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Abstract
Imaging of electronic device surface or sub-surface electromagnetic fields under operating conditions is important for device design and diagnosis. In this study, we proposed a method to characterize specific magnetic field properties of electromagnetic devices at micron-scale using a solid-state quantum sensor, namely diamond nitrogen-vacancy (NV) centers. By employing a wide-field magnetic field measurement technique based on NV centers, we rapidly obtain the first-order magnetic field distribution of anomalous regions. Furthermore, we approximate the second-order magnetic field (magnetic gradient tensor) using the differential gradient method. To visualize the electromagnetic anomalous regions boundary, we utilize the tensor invariants of the magnetic gradient tensor components, along with their nonlinear combinations. The identification error rate of the anomalous regions is within 12.5%. Additionally, the electromagnetic field of anomalous regions is simulated showing the measurement accuracy. Our study shows that the experimental results are very similar to the theoretical simulation of the electromagnetic field (error: 7%). This work is essential for advancing electromagnetic field characterization of electronic devices and the advancement of quantum magnetic sensor applications.
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1. Introduction
Imaging of the electromagnetic anomalous distribution region (a region of strong magnetic field due to the current concentration effect caused by line structure) of electronic devices under operating conditions is very important and can be used for structural analysis of surface or sub-surface defects on the devices [1,2]. With the continuous shrinkage of the microdevices and the continuous improvement of the manufacturing process, structural defects, crosstalk and circuit defects will seriously affect the normal operation of the devices. Consequently, the testing of microdevices has become a crucial topic. Over the past few decades, numerous microdevice test methods and instruments have been proposed and utilized. Currently, the primary methods for chip testing include optical testing [3], X-ray imaging [4], and flying probe testing [5]. Optical testing uses an industrial camera to photograph the microdevices and then analyze the surface defects through image processing algorithms [3]. Although the optical method is simple and effective, it has limitations due to constraints in the light field, rendering it unable to detect devices sub-surface. X-ray imaging, based on varying absorption characteristics of different materials, allows us to visualize the internal structure or sub-surface of microdevices and identify anomalous regions within the components [4]. This method enables the analysis and image of the entire device, but it can be expensive, time-consuming, and mainly used as a follow-up to determine defects. Flying probe testing facilitates thermal parameter measurements using a computer-controlled moving probe to test microdevices [5]. However, this method requires the attachment of fixtures, which increases the risk of damaging the test sample. We need to find a kind of testing technology that allows for testing under shelter conditions and can detect issues without causing interferences to the sample while the components are running. Therefore, developing a testing method to assess the electronic device's anomalous region in operation is extremely important [6].
Rapidly developing quantum precision sensing technology provides an opportunity for developing and researching next-generation measurement instruments in various fields [7,8]. Quantum sensors based on nitrogen-vacancy (NV) centers exhibit remarkable magnetic detection sensitivity and spatial resolution, even at room temperature, and the quantum state of NV centers can be easily polarized and read by optical means [9–11]. They also offer the capability to achieve vector magnetic field imaging over a wide-field of view, with time scales compressed to as short as ten seconds [12]. Additionally, magnetic gradients are places where the magnetic field domain changes dramatically and can reflect the boundaries of the magnetic field [13–15]. Using the magnetic gradient tensor (MGT) data based on the second-order vector magnetic field provides higher resolution and boundary information than traditional vector magnetic field data [16,17]. The MGT component and nonlinear combination (tensor invariant) could provide more boundary information while neglecting the effects of the magnetic field.
In this paper, we have practically extended the application of the first-order wide-field vector magnetic field measurement technique, which has been successfully used for electromagnetic anomalous region surface and sub-surface inspection of microdevices. To verify the accuracy of the first-order magnetic field measurements, we conducted first-order wide-field vector magnetic field measurements on a sample (width: 200 µm-600 µm) with different currents. The results showed good agreement with the simulation data (error: 8%). Furthermore, we derived the second-order vector magnetic field by employing the differential approximation method to characterize the boundaries of the samples. The results obtained at different currents aligned well with the anomalous region locations and sizes. These findings present a new idea for electromagnetic anomalous distribution inspection by expanding the traditional first-order magnetic field measurement to the second-order, and also expand the application of quantum sensors in the microdevices testing field.
