Deterministic domain wall rotation in a strain mediated FeGaB/PMN-PT asymmetrical ring structure for manipulating trapped magnetic nanoparticles in a fluidic environment

The manipulation of domain walls (DWs) in strain-mediated magnetoelectric (ME) heterostructures has attracted much attention recently, with potential applications in precise and location-specific manipulation of magnetic nanoparticles (MNPs). However, the manipulation ability in these structures is restricted to magnetostrictive circular ring structures only, where the required onion state is metastable, less thermally stable, and cannot be obtained easily. This work investigates the highly shape anisotropic FeGaB magnetostrictive elliptical ring structures of different aspect ratios and trackwidths on the PMN-PT piezoelectric substrate to manipulate fluid-borne MNPs using active control of DWs. The proposed model utilizes the attribute that the required onion state in a magnetostrictive elliptical ring is thermally stable and easily obtained compared to magnetostrictive circular ring structures. By varying the trackwidth of elliptical rings, nucleated DWs are rotated at different angles to capture and transport fluid-borne MNPs. Up to a critical trackwidth, DW rotation is predicted by dominant stress anisotropy energy that leads the rotation of DWs and attached MNPs toward the dominant tensile strain direction of PMN-PT with reversibility. Increasing the trackwidth beyond the critical trackwidth caused a complete 90° rotation of DWs and attached MNPs without reversibility and is given by dominant shape anisotropy energy. The fundamental relationship of capture probability with the size and velocity of injected MNPs is also demonstrated. The nucleation and rotation of DWs are predicated using the coupled elastodynamic and electrostatic Finite Difference Method (FDM) micromagnetic model. Dynamics of MNP capture and rotation are envisaged using an analytical model.


Introduction
Manipulation of magnetic nanoparticles (MNPs) 1-3 in a uidic environment with the precision in the limit of a single cell size is essential for location-specic analysis in various lab-on-achip applications. These applications include diverse functionalities in nanobiotechnology, 4,5 nanochemistry, 6,7 nanomedicine 5,8 etc. Although several techniques have been developed previously to manipulate MNPs at the nano-scale, one of the most successful attempts in this direction is reported by controlling the magnetic domain walls (DWs) in nanomagnetic structures, where highly localized magnetic energy density and its gradient can couple to the MNPs. 9,10 In preceding studies, conventional methods incorporating external magnetic eld [11][12][13][14] and current [15][16][17] are investigated widely to control magnetic domains and their DWs. However, these methods are energy inefficient, spatially inaccurate and fall short to perceive the required manipulation. Lately, successful efforts toward this direction are reported using strain-mediated magnetoelectrics (MEs). [18][19][20][21] A ME heterostructure constitutes of magnetostrictive and piezoelectric order parameters. 22 An applied voltage across the piezoelectric material instigates a strain at the heterostructure interface, altering the magnetization in magnetostrictive material due to the Villari effect. 22,23 Although previous research suggests that the control and manipulation of the magnetic DWs using strain mediated MEs provides an ultra-low energy route for MNPs manipulation, still this manipulation ability in strain mediated MEs with magnetostrictive/piezoelectric heterostructure is restricted to magnetostrictive circular ring structures only. These circular ring elements show two distinct states: (i) Onion state, in which two semi-circular domains are separated by two DWs and (ii) Vortex state, in which magnetization is circularly oriented either clockwise or anticlockwise. 19,24 Out of these two states, only the onion state is used to manipulate MNPs due to the high magnetic energy density and its gradient compared to the Vortex state. 9,10,24 However, the Onion state obtained using a magnetostrictive circular ring is metastable [25][26][27] and can not be obtained easily. Also, the obtained onion state is less thermally stable. 28,29 The thermal stability of DWs is dened as the ability of DW to resist the action of external temperature and to maintain its magnetization properties. Despite successful MNP manipulation, the lower thermal stability of the circular ring can restrict their application for critical biotechnology applications, such as photo-thermal therapy, 30 ultrasound hyperthermia treatment, 31 laser-induced hyperthermia 32 etc. This is because, in such applications, heated MNPs (>47°C) are manipulated to reach a target region. 33 Once captured, heated MNPs can distort the magnetization properties of DWs in the magnetostrictive circular ring structure, which could hinder their manipulation from reaching the desired target location. Also, the additional rotation of the DWs in magnetostrictive circular ring beyond in-plane (IP) 45°requires strains in multiple angles, which is achieved using a multielectrode system, [34][35][36] turning the clocking mechanism as well as the fabrication steps complex.
Han et al. 28 demonstrated that the magnetostrictive elliptical ring structures can easily generate the onion state as their shape anisotropy is larger. Also, thermal stability of an onion state in the elliptical ring improves as compared to the magnetostrictive circular ring elements. 28,29 Although preceding research attempts using elliptical ring structures have focused on memory and logic applications only, 37 their high thermal stability can provide more comprehensive benets to critical biotechnology applications related to hyperthermia therapy. [30][31][32] Apart from that, additional rotation of DWs beyond IP 45°can be achieved with a simple two-electrode system utilizing shape anisotropy of the elliptical ring. However, magnetostrictive elliptical ring structure has not been explored so far to replace the magnetostrictive circular ring structure in strain mediated MEs for MNPs manipulation, which could possibly make particle manipulation easier and more energy-efficient.
