Predictive simulations and optimization of nanowire field-effect PSA sensors including screening

The domain Ω as depicted in figure 1 is partitioned into three subdomains Ω_Si,Ω_ox, and Ω_liq corresponding to the three materials silicon, silicon oxide, and the liquid. The silicon nanowire is always covered by a thin layer of silicon oxide and surrounded by the liquid containing the target molecules, i.e., antigens in our case. At the interface Γ between the silicon oxide and the liquid, the charge concentration exhibits a rapidly oscillating spatial structure which leads to a multiscale problem [22]. In the following, the x-direction is always normal to the interface Γ located at x = 0 for notational convenience.

The basic equation for the electrostatic potential is the Poisson equation

$$\begin{eqnarray} &&\fl -\nabla \cdot (\varepsilon (x,y,z)\nabla V(x,y,z))=\rho (x,y,z)\tqs \hspace{-.17pc} \text{in }\Omega , \end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray} &&\fl V(0+,y,z)=V(0-,y,z)\tqs \hspace{4.85pc} \text{on }\Gamma , \end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray} &&\fl {\varepsilon }_{\mathrm{liq}}{\partial }_{x}V(0+,y,z)={\varepsilon }_{\mathrm{ox}}{\partial }_{x}V(0-,y,z)\tqs \hspace{1.3pc} \text{on }\Gamma , \end{eqnarray} \tag{ 1c }$$

where V is the electrostatic potential, ρ is the charge concentration, and ε is the permittivity. The continuity conditions (1b) and (1c) are due to the jump in the permittivity ε between two different materials.

After homogenization of the rapidly oscillating charge concentration in the boundary layer [22], the Poisson equation becomes the effective system

$$\begin{eqnarray} &&\fl -\nabla \cdot (\varepsilon (x,y,z)\nabla V(x,y,z))=\rho (x,y,z)\nonumber\\ &&\hspace{12.7pc} \text{in }{\Omega }_{\mathrm{Si}}\cup {\Omega }_{\mathrm{ox}}, \end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray} &&\fl -\nabla \cdot (\varepsilon (x,y,z)\nabla V(x,y,z))=0\tqs \hspace{.62pc} \text{in }{\Omega }_{\mathrm{liq}}, \end{eqnarray} \tag{ 2b }$$

$$\begin{eqnarray} &&\fl V(0+,y,z)-V(0-,y,z)=\frac{\alpha (y,z)}{{\varepsilon }_{\mathrm{liq}}}\tqs \hspace{-.42pc} \text{on }\Gamma , \end{eqnarray} \tag{ 2c }$$

$$\begin{eqnarray} &&\fl {\varepsilon }_{\mathrm{liq}}{\partial }_{x}V(0+,y,z)-{\varepsilon }_{\mathrm{ox}}{\partial }_{x}V(0-,y,z)=-\gamma (y,z)\nonumber\\ &&\hspace{12.72pc} \text{on }\Gamma , \end{eqnarray} \tag{ 2d }$$

where 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 (2c) and (2d) mean that the rapidly oscillating charge concentration in the surface layer is described by the macroscopic dipole-moment density α and the macroscopic surface-charge density γ after homogenization. The computational significance of the homogenized equation (2) and the interface conditions is that the cumulative effect of the boundary layer can be included in device-level simulations without the need to resolve the molecules in the computational grid, thus decreasing the computational effort by orders of magnitude.

The charge concentration in the boundary layer and hence both the values of γ and α are computed by propka simulations for the biomolecules in question as described below.

