A versatile platform for magnetostriction measurements in thin films

A versatile platform for magnetostriction measurements in thin films

M.Pernpeintner,^1,2,3,a)R. B.Holl€ander,^1,3,b)M. J.Seitner,⁴E. M.Weig,^4,5R.Gross,^1,2,3 S. T. B.Goennenwein,^1,2,3and H.Huebl^1,2,3,c)

1Walther Meißner Institut, Bayerische Akademie der Wissenschaften, D 85748 Garching, Germany

2Nanosystems Initiative Munich, Schellingstraße 4, D 80799 M€unchen, Germany

3Physik Department, Technische Universitat M€ unchen, D 85748 Garching, Germany€

4Department of Physics, University of Konstanz, D 78457 Konstanz, Germany

5Center for NanoScience (CeNS) and Fakultat f€ ur Physik, Ludwig Maximilians Universit€ €at M€unchen, D 80799 Munchen, Germany€

We present a versatile nanomechanical sensing platform for the investigation of magnetostriction in thin films. It is based on a doubly clamped silicon nitride nanobeam resonator covered with a thin magnetostrictive film. Changing the magnetization direction within the film plane by an applied magnetic field generates a magnetoelastic stress and thus changes the resonance frequency of the nanobeam. A measurement of the resulting resonance frequency shift, e.g., by optical interferometry, allows to quantitatively determine the magnetostriction constants of the thin film.

In a proof-of-principle experiment, we determine the magnetostriction constants of a 10 nm thick polycrystalline cobalt film, showing very good agreement with literature values. The presented technique aims, in particular, for the precise measurement of magnetostriction in a variety of (conducting and insulating) thin films, which can be deposited by, e.g., electron beam deposition, thermal evaporation, or sputtering.

I. INTRODUCTION

Nanomechanical systems are an established platform for mass and force detection. In particular, the high quality factors of their vibrational modes¹make them ideally suited for high- precision sensing applications in (nano)biology, medicine, chemistry, and physics.^2–6For example, nanomechanical reso- nators allow for the detection of DNA molecules⁷and atoms,⁸ and nanomechanical resonance spectroscopy has been pro- posed as a versatile tool and extension of conventional spec- troscopy techniques in biology and chemistry.⁹In solid state physics, nanomechanical sensors are utilized for the investiga- tion of material properties of thin films,^10–12 which often significantly differ from those of bulk materials.^13,14One par- ticular aspect is the investigation of externally tunable mate- rial properties as discussed in the field of multiferroics.¹⁵For example, it has been demonstrated that magnetostriction and magnetic anisotropy in a Ga_0.948Mn_0.052As thin film can be precisely investigated using a nanomechanical beam setup.¹⁶

Here, we extend this concept and present a versatile platform for the experimental investigation of magnetostric- tive thin films, which uses a doubly clamped silicon nitride (Si₃N₄) nanobeam covered with a thin layer of the material of interest. This approach allows for the investigation of any magnetostrictive thin film material conducting as well as insulating which can be deposited on a Si₃N₄ nanobeam, using, e.g., electron beam evaporation, thermal evaporation, or sputtering.

In the given sample layout, the magnetic thin film is tightly connected to the underlying nanobeam. An exter- nally applied magnetic field does therefore not cause a measurable deformation of the magnetic film (as for usual magnetostriction measurement techniques); instead, it gives rise to a magnetoelastic stress, which can be read out by monitoring the resonance frequency of the double-layer nanobeam. This allows to deduce the magnetostriction con- stants of the thin film (see also Ref.16).

As a proof-of-principle experiment, we use the pre- sented method to investigate a well-characterized material, a simple cobalt thin film, to allow comparison to literature data obtained with well-established methods. We show that the magnetostriction constants determined with our method are in accordance with literature values, which demonstrates the validity of the new method.

As the thin film deposition is the last step in the sam- ple fabrication process, the ferromagnetic (FM) film is not exposed to etching solution or dry etch reactants, which allows to apply this technique to a broad range of materi- als. Moreover, the measurement sensitivity is expected to be independent of the film thickness, which could be use- ful for the investigation of very thin magnetostrictive films.