Considering that, we have investigated the manipulation of MNPs in a uidic environment using elliptical ring-shaped material on strain mediated MEs. For that purpose, elliptical rings of FeGaB on a 0.5 mm thick single-crystal PMN-PT substrate are simulated. The outer diameter (D1) of the elliptical ring is xed at 1 mm, while trackwidth (t) and outer diameter along the minor axis (D2) are varied. The magnetostrictive material is chosen as FeGaB due to its reasonably large piezomagnetic co-efficient compared to other well-known magnetostrictive materials such as Ni, FeGa, Co etc. 22,38 Also, its lower Gilbert damping coefficient (a) reduces power consumption and improves thermal stability. 39 The PMN-PT with spontaneous polarization along h111i direction is used as a piezoelectric substrate in our model. The reason for choice of such substrate is that their h011i cut shows large IP anisotropic strain upon applying a voltage. 40 The strain prole of PMN-PT as a function of applied voltage is considered linear, as shown in Fig. 1(b). Also, it is assumed that the strain becomes zero upon removal of voltage. Initially, the onion state containing DWs in FeGaB elliptical rings is created. By applying an external voltage across the PMN-PT substrate, these DWs are rotated at different angles depending on the elliptical ring dimension. Subsequently, DW reversibility is examined aer removing an external voltage. The DW formation, rotation and reversibility are predicted by Landau-Lifshitz-Gilbert (LLG) equation coupled with elastodynamics using the micromagnetic simulation platform MuMax3. 41 Next, we magnetostatically coupled the uid borne MNPs to DWs. Once an external voltage is applied, MNPs motion is observed at different angles depending on the elliptical ring dimension. Using an analytical model, the transport dynamics of MNPs is studied. Also, an analytical model is used to calculate the coupling and drag force between DW and MNPs.
2 Theoretical modelling framework

Modeling for domain wall rotation
Our model for controlling the DWs consist of elliptical nanomagnetic rings of FeGaB on PMN-PT substrate, as shown in Fig. 1(a). Our model uses external voltage to instigate the strain in the piezoelectric substrate and assumes 100% strain transfer to magnetostrictive material. The material parameters to model the nano-magnetic rings are at room temperature (300 K). Any thermal uctuation or noise is neglected as the difference between magnetization dynamics at 0 K and raised temperature is insignicant. 42 Landau-Lifshitz-Gilbert (LLG) relation (eqn (1) where A ex and M sat are the exchange stiffness coefficient and saturation magnetization of FeGaB respectively.H d is given bỹ where f represents magnetic potential and satises Poisson equation By applying an external voltage to the piezoelectric substrate, additional stress anisotropy energy (H s ) is produced.H s is given byH where l and Y are magnetostrictive co-efficient and Young's modulus of FeGaB respectively. 3 net is voltage induced biaxial strain transferred to magnetostrictive material and dened as difference between the strain generated along tensile [01−1] and compressive strain [100] direction i.e.
where d 31 , d 32 are the piezoelectric coefficient, V is an external applied voltage and d is thickness of PMN-PT. Here, the piezoelectric coefficients d 31 and d 32 are 771 pm V −1 and −1147 pm V −1 respectively. 44,45 The simulated elliptical ring's outer diameter along the major axis (D1) is 1 mm for all cases. The trackwidth (t) and outer diameter along the minor axis (D2) are varied. Three elliptical rings with aspect ratios (AR-D1 : D2) 1.1 : 1, 1.3 : 1 and 1.5 : 1 with varied trackwidth on a 0.5 mm thick single-crystal PMN-PT substrate are simulated. The thickness of each magnetic ring is 30 nm. As shown in Fig. 1(c) major axis of each magnetic ring is 45°(clockwise) relative to the tensile strain [01−1] direction of PMN-PT. The cell size of the geometries mentioned above is 1 × 1 × 1 nm 3 with the material parameters for FeGaB as given in Table 1.

