J Comput ElectronDOI 10.1007/s10825-016-0922-2

Basis adaptation for the stochastic nonlinear Poisson–Boltzmannequation

Amirreza Khodadadian1 · Clemens Heitzinger1,2

© Springer Science+Business Media New York 2016

Abstract A basis-adaptation method based on polyno-mial chaos expansion is used for the stochastic nonlinearPoisson–Boltzmann equation. The uncertainty in this numer-ical approach is motivated by the quantification of noise andfluctuations in nanoscale field-effect sensors. The methodused here takes advantage of the properties of the nonlinearPoisson–Boltzmann equation and shows an exact and effi-cient approximation of the real solution. Numerical examplesare motivated by the quantification of noise and fluctuationsin nanowire field-effect sensors as a concrete example. Basisadaptation is validated by comparison with the full solution,and it is compared to optimized multi-level Monte-Carlomethod, and the model equations are validated by compari-son with experiments. Finally, various design parameters ofthe field-effect sensors are investigated in order to maximizethe signal-to-noise ratio.

Keywords Poisson–Boltzmann equation · Current ·Biological noise · Polynomial chaos expansion · Biosensor

1 Introduction

A basis-adaptation method for the stochastic nonlinearPoisson–Boltzmann equation is applied. Here the main

B Amirreza Khodadadianamirreza.khodadadian@tuwien.ac.at

Clemens Heitzingerclemens.heitzinger@asu.edu;clemens.heitzinger@tuwien.ac.at

1 Institute for Analysis and Scientific Computing, ViennaUniversity of Technology (TU Vienna), Vienna, Austria

2 School of Mathematical and Statistical Sciences,Arizona State University (ASU), Tempe, AZ, USA

sources of randomness in the stochastic version of the nonlin-ear Poisson–Boltzmann equation are random permittivitiesand random charge concentrations near the sensor surface.Because the number of dimensions describing the randompermittivity and charge concentration is large in realisticapplications, a reduction of the computational complexity isessential and this goal is achieved by the basis-adaptationapproach presented here. The quantity of interest is thecurrent through the transducer subdomain, which can becalculated by a nonlinear functional in the graded-channelapproximation.

The Poisson–Boltzmann equation, whose stochastic ver-sion is the basic model equation here, is one of the mostimportant equations in computational chemistry. The equa-tion models the electrostatic potential in any applicationwhere screening by free ions in a liquid occurs. It alsodescribes screening in a semiconductor. A concrete phys-ical realization of the model equation are affinity- basedfield-effect sensors. In the numerical examples shown here,nanowire field-effect biosensors are simulated, and theexpected values, standard deviations, and signal-to-noiseratios are calculated.

Affinity-based sensors, in particular the recently exper-imentally demonstrated nanowire field-effect sensors, posedemanding numerical problems because of the large num-ber of stochastic dimensions. A cross section through ananowire field-effect biosensor [10,12,15,22,26,27,34,35,42] is shown in Fig. 1, but the same principle also appliesto gas sensors [6,20,31,40]. The sensors are affinity-based,since the target molecules are only detected when they aresufficiently close to the sensor. The current through the semi-conducting silicon nanowire, which is the actual transducer,is measured. The conductance of the nanowire is modulatedby the electrostatic potential, which—in turn—is changedwhen charged target molecules bind to the probe molecules

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Fig. 1 Cross section through a field-effect nanowire sensor, showingsubdomains and boundary conditions. In the electrolyte (Dliq), the ran-dom binding of target molecules to immobilized receptor molecules atthe surface defines molecule subdomain (DM)

functionalized at the surface of the sensor. The number oftarget molecules bound to probes at the sensor surface israndom, since the target molecules bind to and dissociatefrom the probe molecules. The positions of the probe mole-cules, as well as their orientations and the orientations ofthe probe–target complexes are random. This type of noise isoften called biological noise [16,38]. The uncertainty domain(electrolyte and molecule subdomains) is shown in Fig. 1 aswell. Each molecule or each probe molecule adds at least onestochastic dimension.

The concept of nanowire field-effect sensors is very gen-eral and has been applied to the detection of DNA, proteinssuch as tumor markers, and toxic gases such as carbonmonoxide. The sensors offer advantages like miniaturization,high sensitivity, fast response, and low power consumption[12,27,35]. Furthermore, the target molecules do not have tobe marked in field-effect sensors in contrast to the commonlyused fluorescent or radioactive markers in other detectionmethods.

There are various sources of noise and fluctuations inaffinity- based sensors, where the biological noise is one ofthe most prominent ones. Its quantification is essential forunderstanding the signal-to-noise ratio [33] and the detectionlimit of the sensors. Regarding the modeling, the randomnessdue to the biological noise at the sensor surface propa-gates through a partial differential equation (PDE) modeland finally yields the sensor output. The model used in thepresent work, based on the stochastic nonlinear Poisson–Boltzmann equation and a functional for the quantity ofinterest, is described in detail in Sect. 2.

