1. Introduction

Graphene is a two-dimensional (2-D) sheet of sp$^{2}$-bonded carbon atoms with a hexagonal configuration^[1]. Due to its prominent and unique properties in eletronics,it is considered as an excellent nano-electronic material and a promising candidate for high speed transistors^[2],energy storage^[3],capacitors^[4],and anodes in lithium ion batteries^[5]. In the tremendous graphene related applications,the graphene field effect transistor (GFET) is only a little branch. However,it has provided an excellent platform for the detection of various biological and chemical agents^{[6, 7]}. It was firstly published in 2009 on the discovery of GFET's high trans-conductance in an electrolyte solution and its dependence on the pH of the electrolyte^[8]. Since then thousands of GFET-based bio-sensors have been developed. The absorbed protein molecules on graphene could be examined by the conductance's variations,which was enhanced by the increasing concentrations of BSA^[8]. The hybridization of deoxyribonucleic acid (DNA) could be monitored by direct electrical detection based on GFET,and the resolution for the mismatched base was achieved as one base^[9]. Not only bio-molecules but also bacteria and small inorganic chemicals could be specifically detected by these total electronic and solid-state sensing components. The detection of E.coli was realized with a sensitivity of 10 cfu/mL and high specificity^[10]. Gas sensing could also be executed by solution gated GFET,and the lower limits for NH$_{3}$ and CO$_{2}$ were 30 and 4000 ppm,respectively^[11]. Heavy metal ion (Hg$^{2+})$ was discriminated and monitored in real time based on reduced graphene FET^[12]. It has been demonstrated that this kind of GFET sensing platform was a quick sensing,label-free,portable and low-cost bio-electronic system,and it could be developed as a high throughput biochemical analyzer for environment monitoring,food supervising and drug screening.

DNA is a powerful bio-material. It has played an important role in biology,biochemistry and biophysics due to its function of carrying genetic information in all living species^[13]. Lots of research has been carried out to prove that there are relationships between the electrical properties of DNA and its bio-physical structure when it undergoes replication,damage or repair^[14]. Charge transfer along the molecules of DNA is an important micro-mechanism and it is reported to occur in diverse biological platforms^[15]. Furthermore,as a natural nano-structure material,DNA has great potential applications in the field of molecule devices^[16]. Zhu \textit{et al.} fabricated a DNA sensor based on thionine-graphene nano-composite (Th-G) and its detection limit could reach 1.26 $\times$ 10$^{-13}$ M (S/N $=$ 3). With the rapid development of DNA related practical works,the systematical theory to explain the observed or reported phenomena is necessary. So,extensive theoretical studies and experiments were carried out,and it was proved that electron transfer (ET) can occur in the DNA molecule by two principal mechanisms^{[17, 18]},which are "coherent transfer" and "thermal hopping". In a real DNA molecule,the dominant way in the process of ET will depend on its chain length^[19].

The combination of DNA and GFET has given birth to a new kind of DNA sensors,called the aptamer sensor,in which single strand DNA (ssDNA) molecules are selected and grafted on the gate of a GFET^{[20, 21]}. The applications of these DNA sensors are not only in the monitoring of DNA molecules,but also in the recognition of almost all chemical or biological analytes^{[22, 23, 24]}. Lots of effort has been exerted on these GFETs based aptamer biochemical sensors recently. It has been verified by the published theoretical and experimental studies that in these sensing systems the interaction between ssDNA and graphene is the key point for understanding the responding of GFET^[25]. For an electronic engineer,one is more concerned about its electronic performance and its capability of integrating with a modern electronic designing system. Currently,the experimental works or physical mechanisms' studies about GFET-DNA sensors could be extensively referred,but how to simulate their electronic behaviors in the environment of a general electronic designing system is relatively little. So,the emphasis of this work is put on the simulation of GFET-DNA sensors' electronic behaviors by Sentaurus$^{\textregistered}$,which is a popular software for technology computer aided design (TCAD). This work provide a feasible method for the electronic simulation of GFET-DNA sensors,and it has good potential for achieving the TCAD of GFET-DNA sensors.

