Wide-field tomography imaging of a double circuit using NV center ensembles in a diamond

Zhonghao Li; Zhonghao Li; Zhonghao Li; Zhonghao Li; Yanling Liang; Yanling Liang; Yanling Liang; Yanling Liang; Chong Shen; Chong Shen; Chong Shen; Zhenrong Shi; Zhenrong Shi; Zhenrong Shi; Huanfei Wen; Huanfei Wen; Huanfei Wen; Hao Guo; Hao Guo; Hao Guo; Zongmin Ma; Zongmin Ma; Zongmin Ma; Jun Tang; Jun Tang; Jun Tang; Jun Tang; Jun Liu; Jun Liu; Jun Liu; Jun Liu

doi:10.1364/OE.469077

1. Introduction

The development of information technology has led to the wide use of integrated circuits (ICs) and chips in various fields. This is mainly because of their large capacity, low cost, and high reliability. During the operation of a chip, current passes through a circuit and generates a magnetic field. Therefore, magnetic-field imaging can be used to characterize the magnetic-field distribution of ICs and determine their working conditions. This technology can be used to locate and diagnose faults in electronic devices [1].

ICs inspection techniques are primarily implemented using magnetic force microscopy (MFM) [2], electron and X-ray microscopy [3], superconducting quantum interference devices (SQUIDs) [4], and giant magnetoresistance (GMR) magnetometer sensors. MFM is generally used to inspect magnetic devices; however, it requires a long time for data acquisition. Electron and X-ray microscopy can provide a high spatial resolution; however, they require cryogenic cooling, careful sample preparation, and high-vacuum conditions. SQUID microscopy is a scanning magnetic-imaging detection technique. It consists of a superconducting loop circuit with two Josephson superconducting junctions connected in parallel [5]. This provides high sensitivity and spatial resolution [6] and the ability to locate open circuits by detecting an AC signal that propagates in a path. However, this method requires point-by-point scanning to create an image. In addition, it requires cryogenic and vacuum covers to ensure measurement conditions. A SQUID can detect only out-of-plane field components (B_z). A GMR magnetometer is based on the magnetoresistance effect. It consists of an ultrathin, conductive, and nonmagnetic layer (e.g., copper) sandwiched between two ferromagnetic layers. Its application in imaging is limited because of the specificity of its structure and principle [7]. The abovementioned drawbacks make it difficult to image large-size multilayer ICs and reduce the detection efficiency. Furthermore, dense arrangements of ICs are used to reduce power consumption and improve the versatility in operation. Therefore, magnetic-field imaging should have a wide field of view, high resolution, and tomography capabilities.

The rapid development of the quantum precision measurement technology has led to the creation of precision sensing and detection techniques based on nitrogen-vacancy (NV) centers in diamonds [8–11]. NV centers are typically used to obtain the magnetic-field intensity [12–14]. The imaging of magnetic-field distributions using NV centers has been demonstrated [15,16], such as scanning probe microscopy using single NV centers in diamonds [17,18] and wide-field magnetic imaging using NV center ensembles in diamonds [19]. For wide-field magnetic imaging, the ensembles of an NV center in a diamond can provide the vector distribution of a magnetic field at the millimeter scale. Moreover, imaging results can be rapidly obtained without spatial scanning [20]. Therefore, this method has excellent potential for the high-precision, dynamic, and real-time imaging of wide-field magnetic-field distributions. However, the effectiveness of wide-field imaging using NV center ensembles in diamonds has not been systematically reported for multilayer circuits.

This study develops a system for wide-field vector magnetic imaging using NV center ensembles in a diamond. The magnetic-field distribution on the surface of a circuit produced by the lower circuit layer is separated using vector magnetic-field superposition. The inversion of the magnetic-field distribution on the surface of the lower layer is obtained using the inversion model based on the Biot-Savart law. The magnetic-field distribution is measured at different depths of the lower layer to verify the reliability of the experiments. The magnetic-field imaging of the surface of a double circuit is performed at different currents and depths to obtain the inversion law of the circuit tomography. The magnetic-field distribution on the surface of the lower layer of the double circuit is obtained. This method has the potential to provide an effective reference for the detection and troubleshooting of multilayer circuits and ICs and high-precision magnetic imaging at the submicron scale and nanoscale [1,11].

