Impact of triple-material gate and highly doped source/drain extensions on sensitivity of DNA biosensors

Abstract

Gate engineering and highly doped source/drain region have been investigated to design a new DNA sensor for use in biomedical applications based on a double gate (DG) dielectric modulated (DM) junctionless (JL) metal oxide semiconductor field effect transistor (MOSFET) with triple material (TM) gate. Based on the dielectric modulation effect, DNA molecules in the nanogap cavity change due to the charge density of biomolecules, producing a change in the threshold voltage of the device. Analytical and numerical analysis was carried out to reveal the impact of physical parameters on the sensitivity of the proposed biosensor. Various characteristics, such as the surface potential, threshold voltage, and drain current were also investigated. The effectiveness of the proposed TM-DG-DM-JL-MOSFET structure with highly doped source/drain extensions is confirmed by comparison of the results with those for a conventional single-materiel (SM) gate DM-JL-MOSFET, revealing a good improvement in sensitivity and making the proposed structure an attractive solution for use in DNA-based sensor applications.

1 Introduction

The use of DNA in the diagnosis of different diseases has increased rapidly since the completion of the Human Genome Project, enabling research to progress towards bioelectronics applications. This combination of biology and electronics has been promoted by the use of field-effect transistors as transducers. Indeed, their use in biomedical applications is growing rapidly due to their good performance, including rapid detection, miniaturized device size, low cost, and high sensitivity [1,2,3,4,5,6]. Research interest has also been attracted to label-free hybridization technology [7,8,9], and a significant number of investigations have been performed on different types of FET-based biosensors, including extended-gate FETs [10, 11], ion-selective FETs [12,13,14,15], nanowire FETs [16,17,18,19,20,21], and carbon nanotube FETs [22,23,24]. More recently, such work has also focused on the dielectric-modulated FET (DMFET), which is essentially based on the use of a nanogap in a traditional MOSFET structure [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].

A DMFET is a new MOSFET architecture in which biomolecules are fixed in a nanogap cavity located between the gate oxide and the gate [25]. The first application of this technique was based on the shift of the threshold voltage as a detection mechanism [7, 25]. When biomolecules fill the nanogap, they induce a variation in the threshold voltage because of the deviation of the dielectric constant K from unity (corresponding to air) to a certain value (K > 1) [7, 25]. Similarly, a variation in current can be produced via the charge-pumping technique, providing information related to the specific binding process [31, 33, 35]. Several investigations have been carried out on dual- and three-material gate MOSFETs [40,41,42,43,44,45,46,47,48,49,50] with different metal work functions.

The aim of this study is to determine the impact of gate engineering on such a molecular biosensor. The present work considers the design of a novel long-channel DG-DM-JL-MOSFET with a three-material gate and highly-doped source/drain extensions for use in biomedical applications. To investigate systematically the impacts of gate engineering and the highly doping source/drain regions on the sensitivity of the DG-DM-JL-MOSFET device, an analytical model for the surface potential and threshold voltage was developed then verified using numerical simulations with ATLAS-2D simulator. The results are then compared with those for a classical single-material gate structure [37] of similar dimensions, clearly revealing that the proposed design exhibits excellent sensitivity that makes it a potential candidate for use in future medical applications.

2 Device structure and operating mode

Figure 1 depicts the architecture of the DG-TM-DM-JL-MOSFET with highly doped source/drain extensions. The channel body is less heavily doped then the source/drain regions, so the concentration distribution is npp/np/npp. Besides, the device structure is divided into three regions, namely I, II, and III. Regions I and III contain cavity regions created in the gate oxide under gate 1 and gate 3, respectively, with metals M1 and M3; while region II contains gate oxide under gate 2 with metal M2. The lengths of M1, M2, and M3 are L₁, L₂, and L₃, respectively. The work function of M3 ($\phi_{{{\text{M}}3}} = 4.5$) is lower than that of M2 ($\phi_{{{\text{M}}2}} = 4.7$), which is lower than that of M1 ($\phi_{{{\text{M}}1}} = 5.1$). In the proposed device, M1 is towards the source side, while M3 is towards the drain side. The SiO₂ and TiO₂ in the gate oxide aim to improve the performance of the device and reduce the fringing electric field by the deposition of a high-k material directly on silicon [45]. The dimensions of the proposed device are presented in Table 1.

