Exploring a novel cantilever design for enhanced sensitivity in magnetic resonance force microscopy

Sensitivity in Magnetic Resonance Force Microscopy

Rembrandt Donkersloot

2016

Exploring a Novel Cantilever Design for Enhanced

Sensitivity in Magnetic Resonance Force Microscopy

Master Thesis

submitted in partial fulfilment of the

requirements for the degree of

Master of Science

in

Physics

Author

Rembrandt Donkersloot

Kamerlingh Onnes Laboratory, Leiden University PO Box 9504, 2300 RA Leiden, The Netherlands

Host Institution

Baker Laboratory, Cornell University Ithaca, NY 14850-1301 USA

Copyright c

⃝2016 Rembrandt Donkersloot. All Rights Reserved.

This thesis is written under joint supervision by the home and host

universities, respectively Leiden University and Cornell University.

Home supervisor: prof. dr. ir. T.H. Oosterkamp.

Contents

1 Introduction 1

2 Material and Methods 3

3 Results 7

3.1 Magnetometry study on a micro-sized Co needle . . . 7

3.2 Magnetometry study on a nano-sized Co particle . . . 10

3.3 Preparation of a nanomagnet-tipped cantilever . . . 12

3.4 Discussion . . . 14

4 Conclusions 17

We have investigated the technical feasibility of a novel concept within the scientific field of magnetic resonance force microscopy (MRFM), in which a cantilever is used as a force sensor on the one hand and a radio frequency (rf) source on the other. By driving a magnet-tipped cantilever at a higher resonance mode, the rotational motion of the mag-netic tip generates an oscillating magmag-netic field. In this way the cantilever can serve as an ultra-low dissipative rf source with an rf frequency corresponding to the specific higher mechanical resonance mode of the cantilever. In this work we have tested this idea for an attonewton-sensitivity silicon cantilever with a high magnetic gradient cobalt nanomagnet attached at the cantilever’s free end. Using frequency-shift cantilever magnetometry, we found that the nanomagnet’s remanent magnetisation is BR = 0.83 T. When the

nanomag-net is close enough to a spin-containing sample, we have calculated that the nanomagnanomag-net’s magnetisation - even at zero applied magnetic field - can mechanically generate an rf field of the order mT. This implies that the protocol for adiabatic rapid passage can be con-ducted without the use of an rf wire as a radio frequency source, which eliminates a major dissipation channel that constitutes an obstacle to date for lower working temperatures in high resolution 3D-imaging experiments with MRFM.

Chapter 1

Introduction

State of the art Magnetic Resonance Force Microscopy (MRFM) is known for its ability to image biological tissue in 3 spatial dimensions at high resolution as impressively demon-strated by Degen and co-workers at the IBM Research Division in 2009. Accordingly, they imaged a Tobacco Mosaic Virus with a (5 nm)3 _{resolution which corresponds to a}

sens-itivity of 100 protons[1]. The experiment is based on a measurement protocol known as ’Adiabatic Rapid Passage’ (ARP), in which the proton spins are flipped twice per canti-lever oscillation [2]. An indispensable ingredient for this type of imaging experiment is a radio frequency (rf) field of the order mT which is needed for spin manipulation. However, a conventional rf source such as an rf wire or an rf coil limits the sensitivity of MRFM since the associated rf current generates heat. This is illustrated by the tobacco virus imaging experiment: The heat that is dissipated by the large rf current prevented the researchers to operate at a temperature below 300 mK. Lower temperatures would allow them to image at an even better resolution.

Oosterkamp and co-workers discovered an alternative rf source that does not inhabit the limitation as mentioned above. They noticed that a higher resonance mode of the MRFM force sensor can be used to mechanically generate an rf field [3]. Their force sensor con-sists of an IBM-style cantilever with a spherical NdFeB magnetic particle attached at the cantilever’s free tip. When the cantilever is excited in a higher resonance mode, the mag-netic particle will make a rotational motion and will thereby generate a small oscillating magnetic field. Using this novel method they claimed to have generated an rf field with a magnetic field strength of at least 24 µT [4]. If this result could somehow be improved with a factor of 50, than this ultra-low dissipative rf source would be suitable for ARP at ultra-low temperatures. One possible strategy to achieve this is to use a smaller magnet with a larger field gradient. In this thesis we explore this idea further: We investigate the possibility to attach a nano-sized magnet at the cantilever’s free end. For this we closely colaborate with Marohn and co-workers at Cornell University, who are well known for the fabrication of nano-sized cobalt magnets with a high magnetic field gradient [5]. Subsequently, a relatively larger NdFeB magnet will be mounted above the nanomagnet.