2. Principles and experimental setup

2.1. Experimental principles

NV centers are defects in diamonds, which consist of a nitrogen atom (N) and an adjacent vacancy (V) that replaces two carbon atoms and their bonding in the diamond lattice. NV centers have two different fluorescent charge states: NV° and NV^-. The NV stands for NV^- in this paper [21,22]. When an NV center is irradiated with a 532 nm laser, its energy level is excited from the ground state (³A₂) to the first excited state (³E), as shown in Fig. 1(a). Additionally, two types of spontaneous radiation are observed. The first is red fluorescence emission from the excited state to the ground state, which accounts for most of the process. The other is the intersystem crossover, as shown in Fig. 1(a). The single-linear states (¹A₁ and ¹E) return to the ground state through spin-orbit coupling, which does not produce fluorescence and does not fluoresce. Thus, the fluorescence intensity is plotted as an optically detected magnetic resonance (ODMR) signal curve by scanning the microwave frequency.

Fig. 1. Schematic of the experimental principle. (a) NV center energy levels. (b) ODMR (black) and Lorentz-fitted curve (red) for one pixel.

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When the microwave frequency resonates with the transition frequency between the excited and ground states, the fluorescence intensity is significantly reduced. The m_s= -1 and m_s = + 1 states are degenerated if the NV is in a magnetic field at 0 Gs. Moreover, the ODMR curve shows splitting resonance only at ${\sim}$2.87 GHz. When the magnetic field is not zero, the magnetic field projected on the NV axis will split the m_s= -1 and m_s = + 1 states due to the Zeeman effect and can produce the multi-peaks of the ODMR. This leads to the splitting of the resonance peaks of the ODMR curve.

When an external magnetic field with different components projected on the four NV axes is applied, the simplification of the m_s= -1 and m_s = + 1 states results in the splitting of the peaks of the ODMR curve into eight peaks. The points of the four pairs of peaks contain information about the magnetic-field intensity at the NV center in the direction of the four axes. The ODMR curve with eight peaks is shown in Fig. 1(b).

Based on the vector magnetic-field imaging with NV centers, the surface magnetic-field distribution of a double circuit is obtained when the upper and lower wires are simultaneously energized. If only the upper wire is energized, then the vector magnetic-field distribution is produced on the upper surface. The magnetic-field vectors of the double circuit and upper layer are subtracted according to the principle of superposition and cancellation of trivial vector magnetic fields. Finally, the magnetic-field distribution on the lower layer surface is obtained using the Biot-Savart law. The schematic of the magnetic-field distribution inversion in a double circuit is shown in Fig. 2.

Fig. 2. Schematic of tomography and inversion for double circuit: (a) Schematic of double circuit. (b) Magnetic-field distribution on the surface of double circuit produced by the upper layer. (c) Magnetic-field distribution on the surface of double circuit produced by the lower layer. (d) Magnetic-field distribution on the surface of lower layer after inversion.

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2.2. Experimental setup

The schematic of the experimental setup is shown in Fig. 3. It consists of an optical path system, microwave system, magnetic field system, synchronization control system, and data processing system. The 532 mm laser is radiated on the diamond that fills the field of view of the camera through the optical path system, and the power is approximately 300 mW. Then, the red fluorescence that is spontaneously emitted from the NV center is collected by the camera with a 4×/0.1 (Magnification is 4, and numerical aperture is 0.1) objective. Green light is filtered out using a longpass filter. The diamond sample (Type Ib, Element Six) is produced using high-pressure and high-temperature synthesis processes. The initial nitrogen concentration is <200 ppm, and the ¹³C abundance is approximately 1.1%. The diamond is irradiated with 10 MeV electrons for 4 h to a total dose of 1.8 × 10¹⁸ cm² and then annealed in vacuum for 1.5 h at 850 °C. The NV^- concentration is approximately 3 ppm. The bottom surface of the diamond is parallel to the (001) crystal plane of the unit cell, and its two perpendicular sides are parallel to the (100) and (010) crystal planes of the unit cell.