Previous work on the modeling and simulation of nanowirefield-effect sensors has focused on deterministic equations.The inherent deterministic multiscale problem was solvedin [18]. Existence and uniqueness of a self-consistent modelwas shown in [3], and a FETI algorithm was presented in[4]. A method for calculating the probabilities of the orienta-

tions of various molecules and hence the stochastic processwas developed in [17]. The design of optimal sensors wasdiscussed in [5], and the biological noise was modeled in[37,38]. An effective equation for the covariance was foundin [19] as the result of a homogenization approach. Nanowirefield-effect biosensors were also modeled and simulatedin [22,28–30], where the alternating-current small-signalregime was considered.

The numerical approach developed here can also beapplied to investigate various sources of noise and fluctu-ations in more traditional semiconductor devices such asMOSFETs and FinFETs. Random-dopant effects are oneexample [1,8] and of great importance for integrated circuitsconsisting of nanoscale transistors. Another example is ran-dom telegraph noise (RTN) [23,39]. It is caused by randomtrapping and de-trapping of charges at interface states andalso of great importance for such small devices. By usingstochastic partial differential equations, these effects can bemodeled by stochastic coefficients analogously to the presentwork.

The rest of this paper is organized as follows: In Sect. 2, thestochastic Poisson–Boltzmann equation and the rest of themodel are described explicitly. In Sect. 3, basis adaptationfor this model is introduced in detail. In Sect. 4, optimalmulti-level Monte Carlo is shortly described and serves asa comparison. Numerical results are presented in Sect. 5.Finally, conclusions are drawn in Sect. 6.

2 The model equations

2.1 The stochastic nonlinear Poisson–Boltzmannequation

The main model equation is the stochastic nonlinear Poisson–Boltzmann equation:

− ∇ · (A(x, y,ω)∇u(x, y,ω))

=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

q(Cdop(x, y,ω)+ p(x, y,ω) − n(x, y,ω)) in DSi,

0 in Dox,

ρ(x, y,ω) in DM,

−2ϕ(x, y,ω) sinh(β(u(x, y,ω) − ΦF(x, y,ω))) in Dliq,

(1a)

u(0+, y,ω) − u(0−, y,ω) = α(y,ω) on Γ, (1b)A(0+)∂xu(0+, y,ω) − A(0−)∂xu(0−, y,ω) = χ(y,ω) on Γ,

(1c)

u(x, y,ω) = uD(x, y) on ∂DD, (1d)∇nu(x, y,ω) = 0 on ∂DN. (1e)

Here, u is the electrostatic potential, A is the permittiv-ity function, ϕ is the ion accessibility function, ΦF is theFermi level, ρ describes the fixed (surface) charges of mole-

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cules, and Cdop is the doping concentration. Furthermore,(Ω,Σ, P) is a probability space and ω ∈ Ω is a set of inde-pendent Gaussian random variables. The domain D ⊂ R2 isdepicted in Fig. 1. It is partitioned into the four subdomains,i.e., DSi, (silicon), Dox (oxide), Dliq (liquid), and DM (mole-cule) with the interface Γ between Dox and Dliq. The siliconnanowire is always covered by a thin layer of silicon oxide(Dox) and surrounded by the liquid containing the targetmolecules. The subdomain DM is defined when in the liquidthe charged target molecules bind to the probe molecules atoxide (insulator) surface. The relative permittivities of thesubdomains are ASi = 11.7, Aox = 3.9, AM = 3.7, andAliq = 78.4. Furthermore, q > 0 is the elementary charge,and the constant β is defined as β := q/(kBT ) in terms of theBoltzmann constant kB and the temperature T . In the simula-tions performed here, a thermal voltage of 0.021 V was used.The concentration p(x, y,ω) of positive free charge carriersand the concentration n(x, y,ω) of negative ones are givenby Boltzmann distributions as

p(x, y,ω) = κi exp(

−q(u(x, y,ω) − ΦF)

kBT

), (2a)

n(x, y,ω) = κi exp(q(u(x, y,ω) − ΦF)

kBT

). (2b)

Here, the constant κi is the intrinsic carrier concentration niof the semiconductor (1.5 × 1010 cm−3) or the bulk ionicconcentration [7] in the electrolyte.

The discontinuities or interface conditions in the elec-trostatic potential arise from homogenization [18]. At theinterface Γ between the silicon oxide and the liquid, thecharge concentration exhibits a rapidly oscillating spatialstructure which leads to a multiscale problem [18]. The con-tinuity conditions (1b) and (1c) are due to the jump in thepermittivity A between two different materials. Here, 0+denotes the limit at the interface on the side of the liquid,while 0− is the limit on the side of the silicon oxide layer.The two interface conditions mean that the rapidly oscillatingcharge concentration in the surface layer is described by themacroscopic dipole moment density α and the macroscopicsurface charge density χ [4].

The boundary conditions in (1), as illustrated in Fig. 1, areDirichlet boundary conditions (∂DD) and Neumann bound-ary condition (∂DN). A voltage across the simulation domainin the vertical direction can be applied as well by an electrodein the liquid (solution voltage) and by a back-gate contact atthe bottom of the structure (back-gate voltage). These arealso part of ∂DD. Zero Neumann boundary conditions holdon the Neumann part ∂DN of the boundary.

