Introduction

Microwave devices such as resonators and band-stop and band-pass filters are important components in high frequency communication devices and radars1,2,3. Such devices, in general, are based on low-loss semiconductors, ferroelectrics, or ferrites1,2,3,4,5,6. Tuning the operating frequencies of ferroelectric devices could be accomplished with an electric field E and the devices could easily be miniaturized. The frequency of ferrite-based devices, however, could only be tuned with a source of variable magnetic field H4,5,6,7,8. Thus, ferrite microwave devices cannot be miniaturized or integrated with semiconductor devices due to the need for a solenoid or similar sources to generate a variable H. A significant development in microwave device technologies is the use of a ferrite-ferroelectric composite to facilitate E tuning of the ferrite devices9,10,11,12. The E-tuning is possible due to magneto-electric (ME) effect is such composites13, 14. The piezoelectric strain under an E-field in the composite when transferred to the ferrite manifests as an induced magnetic field and leads to tuning of the ferrite device12,13,14,15. A variety of dual H- and E-tunable ferrite-ferroelectric devices including resonators, filters, and inductors were demonstrated in the past years16,17,18,19,20.

An alternate E-tuning mechanism of ferrite devices aided by non-linear ME effect (NLME) was reported recently21, 22. The NLME effect, first reported in M-type strontium hexagonal ferrite, was observed as tuning of ferromagnetic resonance (FMR) in the ferrite under a DC or pulsed E-field. The E/I-induced changes in the magnetization and anisotropy field due to NLME, determined from FMR data, were found to vary with the input DC power (or E2). Studies on c-plane single crystals of (Sr, Al)- and Ba-hexaferrites also revealed a similar NLME effects for currents either in the c-plane or along the c-axis23, 24. E- tunable resonators and filters of M- and Y-type hexagonal ferrites operating at 12–50 GHz were demonstrated25,26,27.

This report is on the first observation of a similar NLME effect in a spinel ferrite with cubic symmetry. We present here experimental evidence for the effect in a microwave resonator with a single crystal platelet of nickel zinc ferrite. The device operates in the X- to K-bands, thus complements similar resonators and filters that make use of hexaferrites for higher frequencies. Besides, nickel ferrite easily adopts to dopants such as Zn which makes possible tuning of its magnetic parameters for ferrite devices operating at a specific frequency range28. The experiments were carried out with the ferrite in a stripline device structure and a static magnetic field H0was applied to excite magnetostatic surface wave (MSSW) modes in the 8–20 GHz frequency range29, 30. An in-plane DC current applied parallel to H0 resulted in a down shift in the mode frequencies of equal magnitude for all of the modes. The decrease in the mode frequency ∆f was in excess of 400 MHz for a DC power P ~ 400 mW and ∆f was found to vary linearly with P. Data on mode frequency f versus P was used to estimate the changes in the magnetic parameters for the ferrite. Since the frequency shift due to NLME effects is similar to the influence of sample heating, experiments carried out to determine the Joule heating effect indicated a frequency shift that was an order of magnitude smaller than the shift due to NLME effects. Details on our studies are provided in the sections that follow.

Experiment

The material under investigation was a single-crystal platelet of zinc-substituted nickel ferrite with the composition Ni0.7Zn0.3Fe2O4 (NZFO) grown by floating zone technique (and provided by Professor A. Balbashov, Moscow Power Engineering Institute, Russia). It was cut into rectangular slabs in such way that the crystallographic < 111 > direction coincided with the normal to the sample plane. It was then polished down to a thickness of S = 335 μm and to the lateral dimensions a × b = 2.5 × 1 mm2. Finally, conducting electrodes (3 μm of Pt on 45 nm Ti buffer layer) as shown in Fig. 1 were deposited on opposite sides of samples by magnetron sputtering. Those electrodes provided a path for current flow through the NZFO in the (111) plane along the longer side. Figure 1 shows the experimental setup with the NZFO sample placed on top of a microstrip transmission line. The current I is parallel to the static field H, along the Z-direction. The sample is separated from the stripline by an adhesive dielectric layer of thickness D1.

Figure 1
figure 1

(a) Schematic diagram showing the experimental setup with the nickel zinc ferrite sample, Ni0.7Zn0.3Fe2O4 (NZFO), with Pt-electrodes placed on top of a microstrip line. (b) Cross section view of the NZFO samples. S is the sample thickness and D1 is the thickness of an adhesive epoxy layer. The static field H0 and the DC current I are applied along the Z-direction.