2. Device model for GFET-DNA sensor

2.1 Structure of GFET

The structure of a common GFET-based biochemical sensor is depicted in Figure1(a). The explanation is outlined here. First,on the insulated substrate there is a rectangular graphene film,used as the conductive channel. Second,on the two ends of the channel there are two electrodes,called source and drain,respectively. Third,the gate electrode is above the graphene film and across a layer of buffer solution,used as the dielectric layer of the gate. Finally,the electrode of the source is grounded,which is the benchmark for the voltages applied by gate and drain. The voltages applied by gate and drain are named $V_{\rm g}$ and $V_{\rm ds}$,respectively. When there is an electric field along the conducting channel,the current between source and drain ($I_{\rm ds})$ will be formed. Based on Figure1(a),an analytic model for GFET is proposed and simulated based on Sentaurus$^{\textregistered}$. The GFET Sentaurus simulation method is verified by comparing with the experimental data. Then the single strand DNA molecule's effects on GFET are simulated based on the proposed Sentaurus simulation method.

2.2 Working principle of GFET

It has been proved that the conductance of GFET could be modulated by $V_{\rm g}$^[1]. So the $I_{\rm ds}$ could be modulated by $V_{\rm g}$,when $V_{\rm ds}$ is maintained as a constant value. With a given $V_{\rm g}$,the arising and saturation value of $I_{\rm ds}$ could be measured with the increasing of $V_{\rm ds}$. For GFET,no matter whether the positive or negative voltage of $V_{\rm g}$ is applied,the characteristic curves of $I_{\rm ds}$ versus $V_{\rm ds}$ could be observed and it is different from traditional Si based FET^{[26, 27, 28, 29, 30, 31]}. The general explanation for GFET's bipolarity is accepted as: the carrier density and type in the channel can be modulated by the variation of $V_{\rm g}$ (the type of carriers could be electrons,holes or their coexistence)^[32].

There were four working states when different voltages of $V_{\rm g}$ and $V_{\rm ds}$ were applied on GFET^{[33, 34, 35]}. As shown in Figures 1(b) to 1(e),the type of majority carriers in the graphene channel can be determined depending on $V_{\rm g}$ and $V_{\rm ds}$. When $V_{\rm g}$ was controlled at zero,there were no accumulated or depleted charges,as shown in Figure1(b). The carriers in the graphene conducting channel could be drifted by a positive electric field along the channel,then the drift current ($I_{\rm ds})$ is constructed. For Figure1(c),under the negative gate voltage ($V_{\rm g}$ $<$ 0),holes are accumulated and electrons are depleted. The major carriers are holes in the channel of the GFET. When there is a positive electronic field from drain to source,$I_{\rm ds}$ is formed. For $V_{\rm g}$ $>$ $V_{\rm ds}$ $>$~0,as shown in Figure1(d),the major carriers are accumulated electrons and $I_{\rm ds}$ is formed by drifted electrons from source to drain. In the state of $V_{\rm ds}$ $>$ $V_{\rm g}$ $>$ 0,there is a turning point in the channel as given in Figure1(e),which is called the offset point. From this point to drain,electrons are depleted and holes are accumulated. It is opposite from the turning point to source. In this situation,electrons and holes are drifted oppositely to form the current $I_{\rm ds}$.