In crystals of pure silicon, the number density of electronsin the conduction band and of holes in the valence band areequal, and therefore the Fermi level as a function of dopingconcentration can be calculated for electrons and holes as

ΦF = Ec + kBT lnnni

= Ec − kBT lnpni,

where Ec is the minimum energy of the conduction band inthe semiconductor and ni is again the intrinsic carrier concen-tration of silicon. The equation holds true because pn = n2

i .

2.2 The quantity of interest and the graded-channelapproximation

Solving the Poisson–Boltzmann equation (1) yields the elec-trostatic potential on a cross section of the structure as shownin Fig. 1, from which the current through the sensor can becalculated by the graded-channel approximation as in [17].The graded-channel approximation can be derived from thedrift-diffusion equations as follows:

∇ · Jn(x, y,ω) = qR(n(x, y,ω), p(x, y,ω)),

− ∇ · Jp(x, y,ω) = qR (n(x, y,ω), p(x, y,ω)) ,

Jn(x, y,ω) = q(Dn∇n(x, y,ω)

− µnn(x, y,ω)∇u(x, y,ω),

Jp(x, y,ω) = q(−Dp∇ p(x, y,ω)

− µp p(x, y,ω)∇u(x, y,ω)),

where n and p are the electron and hole concentrations; Jnand Jp are their current densities; Dn and Dp are their dif-fusion coefficients; and µn and µp are their mobilities. Inaddition, we use the popular Shockley–Read–Hall (SRH)recombination rate:

R(n(x, y,ω), p(x, y,ω))

:= n(x, y,ω)p(x, y,ω) − n2i

τp(n(x, y,ω)+ ni)+ τn(p(x, y,ω)+ ni).

Two assumptions are necessary for the graded-channelapproximation. The first assumption is that only the drift cur-rent, and not the diffusion current, plays a role. This yieldsthe current density as

J drift := J driftn + J drift

p = −qµnn∇u − qµp p∇u.

The second assumption is that the electric field is constantalong the z-axis of the sensor, i.e., E = (uS − uD)/Z holds,where uS − uD is the potential difference between the twocontacts (the source and the drain) with a distance Z apart.Since E = −∂zu, we hence find

J drift = qE(µnn + µp p).

The total electrical current is now obtained by taking theintegral

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I (ω) := f (u(x, y,ω)) :=∫

J drift dxdy

= qE∫(µnn(x, y,ω)+ µp p(x, y,ω)) dxdy (3)

with n and p from (2) over a cross section of the semiconduct-ing part (DSi) of the sensor. This is the quantity of interestthat is recorded in measurements.

As we have seen, the graded-channel approximation isa simplification of the drift-diffusion–Poisson equations andneglects diffusion. Therefore, the full drift-diffusion–Poissonsystem is a better model for investigations in the subthresholdregime.

2.3 The random coefficients in the stochastic nonlinearPoisson–Boltzmann equation

The biological noise, i.e., the random movement of probe andtarget molecules at or near the surface of any affinity-basedsensor, propagates through a PDE model of the sensor andresults in noise in the sensor output. Biological noise is oneof the major sources of noise and fluctuations in field-effectsensors; it is due to two effects: the random association oftarget molecules with probe molecules functionalized at thesensor surface and their random dissociation from the probemolecules as well as randomness in the orientation of anymolecule bound to a surface. In order to quantify biologicalnoise, the association and dissociation processes were mod-eled in [38], where the expected number and variance of thenumber of target molecules bound to probe molecules werecalculated. The random orientations were modeled in [17],where the probabilities of the orientations were determined.

When any affinity-based sensor is manufactured, the probemolecules are functionalized at the sensor surface [32] at ran-

dom, but fixed locations. In other words, each probe moleculeis a binding site that is occupied by a target molecule witha certain probability that can be determined [38]. Unspe-cific binding of target molecules to the sensor surface in theabsence of probe molecules is also possible. Since this effectis not important in a well-designed sensor, it is not includedhere; however, it can be taken into account in the model in astraightforward manner.

The orientation of a charged molecule with respect to acharged surface is determined by the electrostatic free energyof the system. A method for determining the free energies ofcharged molecules as functions of various parameters waspresented in [17]. First, the electrostatic free energy E(η) iscalculated as a function of the angle η := (η1, η2), whichrepresents the orientation of the molecule with respect to thesurface. In general, each molecule has two degrees of free-dom, although in the case of DNA oligomers their structureleads to a simplification and one angle η := η1 ∈ [0,π/2] issufficient. Then a Boltzmann distribution is used to find theprobability of a configuration as a function of the angle η. Itis given by

P(η) = exp(−E(η)/(kBT ))∫exp(−E(η)/(kBT ))dη

∝ exp(

− E(η)kBT

). (4)