Fig. 1

Structure of TM-DG-DM-JL-MOSFET with highly doped source/drain extensions

Table 1 Dimensions of DG-TM-DM-JL-MOSFET with highly doped source/drain extensions

The aim of this study is to describe the influence of DNA molecules on the characteristics of these biosensors.

Since DNA is negatively charged, due to the phosphate groups present in its backbone [51], it will inject a negative charge density (N_f) at the SiO₂–air interface of the device.

The operating mode of this biosensor is based on the presence or absence of DNA molecules in the cavity regions. When not present in the nanogap cavity, the DNA molecules are replaced by air; this fact implies the absence of the charge density N_f on the one hand, and a dielectric constant of 1 on the other hand. When DNA molecules are injected into the nanogap region, the charge density (N_f) appears and the dielectric constant changes from 1 to K. Therefore, changes occur in the gate capacitance and the flat band of the device, with a corresponding variation in the threshold voltage (v_th). The sensing site of this device is the cavity region created in the gate oxide [7].

3 Surface potential and threshold voltage models

3.1 Surface potential model

The distribution of the surface potential was obtained by solving the two-dimensional (2D) Poisson equations with a parabolic approximation [52] in the silicon channel, divided into three regions:

Region I: 0 ≤ Y≤ t_Si; 0 ˂ X ˂ L₁.

Region II: 0 ≤ Y ≤ t_Si; L₁ ˂ X ˂ L₁ + L₂.

Region III: 0 ≤ Y ≤ t_Si; L₁ + L₂ ˂ X ˂ L.

$$ \frac{{{\text{d}}\varphi_{i} \left( {x,y} \right)}}{{{\text{d}}x^{2} }} + \frac{{{\text{d}}\varphi_{i} \left( {x,y} \right)}}{{{\text{d}}y^{2} }} = - q\frac{{N_{\rm d } }}{{\varepsilon_{\text{Si}} }}, $$

(1)

where i refers to the considered region (I, II or III), q is the electron charge, N_d is the doping concentration of the channel, and $\varepsilon_{\text{Si}}$ is the dielectric permittivity of silicon.

$$ \varphi_{i} \left( {x, y} \right) = A_{0i} \left( x \right) + A_{1i} \left( x \right)y + A_{2i} \left( x \right)y ^{2}, $$

(2)

where A_0i(x), A_1i(x), and A_2i(x) are determined by using the boundary conditions and continuity of the electric flux.

At Si–SiO₂ interfaces, the following hold:

$$ \varphi_{i} \left( {x,0} \right) = \varphi_{\rm Si} \left( x \right), $$

(3-a)

$$ \varphi_{i} \left( {x,\frac{{t_{\text{Si}} }}{2}} \right) = \varphi_{{\rm c}i} \left( x \right), $$

(3-b)

$$ \frac{{\partial \varphi_{i} \left( {x,0} \right)}}{\partial y} = \frac{{C_{i} }}{{\varepsilon_{\text{Si}} }}\left( {\varphi_{\text{Si}} \left( x \right) - V_{\text{gs}} + V_{\text{fbi}} } \right), $$

(3-c)

$$ \frac{{\partial \varphi_{i} \left( {x,t_{\text{Si}} } \right)}}{\partial y} = - \frac{{C_{i} }}{{\varepsilon_{\text{Si}} }}\left( {\varphi_{\text{Si}} \left( x \right) - V_{\text{gs}} + V_{\text{fbi}} } \right), $$

(3-d)

$$ \frac{{\partial \varphi_{i} \left( {x,\frac{{t_{\rm Si} }}{2}} \right)}}{\partial y} = 0, $$

