Nanotesla torque magnetometry using a microcantilever

NANOTESLA TORQUE MAGNETOMETRY USING A MICROCANTILEVER

J.W. van Honschoten, W. W. Koelmans, S. M. Konings, L. Abelmann, M. Elwenspoek

1

MESA+ Research Institute, University of Twente, PO 217, Enschede, The Netherlands *Corresponding author: J.W. van Honschoten, Phone: (+31)534894438, Fax: (+31)534893343,

j.w.vanhonschoten@utwente.nl

Abstract: We present a novel ultrasensitive magnetometry technique using a micromachined magnetic

cantilever that is brought in resonance. The induced magnetic moment generates a torque on the cantilever, thereby effectively stiffening the cantilever spring constant and changing its resonance frequency. Experiments are in good correspondence with the presented analytical model for this frequency shift, predicting the detection of nanotesla magnetic fields.

Keywords: magnetometry, microcantilevers, magnetic sensors

INTRODUCTION

Microcantilevers have been shown to be highly sensitive sensors in a wide variety of applications, like high sensitivity calorimetry [1], chemical reaction rate monitoring [2], optical detection [3], and magnetostrictive magnetometry [4]. In cantilever magnetometry, a magnetic material is deposited on the cantilever. Rossel et al. [5] realized a miniature torque magnetometer in which the magnetic material to be studied is placed on the end of a microcantilever and where the sensor is placed in a large external magnetic field. By applying the magnetic field, a torque is exerted on the material (and hence on the cantilever) which is proportional to the total magnetic moment of the material. In order to measure very small fields, a large magnetic moment of this material is required. Cowburn et al. [6] used a similar technique to measure very low magnetic fields using microcantilevers and were able to detect changes in applied magnetic field of about 10 nT. However, these measurements were also performed in the presence of a large external field, that was flipped continuously to eliminate drift.

Cantilever torque magnetometry has, apart from the direct detection of magnetic fields, various other applications. Examples are the measurement of the magnetic moment of small (anisotropic) samples [5,7-11], detecting the magnetic dissipation and fluctuation in nanomagnets [12], the determination of the magnetoelastic coupling constants of thin films [13], and in situ monitoring of magnetic film thickness [14]. An important application is the measurement of small planetary magnetic fields in microsatellites. Currently, in satellites such as for example the Mars Global Surveyor spacecraft [15-17], fluxgate magnetometers [18] are employed for this purpose. The used magnetometers have an accuracy of several nanotesla’s [17], a power consumption of hundreds of milliwatts and a characteristic size of one centimeter [19]. For future space missions with microsatellites, the demands on the size and power consumption of the sensors are more strict and very small magnetic field sensors are required.

In this paper we present an innovative magnetometry technique that allows for a very small, low-power and ultrasensitive magnetic field sensor. The presented magnetometry is based on the principle of actuating a magnetized cantilever to

resonance. The induced magnetic moment generates a restoring torque on the oscillating cantilever, that effectively stiffens the cantilever and thus changes its spring constant. This results in a shift of its resonance frequency, which can be measured accurately.

Two different types of silicon cantilevers, of 500x100x1 µm and 225x28x3 µm respectively, with cobalt-nickel layers of 150 nm thickness on top, are fabricated and the cantilevers’ shifts in resonance frequencies are determined as a function of applied magnetic field in a Helmholtz coil configuration. We present a model to describe the behaviour of the microcantilevers, and the observed frequency shifts are found to be linearly proportional to the applied field and in correspondence with the theoretical predictions.

Figure 1. SEM photograph of the microcantilever.

