Influence of Measuring Circuit Parameters on the Characteristics of MIS-Capacitor Hydrogen Sensors

Abstract

Using electrophysical models of MIS-capacitor gas-sensing elements, the influence of measuring circuit parameters on the metrological characteristics of hydrogen sensors was investigated. Recommendations for selecting optimal measurement circuit modes are provided, both in general and using sensor elements with a Pd-Ta₂O₅-SiO₂-nSi structure as an example. This article presents the results of an analysis and comparative study of three methods for measuring the capacitance of MISC sensors: (a) the AC bridge with a balance indicator (ACB + BI), (b) the divider method (DM), and (c) the bridge method (BM). The advantages and disadvantages of each method for practical implementation in gas analytical instruments are discussed. Furthermore, experimental data on the long-term stability of MISC sensor characteristics are provided, including the sensor response to hydrogen and the zero-point drift.

1. Introduction

Hydrogen sensors are widely employed in devices and systems designed for monitoring environmental conditions and ensuring explosion safety across various industrial and civil facilities [1,2]. The development of microelectronic sensors and gas analysis systems based on micro- and nano-technologies is regarded as a promising approach for creating compact means of measuring gas concentrations [3]. A critical requirement for such devices is the technological, thermal, and electromagnetic compatibility of their constituent elements, including sensors as well as primary and secondary transducers. Among solid-state gas-sensitive transducers, capacitors and field-effect transistors with a metal-insulator-semiconductor structure (MISC and MISFET) exhibit excellent compatibility with integrated circuit elements [4].

Gas concentration sensors based on MIS-structures have been extensively studied by numerous research groups [5,6,7]. A significant contribution to the development of such sensors has been made by researchers from Linköping University [8,9,10]. Objectively, this sensor type has not yet gained widespread adoption, as its operational range is primarily limited to low H₂ concentrations, from hundreds of ppb to 1–2% vol. This range is currently in lower demand compared to the explosive range. Nevertheless, the tasks of early diagnostics for critical events involving trace hydrogen are becoming increasingly relevant, including breath analysis [11,12], early detection of technical equipment malfunctions [13,14] and fire precursors [15,16,17], as well as mineral exploration [18,19], etc. The main advantages of the proposed sensor type are the long-term stability of the sensitive elements’ characteristics and the high reproducibility of their fabrication process, which is compatible with integrated circuit (IC) technology. The development of artificial intelligence (AI) and the resolution of the selectivity problem may ultimately facilitate the broader adoption and practical application of such sensors [20,21,22].

At the National Research Nuclear University MEPhI, sensor elements based on MISC and MISFET with Pd and Pt gates have been developed. These sensors are designed for the detection of hydrogen-containing gases (H₂ [23], NH₃ [24], H₂S [25]) and are based on multilayer structures such as SiO₂-Si, Si₃N₄-SiO₂-Si, and Ta₂O₅-SiO₂-Si [3,25,26]. Experimental studies have demonstrated that hydrogen sensors utilizing a sensitive element with a Pd-Ta₂O₅-SiO₂-Si structure on a silicon MOS crystal exhibit superior performance characteristic. Furthermore, the influence of various factors on the metrological and operational properties of these sensors has been investigated. It has been shown that these characteristics are dependent on design and technological parameters, chip temperature, operating time, and environmental conditions [23,24,27,28].

The aim of this study is to evaluate the influence of the electrical operating modes of measurement circuits on the performance characteristics of MISC-based hydrogen sensors. The investigation is conducted in a general context and illustrated by a specific example of a sensor utilizing a Pd-Ta₂O₅-SiO₂-nSi structure. For the interpretation of the obtained experimental data, updated physical models and equivalent circuit diagrams were employed. These models take into account the influence of various external and internal factors. It is anticipated that the proposed models and gas concentration measurement techniques will serve as a foundation for predicting the performance of sensors with more complex circuit topologies, including those incorporating one or multiple MISC elements.

2. Initial Data and Performance Models of MISC Hydrogen Sensors

Hydrogen sensors can be implemented with either a vertical or planar structure of the sensing element. Simplified schematic diagrams illustrating the structure of the MISC sensing element in the cross-section view of the device and its equivalent electrical circuit are presented in Figure 1. The maximum sensitivity of sensors based on MISC structures is observed at elevated temperatures of active region 8 (380–500 K) (see Figure 1a,b). The heating and stabilization of the operating temperature are provided by integrated heating and temperature-sensitive elements within the sensor design [24]. Furthermore, it should be noted that the electronic circuits employed for measuring gas concentration and maintaining temperature stability have a significant impact on the metrological characteristics of the sensor [29].

