Magnetic field detection with single mode spin wave interference in asymmetric structure

Abstract

The magnetic field detection based on the interference phenomenon of surface-mode spin waves has been demonstrated in yttrium iron garnet (YIG) thin films, where the asymmetric arrangement of two excitation sources and one detection antenna allows for the field detection with a simple YIG strip structure that does not require microfabrication. The magnetic field can be detected by observing changes in the amplitude of the standing wave at the detection position, which result from alterations in the wavenumber of the excited spin wave caused by variations in the magnetic field. Time-domain measurements confirmed that the interference signal of the spin wave changed with the magnetic field. The induced electromotive force yielded a change of approximately 7 mV for a magnetic field change of ± 0.13 mT, resulting in a sensitivity of 24–25 V/T. The sinusoidal interference calculation using the wavenumber change due to a small magnetic field derived from the dispersion relation of spin waves agrees with the experimental results. This suggests that the mechanism of magnetic field detection is the wavenumber change due to the magnetic field.

Introduction

Spin waves represent collective excitation in magnetic materials, where the precession phase of localized spins propagates spatially. Compared to electric currents, spin waves possess two distinct information channels—amplitude and phase. This key difference allows spin waves to propagate even in insulators because their propagation does not inherently involve electric charge transport^1–4. Spin waves are promising due to factors such as high information density and low-power information processing. This unique advantage has spurred extensive research into spin-wave-based logic devices. Previous studies have explored the implementation of various logic gates^5–10, including reservoir computing architectures^11,12. Magnetic field sensing is another compelling application of spin waves. As a result of their resonant nature, spin waves exhibit a change in amplitude when an external magnetic field near the resonance frequency experiences slight variations. This sensitivity has been exploited in the development of magnetic sensors, as reported in several studies^13–20. Additionally, numerous studies have proposed methods that utilize the wavelike properties of spin waves, particularly the phenomenon of wave interference²¹. This technique mainly relies on the principle that the interference pattern generated by two interacting spin waves is highly susceptible to changes in the external magnetic field^6,22–24. The magnitude of the field directly affects the evolution of the interference pattern.

Balynsky et al. proposed a novel magnetic sensor design that exploited the interference between two distinct spin wave modes: magnetostatic surface spin waves (MSSWs) and backward volume magnetostatic spin waves (BVMSWs)^22,23. For a fixed excitation frequency ω (omega), the wavenumber k of MSSWs exhibits a negative dependence on the external magnetic field H while that of BVMSWs demonstrates a positive dependence. This unique interplay between the wavenumbers of the two modes forms the basis of a highly sensitive magnetic field sensor. The principle hinges on the phase difference between the interfering spin waves. This phase difference is determined based on the wavenumbers (k_MSSW and k_BVMSW) and the propagation distance (l) of each wave mode because the phase of each wave is given by . MSSW and BVMSW cannot be simultaneously excited in an infinite plane of the magnetic medium because the dispersion relations have no cross-points except for k = 0 (corresponding to the uniform ferromagnetic resonance; FMR). However, by utilizing a shape magnetic anisotropy of micro-scaled wire, the simultaneous excitation of propagating (k ≠ 0) MSSW and BVMSW was achieved^25,26. In a previous study using single-crystal YIG without a feedback system, the magnetic field sensitivity was experimentally reported to be approximately 10 V/T²².

This study presents a new magnetic detection that solely utilizes MSSWs (namely single mode detection), rather than employing both BVMSWs and MSSWs. This approach offers the potential for a more compact and simpler architecture. We exploit the dependence of the MSSW phase () on its propagation length (l). By introducing a controlled path length difference (l₁ ≠ l₂), a magnetic field change for the two spin waves induces a measurable phase difference . To achieve this, the design incorporates an asymmetric configuration for the excitation and detection antennas. This contrasts with previously reported magnetic field detection using spin-wave interference. This method enables detection operation in strip-shaped materials without require microfabrication. The time-domain observation method^27,28 was employed to examine the interference of the MSSWs and the resulting response of the interference signal to an applied small magnetic field changes. The obtained sensitivity was about 24–25 V/T. Numerical calculations suggest that the experimental results are due to a change in the wavenumber of the spin wave due to small magnetic field changes.

