1 Introduction
Silicon nanowire sensors (SiNW) duan2013complementary ; cui2001nanowire (see Figure 1) are promising devices used to detect the presence or concentration of different biological species, such as cancer cells zheng2005multiplexed , DNA and miRNA molecules he2015label ; hahm2004direct , and proteins wang2005label . The sensors are being developed for the early detection of cardiovascular diseases chua2009label , prostate cancer baumgartner2013predictive , breast cancer shashaani2016silicon , gastric cancer lee2010measurements , flu shen2012rapid , and uric acid in human blood guan2014highly . In the sensors, the target molecules such as biomarkers bind selectively to the recognition elements, e.g., antibodies or aptamers. The semiconductor transducer converts the potential change due to the analyte molecules into a measurable electrical signal, i.e., a current or voltage change mu2015silicon . The biosensors are interesting candidates for biomarker detection since the sensors are reliable, labelfree, inexpensive, highly sensitive, and have short operation time khodadadian2017optimal .
The binding of target molecules to the receptors changes the charge concentration, which modulates the transducer. The driftdiffusionPoissonBoltzmann (DDPB) system is a comprehensive set of equations to model the device. The PoissonBoltzmann equation enables us to calculate the charge concentrations around biomolecules and the effect of the charged target molecules on the nanowire, while the driftdiffusion equations model the charge transport of the carriers through the nanowire. In the biosensors, there are many sources of noises, e.g., random movement and binding of the target molecules khodadadian2017optimal ; khodadadian2016basis . There are also random dopant fluctuations (RDF) taghizadeh2017optimal . The stochastic version of the equation system can be used to investigate these effects. Also, existence and uniqueness theorems for the stochastic equations are given in taghizadeh2017optimal .
For solving the stochastic DDPB system, efficient numerical methods are finite elements (for space discretization) and the Monte Carlo technique (for the stochastic dimensions). In taghizadeh2017optimal ; khodadadian2018three , the authors developed a multilevel Monte Carlo finite element method (MLMCFEM) to solve the equation with little computational effort. A further complexity reduction has been been achieved by replacing the random points by quasiMonte Carlo points khodadadian2018optimal . Also, by using threedimensional simulations khodadadian2018three ; baumgartner2013predictive we can perform realistic simulations.
In order to obtain good agreement with experiments, a robust estimation of unknown model parameters, e.g., diffusion coefficients, the charge and density of molecules, doping concentration, etc., is essential. In classical inversion methods, one strives to minimize the distance between measurement and simulation to estimate the unknown parameters. Minimizing the difference between measurement and simulation always yields a value, but its sensitivity must be assessed separately. This is not a trivial manner especially in the case of illposed nonlinear problems. The great advantage of the Bayesian approach followed here is that it yields the probability density of the unknown parameters.
In Bayesian inversion dashti2017bayesian the solution of the inverse problem is the posterior density giving the distribution of the unknown parameter values based on the sampled observations smith2013uncertainty
. Markovchain Monte Carlo (MCMC) is a popular method to calculate the distribution. In this method, a Markov chain is constructed whose stationary distribution is the sought posterior distribution in Bayes theorem.
The MetropolisHastings (MH) algorithm smith1993bayesian is one of the most common techniques among the MCMC methods since it is simple and sufficiently powerful for many problems (specifically when the parameters are not strongly correlated). In order to estimate the posterior distribution, in each iteration, we propose a new candidate parameter value based on the current sample value according to a proposal distribution. Then, we calculate the acceptance ratio and decide whether the candidate value is accepted or rejected. The acceptance ratio points out how probable the new candidate value is with respect to the current sample.
The MH algorithm has some drawbacks, e.g., the proposal covariance must be manually tuned and has high autocorrelation andrieu2003introduction . To overcome these deficiencies, instead of using a fixed proposal distribution in each iteration, we update the distribution according to the available samples (adaptive Metropolis). This approach is useful since the posterior distribution is not sensitive to the proposal distribution. The adaptive method can be modified additionally by combining it with delayed rejection yielding the DRAM algorithm. In this algorithm, an alternative for the rejected candidate is proposed and the probability of this conditional acceptance is corrected mira2001metropolis . Upon rejection in the MH algorithm, instead of retaining the current position, a secondstage move is proposed haario2006dram . The method is noticeably advantageous when sampling from highdimensional conditional distributions zuev2011modified .