In the equivalent circuit shown in Figure 1c, the capacitors correspond to the following capacitances:

• C_i of the thin dielectric film;
• C_in of the inversion layer;
• C_D of the space-charge region (depletion layer);
• C_ss effective capacitance of the surface states at the dielectric–semiconductor interface.

The resistors R_s and R_ss represent the total resistance of semiconductor regions 5 and 6 and the resistance that determines the switching time (τ_ss = C_ssR_ss) of the surface-state charges with density Q_ss, respectively.

Studies have shown that the general operating principle of hydrogen-sensitive MISC sensors is independent of their structural type and manufacturing technology. Hydrogen exposure induces physicochemical reactions at the various interfaces of the working electrode: at the gas/metal and metal/dielectric boundaries, as well as within the bulk of the metal and dielectric. These processes depend on the gas concentration N and lead to changes in the work function difference potential φ_ms and the effective charge density in the dielectric, Q_te. As a result, the capacitance-voltage (CV) characteristic (the dependence of capacitance C on voltage V) shifts along the voltage (V) axis parallel to its original position, as illustrated in Figure 2.

We examined the quantitative changes in the CV parameters of MISCs based on n-type silicon as a function of various factors. Subsequently, for the operating voltage V = V₁₂ < 0, the capacitance C can be represented, in the general case, as a function of the surface potential φ_s using Equations (1) and (2):

V(φ_s) = φ_ms − [Q_te + Q_ss(φ_s) + Q_s(φ_s)]/C_i + φ_s = V₀(φ_s) − ΔV(N)

(1)

C(φ_s) = C_i[C_in(φ_s) + C_D(φ_s) + C_ss(φ_s)]/[C_i + C_in(φ_s) + C_D(φ_s) + C_ss(φ_s)]

(2)

where, in the inversion region (φ_s > φ_s0), the effective charge density Q_s(φ_s) in n-Si is calculated as

Q_s(φ_s) = Q_d(φ_s) + Q_in(φ_s) ≈ a·C_i{φ_s + φ_T exp[(φ_s − 2φ_s0)/φ_T]}^1/2

The transfer function models V(N) are defined as approximations of the averaged experimental dependences V_ki(N_ki), based on the data of m complete i-th responses, each measured n times, where k = 1, 2, …, n and i = 1, 2, …, m [28]. Typically, to determine the transfer function for a specific sensor series, m complete responses are obtained from a sample of n sensor units (here, indices k and i denote the ordinal numbers of the sensor samples and responses, respectively). If the absolute errors of the measuring instruments are ΔN_ki and ΔV_ki, then the true values of each (ki)-th measurement are located on the V(N) graph within a rectangular field with sides 2ΔN_ki and 2ΔV_ki, centered at the point (N_ki, V_ki), as shown in Figure 3.

The sensitivity of the CV characteristic to hydrogen and external factors {z_j} acting on the MISC (background gases, temperature, radiation) is manifested provided that the overlap coefficient K_C = C_B/C_A > 1 (Figure 2) and the slope g_C = d_C/d_V exceeds the ratio of the measurement errors, i.e., g_C > ΔC_C/ΔV_V, where ΔC_C and ΔV_V are the absolute errors of capacitance and voltage measurement, respectively. The dependence V(N, C_i) determines the hydrogen sensitivity of the MISC, which is quantified by the voltage sensitivity coefficient S_V = d_V/d_N. To mitigate the influence of the stray capacitances C_ss and C_in in MISC sensors, the operating region is selected based on high-frequency CV characteristics (f > 50 kHz) in the depletion and weak inversion modes, where the surface potential is constrained to the interval |φ_s| ∈ (0: 2φ_s0]. Consequently, the dependencies |V₀(φ_s)|, ΔV(N) and C(φ_s) are represented as

|V₀(φ_s)| = φ_s + a{φ_s + φ_T·exp[(φ_s − 2φ_s0)/φ_T)]}^1/2 + b(φ_s0 − φ_s) + φ_ms0 − Q_te0/C_i

(3)

