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Ultrasensitive integrated circuit sensors based on high-order non-Hermitian topological physics
Ultrasensitive integrated topological circuit sensors
https://orcid.org/0000-0002-5281-2890
Deng Wenyuan Conceptualization Formal analysis Investigation Methodology Software Validation Visualization Writing - original draft Writing - review & editing 1 †
Zhu Wei Data curation Investigation Methodology Project administration Software Validation 2 *†
https://orcid.org/0000-0003-1655-5249
Chen Tian Conceptualization Formal analysis Methodology Validation Writing - original draft Writing - review & editing 1
https://orcid.org/0000-0001-6933-5261
Sun Houjun Conceptualization Funding acquisition Project administration Resources Supervision 2 *
https://orcid.org/0000-0002-7725-8814
Zhang Xiangdong Conceptualization Data curation Funding acquisition Investigation Methodology Project administration Resources Validation Visualization Writing - original draft Writing - review & editing 1 *
1 Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements of Ministry of Education, Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China.
2 Beijing Key Laboratory of Millimeter Wave and Terahertz Techniques, School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China.
* Corresponding author. Email: zhangxd@bit.edu.cn (X.Z.); sunhoujun@bit.edu.cn (H.S.); zhuwei@bit.edu.cn (W.Z.)
† These authors contributed equally to this work.

20 9 2024
18 9 2024
10 38 eadp690507 4 2024
12 8 2024
Copyright © 2024 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
2024
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https://creativecommons.org/licenses/by-nc/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

High-precision sensors are of fundamental importance in modern society and technology. Although numerous sensors have been developed, obtaining sensors with higher levels of sensitivity and stronger robustness has always been expected. Here, we propose theoretically and demonstrate experimentally an alternative class of sensors with superior performances based on exotic properties of high-order non-Hermitian topological physics. The frequency shift induced by perturbations for these sensors can show an exponential growth with respect to the size of the device, which can grow well beyond the limitations of conventional sensors. The fully integrated circuit chips have been designed and fabricated in a standard 65–nanometer complementary metal-oxide semiconductor process technology. Not only has the sensitivity of systems less than 10−3 femtofarad been experimentally verified, but these systems are also robust against disorders. Our proposed ultrasensitive integrated circuit sensors can have a wide range of applications in various fields and show an exciting prospect for next-generation sensing technologies.

The high-order non-Hermitian topological electronic chip with robustness is fabricated for the ultrasensitive sensors.

http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 12234004 http://dx.doi.org/10.13039/501100013290 National Key Research and Development Program of China Stem Cell and Translational Research 2022YFA1404900
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pmcINTRODUCTION

Sensors with high precision play an important part in many aspects of daily life. There are various schemes for the construction of sensors relying on different physical mechanisms (1–4). Most sensors rely on resonant structures, where the shifting and splitting of frequency spectra are always used to identify the external perturbation. For example, the photonic microcavity sensor with an ultrahigh quality factor can be used to monitor the change in background refractive index, and the label-free detection of single molecules can be realized (5–7). The optomechanical transducer can be used as an ultrasensitive detector of weak incoherent forces (8). In particular, electronic sensors can also offer excellent performances in monitoring multiple environmental parameters (9–15). Recent advances in the fields of non-Hermitian physics have revealed that enhanced sensitivity can be achieved using a type of degenerate point, known as exceptional point (16–25). Furthermore, non-Hermitian topological sensors relying on the anomalous sensitivity to one-dimensional boundary states have also been proposed theoretically and demonstrated experimentally (26–28). Although numerous sensors have been developed, there is a continuous demand for sensors with increased sensitivity to detect signals that were previously undetectable. In addition, the robust property of sensors has always been a pursuit, which enables them to work in special environments.

On the other hand, higher-order topological phases in Hermitian and non-Hermitian systems have been strongly studied in recent years due to their unconventional physical properties (29–46). In particular, higher-order non-Hermitian skin effects have also been revealed in some non-Hermitian systems (47, 48). They lead to types of boundary physics, which are in contrast to skin modes in the one-order non-Hermitian systems. These higher-order skin effects originate from intrinsic non-Hermitian topology protected by spatial symmetry. The question is if these higher-order topological physics can be used to realize the sensors with higher levels of sensitivity and stronger robustness.