Figure 2 shows a concrete example and the data used in thenumerical examples in this paper. Electrostatic free energiesfor single- and double-stranded DNA oligomers of differentlengths are shown. As the charge density of dsDNA is twicethat of ssDNA, the binding energies of double-stranded DNA(dsDNA) oligomers are higher than their single-stranded(ssDNA) counterparts. The length of the oligomers decreasesthe free energy mostly for lower angles, i.e., for horizontalconfigurations, where electrostatic interactions with the sur-

Fig. 2 Electrostatic freeenergies of various ssDNA anddsDNA oligomers as a functionof angle for a surface charge of−0.5 q nm−2 in a liquid with anionic concentration of 30 mM.The angles are defined withrespect to the surface, where 0means a molecule parallel to thesurface and π/2 means amolecule perpendicular to thesurface (as indicated in Fig. 1).Data were obtained from [17]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

Rotation angle

Ene

rgy

[J]

ssDNA 12 base pairs

dsDNA 12 base pairs

ssDNA 20 base pairs

dsDNA 20 base pairs

ssDNA 24 base pairs

dsDNA 24 base pairs

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Rotation angle

PDF

0.2 q/nm2, 30 mM

0.5 q/nm2, 100 mM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rotation angle

CD

F

0.2 q/nm2, 30 mM

0.5 q/nm2, 100 mM

Fig. 3 Probability density function (left) and cumulative distribution function (right) of random orientation of 12-base pair ssDNA oligomers inan aqueous solution with different ionic concentrations and surface charges. The proteins rotate between 0 (horizontal) and π/2 (vertical) degreesat the surface

face are stronger. The orientations also depend on the ionicconcentration of the liquid, as the effect of Debye screeningis significant. Figure 3 shows the probability density function(PDF) and cumulative distribution function (CDF) of randomorientation of various DNA oligomers bound to the nanowiresurface. The figure indicates the Gaussian distribution ofrandom variables. In summary, the coefficients A(x, y, .),ρ(x, y, .) and ϕ(x, y, .) in (1) are random variables, i.e.,A(x, y,ω), ρ(x, y,ω), and ϕ(x, y,ω) depend on an ele-ment ω = (ω1,ω2, . . . ,ωn) of the underlying probabilityspace (Ω,Σ, P). The sample space Ω describes the loca-tions of the probe molecules, the state of the probe molecules(bound to a target molecules or not), and the orientations ofthe probe molecules and probe–target complexes. The mod-eling and simulation results in [17,38] yield the probabilitymeasure P .

We assume that the number of probe molecules per unitarea follows a Poisson distribution. Once the number N ofprobe molecules or binding sites and their positions are fixed,the sample space is

Ω =(0, 1 × [0,π/2]

)N,

where 0 denotes the absence of any target molecule, 1 thepresence of a target molecule, and η ∈ [0,π/2] is the angleof molecule.

3 Basis adaptation

It is clear from Fig. 1 and Sect. 2 that the quantity of inter-est I in (3) is a nonlinear functional of the potential u, the

solution of the stochastic semilinear elliptic equation (1). Fur-thermore, in order to obtain a reasonable estimation of the(expected value of) current, several evaluations are neces-sary, which are computationally speaking expensive. Thesefacts motivate the use of basis adaptation [11] to solve (1).

3.1 Polynomial chaos expansion (PCE)

Generalized polynomial chaos (GPC) seeks to represent anapproximation of a random function by a set of random vari-ables and orthogonal polynomials. It estimates coefficientsfor known orthogonal polynomial basis functions based on aset of response-function evaluations using sampling, tensor-product quadrature, or Smolyak sparse-grid approaches [13].The general idea of the expansion is to transfer randomnessto the basis functions and to use Hermite polynomials asbasis functions. A stochastic function like u ∈ L2(D,Ω)

can be explained as its Wiener–Hermite polynomial chaosexpansion [41]:

u(x,ω) :=∑

β

uβ(x)Φβ(ω). (5)

Here, Φβ is an orthonormal Hermit polynomial, β =(β1,β2, . . . ,βn) is a vector of n nonnegative integers whichdescribes the order of polynomials, x ∈ D, and ω is a vec-tor of uncorrelated Gaussian random variables. In order toapproximate the series (5), we truncate it to a finite numberof terms for the sake of computation

u(x,ω) :=∑

β∈Mk

uβ(x)Φβ(ω), (6)

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where the truncated index set

Mk = β = (βi , i ≥ 1 | βi ∈ 0, 1, . . . , k

is used for the series. The function can be characterized byits expected value with respect to the probability space as

E(u(x,ω)) =∫

Ωu(x,ω)µ(ω)dω, (7)

where µ(ω) is the probability density function (PDF) cor-responding to random variable ω. By the Cameron–Martintheorem, GPC approximation of (6) is fully converged in L2

norm if

limk→∞

∫

Ω

⎛

⎝∑

β∈Mk

uβ(x)Φβ(ω) − u(x,ω)

⎞

⎠µ(ω)dω = 0.