In the sensor design process, we use the PDEbased model to simulate the device characteristic. However, a very good estimation of the model parameters enables us to have a more efficient simulation and predict the sensor electrical behavior in different situations (e.g., subthreshold and linear regimes). In this work, the main aim is to propose an efficient computational method to estimate physical parameters of the sensors that cannot be measured directly or only with great experimental efforts punzet2012determination . For instance, regarding the target molecules, a reliable estimation of the surface charge (due to their binding to the receptors) or their reaction with the probe molecules cannot be achieved easily. Similarly, a reasonable estimation of the number of target molecules bound to the receptors cannot be achieved experimentally. However, in order to have an exact simulation, these parameters are crucial. To that end, we use the DRAM algorithm to calculate the posterior distribution of various unknown parameters. The extracted information will help us to improve the simulation quality since as a biosensor and a transistor the necessary parameters are determined efficiently.
The outline of the paper is as follows. In Section 2, we introduce the stochastic driftdiffusionPoissonBoltzmann model and explain how it can be used to model all charge interactions in the device. In Section 3, we present the MH algorithm and its modification by covariance adaption and delayed rejection. In Section 4, we use the PDE model and the DRAM technique to estimate the unknown parameters of a nanowire sensor. In order to do this, we first validate the transport model by comparison with experimental data and then use these experimental data in the Bayesian inversion. Finally, the conclusions are drawn in Section 5.
2 The macroscopic model equation
In this section, we review a complete mathematical model to understand the physical behavior of nanowire sensors where the details are explained in the authors previous papers, e.g., baumgartner2013predictive ; khodadadian2018optimal ; khodadadian2017optimal . It consists of the PoissonBoltzmann equation to model the electrolyte and the driftdiffusionPoisson system to model the charge transport in the semiconducting part.
Here we assume is the sensor domain (our computational geometry) which is partitioned into three regions with their physical characteristics. The subdomain (silicon nanowire) is the transducer insulated by (), the second subdomain. In , the electrolyte contains cations and anions; therefore, the PoissonBoltzmann equation holds.
In order to describe the sensor electrostatic interactions, we use the stochastic PoissonBoltzmann equation
(1) 
where is the dielectric (permittivity) function with the relative permittivities of the materials assumed to be constant and equal to , , and . In this equation, and belongs to which is the probability space. Moreover, is the electrical potential, is the elementary charge, is the doping concentration, considers the molecules charge, is the ionic concentration (holds for a symmetric electrolyte of monovalent ions), , and is the Fermi level. The concentration of electrons and holes are given by a Boltzmann distribution as
(2)  
(3) 
where is the Boltzmann constant and is the temperature.
The interface conditions in the electrostatic potential arise from homogenization heitzinger2010multiscale . The interface conditions are
(4)  
(5) 
where the interface is . Here, indicates the limit at the interface on the side of the silicon oxide, while
is the limit on the side of the aqueous solution. The two interface conditions indicate that the rapidly oscillating charge concentration in the surface layer is described by the macroscopic dipole moment density
and the macroscopic surfacecharge density .For this model, the boundary conditions are Dirichlet boundary conditions () and Neumann boundary condition (). A voltage across the simulation domain in the vertical direction can be applied by an electrode in the liquid (solution voltage) and by a backgate contact at the bottom of the structure (backgate voltage). At the source and drain contacts, the Dirichlet boundary conditions are respectively (source voltage) and (drain voltage). Additionally, zero Neumann boundary conditions hold on else everywhere (the Neumann part of the boundary).
