Quantitative Profilometric Measurement of Magnetostriction in Thin-Films

A DC non-contact method for measuring the magnetostrictive strain in thin-films is demonstrated, achieving a state-of-the-art sensitivity of 0.1 ppm. In this method, an optical profilometer is used to measure the curvature induced in a magnetostrictively coated coverslip under a DC field through phase-sensitive interferometry. From this the magnetostrictive stress and strain are calculated using Stoney's formula. This addresses limitations of conventional techniques that measure magnetostriction based on the deflection of a cantilever under an AC field, which require complex dedicated set-ups and are sensitive to vibrational noise. Further, it reveals information about the anisotropy of the film and allows for the possibility of measuring multiple samples simultaneously. The theoretical sensitivity limits are derived, predicting a shot-noise-limit of 0.01 ppm. The method is implemented to measure the magnetostrictive hysteresis and piezomagnetic coupling of thin-film galfenol. Degradation in film performance is observed above a thickness of 206 nm, alongside a change in coercivity. This prompts investigation into the growth and optimization of galfenol films for use in devices.


I. INTRODUCTION
In recent years magnetostrictive materials have been the focus of significant research due to their potential applications in fields ranging from smart materials [1][2][3] to sensing [4][5][6][7][8] and actuation [9,10].One of their primary appeals for technological applications is that standard fabrication techniques, such as sputtering and liftoff, can be used to produce magnetostrictive thin-films.This enables scalable, low cost manufacturing of magnetostrictive devices.It is important to develop efficient and accurate methods to characterize these films.
Current techniques to measure magnetostriction in thin-films can be categorized as either direct measurements, measuring the strain induced by a magnetic field (magnetostrictive effect), or inverse, measuring the change in the magnetic properties of the material in response to an applied strain (inverse magnetostrictive effect) [11].Among direct methods, the magnetostriction of thin-films is typically measured based on the deflection of a magnetostrictively coated cantilever, using capacitive or optical techniques [12].This requires careful mounting of the sample along with specialized set-ups that use AC fields, making them susceptible to vibrational noise and pick-up [13].To avoid these requirements, several techniques that measure the deflection under a DC field have been developed using atomic force microscopy (AFM) [14][15][16] and nano-indentation systems [17].These however are single point contact measurements, which risk damage to the films and would require position scanning to obtain information about their spatial profile.
Here we present a non-contact DC method to characterize thin-film magnetostriction.The method uses optical profilometry to measure the curvature induced in a magnetostrictively coated sample by an applied DC field, from which the magnetostrictive strain is calculated using an appropriate model.This is analogous to the wafer bow method which is applied to determine the intrinsic stress of thin-films on silicon wafers [18][19][20].Our method also provides the 2D profile of the bowed film without spatial scanning, this can reveal information about the anisotropy of the film and allows for the possibility of measuring several samples simultaneously.No specialized set-up is required, relying on readily available laboratory equipment.We show that state-of-the-art sensitivity can be achieved, resolving magnetostrictive strains down to ∼ 0.1 ppm.Moreover, the samples are compatible with other standard film characterization techniques, such as magneto-optical Kerr effect (MOKE) microscopy, x-ray diffraction (XRD) analysis, reflectometry and ellipsometry.This allows for magnetostrictive characterization along with the characterization of multiple other important material properties.