(3-e)

where $\varphi_{\text{Si}} \left( {\text{x}} \right)$ is the surface potential, $\varphi_{\text{c}i} \left( {\text{x}} \right)$ is the central potential, $V_{\text{gs}}$ is the gate-to-source voltage, and ${\text{V}}_{\text{fbi}}$ denotes the flat-band voltage, given by

$$ V_{{{\text{fb}}1}} = \phi_{\rm M1} - \phi_{\text{Si}} - \frac{{qN_{\rm f} }}{{C_{\text{bio}} }}, $$

(4-a)

$$ V_{\rm fb2} = \phi_{\rm M2} - \phi_{\rm Si}, $$

(4-b)

$$ V_{{{\text{fb}}3}} = \phi_{\rm M3} - \phi_{\text{Si}} - \frac{{qN_{\rm f} }}{{C_{\text{bio}} }}, $$

(4-c)

$$ \phi_{\rm Si} = \chi_{\rm Si} + \frac{{E_{\rm g} }}{2}, $$

(5)

$$ C_{\rm bio} = \frac{{\varepsilon_{\rm bio} }}{{t_{\rm bio} }}, $$

(6-a)

$$ C_{1} = C_{3} = \frac{{C_{{{\text{bio}} \cdot C_{{{\text{SiO}}_{2} }} }} }}{{C_{\text{bio}} + C_{{{\text{SiO}}_{2} }} }}, $$

(6-b)

$$ C_{{{\text{SiO}}_{2} }} = \frac{{\varepsilon_{{{\text{SiO}}_{2} }} }}{{t_{{{\text{SiO}}_{2} }} }}, $$

(6-c)

$$ C_{2} = C_{\text{ox}} = \frac{{\varepsilon_{\text{ox}} }}{{t_{\text{ox}} }}, $$

(6-d)

with $t_{\text{ox}} = t_{\text{ox}1} + t_{{ \text{ox}2}} \frac{{\varepsilon_{{{\text{ox}}1}} }}{{\varepsilon_{{{\text{ox}}2}} }}$.

C_bio is the capacitance of the cavity region, and C_i is the gate capacitance per unit area of the gate dielectric of the TM-JL-DG-MOSFET. N_f and $\varepsilon_{\text{bio}}$ are, respectively, the charge density and the permittivity of the DNA molecules.

$$ A_{0i} \left( x \right) = \varphi_{\rm Si} \left( x \right), $$

(7-a)

$$ A_{1i} \left( x \right) = \frac{{C_{i} }}{{\varepsilon_{\text{Si}} }}\left( {\varphi_{\text{Si}} \left( x \right) + V_{\text{fbi}} - V_{\text{gs}} } \right), $$

(7-b)

$$ A_{2i} \left( x \right) = - \frac{{C_{i} }}{{t_{\rm Si} \varepsilon_{\rm Si} }}\left( {\varphi_{\rm Si} \left( x \right) + V_{\mathrm{fbi}} - V_{\rm gs} } \right). $$

(7-c)

Substituting the expressions for $A_{0i}(x), A_{1i}(x), \text{ and } A_{2i}(x)$ into (2), Eq. (1) leads to a one-dimensional second-order differential equation:

$$ \frac{{\partial^{2} \varphi_{\text{Si}} \left( x \right)}}{{\partial x^{2} }} - \alpha_{i} \varphi_{\text{Si}} \left( x \right) = \beta_{i}, $$

(8)

where

$$ \alpha_{i} = \frac{{8C_{i} }}{{4t_{\text{Si}} \varepsilon_{\text{Si}} + C_{i} t_{\text{Si}}^{2} }}, $$

(8-a)

$$ \beta_{i} = - \frac{{qN_{\rm d} }}{{\varepsilon_{\text{Si}} }} + \alpha_{i} \left( {V_{\text{fb}i} - V_{\text{gs}} } \right). $$

(8-b)

Hence Eq. (8) has a general solution of the form

$$ \varphi_{\rm s1} \left( x \right) = D_{1 } \mathrm{e}^{{ - \sqrt {\alpha_{1} } x}} + E_{1} \mathrm{e}^{{ - \sqrt {\alpha_{1} } \left( {L_{1} - x} \right)}} - \sigma_{1}, $$

