Influence of the piezoelectric material on the signal and noise of magnetoelectric magnetic field sensors based on the delta-E effect

Detecting small amplitude and low-frequency magnetic fields is of high interest in the field of biomagnetics and opens promising prospects for potential medical diagnostics and therapies.^1–3 Thin-film magnetoelectric composite sensors have been investigated for such purposes.^4–7 They can be downscaled with standard microelectromechanical system (MEMS) technology and are compatible with complementary metal–oxide–semiconductor (CMOS) technology. Combining with the possibility of room-temperature operation could make them attractive alternatives^8,9 to traditionally used sensor types¹⁰ and atomic magnetometers^11,12 in the future.

Detection limits in the range of few $pT/Hz$¹³ can be achieved with the direct magnetoelectric effect yet only in a small bandwidth of a few Hz around the resonance frequency of the device. One way to overcome this hurdle is the utilization of modulation techniques such as the delta-E effect read-out scheme.¹⁴ It has widely been demonstrated in numerous studies^15–27 and has shown the potential of detecting low-frequency magnetic fields with typical signal amplitudes in the pT and nT regimes.

MEMS cantilever delta-E effect sensors were experimentally and theoretically analyzed regarding geometry and sensitivity,²⁸ the influence of the electrode design,²⁶ frequency effects,²⁹ signal and noise,^30,31 and how it is influenced by the quality factor.³² Recently, they enabled the detection of magnetically labeled cells.³³ Such delta-E effect sensors consist of magnetoelectric thin-film composites with a soft-magnetic material and a piezoelectric layer. Whereas macroscopic delta-E effect sensors^22,27,34 often use PZT as a piezoelectric material, the established material in MEMS technology is aluminum nitride (AlN).

Recently, it was shown that the addition of scandium can increase the piezoelectric coefficient by more than a factor of 2.^35,36 This caused a corresponding increase in the signal in direct magnetoelectric operation³⁷ by a factor of ∼2. Because the sensor intrinsic thermal–mechanical noise in resonance increases accordingly, no improvement in the limit of detection (LOD) was achieved.

In contrast to conventional magnetoelectric sensors, the signal and noise of delta-E effect sensors behave differently and have not been investigated with respect to the properties of the piezoelectric material, yet.

In this work, we identify and discuss the influence of the piezoelectric material on the sensing characteristics of delta-E effect magnetic field sensors. As an example, two sensors with AlScN and AlN as the respective piezoelectric material are compared. After introducing the sensors in section Sensor Design and Materials, the delta-E effect read-out concept is reviewed to define the quantities that are used throughout this paper. Measurement results are then presented and analyzed using an electromechanical finite element model and a signal-and-noise model. Finally, we focus the analysis on the influence of the piezoelectric coefficients on the signal and noise to identify general trends for the LOD.

A schematic cross section of the sensors investigated here and a photograph are shown in Fig. 1. The sensors consist of a 5 µm thick poly-silicon cantilever with targeted dimensions of 1000 × 200 µm². It is sandwiched between two SiO₂ layers with thicknesses of 990 nm (bottom) and 650 nm (top). A 1 µm thick piezoelectric layer (AlN or Al_0.73Sc_0.27N) on top is covered by a 1 µm thick Si₃N₄ passivation layer. The piezoelectric layer is contacted via a 120 nm thin Pt/Ti bottom electrode that extends over the whole cantilever. From the top, through the Si₃N₄ layer, a 4 µm thick Au contact line is connected to a 150 nm thin Mo top electrode. A 2 µm thick magnetostrictive layer of amorphous (Fe₉₀Co₁₀)₇₈Si₁₂B₁₀ (FeCoSiB) is deposited on Si₃N₄, also covering the Au contact line. A magnetic DC field (≈10 mT) was applied during the deposition to induce a magnetic easy axis along the short axis of the cantilever. In a final step, the cantilevers are encapsulated with a wafer-level AuSn transient-liquid-phase (TLP) bonding.³⁸ Except for the deposition process of AlN and Al_0.73Sc_0.27N layers, the design and the fabrication steps are identical for both sensors. In this work, the piezoelectric layers were grown by a reactive pulsed direct current co-sputter deposition, described in Ref. 35. An overview of the layer dimensions is given in the supplementary material, and a detailed description of the fabrication process can be found in Ref. 39. In all following discussions and simulations, we consider the cantilevers to be oriented lengthwise in the x₁-direction and widthwise along the x₂-direction of a right-handed Cartesian coordinate system.