Results

Concept

First, the concept of magnetic field detection using the interference of only MSSW is introduced., The dispersion relation for MSSWs propagating within a thin film of thickness, d, can be expressed as follows²⁹ under the condition that the wavelength of MSSW l (= 2p/k) significantly exceeds d (2pd/l = kd < < 1):

1

where f is the spin wave excitation frequency, g is the gyromagnetic ratio, H₀ is the bias magnetic field, and M_s is the saturation magnetization. By reorganizing Eq. (1), regarding k, we obtain the following equation:

2

According to Eq. (2), k sensitively reflects the change in H₀ (see Fig. 1a). For example, when an MSSW is excited in a material with a saturation magnetization of µ₀M_s = 175 mT and film thickness of 10 μm with f = 3.45 GHz under a bias field of 60.0 mT, k changes from 7549 to 7684 m⁻¹ under the magnetic field changes by ± 0.3 mT.

Fig. 1.

(a) Change of wavenumber of MSSW by small magnetic field. (b) Experimental setup.

To achieve magnetic field detection using MSSW interference, two MSSWs are excited at both ends of a strip of magnetic material. As a result of the interference of the two MSSWs, a standing wave is formed. When the magnitude of the magnetic field changes, the wavenumber of the MSSWs changes. As a result, the position of the standing wave’s belly changes. This spatial displacement results in a change in amplitude at a fixed detection point. However, since the phase of the two MSSWs is constant and independent of the magnetic field at the center of the standing wave, no change in amplitude due to the magnetic field is obtained. Therefore, to achieve ground detection, it is important to have an asymmetric structure where the detection point is shifted from the center of the standing wave.

Experimental setup

This section outlines the experimental methods employed in this study. Figure 1b illustrates the experimental setup. A YIG film with a composition of Y₃Fe₅O₁₂, thickness of 10.1 μm, and width of 2 mm in a strip configuration served as the spin wave propagation medium. The YIG film was fabricated using liquid-phase epitaxy on a Gd₃Ga₅O₁₂ (111) single-crystal substrate. Spin wave excitation was achieved by combining a static-bias magnetic field generated by a permanent magnet with an alternating magnetic field induced by an AC current passing through the excitation antenna. Spin-wave detection was achieved by measuring the AC electromotive force induced in the detection antenna, resulting from the magnetic flux variation due to the propagating spin wave. The antennas were fabricated from copper on a glass-reinforced epoxy laminate substrate and comprised two excitation elements (Left and Right) and a single detection element for spin-wave interference. The width of excitation and detection antennas were 0.450 mm and 0.075 mm, respectively. However, the center-to-center distances between the Left and Right excitation antennas were 5.0 mm and 3.0 mm, respectively, as illustrated in Fig. 1b. An in-plane perpendicular-bias magnetic field was applied to the YIG stripe using a neodymium permanent magnet. The magnitude of the bias magnetic field, measured using the Hall sensor, was µ₀H = 49.2 mT. Here, µ₀ is the permeability of vacuum. The direction of the magnetic field was defined as positive from the bottom to the top. The small magnetic field to be detected was generated by applying a DC current to a handmade coil placed between the permanent magnet and the YIG. The magnitude of the small magnetic field can be controlled in the range of ± 0.13 mT.

Microwave current (RF) was injected into the excitation antennas to generate magnetostatic surface spin waves (MSSWs). The RF signal, sourced from a signal generator (Hewlett-Packard 83732 B), was divided using a directional coupler (Agilent 87300 B) and fed into the two excitation antennas. A phase shifter (ARRAinc 6425E) and an attenuator (Agilent 8494B) were employed to regulate the amplitude and phase of the excitation signals, with the attenuator set to a fixed value of -7 dB. The detection antenna was connected to an oscilloscope for time-domain measurement of the induced electromotive force generated by the propagating spin wave.

