubic anisotropy, which is dominant below ~30 K, as reported by others [29][30][31] . However, in the free-standing GaMnAs structure, this feature along the < > 110 direction in the longitudinal resistance appears only below 15 K. This reduction in the resistance change along < > 110 can be attributed to a reduction in the in-plane cubic magnetic anisotropy. To quantify the magnetic anisotropy, the angle dependency of planar Hall resistivity can be analysed by the Stoner-Wohlfarth model [32][33][34] . At T = 10 K and H = 1.2 T, fitted values for cubic anisotropy and uniaxial anisotropy are 1151 Oe and 325 Oe for the reference and 380 Oe and 697 Oe, respectively, for the suspended sample. Similarly, from AHE, at T = 10 K and H = 1.2 T, we find the out-of-plane uniaxial components of the anisotropy fields to be 1767 Oe for the reference and 99 Oe for the suspended sample (see METHODS).
The anomalous Hall effect and planar Hall effect measurements suggest that mechanical strain can be utilized to manipulate the magnetic anisotropy of the GaMnAs epilayer. The origin of this manipulation is the magnetic anisotropy change due to strain. The magnetic easy axis of GaMnAs tends to stay in-plane for compressive strain and out-of-plane for tensile strain 35 . GaMnAs on AlGaAs possesses compressive strain due to the lattice www.nature.com/scientificreports www.nature.com/scientificreports/ mismatch between the two layers. However, this strain, which is determined during growth, can be modulated by a mechanical strain modulation, resulting in the magnetic anisotropy changes shown in Figs 2 and 3.
The reduction in the in-plane magnetic anisotropy can be attributed to the free-standing Hall bar structure, which is isolated from the GaMnAs layer under compressive strain. Since the Hall bar structure is supported solely by a soft metal electrode as noted above, the isolated GaMnAs structure can be relaxed freely: any possible strain evolved during the restoration of the lattice is absorbed by the soft metal electrode. Without the removal of GaMnAs beneath the metal electrode, the free-standing structure is no longer free from evolving strains: lattice restoration in the beam segment may cause further compression towards both ends of the beam, possibly inducing more compressive strain in semi-rigidly supported Hall probes.
Magnetotransport measurements in the freely suspended buckled structure. Strain induction in the semi-rigidly supported free-standing GaMnAs structure is further investigated in a multiple Hall bar structure. The multiple Hall bar structure is patterned with an identical e-beam lithography technique but is no longer isolated from the GaMnAs layer. The multiple Hall bar, with dimensions (w × l × h) of 2 µm × 30 µm × 175 nm, is designed to contain six equally spaced (s = 6 µm) transverse Hall probes (P A-B,F , P C-E ) (with a width of 1 µm), as shown in Fig. 4(a). P A serves as a unsuspended Hall probe (reference), and P C-E serves as suspended Hall probes with semi-rigid supports, while P B and P F are located at the clamped ends. Contrary to the suspended Hall bar structure with a soft metal support, the structure with the semi-rigid support exhibits mechanical buckling phenomena in the second mode, as characterized by non-contact optical microscopy ( Fig. 4(b)). At both ends of the suspended part, the "edge structures" are placed where most of the structure is unsuspended (P B&F ). Hall probes P C and P E are structures with the most vertical displacement, with Hall probe P C buckled up and Hall probe P E buckled down. Hall probe P D , placed between P C and P E , is an inflection point in the whole structure.
Mechanical buckling is a highly nonlinear phenomenon, sensitive to initial conditions. Buckling of the suspended structure can be explained in terms of the Euler critical load 36 . Using the mechanical buckling model of a beam and the corresponding differential equation, the critical load is = π P n , cr EI L 2 eff 2 2 , where n is a buckling mode and L eff is the effective length dependent on each end. As we have a beam with both ends fixed, L eff is half of the total length. The critical stress corresponding to the critical load is σ cr = P cr /A (A is the cross-sectional area). With the same dimension as the suspended part of the Hall beam (2 μm × 24 μm × 175 nm for w × l × h), we calculate the critical stress corresponding to the critical load σ cr to be 913.9 kPa for the second buckling mode, and the corresponding strain is 0.00107%, which is much smaller than the strain due to the lattice mismatch. Therefore, it is reasonable for the Hall beam structure to buckle.
A simple mass-spring model can be utilized to predict the induced strain in each probe of the buckled structure, with beams as springs and probes as masses (Fig. 4(c)). As noted above, when the GaMnAs beam structure is freely suspended, the beam relaxes and the probes serve as semi-rigid supports. With compression acting on a Hall probe with point symmetry, all the compressions are applied to the Hall probe (P D ). In contrast, when this point symmetry is broken (P C&E ), all the compressions cannot be applied, but compressions can be applied that are larger than that before relaxation (P A , P B&F ).
Magnetotransport measurements are performed to test our physical model. For temperatures from 5-300 K, we measure the transverse resistances or the AHE (ρ xy ) as a function of the applied magnetic field (H ⊥ ). Figure 4(d) plots the AHE for each probe: the reference probe P A , P B and P F at the suspended beam clamps, the off-centre suspended probes P C and P E , and the central suspended probe P D . P A displays a typical response for GaMnAs, with suppression of the AHE near T C . For the asymmetrically buckled GaMnAs Hall beam, we observe a progressive and symmetric suppression of the anomalous Hall response away from the beam centre. This symmetry in the responses of the off-centre probes can be seen more clearly in Fig. 4(e). Comparing the reference probe P A or the probes at the beam clamps (P B and P F ) to the off-centre suspended probes (P C and P E ) and the central suspended probe (P D ), we see a systematic suppression of σ xy (H ⊥ ) as we progress towards the probe under the highest strain. The behaviour and values of the strain dependence agree well with theory 20,21,23 .
A means to reversibly observe changes in the AHE corresponding to a particular mechanical state would give great credence to our physical model and the strain-dependent and mechanically controlled AHE in GaMnAs.