(9-a)

$$ \varphi_{\rm s2} \left( x \right) = D_{2 } \mathrm{e}^{{ - \sqrt {\alpha_{2} } \left( {x - L_{1} } \right)}} + E_{2} \mathrm{e}^{{ - \sqrt {\alpha_{2} } \left( {L_{1} + L_{2} - x} \right)}} - \sigma_{2}, $$

(9-b)

$$ \varphi_{\rm s3} \left( x \right) = D_{3 } \mathrm{e}^{{ - \sqrt {\alpha_{3} } \left( {x - L_{1} - L_{2} } \right)}} + E_{3} \mathrm{e}^{{ - \sqrt {\alpha_{3} } \left( {L - x} \right)}} - \sigma_{3}. $$

(9-c)

The coefficients D_i, E_i, and $\sigma_{i}$ are detailed in “Appendix A.”

3.2 Threshold voltage model

The threshold voltage is the value of V_gs at which $\varphi_{\rm s1,min}$ = 2$\Phi_{\rm F}$, where $\Phi_{\rm F} = V_{\rm t} \ln \frac{{N_{\rm d} }}{n_{\rm i}}$ is the variance between the intrinsic and extrinsic Fermi levels [44].

In the case of the TM-DG structure and due to the coexistence of metal gates M1, M2, and M3 (with different work functions), the minimum of the surface potential $\varphi_{{\rm s}\left( i \right),\hbox{min} }$ is solely determined by the metal gate with highest work function. The position $x$_min of the minimum surface potential $\varphi_{{\rm s}\left( i \right),\hbox{min} }$ can be determined by solving $\frac{{{\text{d}}\varphi_{s1} }}{{{\text{d}}x}} = 0$. Thus, we obtain

$x_{\hbox{min} } = \frac{1}{{2\sqrt {\alpha_{1} } }}\ln \left( {\frac{{D_{1} }}{{E_{1} .\mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }}} \right)$. Hence,$\varphi_{{\rm s}\left( i \right),\hbox{min} }$ is determined as

$$ \varphi_{{\rm s}1,\hbox{min} } = 2\sqrt {D_{1} E_{1} \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} } - \delta_{1}. $$

The final expression for the threshold voltage is therefore

$$ V_{{\rm th},i} = \frac{{ - G_{i} + \sqrt {G_{i} - 4HK} }}{2H}, $$

(10)

where

$$ G_{i} = m_{1} n_{2} + m_{2} n_{1} - 2p_{i}, $$

(10-a)

$$ H = 1 + n_{1} n_{2}, $$

(10-b)

$$ K = m_{1} m_{2} + \rho^{2}. $$

(10-c)

The coefficients m_i, n_i, and $p_{i}$ are detailed in “Appendix B.”

This mathematical model for our device was verified using ATLAS 2-D simulator, where all simulations were realized at room temperature (300 K) [52]. The physical effects used to approximate the numerical model in the analytical model are summarized in Table 2.

Table 2 Models and methods used for the TCAD simulations of the TM-JL-DG-MOSFET with highly doped source/drain extensions

Note that quantum–mechanical effects (QMEs) were neglected during the simulations and modeling, as they start to dominate for devices having t_Si below 5 nm [40].

4 Results and discussion

To validate the proposed structure, many simulations were carried out, investigating the impacts of the charge density of DNA molecules and the device dimensions on the surface potential, threshold voltage, sensitivity, and I_on/I_off ratio.

Figure 2 shows the shape of the surface potential as a function of position along the channel. These simulations were performed using three gate materials with different work functions and different amounts of DNA molecules. The figure clearly shows that the surface potential decreases under the nanogap cavity (the region where the DNA molecules are injected), whereas there is no deformation in the rest of the channel region where the cavity is not formed.

Fig. 2

Shape of surface potential of TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of charge density (drain voltage V_ds = 1 V, V_gs = 0 V)

The relation between the surface potential and the charged biomolecules can be explained based on the changes in the flat-band voltage (∆V_fb) in the cavity region, which depend on the charged biomolecules (N_f) as follows: $\Delta V_{\rm fb} = \frac{{q N_{\rm f} }}{{C_{\rm gap} }}$.