NdFeB magnet Cantilever in higher mode Nano magnet rf μ M m x z

B

m

m

α

m

Figure 1.1: Illustration of the prin-ciple in which an MRFM cantilever serves as an ultra-low dissipative rf source. Here, the cantilever is depic-ted in spam-geometry, in which the magnetic moment is parallel to the cantilever motion. A higher reson-ance mode of the cantilever will let a nanomagnet rotate with an angle

α. The magnetisation m of this nanomagnet can be decomposed in a static field mx and an

oscillat-ing rf field (mz) that oscillates with

the frequency of the higher reson-ance mode. The bigger magnet (Nd-FeB particle) with magnetisation

M provides the static field B0 and

is needed for cantilever motion de-tection via a SQUID read-out meas-urement. Furthermore, the big-ger magnet’s magnetic field might also ’boost’ the magnetisation of the nanomagnet.

Here, the NdFeB magnet serves a three-fold purpose 1_{: (1) The NdFeB particle provides}

the static field B0 and (2) provides the magnetic field that is needed for cantilever motion

detection via a Superconducting Quantum Inteference Device (SQUID). Thirdly (3), the NdFeB particle’s magnetic field might be necessary for magnetisation of the nanomagnet in the case that the nanomagnet’s remanence magnetisation is too low. An important part of this thesis is confined to the latter, in which we use frequency-shift cantilever magneto-metry to characterise the nanomagnet’s magnetisation with respect to an applied magnetic field [7]. Chapter 2 describes the working principle and corresponding experimental set-up of cantilever magnetometry and we will provide the results of the magnetometry study in chapter 3. Taken together with the results from some computer simulations we will discuss the feasibility of this cantilever to serve as an ultra-low dissipative rf source in ARP MRFM imaging experiments.

1_{This ’double-magnet cantilever’ is specifically designed for the Oosterkamp group who work without}

the use of an external magnetic field. On top of this, in stead of using optical inteferometry to detect the cantilever motion the group uses a SQUID. Both modifications were crucial to allow MRFM experiments at temperatures as low as 10 mK [6].

Chapter 2

Material and Methods

In this chapter we provide a detailed description of the in-house developed magnetometry set-up and the working principle of frequency-shift cantilever magnetometry. Figure 2.1 shows the heart of this set-up. A small magnetic sample under study is attached at the free end of an ultra-sensitive force sensor (cantilever). By measuring the interaction between the cantilever and an applied magnetic field, one is able to infer the magnetic properties of the sample. For this it is crucial to measure the cantilever motion over time, which can be done very accurately by using a laser detection set-up. Accordingly, a simple Fourier analysis can be applied to extract relevant interaction parameters such as the cantilever’s resonance frequency f0.

Instrumentation

In our study we use an attonewton-sensitivity silicon cantilever [8] as a force sensor. These cantilevers are in-house fabricated [9, 10, 11] with length, width and thickness 200 µm, 5.5 µm, 0.34 µm, respectively. The cantilever motion is detected using optical inteferro-metry. A silicon chip containing the cantilever is glued on a small stage at the end of a vacuum probe. Using an optical microscope, the cantilever’s reflecting pad (size: 30 µm) is aligned with respect to an optical fiber through which we sent laser 1 _{light (λ = 1321.8}

nm). The light that is reflected at the cantilever interferes with light that is reflected at the end of the optical fiber. For the latter it is necessary to make a proper cleave at the end of the fiber to create a surface perpendicular to the wire. We use a 90:10 fiber optic coupler to allow the interference signal to be detected by an interferometer 2. This signal is optimised by phase-shifting the laser’s wavelength, see figure 2.2 top left. Subsequently, the vacuum probe is closed and is brought into high vacuum (< 10−5 mbar). Now one should be able to measure the cantilever’s thermal spectrum/Brownian motion, which can be analysed to extract the cantilever’s resonance frequency and stiffness. This signal can

1_{QPHOTONICS, LLC Model: QFLD-1300-1S}

Co nano magnet laser fiber cantilever reflecting pad inteferrometer outer layer glass fiber |μ0H| -6T +6T x z μ0H

Figure 2.1: Overview of the frequency-shift magnetometry set-up. The magnetic properties of a small magnetic sample can be studied by attaching the sample at the end of a sensitive cantilever and measuring the response of the cantilever’s resonance frequency f to an applied magnetic field B = µ0H ˆz. In our set-up, the maximal field amplitude we can

apply is 6 T. The cantilever motion is detected with optical inteferometry.