Results and discussion

Spin wave interference

Initially, the characteristics of two individual MSSWs were examined. Figure 2 illustrates the induced electromotive force (EMF) within the detection antenna when a 3.072-GHz RF pulse with a duration of 50 ns and power of 31.6 mW is transmitted through the excitation antennas using a signal generator and an RF amplifier. The temporal origin, denoted as t = 0, corresponds to the initiation of AC current flow from the signal generator to the excitation antenna. Figure 2a and b illustrate the induced EMFs for the left and right excitation antennas, respectively. A distinct pulse waveform SW_L was observed during 30–80 ns when only the left excitation antenna was activated. The disappearance of this pulse waveform following the removal of the permanent magnet and subsequent zeroing of the external magnetic field (data not shown) confirmed its association with the MSSW excited by the left antenna. Conversely, when the right-excitation antenna was utilized for excitation, as illustrated in Fig. 2b, two pulse-waveforms were observed during 2–40 and 40–90 ns. Figure 2c illustrates the waveform obtained under the conditions of the removed permanent magnet and zero external magnetic field. Only the second pulse vanished, leaving the initial waveform intact within the 0–50 ns interval, which means that the second waveform was attributed to the spin-wave component SW_R. Conversely, the first waveform, detected in the presence and absence of a magnetic field and characterized by a rapid arrival at the detection antenna, was attributed to an electromagnetic wave component originating from the right excitation antenna. The exclusive detection of the electromagnetic wave from the right antenna was attributed to the higher input power to this antenna, leading to the simultaneous arrival of the electromagnetic wave and the right MSSW within the 40–50 ns timeframe, resulting in interference and distortion of the right MSSW waveform.

Fig. 2.

Pulse signals detected at the detection antenna under (a) input only from Left antenna, (b) input only from Right antenna, (c) input only from Right antenna without applying external magnetic field H_ex. Pulse signals detected when inputs are applied from both Left and Right excitation antenna with phase difference (d) Df = 0.00, (e) 0.64p, and (f) 0.96p.

The spatial separation between the spin waves relative to the detection antenna manifested as a temporal difference in the signal arrival times. The wave packet centers of the MSSWs propagating from the left and right excitation antennas arrived at 57 and 68 ns, as illustrated in Fig. 2a and b, respectively. This difference is due to the 2 mm difference in propagation distance between SWL and SWR. Assuming an ideal wave packet width of 50 ns and considering the inter-antenna distance, the calculated group velocities for the left and right MSSWs were determined to be 9.4 × 10⁴ m/s and 11.6 × 10⁴ m/s, respectively, indicating close agreement. This experimental determination of group velocity was found to be consistent with the theoretical value of 8.2 × 10⁴ m/s, derived from the MSSW dispersion relation (Eq. 2) using the known physical properties of YIG³⁰ and numerical differentiation at an excitation frequency of 3.072 GHz. This concordance between the experimental and theoretical group velocities further substantiates the identification of the signals as MSSWs originating from the left- and right-excitation antennas. Based on a comparative analysis of Fig. 2a–c, the MSSWs transmitted from the right and left excitation antennas will coexist and consequently interfere within the temporal range of 68–75 ns. Figure 2d–f illustrate EMFs induced in the detection antenna for varying phase differences (Df) of the excitation signals, particularly at Df = 0.0, 0.64, and 0.96p, respectively. As the phase difference was modulated, a substantial reduction in the amplitude (V) of FMEs from 21 mV to 0.1 mV was observed within the 68–75 ns timeframe.

Figure 3 illustrates the dependence of the spin-wave amplitude within the 68–75 ns timeframe on Df. A correlation between amplitude and phase difference can be observed, with amplitude minima and maxima observed near Df = 1.0p and Df = 0, respectively. Given that the individual spin wave amplitudes were approximately 10 mV, the observed amplitude doubling near Df = 0 suggested constructive interference. Conversely, the near-zero amplitude at Df = 1.0p corresponds to destructive interference. To further elucidate this behavior, we considered the interference of two sine waves with amplitudes A₁ and A₂ and identical frequencies. The amplitude of the resulting waveform can be expressed as a function of the phase difference Df between the two sine waves.

Fig. 3.

Amplitude at t = 68–75 ns of the voltage at the detecting antenna when the phase difference Df between the left and right spin waves is changed.

3

As indicated by the dashed curve in Fig. 3, the experimental data exhibit excellent agreement with the theoretical predictions of Eq. (3). This concordance provides compelling evidence for the successful detection of spin wave interference, further confirming the validity of the experimental setup and the analytical techniques employed in this study.