Another parameter that was investigated to assess the impact of biomolecules on the device was the threshold voltage, which is an important sensing parameter in such biosensors. Figure 3 depicts the impact of DNA molecules and the nanogap length on the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped S/D extensions, comparing the results obtained from the developed model with those of the numerical technology computer-aided design (TCAD) simulation.

Fig. 3

a Variation of the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density and several lengths of the nanogap $(\varepsilon_{\text{bio}} = 15,\,t_{\text{Si}} = 10\,{\text{nm}})$. b Variation of the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density and several lengths of the middle gate. c Variation of the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density for several thicknesses of silicon. d Variation of the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the nanogap thickness t_bio and several charge densities $(\varepsilon_{\text{bio}} = 13,\,t_{\text{Si}} = 10\,{\text{nm}})$

Figure 3a depicts the variation of the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions. These simulations were carried out for different charge densities and several nanogap lengths. It can be seen that the threshold voltage decreases as the charge density of the DNA molecules increases.

Figure 3b plots the variation of the threshold voltage of the considered device as a function of the middle gate length for different values of the charge density. Note that the threshold voltage increases as the middle gate length is decreased.

To investigate the effect of the charge of the biomolecules, the variation of the threshold voltage is also plotted in Fig. 3c. This figure shows corresponding plots for various silicon thicknesses and charge density values. Note that the threshold voltage increases as the silicon thickness is decreased.

Figure 3d depicts the variation in the threshold voltage of the TM-DG-DM-JL-MOSFET with highly doped S/D regions as a function of the nanogap thickness. These simulations were carried out for two different charge densities (t_bio). It can be clearly seen that there is a small variation in the threshold voltage as the thickness of the hollow regions is increased.

Sensitivity is an important parameter for selection of biosensors, calculated as the shift of the threshold voltage between the presence and absence of DNA molecules in the hollow regions as

$$ \begin{aligned} & \Delta V_{\text{th}} = V_{\text{th(DNA)}} - V_{\text{th(AIR)}} \\ & \Delta {\text{V}}_{\text{TH}} = {\text{V}}_{{{\text{TH}}\left( {\text{withDNAbiomolecules}} \right)}}\\&\quad - {\text{V}}_{{{\text{TH}}\left( {\text{withoutDNAbiomolecules}} \right)}}. \\ \end{aligned} $$

(7)

Figure 4a, b depicts the sensitivity when DNA molecules are immobilized in the cavity regions of the TM-DG-DM-JL-MOSFET with highly doped extensions.

Fig. 4

a Variation of the sensitivity of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density and several thicknesses of silicon $(K_{\text{bio}} = 13,\,t_{\text{bio}} = 9\,{\text{nm}})$. b Variation of the sensitivity of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density and various lengths of the nanogap cavity $(K_{\text{bio}} = 15,t_{\text{bio}} = 9\,{\text{nm}})$. c Variation of the sensitivity of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for several lengths of middle gate and different values of the charge density $(K_{\text{bio}} = 15,t_{\text{bio}} = 9\,{\text{nm}})$. d Variation of the sensitivity of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for several thicknesses of the nanogap cavity and different values of the charge density $(K_{\text{bio}} = 13)$

Figure 4a reveals that the sensitivity increases as the silicon thickness is decreased (for t_Si = 10 nm, ∆V_th = 0.56 V and for t_Si = 20 nm, ∆V_th = 0.33 V). Conversely, in Fig. 4b, the sensitivity decreases when the length of the cavity region is decreased too; i.e., with N_f = -3 × 10¹² cm^-2, then L_gap = 25 nm gives ∆V_th = 480 mV while L_gap = 20 nm gives ∆Vth = 340 mV.

Figure 4c plots the variation of the sensitivity of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for several middle gate lengths and different values of the charge density. The middle gate length was varied from 30 to 60 nm. The results clearly show that the sensitivity with a longer middle gate was low compared with that obtained for short middle gate length; nevertheless, the functionality of the considered biosensor remains acceptable.