Fig. 3. Schematic of the experimental setup.

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The vector signal generator provides a swept microwave frequency of 2.6–3.2 GHz in steps of 0.3 MHz. A pair of permanent magnets produces a steady-state bias magnetic field of approximately 21 Gs to separate the ODMR curve into eight peaks. The synchronization control system synchronizes the microwave sweep step with the camera exposure time (3 ms). The frame rate of the camera is 30.8 fps, and the time required to collect a dataset is approximately 65 s. The data processing system first extracts the data of the images saved in the camera according to the acquisition time in a sheet-by-sheet manner and saves it as a three-dimensional matrix. The curve obtained using the values at each pixel point of this matrix is saved separately. This is the ODMR curve of the fluorescence intensity at a single pixel point (black line in Fig. 1(b)). The eight peak points are extracted by fitting the curve using an algorithm based on Eq. (4). The difference is obtained by subtracting (two by two) the four peak point groups from the outside to the inside. All peak points are settled through coordinate transformation. Then, the magnetic-field intensity and direction at this pixel point are obtained. Finally, the magnetic-field intensity at all pixel points is combined to obtain the wide-field magnetic-field distribution.

2.3. Magnetic-field tomography

For a given NV orientation, the Hamiltonian of the NV color-centered ground-state electron spin can be expressed as [23,24]

(1)$$H = hDS_z^2 + hE({S_x^2 - S_y^2} )+ g{\mu _B}B. $$

In the first term, h is the Planck constant, and D${\sim}$2.87 GHz is the zero-field splitting parameter of the electron spin in the direction of the NV axis. The second term describes transverse zero-field splitting. In the absence of an external electric field and transverse stress E = 0, S_x, S_y, and S_z represent the spin angular momentum operators of the NV electron with a spin number of 1. The third term is the Zeeman term for the static magnetic field, which can be used for magnetic-field detection. S represents zero-field splitting with S_x, S_y, and S_z as components; g is the Lund factor, which is approximately 2.0; and µ_B is the Bohr magneton. The resonant frequencies of the ODMR curve can be derived as follows [25,26]:

(2)$${f_{res}}_ \pm{=} D \pm g{\mu _B}{B_{NV}}/h,$$

(3)$${B_{NV}} = \frac{{{f_{res}}_ +{-} {f_{res}}_ - }}{{2g{\mu _B}/h}},$$

where f_res+ and f_res- denote the resonant frequencies for m_s = + 1 and m_s = -1, respectively, when the intensity of magnetic-field component B_NV is on the NV axis. It can be shown that the resonant frequency f_res at the NV center has a linear relationship with the magnetic-field intensity. The magnetic-field intensity increases owing to Zeeman splitting. The two resonant frequencies also increase in accordance with the ground state transition of the NV center. Hence, the two peaks of the ODMR spectrum increase linearly.

In the experiment, a circuit board is controlled by operating control software to trigger the start signal. It simultaneously controls the microwave frequency step and image storage. Then, the ODMR curve in the frequency domain is obtained. Owing to the Zeeman effect, the ODMR curve can be constructed to show four pairs of peaks by controlling the magnetic-field direction [27]. At this point, the variation in the four pairs of peak points is obtained by controlling the magnetic-field intensity. The ODMR curve is fitted using the least-squares method of the Lorentz formula, as shown in Fig. 1(b) [28].

(4)$$f = {y_0} + \frac{{2A}}{\pi }\mathop \sum \nolimits_{i = 1}^8 \left( {\frac{\omega }{{4{{({x - {x_c}} )}^2} + {\omega^2}}}} \right).$$

y₀ is the initial height of the ODMR curve, ω is the full width at half maximum (FWHM) of the ODMR curve, and x_c is the peak point of a single peak on the curve. The peak value is y_c = y₀+ 2A/(ωπ).