II. SAMPLE FABRICATION AND EXPERIMENTAL SETUP

We start the sample fabrication with a single-crystalline silicon wafer commercially coated with a 200 nm thick ther- mal oxide and atSiN ¼90 nm thick LPCVD (low pressure chemical vapor deposition) high-stress Si₃N₄ film. We use electron beam lithography, aluminium evaporation, and lift-

a)Electronic mail: matthias.pernpeintner@wmi.badw.de.

b)Present address: Institute for Materials Science, Christian Albrechts Universitat zu Kiel, D 24143 Kiel, Germany.

c)Electronic mail: huebl@wmi.badw.de.

093901-1

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-325898 Erschienen in: Journal of Applied Physics ; 119 (2016), 9. - 093901

https://dx.doi.org/10.1063/1.4942531

off to create an etch mask for the I = 25 Jlm long and 350 nm wide, doubly clan1ped nanomechanical beam. With a reac- tive ion etching step, we pattern transfer the structure to the silicon nitride layer. We subsequently remove the aluminium mask and release the nanobeam with buffered hydrofluoric acid. Finally, we deposit tlirm = lOom of cobalt on the chip using electron beam evaporation. Figure l{a) shows a sche- matic of the Si3N4/Co bilayer nanobeam.

To spectroscopically investigate the sample, we use the tiber interferometer (laser wavelength: 673 run) sketched in Fig. l(b) (see also Ref. 17) operating at room tempemture and in vacuum (p

<

10 ⁴ mbar) to avoid air damping. An electromagnet provides a homogeneous magnetic field at the sample position. ln this setup, any beam displacement is translated linearly into a change in the detected photovolt- age, which allows to investigate the mechanical motion of the nanobeam.¹⁸ Using a piezoelectric actuator, we can reso- nantly drive the flexural out-of-plane motion of the beam.

Employing signal vector analysis we study the mechanical response of the beam as function of the applied actuator drive frequency. Additiona11y, to control the magnetization direction of the nanobeam, we use the static magnetic field provided by the electromagnet, which we are able to rotate in the san1ple plane (i.e., fue x-y-plane).

Ill. MODEL CONNECTING MECHANICAL RESONANCE FREQUENCY CHANGES TO MAGNETOSTRICTION

In this section. we present a model predicting the add-on stress on tlle magnetic layer caused by the magne- tostrictive effect. We start with the magnetostrictive defor- mation of a polycrystalline free-standing FM film with cylindrical shape and transform the corresponding strain tensor to the xyz-coordinate system (where the x-axis is parallel to the nanobeam). Then, we use this result to calculate the magnetoelastic stress in a FM thin film on top of a nanobeam resonator and the induced resonance frequency shift.

(a)

• co

• Si₃N₄

• si0

₂

• si

(b)

<:;"- JJoH

FIG. 1. (a) Schematic of the doubly clamped Si₃N4 nanobeam (green) cov ered with a thin cobalt film (orange). (b) Fiber interferometer used to charac terize the sample in vacuum.

A. Magnetostrictive strain in a free-standing magnetic thin film

First, we consider a free-standing FM fuin film, which is centered around the origin of the xyz-coordinate system (the lab system), as illustrated in Fig. 2 (dashed circle). In a sufficiently large magnetic field, the FM film is fully magne- tized. i.e., all magnetic moments are aligned along fue exter- nal field direction. Due to magnetostriction, fue magnetic film is mechanically deformed along the magnetization direction M/Ms, which also induces a deformation in both orthogonal directions (dependent on Poisson's ratio), as sketched in Fig. 2. The relative contmction/elongation "t .L for the directions along and perpendicular to the magnetiza- tion orientation is given by fue magnetostrictive constants All and A.L, respectively: ¹⁹

and L\W

E.L = - = A.L.

w

For cobalt (as well as iron or nickel), 211

<

0 and A.L

>

0.