Modeling of magnetic nanoparticle manipulation
For predicting the rotation of magnetic nanoparticles (MNPs) in a uid environment using DW rotation, spherical iron oxidebased Fe 3 O 4 nanoparticle is considered, which has a density 5000 kg m −3 , volume susceptibility 800 kA m −1 T −1 and saturation magnetization 4.78 × 10 5 A m −1 . 46 Fe 3 O 4 is considered as it is biocompatible, chemically stable, non-toxic and inexpensive. [47][48][49] In this paper, the term magnetic nanoparticle (MNP) is oen used interchangeably with the nanoparticle. It is assumed that uid is non-magnetic water with density 1000 kg m −3 , permeability 4p × 10 −7 H m −1 and viscosity 0.001 kg m −1 s −1 . 48 Several forces govern the transport dynamics of MNPs due to DW rotation in the magnetic ring, which includes force due to DW stray eld, viscous drag force due to uid viscosity, surfactant force, gravitational, inertia force on nanoparticle etc. For most applications, force due to DW stray eld and viscous drag force are the dominant contributions of total force and other forces can be ignored. 19,50 In this paper, the transport dynamics of a MNP is considered in the non-ow regime, i.e. velocity of the uid is zero. Newton's second law equation is solved to study the nanoparticle motion, as given in eqn (8) Here effective force (F) is vector sum of force due to DW stray eld (F dw ) and uid viscosity (F d ) on nanoparticle. m p and a ⃑ p are mass and acceleration of nanoparticle respectively. As whereṽ p is nanoparticle velocity. In eqn (9)F dw is given bỹ where m o , V p andM p are free space permeability, nanoparticle volume and magnetization respectively.H dw is magnetic eld due to DW stray eld at the center of nanoparticle andM f is uid magnetization. As already discussed uid is non-magnetic (M f = 0), eqn (10) can be written as We have considered linear magnetization for nanoparticle below saturation i.e.M p ¼ where m p is permeability of nanoparticle andH =H dw −H dp , whereH dp ¼M p 3 is self-demagnetization eld of nanoparticle. 50 Using Stokes' lawF d is given bỹ where h andṽ f are the uid viscosity and velocity respectively. r p is nanoparticle radius. As nanoparticle is in non-ow regime (ṽ f = 0), eqn (13) reduces tõ The negative sign in eqn (14) indicates thatF d acts opposite toF dw . Because Brownian motion can inuence the coupling of nanoparticle to DWs when the nanoparticle is sufficiently small, following criteria is used to estimate the critical radius (r c ) of the nanoparticle 51 where jFj, k b and T are the magnitude of effective force, Boltzmann's constant and temperature respectively. Eqn (8) is valid only when r p $ r c . 51 For r p < r c , advection-diffusion equation 52 is used rather than Newton's equation which is beyond the scope of current work. For Fe 3 O 4 in water critical radius is 80 nm at the room temperature. 50 In our work r p = 100 nm, 200 nm and 300 nm; thus, Brownian motion is neglected. We assume that the MNP is injected with the average velocity (ṽ p ) of 0.1 mm s −1 , for which force due to uid viscosity or drag force (F d ) calculated is 0.1884 pN, 0.3768 pN and 0.5652 pN for r p = 100 nm, 200 nm and 300 nm respectively, using eqn (14).
To couple the nanoparticle to DW, jF dw j > jF d j. This condition predicts that as long asF dw surpassesF d during DW rotation, nanoparticles bind to the DW and track their location. This conrms the successful nanoparticle rotation. If jF dj overcomes jF dw j, the nanoparticle does not bind to the DW.

Domain wall initialization
Initially, an external magnetic eld is applied along the minor axis of the magnetic rings and subsequently removed aer saturation. At the remanent state, redistribution of the energies occurs, which leads to a minimum energy density state. As shown in Fig. 2(a), the minimum energy density is non-negative up to a specic trackwidth and then becomes almost zero on increasing the trackwidth further. This variation can be explained by the competition between the exchange and the demagnetization energies, the dominant contributors to the total energy when an external voltage is absent. Exchange energy favours parallel alignment of the magnetization, whereas demagnetization energy favours the magnetic closure domains. 19,22 As shown in Fig. 2(b), for elliptical rings with AR 1.3 : 1 up to 250 nm trackwidth, the required demagnetization energy to rotate the magnetization 180°is high. The initial energy density can not overcome it, which produces an onion state with transverse DWs having a non-negative minimum energy density. For trackwidths greater than 250 nm, required demagnetization energy to rotate the magnetization 180°is low, and the initial energy density can easily overcome it. This causes magnetization to ip over one-half of the ring, producing a vortex state with a negligible minimum energy density. Fig. 2(c) shows the corresponding simulated micromagnetics and PEEM images, where an onion state with transverse DWs up to 250 nm and a vortex state beyond 250 nm is obtained. The micromagnetic images are used to show the spin conguration of the rings whereas simulated PEEM images are used to display the evolution of magnetic domains.
It is to be noted that the application of the initialized external magnetic eld along the major axis of the magnetic rings is not considered as the required external voltage to rotate DWs will be very high because of the high shape anisotropy energy of the elliptical ring.