Here, we propose theoretically an alternative class of sensors with superior performances based on exotic properties of high-order non-Hermitian topological physics; the corresponding integrated circuit sensors are fabricated using a 65-nm complementary metal-oxide semiconductor (CMOS) process technology. On the basis of non-Hermitian topological corner states, we demonstrate extremely weak signal detection characteristics on this platform. Because of the high-frequency oscillation characteristics of the system and advanced nanotechnology, our system can maintain an error of less than 1% at frequencies up to 2 GHz. We also introduced a field programmable gate array (FPGA) module to control the non-Hermitian sensing system for achieving high-precision detection of an unpredicted measurand. Our work paves the way for ultrasensitive sensors with stronger robustness.

RESULTS

Model on high-order non-Hermitian topological sensors

Now, we provide the model and theory of non-Hermitian high-order topological sensors. We first consider two-dimensional second-order cases with a size of LX × LY, as shown in Fig. 1A. The magenta spheres in Fig. 1A represent unit cells; each unit cell consists of two lattices (1 and 2), as shown in Fig. 1B. The marks λx(y) and λxy′  λxy≠λxy′  in Fig. 1B display the nonreciprocal couplings among the lattices and unit cells. By introducing the Fourier transform and Pauli matrices σi (i = x, y, z), the Hamiltonian of the second-order system in k→ space isH=λx+λx′coskxσ0+λy+λy′cosky+iλy−λy′sinkyσx+iλx−λx′sinkxσz(1)

where σ0 is the identity matrix. Under the open boundary condition, we can calculate the energy spectrum of the system, as shown in Fig. 1C. It should be noted that the skin mode appears at zero energy in the complex space spectrum (49, 50), indicated in blue in Fig. 1C. The corresponding density of states (DOS) are plotted in Fig. 1D, where the skin effect can be clearly observed at zero energy. Here, λx=λy=2,λx′=λy′=10−3 and Lx = Ly = 13 are taken. According to a non-Hermitian topological theory (51), the skin effect is topologically protected and conforms to the bulk and boundary correspondence of non-Hermitian systems.

Fig. 1. Second-order non-Hermitian sensing system.

(A) Diagram of a second-order sensing model with the size of LX × LY. The unit cells and system couplings are indicated by magenta balls and lines, respectively. Also, the measurand Γ indicated by the magenta box can be connected to any two unit cells. (B) Part of a second-order sensing model including four unit cells indicated by a black box in (A). The non-Hermitian couplings between cells are indicated by blue (λx(y)) and green ( λxy′ ) arrows. (C) Energy spectrum of a second-order sensing system in complex space where Lx = Ly = 13 and λx=λy=2,λx′=λy′=10−3 . The two degenerate zero-energy modes are specially marked with red dots. (D) DOS of a second-order sensing system. The DOS corresponds to one of the skin modes indicated by a blue dot in (C). (E) Simulation results of the relationship between the sensitivity and system scale where λx=λy=1.9,λx′=λy′=0.1 . The brown line is the numerical calculation result of Eq. 2 with an infinitely small measurand. Other points represent the calculation results of the energy spectrum with different measurands Γ = 10−26,10−24,10−22,10−20, and 10−18. (F) MR for the sensing system within the measurand Γ. The triangle and circular dotted lines display the upper and lower limits of system detection, respectively, with the middle area representing the MR of the system.