During the delta-E operation, the cantilever is electrically excited to oscillate. For that, a sinusoidal voltage u_ex with excitation frequency f_ex and amplitude $ûex$ is applied to the piezoelectric layer. A magnetic field alters the sensor’s mechanical resonance frequency f_r via the delta-E effect,^40,41 which shifts the sensor’s admittance characteristic Y(f_ex) on its frequency axis. An alternating magnetic field consequently modulates the current through the sensor. It is measured as a voltage $ut$ over time t by utilizing a charge amplifier with impedance Z_f. In small signal approximation, it is given by

The magnitude and phase angle of the sensor’s admittance Y are given by |Y| and ϕ, respectively. In Eq. (1), Y₀ ≔ $∣Yfex,μ0H∣$ and ϕ₀ ≔ $ϕfex,μ0H$ at μ₀H = B and f_ex = f_r(B). Hence, each sensor is operated at a defined magnetic bias flux density B and excited at the corresponding mechanical resonance frequency f_r(B). An alternating sinusoidal magnetic test signal $Bact$ with frequency f_ac and amplitude $B̂ac$ will later be applied to characterize the sensors. It modulates the amplitude of u via the amplitude sensitivity S_am and the phase via the phase sensitivity S_pm. They are defined by³² 

with the electrical magnitude sensitivity S_el,am and the electrical phase sensitivity S_el,pm. The dynamic behavior of the sensor can be approximated by using a first-order Bessel filter.¹⁴ The output signal of the charge amplifier is fed into a quadrature amplitude demodulator to obtain the demodulated output signal $ucot$⁠. For small magnetic field amplitudes $B̂ac$⁠, the voltage amplitude spectrum $Ûco(f)$ of $ucot$ is used to define a voltage sensitivity S_V,

at the frequency f = f_ac. The voltage sensitivity can be used to estimate the equivalent magnetic field noise density also referred to as detectivity or limit of detection (LOD),

with the voltage noise density U_co of $ucot$ measured without applying the magnetic test signal B_ac.

All signal and noise measurements were performed with a high-resolution A/D and D/A converter Fireface UFX+ (RME, Germany) in a magnetically shielded setup⁴ with a mu-metal cylinder ZG1 (Aaronia AG, Germany). The AD745-based charge amplifier⁴² has a feedback capacitance of C_f = 33 pF and a feedback resistance of R_f = 5 GΩ.

To analyze the magnetic properties, i.e., the delta-E effect of the two sensors, admittance measurements were performed at various applied magnetic flux densities μ₀H starting close to the negative magnetic saturation at −10 mT. An excitation voltage amplitude of u_ex = 10 mV was used. With a modified Butterworth Van Dyke (mBvD) equivalent circuit model like in Ref. 32, the resonance frequencies f_r and quality factors Q were extracted. The f_r(H) curves are plotted in Fig. 2(a), normalized to the respective maximum values of f_r,max(AlN) = 9119.5 kHz and f_r,max(AlScN) = 7546.8 kHz. Resonance detuning via the delta-E effect has been investigated before²⁸ in similar cantilever sensors and is generally well understood (e.g., Refs. 40 and 41).

In Fig. 2(b), the corresponding derivatives of f_r(H), the magnetic sensitivities S_H are shown. Both sensors show a similar maximum drop in the resonance frequency around 0.75% and maximum magnetic sensitivities of ≈55 Hz/mT (AlN) and ≈37 Hz/mT (AlScN). These values are within the typical range reported for other delta-E effect sensors (e.g., Refs. 16, 23, 26, 27, and 34).