THEORY

To interpret the magnetometry data, we model the vibrating cantilever as a thin beam of length L with a thin film of magnetic material on it. The cantilever is directed along the x-axis and vibrates with a displacement z(x). The magnetic layer has a strong magnetic anisotropy, directed in the direction of the cantilever, along the x-axis, and which is assumed to have a magnetic moment equal to the saturation magnetisation Ms. When the cantilever is placed in

a uniform magnetic field, the induced magnetic moment generates a restoring force on the cantilever that effectively stiffens the cantilever spring constant, thereby influencing the cantilever's eigenfrequency of vibration. To calculate the frequency shift, we consider the magnetic field of strength H that is applied in line with the x-axis. Because of the applied field, the magnetic moment of the material cants away from the original direction of the cantilever by an angle φ, when the

cantilever vibrates. The canting of magnetisation follows from considering the total magnetic energy of the magnet. This magnetic energy is the sum of two terms: the anisotropy energy, which is positive, and the Zeeman energy [12, 21] being negative. The total magnetic energy of the thin layer, per unit length, can then be written as

)

cos(

sin

)

(

_x

=

₂1

µ

₀

_M

2

_A

2

φ

−

µ

₀

_HM

_A

θ

−

φ

E

_{m s s} (1) where A denotes the area of cross section of the film on the cantilever, φ the angle that the magnetic moment of this magnetic material cants away from the original direction of the cantilever due to the applied field, and θ=θ (x) the angle of the applied magnetic field H with the cantilever. Because of the vibrational mode shape of the cantilever, both θ and φ depend on x. Minimisation of Em with respect

to φ yields

)

(

)

(

x

x

s M H H

θ

φ

=

₊ (2)

The vibrational mode shapes of the cantilever can be found from the solutions of the one-dimensional bending equation [22, 23], so that we can write an explicit expression for θ (x). The first flexural mode of the oscillations is the vibrational mode we are specifically interested in. The canting of magnetisation φ(x) depends then on the local angle between the magnetic field and the cantilever θ(x) according to Eq.(2).

The torque τ (x) that a point at place x along the cantilever is subject to, can be found from differentiating the magnetic energy Em, Eq.(1), to θ.

Using the small angle approximation, this torque can be written as

)

(

)

(

x

0

M

2

A

H M_s

x

H s

θ

µ

τ

=

+ (3)

The total torque T that the magnetized cantilever experiences, is then calculated as

( )

H M

( )

L H s L M H H s s s

V

M

dx

x

A

M

T

δ

µ

θ

µ

+ +

=

=

=

2 0 0 2 0

785

.

0

)

(

(4)

with

δ

the deflection at the end and V the total volume of the magnetic material. The last step in Eq. (4) follows from evaluation of the integral over x using the explicit expression for θ(x) for the first vibrational mode.

This torque effectively stiffens the spring constant

k0 by ∆k =1/Leff⋅∂T/∂δ. Here Leff denotes the

effective cantilever length, that differs from the actual length L and depends on the vibrational mode shape. For the first flexural mode,

Leff=0.725L. Then for ∆k

/

k0 << 1, the change in

resonance frequency, ∆ω, is related to the change in spring constant as ∆ω / ω0=

1/2

∆k

/

k0

, so that

( )

s s M H H Lk M

A

₊

=

∆

₀2 0 0

084

.

1

µ

ω

ω

(5)

with ω0 the resonance frequency at H = 0. For the

current dimensions and parameters of the microcantilevers this yields ∆ω/B= 0.33 Hz/mT, 1.79 Hz/mT, and 3.65 Hz/mT, for cantilever type 1,

2 and 3, respectively (see Table 1). We used B =

µoH.

A second important point to consider is the energy dissipation during operation of the cantilever. For dynamic-mode cantilever magnetometry the energy dissipation is an important subject since it determines the limit for the minimum detectable magnetic field and provides insight into the spectral behaviour of the noise. For a damped harmonic oscillator, which is a good model for a vibrating cantilever beam in the absence of magnetic forces, the damping coefficient Γ is directly related to the energy loss due to friction during one oscillation. If, in addition, a magnetic field is applied and the system is subject to magnetic forces, energy is also dissipated because of the canting of the magnetisation in the magnetic material during oscillation [12]. We can therefore write

m Γ + Γ = Γ 0 (6)