By introducing a measurand Γ connected in the above system, as shown in Fig. 1A, the Hamiltonian of the sensing system can be given as H′ = H + HΓ, where HΓ = Γ(|1,1〉〈m, n| + h. c.) represents a weak disturbance and depends on the connection positions of the measurand (h. c. indicates Hermitian conjugation). Here, m and n are the X and Y coordinates, respectively. With the HΓ above, the shift of the system skin mode can be obtained by a leading-order perturbation theory as∆E=〈ψL|HΓ|ψR〉〈ψL|ψR〉≈2−λxλx′χx·λyλy′χy−λxλx′−χx·λyλy′−χyχx+1·χy+1·Γ→λxy>λxy′ and Γ→0CeK·Γ(2)

where K=κx·χx+κy·χy , κxy=lnλxy/λxy′ , χxy=mn−12 , and C=1χx+1χy+1 . The |ψR〉 and 〈ψL| are the right and left eigenvectors of the skin mode, respectively. Detailed derivation for Eq. 2 is given in section S1 of the Supplementary Materials. The sensitivity is generally represented by ∆E/Γ, and the brown line in Fig. 1E shows the theoretical results of sensitivity from Eq. 2 as a function of system size L. Here, the parameters are taken as m = n = L, λx = λy = 1.9, and λx′=λy′=0.1 . It is an exponential relationship between the sensitivity and system scale. Also, the energy spectrum calculation results are represented by different colored points, which are not approximated by Γ → 0. When the scale is constant and Γ is finite, the sensitivity of second-order non-Hermitian sensing systems has a certain saturation effect with finite measurand. The saturation effect is related to the measurand Γ; for example, the saturation positions correspond to Lx = Ly = 17 and Lx = Ly = 23 at Γ = 10−18 and Γ = 10−26, respectively, as shown in Fig. 1E. The saturation effect and the sensitivity of the system itself determine the upper and lower limits of system detection, respectively. The upper and lower limits of the system are also related to the coupling parameters of the system. Figure 1F displays the relationship between the measurand Γ and non-Hermitian strength  λxy/λxy′  at Lx=Ly=13 . As λxy/λxy′=20 , the minimum detectable value is about 10−33 (lower limit); the corresponding maximum detectable value is about 10−29 (upper limit). With the increases in the non-Hermitian coupling strength, a weaker external measurand can be detected, as indicated by the dots (lower limits) and triangles (upper limits) in Fig. 1F. The region between two lines represents the measurement range, which is marked by MR. The MR in Fig. 1F is under special parameters, but we can achieve any MR by adjusting the system scale and coupling parameters. This means that, no matter how weak the measurand is, the required sensitivity can be achieved by expanding the scale of the system.

The above only provides second-order results, but the theory can be extended to third-order and even nth-order cases by introducing more symmetries. In these higher-order cases, much stronger skin effects can be observed, and the sensitivity of the system can be further experimentally improved. The detailed discussions are given in section S2 of the Supplementary Materials. In the following, we explore how to implement the above ultrasensitive sensors in integrated circuit systems.

Integrated circuit sensors with ultrasensitivity based on high-order non-Hermitian topological physics

To flexibly change the system size, we prepare a sensor circuit using a modular design. Figure 2A shows two module units; each corresponds to the theoretical model with a size of 3 × 3, as shown in Fig. 1A. The yellow and orange spheres in the module represent the nodes of the circuit network, which correspond to the lattices of the theoretical model. Also, the nodes are connected through capacitors and buffers (blue and green arrows). Here, the two sensing units are connected and controlled by multiple synchronous switches (SWs). In such a case, the theoretical model with a size of 25 × 25 requires 12 sensing units to be connected in series. In Fig. 2B, we show the nonreciprocal couplings among the nodes in the designed circuit system, which correspond to the model shown in Fig. 1B. The markers 1 and 2 in Fig. 2B also correspond to the lattices 1 and 2 in Fig. 1B. Also, the nonreciprocal couplings (blue and green arrows) are achieved by controlling the buffer where the capacitor C1 and C2 control the nonreciprocal strength in the corresponding direction, respectively. The values of capacitors C1 and C2 correspond to the coupling strengths λx(y) and λxy′ in Fig. 1B. The detailed design of nonreciprocal couplings is provided in Methods. In this way, we can design the circuit network of any size. In addition, to ensure stable operation of the circuit, we have set the corresponding grounding capacitors and inductors for each node, and the buffer has also been specially designed accordingly. The grounding method and the principle of the buffer are provided in section S3 of the Supplementary Materials. For such a circuit network, we can theoretically derive the circuit Laplacian J and prove that it corresponds to the Hamiltonian H of the lattice model described in Eq. 1. Moreover, the resonance frequency of the oscillator circuit also has the one-to-one correspondence to the eigenvalue of the Hamiltonian H (52, 53). Detailed demonstrations of these relations can be found in section S4 of the Supplementary Materials.