Here, we do not operate both sensors at maximum S_H but chose B_AlN = −1.7 mT and B_AlScN = −1.3 mT. At these bias flux densities, S_H ≈ (40 ± 5) Hz/mT for both sensors. The mBvD parameters at B_AlN and B_AlScN are given in the supplementary material. From these parameters, we obtain resonance frequencies of f_r,AlN = 9045.5 Hz and f_r,AlScN = 7508.2 Hz and quality factors of Q_AlN = 570 and Q_AlScN = 635. All following measurements are performed at these bias fields.

In Fig. 3(a), the electrical admittance magnitude $Y$ of AlN and AlScN at B_AlN and B_AlScN is plotted as functions of the excitation frequency f_ex, shifted by the respective f_r. The AlScN sensor shows a significantly increased admittance magnitude |Y| compared to the AlN sensor. From |Y|, the corresponding electrical amplitude sensitivities S_el,am are calculated and plotted in Fig. 3(b). In mechanical resonance, S_el,am of the AlScN sensor is approximately a factor of 8 times higher than S_el,am of the AlN sensor. Averaging over 20 measurements results in mean values and standard deviations of S_el,am(AlN) = − 4.6 ± 0.6 nS/Hz and S_el,am(AlScN) = −38.9 ± 2.6 nS/Hz at f_r. An overview of all mean values is given in the supplementary material.

In Figs. 3(c) and 3(d), the phase angle ϕ and its derivative the electrical phase sensitivity S_el,pm are plotted, calculated with the mBvD model. Because the minima of ϕ are close to f_r, S_el,pm at f_r is close to 0 for both sensors. Hence, the following discussions are focused on the amplitude sensitivities. The factor of 8 in S_el,am might be caused by the different piezoelectric materials. Yet, due to the large difference in the resonance frequencies (≈1.5 kHz) of the two sensors, additional geometrical influences cannot be excluded.

To distinguish geometric from material effects, a detailed finite element method (FEM) model is set up (COMSOL Multiphysics™, v. 5.4). This model solves the mechanical equations of motion⁴³ with linear strains, coupled to the electrostatic equations⁴⁴ via the piezoelectric constitutive relations.⁴⁵ As we deal with small displacements and rather low frequencies, both major approximations are justified. The cantilever’s geometry is approximated with a stratified structure of thin rectangular cuboids of different lengths and widths that correspond to the dimensions of the mask layout. Details on the model and all material parameters^59–71 and layer dimensions are given in the supplementary material.

During the wet etching process that releases the cantilever, the three-layer structure of SiO₂ and poly-Si can be underetched at the anchor. The depth of underetching is represented by the parameter L_etch in the model. The components of the piezoelectric coupling tensor and the values of L_etch are determined from a fit of the model to the data in Fig. 3(a).

With the measured quality factors (Q_AlN = 570, Q_AlScN = 635), the simulations match the measurements and mBvD simulations in Fig. 3 very well. From the fit, we obtain underetch lengths of L_etch = 80.7 µm for AlN and L_etch = 275.0 µm for AlScN. Other factors such as geometric inaccuracies and stress could contribute to L_etch. It is worth mentioning that the difference in resonance frequency cannot be explained by reasonable variations in material parameters or layer thicknesses alone. Hence, they were considered to be fixed parameters during the fit. The other fitting parameters are given in Table I. For the components of the piezoelectric coupling tensor e_ij, we obtain values overall close to ab initio calculations,⁴⁶ yet slightly smaller.

The admittance measurements in Fig. 3 are repeated for an increase in the excitation voltage amplitude $ûex$⁠. Example measurements of |Y| at three different $ûex$ are shown in Figs. 4(a) and 4(b). With an increase in $ûex$⁠, the resonance visible in the admittance magnitude incrementally shifts toward lower frequencies. The difference between the maximum and minimum values decreases, and the curvature around the maximum increases. Whereas the linear mBvD model matched the measurements at $ûex$ = 10 mV very well, it is not sufficient at higher voltage amplitudes. Instead, a nonlinear equivalent circuit model similar to that of Ref. 47 is implemented (supplementary material). It extends the previous model by an additional cubic restoring force with capacitance C₃ in the LCR resonator circuit of the mBvD model. The ratio C₃/C_r of the additional capacitance C₃ and the linear capacitance C_r is used as a quantitative measure for the nonlinearity of the resonator. From the nonlinear mBvD fit, the linear resonance frequencies and the quality factors are extracted.

