Advanced Lab Course. E Effect Sensors. 1 Introduction Objective Prerequisites 1

Transcription

1 Advanced Lab Course E Effect Sensors M214 Stand: Juni 2016 Objective: Introduction to the working principle of E effect sensors and generally important sensor characteristics. Contents 1 Introduction Objective Prerequisites 1 2 Theory Repetition of magnetoelasticity E Effect Working principle of E sensor Sensitivity Amplitude modulation Linearity Other Sensor Parameters Total harmonic distortion (THD) Quality factor and losses Signal to Noise Ration (SNR) and Limit of detection (LOD) Bandwidth 9 3 Experiment Sensor Setup Tasks General remarks Working point SNR (change excitation voltage) THD Bandwidth LOD 12 4 References 13

2 1 Introduction Magnetic measurements are used in various fields of application. Especially in geosciences and aerospace they are extensively used to detect the earth magnetic field or magnetic anomalies. Satellites employ flux gate sensors for reference measurements in positioning devices. But also in consumer devices magnetometers are commonly present, e.g. magnetoresistive sensors in reading heads of hard discs or as positioning devices in smart phones. Simple Hall-effect sensors can be used to find underground power lines. In medicine and diagnostics measurements of biomagnetic fields, e.g. for magnetocardiography (MKG) or magnetoenzephalography (MEG) is accompanied with great advantages compared to their electrical counterparts, like a superior spatial resolution [1] or improved sensor array positioning and less exogenous distortions by the patient due to contactless measurement [2]. In medical diagnostics magnetic measurement is still used rarely, because these signals are characterized by amplitudes below 100 pt and low frequency components [3] in the range of 0.1 Hz Hz [4]. This requires high sensitivity sensors with sufficient band width down to the dc range. Additionally, these fields are orders of magnitudes lower than disturbance sources like the earth magnetic field (B earth 50 µt) or fields originating from power cables (B cable 0.1 µt) [5] and other electronic devices. Performing the measurements in a shielded environment is therefore necessary. State-of-the art magnetic sensor systems with satisfying characteristics are mainly based on super-conducting quantum interference devices (SQUIDs) which require liquid Helium/Nitrogen cooling to maintain superconductivity. Consequently, SQUID measurements are expensive and extensive in handling, preventing bedside diagnostics and wide spread utilization [3]. As a promising new approach E effect sensors utilize a magnetic field induced resonance frequency shift, which arises mainly by the E effect in ferromagnetic materials. The E effect describes the non linearity of Young s modulus, which results in a field and stress dependency of the mechanical properties [6]. 1.1 Objective This Lab course serves as an introduction to the working principle of magnetic field sensors based on the E effect. During this course, measurements of low frequency magnetic fields will be performed to investigate sensor characteristics and to become familiar with its properties. 1.2 Prerequisites As this is an advanced lab course, the students must be familiar with the following topics: Piezo electricity, magnetostriction, magnetic anisotropy, magnetic anisotropy energy, Fourier series/transformation, harmonic oscillator, including resonance frequency, damping, phase shift and amplitude modulation. 1

3 2 Theory 2.1 Repetition of magnetoelasticity The E effect is strongly related to magnetoelasticity, which involves all effects with origin in magnetostriction. In practice, often Joule magnetostriction and inverse Joule magnetostriction is relevant, referring to the anisotropic induction of strain in a direction, relative to the axis of magnetization or vice versa (inverse). The anisotropic magnetostrictive strain λ relative to the direction of magnetization may be described for an isotropic material by the relation [7]: λ = ΔL L = 3 2 λ s (cos² θ 1 3 ) (1) This equation holds under certain condition, with λ measured at an angle θ relative to the saturation magnetization direction in which the saturation magnetostrictive strain λ s is measured. So along direction of magnetization the magnetostriction is λ = 3 2 λ s (1 1 3 ) = λ s (2) And perpendicular to the magnetization: λ = 3 2 λ s (0 1 3 ) = λ s 2 (3) If λ is negative, we refer to it as negative Joule magnetostriction, denoting the contraction of the sample in direction of magnetizsation. Origin of this effect is the coupling of atomic magnetic moments to the orientation of atomic orbitals resulting in a different interatomic distance in the magnetized state. Very schematically this is depicted in Fig. 1. Fig. 1 Schematic of magnetic field induced magnetostriction. A change of length L occurs by applying a magnetic field H that rotates the magnetic moment (indicated by arrows) out of the easy axis (EA) 2