be amplified by several factors using a positive feedback driving circuit as described in the PhD thesis of E.W. Moore [12]. Here, we place a small piezoelectric element close to the cantilever and use the cantilever signal, phase-shifted by 90 degrees, to drive the cantilever. An example of a transfer function of a driven cantilever is shown in figure 2.2 left bottom. We perform a ring-down measurement to accurately measure the cantilever’s quality factor (Q-factor). In this procedure we turn off the piezo actuation and we measure the exponentially decaying cantilever signal in real-time. From a fit to V = Ae−t/τ we can obtain the Q-factor Q0 = πf0τ . An example of a ring-down measurement is shown in figure

2.2 right bottom. After these characterisation measurements the probe is placed inside a cryogenic dewar 3 which contains a 6 T superconducting magnet (inductance: 5.3 H). For specifications on the cryogenic dewar and the magnet we refer to the PhD thesis of T.N. Ng [13]. A magnet power supply 4 _{(43.1 A = 6 T) provides the current for applying a}

magnetic field strength up to 6 T in the direction of the cantilever length. After filling the dewar with liquid nitrogen, the cantilever characterisation measurements as described above can be repeated at T = 77 K. Then, after the liquid nitrogen is replaced with liquid helium, the characterisation measurements are repeated at T = 4.2 K. Finally, one can start the magnetomety experiment by slowly sweeping the magnetic field while measuring the cantilever’s resonance frequency and Q-factor.

3_{Precision Cryogenic Systems, Inc. (Indianapolis, Indiana)}

5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 15 20 25 30 35 40 Reflection voltage (V) Temperature (°C) 20 mA 22 mA 24 mA 26 mA 28 mA 30 mA 10-7 10-6 10-5 10-4 10-3 10-2 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 Amplitude (V 2/Hz) Frequency (Hz) 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 Amplitude (V 2/Hz) Frequency (Hz) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 5 Amplitude (V) Time (s)

Figure 2.2: Example of cantilever characterisation measurements in the magnetometry set-up. Once the cantilever is aligned with respect to the optic fiber, the cantilever signal can be optimised by changing the phase of the laser (left top - RT). After proper alignment and signal optimization the cantilever is brought into high vacuum and the thermomechan-ical position fluctuations (Brownian motion) of the cantilever can be measured (right top - RT, 32 averages). The corresponding thermal spectrum can be analysed to obtain the cantilever’s resonance frequency and stiffness. Using a positive feedback circuit the canti-lever can be driven into resonance, amplifying the canticanti-lever signal by a couple of orders of magnitude (left bottom - RT, 32 averages). By turning off the drive while recording the cantilever displacement amplitude one can extract the cantilever’s Q-factor from fitting (red curve) this data to a single exponentially decaying function (right bottom - 4.4 K, 10 averages).

Frequency-shift cantilever magnetometry: working principle

In frequency-shift cantilever magnetometry, a small magnetic sample with magnetic mo-ment µ is mounted at the free end of a cantilever with length l, while an external magnetic field B is applied. This field exerts a torque τ = µ× B on the cantilever which leads to an effective force F = µBcos(θ)/l at the end of the cantilever, which in turn results in a change in cantilever stiffness km = 2k0(f − f0)/f0, or equivalently a frequency shift of the

cantilever’s resonance frequency f − f0. Here, f0 and k0 represent the natural resonance

frequency and stiffness, respectively. By studying the change in resonance frequency versus applied magnetic field, one is able to determine the magnetic moment of the sample. On top of this, the stiffness dependency on an external field can be used to reconstruct the full magnetic hysteresis curve of the magnetic sample under study. For this we closely follow Ng et al. [14], who use the following equations from the tip field interaction model of Marohn et al. [15] f − f0 ≈ ∆f B∆B B + ∆B + βB (2.1) ∆f = f0 2k ( α l )2 µ (2.2) ∆B = µ0µ ∆N V (2.3)

where µ0 = 4π· 10−7N/A2 is the vacuum permeability, V is the volume of the magnetic

sample and α = 1.377 is a specific constant for a beam cantilever. The term βB is an additional term to correct for any linear background of the cantilever. By rearranging these equations and solving for the sample’s magnetization Bm = µ0µ/V we obtain

Bm = Bk 2B ± 1 2B ( B_k2+4BkB 2 ∆N )1/2 (2.4)

where Bk = kmµ0l2/α2V is the magnetic contribution to the cantilever spring constant,

expressed in units of magnetic field. The difference in the tip’s demagnetization factors ∆N = Nt− Nl can be determined from fitting 2.1 to the measured frequency shift, using

∆f and ∆B as free parameters and subsequently combining equations 2.2 and 2.3 to solve for ∆N .