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Since the DC electric field/current is a key parameter in the measurements, the voltage V versus current I characteristics were measured for the sample and are shown in Fig. 2. The current I varies linearly with V for V < 2 V, corresponding to a resistance R ~ 510 Ω., and a nonlinear variation in I is evident for higher V values.

Figure 2
figure 2

DC voltage V versus current I data for the NZFO sample.

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Results

The NZFO sample was placed on top of a S-shaped microstrip line etched on a 0.02 inch (508 μm) thick RT/Duroid 5880 substrate. A thin adhesive tape of thickness of thickness D1 = 20 μm was used to fix the sample in place. A bias magnetic field H0 was applied along the longer side of the sample for the excitation of MSSW modes in the ferrite. The stripline structure with the NZFO sample was connected to the ports of a vector network analyzer (Agilent PNA E8361A) and excited with microwave power (Figure S-1). The scattering matrix parameterS21for the resonator was measured as a function of the frequency f of the input microwave power. Representative S21 versus f profiles for H = 665 Oe and 2725 Oe are shown in Fig. 3. Depending on H0 values, several MSSW modes were observed in the frequency range 8–16 GHz. The two dominant modes are shown in Fig. 3. Similar measurements were done for a series of H0 values ranging from 400 to 4400 Oe (corresponding to fr ≈ 8–20 GHz) (Figure S-2). Data on the variation of fr with H obtained from such profiles are shown in Fig. 4 (and also in Figure S-3).

Figure 3
figure 3

Transmission loss S21 as a function of frequency f for NZFO sample for an in-plane static magnetic field (a) H = 665 Oe and (b) 2725 Oe. The profiles show two dominant magnetostatic surface wave (MSSW) modes.

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Figure 4
figure 4

MSSW mode frequency fr as a function of the applied magnetic field H0 for the two dominant modes as in Fig. 3.

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Following the measurements of fr versus H0, studies were carried out on the tuning of the resonator mode frequency with a DC E/I for constant H0 values. A DC voltage single pulse, 1–5 s in duration, applied to the electrodes (shown in Fig. 1), established a current parallel to H0. Successive pulses were applied after a time interval of 1 min to minimize potential heating of the sample. Figure 5 shows a series of S21 versus f profiles for the two dominant MSSW modes for H0 value of 2045 Oe and I = 0–20 mA. The E/I-induced shift in the mode frequency ∆fr is also shown for the two modes as a function of the input power P. It is clear from the results that ∆fr for the two modes are approximately the same for a specific P-value. Both modes show a decrease in the frequency shift with increasing input power and their dependence on P is linear. Similar linear variation in ∆fr was measured for a series of bias magnetic fields as shown in Fig.S-4–S-7 in the supplement. From now on we, therefore, focus only on results obtained for the higher frequency MSSW mode.

Figure 5
figure 5

(a) Profiles of S21 versus f showing the current tuning of the dominant resonance modes in the NZFO platelet for H = 2045 Oe for I = 0–20 mA. (b) Variation in the I-induced shift ∆fr in the lower (squares) and higher (circles) frequency modes with the input DC power P.

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Results on the I-tuning of the high frequency mode are shown in Fig. 6 for H0 = 1355 Oe and 3440 Oe and the variation in the mode frequency shift ∆fr with P is shown in Fig. 7 for a series of H0 values. It is evident that the measured decrease in ∆fr with P is independent for H0 value. It is also obvious from the data in Fig. 7 that the E-tuning of the mode frequency is due to nonlinear ME effects and ∆fr is proportional to E2 (and not just E). The data also allows one to extract the efficiency of tuning, the slope Δfr/∆P, for meaningful comparison with other materials or electrically tunable ferrite devices. The tuning efficiency is Δfr/∆P ~ 4.1 MHz/mW for the NZFO resonator. Comparison of the tuning efficiency with similar ferrite devices are provided in later sections.

Figure 6
figure 6

Representative data for current tuning of the high frequency MSSW modes in (111) NZFO resonator for the current flowing parallel to in-plane magnetic field H0.

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Figure 7
figure 7

Shift in the MSSW mode frequency versus applied electric power measured for a series of magnetic bias field values.