We now model the semiconductor. In Ω_Si, the drift-diffusion-Poisson system

$$\begin{eqnarray} -\nabla \cdot ({\varepsilon }_{\mathrm{Si}}\nabla V)&=&q(p-n+{C}_{\mathrm{dop}}), \end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray} \nabla \cdot {J}_{n}&=&R(n,p), \end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray} \nabla \cdot {J}_{p}&=&-R(n,p), \end{eqnarray} \tag{ 3c }$$

$$\begin{eqnarray} {J}_{n}&=&{D}_{n}\nabla n-{\mu }_{n}n \nabla V, \end{eqnarray} \tag{ 3d }$$

$$\begin{eqnarray} {J}_{p}&=&-{D}_{p}\nabla p-{\mu }_{p}p \nabla V, \end{eqnarray} \tag{ 3e }$$

describes the transport of electrons and holes. Here q is the elementary charge, C_dop is the doping concentration, n is the electron concentration, p is the hole concentration, D_n and D_p are the electron and hole diffusion coefficients, μ_n and μ_p are their mobilities, J_n and J_p are their current densities, and R denotes the recombination rate [25]. Here we use the popular Shockley–Read–Hall recombination rate

$$\begin{eqnarray} &&R(n,p):= \frac{n p-{n}_{\mathrm{i}}^{2}}{{\tau }_{p}(n+{n}_{\mathrm{i}})+{\tau }_{n}(p+{n}_{\mathrm{i}})}, \end{eqnarray} \tag{ 4 }$$

where n_i denotes the intrinsic charge density and τ_n and τ_p are the relaxation times of the electrons and holes, respectively. Furthermore, the Einstein relations D_n = U_Tμ_n and D_p = U_Tμ_p are assumed to hold, where U_T is the thermal voltage.

The liquid Ω_liq is described by the Poisson–Boltzmann equation

$$\begin{eqnarray} &&-\nabla \cdot ({\varepsilon }_{\mathrm{liq}}\nabla V)=\mathop{\sum }\limits _{\sigma \in \{ -1,1\} }\eta \sigma \exp (-\sigma \beta V), \end{eqnarray} \tag{ 5 }$$

where η is the ionic bulk concentration and the constant β is defined as β:=q/(kT), where k is the Boltzmann constant and T is the temperature. The right-hand side of equation (5) is the sum over all valences σ of ion species so that the set { − 1,1} corresponds to a 1:1 electrolyte such as Na⁺Cl⁻, for example.

At the source, drain, and backgate contacts, Dirichlet boundary conditions

$$\begin{eqnarray} V\vert _{\partial {\Omega }_{\mathrm{S}}}={V}_{\mathrm{S}},\hspace{1.6pc} n\vert _{\partial {\Omega }_{\mathrm{S}}}&=&{n}_{\mathrm{S}},\hspace{1.65pc} p\vert _{\partial {\Omega }_{\mathrm{S}}}={p}_{\mathrm{S}}, \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} V\vert _{\partial {\Omega }_{\mathrm{D}}}={V}_{\mathrm{D}},\hspace{1.47pc} n\vert _{\partial {\Omega }_{\mathrm{D}}}&=&{n}_{\mathrm{D}},\hspace{1.47pc} p\vert _{\partial {\Omega }_{\mathrm{D}}}={p}_{\mathrm{D}}, \end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray} \hspace{6pc} V\vert _{\partial {\Omega }_{\mathrm{G}}}&=&{V}_{\mathrm{G}} \end{eqnarray} \tag{ 6c }$$

hold, as well as at the electrode in the liquid if one is present [26]. Zero Neumann boundary conditions are used everywhere else.

In order to calculate the charge of PSA, we use the structure from the Protein Data Bank (see the left-hand part of figure 2) and compute the charge state of the protein by using the propka algorithm [27–30]. The charge of PSA as a function of pH value obtained in that manner is shown in the right-hand part of figure 2.

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Since we compare the simulations with measurements performed at a pH value of 9 below, we note a total charge of PSA −15 q at this pH value. Using a footprint of 10 nm for the functionalized molecules, which is achievable in experiments [32], this results in an unscreened surface charge of −1.5 q nm⁻².

The liquid surrounding the nanowire sensors is an ionic solution. Therefore, the effective charge of the biomolecules giving rise to the field effect is clearly lowered due to Debye screening [33]. The screening decreases exponentially with distance. More precisely, it decreases as exp(−x/λ_D), where the Debye length λ_D is defined as

$$\begin{eqnarray} &&{\lambda }_{\mathrm{D}}:= \frac{1}{\sqrt{4 \pi {l}_{\mathrm{B}}\mathop{\sum }_{\mathrm{i}}{\rho }_{\mathrm{i}}z_{\mathrm{i}}^{2}}}. \end{eqnarray} \tag{ 7 }$$

Here l_B = 0.7 nm is the Bjerrum length, and ρ_i and z_i are the concentration and valence of the ion species i.