(8)

The general idea of our basis adaptation is rotating randomvariables and transferring them from uncorrelated to corre-lated random variables. To this end, we define an isometryL : Rn → Rn to define new basis. Therefore, the rotatedrandom variables are defined as

γ := Lω. (9)

The solution of (1) under this isometry can be obtained by

u(x, γ ) =∑

β∈Mk

uLβ (x)Φβ(Lω). (10)

The calculation of the coefficients of the series is the mainpoint which can be defined as

uLβ (x) =∑

β,δ∈Mk

uδ(x)⟨Φβ(ω),Φ

Lδ (ω)

⟩

=∑

β,δ∈Mk

uδ(x)⟨Φβ(ω),Φδ(γ )

⟩. (11)

The inner product in the Hilbert space is characterized byweight function as the probability density function relativeto the Gaussian measure [41]. An important property of thepolynomials which will be used in the following estimationsis their orthogonality with respect to the Gaussian probabilitymeasure. Therefore, the orthogonality of polynomial basiscan be used to simplify (11) as follows:

uLβ (x) =∑

β∈Mk

uβ(x)⟨Φβ(ω),Φβ(γ )

⟩. (12)

The isometry L still needs to be constructed in a suitablemanner to transfer one basis (ω) to other basis (γ ). To con-struct the isometry L, we use quadratic adaptation [36]. Asthe first step, we define a multi-index qi = (0, . . . , 1, . . . , 0)where only the i th point is 1 and other elements are zero. Thematrix L is given by

LT DL = B, (13)

where the diagonal elements of B are bii = u2qi√2

and the

rest of entries are bi j =uqi j√

2. After this calculation, L and

D, which are, respectively, eigenvectors and eigenvalues ofmatrix B, can be calculated.

3.2 Application to the stochastic nonlinearPoisson–Boltzmann equation

In the application considered here, the random process islocalized near the manifold that describes the surface of thesensor. Physically speaking, the free ions in the aqueous solu-tion screen the effect of the charges of the biomolecules,whose movement is random. The distance where the fieldeffect is still significant is given by (a multiple of) the Debyelength. Mathematically speaking, the semilinear term in (1)results in an exponential decay of the solution away froma point charge. Therefore, the important uncertainty area ofbiomolecules is at the surface.

In Monte-Carlo sampling, the coefficients of (6) are esti-mated as follows:

uβ(x) =1M

M∑

j=1

u(x,ω( j))Φβ

(ω( j)), (14)

where M is the number of evaluations and β ∈ Mk . For thefluctuation of n target molecules at the surface, we calculate(10) as an approximation of (1) using the following formula:

uL(x, γ ) = uL0 (x)+n∑

i=1

uLi (x)γi +n∑

i=1

uLi i (x)γ 2i − 1√

2

+∑

β∈Mkk≥3

n∑

i=1

uLβi Φβ(γi ), (15)

where the rotated random variables are as (9) and the coef-ficients are calculated by (12) and (14). The main advantageof the basis adaptation compared to full-dimensional vari-ables (ω) is that with transformation of the chaos to Gaussianprocess, the cost of calculation decreases dramatically. Infact, in spite of truncation of (15), e.g., second-order adaptedseries, very good estimation of (1) can be achieved. In thiswork, the main goal is the calculation of electrical current

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(quantity of interest). Hence, with respect to the transforma-tion, it can finally be obtained by the following equation:

I (γ ) = f(uLβ (x, γ )

). (16)

4 Multi-level Monte-Carlo method

We have also devised a multi-level Monte-Carlo (MLMC)method for this problem. Since numerical results comparingMLMC and basis adaptation will be shown in Sect. 5, a sum-mary of our MLMC approach is given here. The multi-levelapproach to the Monte-Carlo method was introduced for sto-chastic ordinary differential equations first [14] and has beenapplied to stochastic partial differential equations since then[2,9].

The basic idea of MLMC is to use levels ℓ ∈ 0, . . . , Lwith grids of different fineness hl in order to approximate theexpectation E[u] of the solution u and to reduce the varianceby combining the solutions at various levels. It is straightfor-ward to see that

E[uhL ] =L∑

ℓ=1

E[uhℓ − uhℓ−1 ]

holds after defining E[uh0 ] := 0. In multi-level Monte Carlo,the expectation E[uhℓ − uhℓ−1 ] is approximated by a level-dependent number Ml of samples, i.e., the estimator

EMLMC[u] :=L∑

ℓ=1

EMC[uhℓ − uhℓ−1 ]

is used, where EMC is the usual Monte-Carlo estimator. Forthe mesh refinement, we define the mesh size in level ℓ as

hℓ =h0

2ℓ,

where h0 denotes the mesh size of the coarsest triangulation.If an inequality of the form

∥V[uhℓ − uhℓ−1 ]∥L2(D) ≤ C20h

αℓ−1 ∃C0 ∈ R+ (17)

holds for the variance and an inequality of the form

∥E[uhL − u]∥L2(D) ≤ C21h

βL ∃C1 ∈ R+ (18)

holds for the spatial discretization, then it can be shown thatthe estimate

MSE(u) ≤ C20

L∑

ℓ=0

M−1ℓ hα

ℓ + C21h

βL (19)

holds for the mean square error.