We define a second coordinate system xy'z', which is rotated relative to the xyz-system by an angle of ¢ around the z-axis. Here, ¢ denotes the angle between the x-axis and the magnetization direction and <I> is the angle between the x-axis and the external magnetic field orientation, as shown in Fig. 2. The xy'z' -coordinate system represents the natural system for the magnetostriction. ln tills frame of reference, the magnetization M is along the

x

-axis and the magneto- strictive strain tensor is given by

Using the rotation tensor R, which maps the xyz-coordi- nate system to the xy' z' -coordinate system

y' y

FIG. 2. Definition of the xyt coordinate system which is rotated by 4> with respect to the xyz coordinate system, with t II z. The magnetization is along the

x

axis, while the nanobeam is parallel to the x axis. Aligning the mag netization in a free standing magnetosnictive thin fihn (for iUusLTation pur poses we have chosen a cylindrically shaped film here) causes a sLTain l₁ (l.L) along the

x

^(f and i) axis.

we obtain the components of the strain tensor in the xyz- coordinate system of the nanobeam

^mag¼R^T0magR

¼

k_kcos²/þk?sin²/ ðk?k_kÞcos/sin/ 0 ðk?k_kÞcos/sin/ k?cos²/þk_ksin²/ 0

0 0 k?

0 BB B@

1 CC CA:

B. Magnetoelastic stress in a magnetic thin film on top of a substrate

In case of a thin magnetic film deposited on a substrate, the shared interface imposes a boundary condition on the FM. More precisely, the geometric dimensions of the FM are fixed inx- andy-direction, which means the effective strain along these axes vanishes. We describe this by introducing an additional strainbso that the net strain

^net¼^magþ^b vanishes along thex- andy-direction:

^net;x¼^net;y¼0:

In a more intuitive picture, magnetostriction changes the equilibrium dimensions of the film. Along the magnetization direction, e.g., the equilibrium length of the film is reduced (k_k<0). However, the boundary conditions require an unchanged length, which results in a tensile stress in the FM film along thex-axis. Analogously, magnetostriction creates a compressive stress in the FM along the y-direction.

Perpendicular to the film plane, the FM is free to expand.

Thus, the strain applied by the boundary conditions creates the stress

r^mag¼C^b; (1) with the elasticity tensorC(see Ref.20).

In Voigt notation (see, e.g., Ref.20),bis

b ¼ ^b;xx

^b;yy

^b;zz

2^b;yz

2^b;xz

2^b;xy

0 BB BB BB BB B@

1 CC CC CC CC CA

¼

kkcos²/þk?sin²/ k?cos²/þk_ksin²/

^b;zz

0 0

2ðk?kkÞcos/sin/ 0

BB BB BB BB B@

1 CC CC CC CC CA

; (2)

and the elasticity tensorCis given by

C¼

kþ2l k k 0 0 0 k kþ2l k 0 0 0 k k kþ2l 0 0 0

0 0 0 l 0 0

0 0 0 0 l 0

0 0 0 0 0 l

0 BB BB BB BB B@

1 CC CC CC CC CA

; (3)

with the shear modulus l, the Lame constant k¼ ð2l² ElÞ=ðE3lÞ, and Young’s modulus E (see Refs. 20 and21).

As the FM film can expand freely in z-direction, the stress componentr^mag;zzhas to vanish. This results in

^b;zz¼k_k 1 E 2l

; (4)

where we have used l¼E=ð2ð1þÞÞ (see Ref. 20) and k_k¼ k?(as volume magnetostriction can be neglected in first order¹⁹). Here,denotes the Poisson ratio of the magne- tostrictive material.

Using Eqs.(1) (4), we obtain the following expression for the stressr^magin the magnetic film:

r^mag¼ rmag;xx

rmag;yy

rmag;zz

0

@

1 A¼

Ekkcos²ð/Þ Ekksin²ð/Þ

0 0 B@

1 CA:

C. Effective stress in the double-layer nanobeam To obtain the effective stress present in the double-layer beam, we take into account the stress in the silicon nitride, rSiN, as well as in the magnetic thin film, r^totfilm. For a double-layer system, the effective stress along the beam direction is given by^22,23

reff¼r^SiNtSiN þr^totfilm;xtfilm

tSiN þtfilm :

Note that the total stress in the magnetic layer contains stress contributions from the fabrication process as well as the magnetoelastic stress,r^totfilm¼r⁰filmþrmag.