4 2.2 E Effect The E effect is fundamental for the working principle of our sensor. It can be defined as the deviation of Young s modulus from Hooks law. Instead of a linear stress strain response a nonlinear curve is observed. This phenomenon occurs due to stress induced anisotropy, which rotates the magnetization while the material is strained. The rotated magnetization results in an additional magnetostrictive strain, which adds up to the linear elastic Hookean strain. If we define the Young s modulus as the slope of the stress strain curve, it is therefore not equivalent to the interatomic spring constant [8]. Not only stress, but also a magnetic field can rotate the magnetization of a ferromagnetic material. Thereby the non linearity of the stress strain curve is a function of applied field, too. A simple example is provided in Fig. 2. In magnetic saturation (H = H sat ) an applied stress cannot induce an additional magnetostrictive strain, therefore the stress strain curve is linear and equal to the Hookean response. Below magnetic saturation (H = 0), stress can induce magnetostrictive strain, which results in a nonlinear stress strain curve. With increasing stress the material is successively magnetized until it is saturated by the applied stress. Then the stress strain curve is linear again and the Young s modulus is constant. Fig. 2: Stress-Strain curves for a tensile test with a sample magnetized up to saturation in stress direction (H = H sat ) and with no magnetic field applied (H = 0). Since no additional magnetization can occur in magnetic saturation the behavior is linear for the first case. If no magnetic field is applied the magnetization continuously rotates with increasing strain by inverse magnetostriction, resulting in an increasing Young s modulus. Depending on the material, magnetostrictive strains are in order of 10 6 up to 10 3, where the largest strains are observed for rare earth elements [9]. Consequently, the effect is only relevant for small deformations, because large strains saturate the material magnetically. The principle course of Young s modulus with magnetic field can be qualitatively derived from the hysteresis curve, if the influence of stress is known. Consider the hard axis magnetization process that is completely dominated by the rotation of magnetization. In this case the absolute M of the magnetization vector M equals the saturation magnetization M s. The 3

5 component M of M, which is measured in direction of the applied magnetic field is given by the direction cosine of M to that direction, so it is: cos θ = M M s (4) From (1) the magnetostrictive strain is proportional to the square of magnetization: λ = 3 2 λ s (cos² θ 1 ) M² (5) 3 The total strain ε that occurs for a certain applied stress σ can be composed of a superposition of stress induced magnetostrictive strain λ and linear Hookean strain e: ε = e + λ (6) Therewith an effective Young s modulus E can be defined as the derivative of applied stress with respect to the total strain ε. Using the inverse, it is: 1 E = ε (e + λ) = σ σ (7) Note that e/ σ is simply the constant Hookean Young s modulus E m, which can be measured in saturation. Substituting E m and λ yields: 1 E = ε σ = 1 + λ E m σ 1 + M2 E m σ (8) Consequently E is reduced relative to E m, if M²/ σ > 0. A schematic course of M(H) is sketched in Fig. 3 for a hard axis magnetization at σ = 0 and σ > 0. Fig. 3: Schematic course of magnetization for different applied stresses. A tensile stress tilts the magnetization curve due to a reduced effective anisotropy energy in stress direction. This results in a positive M². Because M 2 > 0 reduces E, the expected course of Young s modulus as function of magnetic field is similar to M 2 as indicated by the dashed line. 4