It is additionally possible to optimize the fineness of thegrids, the number of samples at each level, and the numberof levels. More precisely, one considers the minimizationproblem

minimizeL∈N,Mℓ,hℓ∈R+

L∑ℓ=0

Mℓh−γℓ

subject to C20

L∑ℓ=0

M−1ℓ hα

ℓ + C21h

βL ≤ ϵ.

(20)

Here, L ∈ N, the terms h−γℓ denote the average cost at level ℓ,

and ϵ is the given total error to be achieved. Then this min-imization problem can be solved by numerical optimizationmethods, e.g., interior point. This yields the optimal MLMCmethod for a given spatial discretization method such as FEMof a given order, a given implementation, and a given error.Here, first of all we take advantage of multi-level MonteCarlo, i.e., variance reduction of random variables. Secondly,with optimized hierarchies of mesh size and number of sam-ples, the lowest computational complexity (work) is achievedfor a given ϵ. It provides an efficient alternative of MonteCarlo.

5 Numerical results

In order to assess the efficiency of the basis-adaptationmethod devised and also to compare it to the MLMCapproach, the structure in Fig. 1 was considered with real-istic parameter values. It corresponds to a cross section ofa nanowire field-effect biosensor. To verify our stochasticmethods, the statistical parameters of quantity of interests(current) are compared. To this end, we compare the expectedvalue of current (E(I )) with respect to different physicalparameters, e.g., ionic concentration, doping concentration.Moreover, we define signal-to-noise ratio (SNR) of currentas

SNR(I ) := E(I )√Var(I )

,

where Var(I ) indicates the variance of quantity of interest.The ratio enables us to study the effect of the parameters onthe biological noise, i.e., random movement of biomoleculesat the surface.

When quantifying noise and fluctuations in sensors, vari-ous situations where different types of molecules are presentshould be considered. Probe molecules are functionalizedat the sensor surface [32] when the sensor is manufacturedand thus they are responsible for selectivity. In the case ofDNA sensors, the probe molecules may be single-strandedpeptide-nucleic-acid (PNA) oligomers, which are uncharged,or ssDNA oligomers, which are highly charged. The probe

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−4 −3.5 −3 −2.5 −210

−13

10−12

10−11

10−10

10−9

Back−gate voltage [V]

E [I

] (A

)

full−dimensional (ω)

basis adaptation (γ)−2nd order

basis adaptation (γ)−3rd order

experimental data

−4 −3.5 −3 −2.5 −210

−11

10−10

10−9

10−8

Back−gate voltage [V]

E(I

) [A

]

full−dimensional (ω)

basis adaptation (γ)−2nd order

basis adaptation (γ)−3rd order

experimental data

Fig. 4 Comparison of measured and simulated current (expected value) in respect of different back-gate voltages for 60-nm (left) and 100-nm(right) widths of transducer. The simulations are performed for the reference structure, where the back-gate voltage was varied

0.10.5 1 3 5 10

x 1016

10−11

10−10

10−9

10−8

10−7

Doping concentration [cm−3

]

E(I

) [A

]

basis adaptation (γ)− 60nm

full−dimensional (ω) − 60nm

basis adaptation (γ) − 100nm

full−dimensional (ω) − 100nm

0.1 0.5 1 2.5 5 10

x 1016

100

101

102

103

104

105

106

Doping concentration [cm−3

]

SNR

(I)

basis adaptation (γ)− 60 nm

full−dimensional (ω)− 60 nm

basis adaptation (γ)− 100nm

full−dimensional (ω)− 100nm

Fig. 5 Expected value of current and its signal-to-noise ratio as functions of doping concentration for 60-nm and 100-nm-wide devices. Here, theionic concentration is 30 mM; the surface charge is −0.5 q nm−2; the oligomer length is 12 base pairs; VBG = −3 V; and the thermal voltage is0.021 V

molecules move randomly, although there are preferred ori-entations, which have been calculated [17]. Target moleculesbind to the probe molecules and are detected by the fieldeffect modeled here. In the case of a DNA sensor, ssDNAoligomers bind to ssDNA or PNA oligomers. Binding andunbinding are stochastic processes [38]. Here we consider theleading example of a DNA sensor with ssDNA oligomers as

probe molecules and ssDNA oligomers as target molecules.The double-stranded probe–target complex carries twice thecharge of a single strand. These considerations give the sto-chastic coefficients in (1).

The sizes in the various subdomains in Fig. 1 were deter-mined by the following considerations. As found in [5], theoptimal width of the silicon transducer is certainly smaller

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10 25 50 75 100 125 1500.5

1

1.5

2

2.5

3

3.5

x 10−11

Ionic concentration [mM]

E(I

) [A

]

basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

10 25 50 75 100 125 150

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

x 10−10

Ionic concentration [mM]

E(I

) [A

]

basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

10 25 50 75 100 125 15010

20

30

40

50

60

70

80

Ionic concentration [mM]

SNR

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

10 25 50 75 100 125 150200

300

400

500

600

700

800

Ionic concentration [mM]

SNR

(I)

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

Fig. 6 Comparison of expected value (top) and signal-to-noise ratio ofcurrent (bottom) for 60-nm (left) and 100-nm (right) widths as functionsof ionic concentration. Here, the doping concentration is 1016 cm−3;

the surface charge is −0.5 q nm−2; the oligomer length is 12 base pairs;VBG = −3 V; and the thermal voltage is 0.021 V

than 100 nm, while its precise value depends on otherparameters such as the applied back-gate voltage. Smallergeometries were found to have higher sensitivity.