II. OPTICAL PROFILOMETRY TO CHARACTERIZE THIN-FILM MAGNETOSTRICTION
In our method a sample coated with a magnetostrictive thin-film is placed on a nonmagnetic stage in an optical profilometer.The sample is then imaged and a two dimensional surface profile measurement is taken, from which the intrinsic stress of the film can be calculated.A Veeco Wyko NT1100 profilometer is used, which utilizes phase-shifting interferometry (PSI), offering high resolution for mapping out smooth continuous surfaces.In the NT1100, a white-light beam is filtered to red light (∼633 nm) and passed through an interferometer objective to the sample surface.Within the interferometer a beam splitter reflects half of the incident beam to a reference surface.The beams reflected from the sample and the reference recombine to form interference fringes, separated by an optical path difference (OPD) of λ.An example image of a galfenol film sputtered on a glass coverslip is shown in Fig. (1), the circular rings are interference fringes due to changes in the surface profile of the film.During the measurement, a piezoelectric transducer moves the objective to cause a phase shift between the objective and sample surface.The system records the intensity of the resulting interference pattern at many different relative phase shifts, from which the surface profile is calculated [21].
To measure the magnetostriction of a thin-film a uniform magnetic field is applied using an electromagnet, see Fig. (2).The magnetic field is aligned in-plane with the sample so that no torque is generated and the sample stays in place, with no need for clamping.The field creates a magnetostrictive strain in the material and causes the sample to deform.This deformation is observed in the profilometer, from which the strain can be determined.We use a magnetostrictive film sputtered on a circular glass coverslip.In this regime the film is much thinner than the substrate, which in turn is much smaller than the radius of the coverslip, t f ≪ t s ≪ r s .Further, the deformations observed are small, with a radius of curvature much larger than the radius of the sample, R ≫ r s .In the case of spherical deformation, the magnetostrictive strain and stress can be calculated using Eqs.( 1) and (2) [18,22].Eq. ( 2) is known as Stoney's equation [23] as applied to a circular geometry.For nonspherical deformation the extended Stoney's equations Eqs.(11)(12)(13)(14) can be used.
Here σ B is magnetostrictive stress induced at a given Bfield, E s is the Young's Modulus of the substrate, ν s is the Poisson's ratio of the substrate, t s is the thickness of the substrate, t f is the thickness of the film, κ B = 1 R B is the curvature before the B-field is applied, due to the intrinsic stress, and κ B is the curvature of the film after the B-field is applied, where R denotes the radius of curvature [18].
Interference pattern viewed through the NT1100 profilometer, for a galfenol film sputtered on a glass coverslip, with no magnetic field applied.The interference rings are elliptical, indicating the sample is bowed anisotropically.This could be due to an anisotropic film stress, or due to anisotropy in the substrate, such as nonuniform thickness.

Objective Helmholtz Coil
FIG. 2. Optical profilometer experimental setup for characterizing the magnetostriction of thin-films.The surface profile of the film is measured and the change in radius of curvature is used to determine the strain induced by an applied field.

III. DERIVING THE SHOT-NOISE-LIMIT
To estimate the theoretical precision of a measurement an expression for the uncertainty in the change in strain, is derived, for which any smaller change in strain could not be resolved.We present the derivation in the simplest case of isotropic deformation, however a similar derivation could be followed for any geometry and deformation.

A. Modeling the system
In the experimentally relevant regime where radius of curvature, R, is much greater than the radius of the sample, the profile can be modeled by a paraboloid of the form z =1 2 κr 2 , see Sec. (S.1).The profilometer measures the height of the surface z j,k at positions r j,k for each pixel (j, k), with noise δz j,k .We assume the noise is not correlated between pixels, δz j,k is Gaussian and that the uncertainty is the same for all pixels SD(z j,k ) = SD(z) = ⟨(δz) 2 ⟩.In Sec.(S.3) we show that under these assumptions, for a uniform array of sufficiently dense measurements on a circular sample, the uncertainty in the optimal estimate of κ is given by where r meas is the radius of the measurement area and N meas is the number of measurement points, corresponding to the total number of pixels used to image the sample.To convert this to an uncertainty in the magnetostrictive strain we first consider the uncertainty in the change in curvature as where we have assumed κ B and κ I are uncorrelated and SD(κ B ) = SD(κ I ) = SD(κ).Combining Eqs. ( 2), ( 3) and ( 4) gives a general expression for the uncertainty in the magnetostrictive strain in terms of SD(z), Eq. ( 5) is used to inform our substrate choice, and optimize the precision of a measurement, see Sec. (IV A).