Fig. 2. Illustration and characterization of the second-order nonreciprocal sensing chip.

(A) Simplified schematic diagram of the chip circuit design. The nonreciprocal couplings are indicated by blue and green arrows, and C0 represents the capacitor. The red dashed box represents one of the unit cells in the circuit. The blue dashed box represents the sensing unit of the sensing system, and the sensing units are connected through multiple SWs. (B) Detailed circuit coupling of the sensing system (four unit cells). The unit cell is indicated in magenta, which is formed by two lattices connected by the capacitor C0. The green (C1) and blue (C2) arrows represent nonreciprocal couplings in different directions, respectively. (C) Fully integrated nonreciprocal sensing circuit system. The sensing chip is bonded on a PCB, which provides the power supply, input/output interfaces, and FPGA control ports. The chip is fabricated in a 65-nm CMOS process technology whose core area is 3000 × 3500 μm2. (D) Simulation results of the circuit nonreciprocal strength range. The non-Hermitian strength (C1/C2) of the circuit non-Hermitian couplings corresponds to the non-Hermitian strength λx(y)/ (λxy′)  in quantum models.

Now, we simulate the voltage distribution of the system through the Simulation Program with Integrated Circuit Emphasis (SPICE). The detailed simulation method can be found in section S5 of the Supplementary Materials. We consider the case with 12 sensing units, as shown in Fig. 3A. This 12-unit system has a total of 13 measurable nodes, marked by nodes 1 to 13. The resonance frequency for such an Inductance and Capacitance (LC) oscillation circuit is at f0≈12π2LC1 , which is known as eigenfrequency. If the input signal with f0 is given at node 13, then we can obtain the voltage distribution of the circuit network and the corresponding simulation results indicated by a blue line are shown in Fig. 3B. Here, C1/C2 = 160, C1 = 5 pF, and L = 1 microhenry (μH) and the corresponding eigenfrequency is f0 ≈ 1.59 GHz. It can be seen that the node voltage shows an exponential decreasing trend in the circuit system, which is completely consistent with the high-order skin effect exhibited in the theoretical model. The orange line represents the simulation results of the node voltage at a randomly selected frequency f1. Comparing the results at frequencies f0 and f1, the highest voltage difference between the two cases reaches 66 dB, indicating that the non-Hermitian circuit exhibits a stronger skin effect at the eigenfrequency.

Fig. 3. Skin effect and eigenfrequency measurement.

(A) Schematic diagram of the measurement model. The output/input ports are set at the connection positions between the sensing units, marked as node 1 to N. For example, 12 units have 13 nodes. By controlling the switch through the FPGA, the access position of the external measurand CΓ can be adjusted. Any node can serve as the input and output position for the signal. (B) Skin effect on the nonreciprocal circuit system with C1/C2 = 160, C1 = 5 pF, and L = 1 nH and the input voltage is set as 1 V. The voltage decreases exponentially with the number of units at f0 ≈ 1.59 GHz until a voltage drop of 66 dB. The skin effect will decrease at f1 ≈ 1.27 GHz. (C) Relationship between the eigenfrequency shift (∆f) and number of sensing unit(s). Under the control of the FPGA, a single chip can achieve non-Hermitian sensing in three states: one unit, three units, and six units. By setting the nonreciprocal strength C1/C2 = 300, the ∆f increases with the number of sensor units. (D) Relationship between the eigenfrequency shift and nonreciprocal strength (C1/C2). In the case of C1/C2 = 10 to 200 and six sensing units, the ∆f increases exponentially with the increase in C1/C2. (E) MR of the sensing system under different nonreciprocal strengths. Lines and dots represent the theoretical limit measurement boundary and experimental results, respectively.