In order to validate the basis-adaptation approach, twonanowires with different widths are compared and shown inFig. 4. The geometry of the sensor is shown in Fig. 1, wherethe thickness of the oxide layer is 8 nm and the thicknessof the nanowire is 50 nm. For the simulations, we definea reference structure, where the back-gate voltage (VBG)

is −3 V; the solution voltage is 0 V; uS − uD is 0.2 V;the length of the nanowire is 1000 nm; the ionic concen-tration is 30 mM; the surface charge is −0.5 q nm−2; thedoping concentration is 1016 cm−3; the oligomers consistof 12 base pairs; and the thermal voltage is 0.021 V. Thelength of each base pair is 0.34 nm. Furthermore, the elec-tron mobility is 1000 cm2 V−1 s−1, and the hole mobilityis 100 cm2 V−1 s−1. The model is validated by comparisonwith experimental data (using the same data as in [5]).

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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2x 10

−11

Surface charge density [q/nm2]

E(I

) [A

]

basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9x 10

−10

Surface charge [q/nm2]

E(I

) [A

]

Basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

Basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 131

32

33

34

35

36

37

38

39

40

Surface charge [q/nm2]

SNR

(I)

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1380

390

400

410

420

430

440

450

460

470

Surface charge density [q/nm2]

SNR

(I)

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

Fig. 7 Comparison of expected value (top) and the signal-to-noise ratioof the current (bottom) for 60-nm (left) and 100-nm (right)-wide sensorsas functions of surface charge density. Here, the doping concentration

is 1016 cm−3; the ionic concentration is 30 nm; the oligomer length is12 base pairs; VBG = −3 V; and the thermal voltage is 0.021 V

The good agreement shows that the current calculated bythe graded-channel approximation in Sect. 2.2 works verywell for this kind of device due to its long length. Secondly,the solutions obtained by basis adaptation are compared withfull-dimensional solutions.

The basis-adaptation approximation is obtained usinga second-order polynomial chaos expansion of (15). Thefull-dimensional solution is the benchmark and actually

results from Hermite polynomial chaos expansion up toorder 2. Here, the simulations are for fluctuations of 10 tar-get molecules (n = 10) and 1000 Monte-Carlo replications(M = 1000). The agreement suggests that the implementa-tion is correct. The differences between the full- dimensionalsolutions and the solutions obtained by basis adaptationbecome larger as the absolute value of the back-gate volt-age increases; this is due to the fact that small differences

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12 14 16 18 20 22 241.7

1.8

1.9

2

2.1

2.2

2.3

2.4x 10

−11

Base pair

E(I

) [A

]

basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

12 14 16 18 20 22 243.4

3.6

3.8

4

4.2

4.4

4.6

4.8x 10

−10

Base pairs

E(I

) [A

]

basis adaptation (γ)−ssDNA

full−dimensional (ω)−ssDNA

basis adaptation (γ)−dsDNA

full−dimensional (ω)−dsDNA

12 14 16 18 20 22 2434

36

38

40

42

44

46

48

Base pairs

SNR

(I)

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

12 14 16 18 20 22 24420

440

460

480

500

520

540

560

Base pairs

SNR

(I)

basis adaptation (γ)−ssDNA

MLMC−ssDNA

basis adaptation (γ)−dsDNA

MLMC−dsDNA

Fig. 8 Comparison of mean value (top) through the transducer and thesignal-to-noise ratio of the current (bottom) for 60-nm (left) and 100-nm (right) widths as functions of oligomer length. Here, the doping

concentration is 1016 cm−3; the ionic concentration is 30 nm; the sur-face charge is −0.5 q nm−2; VBG = −3 V; and the thermal voltage is0.021 V

are amplified by the large applied potential. The resultsare shown in Fig. 4. In the basis adaptation, the differ-ence between second-order and third-order of PCE for both60- and 100-nm-wide nanowires was negligible; therefore,second-order is used for all expected value and noise cal-culation in subsequent simulations. The results indicate theefficiency and the exactness of the basis adaptation againstthe full solution. The agreement with experimental data isvery good and again justifies the model equations.

In this work, we also apply MLMC as another effec-tive numerical technique. In different levels, optimized meshsizes and number of realizations are obtained by solvingthe optimization problem (20). The coefficients of (17) and(18) are regarding the convergence of expected value andvariance. The total error consists of discretization (spa-tial) error and statistical errors where the correspondingexponents are α ≈ 3.2 and β ≈ 1.7. The coefficient γ

is related to the dimension of the computational domain

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(D ⊂ R2). The hierarchies are obtained for the lowest costof calculation such that the constraint (MSE) is satisfied forϵ = 0.001.