B. Shot-noise-limit
The precision in the case where the profilometer is operating at the shot-noise-limit can be derived to give the lower bound on the performance that can be achieved.For heterodyne optical profilometers the height of the surface is calculated from the phase change (ϕ) between the measurement and reference signals as [21], where λ is the wavelength of the illumination.In the shot-noise-limit, with small deviations in ϕ from a known mean value, SD(ϕ) shot = 1 √

Npp
, where N pp is the photon count per pixel [24].The uncertainty in z is therefore Combining this with Eq. ( 7) gives the uncertainty in the magnetostrictive strain as Where a uniform photon count across all pixels has been assumed, with the total photon count for a given measurement taken to be N T = N meas • N pp .

C. Piezomagnetic coupling
In our method we directly measure the piezomagnetic coupling as the slope of the strain vs. applied H-field curve ∂λ B ∂H 1 , where H = B applied /µ 0 .For many applications, such as magnetic field sensing, this is the material property which determines the performance of the device [7].To determine ∂λ B ∂H we estimate it as linear over a small region, ∆H, as Assuming there is no uncertainty in H, this gives

IV. CHARACTERIZING THE MAGNETOSTRICTION OF GALFENOL THIN-FILMS
To validate our optical profilometry based method we measured the magnetostrictive hysteresis and piezomagnetic coupling of sputtered Galfenol (FeGa) thin-films.Galfenol is an alloy of iron and gallium that has gathered significant attention in recent years owing to its high magnetostriction, excellent mechanical properties and corrosion resistance [26,27].

A. Substrate choice
To optimize the measurement precision and minimize SD(λ B ), the substrate must be thin, compliant and large enough to fill the field of view of the profilometer, see Eq. (5).In addition, it must be able to withstand the intrinsic stresses of the films.We used 0.1 mm thick, 3 mm diameter glass coverslips, which are well suited to these requirements.Moreover they can easily be used in other characterization techniques and are highly affordable.Other possible substrates could include freestanding membranes, which can be made as thin as 50 nm [28].However the model would need be adapted to optimize for this case.

Theoretical shot-noise-limit
To determine the shot-noise-limit of the magnetostrictive measurement given our set-up and choice of substrate, the number of photons used for a profilometer measurement must be calculated.The profilometer takes six frames of intensity data on a 60 fps camera, from which the phase data is determined and the surface profile is calculated using Eq. ( 6) [21].The illumination power used is 300 nW, corresponding to a total photon count of N meas ≈ 10 11 .Combining this with the system parameters t s = 0.1 mm, λ = 655 nm, r meas = 0.85 mm, provides an estimate the shot-noise-limit using Eq. ( 8).For a 300 nm film this gives SD(λ B ) shot ≈ 0.01 ppm.In our case λ B can be estimated as linear over a 5 mT window for the highest slope regions, see Fig. (4), corresponding to SD ∂λ B ∂H shot ≈ 4 × 10 −3 nm/A.

Measured system noise
To measure the actual noise of the profilometer SD(z), two successive profile measurements of a flat sample were taken.The difference between the profile surface heights was then calculated.The standard deviation of this difference provided SD(z) pro ≈ 0.7 nm, which for the same parameters as the shot-noise calculation, corresponds to SD(λ B ) pro ≈ 0.1 ppm and from this SD ∂λ B ∂H pro ≈ 4 × 10 −2 nm/A.This predicted sensitivity is only around a factor of ten away from the shot-noise-limit, with noise likely dominated by other noise sources such as electronic noise, environmental noise, thermal noise, and speckle noise [29,30].We note that the NT1100 uses a tungsten halogen lamp as a light source, which is filtered to red light.This lamp has significant intensity fluctuations compared to the LEDs used in newer models and the wavelength filter has a larger linewidth than if a laser were used, resulting in more noise.Nevertheless, the predicted strain sensitivity is on par with state-of-the-art methods which achieve strain sensitivities of ∼0.1 ppm using optical [13], capacitive [31], nanoindentation [17], and inverse techniques [32].