6 For σ > 0 the effective anisotropy energy density in stress direction is reduced, which results in a M 2 > 0. From (8) the principle course of Young s modulus is expected to follow qualitatively M 2. This results in a w shaped course as indicated by the dashed line in Fig. 3. Note that M 2 = 0 at H = 0 because the magnetization curve tilts around this point. This is only true for an exact hard axis magnetization process and if no domain wall motion is possible. In all other cases the magnetization can also be changed by a stress at H = 0. Then the center maximum at H = 0 is not at M = 0 but at M < 0. Consequently, this simple qualitative consideration must not be overinterpreted. For a quantitative description, magnetic models must be used to obtain λ/ σ, which is beyond the scope of this manual. 2.3 Working principle of E sensor The sensor consists of a hetero structure of magnetoelastic and piezoelectric material on a silicon cantilevers (see section 3.1 and 3.2 for the detailed setup). By field annealing a uniaxial anisotropy is induced with magnetic easy axis oriented perpendicular to the cantilever. For operating the sensor, a sinusoidal voltage is applied to the piezoelectric layer, exciting the cantilever to oscillate at f 0. Utilizing the same electrodes for readout a current with characteristic amplitude is measured. This resonance curve differs from the course of a typical mechanical resonance. It is called electromechanical resonance. The typical course of amplitude as function of frequency is shown in Fig. 4 (left). Fig. 4: Amplitude of measured current vs. excitation frequency applied to the piezoelectric layer (left); Shift of resonance curve by application of a (DC or AC) magnetic field at constant excitation frequency results in lower measured current amplitude (right) Apparent from Fig. 4 the electromechanical resonance curve obeys a resonance - antiresonance behavior, which indicates the presence of coupled oscillators. The resonator can be modeled by an equivalent circuit of two coupled parallel oscillators (BVD-Model). One is a series LCR circuit to describe the mechanical resonance, the other an RC parallel circuit to describe the piezoelectric contribution [6]. This oscillator circuit exhibits capacitive behavior below the resonance frequency and inductive behavior above. Upon application of a DC magnetic field the effective Young s modulus changes according to Fig. 3 due to the E effect. Thereby the resonance frequency of the device changes and a different amplitude is measured as schematically indicated in Fig. 4 (right). The change of resonance frequency that occurs if E changes can be understood considering a simple harmonic oscillator: 5

7 f 0 = 1 2π k E (9) m with the effective Young s modulus E the spring constant k and equivalent mass m. Consistently the course of resonance frequency with magnetic field (Fig. 5) is similar to the change of Young s modulus, w shaped. This does not change principally when applying an alternating magnetic field. The AC field H ac (t), necessarily results in a time dependent current amplitude, depending on the momentary value of H ac (t) and leads to an oscillation of current amplitude around the amplitude measured without H ac (t). Fig. 5: Resonance frequency of the device as function of applied magnetic field, originating directly from the induced change of Young s modulus. At the working point (WP) the sensitivity of the sensor is at its maximum. It is fixed by the point of highest slope and defines the optimum excitation frequency f r,w and optimum bias field H w. The reduced maximum indicates domain wall motion Sensitivity The two functions in Fig. 4 and Fig. 5 are of fundamental importance, since they determine together the output signal of the sensor for a given input signal. In this context, it is appropriate to define a sensor parameter referred to as the sensitivity S. The sensitivity can be defined as the change (da ) of output signal amplitude with physical input parameter (dh) S = da dh = df R dh da WP f R (H w ) df A(f r,w ) (10) Where df R da is called the magnetic sensitivity and the frequency sensitivity, respectively. dh df R Apparently, the highest change of output amplitude da with change of magnetic field strength 6