Phase diagram
In order to predict the dependency of DW nature to the ring geometry, a phase diagram is plotted in Fig. 3 with varying AR and trackwidth for a xed thickness. Trackwidth corresponding to the red (300 nm) and blue (250 nm) boxed area shows the The nature of the minimum energy density state is observed due to competition between the exchange and demagnetization energies, which are the dominant contribution to the total energy in the absence of an external voltage (b) exchange and demagnetization energy variation against trackwidth for elliptical rings with AR 1.3 : 1 and (c) corresponding simulated micromagnetics and PEEM images. Up to 250 nm trackwidth, the required demagnetization energy to rotate the magnetization 180°is high, producing an onion state with transverse DWs having a non-negative minimum energy density. For trackwidths greater than 250 nm, the required demagnetization energy to rotate the magnetization 180°is low, producing a vortex state with almost negligible minimum energy density. To illustrate magnetization orientation in simulated PEEM images, magnetization contrast is illustrated.
changed nature of DW from onion to vortex when AR is increased from 1.1 : 1 to 1.3 : 1 and 1.3 : 1 to 1.5 : 1, respectively. At these particular trackwidths, the exchange energy starts to dominate compared to the demagnetization energy as the AR is increased. In Fig. 3, the black dotted curve shows the phase boundary between the onion and the vortex states for the geometrical variation of the magnetic ring. For further discussion, the maximum trackwidth which shows the onion state is represented by t o . t o obtained for magnetic rings of AR 1.1 : 1, 1.3 : 1 and 1.5 : 1 has the values of 350 nm, 250 nm and 200 nm, respectively.
It is worth noting that only the onion state can couple to MNPs in a uidic environment because of the nite value of the stray magnetic eld. 9,10,19,24 Thus in subsequent sections, only the onion state, i.e. t # t o for each elliptical ring is analyzed.

Domain wall rotation
Aer applying a voltage to the PMN-PT substrate, DW rotation is examined for each elliptical ring with the onion state (t # t o ). Aer initializing the onion state at pre-stressed condition, each magnetic ring is unstrained (3 net = 0). The external voltage is applied and strain is instigated at the heterostructure interface due to the Villari effect. 22,23 The generated strain modies the stress anisotropic energy that competes with the initialized ring energies to bring it to the minimum energy state. A maximum applied voltage of 400 volts is considered as the IP anisotropic strain beyond this voltage is outside the linear piezoelectric response range of PMN-PT substrate and requires a phase transition. 19,35 Aer ramping up the voltage from 0 volts to 400 volts with 40 V/ns ramp rate, two cases are observed depending on the trackwidth of the magnetic ring. Within the onion state, DW rotation up to 45°is observed up to a critical trackwidth represented by t cr , and complete 90°DW rotation is observed beyond t cr , as discussed next: 3.3.1 In-plane DW rotation up to 45°. For each magnetic ring up to a critical trackwidth (t cr ), the generated strain introduces stress anisotropy energy that competes with the initialized ring energies. It is observed that the change in total ring energy resulting from the stress anisotropy energy outweighs any substantial change to the initialized exchange and demagnetization energies during DW rotation, as shown in Fig. 4(a). As a result, the DWs reorient toward a new easy axis created along the tensile strain direction as FeGaB is a positive magnetostrictive material. 46 The rotated DWs also transitions from the initialized onion state with transverse (O-T) DWs to the onion state having vortex (O-V) DWs, as shown in Fig. 5(ad). Such transitions occur because generated stress anisotropy energy modies the DW magnetization conguration to move it to a more stable state. 19,53 This signies that O-V DWs are stable than O-T DWs when stress anisotropy is the dominant contributor to the total energy. Because of the simulation limit, the exact voltage where O-T to O-V transition occurs can not be determined. As shown in Fig. 5(d-f), DW broadening is also observed when voltage is ramped up as an elastic force is generated due to the stress anisotropy energy. Although it is possible that DW broadening can transform the DW from onion to vortex state completely, 53 it is not observed in our model even  aer applying the maximum voltage. As shown in Fig. 6, increase in the reorientation angle towards tensile strain direction is observed with increase in trackwidth when a maximum voltage of V = 400 volts is applied. Consequently, maximum rotation (d max ) of such stable O-V DWs toward the tensile strain direction is observed at t cr . The observed t cr for magnetic rings of AR 1.1 : 1, 1.3 : 1 and 1.5 : 1 are 300 nm, 150 nm and 50 nm, respectively and the corresponding d max are approximately 41.5°, 27.2°and 13.7°, respectively, being less than the ideal value of 45°relative to the initialized onion state. It is clear that t cr and corresponding d max reduces with increase in AR.