Assuming the protein to be a point charge at a distance l from the nanowire surface, i.e., we can model Debye screening by using a sensitivity factor describing the induced change due to screening between 0 and 1. This sensitivity factor is given by

$$\begin{eqnarray} &&{\Gamma }_{l}\approx 2\frac{r}{r+l}\left (1+\sqrt{\frac{r}{r+l}}\exp \left (\frac{l}{{\lambda }_{\mathrm{D}}}\right )\right )^{-1} \end{eqnarray} \tag{ 8 }$$

for a radial nanowire with radius r [34]. This model has been used for the simulation of nanowire sensors in [35]. In the case of a planar nanoribbon nanowire, we obtain the sensitivity factor

$$\begin{eqnarray} &&{\Gamma }_{l}\approx 2\left (1+\exp \left (\frac{l}{{\lambda }_{\mathrm{D}}}\right )\right )^{-1} \end{eqnarray} \tag{ 9 }$$

for r → ∞ (see also [13]).

For antibody–antigen sensors using a functionalization with anti-CA15-3 (CA15-3 is cancer antigen 15-3), this value is ≈50% for C-termini functionalized antibodies when using a 1 mM bicarbonate sensing buffer and ≈65% when using N-terminal functionalization (for a detailed discussion see [13]). Hence the resulting screened surface charge of PSA at a pH value of 9 and a C-terminal functionalized antibody is around −0.75 q nm⁻².

We can now compute the electrostatic potential V and the concentrations of the charge carriers n and p in a self-consistent loop. After choosing initial values n₀,p₀, α₀, and γ₀, the following steps are performed in a Scharfetter–Gummel type iteration scheme.

• (i)  
  In iteration k + 1, solve the homogenized boundary-value problem

  $$\begin{eqnarray*} -{\varepsilon }_{\mathrm{Si}}\Delta {V}_{k+1}=q({C}_{\mathrm{dop}}+{p}_{k}-{n}_{k})&&\qquad \text{in }{\Omega }_{\mathrm{Si}},\nonumber\\ -{\varepsilon }_{\mathrm{ox}}\Delta {V}_{k+1}=0&&\qquad \text{in }{\Omega }_{\mathrm{ox}},\nonumber\\ -{\varepsilon }_{\mathrm{liq}}\Delta {V}_{k+1}=\mathop{\sum }\limits _{\sigma \in \{ -1,1\} }\eta \sigma {\mathrm{e}}^{-\sigma \beta {V}_{k+1}}&&\qquad \text{in }{\Omega }_{\mathrm{liq}},\nonumber\\ {V}_{k+1}(0+)-{V}_{k+1}(0-)=\frac{{\alpha }_{k}}{\varepsilon (0+)}&&\qquad \text{on }\Gamma ,\nonumber\\ {\varepsilon }_{\mathrm{liq}}\nabla {V}_{k+1}(0+)-{\varepsilon }_{\mathrm{ox}}\nabla {V}_{k+1}(0-)=-{\gamma }_{k}&&\qquad \text{on }\Gamma ,\nonumber\\ {V}_{k+1}={V}_{\mathrm{D}}&&\qquad \text{on }\partial {\Omega }_{\mathrm{D}},\nonumber\\ {\nabla }_{\nu }{V}_{k+1}=0&&\qquad \text{on }\partial {\Omega }_{N} \end{eqnarray*}$$

  to obtain the electrostatic potential V_k+1.
• (ii)  
  Then solve the boundary-value problem