where Γ0 is the damping coefficient of the

cantilever due to mechanical friction while Γm

denotes the damping coefficient as a result of the canting of the magnetisation in the material. If the quality factor Q of the cantilever is measured, Γ0

can be found from Γ0=ω/Q (k0/ω0 2

) with k0/ω0 2

the effective mass, which is constant for all vibration modes [12, 24]. For the magnetic-induced damping, we consider the first term in Eq.(1) representing the anisotropy energy. It has been found by Stipe et al. [12] that the anisotropy energy that is lost during one cycle is proportional to φmax2,

with φmax the maximum angle that the

magnetisation cants as the cantilever oscillates. With

ε

a dimensionless quantity that represents the fraction of the peak anisotropy energy that is lost per cycle, and using the fact that the relation between dissipated energy per cycle and the damping coefficient is ∆E=

π

ω

δ

2

Γ

,one obtains

( )

2 2 2 s eff s M H H L V M m

=

+

Γ

ε

_πω (7)

The parameter

ε

is in principle frequency dependent and has a value of 0.1–0.2 in the frequency range of interest.

As described in the next section, the cantilever vibration is detected optically by a laser beam that is reflected off the backside of the cantilever and directed into a photodiode. In this type of detection, shot noise will occur [20, 23], while also other noise sources related to the optical detection method like fluctuations in the intensity of the laser contribute to the final accuracy of the measurement. However, we concentrate here on the noise spectral density of the cantilever itself. The highest resolution that in principle can be obtained is limited by the mechanical noise of the cantilever due to thermal fluctuations and excitations. For the thermal mechanical noise, the random fluctuations in

δ

have a magnitude given by

0 0 4 2 ω π ω

δ

>=

_QkkBTB

<

(8)

with kB the Boltzmann factor, T the temperature, Q

the quality factor of the system and B_ω the frequency bandwidth of the measurement. In accordance with the fluctuation dissipation theorem, the magnetic dissipation related to the anisotropy energy that we considered above, will also lead to cantilever excitations. Since the spectral density of the involved force variations is given by [20] <F2>= 4Γmk

2

kBT, the equivalent

deflection spectral density of magnitude is <

δ

2>= 4Γmk

2

kBT, which becomes, with Eq. (7), for

small values of H: 2 0 2 0 2 0 2

2

eff B L T Vk k H πω

µ

δ

>=

<

(8)

That is, the magnetic dissipation leads to a noise source with a power spectrum having a 1/f frequency dependence. In the current measurement technique, the shift of the resonance frequency results from the force gradient that the cantilever measures. It was shown before [25-27] that the minimum detectable force gradient for a system at resonance is given by ∂F/∂

δ

=√(k0

2

<

δ

2>/<

δ

0 2

>), with <

δ

2>, as before, the mean square amplitude of the cantilever displacement due to noise, and <

δ

0

2

> the mean-square amplitude of the self-oscillating cantilever. For the minimal detectable frequency shift (∆

ω)

min

we have (∆

ω/ω

0

)

min=1/(2k0)

⋅

(∂F/∂

δ

)

min , so that

we obtain for the smallest detectable change in resonance frequency

( )

(

2

)

2 3 0 0 0 2 0 0 0 min 2 eff B L V H k Q B k T k ω µ ω δ π ωω

=

_{< >} ω

+

∆ (9) Evaluating Eq. (9) for the current situation, we obtain a minimum detectable frequency shift of ∆

ω

min

/2

π

= 1.80⋅ 10

-6

Hz. For the cantilevers 1,2, and 3, this corresponds to a minimum detectable field Bmin of 5.45 nT, 1.01 nT, and 0.492 nT,

respectively. See table 2.