For a highly tensile-stressed nanobeam, the resonance frequency of the fundamental flexural mode is well approxi- mated by xres=2p¼1=ð2lÞpr=q

, where r and q denote pre-stress and density of the nanobeam (see, e.g., Ref.24).

Using effective values for stress and density of the double- layer beam,^22,23we obtain

xresð Þ/ 2p ¼ 1

2l reff

qeff

r

¼ 1 2l

r0r1cos²ð Þ/ qeff

s

; (5) where we have defined r⁰¼ ðr^SiNtSiN þr⁰film;xtfilmÞ=ðtSiN

þtfilmÞandr1¼Etfilmkk=ðtSiN þtfilmÞ.

For jr1=r0j 1, we can Taylor-expand Eq. (5) and obtain for the relative resonance frequency shift

Dxresð Þ/

xres;0 ¼xresð Þ / xres;0

xres;0 ¼ r1

2r0

cos²ð Þ;/ (6) withxres;0=2p¼1=ð2lÞ r0=qeff

p the resonance frequency at /¼90.

We thus expect a cos²ð/Þ-dependence of the resonance frequency as a function of the magnetization direction. By measuring the tuning amplituder¹=2r⁰of the resonance fre- quency, we can deduce the magnetostriction constants kk

andk?.

093901-3

IV. EXPERIMENTAL RESULTS AND DISCUSSION A. Resonance frequency measurements at high magnetic field

To experimentally investigate such a magnetostncuve tuning of a nanomechanical beam, we measure the resonance frequency of the beam in an external magnetic field of con- stant magnitude JloH = 200 mT, which is above the coercive field and the in-plane anisotropy fields of the Co film. 1n this regime, the magnetization is, in a good approximation, aligned along the external field direction(</>~ <I>). We rotate the external field between <I> = -35° and 200°, where

<I> = 0° corresponds to 11oH

II

i (see Fig. 2).

Figure 3 shows the measured homodyne photovoltage as a function of the drive frequency for four different external field directions <I>= -4°,41°₁86°,176°. We observe a clear resonance peak corresponding to the ftmdamental vibrational out-of-plane mode of the nanobean1, whose frequency shifts as a function of the magnetic field orientation. The magneto- strictive frequency shift significantly exceeds the linewidth of the resonance peaks, which is

r

m/2n ~300Hz. ln addi- tion to the resonance frequency shift, we observe a variation of the resonance peak an1plitude when rotating the external magnetic field vector, as one can see in Fig. 3. We attribute this to a slight translational shift of the san1ple position in high magnetic fields. ln this case the laser spot is not centered on the nanobeam any more, which decreases the detected photovoltage an1plitude. The small discrepancy between the resonance frequency at <I>= -4° and 176° is due to thermal drift of the resonance frequency during the measurement.

To analyze the observed field orientation dependence in more detail, we measure the amplitude spectrum of the nano- beanl as a function of <I>, as shown in Fig. 4(a). The experi- mental data con:fum the expected 180°-periodicity of the resonance frequency Wres(<I>). From these data, we extract the maximum resonance frequency shi_ft .1.w,..,_~,rnax :=

Wres(0°) -w"'-~(90°) = 2n X 8.00kHz (see Fig. 4(b)). We observe slight deviations between the measured w.-es(<I>) and the expected cos2(<I>) behaviour, especially around <I>= 45°

and 135°. This is quantitatively understood as the magnetiza- tion for these angles <I> is not perfectly aligned in parallel with the applied magnetic field due to the shape anisotropy of the nanobeam (see Section IV B for details). AdditionaiJy, this quantitative model is compatible with the absence of

20

>

2: 10

>

10.230 10.235 ro/2n (M Hz)

10.240 10.245

FIG. 3. Measured photovolt.age spectra for an external magnetic field of 11oH 200 mT applied along the nanobeam axis (x axis) and at an angle of 45°, 90• and 1800 to the x axis. The angle misalignment of 4• is due to the finite angle resolution in the experiment.

