Magnetic Field Tunable Small-scale Mechanical Properties of Nickel Single Crystals Measured by Nanoindentation Technique

Nano- and micromagnetic materials have been extensively employed in micro-functional devices. However, measuring small-scale mechanical and magnetomechanical properties is challenging, which restricts the design of new products and the performance of smart devices. A new magnetomechanical nanoindentation technique is developed and tested on a nickel single crystal in the absence and presence of a saturated magnetic field. Small-scale parameters such as Young's modulus, indentation hardness, and plastic index are dependent on the applied magnetic field, which differ greatly from their macroscale counterparts. Possible mechanisms that induced 31% increase in modulus and 7% reduction in hardness (i.e., the flexomagnetic effect and the interaction between dislocations and magnetic field, respectively) are analyzed and discussed. Results could be useful in the microminiaturization of applications, such as tunable mechanical resonators and magnetic field sensors.


Results
Design of the magnetomechanical nanoindentation apparatus. Figure 1a shows the new magnetomechanical nanoindentation apparatus comprising a specially designed magnetic field device made of rare-earth permanent magnets (e.g., NdFeB) and pure iron, mechanical loading part, and measurement system. Changing the distance between two poles tunes the magnetic field strength in a sample region. All parts of the apparatus exposed to the magnetic field are fabricated from nonmagnetic materials to avoid the interference between the magnetic field and other apparatus components. Magnetic field-independent tests on standard sample (fused quartz) indicate the feasibility of this apparatus. The details of the design and calibration of the apparatus are listed on Ref. 23.
Characterization of small-scale mechanical and magnetomechanical parameters. Indentations are conducted on Ni(111) single crystal samples with and without an external horizontal magnetic field of 2000 Oe with this apparatus. This magnetic field, which is stronger than the saturated magnetic field strength of 1400 Oe, is used to ensure the magnetic saturation state 24 . Indentations with the magnetic field are first conducted, and then those without the magnetic field. Given the soft magnetic properties (low coercivity and residual magnetization) of nickel, the effect of magnetic hysteresis on the DE values is small 25 and is not addressed in this study. Figure 1b shows the typical indentation load-depth. Fifty to sixty indentations with and without magnetic field are conducted for the statistical averaging to minimize the experimental error. The maximum depth increases by 2.67% on the average upon the application of the magnetic field. Adopting the widely used Oliver-Pharr method 1 , the upper portion of the unloading curves is approximated by the power law relation: P 5 a(h 2 h f ) m , where P and h refer to the load and depth at any time of the indentation, and h f is the residual depth, which increases by 4.14% in the presence of magnetic field. a and m are the fitting constants. The field-induced increase in the power law exponent is small (0.56%). The initial unloading stiffness is obtained by calculating the derivative of the load with respect to the depth at the maximum depth,

4)
Plasticity index y. In tribology, the plasticity index describes the deformation properties of a rough surface in contact with a smooth surface 29 , which is expressed as is a geometric factor. The deformations are predominantly elastic or plastic when y is much less than or greater than unity, respectively.