Figure 4d depicts the variation of the sensitivity of the device under study for several nanogap cavity thicknesses and different values of the charge density. Note from this curve that a slight variation occurs in terms of sensitivity as the nanogap cavity thickness is increased.

Figures 5 and 6 show simulation results for the transfer characteristic and I_on/I_off ratio for different values of the charge density for the TM-DG-DM-JL-MOSFET with highly doped extensions. These simulations were performed using the TCAD simulator.

Fig. 5

Variation of the drain current as a function of the gate voltage for the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different values of the charge density $(K_{\text{bio}} = 15,\,t_{\text{bio}} = 9\,{\text{nm}},\,\,{\text{and}}\,\,t_{\text{Si}} = 10\,{\text{nm }})$

Fig. 6

Variation of the I_on/I_off ratio for different values of the charge density for the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different gate oxide materials (Al₂O₃ and SiO₂ + TiO₂)

Figure 5 plots the drain current as a function of the gate voltage for different charge density values. The results clearly show that, as the charge density of the DNA molecules increases, the on/off currents decrease.

The I_on/I_off ratio is another useful parameter for assessment of the sensitivity of biosensors. Figure 6 shows the impact of the DNA molecule density on the I_on/I_off ratio of the TM-DG-DM-JL-MOSFET with highly doped source/drain extensions for different gate oxide materials, namely Al₂O₃ and SiO₂ + TiO₂. When DNA molecules are immobilized in the cavity region, use of SiO₂ and TiO₂ as gate oxide materials allows an increase in the I_on/I_off ratio compared with the use of Al₂O₃.

5 Comparison of TM-DG-DM-JL-MOSFET with SM-DM-JL-MOSFET

This section compares the sensitivity of the developed device with that of an SM-DG-DM-JL-MOSFET [37] of similar dimensions.

The sensitivity of both devices is presented in Table 3. These results clearly show that the TM-DG-DM-JL-MOSFET with highly doped extensions provides high sensitivity with presence of DNA molecules in the nanogap cavity compared with the SM-DG-DM-JL-MOSFET.

Table 3 Comparison of the TM-JL-DG-MOSFET with highly doped source/drain extensions versus the SM-DM-JL-MOSFET

6 Conclusions

Analytical and numerical modeling studies were carried out to analyze the sensitivity of a DG-DM-JL-MOSFET-based biosensor depending on its physical and electrical parameters. To assess the impact of DNA molecules (essential biomolecules in Nature), gate engineering and highly doped source/drain extensions of the DG-DM-JL-MOSFET were considered. It is noteworthy that this structure is the most useful to achieve the typical biosensing characteristic.

The investigated parameters included the surface potential, threshold voltage, sensitivity, transfer characteristic, and I_on/I_off ratio as functions of the biomolecule charge density and dimensional parameters of the biosensor. The results demonstrate the superiority of the TM-DG-DM-JL-MOSFET over the SM-DM-JL-MOSFET based on a comparison of the sensitivity of each device. The TM-DG-DM-JL-MOSFET with highly doped extensions permits detection of DNA molecules with sensitivity almost 50 % higher compared with the SM-DG-DM-JL-MOSFET structure.

The results presented herein prove the effectiveness of the proposed design over the conventional SM-DM-JL-MOSFET. The insight offered by this investigation makes the TM-DG-DM-JL-MOSFET with highly doped extensions a potential candidate for use in biosensing applications.

Appendices

Appendix A

$$ \sigma_{i} = - \frac{{\beta_{i} }}{{\alpha_{i} }}, $$

$$ D_{1 } = \frac{{(V_{\text{bi}} - \sigma_{1} ) - \left( {V_{1} - \sigma_{1} } \right)\mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }}{{1 - \mathrm{e}^{{ - 2L_{1} \sqrt {\alpha_{1} } }} }} , $$

$$ E_{1 } = \frac{{\left( {V_{1} - \sigma_{1} } \right) - (V_{\text{bi}} - \sigma_{1} )\mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }}{{1 - \mathrm{e}^{{ - 2L_{1} \sqrt {\alpha_{1} } }} }}, $$