  $$\begin{eqnarray*} &&\fl \nabla \cdot ({D}_{n}\nabla {n}_{k+1}-{\mu }_{n}{n}_{k+1}\nabla {V}_{k})=R({n}_{k+1},{p}_{k})\tqs \nonumber\\ &&\text{in }{\Omega }_{\mathrm{Si}},\nonumber\\ &&\fl {n}_{k+1}={n}_{\mathrm{D}}\tqs \text{on }\partial {\Omega }_{\mathrm{D}},\nonumber\\ &&{\nabla }_{\nu }{n}_{k+1}=0\tqs \hspace{.5pc} \text{on }\partial {\Omega }_{N} \end{eqnarray*}$$

  to obtain the electron concentration n_k+1.
• (iii)  
  Then solve the boundary-value problem

  $$\begin{eqnarray*} &&\fl \nabla \cdot ({D}_{p}\nabla {p}_{k+1}-{\mu }_{p}{p}_{k+1}\nabla {V}_{k})=R({n}_{k+1},{p}_{k+1})\nonumber\\ &&\text{in }{\Omega }_{\mathrm{Si}},\nonumber\\ &&\fl {p}_{k+1}={p}_{\mathrm{D}}\tqs \text{on }\partial {\Omega }_{\mathrm{D}},\nonumber\\ &&{\nabla }_{\nu }{p}_{k+1}=0\tqs \hspace{.5pc} \text{on }\partial {\Omega }_{N} \end{eqnarray*}$$

  to obtain the hole concentration p_k+1.
• (iv)  
  Compute α_k+1 = α(V_k+1) and γ_k+1 = γ(V_k+1) or look up the values in a precomputed table.
• (v)  
  Increase k and go to the first step.

The existence, local uniqueness around thermal equilibrium, and smoothness of the solution (V,n,p) have been shown based on a fixed-point theorem and the inverse-function theorem [23].

As aforementioned, we simulate a sensor with the basic geometry depicted in figure 1. The length of the nanowire is 1000 nm, the doping concentration is −10¹⁶ cm⁻³, and the source voltage is V_S =− 0.2 V. In the structures fabricated, the thickness of the silicon nanowire can vary between approximately 40 and 50 nm and the thickness of the silicon dioxide can vary between 8 and 20 nm.

Knowing the geometry, it is possible to extract the electron and hole mobilities and the thermal voltage by comparison with the measured current–voltage characteristics. This is required, because silicon nanowires with such a high surface-to-volume ratio cannot be expected to have bulk properties, e.g., due to surface-roughness effects. Therefore the slope of the current–voltage characteristics and the gap between the characteristics of nanowires of different widths are used to determine the physical properties of the nanowire sensors.

The physical parameters determined in this manner are shown in table 1 and the corresponding agreement between measurement and simulation is shown in figure 3. In figure 3(a), simulated and measured current–voltage characteristics of two devices with 60 and 100 nm wide nanowires, each with a 8 nm thick top oxide, a nanowire thickness of 50 nm, a hole mobility of 100 cm⁻² V⁻¹ s⁻¹, and a thermal voltage of 0.021 V, are shown. We note that there is a substantial difference in the hole mobilities (up to four times larger). Due to loss of convergence of the Scharfetter–Gummel iteration for large negative gate voltages, the current–voltage characteristics are calculated only up to a certain gate voltage. This is due to the fact that the solutions of the drift-diffusion-Poisson system are unique only for applied voltages sufficiently close to thermal equilibrium [23].

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The simulations with the other parameters in rows (b), (c), and (d) in table 1 were compared to the current–voltage characteristic of the device with a 100 nm wide nanowire as well, while it is found that the agreement for the 60 nm wide nanowire is always worse (see figures 3(b)–(d)). This implies that row (a) in table 1 contains the correct parameters for the real-world device. This can be further validated by comparing the measured sensitivity of the sensor with simulated sensitivity values.

In the next step, we therefore consider the sensitivity of the devices in figure 3 with the parameters in table 1 and vary the screened surface charge γ between −0.4 q nm⁻² and −0.8 q nm⁻² as an additional free parameter. The results are shown in figure 4. As expected, the best agreement with the measured sensitivity is found using the best parameters for the current–voltage characteristics, i.e., the slope when changing from 60 to 100 nm wide nanowires is nearly the same and the corresponding screened surface charge is in the range of the screened surface charge computed by propka. To be more specific, the devices with a 14 nm thick top oxide (see (b) and (d) in figure 4) have the same slope when changing from 60 nm wide nanowires to 100 nm wide nanowires, but then the screened surface charge is higher than the unscreened surface charge computed by propka. Therefore this oxide thickness cannot be realistic. For the devices with a 8 nm thick top oxide and a 40 nm thick nanowire, the slope does not match the measured sensitivity, and the corresponding screened surface charge for 60 nm wide nanowires is higher then the unscreened surface charge computed by propka. Therefore this case is also not realistic.