Chapter 3

Results

In this chapter we report on experimentally measured magnetic hysteresis loops of a nano-sized cobalt magnet which is attached at the free end of an ultra-sensitive silicon canti-lever. As a sanity check, a similar experiment has been performed for a cantilever with cobalt evaporated on the edge. The magnetic hysteresis loops are indirectly obtained via frequency-shift cantilever magnetometry. Finally, we report on the fabrication of a canti-lever prototype that is designed for MRFM experiments in which the canticanti-lever serves as a force sensor on the one hand, and an ultra-low dissipative rf source on the other. We conclude the chapter with a brief discussion and outlook, in which we put the results in perspective with computer simulations and other related studies.

3.1

Magnetometry study on a micro-sized Co needle

We have performed a frequency-shift cantilever magnetometry study on an attonewton-sensitivity silicon cantilever where we have evaporated 5.0 nm Ti, 201.5 nm Co, and 18 nm Pt onto the cantilever’s thin edge. The Ti and Co capping layers served to promote adhesion to the silicon cantilever and to mitigate oxidation, respectively. A scanning electron microscopy (SEM) image of the cantilever under study is shown in figure 3.2. For the current study we have used the magnetometry probe as described in chapter 2. The data and analysis is shown in figure 3.1. Here, the following protocol has been used: the dewar magnet current is slowly ramped up while measuring the cantilever’s resonance frequency, until a magnetic field strength B = µ0H of 6 T is reached. Subsequently, we

slowly swept the field to µ0H = 0 T, µ0H = −6 T, µ0H = 0 T and µ0H = 2 T. At

every field step during the forward and backward sweep, a brownian spectrum is recorded to obtain the cantilever’s resonance frequency, and a ring-down measurement is conducted from which we could extract the cantilever’s Q-factor. When necessary, we adjusted the laser temperature and the signal/piezo amplification to optimise these measurements. The full sweep has been performed at 4.2 K and at a pressure of 1.4· 10−6 mbar. As expected,

Figure 3.1: SEM image of the cantilever under study: a 201.5 nm layer of Cobalt is evaporated on the edge, as illustrated by the yellow stripe. The insert image shows the cantilever tip at a finer scale, from which we could verify that the evapora-tion process was successful. The Co layer serves as a magnetic sample for a mag-netometry test experiment: a magnetic field µ0H is applied in the direction of the

cantilever length (hang-down geometry), and the response in resonance frequency is being studied to extract properties of the cobalt in this specific ’micro needle’ geometry, such as the sample’s magnetiz-ation µ0M versus applied field.

the butterfly shaped hysteresis curve at low fields is present, indicating a non-zero remanent magnetization. The corresponding data is analysed further using the procedure for inferring a magnet’s full hysteresis curve from the response of the cantilever resonance frequency f versus magnetic field µ0H, as described in detail in chapter 2. Accordingly, equation 2.1 is

fitted to the change in resonance frequency f − f0 yielding ∆B = 1.82 T and ∆f = 581.2

Hz/T, from which we calculated ∆N = 4.19. From here on we use equation 2.4 to construct the magnetic hysteresis curve. Within our model, we have set V equal to the magnetic needle’s volume: V = 200 µm × 0.2 µm × 0.34 µm = 13.6 µm3_{. We obtained a clear}

hysteresis curve when discarding data points at low field strengths. However, we address three points of concern: (1) The saturation magnetization Bs = µ0Ms = 0.44 T does not

agree with the expected saturation magnetization for cobalt, which is µ0MS,Co = 1.81 tesla

and (2) ∆N does not agree with ∆Nneedle= 1₂, the difference in the tip’s demagnetization

factors for a needle magnet. The origin of these two discrepancies lies in the fact that our analysis model is based on a geometry in which the magnetic sample is attached at the cantilever’s free end, and not along the cantilever edge. We believe that a modified analysis, starting with first principles from Euler-Bernoulli beam theory, might allow us to construct correct hysteresis curves. As a final remark (3), when we do include data points at low field strengths in our analysis, we notice that the hysteresis curve tends to suddenly break from the overall trend, which might be an artefact of equation 2.4 in the limiting case when B and f − f0 both approach zero. Furthermore, from the measured Q-factors, we

observe a decreasing Q-factor with respect to an increasing magnetic field. This behaviour can be well understood when we include the magnetic friction Γm = Γ − Γ (B = 0), into

this picture, where Γ = 2πf m/Q is the cantilever dissipation. For a detailed discussion on this topic, we refer to [14].