8 dh is obtained, when both factors become maximum. The frequency sensitivity, as slope of current amplitude over excitation frequency is maximum at the electromechanical resonance frequency f 0. Therefore, the optimum excitation frequency is chosen to be f r,w = f 0 always. However, since it is a function of H it convenient to choose the magnetic bias field H w at which the magnetic sensitivity is maximum, first. As indicated in Fig. 5 the overall working point (WP) is then defined by H w and f r,w. Thus, in the experimental part we will first determine the resonance frequency and the working point, before other sensor characteristics will be determined. 2.4 Amplitude modulation Operating the sensor, a sinusoidal excitation voltage of (circular) frequency ω is applied to the piezoelectric layer. If no additional time dependent magnetic field is applied, it results in a measured sensor output current A C (t) = A C cos (ωt), presuming linear piezoelectric response. This signal is shown schematically in Fig. 6 (left, blue curve). It is called carrier in terms of signal processing nomenclature. Its amplitude A C is proportional to the amplitude of the excitation signal and a function of the (carrier) frequency as described by the electromechanical resonance curve (Fig. 4). Upon application of a lower frequency magnetic AC field H(t), the amplitude of the carrier signal A C is modulated by a signal A inf (t) that originates from the field. The resulting new, modulated signal A(t) is measured at the sensor output. It is schematically depicted in Fig. 6 (right). Fig. 6: Carrier signal (blue) and information signal (red) (left) superpose according to (11) and form the modulated signal (right). The modulated signal A(t) is mathematically obtained by application of (11): A(t) = [A C + A inf (t)] cos(ωt) = A (t) cos(ωt) (11) where A C (t) = A C cos(ωt) is the carrier signal and A inf (t) the periodic modulating (information) signal. Details about A inf (t) are discussed in the next chapter. However, it should be evident that A inf (t) is simply the additional amplitude at time t, added to the amplitude of the carrier signal. This results in a time dependency of the amplitude A (t) of the measured signal A(t). To point it out: The information signal A inf (t) is a direct consequence of the applied alternating magnetic field H(t). They are related by the materials physics, but are still completely different quantities and may generally differ a lot regarding their 7

9 respective appearance in time and frequency domain. According to (10) it is at the working point: da inf (t) = da (t) = S dh(t) (12) For small magnetic fields at the working point, the sensitivity S is approximately constant, thus integrating yields: A inf (t) = ΔA (t) S H(t) (13) If S is independent of H(t), then A inf (t) is a linear function of H(t). Referring to (13), this is approximately valid for small fields, if the course of f R (H) and A (f R ) can be approximated as linear functions around the respective working point. The influence of the linearity of the information signal in H(t) on the output signal A(t) is discussed in the subsequent section Linearity Consider the frequency domain of A(t) by taking (11) and assuming A inf (t) to be a pure cosine function: Using A(t) = [A C + A inf cos(θt)] cos(ωt) = A C cos(ωt) + A inf cos(θt) cos(ωt) (14) it follows: cos(θt) cos(ωt) = 1 (cos([ω θ]t) + cos([ω + θ]t)) (15) 2 A(t) = A C cos(ωt) + A inf 2 (cos([ω θ]t) + cos([ω + θ]t)) (16) According to (16) the signal in frequency domain has to be composed of at least three peaks shown in Fig. 7 (left). One center peak at the frequency ω corresponding to the carrier signal and two side peaks at [ω θ] and [ω + θ] from the modulating signal A inf (t). In the general case, however, A inf (t) is not a pure sine function due to nonlinearities in S. Expressing it as a Fourier series will result in sinusoidal contribution of higher order. Thus, additional peaks occur in the spectrum as pictured in Fig. 7 (right). Fig. 7: Power density spectrum of a carrier signal with frequency ω modulated by an information signal with frequency θ (left) and additional harmonics from nonlinearities of the modulating signals in H (right) 8