2. Principles and experimental setup
2.1 Experimental setup
The NV center is a special kind of point defect in diamond, which is composed of a nitrogen (N) atom and an adjacent vacancy (V) [18]. The NV center for quantum applications is mainly negatively charged NV centers. The NV center exhibits a ground state triplet (3A2). When pumped with a green laser, the NV center will jump from the ground state to the excited state (3E) and emit red fluorescence (637-800 nm), and the quantum state (ms = 0, ms=±1) of the NV center can be read from the fluorescence intensity for the presence of the metastable state (1A, 1E) [9]. At the same time, the quantum state of the NV center can be manipulated by a microwave field, and by scanning the microwave frequency in a specific range, we can obtain the fluorescence intensity as a function of the microwave frequency, which is called optically detected magnetic resonance (ODMR). The ms=±1 state of the NV center experiences Zeeman splitting under an external magnetic field, resulting in two peaks in the ODMR spectral line. Thus, the magnitude of the applied magnetic field along the NV axis can be solved by recording the peak shift. Since there are four axial directions in the NV ensemble diamond, using specific direction magnetic field will generate eight peaks in the ODMR spectrum. This enables the measurement of vector magnetic field [19].
The experiment system consists of three modules: the optical path module, the microwave module, and the synchronous acquisition system, as shown in Fig. 1(b). A high-power (1 W) 532 nm laser (Changchun New Industries Optoelectronics Technology, MGL-III-532-1W) is employed in the optical system to ensure sufficient polarization efficiency. To control the light intensity, the emitted laser beam passes through a half-wave wavelength (1/2λ) and a polarized beam splitter (PBS) cube. It is then expanded using a lens group and directed toward the diamond through a dichroic mirror and an objective (10×, 0.25 NA). The objective is adjusted to provide a wide-field of view by modifying the distance between the objective and the diamond. The fluorescence emitted from the diamond is filtered and collected by a camera (Thorlabs, CS505MU) and a photodetector (Thorlabs, APD4303A2/M). The photodetector, connected to an oscilloscope (Tektronix, DPO5204B), is used to detect changes in fluorescence intensity. In the microwave system, a microwave signal from a microwave source (Rohde & Schwarz, SMA100B) passes through a power amplifier to provide sufficient ODMR contrast and is radiated onto the diamond via an antenna. The camera’s exposure time is 1.5 ms with a storage time of 4.5 ms, resulting in a camera frame rate of 166 frames per second (FPS). An arbitrary waveform generator (Tektronix, AWG 5204) sends synchronization control signals to the camera and the microwave source every 6 ms. This control ensures the camera’s acquisition starts at the proper time and the microwave source steps in frequency accordingly.
Fig. 1. (a) The diagram of NV center’s energy level. (b) Schematic diagram of the experimental setup. (c) The typical ODMR spectral lines.
Figure 1(b) shows a sample of our designed and fabricated defective circuit with a width of 200 µm at the center of the defect and 600 µm on both sides. The substrate material used is a fiberglass-reinforced epoxy laminate (FR-4), and the conductive lines are made of copper with a thickness of 35 µm.
The diamond used in the experiment is a type Ib single-crystal high pressure and high temperature (HPHT, size: 4 mm × 4 mm × 0.5 mm) diamond purchased from Element Six. It has an initial nitrogen (N) concentration of about 200 ppm. It was subjected to high-energy electron irradiation at 1 MeV (dose: 1.8 × 1018 cm2) and annealed at 800 °C for four hours (NV density: about 2.5 ppm). The diamond's top surface (sensing surface) is parallel to (100) crystal orientation.