3.1 Magnetometry study on a micro-sized Co needle 9 8200 8300 8400 8500 8600 8700 8800 8900 9000 9100 -6 -4 -2 0 2 4 6 fres (Hz) µ0H (T) 8250 8260 8270 8280 8290 8300 8310 -0.1 -0.05 0 0.05 0.1 fres (Hz) µ0H (T) 0 100 200 300 400 500 600 700 800 0 1 2 3 4 5 6 Frequency shift f -f0 (Hz) µ0H (T) -0.4 -0.2 0 0.2 0.4 -6 -4 -2 0 2 4 6 µ0 M (T) µ0H (T) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.1 -0.05 0 0.05 0.1 2000 4000 6000 8000 10000 12000 14000 -6 -4 -2 0 2 4 6 Q µ0H (T) 0 5 10 15 20 25 -6 -4 -2 0 2 4 6 Γm (pN sm -1) µ0H (T)

Figure 3.2: Results of a cantilever magnetometry study on a micro-sized Co needle. Re-sponse of the cantilever resonance frequency f versus an applied magnetic field µ0H (top

left). The red triangles represent the forward sweep whereas the black circles represent the backward sweep. At low fields, the butterfly shaped hysteresis curve is observed as expec-ted, indicating a non-zero remanent magnetization. An analysis of the frequency shift data (center left) yields relevant parameters which are used for inferring the Co magnet’s full hysteresis curve (center right). Here, the insert graph shows the magnetization at low field strengths. The Q-factor and magnetic friction Γm (bottom) show a clear dependence on

3.2

Magnetometry study on a nano-sized Co particle

The following study describes the results of a frequency-shift magnetometry study on a nano-sized cobalt magnet with dimensions 1.5× 0.1 × 0.1 µm (length, height and width respectively). The nanomagnet is in-house fabricated using a protocol for the fabrication of nickel-tipped chips [11], which is modified for cobalt according to the Methods section of reference [16]. For attachement of the nanomagnet to the cantilever’s tip we used the protocol of Longenecker et al. [16]. Figure 3.3 shows an example of an SEM image of a nanomagnet attached at the free end of a cantilever. The experiment is performed using the custom build magnetic resonance force microscope as described in detail in reference [17]. We followed a similar measurement procedure as reported in the previous section, ramping the field up to 5 T and performing a reverse sweep +5 T → −5 T and subsequently a forward sweep −5 T → +5 T, while monitoring the cantilever’s resonance frequency with a frequency counter. At every field-step, a ring-down measurement is performed from which the cantilever’s Q-factor could be extracted. The complete sweep is conducted at 4.2 K and a pressure of 1· 10−5 mbar. The Q-factor and resonance frequency dependency on the field shows comparable behaviour as observed for the Co micro needle, where we again find the butterfly shaped hysteresis curve at fields of the order of mT. Using the same analysis model as before, we were able to fit the frequency shift to equation 2.1, yielding ∆B = 0.89 T, ∆f = 0.579 Hz/T, from which we calculated ∆N = 0.47. The latter parameter is used to compute the magnetic hysteresis curve, as shown at the bottom of figure 3.4. The curves indeed seem to saturate around 1.8 T magnetization, which agrees well with the theoretical value for the saturation magnetisation of cobalt: Bs = 1.81 T.

However, a very striking observation is the absence of any remanent magnetization: The µ0M versus µ0H curve shows no magnetization when we return to zero external field.

This observation is remarkable since the cobalt-alloyed magnet is known to exibit a high remanent field. In section 3.4 we will discuss this issue in more detail.