As demonstrated in Figs. 4(a) and 4(b), the nonlinear model matches the measurements for both sensors very well. With an increase in the voltage amplitude $ûex$⁠, the admittance magnitude |Y| becomes increasingly asymmetric. The electrical resonance and antiresonance peaks shift to lower frequencies, and the difference in their admittance magnitude decreases. These effects are significantly more distinct in the AlScN sensor. Accordingly, the nonlinearity increases with $ûex$ and is more than an order of magnitude larger in the AlScN sensor [Fig. 4(c)].

Such resonator nonlinearities are expected to emerge from the nonlinearity of the magnetostrictive stress–strain relation (e.g., Refs. 48–50). During sensor operation, the oscillating stress in the magnetic layer induces an alternating magnetization, which is accompanied by a magnetostrictive strain. With an increase in $ûex$⁠, the amplitude of stress and induced magnetostriction increases, and the nonlinearity becomes visible for both sensors, as shown in Fig. 4. Yet, the amplitude of the oscillating stress does not only depend on $ûex$ but also on the piezoelectric material. AlScN exhibits significantly larger electromechanical coupling factors compared to AlN. The larger coupling causes a larger stress amplitude and results in a larger nonlinearity at the same $ûex$⁠, as visible in Fig. 4. A strong magnetic bias field (≈80 mT) was applied to fix the magnetization and thereby suppress the change in the magnetostrictive strain during the oscillation of the cantilever. As expected, the nonlinearity vanishes, which confirms its magnetic origin (supplementary material). It is worth noting that the nonlinear stress–strain relation is not only an inherent property of magnetostrictive materials⁴⁵ but also the origin of the magnetic field dependent delta-E effect.^28,40,51,52 Consequently, the delta-E effect sensor concept is necessarily connected with nonlinearities.

Because the nonlinearity changes the admittance characteristic, it can potentially reduce the electrical sensitivity. How exactly the electrical sensitivity changes with $ûex$ depends also on the operation frequency f_ex chosen. In the present case, we observe a reduction in S_el,am at f_ex = f_r by more than 99% (AlScN) and 85% (AlN) with $ûex$ (supplementary material). Besides the nonlinearity, the quality factor Q can strongly influence the admittance and the sensitivity of delta-E effect sensors.³² A reduction in Q with an increase in $ûex$ was reported previously^24,31 with similar magnetoelastic resonators and could be explained with loss from stress-induced magnetic domain activity.⁵³ In our case, the quality factors of both sensors decrease by up to 40% ± 5% with $ûex$⁠. Consequently, we expect a significant contribution of Q to the reduction in the electrical sensitivity.

In the following, the mean resonance frequency as a function of $ûex$ is used as an excitation frequency for operating the sensor at larger voltage amplitudes. Signal and noise are measured 20 times, respectively, at each $ûex$⁠. During the signal measurements, a sinusoidal magnetic test signal with an amplitude of $B̂ac=$ 1 µT and a frequency of f_ac = 10 Hz is applied. From the measurements, we obtain the voltage sensitivity S_V and the limit of detection (LOD) [Eqs. (3) and (4)] at a frequency of 10 Hz. The resulting mean values and standard deviations are plotted in Fig. 5.

A linear signal model based on Eq. (1) is set up like that in Ref. 32 to calculate the expected voltage sensitivity S_V. As input we use the admittance data obtained from the measurements at $ûex$ = 10 mV. With the same input dataset, we estimate S_V at larger $ûex$⁠. Because the sensitivity changes with $ûex$ and nonlinearities are present (Fig. 4), the simulations at larger $ûex$ represent rather ideal reference values than predictions.