The frequency range of the microwave source is set from 2670 MHz to 3090 MHz, with a frequency step of 0.3 MHz. The camera has a pixel range of 600 × 1000 with a pixel size of 3.54 µm. The image size is reduced to 150 × 250 after pixel merging to improve signal-to-noise ratio (SNR). It takes approximately 7.8 s to acquire the signal once and about 390 s to repeat it 50 times in the experiment to reduce the effect of ambient noise. The laser is kept continuously during the experiment, and the microwave source and camera are controlled to achieve a wide-field ODMR signal acquisition. To measure the eight-peak ODMR signal, an external auxiliary magnetic field is first applied to ensure that all four directions of the NV axis in the diamond have magnetic field components [20,21]. Then, a spatial Cartesian coordinate system is established to calculate the magnetic field felt by each axis, which allows for obtaining the vector of the applied magnetic field. Figure 1(c) shows a typical ODMR signal with eight peaks. Next, a current is applied to the test sample, causing the resulting magnetic field to disturb the initial bias magnetic field and generate a new ODMR signal through the Zeeman splitting. The new ODMR signal is then measured, and the magnetic field obtained from the solution is combined with the bias magnetic field to calculate the vector magnetic field generated by the current. Moreover, the magnitude of the current can also be calculated using the Biot-Savart law to assess the reliability of the results [19].
2.2 Experimental principles
Neglecting nuclear spin interaction and considering zero strain, the ground state Hamiltonian of the NV center can be expressed as follows [10]:
where D∼2.873 GHz is the zero-field splitting term and ${S} = \{{{{S}_{x}},{{S}_{y}},{{S}_{z}}} \}$ is thespin operator. ge represents the Landé factor, µB is the Bohr magneton, and BNV is the magnitude of the magnetic field component parallel to the NV axis. The frequency resonance peak corresponding to the ground state degeneracy by the Zeeman effect after the application of a magnetic field can be expressed as:Thus, the applied magnetic field can be solved by the ODMR resonance peaks:
Since the ensemble NV centers consist of four directions of the NV axis, the vector magnetic field can be calculated based on the orthotetrahedron structure of the NV centers and the crystal axis direction [19]. This is achieved by splitting the eight-peak by applying an external magnetic field. After using an external magnetic field, the vector magnetic in the laboratory coordinate system can be represented by the changes in the magnetic field experienced by the four NV axes, as shown in Fig. 2(a).
After being measured using Eq. (4), the wide-field vector magnetic field can be expressed as:
In principle, the MGT can be expressed as the partial derivatives of the vector magnetic field, usually denoted by the G-matrix [13,22–24]:
Since the magnetic field has zero divergence and curl, there are only five independent variables in the G-matrix [24]:
For the MGT data, Bxx and Byy determine the extent of the boundaries of the magnetic field, Bxy determines the boundary points of the magnetic field, Bxz determines the boundary along the y-direction, and Byz determines the boundary along the x-direction. Using a single magnetic gradient tensor component for magnetic field boundary identification will lose the useful information in the other components. To avoid this situation, the nonlinear combination of the components of the magnetic gradient tensor can be used for boundary identification. Since the magnetic gradient tensor has three rotational invariants (tensor invariants), which are not affected by the rotational transformation of the coordinate system and at the same time combine the different components of the magnetic gradient tensor, the magnetic field boundaries can be identified more accurately. The tensor invariants can be expressed as following formula [25].
Differential estimation is generally used since the magnetic tensor data cannot be acquired directly. We can quickly acquire the MGT information by taking advantage of the wide-field imaging of the NV centers. In practical applications, a differential approximation is used to obtain MGT information, as shown in Fig. 2(b):
To analyze the error, it is necessary to convert the current to a magnetic field using the Biot-Savart law. The magnetic field generated by an energized long straight wire can be expressed as [26]:
where µ0 is the magnetic permeability, I is the magnitude of the current flowing through the wire, and d is the distance of the wire from the observation point.3. Experimental results and analysis
The wide-field vector magnetic field distribution of the bias field (current: 0 A) was first measured during the experiment. Then, the magnetic field distribution was measured again after changing the current strength. The magnetic field distribution corresponding to each current was calculated according to Eq. (4). It is important to note that the direction of the current was kept constant throughout the experiment, and only the magnitude of the current was changed.