ΔV(N, C_i) = Δφ_ms(N) − ΔQ_te(N)/C_i

(4)

C(φ_s) = C_iC_D(φ_s)/[C_i + C_D(φ_s)] = C_i/[1 + F_C(V)]

(5)

F_C(φ_s) = 2{φ_s + φ_T·exp[(φ_s − 2φ_s0)/φ_T]}^1/2/{a + a·exp[(φ_s − 2φ_s0)/φ_T]}

(6)

where φ_s ≡ |φ_s|, C_i = C_max = ε₀ε/d, a = [2ε₀ε_sqN_D]^1/2/C_i and b = qN_ss/(φ_bgC_i) are the design and technological parameters, generally designated as {p_k}. Electrophysical models (3)–(6) were used for the quantitative analysis of the MISC sensors’ characteristics with the Pd-Ta₂O₅-SiO₂-nSi structure, the parameters of which are presented in

Table 1. Parameters of the MISC with a Pd-Ta₂O₅-SiO₂-nSi structure and the models used.

Designations	Parameters	Average Values
ε1, ε2 and εs	relative permittivity of Ta2O5, SiO2 and Si	25, 4 and 12
ND	donor concentration in Si	5 × 1015 cm−3
d1 and d2	thickness of Ta2O5 and SiO2	90 nm and 80 nm
ε0	relative permittivity of vacuum	8.85 × 10−12 F/m
k	Boltzmann constant	1.38 × 10−23 J/K
q	electron charge	1.6 × 10−19 C
d	thickness of thin dielectric (d1 + d2)	170 nm
se	effective area of the working electrode	0.01 cm2
ε	effective permittivity (dε1ε2)/(ε1d2 + ε2d1)	7.2
Ci	specific capacitance of a thin dielectric (ε0ε)/d	37 nF/cm2
a	charge parameter in Si [(2q·ε0∙εs∙ND)1/2/Ci]	1.1 V1/2
b	charge parameter Qss at Nss = 1012 cm−2 [qNss/(φbgCi)]	0.43
φms	work function difference potential between Pd and Si	φms0 = 85 mV
T	MISC chip temperature	400 K
φT	thermal potential (kT/q) at 400 K	33 mV
φgb	band gap potential in Si	1.08 V
φs0	donor level potential [φTln(ND/ni)]	0.34 V
φs	surface potential [φ(SiO2-Si) − φF]	(0.05…0.8) V
Qte and Qss	charge density in the dielectric and at the SiO2-Si interface	(2…100) nC/cm2
Nss	density of surface states at the boundary SiO2-Si	(1011…1013) cm−2

. Accurate modeling of the CV characteristics is possible only by numerical methods. By specifying the values of the surface potential φ_s, one can calculate V(φ_s) using Equation (3), determine the corresponding values of the inverse function φ_s(V), and, by substituting these into Equations (5) and (6), obtain the dependences C[φ_s(V)]. The potential φ_ms0, the capacitance C_i, and the dependence F_C(φ_s) can be calculated theoretically, whereas the dependence ΔV(N) and the charges Q_te0 and Q_ss0 are determined experimentally.

The electrophysical models of hydrogen sensitivity proposed in the first part of the article in general form can serve as a basis for the development of “more complex designs” (for example, sensors with several MISC elements) and can be integrated into computer-aided design systems. For example, a sub-circuit schematic of model implementation using a VCVS (voltage-controlled voltage source φ_s) that connects in series with the gate VG of the standard MIS-structure done in a Verilog-A-model [30].

3. Results

In addition to the conversion function V(N) and the differential sensitivity S_V = dV/dN, the main metrological characteristics of the MISC include the following:

• Sensitivity: S_i = ΔV_i/ΔN_i
• Sensitivity Threshold: N₀ = ΔV_V/S_Vm
• Measurement Range: ΔN_m ∈ [N₀; N_m]
• Absolute Error Δ(N) and Relative Error δN = 100% × Δ(N)/N
• Operating Conversion Range: ΔN₁₂ ∈ [N₁; N₂] for a given maximum error δN_m
• Response Speed (Bandwidth): determined by the rise (τ_0.9) and fall (τ_0.1) times of the voltage V_i(t) in response to a concentration pulse N_i

For the studied MISC, in the concentration range from a few ppm to 1 vol.%, the dependencies of the response signal ΔV(N) and sensitivity S_V(N) can be approximated as follows [25]:

ΔV(N) = ΔV_m [1 − exp(−k_N N)] and S_V = k_N ΔV_m exp(−k_N N)

(7)

where ΔV_m = ΔQ_tem/C_i − Δφ_msm. The values of the parameters ΔV_m, ΔQ_tem, C_i, Δφ_msm, and k_N were determined experimentally to be (0.5 ± 0.02) V, (14.4 ± 0.5) nC/cm², (37 ± 1) nF/cm², −(0.11 ± 0.005) V, and (8.0 ± 0.3) (%)⁻¹, respectively.