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Discussion

Linear ME effects mediated by mechanical deformation have been studied extensively in composites of ferrites and ferroelectrics12,13,14. Due to symmetry considerations such linear ME effects, however, are forbidden in ferrites with collinear magnetic ordering. A theory for the NLME effects observed in both hexagonal21,22,23,24,25,26,27 and spinel ferrites is lacking at present. One may attribute the NLME to weakening of the magnetic interactions due to the exclusion of electrons involved in carrying a current in the ferrite under a DC electric field. Even though stoichiometric spinel ferrites are expected to be insulators, divalent Fe are generally present in them due to impurities. Hopping type conduction in the ferrites, therefore, leads to a semiconductor like behavior. Under an applied DC voltage, electrons in Fe sites involved in carrying a current are likely to be excluded from contributing to super-exchange interactions. Weakening of the magnetic interactions will lead to a decrease in the overall magnetization of the ferrite. That exclusion will be in effect for during the time electrons hop from one iron site to another. One may, therefore, attribute the decrease in the magnetization that manifests as a decrease in the frequency of resonance modes to weakening of magnetic exchange interactions and consequent decrease in the magnetization.

We first discuss a model for the MSSW modes and their E-tuning in a uniformly magnetized ferrite platelet enclosed between conducting metal layers, with ferrite to metal spacings equal to D1 and D230. The explict dispersion equation for this case is given by Eq. (6) in Ref.30 which is:

$$\begin{aligned} & (1 + \kappa )(1 + \kappa + \eta^{2} ) - \left( {\nu - \sqrt {1 + \eta^{2} } \tanh \left( {\frac{{D_{1} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right)\left( {\nu + \sqrt {1 + \eta^{2} } \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right) \\ & \quad + (1 + \kappa )\sqrt {1 + \eta^{2} } \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} \left( {\tanh \left( {\frac{{D_{1} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right) + \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right)\coth \left( {\varsigma \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} } \right) = 0 \\ \end{aligned}$$
(1)

where \(\kappa = \Omega_{H} /(\Omega_{H}^{2} - \Omega^{2} )\), \(\nu = \Omega /(\Omega_{H}^{2} - \Omega^{2} )\), \(\Omega_{H} = H_{0} /\left( {4\pi M_{0} } \right)\), \(\Omega = \omega /\left( {\gamma 4\pi M_{0} } \right)\), \(\xi = k_{z} S\), \(\zeta = k_{y} S\), and \(\eta = \xi /\zeta = k_{z} /k_{y}\). Here 4πM0 is the saturation magnetization of the ferrite, γ = 2.8 MHz/Oe28 is the gyromagnetic ratio. The coordinate system used here is shown on the Fig. 1.

In our experimental set-up the bottom metal layer is the central stripe of the S-shaped microstrip line (Fig. 1) of width same as the resonator width b. The distance between the bottom metal layer and the ferrite is 20 μm which is the thickness of the adhesive layer. The top metal layer is formed by the Pt electrodes that partially cover the sample surface. In essence, the ferrite sample consists of two regions with different MSSW dispersion characteristics \(\omega (k)\). The first one is the middle area with only the bottom metal screen. The dispersion equation for this region can be obtained from Eq. (1) by substituting the screen spacing D1 = 20 μm and D2 = ∞, respectively:

$$\begin{aligned} & (1 + \kappa )(1 + \kappa + \eta^{2} ) - (\nu - {\text{sgn}} (\varsigma )\sqrt {1 + \eta^{2} } )\left( {\nu + \sqrt {1 + \eta^{2} } \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right) \\ & \quad + (1 + \kappa )\sqrt {1 + \eta^{2} } \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} \left( {{\text{sgn}} (\varsigma ) + \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right)\coth \left( {\varsigma \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} } \right) = 0 \\ \end{aligned}$$
(2)

The second region is the ferrite under the Pt electrodes. This case is described by conditions D1 = 20 μm and D2 = 0, which leads to the following dispersion equation29, 30

$$\begin{aligned} & (1 + \kappa )(1 + \kappa + \eta^{2} ) - \nu \left( {\nu + \sqrt {1 + \eta^{2} } \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)} \right) \\ & \quad + (1 + \kappa )\sqrt {1 + \eta^{2} } \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} \tanh \left( {\frac{{D_{2} }}{S}\varsigma \sqrt {1 + \eta^{2} } } \right)\coth \left( {\varsigma \sqrt {1 + \frac{{\eta^{2} }}{1 + \kappa }} } \right) = 0 \\ \end{aligned}$$
(3)

[all the notations are the same as in Eq. (1)]. The above equations are valid for an isotropic ferromagnet and thus can be applied in practice only in the situation when the anisotropy field is negligible in comparison to both the static magnetization and H0 which is true for the NZFO used in this study28. The presence of the metal screen leads to a pronounced nonreciprocity of the MSSW propagation, namely, the waves with the opposite directions of the wave-vector will have different frequencies: \(\omega (k) \ne \omega ( - k)\). That fact strongly influences the resonator modes since they are usually formed by the interference of the counter-propagating waves. An example for the nonreciprocal dispersion characteristics of the MSSW in both regions of the resonator is shown in the Fig. 8.