The measurement results at a current of 0.5 A are shown in Fig. 3(a)-(d). To determine the measurement results’ reliability, we simulated the magnetic field distribution generated by the electromagnetic sample using the electromagnetic simulation software (COMSOL Multiphysics, distance: 0.22 mm), as shown in Fig. 3(e)-(h). The simulation of the vector magnetic field and magnetic field synthesis greatly agrees with the measured results. The highest value of the total magnetic field simulation at a current of 0.5 A is 3.26 Gauss (Gs), while the measured result is 3.04 Gs (error: 0.22 Gs). Furthermore, the overall trends of the magnetic field simulation and measurement results exhibit high consistency, demonstrating the accuracy of the NV wide-field vector magnetic field measurements. In both the experiment and the simulation, the direction of the current is toward the right (y direction). For the width at the anomalous regions, 200 µm, the light field image measurement is 199.68 µm, and the magnetic induction intensity measurement is 216.32 µm (error: 8%).
Fig. 3. The measurement and simulation results of the vector magnetic field distribution at 0.5 A current intensity. (a)-(d) The measurement results (unit: Gs). (e)-(h) The simulation results (unit: Gs).
From the results in Fig. 3, it can also be found that each component of the vector magnetic field has its significance. The Bx component indicates the four points where the wire is narrowed from wide to narrow; the absolute value of the By component suggests the magnitude of the magnetic field generated when the wire is of different widths. The Bz component indicates the direction of the current in the wire. The vector magnetic field distribution shows that the By component is a representative data source for the diagnosis of electromagnetic anomalous region. It can be used as a reference for analyzing electromagnetic anomaly from the vector magnetic field because the current direction is consistent with the Y direction. The current direction also affects the results of electromagnetic anomaly diagnosis. The relationship between the current's direction and the magnetic field's direction can be described using Ampere’s law. This is consistent with the experimental results.
Figure 4(a)-(h) shows the difference in magnetic field strength corresponding to different current intensities. Figure 4(i) shows the composite plot of the region with maximum magnetic field strength extracted from the measurement results, Biot-Savart law, and simulation results. The results show that the vector magnetic field measurements exhibit precision within the current range between 0 and 0.8 A. However, the vector magnetic field is difficult to distinguish at higher currents due to the simple merging of the resonance peaks.
Fig. 4. (a)-(h) The magnetic field varies with current magnitude. (i) Measurement and simulation results at different currents magnitude, the d in Biot-Savart simulation is 0.33 mm.
The results obtained from By component at different currents have a common tendency that the absolute value of the magnetic field at the anomalous regions becomes stronger (0.71 Gs-4.78 Gs), and the magnetic field on the left and right sides of the anomalous regions is symmetrically weaker than in the center (0.38 Gs-2.68 Gs). This could be attributed to the narrowing width and reduced cross-sectional area of the intermediate anomalous regions, leading to an increase in the conductor's current density per unit volume. In contrast, the magnitude of the current passing through the intermediate anomalous regions remains constant. As a result, the magnetic field intensifies at the intermediate anomalous regions, giving rise to a series of issues, such as increased heat generation. These findings demonstrate that the By component can detect some electromagnetic anomalous regions, but this is only characterized at the intensity level, and there is no boundary identification at the anomalous regions. It is necessary to use deeper information or apply image processing algorithms to obtain accurate boundaries of electromagnetism anomalous regions.
We processed the vector magnetic field components at 0.5 A. for further electromagnetic anomalous analysis according to Eqs. (7) and (9) to obtain the five components of the magnetic gradient tensor information, as shown in Fig. 5.
Fig. 5. Results for the five MGT independent components of the magnetic gradient tensor (unit: a.u.). (a) Bxx component. (b) Bxy component. (c) Bxz component. (d) Byy component. (e) Byz component.