N(ΔV) = (1/k_N) ln[ΔV_m/(ΔV_m − ΔV)] = 0.125 ln[1/(1 − 2ΔV)]

(8)

In electrophysical models (1)–(6), capacitance and charge values are represented as specific values (per unit area), which does not affect the qualitative analysis of the CV characteristics. Models (3)–(8), parameters {p_k}, and capacitance/voltage measurement errors are key quantities for the quantitative assessment of the influence of circuit electrical modes on the metrological characteristics of MISCs.

Two approaches can be used to determine the N concentration based on the leftward shift of the CV characteristic by ΔV(N) (Figure 2,

Table 2. Approaches to determining the concentration N based on the shifts of the CV characteristic.

Scheme #	Control Parameters	Output Values	Relative Sensitivities
1	C0 ∈ (Cmin + Δ(C); Cmax − Δ(C))	ΔV(N) = ΔVm [1 − exp(−kN N)]	S1 = kNΔVm exp(−kN N)/V0
2	V0 ∈ [VA; VB)	ΔC(N) = C(V0 − ΔV(N)) − C(V0)	S2 = gC kNΔVm × exp(−kN N)/C0

):

• Measure ΔV at a constant capacitance C₀ ∈ [C_min + ΔC_C; C_max − ΔC_C] using scheme 1 (Figure 4a);
• Measure ΔC at a constant voltage V₀ ∈ [V_A;V_B) (Figure 2) using scheme 2 (Figure 4b).

At T = 400 K, the average parameter values are Q_te0 ≈ 20 nC/cm², N_ss≈ 10¹¹ cm⁻², a = 1.11 V^1/2 and b = 0.43. According to the data in Figure 3, the real concentration values are expressed as N_ip = N_i ± ΔN_i. Assuming that the instrumental error ΔN_ki is constant for all samples and equal to ΔN_N, the absolute error ΔN_i is determined as

ΔN_i = ΔN_N + ΔV_i/│S_i│

(9)

Of practical interest is the relative error

δN(N) = 100%·{[ΔN_N·│S_d│ + Δ(V)]/(N·│S_d│)}

(10)

where Δ(V) denotes the total absolute error, which may be a function of the parameter N. This value comprises the instrumental error ΔV_V, errors induced by external influencing factors ΔV_Z, and the spread of circuit parameters ΔV_ep. The minimum attainable value of Δ(V) is defined by the ΔV_V.

According to Equation (7), the intrinsic differential sensitivity to hydrogen, S_V, is a function of the parameters ΔQ_tem, C_i, Δφ_msm and k_N. These parameters are, in turn, determined by the initial characteristics of the MISC, namely the sets {z_j} and {p_k}. Notably, the value of S_V is independent of the electrical operating modes of the measuring circuit.

In circuit 1, an increase in the parameter N at a fixed value of C₀ causes the operating point (C₀, V₀) to shift leftward. This shift occurs from the initial range │V₀│ ∈ (V_A; V_B) by a magnitude ΔV ∈ (0 V; 0.5 V) (see Figure 2). For instance, at C₀ = 26 nF/cm², the operating voltage is │V₀│ = 0.4 V. Furthermore, an increase in C₀ leads to a decrease in │V₀│ and a corresponding increase in the relative sensitivity of the circuit. The sensitivity S₁ lies within the following range: S₁ ∈ [3e^−8N, 20e^−8N] vol⁻¹%. The instrumental error of the circuit, Δ(V), is either constant or, in the presence of a relative voltage measurement error δV, exhibits a dependence on N according to the following expression:

Δ(V) = 0.01·δV{V₀ + 0.5[1 − exp(−8N)]}

If the instrumental error is ΔV_V = 1 mV, the sensitivity threshold is N₀ = 2.5 ppm. The maximum measurement range is defined as ΔN_m ∈ [2.5 ppm;1.55 vol.%]. For hydrogen in air, the conversion factor is approximately 1 vol.% ≈ 10⁴ ppm. The working conversion range ΔN₁₂ ∈ [N₁; N₂] for a specified maximum error δN_m is determined from the functional dependence δN(N), as illustrated in Figure 5 and Figure 6.