Figure 8
figure 8

Nonreciprocal dispersion characteristic of MSSW in an isotropic ferrite layer calculated using Eqs. (1) and (2) and for H0 = 3000 Oe, 4πM0 = 5000 G and ξ = 1. Note, that for the positive values of wavevector (ky > 0) the dispersion curves for D2 = 0 and D2 = ∞ are nearly identical (frequency difference is less than 150 MHz for the given parameters).

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In order to determine the resonance frequencies of the ferrite one needs to define the in-plane wave-vector set by the geometric dimensions of the resonator. The sign of longitudinal wave-vector kz (or \(\xi\)) (with respect to the H0 direction) does not affect the dispersion Eqs. (2) or (3) and therefore it can be calculated as

$$k_{z} = m\frac{\pi }{d},m = 0,1,2...$$
(4a)

Here d is the length of the metal electrode (d ≈ 0.7 mm in our case) or the distance of metal-free region between the electrodes. On the other hand, the transversal wave-vector \(k_{y}\) due to the nonreciprocity of MSSW should be evaluated from the condition31

$$(k_{y}^{ + } + |k_{y}^{ - } |)b = 2\pi n,\,\,n = 0,1,2...$$
(4b)

where \(k^{ \pm }\) are the positive and negative solutions of the Eq. (2) or (3) and b is the width of the sample. The modes represented by Eqs. (4a) and (4b) will therefore be labeled as (m, n) modes. The observed FMR frequency variation is given by

$$\Delta f_{r} = \frac{{\partial f_{r} }}{{\partial (4\pi M_{0} )}}\Delta (4\pi M_{0} )$$
(5)

Assuming that Δ(4πM0) is linearly proportional to the applied power21,22,23,24, the expected shift Δfr also should be proportional to P which agrees with the results in Fig. 7. One may then use the above theory to extract the NLME induced variation in the saturation magnetization. For the nickel-zinc ferrite of the given composition 4πM0 ≈ 5000 G whereas the cubic anisotropy field Ha is less than 100 Oe28. Therefore, the condition 4πM0 >  > Ha is fulfilled and use of Eqs. (2), (3) is justified.

The higher frequency mode shown in Fig. 3 was identified as (1, 1) mode of the metalized ferrite region. We used this mode to estimate the NLME induced changes in magnetization from ∆fr versus P data in Fig. 7. The following procedure was used for the estimate. For specific fr and 4πM0 ≈ 5000 G, γ = 2.8 MHz/Oe, and the dimensions of the resonator (d, b, S) the wave-vector values \(k^{ \pm }\) were calculated from Eq. (2). Then the values of \(k^{ \pm }\) were substituted into Eq. (4b) and checked whether this equation was satisfied. If not, then the procedure was repeated for series of values of 4πM0 until the wave vector values satisfied both Eq. (3) and Eq. (4b). This procedure was repeated for fr corresponding to different values of P to determine the variation of 4πM0 with P. The results on 4πM0 versus P for a series of H0 values are shown in Fig. 8. We took only the data for the largest values of external bias field since in this case the influence of static demagnetization field (that was not taken into account in Eq. (1)–(3) ) is expected to be minimal. One notices a linear decrease in 4πM0 at an average rate of − (2.50 ± 0.12) G/mW for all of the H0 values. The deviation in 4πM0 at P = 0 from the expected value of 5000 G may be attributed to the contribution from the cubic anisotropy field Ha ~ 100 Oe29. Additionally, one may expect a variation in the cubic anisotropy field on E32 due to NLME effects which was not considered in the model discussed here.