As shown in Fig. 4(a), we define the laboratory coordinates on the diamond surface as the XOY plane with the Z axis perpendicular to the surface. The unit vector corresponds to the laboratory coordinates in the direction, where the axial direction of each NV center is as follows [28–30]:

(5)$$\left\{ {\begin{array}{{l}} {\overrightarrow {{u_1}} = \left( { - \sqrt {\frac{2}{3},} 0,\sqrt {\frac{1}{3}} } \right) = \frac{1}{{\sqrt 3 }}\left( { - \sqrt 2 ,0,1} \right)}\\ {\overrightarrow {{u_2}} = \left( {\sqrt {\frac{2}{3},} 0,\sqrt {\frac{1}{3}} } \right) = \frac{1}{{\sqrt 3 }}\left( {\sqrt 2 ,0,1} \right)}\\ {\overrightarrow {{u_3}} = \left( {0,\sqrt {\frac{2}{3},} \sqrt {\frac{1}{3}} } \right) = \frac{1}{{\sqrt 3 }}\left( {0,\sqrt 2 ,1} \right)}\\ {\overrightarrow {{u_4}} = \left( {0, - \sqrt {\frac{2}{3},} \sqrt {\frac{1}{3}} } \right) = \frac{1}{{\sqrt 3 }}\left( {0, - \sqrt 2 ,1} \right)} \end{array}} \right.. $$

Fig. 4. (a) Direction of the bias magnetic field relative to the NV center axes; (b) Principle of the magnetic-field inversion.

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Here, $\overrightarrow {{u_1}} $, $\overrightarrow {{u_2}} $, $\overrightarrow {{u_3}} $, and $\overrightarrow {{u_4}}$ are the unit vectors of the four axes of the NV center. The external vector magnetic field is defined as $\overrightarrow {{B_n}} $. $\overrightarrow {{B_1}} $, $\overrightarrow {{B_2}} $, $\overrightarrow {{B_3}} $, and $\overrightarrow {{B_4}} $ are the magnetic-field components in the four NV-axis directions, which are measured using the NV center. They are expressed as follows:

(6)$$\left\{ {\begin{array}{{c}} {\overrightarrow {{B_1}} = \overrightarrow {{B_n}} \cdot \overrightarrow {{u_1}} }\\ {\overrightarrow {{B_2}} = \overrightarrow {{B_n}} \cdot \overrightarrow {{u_2}} }\\ {\overrightarrow {{B_3}} = \overrightarrow {{B_n}} \cdot \overrightarrow {{u_3}} }\\ {\overrightarrow {{B_4}} = \overrightarrow {{B_n}} \cdot \overrightarrow {{u_4}} } \end{array}} \right.. $$

B₁, B₂, B₃, and B₄ denote the modes of $\overrightarrow {{B_1}} $, $\overrightarrow {{B_2}} $, $\overrightarrow {{B_3}} $, and $\overrightarrow {{B_4}} $, respectively, which are calculated sequentially by four pairs of peaks from outside to inside of the ODMR curve. B_x, B_y, and B_z are the modes of the magnetic-field components to be measured in the laboratory coordinate system, and they are expressed as

(7)$$\left\{ {\begin{array}{{l}} {\sqrt 3 {B_1} ={-} \sqrt 2 {B_x} + {B_z}}\\ {\sqrt 3 {B_2} = \sqrt 2 {B_x} + {B_z}}\\ {\sqrt 3 {B_3} = \sqrt 2 {B_y} + {B_z}}\\ {\sqrt 3 {B_4} ={-} \sqrt 2 {B_y} + {B_z}} \end{array}} \right.. $$

Equations (5), (6), and (7) are combined to obtain B_x, B_y, and B_z, as follows:

(8)$$\left\{ {\begin{array}{{l}} {{B_x} = \frac{{\sqrt 6 }}{4}({{B_2} - {B_1}} )}\\ {{B_y} = \frac{{\sqrt 6 }}{4}({{B_3} - {B_4}} )}\\ {{B_z} = \frac{{\sqrt 3 }}{4}({{B_1} + {B_2} + {B_3} + {B_4}} )} \end{array}} \right.. $$

The intensity $|{\overrightarrow {{B_n}} } |$ of the external magnetic field can be calculated as