The stochastic equation (1) makes it possible to calculatehigher-order moments and in particular the signal-to-noiseratio, an important characteristic value. In the following, wediscuss how various important device parameters influencethe quantity of interest, i.e., the current, and the signal-to-noise ratio. These simulations cover a large part of theparameter space and hence answer the question whether basisadaptation is generally useful or only for the choice of para-meters in the previous figures.

The first parameter investigated here is the doping concen-tration. Currents were calculated for doping concentrationsbetween 1015 and 1017 cm−3 and are shown in Fig. 5. Theresults show that both the current and the signal-to-noise ratioincrease for 60- and 100-nm-wide transducers as the dopingconcentration increases. This means that higher doping con-centrations not only increase the total current, but they alsoimprove the signal compared to the noise, yielding a bettersensor.

The second parameter considered here is the ionic con-centration. It plays a crucial role, since a higher ionicconcentration leads to higher screening and thus reduces theeffectiveness of field-effect sensing. On the other hand, acertain ionic concentration (at least about 10 mM) is usu-ally necessary—especially in the case of DNA—to enableprobe–target binding. A Monte-Carlo algorithm to find theconcentration of (counter) ions around biomolecules wasdeveloped in [7], and the ramifications of screening and howto overcome them were discussed in [21,24]. For a typicalsalt concentration of 100 mM, the Debye length is approx-imately 1 nm and any charge is completely screened at adistance of only a few nanometers.

The numerical results in Fig. 6 show that the effectof the intrinsic charges of the target molecules on chargetransport in the transducer is still measurable. Furthermore,the wider nanowire is more affected by screening withinthe semiconductor. Additionally, the signal-to-noise ratio inthe 100-nm-wide transducer is increased by a factor of 10compared to the 60-nm-wide device. Here, the ionic concen-tration varies between 10 and 150 mM.

The third parameter investigated here is the surface (fixed)charge of molecules. The surface charge is important sinceit determines the operating point of the sensor, which hasa crucial influence on the sensitivity of the sensor [5]. Fur-thermore, a more negative surface charge repels target DNAoligomers, again emphasizing the necessity of consideringthe system in a self-consistent manner. In Fig. 7, the differ-ence between ssDNA and dsDNA oligomers is clearly seenin both 60- and 100-nm-wide devices, as expected. The widthof the device has a strong influence on current and signal-to-noise ratio. The current is about twenty times larger in the

wider device, while the signal-to-noise ratio is approximatelytwelve times larger.

The fourth and final parameter varied here is the length(and hence charge) of the DNA oligomers. The oligomersin the reference structure considered here are 12-base pairslong. However, depending on the application, the lengthof the probe and target oligomers may be varied. Figure 8illustrates that increasing the oligomer length increases thecurrent as well as the signal-to-noise ratio.

6 Conclusions

In all the numerical results, a very good agreement betweenbasis adaptation and the full solution was found. This istrue for a wide range of parameter values: doping concentra-tion, ionic concentration, surface charge, and size and chargeof molecules which were varied over large intervals. Thesenumerical results show the effectiveness of basis adaptationfor the stochastic nonlinear Poisson–Boltzmann equation.

The effectiveness of basis adaptation is due to the trans-formation of random variables and the properties of solutionsof the (stochastic) nonlinear Poisson–Boltzmann equation. Inparticular, it is known that pointwise estimates hold for thistype of semilinear problem [3, Lemma 3.2], [25], justifyingthe transformation of the basis of stochastic process.

Basis adaptation was also compared to a multi-levelMonte-Carlo method as an example of another modernnumerical approach for computing expected values of solu-tions of stochastic partial differential equations. In thisproblem, multi-level Monte Carlo performs well, but not asgood as the basis adaptation. This is mainly due to the factthat the particular multi-level Monte-Carlo method used heredoes not take into account the special structure of the modelequation, whereas the basis adaptation by the rotated ran-dom variables does. Ideas from the multi-level Monte-Carloapproach adapted to the problem at hand may increase theperformance considerably specifically when the mesh sizesand number of evaluations are optimized.

Finally, an efficient numerical method for this equationmakes it possible to perform simulations of realistic struc-tures. Regarding the application of nanowire field-effectsensors, where the nonlinear Poisson–Boltzmann equationplays an important role as a fundamental model equation,a very good agreement of the basis-adaptation method withmeasurements was found. The stochastic nonlinear Poisson–Boltzmann equation makes it possible to include noise andfluctuations in the modeling, and the numerical results showhow the various parameters affect the size of the signal andthe signal-to-noise ratio, an important characteristic value ofsensors. Also, we compared the influence of the width of twodevices, where it was found that the 100-nm-wide sensor hasa larger SNR than the 60-nm-wide one.

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Acknowledgments The authors acknowledge the support by FWF(Austrian Science Fund) START Project No. Y660 PDE Models forNanotechnology. The authors also would like to appreciate Prof. RogerGhanem (University of Southern California) for useful discussionsabout polynomial chaos expansion.

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