A comparison of measured and modeled voltage sensitivity is given in Fig. 5(a). At $ûex=10mV$⁠, the modeled S_V = 0.97 V/T (AlN) and S_V = 7.9 V/T (AlScN) match the measurement of S_V = (0.84 ± 0.4) V/T (AlN) and S_V = (7.0 ± 1.9) V/T (AlScN) very well. The simulations reproduce the factor of ∼8 between S_V(AlScN) and S_V(AlN) seen in the measurements. As expected from Eq. (1), the modeled S_V increases linearly with $ûex$⁠. In contrast, a deviation from this linearity is present in the measurements. After an initial drop of the measured S_V(AlN) by a factor of 2 at $ûex$ = 20 mV, relative to the simulation, it continues increasing approximately linearly. In contrast, S_V(AlScN) continuously decreases relative to the reference, until deviations of almost a factor of 8 are present at large $ûex$⁠.

Such deviations are expected due to the altered admittance and reduced electrical sensitivity. Additional effects not considered in the model might contribute to the behavior of S_V with $ûex$⁠. The magnetic susceptibility can change with $ûex$⁵⁴ and potentially alter the magnetic sensitivity.²⁴ Overall, the behavior of S_V and the output signal with $ûex$ is complex, and a detailed analysis of all potential contributions is beyond the scope of this paper.

The measured voltage noise density U_co at f_ac = 10 Hz is plotted in Fig. 5(b). At small $ûex$⁠, it is approximately constant and similar for both sensors. At around $ûex$ = 100 mV, U_co(AlScN) increases significantly with $ûex$⁠. Notably, larger amplitudes are required to reach this regime with the AlN sensor. In this regime, U_co(AlN) is approximately a factor of 8 times smaller than U_co(AlScN).

Comparing the data with results from a previously developed noise model³⁰ suggests that thermal–electrical noise of the charge amplifier dominates the noise floor at small $ûex$⁠. The slightly different noise densities of the two sensors in this regime are expected from the different sensor capacities that alter the amplification factor of the charge amplifier. The increase in noise at larger $ûex$ was previously explained with magnetic noise, induced by alternating stress during the oscillation of the cantilever.^30,31 In magnetically modulated cantilevers, the noise has been linked directly to magnetic domain activity,⁵³ which might similarly apply here.

In Fig. 5(c), the LOD estimated with S_V and U_co is plotted as a function of the excitation voltage amplitude $ûex$⁠. Overall, a factor of 8 in LOD is gained with the AlScN sensor, but only at small $ûex$ where the thermal–electrical noise is dominant. Both LOD curves reach a minimum roughly at the voltage amplitudes where the magnetic noise starts to dominate the noise floor. With approximately $40±5nT/Hz$⁠, the minimum LOD at 10 Hz of both sensors is identical.

It might be possible to reduce the thermal–electrical noise level using optimized amplifiers^32,42 and suppress the magnetic noise with advanced magnetic multilayers.^55,56 If magnetic noise can be sufficiently reduced, the thermal–mechanical noise of the resonator represents the limiting sensor intrinsic noise source at room temperature.^31,57 It depends on the geometry and the electromechanical properties of the resonator and, therefore, also on the piezoelectric material.^30,58 In the following, we extend our discussion to this fundamental noise limit to gain more general insights into the signal, noise, and minimum LOD as functions of the piezoelectric material properties.