In circuit 2, an increase in N causes the operating point (C₀, V₀) to shift upward by ΔC ∈ (0;ΔC_m) at a constant │V₀│ ∈ (V_D;V_B). The maximum shift ΔC_m is reached at │V₀│= V_A (see Figure 2). The value of ΔC_m depends on the “swing” of the operating region of the CV characteristic, defined as ΔV_p = (V_B − V_A) ≈ 2.5 V, and on ΔV_m = 0.5 V. For the MISC under consideration, where ΔV_m < ΔV_p, the maximum value is ΔC_m ≈ 4 nF/cm² at V_A = −1.4 V. Conversely, for sensors with ΔV_m > ΔV_p, the value of ΔC_m = (C_i − C_min) ≈ 22 nF/cm² and is determined by the initial parameters {p_k}.

For circuit 2, the optimal initial coordinates of the operating point are V₀ = −1.4 V and C₀ = C_min + ΔC_C = 15.15 nF/cm².

With an increase in the magnitude of │V₀│, both the capacitance C₀ and the slope g_C decrease, while the relative sensitivity of the S₂ circuit increases, reaching a maximum value of S_2max= 0.27g_C·exp(−8N) vol⁻¹%. This increase does not exceed the relative sensitivity S₁ within the operating range of the CV characteristics, as g_C < C₀/V₀ holds true. The error of the measuring circuit, ΔC_C, is either constant or depends on δC according to the relation ΔC_C = 0.01δC·C₀. The error Δ(V), as defined by Equation (10), is given by Δ(V) = ΔC/g_C and depends on the parameter N. This error influences the conversion limits of the MISC sensor.

Each value of φ_s corresponds to a capacitance difference ΔC(φ_s) = C₂(φ_s) − C₁(φ_s), which is equal to the integral ∫S_Cdφ_s (see Figure 7). This difference defines the capacitive sensitivity S_C = d(ΔC)/dφ_s and the slope g_C = S_C(dφ_s/dV). The dependences S_C(φ_s) and g_C(φ_s) are determined by the initial parameter set {p_k}. The minimum, average, and maximum values of the capacitive sensitivity are S_C0 = 3 nF/(V·cm²), S_C ≈ 24 nF/(V·cm²), and S_Cm ≈ 70 nF/(V·cm²), respectively.

Analysis of the obtained data reveals that the metrological characteristics of the specific type of MISC sensor are influenced by its intrinsic hydrogen sensitivity S_V(N) and the errors associated with the measurement circuitry. During modeling, the analytical function S_V(N) is determined by approximating the averaged experimental dependences ΔV(N), while taking into account the uncertainties of the model parameters.

The boundaries of the MISC conversion range depend on the following:

• The chosen approach to determining the N concentration based on the ΔV(N) shifts in the CV characteristic;
• The initial coordinates of the operating point (C₀, V₀);
• The instrumental errors of the measuring circuits and the error of Δ(V).

In practice, experimental or production-scale devices with specified metrological characteristics are employed to measure capacitance and CV characteristics. These instruments operate on various physical principles (Figure 8) [31]. For preliminary studies of MISC sensors under development, when their characteristics are still unknown, AC bridge circuits equipped with a balance indicator (ACB + BI) are utilized, as illustrated in Figure 8a. The wide ΔV measurement range is the main advantage of this circuit. However, the complexity of the circuit structure and calibration, as well as the stringent requirements for the ND offset adjustment, limit its applicability in practical hydrogen sensors.

Common methods for measuring capacitance include the divider method (DM) and the bridge method (BM) (Figure 8b,c). In all cases, the measured capacitance values are not specific (i.e., normalized) but absolute values, which depend on the working electrode area (for the MISC under study, C_i = 370 pF).

A circuit based on the DM is recommended for measuring gas concentrations at the maximum allowable concentration (MAC) level in residential and occupational areas. While a BM circuit provides the maximum sensitivity for the MISC sensor, it inherently limits the measurable concentration range. It should be noted that an increase in the range of measurable capacitances and concentrations leads to a reduction in sensitivity. In this regard, the DM occupies an intermediate position, offering a balanced trade-off between sensitivity and the measurement range.