3.3.2 In-plane 90°DW rotation. As the trackwidth increases beyond t cr , complete IP 90°rotation is observed. Similar to case (3.3.1), DWs start rotating toward the tensile strain direction and transform from an initialized O-T to an O-V state upon increasing the voltage. The O-V DWs rotate up to 45°angle relative to the initialized state as total ring energy follows the stress anisotropy energy term, and the change in exchange and demagnetization energies remains almost negligible, as shown in region A of Fig. 4(b). Interestingly, upon further increasing the voltage, stress anisotropic energy does not increase further and the total ring energy follows the demagnetization energy term, as shown in region B of Fig. 4(b). Consequently, DWs reorient towards the easy axis of the magnetic ring as shown in the simulated PEEM snapshots in region B of Fig. 4(b). This signies that the maximum stress anisotropic energy generated by the magnetic ring is limited 19 and the potential d max possible due to that is 45°only. This signies that additional rotation beyond 45°is not attributed to stress anisotropy energy. Instead, DW rotates beyond 45°as the shape anisotropy energy that originates from the demagnetization energy 54 becomes the dominant contribution to the total energy. Consequently, a favourable energy term is created along the easy axis of the magnetic ring, and a complete IP 90°rotation is observed, as shown in region C of Fig. 4(b). Moreover, during this reorientation process, the rotated DW again transitions to the initial O-T state from the intermediate O-V state. Such transition occurs as soon as total ring energy starts following the exchange energy term that favours parallel alignment of the magnetization, 19 and the change in stress anisotropy and demagnetization energies remains almost negligible, as shown in the simulated PEEM snapshots in region C of Fig. 4(b). This result is critical as the additional rotation of the DWs beyond IP 45°requires strains in multiple angles which is conventionally achieved by using a multi-electrode system 34-36 as the effective stress anisotropic eld described in eqn (7) is maximum at ±45°( or ±135°) and starts reducing aer that. 55 As the proposed model employs elliptical magnetic rings, additional rotation of the DWs beyond IP 45°is obtained by utilizing the shape anisotropy of the magnetic ring.

Domain wall reversibility
The DW reversibility is also investigated aer removing the external voltage. The external voltage is ramped down to 0 from 400 volts with a 40 V/ns ramp rate. This gives a different nal DW state depending on the trackwidth of the magnetic ring. Up to t cr , we observe stable O-V DWs return to the initial position, as shown in Fig. 7(c and f). A complete reversal is observed because with ramping down of voltage, the associated stress anisotropy energy reduces at the same rate. As stress anisotropy energy is dominant in this case, its reduction imparts a driving force towards the initial unstrained state. However, the nature of rotated DWs remains O-V even aer the complete removal of the external voltage, as shown in Fig. 7(c and f).  On the other hand, no reversibility is observed when the trackwidth increases beyond t cr as shape anisotropy energy is dominant in this case, as shown in Fig. 8(c and f). With reduction in voltage, even if the stress anisotropy energy imparts a driving force towards the initial state, its impact will be negligible. Also, the nature of DW remains same as the initial unstrained state. However, DW broadening is observed when the voltage is removed. This broadening signies the presence of a remanent strain. 19 Table 2 shows a summary of the results obtained.

Magnetic nanoparticle-domain wall interaction
Based on the size-dependent DW rotation capabilities achieved for different magnetic rings, MNPs rotation is predicted. For this, the interaction of the DW and the injected MNP is analyzed rst. As described in Section 2.2, force due to the stray magnetic eld (H stray ) is the dominant contributor to total magnetic force generated by a magnetic ring,H stray for each magnetic ring is calculated to quantify the magnetic force. Initially, an external magnetic eld is applied along the minor axis of the ring to reach magnetic saturation. Consequently, the magnetic ux around the ring is given bỹ B = m 0 (H external +H stray (16) whereH external is an external applied magnetic eld andH stray is the stray magnetic eld.H stray depends upon the material and shape of the magnetic element. 24 The source of generated stray magnetic eld is demagnetization eld. 17,57,58 Using eqn (3) and (4),H stray is calculated as shown in Fig. 9(b). It is clear from Fig. 9(b) that for wider trackwidth magnetic ring (t > t o ) exhibiting vortex state, magnitude of the stray magnetic eld (H stray =H vortex z 0) is almost negligible. Contrastingly, for narrow trackwidth magnetic ring (t # t o ) exhibiting onion state, the stray magnetic is nite (H stray =H dw s 0). Since for a vortex stateH stray z 0, MNPs can not couple to a vortex state. In contrast, the onion state can couple to MNPs because of nite value ofH stray . The onion state separates two oppositely aligned magnetization regions or spin blocks (1) Head to Head (HH) and (2) Tail to Tail (TT) DWs. The peculiarity of these spin blocks is that they act as a magnetic pole where HH and TT DWs represent the north and south poles respectively. 17,58-60 Consequently, these DWs initiate an inhomogeneous stray magnetic eld with radial distribution, as shown in Fig. 9(a). For a HH DW, the direction of the stray magnetic eld is outward, having a positive out-of-plane component. On the other hand, a TT DW creates a stray magnetic eld directed inwards with a negative out-of-plane component. The stray magnetic eld (H dw ) generated by HH DW creates an attractive potential well with magnetostatic energy (E) given by where m o , m andH dw are free space permeability, magnetic moment of a nanoparticle and stray magnetic eld generated by the ring respectively. Because commercial MNPs are superparamagnetic, it is assumed thatH dw can not saturate these nanoparticles i.e.  where c is the magnetic susceptibility of a nanoparticle. This modies eqn (18) to From eqn (20), it is clear that E depends onH 2 dw . As Fig. 9(b) shows the relation betweenH dw and trackwidth, a direct relation between trackwidth and magnetostatic energy (E) can be made, which indicates that E changes with trackwidth in the same manner asH dw changes with trackwidth.