Figure 4.
Asymmetrically buckled multiple Hall beam structure. An asymmetrically buckled, micron-sized, GaMnAs Hall beam is realized by micromachining techniques. Along the beam, this structure contains six equally spaced (6 μm) pairs of transverse electrical leads: P A , which is the reference probe; P B and P F at the clamped ends of the suspended beam; and P C , P D , and P E on the buckled GaMnAs beam. (a) A false-colour SEM micrograph of the presented sample. (b) The height profile of the buckled beam, as measured by a non-contact optical profilometer after processing. Buckling occurs because the underlying AlGaAs epilayer is selectively removed from underneath the GaMnAs/GaAs epilayers, which are compressively strained when grown. (c) Depicts our physical model of the buckling process and the resulting strain variations in probes P C-E . (d) plots the resistivity at various temperatures. The curves represent the reference probe P A ; the probes P B and P F at the clamped ends; the pairs of probes P C and P E symmetrically placed around the centre of the buckled beam; and finally the central probe P D . The reference probe exhibits a prototypical GaMnAs anomalous Hall response. The transverse resistivities at the beam clamps of P B and P F display behaviours similar to the Hall response and similar to each other. The three suspended probes P C , P D , and P E show significant suppression, which is strongest at the centre of the buckled beam. (e) plots σ xy (H ⊥ ), revealing a drastic decrease in the anomalous Hall conductivity in the suspended probes. (2019) 9:13633 | https://doi.org/10.1038/s41598-019-50115-1 www.nature.com/scientificreports www.nature.com/scientificreports/ Thus, we realize free-standing simple Hall cross structures with a length of 11 µm and widths of 1 µm and 2 µm. As mentioned previously, mechanical buckling is a nonlinear phenomenon; after processing, we find that some crosses have buckled, while others have not. We then simultaneously measure ρ xx and ρ xy as a function of H ⊥ of the suspended Hall cross at various temperatures below T C , as shown in Fig. 5. For a particular initially buckled sample as verified by SEM, we first see a suppressed AHE response, (ρ xy ) Low . After several thermal cycles between room temperature and T = 5 K, the AHE response changed drastically, (ρ xy ) High , while the magnetic field dependence of ρ xx was unaffected. After the change in the AHE response, we found that the Hall cross mechanical state changed to an unbuckled state, again from SEM ( Fig. 5(a,b)

insets).
With large differences between (ρ xy ) High = '0' and (ρ xy ) Low = '1' , highly scalable, robust, non-volatile, low-dissipation binary mechanical sensor elements 17 can be realized 37 , in which the mechanical state can be robustly read electronically, a non-trivial realization in other proposed nanomechanical memory elements such as those based on carbon nanotubes 17 . Further incorporation of nanomechanical actuators such as electrostatic-driven comb drives allows for dynamic avenues to control axial strain and the buckling process 38 .

Summary
In summary, we have fabricated three types of suspended GaMnAs Hall bar structures to obtain a varying strain. We observe an in-plane magnetic anisotropy reduction via distinct changes in the PHE and AHE responses as the strain is relaxed in an isolated GaMnAs suspended structure. In the buckled GaMnAs multiple Hall bar structure, transverse transport measurements show a suppression of the AHE that varies symmetrically about the centre of the buckled beam, supporting the interpretation of the strain-induced SOI modifications. We demonstrate a two-level AHE state corresponding to a bi-stable mechanical state of a simple, scalable GaMnAs Hall cross structure.

Methods
To quantify the magnetic anisotropy field, we assume a coherent rotation of the magnetization in GaMnAs, according to the Stoner-Wohlfarth model with the free energy density described below [39][40][41] . www.nature.com/scientificreports www.nature.com/scientificreports/ The magnetization angle can be derived from the normalized planar Hall resistance, which is expressed as [41][42][43] where t is the thickness of the film, ϕ is the angle between the magnetization and current, and k is a constant related to the AMR. By fitting the data measured at 10 K with an applied field of 1. The magnetization angle can be obtained by normalizing the angle dependence of the anomalous Hall resistance, which is described as [41][42][43] : xy s 0 where t is the thickness of the film, R s is the anomalous Hall coefficient, and θ is the angle between M and I. By fitting the data measured at 10 K with an applied field of 1.2 T, the out-of-plane uniaxial components of the anisotropy fields π π = − ⊥ M M H (4 4 ) eff 2 in the reference sample and suspended sample are found to be 1767 Oe and 99 Oe, respectively. This depicts that the out-of-plane uniaxial anisotropy field (H 2⊥ ) tends to increase as the compressive strain relaxes.

Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author upon reasonable request.