The Bxx and Bxy components provide limited valid information, probably because the Bx component has only four dots of valid information and is affected by noise. The boundaries in the two boxes in the Bxz component show information about the boundaries underneath the anomalous regions. The Byy component has the strongest ability to define the boundaries and can visualize the boundaries of the magnetic field quite well. The black boxes 1 and 3 on both sides show the boundary information at the width of the two sides; the black box 2 in the middle has negative values above and positive values below, and in the middle is where the anomalous regions are located. The Byy component has an anomalous region spacing of 224.64 µm (error: 12.5%), estimated from black box 2, and a distance of 607.36 µm from black box 1. The width of the measurement in black box 1 in the Byz component is 183.04 µm (error: 8%), and black box 2 is probably due to an anomalous boundary due to stray fields. However, it is not easy to observe the anomalous regions boundaries directly in Byy-component and Byz-component images, which are easily affected by stray fields and noise.
The I1 and I2 images calculated according to Eq. (8) are shown in Fig. 6. There are two boundary lines faintly in black box 1 in the I1 image, but the values are particularly small and can be ignored. The black box 2 clearly shows the boundary of the anomalous regions, which is different from both By and Byy. By and Byy images will be all the boundaries of the magnetic field to indicate the subsequent processing to get the anomalous location. However, it can be done to identify the anomalous regions, or it could be more intuitive and accurate than the I1 image. The I1 image of the black box 2 is identified by the anomalous regions’ width of 183.04 µm-224.64 µm. The actual recognition of anomalous regions can be directly based on the I1 image of the obvious region directed to find the anomalous regions location for analysis. The I0 image is not shown here as it is constant 0. No useful information is shown in the I2 image either, and it is not discussed here.
Fig. 6. Tensor invariant electromagnetic anomalous regions identification results at 0.5 A current intensity (unit: a.u.). (a) I1 component. (b) I2 component.
Figure 7 shows the recognition of anomalous regions in the I1 image at different currents. The tensor invariants, the vector magnetic field components, and the MGT components are not recognized at 0.1 A and 0.2 A. In the case of currents above 0.3 A, there is a clear difference, which becomes more obvious with the current increase.
The vector magnetic field components, MGT components, and tensor invariants have their recognition characteristics in different situations, and the corresponding images with advantages can be selected for electromagnetic anomalous regions diagnosis depending on the situation.
Simultaneously, our proposed MGT imaging electromagnetic detection method does not conflict with existing electromagnetic detection methods. There are several vector imaging methods for electromagnetic detection based on NV center [19,20,27–29], while our method is optimized on the basis of vector detection. Therefore, in this paper, we focus on the acquisition method of tensor data and the advantages of comparing vector detection under the same experimental platform.
4. Conclusion and outlook
Characterization techniques of electromagnetic anomaly can provide a reference for the operating condition of the microdevices. Traditional methods rely on image processing techniques to determine the anomalous region of electromagnetic, which are affected by data and environment. In this paper, the method comprehensively determines the boundary of the electromagnetic anomalous region through different components of the magnetic gradient tensor and nonlinear combination information, which avoids useless information and highlights the boundary of the electromagnetic anomalous region. Utilizing quantum magnetic sensors for the anomaly diagnosis of electromagnetism can give full play to its advantages of high sensitivity, high resolution, and fast wide-field vector magnetic field detection. The electromagnetic anomalous region diagnosis is analyzed by vector magnetic field, magnetic gradient tensor component, and tensor invariant of electromagnetic defect samples with no more than 12.5% error, unaffected by light field and stray magnetic field. Moreover, this paper also indicates the advantages and disadvantages of the light field, vector magnetic field component, magnetic gradient tensor component, and tensor invariant for electromagnetic anomalous region diagnosis, which provides ideas for applying quantum magnetic sensors to electromagnetic diagnosis.
Funding
Joint Funds of the National Natural Science Foundation of China (U21A20141); National Natural Science Foundation of China (52275551); Shanxi Scholarship Council of China (2021–117).
Disclosures
The authors declare no conflicts of interest related to this paper.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon reasonable request.
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