(9)$$|{\overrightarrow {{B_n}} } |= \sqrt {B_x^2 + B_y^2 + B_z^2} . $$

When the double circuit is energized, B_x1, B_y1, and B_z1 are the components of the magnetic field produced by the double circuit in the X, Y, and Z directions. Similarly, B_x2, B_y2, and B_z2 are the components magnetic field produced by the upper layer circuit in the X, Y, and Z directions. These components are subtracted to obtain B_xx, B_yy, and B_zz, which are the components of the magnetic field produced by the lower layer circuit in the X, Y, and Z directions:

(10)$$\left\{ {\begin{array}{{l}} {{B_{xx}} = {B_{x1}}\; - \; {B_{x2}}}\\ {{B_{yy}} = {B_{y1}}\; - \; {B_{y2}}}\\ {{B_{zz}} = {B_{z1}}\; - \; {B_{z2}}} \end{array}} \right.. $$

Thus, the magnetic-field intensity, B_nn, produced by the lower layer on the surface of the double circuit is

(11)$${B_{nn}} = \sqrt {B_{xx}^2 + B_{yy}^2 + B_{zz}^2}.$$

As the IC is a double circuit, the magnetic-field distribution on the upper layer is the surface magnetic-field distribution of the circuit wire. The magnetic-field distribution on the surface of the double circuit produced by the lower layer has a fixed depth with respect to the surface of the lower layer (e.g., 0.32 mm). Thus, the inversion of the magnetic-field distribution for the lower layer is through the magnetic-field distribution on the surface of the double-circuit produced by the lower layer. The thickness of the copper foil of the wire in the upper and lower layers is approximately 35 µm. To facilitate the calculations, the copper foil is regarded as an ideal infinitely long metal plate with a line width of a = 0.4 mm and a uniform through the current of I = 0.5 A. The coordinate system is established, and the plate is divided into numerous infinitely long straight wires, as shown in Fig. 4(b). The width of each wire is dx, and current i is given by

(12)$$i = \frac{I}{a}dx. $$

According to the Biot-Savart law, the magnetic field at a point on each ideal long straight wire is calculated as

(13)$$dB = \frac{{{\mu _0}i}}{{2\pi x}} = \frac{{{\mu _0}Idx}}{{2\pi a\sqrt {{d^2} + {{(r - x)}^2}} }}. $$

µ₀ = 4π×10⁻⁷ mT/A is the vacuum permeability, I is the wire current, x is the distance from the point to the long straight wire, d is the perpendicular distance from the point to the metal plate, and r is the perpendicular distance from the point to the z-coordinate axis.

Thus, the magnetic-field intensity, B(d), at a point on the plane at perpendicular depth d is from the metal plate is

(14)$$\begin{aligned} B(d) &= \int_{ - \frac{a}{2}}^{\frac{a}{2}} {\frac{{kI}}{{a\sqrt {{d^2} + {{(r - x)}^2}} }}dx} \\& \textrm{ } = \frac{{kI}}{a}\ln \frac{{\sqrt {{d^2} + {{(\frac{a}{2} - r)}^2}} + \frac{a}{2} - r}}{{\sqrt {{d^2} + {{(\frac{a}{2} + r)}^2}} - \frac{a}{2} - r}} \end{aligned}.$$

Here, f(d) is defined as

(15)$$f(d) = \ln \frac{{\sqrt {{d^2} + {{(\frac{a}{2} - r)}^2}} + \frac{a}{2} - r}}{{\sqrt {{d^2} + {{(\frac{a}{2} + r)}^2}} - \frac{a}{2} - r}}.$$

As B_nn and B_x in Fig. 8(e) and (f) are 480 × 270 two-dimensional matrices, for each point in the matrix of B_nn is calculated based on Eq. (11). Finally, each point in the matrix after the inversion can be obtained as:

(16)$${B_x}(i,j) = \frac{{f({d_0})}}{{f({d_1})}} \cdot {B_{nn}}(i,j),$$

where B_x(i, j) is the magnetic-field distribution on the surface after the inversion of the lower layer and B_nn(i,j) is the magnetic-field intensity on the wire surface after the magnetic-field tomography has been determined. Here, d₀ is the distance between the NV center and the circuit surface close to the lower layer and d₁ is the distance between the lower layer and NV center. The points on the entire plane are solved individually according to Eq. (16), and then the magnetic-field distribution is determined after the inversion of the lower layer.