"N 10.240

J: 24

e "

^10.235

£:1 a ¹²

10.230

0

"N 10.240 J:

e

"

^10.235

N

-

_ae

.

10.230

0 45 90 135 180

<1> (deg)

FlG. 4. Resonance frequency of the fundamental flexural mode as a function of the external magnetic field orientation <I> for !loH 200 mT_ (a) shows the measured photovoltage Vas a function of drive frequency and external field direction. In (b), the fitted resonance frequency co ... (<l>) is compared to the expected cos²(<1>) behaviour.

crystalline magnetic anisotropies in our nanobeam, warrant- ing the assumption of a polycrystalline cobalt film.

To determine the magnetostriction constants 211 ^{and 2j_}

from the experimental data, we first calculate the static pre- stress cr₀ in the nanobeam. With Wres(</> = 90°)/2n = 1/(2/)

J

cro/ Peff• we get cro = 892 MPa. Here, we have used the effective density Petr=34l0kgm ³, which we determine from PsiN = 2800kgm ³ (see Ref. 1) and Peo = 8900kgm ³ (see Ref. 25) using Perf= (PsiN tsiN

+

Peotr.bn)/(tsiN

+

tmm) (see Ref. 22). With Eq. (6), the Young's modulus of cobalt E = 175 GPa (see Ref. 26) and the relation 2j_ = -v211, the observed maximum resonance frequency shift of dW,.es,max

= 2n x 8.00kHz coo·esponds to a magnetostriction constant of 211 ^{= -79.7 X lO 6 (2_i = 27.9 X 10 6).}

B. Impact of imperfect magnetization alignment on the resonance frequency shift

To study the deviation of the measured Wres(<I>) from the expected cos²(<I>) behaviour in more detail, we repeat the above measurement for lower magnetic fields. Whereas for

Jl<JH = 200 mT the magnetization M is roughly aligned

along JloH, this is not the case for smaller external fields where the magnetic anisotropy becomes an issue.

To predict the magnetization direction, we use a Stoner- Wolfarth approach and assume a saturated magnetization state IMI = Ms (see. e.g .. Ref. 19). To determine the equilib- rium direction of M as a function of the external field orien- tation, we start with the free energy density Ftot. containing Zeeman energy density <md shape anisotropy. The first is given by FZeem:m = -11oM· H (see Ref. 19), the latter is Faniso = (J1o/2)M · N · M, where N denotes the demagnetiza- tion tensor.²⁷ We approximate the cobalt thin film on the Si₃N₄ nanobeam as an ellipsoid with axis lengths/, w (width of the nanobeam) and lfilm· The corresponding demagnetiza- tion tensor has diagonal form with the components Na ~ 0, Nyy = 0.03 and N zz = 0.97 (see Ref. 28). Note that we neglect crystalline anisotropy contributions to Ftot, assuming that they average out in a polycrystalline film.²⁹TI1e contribution

093901-5

of magnetoelastic energy to Ft<x is about two orders of mag- nitude smaller than the contribution of shape anisotropy and can therefore be neglected.

For the present experimental geometry, external magnetic field and magnetization are in the x-y-plane (see Fig. 2). Thus, the total free energy density is given by

In Figs. 5(a) 5(c), we plot the total free energy density as a function of the external magnetic field orientation and the magnetization direction for the parameter values M₅

= l 167 kA/m (measured by SQUID rnagnetometry for a sim- ilar Co thin film), Nyy=0.03, and llfJH = 200mT, IOOmT and 25 mT. The minima, which determine the equilibrium magnetization direction, are highlighted by green lines. If the external field significantly exceeds the demagnetization field

llf)Hdemag = (JI()/2)M₅Nyy ~ 22 mT, the magnetization is approximately pamllel to the external field, i.e., ¢ ~<I>.

Nevertheless, even for ll{)H = 200 mT. there are slight devia- tions between the calculated energy density minima and the ideal case ¢ = <I> (indicated with the dashed blue line in Fig.

5(a)). For small external fields ll{)H$jl{)HOOn1Jg. the shape ani- sotropy of the thin cobalt stripe forces the magnetization direction towards thex-axis. as Fig. 5(c) points out.