C. Sample preparation
Magnetron DC sputtering was used to deposit Fe 81 Ga 19 films, with a 3 nm Ti adhesion and a 3 nm Cu seed layer.The Ti/Cu/Fe 81 Ga 19 films were sputtered in thicknesses of 206 nm, 285 nm, 419 nm, 498 nm and 966 nm onto 3 mm diameter, 0.1 mm thick borosilicate glass coverslips (Electron Microscopy Services 72296-03).The sputtering was done in a 2 mTorr argon atmosphere with a power of 150 W (DC).

Single measurements
The samples were placed on a nonmagnetic stage, 3Dprinted from polylactic acid (eSUN PLA+), in the optical profilometer (Veeco Wyko NT1100), with bipole helmholtz magnetic coils (Evico Magnetics GmbH type ifw8.00.00.48) used to apply a magnetic field.To observe the deformation induced by a given B-field in a sample, the difference of the profiles with and without a field applied were taken and examined.The surfaces obtained from these difference measurements were not symmetric, indicating anisotropy in the magnetostrictive strain.To account for this the extension to Stoney's formula given by [33,34] was used, for which where ∆κ B|| = κ B|| − κ I|| and ∆κ B⊥ = κ B⊥ − κ I⊥ are the change in curvature induced in the directions parallel and perpendicular to the B-field.To calculate these and determine the strain an elliptic paraboloid of the form z = 1 2 κ x x 2 + 1 2 κ y y 2 was fit to the difference measurements, from which we obtain ∆κ B⊥ = κ x and ∆κ B|| = κ y , as derived in Sec.(S.2).A typical fit is shown in Fig. (3) for the deformation of a 966 nm film by a 36.5 mT field.For this example changes in curvature of ∆κ B|| = −24.79± 0.02 km −1 , ∆κ B⊥ = −19.11± 0.02 km −1 were measured, corresponding to magnetostrictive stresses of σ B|| = −3.238± 0.002 MPa, σ B⊥ = −2.693± 0.002 MPa, and strains of λ B|| = −42.77± 0.04 ppm, λ B⊥ = −32.97± 0.03 ppm.Here the negative sign indicates a compressive stress, due to the expansion of the film under a magnetic field.This opposes the tensile intrinsic stress of the films which is which is typically ∼ 350 MPa.
The absolute values of saturation strain λ S for thinfilm galfenol have been previously measured in the range of 35-120 ppm [26,35].We did not reach saturation within our measurement range, as can be seen in Fig. (4), and expect the magnetostriction value at 36.5 mT to be slightly lower but of a similar magnitude to saturation values.With this considered our result appears consistent with literature values.The strain uncertainties for this measurement are similar to, but better than, the previously reported state-of-the-art of 0.1 ppm, achieving values of 0.03 -0.04 ppm.This is close to the uncertainty of 0.02 ppm predicted by Eq. ( 5) for a 966 nm film, given the measured system noise described in Sec.(IV B 2).The difference is likely due to imperfections in the film.In addition the optimal weighting function described in Sec.(S.3) used to derive Eq. ( 5) has not been used, with all points weighted equally to obtain the fit in this example.

FIG. 3.
A fit of the form z = 1 2 κxx 2 + 1 2 κyy 2 to a measurement of the deformation of a 966 nm film by a 36.5 mT field.Each point corresponds to the change in film height obtained from a given pixel.For clarity only every 1 in 10 pixels is shown such that the number of measurement points is reduced by a factor of 10.The purple arrow indicates the direction of the B-field.The z = 1 2 κxx 2 and z = 1 2 κyy 2 cross section curves are shown on the 3D surface in (a) and in 2D in (b) and (c).