5)
Resonant frequency of structures f 0 . In structural mechanics, the resonant frequency of structures (e.g., rods and beams) is a function of the Young's modulus. For example, the resonant frequency of a cantilever film or beam is proportional to the square root of Young's modulus, i.e., f 0 ! ffiffiffiffiffiffiffiffiffiffi ffi E H ð Þ p . A magnetic field-dependent Young's modulus regulates the resonant frequency of nano-and microstructures. 6) Magnetomechanical coupling coefficient k. This coefficient is defined as the ratio of the output mechanical energy to the input magnetic energy 30 and can be expressed as k 5 (1 2 E 0 /E s ) 1/2 , where E 0 and E s are the moduli of the sample in the absence and presence of the saturated magnetic field, respectively. A smaller E 0 /E s implies larger k and higher energy conversion efficiency. Table 1 shows the small-scale properties determined using the above equations in this paper. These properties (Young's modulus E, indentation hardness H IT , yield strength Y, plastic index y, and the resonant frequency of microcantilever beam f 0 ) change remarkably with the saturated magnetic field. The local magnetomechanical coupling coefficient of k 5 0.49 is obtained, which is higher than that at the macroscale 15 (from 0.13 to 0.17). The ratio of indentation hardness to effective modulus H IT /E eff and the ratio of the irreversible work to the total work of the indentation process W p /W tot are also analyzed because these parameters are important in tribology and fracture mechanics [26][27][28][29] .
Empirical relationship between energy dissipation and hardnessto-modulus ratios with and without a saturated magnetic field. We briefly introduce the empirical relationship between energy dissipation and hardness-to-modulus ratios determined via dimensional analysis and finite element calculations, H IT /E eff 5 f(W p /W tot ), which was proposed by Cheng and Cheng 31 . The relationship is based on the assumption of elastic-plastic materials with uniaxial constitutive equations given as s~E e, for eƒY=E where s and e are the stress and strain, respectively, K is the strength coefficient, and n is the work-hardening exponent. The solid is elastic below the yield strength Y. E and v are used to characterize the deformation. The solid is work-hardening after yielding.
Consequently, E, Y, n, and v are sufficient to describe the stressstrain relationship. Figure 3 plots the finite element calculations (hollow squares) according to Ref. 31, which correspond to elasticplastic materials with Y/E 5 0-0.006, v 5 0.2-0.4, and n 5 0-0.5. Fitting the calculation points yields an empirical equation Published experimental data for three common metals (W, Cu, and Al as blue solid dots from left to right in Figure 3) are also presented 31 , which show consistency with the fitting line. However, the validity for ferromagnetic materials with strong magnetomechanical coupling properties is yet to be examined. In the present study, the experimental data of the ferromagnetic nickel single crystal in the absence and presence of an external saturated magnetic field are plotted (red five-pointed stars from left to right in Figure 3   dimensionless relationship for ferromagnetic materials is feasible. The deformation and energy dissipation behavior of nickel can be regulated in line with this relationship by varying the external magnetic field, which is a noncontact method. Table 1 shows that the magnetic field changes all the eight properties of nickel. However, only E and H IT are independent. The following reasons for the magnetic field dependence are proposed. Magnetic field-induced increase in the indentation elastic modulus. The variation in E between the saturated and demagnetized states is called the DE effect given as DE/E 0 5 (E s 2 E 0 )/E 0 , where E s and E 0 are the Young's moduli for saturated and demagnetized states [12][13][14][15] , which can be explained from the perspective of the micromagnetic energies. On the one hand, ferromagnetic materials possess spontaneous magnetization because of the strong exchange interaction between the spins below the Curie temperature. Positive exchange interaction yields parallel arrangement of the magnetic moments of the atoms and creates a single domain state. On the other hand, the occurrence of a multi-domain state should reduce the magnetostatic energy of the whole sample. The magnetization of the domains orients in the directions of the easy axes to decrease the anisotropic energy. The orientation of the domains with the spontaneous strain will also change by an applied stress field because of the mechanical anisotropy that elongates or contracts the sample in certain directions. The total reversible strain comprise the purely elastic and domain rotation strains, in which e total 5 e elast 1 e domain . All the magnetic moments arrange in the same direction as the external magnetic field in the case of a highly saturated state to decrease the external magnetic energy. The mechanical disturbance cannot induce domain rotation, so that no domain rotation strain exists upon application of mechanical loading. Here, the reversible deformation is purely elastic, e total 5 e elast . Therefore, the Young's moduli of the demagnetized and saturated states can be expressed as E 0 5 s/(e elast 1 e domain ) and E s 5 s/e elast , respectively. Hence, The domain rotation strain at the macroscale is attributed to the converse magnetostrictive effect, which describes the stress-induced rotations of the magnetic domains: However, the stress gradients can be much larger at the nano/ microscale than that at the macroscopic scale because of the existence of small-scale inhomogeneities, dislocations, or strain relaxations [32][33][34] . The flexomagnetic or flexoelectric effect, a stress gradient-induced magnetization or polarization, has an important function. Therefore, the proposed macroscale model should be modified at small scale. The stress/strain gradient or the breakage of crystal lattice symmetry changes the magnitude and the orientation of spontaneous magnetization 35 The second term on the right side of Equation (4)  Consequently, the small-scale value DE I E I 0~3 1% is larger than the macroscale value (from 1% to 20%), which depends on the grain size or the internal stress. Larger the grain sizes imply larger DE. We use the maximum value of 20% at the macroscale for a nickel single crystal given that the samples possess various grain sizes provided in previous studies. Thus, the value of the flexomagnetic contribution to the smallscale DE effect can be tentatively estimated. The stress/strain gradient field inside the samples at the macroscopic experiments is negligible compared with that at small-scale. The ratio of converse magnetostrictive strain to purely elastic strain at both macro-and small-scale is approximately given as e I elast <e l =e elast ð6Þ Equations (2) to (6) yield the ratio of the flexomagnetic strain to the converse magnetostrictive strain: Statistically, the strain resulting from the flexomagnetic effect can be as much as 50% of the strain from the converse magnetostrictive effect on the average. Figure 4 shows a schematic of the domain switches during unloading. The red arrows indicate the magnetization orientation of the domains, and the long axis direction of blue ellipses is the spontaneous strain direction. The spontaneous strain is particular to the spontaneous magnetization because nickel is a negative magnetostrictive material. Eight ,111. easy magnetization directions in the single crystal nickel exist theoretically, resulting in the domain walls of 180u, 109.47u, and 70.53u. Only two assumed easy axes are plotted in Figure 4 with domain walls of 180u and 90u as a two-dimensional simplification. Branched structures of the volume domains should exist near the surface 24 , but are neglected in this study to simplify the analysis. The unloading deformation of the demagnetized state comprised a purely elastic part and a reversible domain rotation part, whereas that of the saturated state is purely elastic. Thus, the reversible displacement of the deformed surface around the indenter tip in the saturated state is smaller than that in the demagnetized state because of the absence of the reversible domain rotation strain. Figure 4a indicates the indentation-induced magnetic anisotropy by the net magnetization in the perpendicular direction. Sort et al. 37 found the indentation-induced perpendicular magnetic anisotropy and the enhanced magnetic properties in ferromagnetic metallic glass in the absence of an external magnetic field.
Magnetic field-induced reduction in indentation hardness. Figure 2 and Table 1 show that the magnetic field reduces the indentation hardness and yield stress by 6.77%. The magnetoplastic effect should be related to the dislocation generation, movement, and interactions. From the perspective of energy, the dislocations require extra energy to pass through the magnetic domain walls in the absence of a saturated magnetic field. By contrast, most of the domain walls vanish and the mobility of dislocations increases in the presence of a saturated magnetic field. Therefore, plastic deformation becomes easier in the saturated state. This observation is similar to the report of Bose on the magnetic field-dependent fatigue life of steel 16 . The dislocation structures of niobium after 5% uniform elongation with and without 1500 Oe magnetic field are observed with a transmission electron microscope 38 . The magnetic field increases the mobility and decreases the reactions of dislocations, which provides a direct evidence for the interaction between the magnetic field and the dislocations. The variation in the magnetic ordering, which can interact directly with the external magnetic field, in the dislocation region is also possible 39 . The plasticity of materials is intensified from the interactions between the magnetic field and the dislocations. Therefore, the hardness and the yield stress decrease, and the energy dissipation ratio increases.