Next, based on the above analysis, the MNP capture using DW is predicted. MNPs are captured whenH dw prompts a magnetic moment in MNPs. This creates an attractive potential well with magnetostatic energy (E) localized at the center of HH DW, as shown in Fig. 9(c) and (d). Throughout our model, we assume that the center of the MNP is in close proximity to the magnetic ring with a constant height of 200 nm. As a result, force due to DW stray eld attracts MNP towards HH DW. The value of this attractive force (F dw ) is given by eqn (11). As the onion state is observed for t # t o ,F dw for different AR elliptical rings having t # t o for a range of nanoparticle radius (r p ) is calculated, as shown in le side ordinate of Fig. 9(g-i). As the initializedH dw is higher for intermediate AR (1.3) as compared to magnetic rings with AR 1.1 and 1.5, it is observed   ,F dw also increases with an increase in r p .
As mentioned in Section 2.2, as long asF dw surpassesF d (-jF dw j > jF d j), nanoparticles bind to the HH DW, as shown in Fig. 9(e). On the contrary, ifF d overcomesF dw , the nanoparticle does not bind to the HH DW ( Fig. 9(f)). As MNPs are injected with average velocity (ṽ p ) of 0.1 mm s −1 , nextF dw is compared with the drag force (F d ). From eqn (14)F d calculated is 0.188 pN, 0.377 pN and 0.565 pN for r p = 100 nm, 200 nm and 300 nm respectively as shown in right side ordinate of Fig. 9(g-i). Table  3 summarizes the comparison ofF dw withF d . Table 3 shows that the capture probability becomes more signicant for bigger MNPs because of largerF dw thanF d . This indicates that the injected average velocity (ṽ p ) should be lower to enhance the capture probability of smaller MNPs provided r p $ r c .

Magnetic nanoparticle manipulation using domain wall rotation
Once the injected MNP is captured in the magnetostatic potential energy well, MNP rotation aer applying a voltage across the PMN-PT substrate is analyzed. Since the emanated stray magnetic eldH dw from the DW could be different at different positions of the track due to DW transition (O-T to O-V or vice versa), a time-dependent stray magnetic eld is calculated. The magnetization prole is utilized to calculate the stray magnetic eld via the scalar magnetic potential using eqn (3) and (4). Based on the trackwidth of the magnetic ring, different stray magnetic eld proles are obtained. This difference is illustrated in Fig. 10(a), which shows a time-varying stray magnetic eld prole for an elliptical ring with AR 1.3 : 1 and trackwidths 150 nm (t # t cr ) and 200 nm (t > t cr ), respectively. As the voltage is applied at T = 0 s, a reduction in the stray eld is observed from the initialized state. This reduction is observed since vortex walls exhibit a lower stray magnetic eld as compared to the transverse walls 59,60 and obtained DW starts transforming from an initialized O-T (Onion-Transverse) to an O-V (Onion-Vortex) state upon increasing the voltage, as shown in Fig. 5. For t # t cr , as the voltage is ramped up further, the generated stress anisotropy energy modies the DW conguration to move to a more stable state. As already discussed, O-V DWs are more stable than O-T DWs in this case; the lower stray eld is observed in the nal state. On the other hand, for t > t cr , as the voltage is ramped up from T = 0 s, DW initially transforms from an initialized O-  Fig. 10(a). This is observed because transverse walls  exhibit a higher stray magnetic eld as compared to the vortex walls. 59,60 As a result, the nal stray eld is observed almost equal to the initialized state.
Since the continuous motion of the captured MNPs as a DW-MNP bound unit depends on the stray magnetic eld that varies along the track, the minimum value ofH dw from the intermediate state is considered for further analysis. We observe 5-20% reduction in the stray eld from the initialized state to the minimum peak once the voltage is applied. We take a conservative estimate of a 30% reduction in the stray eld. For instance, Fig. 10(b and c) shows variation in the stray magnetic eld calculated for elliptical rings with AR 1.1 : 1 and 1.3 : 1 as a function of trackwidth. The error bar shows the maximum to minimum stray eld variation, where the lower limit shows the minimum stray eld possible during DW rotation. Using eqn (11), the critical force (F dw(critical) ) due to the minimum DW stray eld is calculated. Next, using a model proposed by Bryan et al., 62 the critical MNP transport velocity (ṽ (MNP(critical)) ) is estimated by equatingF dw(critical) with the viscous drag force (F d ) given by eqn (14). As shown in Fig. 10(d),ṽ (MNP(critical)) increases rapidly with MNP radius. The above analysis is crucial since for continuous MNP rotation along DW, the maximum DW speed (ṽ dw =ṽ dw(critical) ) should be less than or at most equal to thẽ v (MNP(critical)) . This is why the minimum stray eld is considered for the complete analysis.