Magnetostrictive hysteresis loops
Following the single measurements the magnetostrictive hysteresis was measured.This was achieved by sweeping the B-field in multiple loops from 0 mT → 37 mT → −37 mT → 0 mT, in intervals of 0.5 mT, cal-culating the magnetostrictive strain at each point.The first loop was used to magnetize the film to a consistent magnetization for the reference profile, and the subsequent loops for measurements.As shown in Fig. (4) for a 206 nm film, the magnetostriction displayed butterfly hysteresis behavior, which is characteristic of magnetostrictive materials [36].
We are able to quantify the piezomagnetic coupling of the film by the slope of these hysteresis loops.For the example in Fig. ( 4) we obtain a maximum piezomagnetic coupling along the direction of the field of ∂λ B|| ∂H = 6.0 ± 0.3 nm/A.For thin-film galfenol this has been measured as approximately 13 nm/A (i.e.1.0 ppm/Oe) for varying sputtering conditions, and thicknesses up to 480 nm [37,38].Hence our result is consistent with literature values, however the films could be optimized for increased performance.
FIG. 4.An example of a magnetostriction butterfly loop obtained for a 206 nm film.Each blue dot represents the strain along the direction of the field measured at a given B-field value, calculated using the method described in Sec.(IV D 1).The red trace interpolates the data, showing the 'butterfly' shape of the curve.As can be seen from the shape of the loop, the magnetostriction does not saturate.This behavior was observed for all film thicknesses.
We found that the transverse magnetostriction is of the same sign as the longitudinal magnetostriction, with a slightly lower magnitude, see Fig (5).If Joule magnetostriction were the dominant effect it would be expected that the film undergo a longitudinal expansion and a transverse contraction [22].This suggests that the field induced changes in the elastic properties of the film, such as the magnetomechanical coupling and ∆E effect may contribute significantly to the shape and magnitude of the deformation of the sample, particularly due to the large intrinsic stress of the films.

Film degradation with thickness
The piezomagnetic coupling decreased with film thickness as can be seen in Fig. (5).Degradation in magnetostriction has been previously observed, for FeGa films sputtered onto glass substrates under a 30 mT deposition field, as the thickness increased from 5 nm to 60 nm [39].This was attributed to an increase in the volume of the nonpreferential polycrystalline arrangement.Changes in the properties and structure of galfenol at larger thicknesses have also been observed.For FeGa films sputtered onto Si with a Ti/Cu seed layer [40] observed an increase in grain size and change in crystallographic texture as the thickness increased from 100 nm and 1000 nm.
To investigate the degradation in performance in our case, we took magnetization measurements using magneto-optical kerr effect microscopy (MOKE) and vibrating sample magnetometry (VSM), see Sec. (S.4).The magnetization hysteresis loops showed a distinct change in shape and coercivity as the samples became thicker, suggesting that the galfenol structure may enter a secondary material phase with lower magnetostriction above 206 nm.We model and present supporting data for this two-layer behavior in Sec.(S.4 B).
FIG. 5.The maximum piezomagnetic coupling of the films parallel and perpendicular to the applied field versus film thickness.The fits are of the form ∂λ B ∂H = t 1 t γ1 + t 2 t γ2, where t is the total thickness of the film, t1, t2 are the thicknesses obtained from the two-layer model described in (S.4 B), and γ1, γ2 are scaling factors fit to the data, proportional to the piezomagnetic coupling of each layer.In this case t1 = 175 nm, t2 = x − t1, γ1,y ≈ 7.0 nm/A γ2,y ≈ 1.7 γ1,x ≈ 6.4 nm/A γ2,x ≈ 1.6 nm/A .
We have presented a new method to measure the magnetostriction of thin-films with state-of-the-art sensitivity using optical profilometry.This method offers benefits over common techniques such as AC field driven cantilever methods.that it is non-contact, allows for magnetostriction under a DC and grants information about the anisotropy of the films.This method could be extended to other smaller substrates such as free-standing membranes, which would allow for multiple substrates to be arrayed and measured simultaneously.Alternatively, larger substrates could be used in combination with a wider field of view objective to further improve the sensitivity.
weighted average of estimates ãj with an optimal weighting function w j gives the best estimate of a, ã = j w j ãj , (S.11) where w j has the property that j w j = 1 .The optimal weighting function is given by Where we have assumed the variance V(z j ) is the same for all measurement points V(z j ) = V(z).So The variance of this estimate is V(ã) = ⟨(ã−⟨ã⟩) 2 ⟩, which substituting Eq. (S.13) and simplifying gives In the case of white noise ⟨δz j δz k ⟩ = ⟨(δz j ) 2 ⟩δ jk = V(z)δ jk where δ jk is the Kronecker delta.Therefore Eq. (S.14) evaluates to (S.15)