Discussion
Nano-and micromagnetic materials are widely used in functional devices. However, the measurement of small-scale mechanical and magnetomechanical properties is still challenging, which restricts both the design of new products and the performance of smart devices. We develop a new magnetomechanical nanoindentation technique, in which the mechanical properties, such as Young's modulus (up to 31%) and hardness (up to 27%), are magnetic field-tunable. However, the results differ from those of the macroscale counterparts. The flexomagnetic effect and the interaction between dislocations and magnetic field are proposed to explain these phenomena. Like that temperature (cooling) can induce symmetry breaking via phase transition, stress/strain gradient resulting from surface/interface, defects, bending/torsion, contact stress concentration, or nanoindentation can also change the local symmetry and contribute to the magnetization of materials [40][41][42] . Proofs for these phenomena have been found in ferromagnetic and antiferromagnetic materials such as cobalt 43 , nickel 44 , and BiFeO 3 6 . Nanoindentation technique has been employed to study the flexoelectric effect in ferroelectric materials, which is similar to our study 45,46 . The interaction between the dislocations and the magnetic field has been experimentally verified in niobium 38 .
Results from this study could be useful in microminiaturization of applications such as tunable mechanical resonators and magnetic field sensors. The magnetic field could control the mechanical, tribological, and structural properties of these devices. However, we encourage additional experiments or calculations to ascertain the feasibility of our proposal for the mechanism, considering that the analyses may not provide a full insight into the effects of the magnetic field on the small-scale mechanical properties. In situ observation via advanced microscopy technique is one of the directions of future research.

Methods
Sample information. The nickel single crystal is provided by MTI Corporation in Hefei, China, and is grown using Bridgman method. The thickness of this sample is approximately 200 mm, and surface roughness of the polished side is less than 0.5 nm. This side is used for nanoindentation measurements.
Magnetomechanical nanoindentation experiments. The mechanical module of the nanoindentation apparatus has maximum load of 1 N, load noise of 35 mN, maximum depth of 20 mm, and depth noise of 8 nm. The performance details of the apparatus are available from Ref. 23. A three-sided Berkovich diamond indenter probe with tip radius of approximately 150 nm is used to penetrate the sample. Each indentation test is performed within 120 s, including 30 s holding time at the peak load to ensure the sufficient loading deformation of nickel from the indentation creep effect. The testing temperature is kept at 23 6 1uC to reduce the thermal drift. Fifty to sixty indentations with and without magnetic field are conducted for statistical averaging to minimize the experimental error. A Gauss meter (Beijing CUIHAIJIACHENG Magnetic Technology Company, CH-1300), which range from 0 kOe to 30 kOe with a resolution of 1 Oe, is used to measure the magnetic field strength. The Oliver-Pharr method is adopted to calculate the Young's modulus and the indentation hardness from the experimental curves. The upper 30% of the unloading curves are fitted with the power law relation: P 5 a(h 2 h f ) m , where h f is the residual depth, and a and m are the fitting constants.