Magnetic field detection

To experimentally validate the magnetic field detecting capability predicated on the principles outlined in the introduction, the modulation of the detection antenna’s amplitude in response to minute magnetic field variations (± 0.13 mT) was investigated. A small magnetic field DH, the target of detection, was generated by applying a current to a hand-made coil. Consistent with the preceding experiment, an RF pulse with a frequency of 3.072 GHz, power of 31.6 mW, and pulse width of 50 ns was introduced. Initial measurements were conducted at Df of 0.84p, where the phase of SW_R lagged that of SW_L by 0.16p compared to that in the destructive interference condition located in Df = 1.0p region identified by fitting. Figure 4 illustrates the EMF waveforms detected by the oscilloscope when SWR and SWL were excited while changing µ₀DH in the range of ± 0.13 mT. As in the preceding analysis, we focused on the amplitude within the 68–75 ns timeframe, corresponding to the spin-wave interference region. When DH is zero, the waveform amplitude was 7.3 mV in that timeframe, as illustrated in Fig. 4b. When µ₀DH was − 0.13 mT, the amplitude dropped to 3.9 mV (Fig. 4a). At µ₀DH of 0.13 mT, the amplitude increased to 10.1 mV (Fig. 4c). Thus, the amplitude varies with the small magnetic field.

Fig. 4.

Time domain signal of the interfering signal when a small magnetic field is superimposed on the bias magnetic field at the input phase difference Df = 0.84 p.

The blue triangles in Fig. 5 illustrate the correlation between the amplitude of the interfering spin wave signal and DH when Df = 0.84p. Within the experimental DH range, the amplitude almost linearly increases with increasing ΔH. The filled and open grey squares in Fig. 5 show the EMF amplitudes when the SW_L and SW_R are excited independently, respectively. The changes in the SW_L and SW_R are insignificant compared to the amplitude changes of the interfering spin wave. Therefore, the amplitude modulation observed in the interfering spin wave may be attributed to the change in the manner in which the two spin waves interfere and not magnetic field variations in the amplitude of the two single-spin waves. The DH dependence of the interfering waves with Df = 1.26p was also measured. The results are illustrated as red circles in Fig. 5. The amplitude decreases monotonically as ΔH increases. In the Df = 1.26 p condition, SW_L leads SW_R by 0.16 p, compared to the destructive condition (Df = π) as shown in Fig. 3. Therefore, the difference in the slope between the results for Df = 0.84 p and Df = 1.26 p could be related to the phase difference between the two spin waves. These results demonstrate that the magnetic field detecting function may be realized using an interference device with asymmetric spin-wave channels. The sensitivities were obtained from the linear fitting of the red and blue plots in Fig. 5, which were 24 and − 25 V/T. In addition, the sign of sensitivity can be controlled by the input phase difference.

Fig. 5.

Variation of the amplitude of the interfering signal (red circles and blue triangles), right spin wave (open squares), and left spin wave (filled squares) for phase difference Df = 0.84p and 1.26p regarding the small magnetic field ΔH. The right axis shows the normalized amplitude by the maximum amplitude of 21 mV at the constructive condition illustrated in Fig. 3.

Discussion

Numerical investigations were performed to elucidate the mechanisms underlying the observed experimental results. The observed phenomena may be attributed to alterations in the interference pattern induced by magnetic-field-dependent variations in the spin-wave wavenumber. In this study, we performed numerical calculations of the interfering sinusoidal waves serving as spin-wave analogs to investigate this hypothesis.

First, we evaluated the case where the SW_R leads the SW_L in phase, as indicated by the blue triangle in Fig. 5. Leveraging the material parameters (µ₀M_s = 175 mT, d = 10.1 μm), experimental conditions (f = 3.072 GHz, µ₀H₀ = 49.2 mT), and the spin wave dispersion relation (Eq. 2), the spin wave wavenumber was calculated to 6978 m^–1 under DH = 0. Two sinusoidal waves with these wavelengths were propagated in opposite directions from a position of ± 4 mm, and their interference was calculated accordingly. The resulting standing-wave pattern is indicated by the black curve in Fig. 6a. Similarly, the wavenumber for DH ≠ 0 was calculated using Eq. (2). As seen in Fig. 6b, a small magnetic field change of ± 0.15 mT predicts a corresponding wavenumber change between 7284 m^–1 and 6673 m^–1. The respective interference waveforms with these wavenumbers were indicated by the blue and orange lines in Fig. 6a. The results demonstrated that the small magnetic field significantly influenced the amplitude of the standing wave at a given detection location. The amplitude at the antenna position (x = – 1.0 mm) plotted against DH is shown by the blue dotted line in Fig. 6c. The calculations reproduce the experimental results illustrated by the blue triangles and show a positive slope versus ΔH.