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This implies that parameter set (a) in table 1 is the best. In the following we therefore consider the device with a 8 nm thick top oxide, a 50 nm thick nanowire, a hole mobility of 100 cm⁻² V⁻¹ s⁻¹, and a thermal voltage of 0.021 V, for which the agreement between measurements and simulations is very good both regarding the current–voltage characteristics (see (a) in figure 3) and the sensitivity for a surface charge of γ =− 0.75 nm⁻² (see (a) in figure 4).

Having validated the model and the simulations using measurements, the device properties can be varied in order to optimize the sensors especially with respect to sensitivity. The simulations also make it possible to easily quantify the effect of changes in device parameters, to quantify the nonlinear dependences between parameters such as the backgate voltage and the geometry of the nanowires, and to determine the optimal operating regime.

We first investigate how the sensitivity is influenced by different physical and geometric device properties. As aforementioned, the standard nanowire device considered here is 60 and 100 nm wide and 50 nm high. The first question is if the sensitivity can be improved by varying the doping concentration and the geometry, i.e., the thickness, width, and length of the nanowire at different gate-voltage levels (see figure 5).

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The sensitivity depends on both the thickness and the width of the nanowire as seen in figure 5(a). Both the 60 nm and the 100 nm wide nanowires show that sensitivity dramatically decreases when the thickness becomes too small. The sensitivity is improved when the wires are thicker unless the gate voltage is too low. But a low gate voltage by itself does not necessarily decrease the sensitivity; in the case of narrow nanowires, we find that a low gate voltage increases the sensitivity (see figure 5(b)).

The length of the nanowire only has a minor influence on the sensitivity. A small increase in sensitivity is found when the nanowires are longer as depicted in the bottom left of figure 5. The doping concentration affects the sensitivity only at high levels of doping and reduces it (see figure 5(c)). Here we find again a correlation between the decrease in the sensitivity and the magnitude of the gate voltage.

A gate voltage that is too negative becomes detrimental. In figure 5, it is seen that a gate voltage of V_G =− 3 V is too large and decreases sensitivity. At this point, the large gate voltage is the major parameter affecting charge transport and the relatively much smaller influence of the molecules is not observed well anymore. This effect is more pronounced in the larger devices (100 nm wide and thicker than circa 50 nm). While the molecules at the surface are sufficient to affect the whole silicon nanowire in the smaller devices, their effect is limited to the Debye length. In the large nanowires, a substantial part is not affected noticeably by the molecules at the surface, but it can be affected by a large gate voltage. Hence the relative influence of the molecules decreases, meaning that the sensitivity decreases.

In figure 6, the crucial influence of the gate voltage on sensitivity becomes clear as well. Since a high gate voltage has more influence on the charge carriers in the nanowire than the typical surface charge of biomolecules, the sensitivity decreases. This can also be understood as a screening effect, i.e., the influence of the backgate voltage on the electrons and holes in the nanowire screens the influence of the surface charge. The clearest examples are the thicknesses of the top oxide and the bulk oxide. In figure 6(a), as the thickness of the top oxide increases, the molecules are farther away from the transducer and their effect and hence the sensitivity decrease. In figure 6(b), as the thickness of the bulk oxide increases, the gate contact is farther away from the transducer, and the detrimental influence of the highly negative gate voltage decreases. The impact of a high gate voltage is also seen in figure 6(c). However, not all voltages applied to the device turn out to decrease the sensitivity: In the case of the source voltage, the sensitivity remains at a certain level as depicted in figure 6(d). These numerical results underline the importance of back-gated devices and the nonlinear influence of the design parameters.

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