B. In 2D
Moving to 2D, the profilometer measures the height of the surface z j,k in a grid of evenly spaced measurements at positions r j,k , across an area of A meas = πr 2 meas , see Fig (S.16) We define f (x j , y k ) = (x 2 j + y 2 k ) 2 with x j = ∆x • j and y k = ∆y • k.Each grid element has area A g = (∆x) 2 = (∆y) 2 = πr 2 meas /N meas , where N meas is the number of measurement points.Together this gives (S.17  Which can be evaluated in polar coordinates with ℓ = Along with magnetostriction measurements we also took magnetization measurements using both a vibrating sample magnetometer (VSM) (Lakeshore 8600 Series) and a Magneto-optical Kerr effect (MOKE) microscope (Evico magnetics).The VSM was used to measure the hysteresis of the films sputtered simultaneously onto silicon chips (0.5 cm × 0.5 cm) while the MOKE was used to measure the hysteresis of the films on the coverslips.
A. VSM curves and two-layer model Fig. (S3) shows the magnetization versus B-field hysteresis curves for galfenol samples of varying thicknesses from 206 nm to 966 nm, measured using VSM.There is a clear qualitative difference between the curves of different thicknesses, whereby, as the film thickness increases the coercivity of the sample increases and a 'kink' in the curve becomes more pronounced.As with the magnetostriction measurements this suggests that as the galfenol layer becomes thicker the structure may enter a secondary phase.
Further insight into this behavior is provided by the MOKE measurements, which in contrast to the VSM measurements that measure the total magnetization of entire film [41], measure primarily the first 10 -20 nm of the film [42].The MOKE hysteresis curves show that at the top of the films with total thicknesses larger than 206 nm there is a layer of material with a coercivity of ∼16 mT, whereas for the thinner films the coercivity measured is ∼6 mT, see Fig. (S4).These observations reinforce the two-layer model and indicate that the first and secondary phases have coercivities of ∼6 mT and ∼16 mT respectively.In Sec.(S.4 B) we present a model for this two-layer behaviour by treating the films as composed of two distinct layers, each with a its own thickness, coercivity, remanence, and saturation magnetization.We find that with these coercivities the model approximately fits the VSM curves as shown in Fig. (S3), with a bottom layer thickness of ∼ 175 nm.

B. Two-layer model
In the two-layer model the total magnetization is taken to be the sum of the magnetizations of each layer scaled by their respective thickness, t 1 & t 2 .The magnetization hysteresis loop of a single layer, reaching the saturation where M + is the equation for the ascending branch, M − is the equation for the descending branch, M S is the saturation magnetization, c is a constant which determines the slope of the hyperbolic tangent, M R is the remanence and B C is the coercivity [43].For the two layer model we take + (B ′′ ) (S.27) − (B ′′ ) , (S.28) with (1) and (2) denoting the first and second layers, B ′ = B cos(θ 1 ), B ′′ = B cos(θ 2 ), where θ 1 and θ 2 are the angles of the magnetization axis relative to the applied B-field for each layer. )

2 j 2 +
FIG. S2.Diagram showing the 2D model of our measurement of the surface height.(a) Shows a grid of Nmeas measurements at positions (xj, yj) in an area of radius rmeas.(b) Shows a zoomed in section, showing each grid element has size Ag = ∆x × ∆y.
obtain the V(ã) in terms of the V(z) asV(ã) ≈ 3V(z) r 4 meas N meas .(S.22)Accordingly the uncertainty in our measurement of κ isSD(κ) ≈ 6 N meas SD(z) r 2 meas .(S.23)S.4.MEASURING THE MAGNETIZATION HYSTERESIS AND INVESTIGATING THE TWO-LAYER BEHAVIOUR FIG. S3.VSM curves showing the magnetization versus B-field hysteresis loops for the different thicknesses.The twolayer model fit described in Sec.(S.4 B) shown for the 966 nm film.