In Fig. 5(d). the measured resonance frequency is plotted as a function of the external field orientation <I> for the field strengths ll{)H

=

200 mT, I 00 mT. and 25 mT. The solid Lines show the modeled resonance frequency behav- iour. based on Eq. (6) and the calculated equilibrium magnet- ization orientation ¢(<I>). ln particular, for Jl{)H = 200 mT, we observe excellent agreement between the experimental data and the model. For small magnetic fields, e.g., II()H

=

25 mT, our simple model explains the resonance fre- quency shift at least qualitatively, reproducing the switching

<1> (deg) <1> (deg) <1> (deg)

90 180 270 0 90 180 270 0 90 180 270

c;

Q)

::8.

...

-1

~

"

^25mT

£:! 10.234

·l

^{10.232 •} _{• 200mT} ^100mT

10.230

0 45 90 135 180

<1> (deg)

FIG. 5. (a) (c) Total free energy density a.~ a function of external magnetic field orientation (IJ and magnetization direction 4> for J.loH 200 mT (a), IOOmT (b) and 25mT (c). The green lines indicate the energy density min ima. The colorbar has been rescaled individually for each graph. (d) Resonance frequency versus external field orientation (IJ for various field Slrength~ ltr/f 200, I 00,25 mT. The symbols represent the experimental data, and the modeled resonance frequency is ploued as solid lines.

of the magnetization at <I>= 110° and the measured maxi- mum frequency shift.

C. Comparison to literature and estimated measurement sensitivity

To compare the magnetostriction constants determined above to litemture values, we calculate ).0 and J..1 in poly- crystalline magnetically satumted cobalt from the crystal magnetostriction constants A.A,B,c,o as described in Ref. 30 and obtain A.11 = -78.4 x 10 ⁶ and A..1 = 27.5 x 10 6. This is in very good agreement with the experimentally deter- mined values. Thus, we conclude that the proposed method conforms with standard magnetostriction measurement tech- niques, which, e.g., use the bending of a cantilever covered with a thin magnetostrictive fi1m,31-³⁴ and is therefore suitable to quantitatively determine the magnetostriction constants of thin films.

ln contrast to cantilever-based experiments, where mag- netostriction causes a bending of the mechanical element, the present approach uses a pre-stressed, doubly clamped nanobeam where the magnetoelastic stress modifies the total stress along the beam axis and therefore changes the reso- nance frequency of the bean1. This stress-to-frequency conversion allows for an effective determination of the mag- netostriction constants via a frequency measurement, which does not rely on a quantitative measurement of the beam dis- placement (as it is the case for cantilever-based techniques).

The high quality factor of pre-stressed Si₃N₄ nanobearn reso- nators1 therefore allows to precisely investigate magneto- striction in thin films.

Although the presented method only allows to experi- mental.ly access one stress direction (i.e., the stress compo- nent along the beam direction), it can be particularly useful for the investigation of very thin (or nanopatterned) magne- tostrictive films with a high precision. The reason is that, as we will show in the following, the experimental uncertainty of the calculated magnetostriction constants does not neces- sarily increase for decreasing film thickness as it is the case for cantilever-based measurement techniques.

To illustrate this, we first calculate the stress-frequency gauge factor, i.e., the change of the resonance frequency of the beam as a function of the stress variation. For the pre- sented sample, this is

2 ( ' ) = 2n x 0.57 Hz/kPa.

O'O I SiN

+

lr.tm

Assuming a frequency measurement precision of bwres

~

r

m/2 (with the linewidth of the resonance

r

m). this allows to resolve a stress variation of c5umag,x = 0.26 MPa. This cor- responds to an experimental uncertainty in the parallel magne- tostriction constant of c5211

=

c5umsv/ E

=

1.5 x 10 ⁶. Using a phase-locked loop (PLL) to tmck the resonance frequency of the beam, however, wouJd increase the frequency resolu- tion significantly, allowing an uncertainty c5.A.R well below