By using the strong skin effect at the eigenfrequency, we can measure the external measurands through the non-Hermitian sensing circuit. Through the SW controlled by the FPGA, we can adjust the position of the measurand CΓ in the circuit, as shown in Fig. 3A. The nodes Vin and Vout near the capacitor CΓ are the input and output nodes for the signal, respectively, and we can measure the eigenfrequency f0′ of the system with the capacitor CΓ. Also, the difference between f0 and f0′ is defined as the eigenfrequency shift ∆f=f0−f0′ . The relationship between ∆f and the number of working sensing units is provided in Fig. 3C, where the lines of different colors represent the SPICE results for different measurands and C1/C2 = 300. As the number of sensing units increases, the eigenfrequency shift increases exponentially, increases to a certain extent, and shows saturation effects, which is completely consistent with the theoretical results in Fig. 1E. Although there are saturation effects, any weaker external measurand can be detected through our designed circuit platform as long as the nonreciprocal coupling strengths and size of the system are taken appropriately. For example, a frequency shift higher than 1 kHz can be obtained for the measurand with CΓ = 10−20 fF by using our designed circuit with 12 sensing units. The detailed discussion is given in section S6 of the Supplementary Materials.

In the experiment, we integrate the designed circuit system onto the chip by using a standard 65-nm CMOS process technology. The detailed process is provided in Methods. The inset I in Fig. 2C shows the microscopic photo of the chip, which contains six sensing units in an area of 3000 × 3500 μm2. The chip is bonded onto a printed circuit board (PCB) through gold wire (II), as shown in Fig. 2C. To test the function of the chip, we also integrate input/output ports (III) on the PCB. In addition, to achieve complete controllability of circuit parameters, FPGA ports (IV) have also been integrated into the PCB, as shown in Fig. 2C. The detailed integration process and design can be found in section S7 of the Supplementary Materials.

The node voltage distributions are measured experimentally at the chip with 12 sensing units under different frequency signals through an oscilloscope spectrum analyzer. Corresponding to the SPICE simulation steps above, we input the signals with f0 and f1 at node 13, and the measurement results are indicated by dots of different colors in Fig. 3B, where the error of the experiment does not exceed 1%. Comparing the experimental results with the corresponding simulation results, the agreements between them are very well. However, there is also a notable deviation between the experiment and theory when the signal strength is below −80 dB. This is due to the influence of environmental thermal noise. When the signal is less than −80 dB, the measured signal is suppressed by the background noise, and the accurate value cannot be measured. However, this phenomenon does not affect the measurement of the eigenfrequency of the system.

Furthermore, we also measure the variation of the eigenfrequency shift with the number of sensing units. As CΓ = 10−3 fF, the measured results are plotted as orange dots in Fig. 3C, where C1/C2 = 300. The pentagonal dots and triangular dots represent the experimental results for CΓ = 0.01 and 0.1 fF, respectively. The good agreements between experimental and simulation results are observed again. However, when the measurand is CΓ = 10−3 fF, the errors of the experiment slightly exceed the simulation results by 5 to 10%. This is because the tested capacitors produced by two reverse electronic control capacitors have an error of 5 to 10%, which is not a problem with the chips we prepared. The detailed description on preparation of the measurand is provided in Methods. If we can have an exact standard capacitor less than 10−20 fF, our prepared chip can also provide good measurement results.