10 2.5 Other Sensor Parameters Total harmonic distortion (THD) The linearity is an important sensor characteristic. As an appropriate measure the (total) harmonic distortion (THD) provides an appropriate quantification for nonlinearities. There are many different definitions, depending on the field of application and area of expertise. For the purpose of this lab course, the harmonic distortion is used and defined as THD db,i = 10 log 10 ( P i P 1 ) = P i,db P 1,dB (17) P 1 is the peak height at the basis frequency and P i the peak height at the i th harmonic in the power density spectrum (Fig. 7). During the lab course this will be determined and discussed with respect to applications Quality factor and losses The Q-factor is a measure for the energy loss of a resonator system at resonance frequency. It is defined by the energy E max stored per cycle by the energy loss E loss per cycle and is defined as: Q = 2π E max E loss f c FWHM (18) The approximation using the peak center frequency f 0 and the full width at half maximum is only valid for small losses. Since the decay time of an oscillation is a function of the losses, the quality factor can be related to the decay constant τ and the number n cycles of periods to decay Q = π n cycles = π τ T = πf 0τ (19) Signal to Noise Ration (SNR) and Limit of detection (LOD) The SNR is a measure for the relative strength of desired signal and defined as the ratio of signal power to mean noise power. Since we measure all the power in db (20) serves as an appropriate definition. SNR db = 10 log 10 P signal P noise = P signal,db P noise,db (20) The magnetic field at which SNR = 1 is referred to as the limit of detection (LOD), since magnetic fields with lower amplitudes cannot be distinguished from the measured background noise Bandwidth For this sensor, the bandwidth f is defined by the frequency, at which the measured amplitude is decreased by 3 db, called the 3dB cut-off frequency f 3dB. f = f 3dB (21) 9

11 3 Experiment 3.1 Sensor The E effect sensor used during the lab course is based on a polycrystalline silicon cantilever with piezoelectric aluminum nitride (AlN) deposited on top. As sketched in Fig. 8 it is connected to the circuit by two gold/platinum electrodes. For the magnetoelastic component, deposited on the bottom, amorphous FeCoSiB is used. As a metallic glass it exhibits a large quality factor, due to less eddy current losses and enhanced elastic properties. Compared with other highly magnetostrictive materials the lack of crystalline anisotropy energy results in a low total anisotropy energy and therefore in a strong E effect [9]. Fig. 8: Structure and composition of the E effect thin film sensors. The cantilever is excited by an alternating external voltage U ex to oscillate. A magnetic easy axis (EA) is induced along the short axis. The mechanical behavior is dominated by the substrate with about 50µm thickness. The thickness of all other layers is in the range of a few µm. This is advantageous due to the large quality factor of the substrate. Because the electric and magnetic response of the sensor originate from the piezoelectric and magnetostrictive layer, their volume fraction is a decisive factor contributing to the frequency sensitivity and the magnetic sensitivity. [6] 3.2 Setup During the experiment, the sensor is placed in the center of two concentrically arranged solenoids, which are used to apply an alternating and a static magnetic field, respectively. Conversion factors between solenoid current and magnetic field are listed in following chart. The complete setup is magnetically shielded with Permalloy foil. Table 1: Conversion factors for solenoids between current and flux density Solenoid Conversion factor /mta 1 DC 7 AC 0.89 As signal source and readout the inputs and outputs of an interface commonly used for audio recording can be triggered by a MatLab script. Details will be provided by the supervisor. As sketched in Fig. 9 output out1 provides the excitation voltage for the piezoelectric layer, whereas out2 is connected to the solenoid to produce the alternating magnetic field. 10

12 Fig. 9: Schematic representation of the experimental setup. Coils and Input/Outputs are connected to the same ground. To apply a magnetic bias field H bias (or H w at the optimum WP) an external power supply is used connected to the second solenoid. Both enclose a tube in which the sensor is placed. The sensors response is recorded as a voltage at Input1, which is measured across the input resistance and therefore it is proportional to the current or the total admittance of the circuit. In the equivalent circuit (Fig. 10), the sensor is represented as a capacitance with admittance adjustable by an external magnetic field. Fig. 10: Equivalent circuit of the sensor - interface connection via output channel 1 (bottom) and input channel 1 (top). The voltage is measured across the input resistance R In1 at the input. 11