10 ⁶. This is comparable to other methods1 _.3t,J2.34 even though the thickness of our magnetostrictive film is only I 0 nm. ln particular, reducing the film thickness further does not necessarily reduce the measurement precision. This is due

to the fact that the quality factor of a highly stressed Si₃N₄ beam covered with a thin film typically strongly depends on the film thickness. For Si₃N₄/Au nanobeams, it has been shown recently that the inverse quality factor is proportional to the film thickness for Au layers between 10 nm and 100 nm as the damping in a highly prestressed silicon nitride film is much lower than in the Au film.²³Therefore, for very thin magnetostrictive films on a highly prestressed Si₃N₄ beam, the resonance frequency measurement uncertainty is proportional to the film thickness, dxres/tfilm. The uncer- tainty dk_k/drmag;x¼2dxresr0ðtfilmþtSiNÞ=ðx^res;0tfilmÞ is therefore in first approximation independent of the film thick- ness (assuming tfilm tSiN). This characteristic makes the proposed technique an ideal platform for the investigation of magnetostriction in thin and ultrathin films.

V. CONCLUSION

In this article, we have proposed a method to quantita- tively investigate magnetostriction in thin films. To this end, we use a Si₃N₄ nanomechanical resonator covered with a thin magnetostrictive film. By measuring the resonance fre- quency of the fundamental vibrational mode of the beam as a function of an external magnetic field, we can deduce the magnetoelastic stress along the beam direction and hence the magnetostriction constants kk and k?. Compared to previ- ously reported methods, the proposed technique does not rely on a quantitative measurement of the mechanical dis- placement, but utilizes a resonance frequency shift caused by magnetostriction. Besides, it offers a measurement precision which is independent of the film thickness. This enables the investigation of ultrathin magnetostrictive films and paves the way to study magnetostriction as a function of the film thickness. The proposed technique can be applied to any conducting or insulating material which can be deposited on a Si₃N₄ nanobeam via electron beam evaporation, thermal evaporation, sputtering, etc. The material under investigation does not have to be etch-resistant as it is deposited on the nano-resonator as the last step of the sample fabrication process.

ACKNOWLEDGMENTS

The authors thank Dr. S. Gepraegs for fruitful discussions and gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft via Project No.

Ko416/18 and SpinCAT GO944/4.

APPENDIX: SURFACE TEXTURE OF THE COBALT THIN FILM AND THE ABSENCE OF CRYSTALLINE ANISOTROPY

According to SEM images, the Co film is flat (i.e., we do not observe grain formation) at a resolution of approx.

10 nm. We have not investigated the texture of the Co film using X-ray diffraction; however, the angular dependence of x^res in our experiments allows to access directly andin situ the magnetic anisotropy parameters of the cobalt film on the Si₃N₄nanobeam. In particular, contributions from crystalline magnetic anisotropy should be prominent at low static

magnetic fields (see above). As all experimental data can be quantitatively modeled considering only contributions from the shape anisotropy of the nanobeam, i.e., the demagnetiza- tion field (see Fig.5), we are confident that there is no signif- icant crystalline anisotropy present in our magnetic thin film.

More precisely, we estimate that the crystalline anisotropy is at least four times smaller than the demagnetization field (22 mT), and thus at least one order of magnitude smaller than the crystalline anisotropy expected for a hexagonal Co lattice (50 mT). Therefore, from a magnetic standpoint we can safely assume the Co film to be polycrystalline.

The surface roughness of the Si₃N₄ film, on which the Co has been deposited, is 2:8A (according to the manufac-˚ turer), which is similar to the lattice constant of Co in a hexagonal lattice (2:5A, resp. 4˚ :0A in˚ c-direction). We therefore believe that the interface is sufficiently flat to guar- antee a homogeneous transfer of the magnetoelastic stress to the Si₃N₄nanobeam at the Si₃N₄/Co interface.

1Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus,Phys. Rev. Lett.105, 027205 (2010).

2C. L. Degen, M. Poggio, H. J. Mamin, C. T. Rettner, and D. Rugar,Proc.

Natl. Acad. Sci. U.S.A.106, 1313 (2009).