1 m

3.2 Magnetometry study on a nano-sized Co particle 11 0⋅100 2⋅10-7 4⋅10-7 6⋅10-7 8⋅10-7 1⋅10-6 -6 -4 -2 0 2 4 6 km (N/m) µ0H (T) -5x10-8 0 5x10-8 1x10-7 1.5x10-7 2x10-7 2.5x10-7 3x10-7 3.5x10-7 4x10-7 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 km (N/m) µ0H (T) 5800 6000 6200 6400 6600 6800 7000 7200 7400 -6 -4 -2 0 2 4 6 Q µ0H (T) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 Frequency shift f -f0 (Hz) µ0H (T) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -4 -2 0 2 4 µ0 M (T) µ0H (T) -200 -150 -100 -50 0 50 100 150 200 -100 -50 0 50 100 µ0 M (mT) µ0H (mT)

Figure 3.4: Results of a cantilever magnetometry study on a nano sized Co magnet. Re-sponse of the cantilever stiffness km versus an applied magnetic field µ0H (top left). The

red triangles represent the forward sweep whereas the black circles represent the backward sweep. At low fields, the butterfly shaped hysteresis curve is observed as expected, indic-ating a non-zero remanent magnetization. The Q-factor (center left) shows a comparable behaviour as observed for the magnetometry experiment described in the previous section. An analysis of the frequency shift data (center right) yields relevant parameters which are used for inferring the Co magnet’s full hysteresis curve (bottom left). Unfortunately, no remanent magnetization is apparent from this hysteresis curve (bottom right).

3.3

Preparation of a nanomagnet-tipped cantilever

In this section we report on the first attempt to attach a nanomagnet-tipped silicon chip1

containing a high-gradient cobalt nanomagnet onto an ’IBM-style’ single crystalline silicon cantilever as batch-fabricated by Chui et al. [18] at IBM Almaden Research Center. We note that the IBM-style cantilevers differ a factor 10 to 100 in stiffness compared to the in-house fabricated cantilevers from Cornell University with similar dimensions: The stiffness of a typical ’Cornell-style’ cantilever is of the order mN/m, whereas the IBM-style canti-levers are much softer. Since the Q-factors of both canticanti-levers do not differ significantly, we expect that a nanomagnet-tipped IBM cantilever would serve as a more sensitive force sensor within MRFM experiments than using a nanomagnet-tipped Cornell cantilever. For reasons as explained in chapter 1, we have worked with a 200 µm long IBM-style canti-lever with a NdFeB spherical ferromagnetic particle (diameter 3.4 µm) mounted at 12 µm from the cantilever tip, as prepared in an in-house fabricated nano manipulator [19] in a commercial SEM2 _{at the Huygens Laboratory, Leiden University. Within this process, we}

approached the cantilever’s tip to a NdFeB powder, and searched for a magnetic particle with suitable diameter, see the left top image in figure 3.5. The magnetic particle is at-tached to the cantilever in an Electron Beam Induced Deposition (EBID) process with a Pt(PF3)4 precursor gas [6]. Subsequently, we transferred the cantilever to the electron microscopy facility of the Cornell Center for Materials Research, where we used the same protocol as described by Longenecker et al. [16] to attach a nanomagnet-tipped silicon chip [5] to the cantilever’s free end. We worked with a FEI’s DualBeam system in which we simultaneously imaged the cantilever with an electron beam and a focused ion beam (FIB), a feature that allowed us to attach the magnetic chip without observing the nano magnet directly with the ion beam, preventing FIB damage on the magnet. Here, we briefly sum-marise the procedure: We start with a substrate containing an array of 10 by 10 magnetic chips. After choosing a chip containing a nanomagnet with suitable dimensions, a probe needle is attached to a tab connected to the silicon chip containing the nano magnet. Subsequently, the magnetic chip is milled from the substrate. After a lift-out, the chip is glued to the cantilever’s free end using FIB-assisted platinum deposition. Finally, the part of the chip containing the probe needle is milled away without damaging the nearby cantilever. Although we worked at the lowest possible beam current, the soft cantilever deforms dramatically after seconds of FIB imaging, an effect that is not observed for the Cornell-style cantilevers, which made it an extremely difficult task to properly attach a magnetic chip to the cantilever. Yet, after several attempts we managed to successfully attach a nanomagnet to a cantilever with a distance of 14 µm between the nanomagnet and the NdFeB particle. After exposing the cantilever for several minutes to a 1 T mag-netic field in the direction of the cantilever length, the cantilever ’bend back’ up to a point where the reflecting pad was straight again. Unfortunately, the cantilever did not survive the subsequent characterization experiment.

1_{Magnetic chip dimensions (maximum): 21 µm long and 9 µm wide.} 2_{FEI Company, Nova NanoSEM 200}

3.3 Preparation of a nanomagnet-tipped cantilever 13

Figure 3.5: SEM images of different steps within a process in which a NdFeB spherical particle and a magnetic chip containing a nano-sized Co magnet are attached to an IBM-style cantilever. A NdFeB particle with suitable dimensions is mounted around 10−15 µm away from the cantilever’s free end (left and right top), leaving enough space for the attachment of the magnetic chip (bottom left). After milling away the probe needle we are left with the final product, although the ion beam causes the cantilever to bend permanently (right bottom). This bending artefact can be corrected by exposing the cantilever to a strong magnetic field.