Fig. 6.

Numerical calculation results of spin wave interference. The micro magnetic field response of (a) the interfering spin waves where SW_R leads SW_L in phase, (b) the wavenumber calculated from dispersion relation, and (c) the amplitude of interfering spin waves at antenna position (x = – 1.0 mm).

Next, we evaluate the case wherein the SW_R leads the SW_L in phase, as indicated by the red circles in Fig. 5. The calculation results demonstrated by the red dotted line in Fig. 6c reproduce the experimental results shown by the red circle, and the slope of the DH dependence is negative. These calculated results are consistent with the experimental results, suggesting that the micromagnetic field response of the amplitude observed in the experiment may be attributed to the change in the spin wavenumber. The change in wavelength owing to the small magnetic field is approximately 40 μm, making it challenging to detect using optical methods.

A brief comparison between this magnetic field detection experiment and previous reports is provided. While there have been studies on magnetic field detection using spin-wave interference, these typically involve two different spin-wave modes and are constrained by the necessity for micro-shaping of YIG and the limited conditions under which both spin waves can be excited^22–24. For instance, MSSW and BVSW cannot be excited simultaneously in an infinitely wide plane due to the absence of a common excitation frequency. As a result, microfabrication is undertaken to achieve the same excitation frequency through anisotropy. However, the frequency range that can be excited simultaneously remains narrow (e.g., the frequency range is a few mT in Ref.²²), thus limiting operational conditions. Conversely, our method benefits from using single mode spin wave, which allows for functionality with a simple strip of YIG and enables operation under any bias field.

In contrast, our method benefits from using solely single mode spin waves, which allows for functionality with a simple strip of YIG and enables operation under any bias field.

Finally, we discuss the magnetic detectivity of our method. Detectivity D (T/) is expressed as follows:

Here, S (V/T) is the sensitivity, and S_V² (V²/Hz) is the noise spectral power density. In recent years, numerous studies on magnon noise have been conducted^23,31,32. To evaluate the noise of magnon interference signals, it is necessary to extract magnon-only signals. However, in this study, the S_V could not be measured due to the electromagnetic radiation from the excitation antenna, as shown in Fig. 2c. The spin wave signal was extracted only for a short time using the group velocity difference between the electromagnetic wave and the spin wave. It was challenging to measure the spin wave interference for a long duration, which is necessary for noise measurement, especially at low frequencies below kHz. To enable noise measurements, optimizing the antenna circuit and improving the spin excitation method using an alternating magnetic field are required to disregard the effect of electromagnetic waves. These aspects will be the focus of future research. In this context, a rough estimate of the detection capability is mentioned based on past reports. Balynsky et al.²³ highlighted that the lowest level of intrinsic noise for a YIG spin wave is approximately 10^− 16 V²/Hz. Based on the value and sensitivity obtained in this study, the detectivity was estimated to be D ~ 0.4 nT/. This value is comparable to that of tunnel magnetoresistance (TMR)-based sensors studied in spintronics. For instance, the detectivity of a single tunnel magnetoresistance element without array structures and magnetic flux concentrators (MFC) is approximately 0.1 nT/^33,34. With advances in garnet thin-film miniaturization technology^10,35–37, arrays of spin-wave-based magnetic sensors will be possible. Combined with FMC technology, their sensitivity can be increased to the picotesla level, similar to those of TMR-based sensors.

In conclusion, we demonstrated a magnetic field detection using MSSW interference in YIG thin films. The concept of spin-wave interference with an asymmetric structure allows us to operate with only one type of spin wave, the MWWS, without any geometric modification of the YIG. The spin wave interference amplitude changes almost linearly for ± 0.13 mT magnetic field change, demonstrating a magnetic field detecting capability with a sensitivity of 24–25 V/T. The results indicated that the sign of the magnetic field sensitivity could be controlled by the phase of the two interfering MSSWs. Numerical calculations demonstrated that the experimental results can be explained by the wavenumber variation of the interference wave owing to the magnetic field. The obtained results shown a detectivity similar to that of a TMR magnetic sensor. Therefore, further improvement in sensitivity may be possible by arraying the elements and using MFC.