Next, we consider the driftdiffusionPoisson equations to model the charge transport through the semiconducting nanowire. In the transducer , the stochastic driftdiffusionPoisson system taghizadeh2017optimal
(6a)  
(6b)  
(6c)  
(6d)  
(6e) 
holds. Here, and indicate the current densities of the carriers, and are the diffusion coefficients, and are the mobilities, and is the recombination rate. We use the ShockleyReadHall (SRH) recombination rate, which is defined as
(7) 
where is the intrinsic charge density and and are the lifetimes of the free carriers. Finally, the total current density is and the total electrical current
(8) 
is obtained by calculating this integral over a crosssection of the transducer.
In order to include the biological noise, we consider an association/dissociation process tulzer2014fluctuations . The association and dissociation processes of target molecules at the surface can be described by the reaction process
(9a)  
(9b) 
where (9) describes association and dissociation of the probetarget complex at the surface. In other words, the binding of target molecules (targetmolecule concentration) to probe molecules (probemolecule concentration), thus forming the probetarget complex (probetarget concentration). The reaction equations provides sufficient information about the number of bound molecules to the receptors (complex at the surface).
3 MetropolisHastings algorithm
In this section, we briefly introduce the Bayesian inversion approach and explain how this technique can be implemented to estimate the unknown parameters. First we use the statistical model
(10) 
where , and
are random variables representing the measurement, the estimated current by the model (here the driftdiffusionPoisson system (
8)), and the measurement error, respectively. The measurement error is a realization of , where is a fidelity parameter that indicates the level of measurement error. For a given value of the parameter and the corresponding observation (the measurement) we assume that is a Lebesgue density. Then, we define the conditional densitysuch that
(11) 
where is the prior probability density stuart2010inverse . Using the measured value , Bayes Theorem yields the posteriori distribution as
(12) 
In (12) the denominator is a normalization constant and its explicit calculation is computationally expensive. Therefore, we sample the posterior distribution without the knowledge of the normalization constant yielding
Now we assume that errors are iid and . The likelihood function is therefore
(13) 
where
(14) 
indicates the simulation error with respect to the parameter .
When we have enough information about the posterior distribution in the most straightforward situation, we can directly sample from it. However, in most cases, we do not have sufficient knowledge about the distribution or it is not possible to sample from it due to highdimensionality or complexity. To overcome this problem, the MH algorithm can be used. A summary of the algorithm is given in Algorithm 1.
In the algorithm, the proposal distribution is fixed; therefore, the rejection rate can be very high. Using an updated covariance matrix for the proposal distribution (applying the learned information about the posterior) allows us to increase the acceptance ratios since they accelerate the rate at which information regarding the posterior is incorporated smith2013uncertainty . Also, to enhance efficiency, the adaptive algorithm is combined with a delayed rejection technique. To that end, using the information of the rejected proposal, a new candidate is proposed and is rejected or accepted based on a suitably computed probability haario2006dram . A summary of the DRAM method is given in Algorithm 2.
In Algorithm 2, where is the
dimensional identity matrix. In order to use a narrower proposal function (compared to the first proposal) we employ
. The covariance function is calculated as(15) 
where . We refer the interested reader to haario2006dram for more details.
The main aim of using Bayesian inversion in this work is matching the electrical current obtained by the driftdiffusion model (8) by the experimental measurements at different gate voltages. To that end, we provide a list of desirable unknown parameters (with their relative equations) and explain why they should be determined precisely.

Surface charge density (), in the stochastic PoissonBoltzmann equation describes the charge due to binding the target molecules to the receptors at the sensor surface, i.e., . A reliable estimation due to the unknown area of probetarget molecule (therefore the surface charge estimation) cannot be easily estimated.

Doping concentration () in the stochastic PoissonBoltzmann equation and driftdiffusion equations (in ) is another influential parameter. Generally, it is an average number of dopants, and the precise concentration is not extracted usually. The exact doping density makes the simulation more reliable.

Electron mobility () and hole density () used in the stochastic driftdiffusion equation. As we already know, the sensor acts as a transistor; therefore, its electrical behavior in the subthreshold and linear regime is crucial. Einstein relations, i.e., and ( is the thermal voltage) gives us good information about the diffusion coefficients (used in the subthreshold conduction).