The chip we prepared not only flexibly controls the system size through switches but also allows us to adjust the nonreciprocal coupling strength of the circuit freely through the FPGA. The nonreciprocal coupling strength is roughly defined as C1/C2 between two nodes, here C1/C2 > 1. Because the nonreciprocal coupling strength is expressed by the capacitors on the chip and is affected by the connected buffer, it has a range of variation, as shown in Fig. 2D. To control the invariance of other parameters, we take C1/C2 from 10 to 200 in the experiment, where the system consists of six sensing units and L = 1 μH. Figure 3D shows the experimental measurement results under different values of C1/C2, and it can be seen that the eigenfrequency shift of the system increases exponentially with the increase in the nonreciprocal coupling strength. Also, as the measurand increases, the saturation effect gradually appears. This can be clearly seen from the experimental results of the system’s range, as shown in Fig. 3E. Here, points and lines represent experimental and simulation results, respectively. Also, the lower and upper limits represent the simulation results under ∆f = 1 Hz and saturation conditions, respectively. As the nonreciprocal strength changes, the MR of the system exponentially changes, which corresponds to theoretical results shown in Fig. 1F. Under this characteristic, the control design of the FPGA greatly improves the MR of the nonreciprocal sensing system.

The chip we prepared not only has a very high sensitivity but also has robust characteristics. To verify the robustness of the circuit system, we introduce disordered cross-talk (DC), which is more than 1000 times the actual noise intensity in normal situations. In the experiment, we additionally input the DC at the signal input node and define the percentage ratio of DC to the main signal as the strength of DC, such as 10, 25, and 50%. Figure 4A shows the experimental results of the voltage skin effect with 50% DC. Here, the parameters are taken identically with those in Fig. 3B. When the signal frequency is at the eigenfrequency f0, the skin effect in a nonreciprocal system is extremely robust. Even driven by the signal with 50% DC, the skin effect voltage of the system remains basically unchanged, as indicated in blue (0 DC) and green (50% DC) in Fig. 4A. The corresponding experimental working frequencies f0 and f1 are also marked in the figure. Compared to the situation of eigenfrequency, at the frequency of f1, the working signal has been completely distorted, and the skin effect no longer exists. The corresponding sensing experiment results are shown in Fig. 4B. It can be seen that, under the influence of DC, the sensing performance of the system is not greatly affected and still maintains the characteristic of exponential sensitivity. The inset in Fig. 4B shows the corresponding admittance spectrum, indicating that the zero mode of the system is not affected by the DC. The reason that our designed chip has such good robust characteristics is not only because of the high-order topology characteristics but also because of the band-stop effect of the LC circuit itself. In Fig. 4C, we provide experimental measurement results of the band-stop effect. Three panels in Fig. 4C display the experimental measurement results with DC = −20, −7.22, and −3.61 dB. Strong band-stop effects appear near the eigenfrequency f0 for all cases. This effect helps us reduce the noise during the measurement of eigenfrequency and improve system accuracy and robustness. A detailed noise analysis and its mitigation can be found in section S8 of the Supplementary Materials.

Fig. 4. System DC test results.

(A) Skin effect under DC influence (50%). Here, C1/C2 = 160, C1 = 5 pF, and L = 1 μH and the input voltage is set as 1 V. The skin effect remains unchanged at f0 ≈ 1.59 GHz and is not affected by DC interference, which is protected by topology. At f1 ≈ 1.27 GHz, the system is greatly affected by the DC, and the signal tends to become chaotic. (B) System eigenfrequency shift under DC influence (50%). There will be a deviation of less than 5% in the eigenfrequency shift when DC = 50%. The inset represents the energy spectrum solution of the system admittance matrix in this case. (C) Signal noise measurement results (input voltage is set as 1 V). The circuit parameters are C1/C2 = 160, C1 = 5 pF, and L = 1 nH, and the theoretically eigenfrequency is 1.59 GHz. The red part represents the frequency sweep results of the DC. Here, the cross-talk test is set as DC = −20, −7.22, and −3.61 dB.

