Impact of Gouy-Chapman-Stern model on conventional ISFET sensitivity and stability

Utilizing Gouy-Chapman-Stern model can improve ISFET sensitivity and stability using Stern layer in direct contact with electrolyte in ISFET sensing window. However, this model remains a challenge in mathematical way, unless it’s re-applied using accurate simulation approaches. Here, we developed an approach using a commercial Silvaco TCAD to re-apply Gouy-Chapman-Stern model as ISFET sensing membrane to investigate its impact on sensitivity and stability of conventional ISFET. Sio2 material and high-k Ta2O5 material have been examined based on Gouy-Chapman and Gouy-Chapman-Stern models. Results shows that the ISFET sensitivity of SiO2 sensing membrane is improved from ~38 mV/pH to ~51 mV/pH and the VTH shift stability is also improved. Additionally, the results indicate that the sensitivity of Ta2O5 is 59.03 mV/pH that hit the Nearnst Limit 59.3 mV/pH and achieves good agreements with mathematical model and previous experimental results. In conclusion, this investigation introduces a real validation of previous mathematical models using commercial TCAD approach rather than expensive fabrication that paves the way for further analysis and optimization.

This paper extends the previous investigations [16] that aimed at developing a different Models for simulation of ISFET operations. Particularly, in this work we examine more closely the links between IC design simulation and previous mathematical models have not be clearly linked and for more accurate analysis for further optimization.

Material and Methods 2.1. Mathematical Model
In the original structure of an ISFET, the gate oxide is in direct contact with the electrolyte solution as shown in Figure 2 (a), acting as a sensing dielectric. The Stern layer is a modified version of the ISFET in which the sensing layer is separated from the gate oxide by using an extended conductive layer Figure 2 (b) after the gate oxide is covered by the electrolyte solution, creating a more robust structure for extended efficacy in the solutions. Through the effective coupling capacitance between the sensor surface and Floating Gate (FG), the surface potential (ψo) modulates the potential of the FG, and therefore there will be a corresponding shift in the threshold voltage (VT) of the sensor [17][18][19][20][21]. Therefore, from the Site-binding model, the charge density can be expressed by [19,20,22]: where; + is the H + activity calculated by ( + = 10 − ); q is the elementary charge; Nsil is the density of the available sites and Ka; Kb represents the intrinsic dissociation constants, and the Nsil, Ka and Kb, are oxide layer dependent.
Based on "charge density", the charge on the electrolyte side of a double layer (σDL) is the same value, but is a negative charge. Therefore, this charge can be calculated from the integral double layer capacitance (Ci) and the surface potential [23,24]: Therefore, by solving (1) and (2), we can demonstrate the relation between + and ψo parameters. According to the Boltzmann distribution model for the H+ ions [9,24], the pH value at the sensor surface is [9]: T and k represent the absolute temperature and Boltzmann constant, respectively. The subscript S and B denote the pH at the sensor surface and in the bulk solution, respectively, and ψo is the potential drop across the diffusion layer. In (3), the intrinsic buffer capacity (βint) is defined as the ability to collect a charge at the sensor surface (σ0) due to the change in surface pH (pHs) [9]: Differentiating (3) with respect to the bulk pH, and using (6), the surface potential sensitivity to the pHB can be derived as [9] ∆ 0 here, α is a positive sensitivity (dimensionless parameter). Therefore, the sensitivity of the potential at the sensor surface and the corresponding change of the sensor threshold voltage to the bulk pH are limited to 59.3 mV/pH (Nernst limit). In the Stern model, the relationship between the diffusion layer potential ψo (previous calculations) and the Stern potential ψs can be expressed by: where ° is the permittivity of free space and its relative permittivity, respectively; is the number concentration of each ion in the bulk, z is the magnitude of the charge on the ions and represents the integral capacitance of the Stern layer.

TCAD Simulation
Commercial TCAD allows users to introduce bias-dependent surface charges in the form of interface donor or acceptor traps. The challenge is simulating the updated surface charge density equation described by in the electrolyte pH change model [19]. To introduce this equation to the simulator, interface trap statements are utilized to mimic the surface charge accurately, as follows [21]: INTTRAP <type> E. LEVEL= <r> DENSITY= <r> <capture parameters> "INTTRAP" activates interface defect traps at discrete energy levels within the bandgap of the semiconductor and sets their parameter values. Device physics has established the existence of three different mechanisms, which add to the space charge term in Poisson's equation in addition to the ionized donor and acceptor impurities" [21]. Interface traps will add space charge directly into the right-hand side of Poisson's equation. To calculate the trapped charge in Poisson's equation, the total charge value is defined by the following: where + and − are the densities of ionized donor-like and acceptor-like traps, respectively. DENSITY and its probability of ionization are represented as FtA and FtD, respectively. For donor-like and acceptor-like traps, the ionized densities are calculated by the following equations: where FtA and FtD are given by the following equations: where SIGN is the carrier capture cross sections for electrons and SIGP holes. The thermal velocities for electrons and holes are and , respectively. For donor-like traps, the electron and hole emission rates, For example, the acceptor interface trap statement and its parameters are as following: considering all equations mentioned above, we can rewrite the sit-binding model (1) based on TCAD model. We first assume that acceptor and donor traps exchange carriers only with the conduction and valence band of the semiconductor representing the electrolyte, respectively. Hence, we can rewrite (7) in terms of TCAD model as follows: for more details about TCAD simulation and modelling, the previous work was well described all modelling methodologies [25].