PTdensity presented in the reaction equations (9) which give us the density of the probetarget complex at the sensor surface. Considering the surface area, we can predict how many target molecules absorb precisely to the receptors.
Finally, there is no correlation between the abovementioned unknown parameters.
4 Parameter estimation and model verification
In this section, we use the DDPB system to model the electrical behavior of the sensor whose 3D schematic diagram is shown in Figure 1. Regarding the geometry, the nanowire length is , and its width and thickness are and , respectively. The insulator (silicon dioxide) thickness is , the distance between the nanowire and the boundary is . In the simulations the thermal voltage is and the sourcetodrain voltage is . The experimental data for the mentioned device are taken from baumgartner2013predictive for the nanowire fieldeffect PSA sensor. Furthermore, we assume that the measurement has 1% error.
Prostatespecific antigen (PSA) is an important biomarker widely used to diagnose prostate cancer. Here we develop a sensor to detect the protein 2ZCH (https://www.rcsb.org/structure/2ZCH). The charge of PSA is a function of pH value, where the PROPKA algorithm li2005very is applied to estimate the net charge shown in Figure 2. We perform the simulations with measurements performed at a pH value of 9; we note a total PSA charge of at this pH value.
In Ptype semiconductors, applying a positive gate voltage depletes carriers and reduces the conductance, while applying a negative gate voltage gives rise to an accumulation of carriers and increases device conductivity. In fieldeffect biosensors, the PSA target molecules carry negative charges (see Figure 2) which act as a negative gate voltage. Since we use a Ptype (borondoped) semiconductor as the transducer, the accumulation of holes increases the conductance as well.
The doping concentration is another important physical parameter in semiconductor devices. In nanowire sensors, on the one hand, a higher concentration increases device conductivity, while, on the other hand, a higher concentration decreases the sensor sensitivity. In other words, when the doping concentration is high, the nanowire is mostly affected by the dopant atoms and the effect of the charged molecules on the sensor response decreases. Therefore, the optimal doping concentration in the device design process is essential.
From now on, we use the DRAM algorithm to estimate the important unknown parameters where in all cases number of samples are used. The first study is the molecule charge density in the sense that the prior knowledge is . Figure 3 shows the posterior distribution and prior distribution of the molecule charge density where the acceptance rate of 67.7% is achieved. Due to the obtained results by the PROPKA algorithm and using a Ptype semiconductor we employ the (Gaussian) proposal distribution between and . The results point out that most of the accepted proposals are around the prior knowledge (its expected value is ). The posterior distribution indicates that the probability of positive charges is negligible, which agrees very well with the transducer structure (Ptype nanowire). Finally, the narrower shape of posterior distribution (compared to the proposal) indicates the Bayesian inversion efficiency.
In the next case, we simultaneously consider the effect of molecule charge density and doping concentration. In other words, the proposal consists of two suggestions for the parameters (twodimensional Bayesian estimation). In order to obtain the posterior distribution, and as the prior knowledge are applied in the simulations. We study the effect of molecule charge density from to and of the doping concentration varying between and . Figure 6
shows the posterior and prior (again Gaussian) distribution of doping concentration and molecule charge density where the acceptance rate is 62.3%. Here,
and as the expected values of the unknown parameters have been obtained. Similar to the first case, the probability of positive charge density is very low.As we already mentioned, higher doping concentration increases the device conductivity; however, it decreases the sensor sensitivity. In other words, in high doping concentrations, (current with molecule) tends to khodadadian2017optimal (current without molecules) since the effect of doping of the transducer is much more pronounced than the charged molecules. Therefore, the chosen doping range strikes a balance between selectivity and conductivity. As the figure shows, the most of accepted candidates are between and and the probability of is negligible, which confirms the doping effect on the sensitivity.