Voltage driven DW speed (ṽ dw ) is estimated using eqn (21) by geometrical consideratioñ represents quarter of the elliptical ring circumference with D1 and D2 as major and minor axis of the ring (Fig. 1(c)). A is given by Quarter of the ring circumference is considered since maximum rotation possible is 90°. f drive represents the voltageinduced DW rotation frequency once voltage is applied and given by where T drive represents the rate at which external voltage across PMN-PT is ramped up. Since simulations do not run in realtime, f drive is analyzed analytically. Fig. 10(e) illustrates one such case where MNP trajectories due to DW rotation as a function of f drive for an elliptical ring with AR 1.1 : 1 and trackwidth 350 nm is considered once an injected MNP of radius 300 nm is captured. Using Fig. 10(d)ṽ (MNP(critical)) is obtained as 1.09 mm s −1 . Since for a continuous MNP rotation as a DW-MNP bound unitṽ dw should be less than or at most equal to theṽ (MNP(critical)) , using eqn (21) f drive = f critical is estimated consideringṽ dw =ṽ (MNP(critical)) . It is clear that for f drive # f critical continuous motion of the captured MNPs as a DW-MNP bound unit will be observed, as shown in curves I, II and III of Fig. 10(e). On the other hand, if f drive is made slightly higher than f critical (f drive z f critical ) a piecewise MNP rotation will be observed, as shown in curve IV of Fig. 10(e). This is because, in this case, MNP will be decoupled once DW starts rotating sincẽ v dw is slightly larger thanṽ (MNP(critical)) but immediately recoupled to a new DW position due to the generation of continuous DW attractive potential well train. This suggest a piecewise MNP rotation. As DW speed increases further, the probability of piecewise MNP rotation decreases, as shown in curve V of Fig. 10(e). For very large DW speed i.e. f drive [ f critical , successful MNP rotation is not possible sinceṽ dw [ṽ (MNP(critical)) , as shown in curve VI of Fig. 10(e).
Once the MNP is trapped in the magnetostatic potential energy well and f drive # f critical , it follows the DW track along the circumference of an elliptical ring aer applying a voltage across PMN-PT substrate. As previously mentioned, up to t cr , the DW reorientation angle increases with increase in trackwidth and maximum rotation (d max ) towards tensile strain direction occurs at t cr . Also, DW returns to the initial position upon removing an external voltage. As expected in this case, the reorientation angle of the captured MNP increases with increase in the trackwidth. At t cr , MNP can rotate maximum towards tensile strain direction and return to the initial position once an external voltage is removed, as shown in Fig. 11. On the other hand, beyond t cr , a favourable energy term along the easy axis of the magnetic ring is observed because of dominant shape anisotropy energy. Due to this, DW completes IP 90°rotation and does not return to the initial position upon removing external voltage. Consequently, captured MNPs complete IP 90°r otation with no reversibility, as shown in Fig. 12.

Estimation of the energy dissipation
Next, we estimated the energy dissipation by the magnetostrictive elliptical ring structure having a simple two-electrode system. We compared it with the magnetostrictive circular ring structure, which generally employs a multielectrode system with patterned top electrodes with the common ground for the additional rotation of the DWs beyond 45°. 3.7.1 Energy dissipation for elliptical ring structure. The strain distribution in piezoelectric with fully covered top and bottom electrodes (two-electrode system) relies on piezoelectric coefficients. 63 As already described in Section 3.3, we have considered a single crystal PMN-PT piezoelectric substrate with spontaneous polarization along h111i direction. With fully covered two electrodes, its h011i cut shows large IP anisotropic strain upon applying a voltage (V) with piezoelectric coefficients d 31 and d 32 . 40 Since outer diameter of the simulated elliptical ring along the major axis (D1) is 1 mm for all cases, 1.1 mm × 1.1 mm × 0.5 mm, substrate dimension is sufficient considering a single device operation. Once the voltage is applied across the piezoelectric, energy dissipation (E d ) is given by where C p is the net piezoelectric capacitance. Since the dielectric permittivity (3 R ) of d = 0.5 mm PMN-PT is in order of z1000, other line capacitances are neglected. 45 Also, since internal dissipations due to Gilbert damping are negligible, these effects are also not considered. Thus, net piezoelectric capacitance will be where 3 R is free space permittivity and A = 1.1 mm × 1.1 mm is cross-sectional area of PMN-PT. Thus, calculated maximum energy dissipation using eqn (24) is 1.7 pJ at 400 V. 3.7.2 Energy dissipation for circular ring structure. Ideally, for a magnetostrictive circular ring, the maximum DW rotation possible due to a two-electrode system is 45°. Thus, a multielectrode system (patterned top electrodes with the common ground) is generally employed for the additional rotation of the DWs beyond 45°. [34][35][36] In this case, pattern electrodes generate an axial strain of tensile nature along the line joining the electrodes upon applying a positive voltage. 31 Note that since the poling direction of PMN-PT considered is 011 (+z direction), tensile strain is generated along the line joining the electrodes upon applying a positive voltage. For an opposite-poled PMN-PT, the nature of axial strain would be compressive for the positive applied voltage. We performed Finite Element Analysis (FEA) using COMSOL 64 to obtain the strain prole of the PMN-PT substrate when patterned electrodes are used. To get maximum tensile strain, a maximum voltage of 400 V is given. It is observed that the minimum cross-sectional area of one electrode required is at least z660 nm × 660 nm to achieve the maximum axial tensile strain. The gap between the edges of the two electrodes is considered 1.01 mm, so a circular ring having the same dimensions (diameter = 1 mm) as that of an elliptical ring considered could be placed between the electrodes, although the gap required would be larger experimentally, which will reduce the axial strain.