3.4

Discussion

Based on the experimentally determined magnetic hysteresis loops of the Co nanomagnet as shown in the bottom graphs of figure 3.4, we are left with a vanishing magnetisation at µ0H = 0 T. However, for the cantilever design we described in the previous section,

the nanomagnet experiences a small magnetic field from the NdFeB magnet. This field strength obviously depends on the relative position of the magnetic particle with respect to the nanomagnet. In retrospect, we would like this distance to be as small as possible. In order to reduce the nanomagnet-NdFeB particle distance for a succeeding ’double-magnet cantilever’ fabrication, we suggest that the cantilever should first be prepared with a nan-omaget tip and subsequently with a NdFeB particle. In this way we are not limited by physical interference of the magnetic chip. Moreover, figure 3.6 shows what field strenghts we can expect in a plane below a 3 µm sized NdFeB particle in the spam-geometry. Here, we assumed a remanent magnetisation of 0.77 T, as determined experimentally in reference [20]. The calculations are computed with a finite element analysis (COMSOL Multiphys-ics) and agree well with the analytic solution for a magnetic field of a perfect sphere along the magnetization axis, see the top left graph in figure 3.7. The simulation shows that for a separation distance of 1 µm we can expect around 90 mT, which in turn corresponds to at least 150 mT magnetisation of the nanomagnet.

As briefly mentioned in section 3.2, the magnetic hysteresis loops at low magnetic field strengths show unexpected features whose origin is not yet understood. This might imply that the underlying analysis is false: taken together with data points within the mT-range that need to be discarded, we cannot claim to have determined the nanomagnet’s magnetisation at low field strengths via the reconstructed hysteresis loops and further investigation is therefore crucial. However, according to the same tip-field interaction model on which the hysteresis loop reconstruction is based [15], the remanent magnetisation can be directly extracted from the frequency shift within the low field range:

f− f0 = f0 2k0 α l 2 µµ0H (3.1)

Oosterkamp and co-workers used the same analysis to probe the magnetic moment of a FePt micromagnet [21]. Using the above equation, we can determine the magnetic moment of the Co nanomagnet at zero field, by using a linear fit for the resonance frequency dependency on the field, see the left graph in figure 3.7. From the slope (backward sweep: 0.795 Hz/T, forward sweep: 0.750 Hz/T) we determined that the remanent magnetization equals BR = µ0MR = µ0µ/V = 5.12 T. This value is not physical, since this would

imply that the remanent field is much larger than the saturation field. We contribute this inconsistency to the unreliable way we estimated the spring constant of the cantilever through the Brownian motion.

3.4 Discussion 15 x y z M 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 B (T) x (µm) R = 1.5 µm R = 2.0 µm x z R B -4 -2 0 2 4 -4 -2 0 2 4 z = 1 µm y (µ m) x (µm) 7.2 18 29 40 51 62 72 83 94 B (mT) -4 -2 0 2 4 -4 -2 0 2 4 z = 2 µm y (µ m) x (µm) 6.1 9.9 14 18 21 25 29 33 37 B (mT) -4 -2 0 2 4 -4 -2 0 2 4 z = 5 µm y (µ m) x (µm) 3.1 3.5 3.9 4.3 4.8 5.2 5.6 6.0 6.5 B (mT) -4 -2 0 2 4 -4 -2 0 2 4 z = 10 µm y (µ m) x (µm) 1.02 1.06 1.11 1.15 1.19 1.23 1.28 1.32 1.36 B (mT)

Figure 3.6: Finite element analysis of the magnetic field strength of a NdFeB spherical particle (3 µm diameter) magnetised in the spam-geometry with saturation magnetisation

Bs = 0.77 T. The simulation (circles/squares) agrees well with the analytic solution (solid

lines) for a magnetic field along the magnetisation axis (left top). The calculated field strengths at several planes perpendicular to the particle are shown.

4778.44 4778.46 4778.48 4778.5 4778.52 4778.54 4778.56 -40 -20 0 20 40 fres (Hz) µ0H (mT) M M x spam hang-down 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 B (mT) x (µm) hang-down spam

Figure 3.7: Probing the magnetic moment of the nanomagnet using an alternative method. A fit of equation 3.1 to the cantilever’s resonance frequency versus an applied field at low field strenght’s yields a remanent magnetisation of BR = 0.83 T (left). A finite element

analysis is used to calculate the field of this nanomagnet along its long symmetry axis (right). The simulation is performed for two different orientations of the magnetisation.