The magnetic-field distributions produced by the upper layer surface of the circuit for different currents are shown in Fig. 5(a). At the position of the wire, a weak magnetic field is detected at a current of 50 mA. When the current increases, the magnetic-field intensity increases and becomes more pronounced in the wire portion of the corresponding magnetic-field distribution. The magnetic-field distribution obtained at the maximum current is clearly influenced by the currents in the other wires of the double circuit. This also leads to asymmetry between the magnetic-field intensities on the left and right sides of the wires.

Fig. 5. (a) Images of magnetic-field distribution on the upper layer surface for different currents. (b) Magnetic-field intensity measured for different currents (green); simulation results of magnetic-field intensity for different currents (red).

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In Fig. 5(b), the green points show the magnetic-field intensity for different currents and the straight red line represents the simulation results. The magnetic-field intensity on the upper layer surface is proportional to the current, which is consistent with the Biot-Savart law for long straight wires (B = µ₀I/2πd). The results show that the magnetic-field distribution on the layer surface can be accurately measured using the system.

The magnetic-field distribution on the circuit surface produced by the layers with different depths is measured to demonstrate the feasibility of the tomographic measurement. To ensure the accuracy of the measurement, circuit boards with different thicknesses are prepared to control the distance between the imaging surface of the diamond and the surface layer of the circuit. Figures 6(a) and 6(b) show the magnetic-field distribution induced by the lower layer for depths of 0-0.12 mm. As the depth increases, the magnetic field gradually spreads, and the magnetic-field intensity induced by the diamond gradually decreases. When the depth increases, the signal-to-noise ratio decreases, and the diamond is increasingly disturbed by other wires around the circuit and the ambient magnetic field. Moreover, the width of the magnetic-field distribution increases. This is because, as the distance between the diamond and layer surface increases, the decay factor of the pixels at different locations in the field of view are different. This leads to different magnetic-field decays at different pixels, which are large in the middle and small at the sides. The feasibility of the tomographic measurement is verified through these results.

Fig. 6. (a) Images of the magnetic-field intensity at different depths from the circuit surface. (b) Simulated magnetic-field intensity at different depths from the circuit surface. (c) Magnetic field measured at different depths from the circuit surface (black); simulation results of magnetic-field intensity at different depths (red).

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The magnetic-field intensity of one pixel point in the middle of the wire is measured to ensure the accuracy of the measurement, as shown in Fig. 6(c). The simulation results are in good agreement with the measurement results, and the absolute error is less than 0.1 Gs. This shows that the decay process is inversely proportional to distance d, which is in agreement with the curve fitted using the Biot-Savart law. Therefore, the measurement results show that the system has potential for use in magnetic tomography.

The tomography imaging of the surface layer of the double circuit is performed using the NV centers. The upper and lower wires are prepared using the same material with the same width, thickness, and current (0.5 A). The depth of the lower layer wire is 0.32 mm. The imaging results for the magnetic-field distribution on the circuit surface are shown in Fig. 7(b). These results are consistent with the simulation results shown in Fig. 7(c).

Fig. 7. Magnetic-field imaging results and simulations. (a) Field of view; the axes are the set spatial axes, with the X-axis facing right, and the Y-axis facing up. (b) Modes of magnetic-field intensity under double-circuit energization. (c) Magnetic-field intensity distribution in the field of view was obtained using the algorithm; the magnetic-field intensity distribution map (F_n) was obtained via simulation. (d) Modes of magnetic-field intensity under the double circuit are energized by only the upper layer. (e) Modes of magnetic-field intensity under double circuit are energized by only the lower layer.

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The magnetic field produced by the lower layer of the double circuit is obtained on the basis of the vector superposition principle in Eqs. (10) and (11), and the results are shown in Fig. 8(c)-(e). These results are consistent with the results shown in Fig. 7(e) and simulations in Fig. 8(d), thereby proving that the principle of the separation algorithm is reliable. Magnetic-field inversion is achieved using Eq. (16) by utilizing the magnetic-field distribution of the separated lower layer on the surface layer. This is consistent with the simulation results in terms of the intensity and spatial distribution. It is proved that the system can realize the inversion of the surface magnetic field at a depth of 0.32 mm, as shown in Figs. 8(d)-(g).