The boundary values of the conversion ranges of the MISC hydrogen sensor for various circuit configurations and error margins are exemplified in

Table 3. Boundaries of the MISC sensor conversion range for various measurement errors.

Δ(V)	N0, ppm	N1, ppm	N2, ppm	Nm, ppm	δNmin, %
0.01·│V(N)│	9	295	2300	7400	3.17
5 mV	140	355	3150	8100	2.72
1 mV	4.5	95	6000	10,450	0.54
0.6 mV (ΔCC = 1 pF)	2.5	65	8200	13,300	0.43

.

The selection of circuit design, the method for evaluating hydrogen sensitivity, and the optimal electrical conditions indirectly influence the metrological and operational characteristics of MISC hydrogen sensors. This influence is mediated by instrumental and statistical errors inherent in the measurement of CV characteristics. Consequently, a reduction in these errors leads to an expansion of the sensor’s conversion range.

4. Experimental Validation

Tests of electronic boards, fabricated using various capacitance measurement techniques, were conducted with the same MISC hydrogen sensor operating at a temperature of 100 °C (370 K). The relevant settings and characteristics of the boards are presented in

Table 4. Configuration parameters and performance characteristics of the electronic boards.

Comparison Parameters	Capacitance Measurement Method
	ACB + BI	DM	BM
F, kHz	8	20	18
A, mV	1500	1000	200
intrinsic noise of the electronic board *, ±pF	0.05	0.1	0.13
noise of the readout circuitry Cnoise, ±pF	0.15	0.2	0.26
measurement range, pF (±δ, %)	C0 ± 50 (±1) 400…1200 (±5) 300…5000 (±11)	500…2000 (±1) 300…5000 (±30)	C0 ± 65 (±1)

* Values obtained using NP0 capacitors.

. The experimental MISC sensor was fabricated using a well-established technology, which is described in greater detail in our previous publications [32].

To investigate hydrogen sensitivity, a dynamic gas mixing setup was employed, utilizing cylinders containing air and a test gas mixture of air + H₂ (0.1 vol%). After mixing, the resulting gas mixture was supplied to the chamber of the MISC sensor, which had a volume not exceeding 0.1 L. A photograph of the experimental setup is presented in Figure 9. Cylinders with hydrogen (JSC MGPZ, Moscow, Russia) were diluted with clean air using a Mikrogaz generator-diluent (JSC Intera, Kirovo-Chepetsk, Russia)

The accuracy of setting and maintaining the V₀ values on the boards was no worse than ±0.1 mV. For the operating temperature of the MISC sensor, the accuracy was no worse than ±0.1 °C. Additionally, prior to testing with the MISC sensor, the accuracy of capacitance measurements performed by the electronic boards was verified. For this purpose, temperature-stable NP0 ceramic capacitors (±30 ppm/°C) were used, and their capacitance was monitored with an AMM-1130 multimeter (Aktakom Ltd., Moscow, Russia) (±1%). The results are presented in Figure 10.

As shown in Table 4, the capacitance measurement range of the BM circuit is significantly narrower compared to the ACB + BI and DM circuits and is determined by the reference capacitance C₀ (see Figure 2). Consequently, while the BM circuit enables high-accuracy measurement of the MISC sensor capacitance, this is achievable only within a limited range of ΔC = 130 pF, which substantially restricts its practical applicability. It is worth noting (see Figure 10) that the measurement error for the ACB + BI and DM circuit boards also depends on the measurement range. The highest accuracy for the DM circuit board is attained when calibrating the operating capacitance within a range not exceeding 1500 pF; beyond this limit, the error increases by 8% for every additional 1000 pF.

The CV characteristics of the control MISC sensor, measured using the electronic boards, are presented in Figure 11a. Note that the sensor’s CV characteristics were measured in clean air and under 5 ppm hydrogen over the entire bias voltage range. The sections are highlighted in bold for clarity to avoid cluttering the graph while clearly indicating the position of the operating point (V₀) and the magnitude of the gas-induced shift (ΔV).