With the FEM model, we calculate the sensor admittance as a function of the piezoelectric charge–stress coefficient tensor d = x · d_AlScN. Here, d_AlScN = $eAlScNCAlScN−1$ is the piezoelectric charge–stress coefficient tensor of AlScN. It is given by the double-dot product of the inverse stiffness tensor $CAlScN−1$ and the piezoelectric coupling tensor e_AlScN. In the simulation, we use the geometry and all other material properties of the AlScN sensor and vary only the scaling factor x. The resulting amplitude sensitivity S_am(x) satisfies a quadratic fit very well [Fig. 6(a)]. Because S_am is the dominating sensitivity at our excitation frequency f_ex = f_r, also S_V(x) is approximately quadratic [Fig. 6(b)]. In Fig. 6(c), the modeled voltage noise density U_co(x) is plotted, considering only thermal–mechanical noise. It is calculated with a previously developed model^32,57 from the simulated admittance curves. Consistent with analytical estimations,³⁷ the thermal–mechanical noise increases linearly with x. Consequently, the LOD estimate [Eq. (4)] improves LOD ∝ 1/x if thermal–mechanical noise is dominant. In contrast to the direct ME detection, we excite the resonator electrically in the delta-E operation scheme. Hence, the deflection magnitude of the resonator increases linearly with d and contributes to the quadratic behavior observed in the sensitivities and the output signal amplitude. For the given geometry and resonance mode, the quadratic response is dominated by components d₃₁ and d₃₃ of d. Details are given in the supplementary material.

Because S_V(x) ∝ x², we expect LOD ∝ 1/x² for constant thermal–electrical noise. At first glance, this seems to be contradictory to the factor of 8 found experimentally. Additional FEM simulations reveal that this is mainly caused by the differences in the quality factor, stiffness tensor, and geometry of the sensors. Whereas the reduced underetching of the AlN sensor increases S_V compared to the AlScN sensor, the lower quality factor and different stiffness tensors have the opposite effect. Simulations with the identical model geometry and quality factor show that AlScN as a piezoelectric material results in ≈7.5 times higher electrical sensitivity and voltage sensitivity compared to AlN.

In summary, we identified the influence of the piezoelectric material on the sensing characteristics of delta-E effect sensors. Experiments were performed on AlScN- and AlN-based cantilever sensors of identical design. The results were analyzed regarding sensitivity, noise, linearity, and limit of detection (LOD). The measurements are supported by an electromechanical finite element and a signal-and-noise model.

Within the simulated parameter range, both the electrical magnitude sensitivity S_el,am and the voltage sensitivity S_V scale quadratically with the dominating components d_ij of the piezoelectric coefficient tensor d. In the regime of intrinsic thermal–mechanical noise, the LOD ∝ 1/d_ij because the thermal–mechanical noise ∝ d_ij. Consequently, our delta-E effect sensors could benefit from an increase in the piezoelectric coefficients much stronger than conventional magnetoelectric sensors, where the LOD is constant due to a merely linear increase in the output signal amplitude.³⁷ 

Considering not only d_ij but also the complete piezoelectric material properties, the model predicts a 7.5× enhanced S_el,am and S_V for the AlScN sensor. In the present case, the two analyzed sensors additionally differ in geometry and slightly in their quality factors. In total, this results in an 8× improved voltage sensitivity and LOD measured with the AlScN sensor. This is well reproduced by the model. The improvement in LOD is limited to small excitation amplitudes where thermal–electrical noise is dominant. With an increase in the excitation amplitude, the factor of 8 decreases and resonator nonlinearities of magnetic origin occur. They can be described and quantified with a nonlinear equivalent circuit model and are up to an order of magnitude larger in the AlScN sensor. Overall, the minimum LOD does not change due to an earlier onset of magnetic noise in the AlScN sensor. This result supports theoretical magnetic noise considerations in Ref. 31 experimentally. Using exchange bias multilayers as in Refs. 55 and 56, it might be possible to reduce the magnetic noise in the future and push its onset to higher excitation voltage amplitudes. The large improvement in the output signal amplitude makes AlScN a promising candidate for tapping into new kinds of resonance modes that are otherwise poorly excited.

See the supplementary material for the material parameters used, details on the models, the fitting parameters found, and additional measurements.

We thank the German Research Foundation (DFG) for funding this work through the Collaborative Research Centre CRC 1261 “Magnetoelectric Sensors”: From Composite Materials to Biomagnetic Diagnostics. The authors thank Lars Thormählen for the deposition of the magnetic layers.

The data that support the findings of this study are available from the corresponding author upon reasonable request.