Result and Discussion
Models and simulation results have been shown in this section. In modeling part, Figure 3 examine the difference between Gouy-Chapman and Gouy-Chapman-Stern models using (8). The sensitivity parameter (α) (Dimensionless parameter) have been shown the sensitivity of surface potential changing based on range of pH bulk [21]. The value of a varies between 0 and 1 depending on the intrinsic buffer capacity and the differential capacitance [8]. Her we converted the values as proportions values from 0 to 59.3 that is the value of top Nearnst level. For a sensitivity close to the theoretical maximum, α approaches 59.3, the intrinsic buffer capacity should be high, and the differential capacitance should be small, as can be concluded from (8). A sensitivity close to zero can theoretically be derived when the intrinsic buffer capacity approaches zero. This shows that the Gouy-Chapman model remain challenge the Roll-off problem that already compensated by Gouy-Chapman-Stern models as we shown in Figure 3. The theoretical surface potential shift based on range of pH bulk from pH2-pH12 were observed in Figure 4. The (7) and (9) are the expression for the sensitivity of the electrostatic potential to changes in the bulk pH [9]. As we shown the Sio2 sensing membrane as Stern layer (9)  and stability comparing with normal silicon dioxide Which is known because high dielectric constant of high-k material. However, from mathematical models is very clear that using of Gouy-Chapman-Stern model enhance the ISFET sensitivity as well as achieve stable shift in surface potential through pH bulk change. On the other side, TCAD Simulaation results process is contribute same as modeling results. Table 1 and Table 2 are the TCAD simulation parameters.   Figure 5 introduce the shifting in threshold voltage (VTH) in range of pH bulk fron 2→12. indeed Shifting in threshold voltage sensitivity is the negetave values of shifting in surface potential based on the following expersion [24]:

∆VTH=-∆ψ
However, the ISFET senssitivity and stability of conventional isfet for Sio2 and Ta2O5 as a sensing membrane contribute very well comparing with ISFET whithout Stern layer as shown in Figure 5. As shown the linirity of Stern-ISFET is higher than ISFET whithout Stern as well as senstivity that consistent with behaviour of Figure 4. For more analysis, the average sensvity of conventinal Stern-ISFET for Ta2O5 sensing membarane is compared with the theoratical snestivity based on Nearnst equation observed in Figure 6. As shown the Ta2O5 is hit the Nearnst limit and the stability also contributed in acceptible way. The importantant aim is the linked between mathmatical model or physical interpretations and ISFET transfer charactrstics is introduced in Figure 7. In this Figure the real charactrization of pH bulk change with surface potential shifiting were demostrated. As shown, the drain current is slitly change with pH range from 1→14. Consequence, the main objective of this work is achived by linking the models whith real charactrization is clearly demostrated.

Discussion
In contrast to the wide existence and importance of mathematical models for developing conventional ISFET sensitivity and ISFET concept as general, the link between these models and electrical characteristics simulation (i.e. TCAD) rather than expensive fabrication remain challenge. In this study, we adapted previous mathematical models and re-applied these models using commercial TCAD. We emphasise these models results and achieve good agreement with models as well as real experimental results for more validation. The above results show that utilizing of Gouy-Chapman-Stern model will increase the conventional ISFET sensitivity and stability of normal silicon dioxide as well as high-k materials comparing with Gouy-Chapman model. For ISFET sensitivity with Sio2 sensing membrane is raised from ~38 mV/pH to ~51 mV/pH, And, Ta2O5 is hit the Nernst Limit (59.3 mV/pH) by value of 59.03 mV/pH. Furthermore, using of Stern Layer is compensate the Roll-off problem in normal model by enhancement of threshold voltage shifts linearity to reach 99.9% and 99.999% in Sio2 and Ta2O5 sensing membranes, respectively.
Further studies should investigate the performance analysis of commonly used high-k materials using same approach. Although the simulation approach still considered not real fabrication and measurements, this study opens new directions for further analysis and optimization prior the real and cost-effective fabrication way.

Conclusion
In this work we examine more closely the links between IC design simulation and previous mathematical models have not be clearly linked and for more accurate analysis. we adapted previous mathematical models and re-applied these models using commercial TCAD. ISFET sensitivity with Sio2 sensing membrane is raised from ~38 mV/pH to ~51 mV/pH, And, Ta2O5 is hit the Nernst Limit (59.3 mV/pH) by value of 59.03 mV/pH. Furthermore, using of Stern Layer is compensate the Roll-off problem in normal model by enhancement of threshold voltage shifts linearity to reach 99.9% and 99.999% in Sio2 and Ta2O5 sensing membranes, respectively. We anticipate this study opens new directions for further analysis and optimization prior the real and cost-effective fabrication way.