It is assumed that the initial DW position is 45°(anticlockwise) from the electrode pair A1A2. First A1A2 electrode pair is given 400 V, as shown in Fig. 13(a). Consequently, tensile strain (3 [01−1] ) is generated along the line joining the electrode, as shown in Fig. 13(b). The value of compressive strain (3 [100] ) is 0m3. Although the generated strain is spatially varying, we have assumed its maximum value for simplication. At 400 V, a maximum net strain (3 net z 600m3) of tensile nature is generated that approximately matches the tensile strain prole of the PMN-PT given in Fig. 1(b). Using eqn (25), the capacitance of single electrode C A1 = C A2 is 771aF. Since the applied voltage (400 V) and common ground are identical for both electrodes, C A1 and C A2 are parallel. Thus, net capacitance is C A1 + C A2 = 1542aF, when one electrode pair is activated. Therefore, the energy dissipated for a maximum 45°DW rotation is 1.2 pJ, based on eqn (24). For an additional DW rotation of 45°, again, 1.2 pJ energy will be dissipated once the B1B2 electrode pair is activated. Thus, the total energy dissipated by PMN-PT per 90°D W rotation is ideally 2.4 pJ. It should be noted that the generated maximum axial tensile strain (z600m3) is almost 40% compared to the maximum biaxial strain (z1500m3) generated due to fully covered top and bottom electrodes. Thus, a complete 90°DW rotation might not be possible using a fourelectrode system, and additional electrode pairs would probably be required. Consequently, the actual energy dissipation could be more substantial. This clearly indicates that an elliptical ring can make particle manipulation more energy-efficient using a simple two-electrode system.

Conclusion
In conclusion, we have illustrated the remote and precise manipulation of MNPs in the uidic environment. The manipulation mechanism relies on voltage-driven DW rotation in an elliptical-shaped ferromagnetic ring using strainmediated MFs and their magnetostatic potential energy well  trapping of uid-borne MNPs. We observed DW transition from an initial metastable O-T state to a stable O-V state up to a critical trackwidth (t cr ) when an external voltage is ramped up from 0 volts. These stable O-V DWs reorient towards the tensile strain direction aer ramping the voltage further, and for t = t cr the maximum reorientation towards tensile strain direction is observed. Finally, an external voltage is ramped down to 0 volts, and it is observed that stable O-V DWs return to the initial position. t cr observed for magnetic rings of AR 1.1 : 1, 1.3 : 1 and 1.5 : 1 is 300 nm, 150 nm and 50 nm respectively, whereas corresponding d max are approximately 41.5°, 27.2°and 13.7°. On the other hand, we observed DW transition from an initial O-T state to an intermediate O-V state for trackwidth greater than t cr when an external voltage is ramped up from 0 volts. The intermediate O-V DWs become metastable and convert to a stable O-T state aer ramping up the voltage further, and complete IP 90°rotation is observed. Finally, an external voltage is ramped down to 0 volts, and it is observed that stable O-T DWs do not return to the initial position. This result is signicant because the additional rotation of the DWs beyond IP 45°in a circular ferromagnet disc or ring using MFs requires strains in multiple angles, which is earlier achieved using a multielectrode system. As we have used an elliptical-shaped ferromagnetic ring, shape anisotropy creates a favourable energy term along the easy axis of the magnetic ring, and an additional rotation beyond IP 45°c an be achieved for trackwidth greater than t cr using two electrode system only. Using an analytical model, it is demonstrated that uid-borne MNPs are magnetostatically coupled to DWs and for f drive # f critical continuous MNP rotation is possible. It is observed that the capture probability becomes more signicant for bigger MNPs, and to enhance the capture probability of smaller MNPs, injected average velocity should be lower. It is predicted that up to t cr , MNPs can rotate maximum IP 45°with f drive # f critical and return to the initial position once an external voltage is removed. On the other hand, continuous IP 90°MNP rotation with no reversibility beyond t cr is observed when f drive # f critical . Lastly, it is demonstrated that an elliptical ring can make particle manipulation more energy-efficient using a simple twoelectrode system than a circular ring structure having multielectrode system. The geometry-dependent manipulation capabilities of different AR elliptical-shaped ferromagnetic rings using strain-mediated MFs described above may be implemented for future energy-efficient compact lab-on-a-chip hyperthermia therapy applications. Also, the above work can be extended to manipulating biomolecules, proteins, DNA, cells etc., which can be used for various medical and biological applications.

Conflicts of interest
There are no conicts to declare.