The electron and hole mobilities have a similar dependence on doping. For low doping concentrations, the mobility is almost constant and primarily limited by phonon scattering. At higher doping concentrations the mobility decreases due to ionized impurity scattering with the ionized doping atoms. In arora1982electron , an analytic expression for electron and hole mobility in silicon as a function of doping concentration has been given. In the previous cases we used the electron/hole mobility according to the Arora formula, i.e., and for . Now in order to validate this empirical formula, we consider the mobilities as the other unknown parameters. Figure 11 illustrates the posterior distribution of four physical parameters, where , , and are found as the expected values. The obtained mobilities also confirm the Arora formula. Again, the (Gaussian) prior distribution is shown, and for this estimation, the acceptance rate of 59.8 % achieved.
We consider probetarget binding in the equilibrium and use a receptor concentration of . In practice, a good estimation of the number of bound target to the probe molecules cannot be achieved easily. Here we study the density between and molecules per square centimeters (the Gaussian prior distribution). Figure 12 shows the density estimation where the rest of (four) unknown parameters are according to the extracted information by the posterior distributions. As shown, the mean of density is and 75.8% is the acceptance rate.
As we already mentioned, using reliable prior knowledge (good guess) by employing PROPKA algorithm, Arora formula and good approximation of doping density enables us to provide an efficient posterior distribution. This fact gives rise to uncertainty reduction of parameters and also a good acceptance rate is achieved. Now we study the effect of the prior distribution on the marginal posterior. To that end, for molecule charge density, we assume it varies between and (with the mean of ), the doping concentration changes from to (the mean is ) and regarding the mobilities, and are chosen. Using the mentioned proposals lead to the mean value of , , and . As a noticeable difference with the previous estimation, in spite of the convergence, the acceptance rate reduced to 32.3 %.
Now we employ the estimated parameters (posterior distribution) to calculate the electrical current. We have obtained two posterior distributions first based on the empirical formulas and second according to not good guesses. Figure 18 shows the simulated current as a function of different gate voltages for both posterior distributions and compared it with the experiments. These results validate the effectiveness and usefulness of the Bayesian inference since using the DRAM algorithm leads to an excellent agreement between the measurement and the simulation. However, the posterior distribution with a reasonable guess gives rise to a more exact electrical current.
The measurement error affects the Bayesian inversion as well. We note that in addition to the measurement error, the spatial discretization error (finite element discretization) and the statistical error (Monte Carlo sampling) might be part of . Here, we study the influence of on the complex density. Figure 21 shows the posterior distribution for two more measurement errors, namely and . Also, the distribution for was already illustrated in Figure 12. The results point out that a smaller error gives rise to a narrower distribution, while a larger error makes it wider, as expected.
5 Conclusions
In sensor design, reliable information about different device parameters is crucial. The stochastic DDPB system is a useful computational system to model the electrical/electrochemical behavior of nanowire sensors. The model enables us to study the effect of different influential parameters, e.g., molecule charge density, diffusion coefficients, doping concentration, gate voltage, etc. We have used Bayesian inversion to provide a reliable estimation of the parameters. More precisely, the MCMC method (DRAM algorithm) has been used to obtain the posterior distribution of the physical device parameters.
In this paper, we have first validated the DDPB system with the experimental data and then applied the DRAM algorithm to estimate the parameters. In order to study the effect of charged molecules on the nanowire, we estimated the molecule charge density. Then, we considered the effect of doping concentration (in a twodimensional Bayesian inversion). Here, due to the narrow probability density for each parameter, reliable information can be extracted. In addition to the mentioned physical parameters, we studied the effect of electron and hole mobilities in addition to the previous unknowns (in a fourdimensional Bayesian estimation) and provided their posterior distributions simultaneously. In the most complicated simulation, we have estimated the probability density of the concentration. The results enable us to determine the device and molecule properties at the same time.
Finally, we have applied the results obtained by Bayesian inference to the DDPB system and again simulated the device current. The results point out that compared to the previous simulations, the agreement with the experimental data has improved, which indicates the effectiveness of the DRAM technique. The results show that Bayesian inversion is a promising technique and has significant capabilities in the design of various sensors and nanoscale devices as well as in interpreting measurement data and assessing its quality.
6 Acknowledgments
The authors acknowledge support by the FWF (Austrian Science Fund) START project No. Y660 PDE Models for Nanotechnology.
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