EXPERIMENTS

Three types of cantilevers were fabricated, starting from (non-magnetic) single crystal silicon cantilevers of different dimensions on which a magnetic layer was deposited. For this coating, a cobalt nickel alloy (80 : 20) was chosen, because of the high saturation magnetization of this material in combination with its wear- and corrosion resistance. Table 1 lists the dimensions and parameters of the three series of cantilevers. The set-up to perform the magnetometry measurements on these cantilevers consisted of an optical deflection and detection system, a vacuum chamber, a cantilever holder and a Helmholtz coil. For the optical deflection detection, the laser beam was focused via a set of mirrors and lenses on the cantilever, reflected, and detected on a four quadrant photodiode by means of a differential measurement technique. A lock-in amplifier was used to drive the cantilever vibration by means of a piezo element. To achieve a low damping (large ‘Q-

Table 1. Cantilever dimensions (width b, length l and height h) and layer thickness tL for the cantilevers types 1, 2 and 3. The used amplitude yosc is a maximum for linear behavior.

b

h

l

L

t

(1)

For the cantilevers with partial coverage, the effective layer thickness is calculated by including the covered area.

Table 2. Sensitivity of the three cantilever types, and the minimum detectable field Bmin based on the thermal noise of the cantilevers only, according to the fluctuation dissipation theorem (Eq. (9)).

Sensitivity B ω π ∆ 2 1 (103 Hz/T) 0.33 1.79 3.65 (nT)

factor’) of the resonating cantilever, the cantilever is placed into a vacuum chamber in which the pressure was reduced down to 0.01 mbar or less, with a thick glass window on top, transparent for the used laser wavelength. An external magnetic field was generated by a pair of aligned Helmholtz coils, and could be tuned accurately by varying the current through the coil wires. A typical resonance curve of the cantilever vibration that was thus measured, together with the shifted curve due to the applied magnetic field, is shown in the inset of figure 2. To investigate possible drift in the measurements, consecutive frequency sweeps are performed at different field strengths. As the overlapping curves in Fig. 3 illustrate, the experiment’s stability was high: the difference in measured resonance frequency at the first and last sweep, at the original field strength, is below 0.01 Hz. The accuracy of the entire measurement is limited by the optical detection system, which can in principle be significantly improved. The noise caused by the used optical deflection detection with the laser beam, the photodiode and electronic amplifier largely surpasses the thermal cantilever noise calculated above.

Figure 2 shows the measured frequency shift for cantilever 2 as a function of field strength. Graphs for the other cantilever types are similar. Clearly, the dependence is in very good approximation linear, as expected from theory. Besides, the quantitative dependence of ∆

ω

as a function of applied B-field is also in correspondence with theoretical calculations, as illustrated by the calculated linear dependence of 1.79 Hz/mT in the figure (dotted line).

5190

5195

5200

5205

5210

5215

-5

0

5

10

B (millitesla)

R

e

s

o

n

a

n

c

e

F

re

q

u

e

n

c

y

(

H

z

)

Figure 2. Measured resonance frequency of the microcantilever vs. B-field. The theoretical dependence is represented by the dotted line. The inset shows a typical resonance curve as a function of frequency : for cantilever 2 at a field of 193 mT.

Figure 3. Three consecutive frequency sweep measurements, on cantilever 1, where the second sweep was performed at a difference in field strength of 500 µT, causing a shift in resonance frequency of 0.17 Hz. The measurement was performed at a pressure of 0.019 mbar and a cantilever driving signal of 4mV rms.

SUMMARY AND CONCLUSIONS

We fabricated three different microcantilevers of varying thickness and length with a thin magnetic layer, that were actuated piezo-electrically and placed in an external magnetic field. The shift in their resonance frequencies due to the applied field could accurately be measured using an optical beam deflection detection method.The observed frequency shift were found to be in good qualitative and quantitative correspondence with the analytical model. Magnetic fields of a few µT could be detected, where the resolution was limited by the detector noise. However, on account of the limitation from the cantilever thermal noise only, the detection of fields down to 1 nT becomesfeasible.

We have thus fabricated an ultrasensitive magnetic sensor of micrometer size, that allows for applications in for example microsatellites, especially in arrays of cantilevers.

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