3K. Eom, H. S. Park, D. S. Yoon, and T. Kwon, Phys. Rep.503, 115 (2011).

4A. Boisen, S. Dohn, S. S. Keller, S. Schmid, and M. Tenje,Rep. Prog.

Phys.74, 036101 (2011).

5J. L. Arlett, E. B. Myers, and M. L. Roukes,Nat. Nanotechnol.6, 203 (2011).

6J. Moser, J. Guttinger, A. Eichler, M. J. Esplandiu, D. E. Liu, M. I.

Dykman, and A. Bachtold,Nat. Nanotechnol.8, 493 (2013).

7R. Mukhopadhyay, M. Lorentzen, J. Kjems, and F. Besenbacher, Langmuir21, 8400 (2005).

8J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, Nat. Nanotechnol.7, 301 (2012).

9P. A. Greaney and J. C. Grossman,Nano Lett.8, 2648 (2008).

10S. Schmid, K. D. Jensen, K. H. Nielsen, and A. Boisen,Phys. Rev. B84, 165307 (2011).

11R. B. Karabalin, L. G. Villanueva, M. H. Matheny, J. E. Sader, and M. L.

Roukes,Phys. Rev. Lett.108, 236101 (2012).

12T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig,Phys.

Rev. B89, 100102 (2014).

13A. J. Sorensen,Phys. Rev.24, 658 (1924).

14F. Richter,Mechanical Properties of Solid Bulk Materials and Thin Films (TU Chemnitz, 2010).

15N. A. Spaldin and M. Fiebig,Science309, 391 (2005).

16S. C. Masmanidis, H. X. Tang, E. B. Myers, M. Li, K. De Greve, G.

Vermeulen, W. Van Roy, and M. L. Roukes,Phys. Rev. Lett.95, 187206 (2005).

17M. Vogel, C. Mooser, K. Karrai, and R. J. Warburton,Appl. Phys. Lett.

83, 1337 (2003).

18N. O. Azak, M. Y. Shagam, D. M. Karabacak, K. L. Ekinci, D. H. Kim, and D. Y. Jang,Appl. Phys. Lett.91, 093112 (2007).

19S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, 1997).

20R. Gross and A. Marx,Festkorperphysik(Oldenbourg, 2012).

21S. Hunklinger,Festkorperphysik(Oldenbourg, 2009).

22F. Hocke, M. Pernpeintner, X. Zhou, A. Schliesser, T. J. Kippenberg, H.

Huebl, and R. Gross,Appl. Phys. Lett.105, 133102 (2014).

23M. J. Seitner, K. Gajo, and E. M. Weig,Appl. Phys. Lett.105, 213101 (2014).

24S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G.

Craighead,J. Appl. Phys.99, 124304 (2006).

25N. N. Greenwood and A. Earnshaw,Chemie der Elemente(Wiley VCH Weinheim, 1988).

26H. Masumoto, H. Saito, and M. Kikuchi, Sci. Rep. Res. Inst., Tohoku Univ., Ser. A19, 172 (1967); available athttp://ir.library.tohoku.ac.jp/re/

bitstream/10097/27368/1/KJ00004197159.pdf.

27A. H. Morrish,The Physical Principles of Magnetism(Wiley, 2001).

28A. Brandlmaier, “Magnetische Anisotropie in dunnen Schichten aus Magnetit,” M.S. thesis, Technische Universitat Munchen, 2006.

29M. Weiler, A. Brandlmaier, S. Geprags, M. Althammer, M. Opel, C.

Bihler, H. Huebl, M. S. Brandt, R. Gross, and S. T. B. Goennenwein,New J. Phys.11, 013021 (2009).

30W. P. Mason,Phys. Rev.96, 302 (1954).

31E. Klokholm,IEEE Trans. Magn.12, 819 (1976).

32A. C. Tam and H. Schroeder,J. Appl. Phys.64, 5422 (1988).

33E. du Tremolet de Lacheisserie and J. C. Peuzin,J. Magn. Magn. Mater.

136, 189 (1994).

34M. Weber, R. Koch, and K. H. Rieder, Phys. Rev. Lett. 73, 1166 (1994).

093901-7