Fortunately, we found an alternative direct method to calculate the remanent field:

µR= lim b→0µ = 2k0 f0 ( l α )2 df dB (3.2) µs = 2k0 f0 f0 ( l α )2 ∆f (3.3) BR Bs = µR µs = df dB∆f (3.4)

For Bs = 1.81 T, _dBdf = 0.795 Hz/T and ∆f = 0.579 Hz/T we obtain a remanent field

of BR = 0.83 T. Since we do not us the cantilever’s spring constant k0, we believe that

this analysis is more reliable as the method discussed above. The graph at the right in figure 3.7 shows a simulation (COMSOL Multiphysics) of what field strength we can expect along the long symmetry axis of a 0.83 T magnetised nanomagnet in spam and hang-down geometry. The result is promising: For instance, at a distance of 25 nm we can expect around B = 25 mT for the hang-down geometry and B = 15 mT for the spam-geometry. Since the magnetic field as well as the field gradient (of the order 105 _{T/m) is larger for}

the hang-down geometry, we advise to use this orientation. Assuming a rotation angle

α∼ 0.1◦ and using the equations from [4] to calculate Brf we obtain an rf field strength of

the order mT. This field is sufficiently large to enable ARP MRFM in which the cantilever at a higher resonance mode serves as (the only) rf source.

Chapter 4

Conclusions

To conclude, we have performed a frequency-shift cantilever magnetometry study on an attonewton-sensitivity cantilever with a nanomagnet attached at the cantilever’s free end. As a sanity check, we have performed a similar experiment using a cantilever with cobalt evaporated on the thin edge. For both studies we were able to indirectly determine the magnetic hysteresis loops, although ambiguities exist about their reconstruction from the frequency dependency on the applied magnetic field. Regarding the first study however, we were able to directly extract the nanomagnet’s remanent magnetization of BR= 0.83 T

from the cantilever’s response in resonance frequency at low field strengths. Furthermore, when driving the cantilever at a higher resonance mode, we predict that the nanomagnet’s magnetization is sufficiently large such that an adiabatic rapid passage experiment without the use of an additional rf source such as a rf wire should be possible. Finally, we have shown that it is possible - although very difficult to achieve - to attach a magnetic chip containing a high gradient field cobalt nanomagnet to a cantilever which is much softer than the cantilevers used by Longenecker et al. in 2012. This softer cantilever also contained a magnetic particle close to the cantilever tip. Since the nanomagnet exhibits a high remanent field, this novel method for the mechanical generation of an rf-field does not exclude SQUID-based MRFM that is not compatible with large external fields. The latter is an important result for the Oosterkamp group, who work at low magnetic field strengths with softer, IBM-style magnet-on-cantilevers in combination with a SQUID-based detection of the cantilever’s position.

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I kindly thank the Dan Ralph group for lending us the magnet power supply. We thank Professor Jeevak Parpia for gifting us the 6 T superconducting magnet. We acknowledge the electron microscopy facility of the Cornell Center for Materials Research. The travel costs, visa expenses and other administrative costs that were associated with my stay in Ithaca are financed by scholarships from the Leids Universitair Fonds (LUF LISF grant) and Leiden University (Lustra+ scholarship). More importantly, I would like to express my deep gratitude to my colleague Hoang Long Nguyen, with whom I had the great pleas-ure to work with at Cornell University. Hoang helped me rebooting the magnetometry set-up, a technical challenge that constituted the major part of my master project. Fur-thermore, Hoang prepared the cobalt needle cantilevers and we contributed equally to the corresponding magnetometry experiment. Furthermore, I would like to thank Pamela Nasr for fabricating the cantilevers and for the attachment of the cobalt nanomagnet onto the IBM-style cantilever, a tough task that required lots of patience. Moreover, Martin de Wit helped me preparing a batch of IBM-style cantilevers decorated with a NdFeB particle. I would further like to thank Corinne Isaac for all her advise and support. Corinne also helped me with the magnetometry experiment on the cobalt nanomagnet, which was con-ducted with her experimental set-up. Finally, I would like to thank my supervisors, Tjerk Oosterkamp and John Marohn, for their advise and involvement into this project. I am very grateful that I could experience this internship abroad, and I am thankful to the Marohn group for their warm hospitality.