$$ D_{2} = \frac{{(V_{1} - \sigma_{2} ) - \left( {V_{2} - \sigma_{2} } \right)\mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }}{{1 - \mathrm{e}^{{ - 2L_{2} \sqrt {\alpha_{2} } }} }} , $$

$$ E_{2 } = \frac{{\left( {V_{2} - \sigma_{2} } \right) - (V_{1} - \sigma_{2} )\mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }}{{1 - \mathrm{e}^{{ - 2L_{2} \sqrt {\alpha_{2} } }} }}, $$

$$ D_{3} = \frac{{\left( {V_{2} - \sigma_{3} } \right) - (V_{\text{bi}} + V_{\text{ds}} - \sigma_{3} )\mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }}{{1 - \mathrm{e}^{{ - 2L_{3} \sqrt {\alpha_{3} } }} }}, $$

where V_bi is the built-in potential, V₁ and V₂ are intermediate potentials, obtained by maintaining the continuity of the potential at the interface between region I, II, and III. Thus, we obtain

$$ A_{1} V_{1} + B_{1} V_{2} = F_{1}, $$

$$ A_{2} V_{1} + B_{2} V_{2} = {\text{F}}_{2}, $$

where

$$ A_{1} = \frac{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} + 1}}{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} - 1}} + \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}}, $$

$$ B_{1} = \frac{ - 2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }}, $$

$$\begin{aligned} F_{1} &= V_{\rm bi} \left( {\frac{2}{{\mathrm{e}^{{L_{1} \sqrt {\alpha_{1} } }} - \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }}} \right)\\ & \quad + \delta_{1} \left( {\frac{2}{{\mathrm{e}^{{L_{1} \sqrt {\alpha_{1} } }} - \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }} - \frac{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} + 1}}{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} - 1}}} \right)\\ & \quad + \delta_{2} \left( {\frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} - \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}}} \right),\end{aligned} $$

$$ A_{2} = \frac{ - 2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }}, $$

$$ B_{2} = \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}} + \frac{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} + 1}}{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} - 1}}, $$

$$\begin{aligned} F_{2} &= (V_{\rm bi} + V_{\rm ds} )\left( {\frac{2}{{\mathrm{e}^{{L_{3} \sqrt {\alpha_{3} } }} - \mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }}} \right)\\&\quad + \delta_{2} \left( {\frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} - \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}}} \right)\\&\quad + \delta_{3} \left( {\frac{2}{{\mathrm{e}^{{L_{3} \sqrt {\alpha_{3} } }} - \mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }} - \frac{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} + 1}}{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} - 1}}} \right).\end{aligned} $$

Using Crammer’s rule, it is possible to obtain the values of the intermediate potential V₁ and V₂ at the interfaces of regions I, II, and III as follows:

$$ V_{1} = \frac{{\left| {\begin{array}{*{20}c} {{{F}}_{1} } & {{{B}}_{1} } \\ {{{F}}_{2} } & {{{B}}_{2} } \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {A_{1} } & {{{B}}_{1} } \\ {A_{2} } & {{{B}}_{2} } \\ \end{array} } \right|}} = \frac{{{{F}}_{1} {{B}}_{2} - F_{2} {{B}}_{1} }}{{A_{1} {{B}}_{2} - A_{2} {{B}}_{1} }} , $$

$$ V_{2} = \frac{{\left| {\begin{array}{*{20}c} {A_{1} } & {{{F}}_{1} } \\ {A_{2} } & {{{F}}_{2} } \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {A_{1} } & {{{B}}_{1} } \\ {A_{2} } & {{{B}}_{2} } \\ \end{array} } \right|}} = \frac{{A_{1} {{F}}_{2} - F_{1} {{A}}_{2} }}{{A_{1} {{B}}_{2} - A_{2} {{B}}_{1} }}. $$