The discussion of ultrasensitive integrated circuit sensors

The designed and fabricated circuit sensors (chips) above only focus on the second-order non-Hermitian case. The design and fabrication can be extended to third-order, fourth-order, and even higher-order cases. The sensitivity and robustness of the systems increase exponentially with the order of the system. These high-order non-Hermitian sensors have a strong detection advantage for capacitor front-end, which are widely used in our daily lives. At the same time, for the construction of this sensing circuit, any type of circuit coupling component can be selected, such as inductive coupling with capacitive grounding in LC oscillation circuits or resistive coupling in Resistance, Inductance and Capacitance (RLC) oscillation circuits. In addition, the strong topology protection characteristics enable the chip to have a high success rate during the preparation process. This is because its higher-order symmetry brings higher performance stability, allowing the system to still have exponential sensitivity improvement even when some parts are damaged.

DISCUSSION

In summary, we have theoretically proposed an alternative class of sensors with superior performances based on the exotic properties of high-order non-Hermitian topological physics. The corresponding integrated electronic platform has been fabricated using a 65-nm CMOS process technology. On the basis of such a platform, we have demonstrated that the designed sensors have not only an extremely high sensitivity but also strong robust properties. This means that they can work in various extreme complex environments. At the same time, the 65-nm CMOS process we use makes the system highly integrated and can be controlled by the FPGA, greatly improving the practicality of the system.

METHODS

Nonreciprocal couplings brought by the buffer

The buffer in Fig. 2 consists of a multistage amplifier, capacitors, resistors, and other devices (section S3). While having the voltage following effect, the buffer also provides extremely strong load carrying capacity. As a result, the forward (output) impedance of the buffer approaches 0, and the reverse (input) impedance can be considered infinite. On the basis of the above characteristics, the circuit Laplace matrix (admittance matrix) of the buffer is as followsJω=01Zoutω1Zinω0→0∞00(3)

where Zin and Zout represent the input and output impedances at both ends of the buffer, respectively. To control the nonreciprocal couplings strength caused by the buffer, we can connect the capacitor C1 to the output port. At the same time, when the buffer is working with the capacitor C1, the input impedance of the buffer will also change according to the value of the devices on the output (capacitor C1), from infinity to minimum. We define the input impedance currently as C2. Here, the Laplace matrix of the system is as followsJω=01Zoutω+iωC1iωC20→0iωC1iωC20(4)

Through the above method, we can achieve nonreciprocal couplings of circuit systems, corresponding to non-Hermitian models. Meanwhile, there is a linear relationship between capacitors C1 and C2, and their linear coefficients are related to the internal structure of the buffer.

Chip implementation and fabrication

All CMOS devices were prepared in Cadence Virtuoso (an industry-standard design tool for front-end circuit design). All designs satisfy the standard CMOS manufacturing rules of the Taiwan Semiconductor Manufacturing Company’s (TSMC’s) commercial 65-nm CMOS process, with physical verification performed using Mentor Graphics Calibre (an industry-standard design tool for back-end layout design). We used the Metal Oxide Semiconductor Implementation Service for fabrication.

Preparation of the measurand CΓ

In the 65-nm CMOS process, it is very difficult to prepare capacitors with capacitance values less than 1 fF. In our experiment, we used two variable capacitor matrices CVM1 and CVM2 for reverse connection. Under voltage control, the transformation range of each variable capacitor is from 0 to 60 pF. When the voltage difference is 1 mV, a capacitance of 0.001 fF can be obtained. The capacitance generated by this method is greatly affected by voltage fluctuations, with an error of 5 to 75%. Because this error is not a problem with the sensing system, in the experiment, we selected a stable fluctuation period (with an error of 5 to 10%) through the voltage spectrum for experimental measurement. Within the allowable range of error, the agreement between experimental and simulation results are very well.

Acknowledgments

Funding: This work was supported by the National Key R&D Program of China (2022YFA1404900) (X.Z.) and the National Natural Science Foundation of China (no. 12234004) (X.Z.).

Author contributions: W.D. finished the theoretical scheme with the help of T.C. W.Z. finished the design of the chip under the supervision of H.S. W.D. finished the experimental measurements with the help of W.Z. X.Z. initiated and designed this research project.

Competing interests: The authors declare that they have no competing interests.

Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

Supplementary Materials

This PDF file includes:

Sections S1 to S8

Figs. S1 to S16
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