Methods

Sample preparation

Y₃Fe₅O₁₂ films, fabricated using liquid-phase epitaxy on Gd₃Ga₅O₁₂ (111) single-crystal substrates purchased from INNOVENT Technologieentwicklung GmbH, were used to propagate the spin waves. The thicknesses of the single-crystal substrate and YIG were 0.5 mm and 10.1 μm, respectively. The substrate was 2 mm (width) and 15 mm (length). The edges were placed at an angle of 45 °to suppress the reflection of spin waves.

Measurement setup

The antennas for spin-wave excitation and detection were not formed directly on the YIG but on a printed circuit board to be placed directly under the YIG. The printed circuit board comprised a MEGTRON7 (Panasonic Industry Corp.). Two excitation antennas were placed parallel to the surface with a detection antenna between them. A thin copper film is formed on the rear surface. The distance between the left excitation antenna and the detection antenna was 3 mm. The distance between the detection antenna and the right excitation antenna was 5 mm. The antennas comprised 18 μm thick Cu. The width of the excitation and detection antennas were 0.450 and 0.075 μm, respectively.

The end of the electrode on the printed circuit board was connected to the terminal of the Hirose L-bend SMA receptacle using a solder. A rectangular aluminum block with screw holes was prepared as an enclosure, on which the PCB and SMA connected to the enclosure were placed. The copper plate on the back of the printed circuit board and the enclosure make contact over a large area, thereby providing electrical continuity.

The printed circuit board was placed above the printed circuit board and secured such that the YIG film faced the printed circuit board. Double-sided tape was used to secure the printed circuit board and YIG, including the tape from the board side. Since the output voltage increases when the distance between the antenna and YIG is insignificant, the double-sided tape must be thin. The rectangular YIG film was mounted in a direction where the longitudinal direction of the film is perpendicular to the parallel excitation and detection electrodes, as illustrated in Fig. 1b.

Control of the micromagnetic field of approximately 0.1 mT was achieved by applying a current to a handmade coil placed between the sample and the permanent magnet. The bobbin of the handmade coil was fabricated using a 3D printer. The dimensions of the bobbin were φ90 mm (outer diameter), 60 mm (inner diameter), and 5 mm (thickness) to fit between the gaps of the electromagnets. Enameled wire (conductor outer diameter 0.1Φmm, enamel outer diameter 0.12 mm) was wound 100 times around the bobbin to form a coil. This coil was placed between the sample and the permanent magnet. A DC current source (YOKOGAWA 7651) was connected to the coil, and a current of – 40 to 40 mA was applied to control the magnetic field from − 013 to + 0.13 mT. The applied current and generated magnetic field were calibrated using Hall sensors.

Numerical calculations

To mimic the experiment, numerical calculations were performed for the interference between a sinusoidal wave propagating in the negative direction from x = 4 mm (SW_R) and a sinusoidal wave propagating in the positive direction from x = – 4 mm (SW_L). The amplitudes and wavelengths of the two sinusoidal waves were identical. The analysis focused on the amplitude at x = – 1 mm, where the detection antenna is located. Standing waves are formed by the interference of two waves. The amplitude at x = – 1 mm depends on the phase difference between SW_R and SW_L. The phase difference in the calculations was used to reproduce the experimental measurements. The phase difference in the calculation was adjusted to ensure that the calculated amplitude at the detection antenna position matches the amplitude at x = – 1 mm obtained in the experiment at ΔH = 0. Here, we used the amplitude obtained from the experimental measurement normalized by the maximum amplitude of 21 mV under the construction conditions shown in Fig. 3. The most suitable phase difference for reproducing the experimental results is 0.79π. The standing wave pattern at that time is indicated by the black curve in Fig. 6a. This phase difference of 0.79π differs from that of 0.84π set by the phase shifter in the experimental measurements. The phase difference between the input signal and excitation antenna deviates from the experimental setting of the phase shifter owing to the cable length difference between the two input signals and the phase delay caused by the attenuator.

Author contributions

Y. N. and K. S. planned and designed the experiments. Y.N. conducted the experiments and analyzed the results. All the authors discussed the results and wrote the manuscript.

Data availability

The data measured and/or calculated during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.