Appendix B

$$ m_{1 } = 2 \times \mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} \left( {\frac{{\left( {V_{\rm bi} + r_{{i}} } \right)\mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} - \left( {V_{s1} + r_{{i}} } \right)}}{{1 - {\text{e}}^{{ - 2{{L}}_{1} \sqrt {\alpha_{1} } }} }}} \right), $$

$$ m_{2 } = 2 \times \frac{{\left( {V_{\rm bi} + r_{{i}} } \right) - \left( {V_{s1} + r_{{i}} } \right)\mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} }}{{1 - {\text{e}}^{{ - 2{{L}}_{1} \sqrt {\alpha_{1} } }} }}, $$

$$ n_{1} = 2\times \mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} \left( {\frac{{1 - V_{s2} - \mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} }}{{1 - {\text{e}}^{{ - 2{{L}}_{1} \sqrt {\alpha_{1} } }} }}} \right), $$

$$ n_{2} = 2\times \frac{{\mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} - V_{{s2\mathrm{e}^{{ - {{L}}_{1} \sqrt {\alpha_{1} } }} }} - 1}}{{1 - {\text{e}}^{{ - 2{{L}}_{1} \sqrt {\alpha_{1} } }} }}, $$

$$ V_{\rm s1} = \frac{{{{C}}_{11} {{B}}_{2} - C_{21} {{B}}_{1} }}{{A_{1} {{B}}_{2} - A_{2} {{B}}_{1} }}V_{s2} = \frac{{C_{12} B_{2} - C_{22} B_{1} }}{{A_{1} B_{2} - A_{2} B_{1} }}, $$

$$ p_{i} = \frac{{qN_{\rm f} }}{{C_{i} }} + V_{\mathrm{fbi}} + 2\left( {V_{t} ln\frac{{N_{\rm d} }}{n_{\rm i}}} \right), $$

$$\begin{aligned} C_{11} &= \left( {\frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} - \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}}} \right)r_{2}\\ &\quad + \left( {\frac{2}{{\mathrm{e}^{{L_{1} \sqrt {\alpha_{1} } }} - \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }} - \frac{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} + 1}}{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} - 1}}} \right)r_{1}\\&\quad + V_{\rm bi} \left( {\frac{2}{{\mathrm{e}^{{L_{1} \sqrt {\alpha_{1} } }} - \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }}} \right), \end{aligned} $$

$$ \begin{aligned} C_{21} &= \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}} - \frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} + \frac{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} + 1}}{{\mathrm{e}^{{2L_{1} \sqrt {\alpha_{1} } }} - 1}}\\&\quad - \frac{2}{{\mathrm{e}^{{L_{1} \sqrt {\alpha_{1} } }} - \mathrm{e}^{{ - L_{1} \sqrt {\alpha_{1} } }} }},\end{aligned} $$

$$ \begin{aligned} C_{12} &= \left( {\frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} - \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}}} \right)r_{2}\\&\quad + \left( {\frac{2}{{\mathrm{e}^{{L_{3} \sqrt {\alpha_{3} } }} - \mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }} - \frac{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} + 1}}{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} - 1}}} \right)r_{3}\\&\quad + \left( {V_{\rm bi} + V_{\rm ds} } \right)\left( {\frac{2}{{\mathrm{e}^{{L_{3} \sqrt {\alpha_{3} } }} - \mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }}} \right),\end{aligned} $$

$$ \begin{aligned} C_{22} &= \frac{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} + 1}}{{\mathrm{e}^{{2L_{2} \sqrt {\alpha_{2} } }} - 1}} - \frac{2}{{\mathrm{e}^{{L_{2} \sqrt {\alpha_{2} } }} - \mathrm{e}^{{ - L_{2} \sqrt {\alpha_{2} } }} }} + \frac{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} + 1}}{{\mathrm{e}^{{2L_{3} \sqrt {\alpha_{3} } }} - 1}}\\&\quad - \frac{2}{{\mathrm{e}^{{L_{3} \sqrt {\alpha_{3} } }} - \mathrm{e}^{{ - L_{3} \sqrt {\alpha_{3} } }} }}, \end{aligned} $$

$$ r_{i} = - \frac{{{{qN}}_{\text{d}} }}{{\varepsilon_{\text{Si}} }} + V_{\mathrm{fb}i} ,\quad i = 1,2,3. $$