In the course of the measurements the DC voltage/current was applied as short pulses to the nickel ferrite sample in order to eliminate the effect of Joule heating of the sample and its contribution to any changes in fr. The pulses were of duration 1–5 s. It is important to investigate the effects of sample heating compared to the influence of NLME effects. Additional experiments were done to separate these two effects on the decrease in the mode frequency under a DC current. Although the data provided in Figs. 3, 4, 5, 6, 7 and 9 (in the revised version) are for pulsed currents of duration 1–5 s, we examined the effects of sample heating under a continuous DC current. The current, corresponding to input power P = 55 mW and 103 mW, was first turned on for a duration of 10 min and the resonance frequency fr was recorded at regular time intervals. The current was then turned off after 10 min, and data on fr versus t were recorded for the next 10 min. The results on ∆fr versus t are shown in Fig. 10. For both DC powers, ∆fr shows an abrupt decrease by 180 MHz for P = 55 mW and by 410 MHz for P = 103 MHz as soon as the current is on and then shows a further decrease by 5–15 MHz over the 10 min period that the current was present. As soon as the current is turned off, fr shows an abrupt increase and ∆fr slowly decays to zero during the next 10 min of cooling. It is therefore evident from these experiments that Joule heating results only in 4–8% of the overall shift due to NLME effects.

Figure 9
figure 9

Calculated dependence of the nickel-zinc ferrite saturation magnetization \(4\pi M_{0}\) on the applied dc electric power P.

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Figure 10
figure 10

Data on Magnetic mode frequency shift under a prolonged application of a continuous DC current for the input power (a) P = 55 mW and (b) 103 mW.

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The ferrite platelet in our experiments had a large surface area-to-volume ratio that is favorable in terms of effective heat exchange. The top surface was exposed to air and the bottom surface with a thin adhesive layer is on top of the thermally conducting copper microstrip line. If the sample were thermally isolated and under a prolonged application of a continuous DC current, the temperature (and thus mode frequency) should continuously change with time. But data in Fig. 10 shows that this is not the case. Next, such thermally isolated sample should remain at high temperature after the current was switched off and thus the resonance frequency should vary slowly after the power being switched off. But, fr shows an immediate jump, and regains its the initial value as the sample cools down. It is safe to conclude from the data in Fig. 10 that any contribution due to sample is small compared to NLME effects.

The nature of strain mediated nonlinear ME effects in ferrite-ferroelectric composites was studied in detail and well understood. But there have been very few reports so far on NLME in spinel or hexagonal ferrites21,22,23,24,25,26,27. We compare the strength of NLME in the spinel ferrite with similar reports for hexagonal ferrites. As mentioned earlier, the MSSW mode frequency tuning efficiency from Fig. 7 for the NZFO resonator Δfr/∆P ≈ − 4.1 MHz/mW (or alternatively, ΔP/Δfr = − 0.25mW/MHz). In the case of M-type barium hexaferrite Δfr/ΔP ≈ + 0.27 MHz/mW25. The efficiency was + 0.36 MHz/mW for Al-substituted M-type strontium hexaferrite11 and − 14.2 MHz/mW for Y-type hexaferrite26. Hence, the tuning efficiency for NZFO is higher than reported values for M-type hexaferrites, but not as high as for Y-type hexaferrite.

Conclusions

Evidence for electric field/current-induced NLME effects at room-temperature in single crystal nickel zinc ferrite resonator were obtained from studies on frequency tuning of MSSW modes at 8–20 GHz. It was found that the application of an in-plane DC current resulted in a down-shift in the MSSW mode frequencies that was linearly dependent on the DC electric power. This proportionality to P implies a nonlinear, quadratic in electric field E, nature of the observed ME effect and is similar to our earlier observations in M- and Y-type hexaferrites. The shift in mode frequencies due to NLME was independent of the mode frequency and was attributed to a reduction in the saturation magnetization of the ferrite. Using a rigorous theory for the MSSW modes in the ferrite slab with metal shielding, we were able to determine the variation of saturation magnetization with applied DC power. It was found that in all cases the magnetization decreased linearly with P at a rate of − 2.50 G/mW. The NLME related frequency tuning efficiency was 4.1 MHz/mW which is higher than the reported values for M-type hexaferrites with uniaxial anisotropy, but smaller than in Y-type hexaferrite with easy plane anisotropy. For the traditional H-tunable YIG filter the frequency tuning is linear with respect to the current in the solenoid coil and the tuning efficiency was 4–4.5 MHz/mW in the 10–18 GHz range33. The results in this study indicate the utility of the NLME phenomenon in the spinel ferrite for planar, energy efficient electrically tunable microwave signal-processing devices that can be miniaturized and integrated with semiconductor devices.