Due to differences in the measurement signal parameters implemented in the electronic boards, the shape of the CV curve for the same sensor varies depending on the measurement scheme. The operating points for each case were selected according to the method described above (see Figure 2), after which the dynamic characteristics of the MISC sensor were investigated in response to a hydrogen concentration of 5 ppm. The results are represented in Figure 11b and

Table 5. MISC sensor hydrogen sensitivity data.

Comparison Parameters	Capacitance Measurement Method
	ACB + BI	DM	BM
operating point coordinate V0, mV	−500	−200	−400
CV characteristic shift of ΔV, mV	−74	−70	−50
capacitance change ∆C, pF	64	70	128
hydrogen sensitivity SC(N), pF/ppm	12.8	14	25.6
response speed τ0.9; τ0.1; τfull, min	9; 14; 25	6; 9; 20	8; 11; 30
calculated value of the detection limit NLOD_H2, ppb	35	43	30.5

Designations: τ_full—characteristic time for the sensor readings to return to their original values after removal of hydrogen exposure; N_{LOD_H2} = (3∙C_noise)/S_C (see Table 4).

. Note that the greater capacitive response of the sensor, observed during measurements on the electronic board using the BM, is attributed to the steeper slope of its CV characteristic (∆C/∆V). Consequently, the magnitude of the capacitance change (∆C = 128 pF) under 5 ppm H₂ is twice as large as that obtained with the other two methods, while exhibiting a significantly smaller shift in the bias voltage (∆V = −50 mV).

Influence of measuring circuit parameters on the characteristics of MIS-capacitor hydrogen sensors also must be compared with the accuracy of the sensors’ response to hydrogen concentration during its lifetime, particularly noting the gas sensor characteristics that are of practical importance for implementation. These include long-term stability of characteristics, as well as the influence of interfering factors, such as humidity and background gases, to which the sensors may exhibit cross-sensitivity. Experimental data on the long-term stability of sensitivity, obtained using the BM capacitance measurement method, are presented in Figure 12.

The experiment involved the continuous operation of a batch of eight hydrogen sensors for over three years at a temperature of 100–130 °C, with periodic verification of their operating parameters, specifically hydrogen sensitivity. Hydrogen measurements were conducted in static mode. A 0.5% vol. H₂ Standard Gas Mixture was injected into a sealed 1 L container filled with room air using a 1 cm³ medical syringe. This resulted in a test H₂ concentration of 5 ppm. As shown in Figure 12a, the sensitivity of the majority of sensors exhibited a slight decrease relative to the initial level, typically stabilizing within the first six months of operation. Observations of the zero-reading drift (Figure 12b) revealed periodic fluctuations on a monthly scale, likely attributable to gradual changes in environmental parameters. As can be seen, the fluctuations in sensor readings did not exceed 10–20 pF, which, according to the sensor’s preliminary calibration, is equivalent to a nominal change in ambient hydrogen concentration of no more than ±1 ppm.

The issues of humidity influence, its compensation algorithms, and cross-sensitivity have been studied in detail and described in our previous work [32,33].

5. Conclusions

This paper presents an analysis of the methods and tools for evaluating the metrological performance of MISC sensor devices. This study is based on electrophysical models of the electrical characteristics of metal-insulator-semiconductor capacitive sensors. As a case study, this paper examines hydrogen sensors with a Pd-Ta₂O₅-SiO₂-nSi structure. Furthermore, recommendations are provided for the selection of optimal circuit configurations to determine the operating characteristics of sensors for detecting hydrogen-containing gas concentrations.

The bridge capacitance measurement circuit has been demonstrated to be well-suited for laboratory applications where high accuracy and sensitivity are of paramount importance. For the determination of gas concentrations at the MAC level in residential and occupational environments, a circuit based on the divider method is recommended. While the bridge circuit provides maximum sensitivity for the MISC sensor, it inherently limits the measurable range of gas concentrations. An increase in the range of measurable capacitances and corresponding concentrations results in a reduction in sensitivity. The divider method offers a compromise, occupying an intermediate position in terms of both sensitivity and measurement range.

The proposed models and methodology for studying the characteristics of MISC-based hydrogen sensors can serve as a foundation for the mathematical support required in the design of gas analyzers utilizing various types of MISCs. This includes the development of more complex designs and the prediction of their performance. Furthermore, engineering-physical models, parameterized with specific numerical values for given MISC structures, can be integrated into computer-aided design (CAD) systems for integrated gas concentration sensors.