13 3.3 Tasks General remarks All graphs have to be scaled appropriately. All results should be discussed. Plots including magnetic field amplitudes should be plotted in mt. All voltages in MatLab are denoted as voltages U rel relative to the maximum output voltage. Therefore amplitudes > 1 must not be applied to avoid overdriving the system! The necessary conversion factors will be provided by the supervisor, as well as all necessary factors for calculating the magnetic field amplitudes. All amplitudes without units are relative amplitudes Working point Record the resonance curve for different applied H bias fields. Therefore, apply an excitation voltage of U exc = 0.2 and increase the bias voltage U bias from 0 V up to 12 V. a) Measure the resonance curve for each voltage step and denote the approximated resonance frequency. b) Plot f r vs. U bias during the measurement to determine the working point c) Now apply H ac frequency f ac = 10 Hz and an amplitude A ac = Record a spectrum and adjust U bias until the spectrum is mirror symmetric around the center peak. Perform all subsequent measurements at the working point you determined above SNR a) Apply U exc = 0.01 and U exc = 0.05 and measure the resonance curve for both excitation amplitudes. b) Record spectra with U exc increased from 0.1 up to 0.9. Measure the peak height of the first harmonics relative to the noise level. Evaluation: (a) Calculate the frequency sensitivities and compare. (b) Plot SNR over excitation amplitude and determine the optimum excitation amplitude THD Increase A ac from 0.1 up to 0.9 and measure the spectra. Record the spectra and peak values of all occurring harmonics and the respective noise level. Evaluation: Calculate the THD for each measurement and plot it vs. the magnetic field Bandwidth Start with a H ac frequency of 1 Hz and increase it stepwise to 200 Hz. Determine the peak height of the first sidebands and respective noise level from the spectrum for each frequency. Evaluation: Plot the signal amplitude vs. f ac and determine the bandwidth. Use an appropriate scale! LOD Measure the signal amplitude for decreasing amplitude of H ac by recording frequency spectra for each amplitude value. Evaluation: Plot the SNR vs. amplitude of H ac and determine the LOD 12

14 4 References [1] P. W. Macfarlane, A. van Oosterom, O. Pahlm, P.Kligfield, M. Janse, and J. Camm, Eds., Comprehensive Electrocardiology. Springer, [2] H. Koch, Recent advances in magnetocardiography, Journal of Electrocardiology, vol. 37, Supplement, pp [3] J. Clarke and A. I. Braginski, The SQUID Handbook Vol II: Applications of SQUIDs and SQUID Systems. Weinheim: Wiley VCH, 2006 [4] S. Baillet, J. C. Mosher, and R. M. Leathy, IEEE Signal Process. Mag. 18, [5] O. Dössel, Bildgebende Verfahren in der Medizin,,Springer-Verlag; Kap [6] S. Zabel, C. Kirchhof, E. Yarar, D. Meyners, E. Quandt, and F. Faupel, Phase modulated magnetoelectric delta-e effect sensor for sub-nano tesla magnetic fields, Applied Physics Letters 107, (2015) [7] R. C. O Handley, Modern Magnetic Materials Principles and Applications, Chapter 7.1 Wiley, 2002 [8] R. C. O Handley, Modern Magnetic Materials Principles and Applications, Chapter 7.4 Wiley, 2002 [9] A. Ludwig and E. Quandt, Optimization of the E Effect in Thin Films and Multilayers by Magnetic Field Annealing, IEEE Transactions on Magnetics, Vol. 38, No. 5, 2002 [10] J. D. Livingston, Magnetomechanical properties of amorphous metals, Phys. Stat. Sol., vol. A 79, pp ,