Fig. 8. (a) Magnetic fields (F_x, F_y, and F_z) of the wires along the X, Y, and Z axes on the double circuit obtained via simulation using finite element software. (b) Components B_x1, B_y1, and B_z1 were obtained by conversion from the NV four-axis coordinate system to the spatial three-axis coordinate system. (c) Triaxial magnetic fields (B_xx, B_yy, and B_zz) when the double circuit is energized and the upper layer is energized alone after subtracting the magnetic-field vectors. (d) Simulated magnetic-field distribution after vector subtraction of the double circuit and upper layer. (e) Modes of the magnetic-field intensity after subtraction of the magnetic-field vector when the double circuit is energized by only the upper layer, corresponding to Fig. 7(e) in the ideal case. (f) Inversion of magnetic-field distance on the surface of the lower energized wire of an IC. (g) Simulated magnetic-field distribution on the lower layer after inversion.

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Finally, the magnetic-field intensities are extracted from a column of pixels on the cross section of the surface magnetic-field distribution before and after inversion, as shown in Fig. 7(d) and Figs. 8(e)-(f). The corresponding distribution curve is shown in Fig. 9. The FWHM of the curve after inversion and measured FWHM are 0.54 mm and 0.5 mm, respectively. The maximum magnetic-field intensity obtained by inversion is 8.73 Gs, and the measured intensity is 8.7 Gs. Compared with the measurements of the upper layer, the difference in the maximum magnetic-field intensity of inversion is approximately [(8.73-8.7)/8.7] × 100% ≈ 0.4% and the difference in the magnetic-field distribution of inversion is approximately [(0.54-0.5)/0.5] × 100%= 8%, where the depth of the lower layer is 0.32 mm. This shows that the inversion results are in good agreement with the measurement results. We confirm that the magnetic-field distribution on the lower layer can be accurately inverted using our model and equation. This shows the reliability of tomography imaging using the NV centers of diamonds.

Fig. 9. Magnetic-field variation curve for a column/row in the magnetic-field intensity distribution chart. The black curve is the magnetic field before the inversion, the red curve is the magnetic field after the inversion, and the blue curve is the measured surface magnetic field of the wire.

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3. Conclusion

The wide-field tomography imaging of a double circuit using the NV center ensembles in a diamond is verified using a vector tomography model. First, the vector magnetic-field calculation, separation, and inversion models are established on the basis of the NV centers and Biot-Savart law. Second, an accurate characterization of the magnetic-field distribution on the upper energized wire surface of the circuit is realized with a field of view of 2.42 mm × 1.36 mm. The magnetic-field intensity is 0-9 Gs for different currents. Then, different depth conditions, magnetic-field separation, and distance diffusion models are obtained under laboratory conditions. The magnetic field of the double-circuit wire is separated according to the principle of vector superposition. The inversion of the magnetic field distribution after the lower separation is carried out according to the Biot-Savart law. The tomographic analysis of the magnetic-field distribution on the surface of the double circuit is performed. The system can measure the magnetic-field intensity in a range of 0-9 Gs and invert the magnetic-field intensity from 0 mm to 0.32 mm depth. The maximum relative error in the magnetic-field inversion is approximately 0.4%, and the spatial resolution is 5.04 µm. The results show that the magnetic-field distribution on the surface of the double circuit is accurately measured while separating and inverting the magnetic field of the lower layer of the circuit. In the future, this can provide a new method for the functional detection and fault diagnosis of multilayer ICs based on the quantum sensing technology.

Funding

Special Fund for Research on National Major Research Instruments and Facilities of the National Natural Science Foundation of China (51727808); National Natural Science Foundation of China (51821003, 51922009, 62103385, 62175219); Key Laboratory of Shanxi Province, China (201905D121001); Fund for Shanxi “1331 Project” Key Subjects Construction, China.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Wide-field tomography imaging of a double circuit using NV center ensembles in a diamond

Abstract

1. Introduction

2. Principles and experimental setup

2.1. Experimental principles

2.2. Experimental setup

2.3. Magnetic-field tomography

3. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (16)

Optics Express