2.3 The electrical properties of DNA molecules

The DNA molecules are a kind of biological macromolecules built from repeating sugar molecules and phosphate groups. Each sugar group is attached to one of the four bases,which are guanine (G),cytosine (C),adenine (A) and thymine (T). The negative charge of phosphate groups make single- and double-strand DNA negatively charged. Double-strand DNA (dsDNA) consists of two strands of DNA around each other and hydrogen bonds between the bases coupling the two strands together. The dsDNA is more negatively charged than the ssDNA since the amount of phosphate groups in the dsDNA is higher than in the ssDNA. The experimental and theoretical works have been carried out and led to substantial clarification of the charge-transfer mechanism in DNA^{[36, 37, 38, 39]}. The dominant mechanism falls into two ways^{[19, 40]}: one way is that the charge can transfer from the donor to the acceptor groups by the tunnel effect,and the charge-transfer rate will sharply decrease with the length of the DNA molecule chain increasing; the other way is that the charge can transfer by "hopping" from a base pair to another,and in this case the charge-transfer rate is weakly dependent on the distance between donor and acceptor groups. Moreover,the process of charge transfer in DNA molecules could be affected by many factors including the length of DNA molecules,the sequence of base pairs,the internal energy of donor and acceptor,the ambient environment temperature and so on^{[18, 41]}.

Due to a high surface area,graphene has a strong biocompatibility^{[42, 43, 44]}. There is an interaction ($\pi$-$\pi)$ between DNA molecules and the surface of graphene which results in a remarkable influence on the carriers' density of the GFET by transferring electrons from the DNA to the graphene film. Then the channel conductance of the GFET can be affected consequently^[45]. It means that the channel conductance of the GFET will be affected by changing the density of DNA molecules immobilized on the surface of the graphene channel^[46]. So it is a reasonable inference that $I_{\rm ds}$ will change when ssDNA is immobilized on the gate surface of GFET and $V_{\rm ds}$ and $V_{\rm gs}$. Furthermore,since there are more charges on the molecules of dsDNA than ssDNA,and ET in the structure of dsDNA is more easier than in ssDNA^[19],another speculation could be formed that more electrons will transfer from dsDNA to graphene film when immobilized ssDNA molecules are hybridized with its complementary partner.

2.4 Model of GFET's DNA sensing in Sentaurus

There are a large number of physical analytical models in Sentaurus. It is a popular CAD tool in micro-electronic devices' design^{[47, 48]}. Graphene is a novel nanomaterial and its parameters of properties need to be defined in Sentaurus,such as the effective mass of electron and hole,mobility at room temperature and band gap^[49]. In the environment of Sentaurus,GFET is modelled firstly and its sensing for ssDNA immobilization and the following hybridization are simulated. The modelling method is outlined here.

Since the conducting channel of GFET is graphene,the concentration of carriers in the intrinsic semiconductor is summarized as:

$$\begin{align} n_{\rm i} (T)=\sqrt {N_{\rm C} (T)N_{\rm V} (T)} \exp \left[-\frac{E_{\rm g} (T)}{2kT}\right], \end{align}$$

(1)

where $T$ is the lattice temperature,$E_{\rm g}(T)$ is the energy of the band gap,$k$ is Boltzmann factors,$N_{\rm c}(T)$ and $N_{\rm v}(T)$ are the effective density-of-state (DOS) for electrons and holes,respectively. The forms and parameter values of $E_{\rm g}(T),$ $N_{\rm c}(T)$ and $N_{\rm v}(T)$ are listed in Table1.

By default,a "band gap narrowing" model is activated with the Bennett Wilson model. Since in the process of simulation,graphene has been undoped,the Bennett Wilson model needs to be turned off. Graphene possesses a stable structure without atomic defects; even if under the action of an external force,the arrangement of carbon atoms is not changed. The carriers of graphene move on their own orbit,they are not affected by lattice defects. Thus the constant mobility model is selected in the process of simulation and defined as:

$$\begin{align} u_{\rm const} =u_{\rm L} \left(\frac{T}{300 {\rm K}}\right)^{-\lambda }, \end{align}$$

(2)

where $u_{\rm L}$ $=$ 5000 cm$^{2}$/(V$\cdot$s) is the mobility due to bulk photon scattering,$\lambda$ $= -1$ is a correlation index.

The model for DNA detection was proposed by HKF Abadi^[50]:

$$\begin{align} I_{\rm d} =\frac{3q^2(3\pi d^3t^3k_{\rm B} T)^{0.5}}{hL}[J-0.5(\eta )+J-0.5(-\eta )]\times \frac{\alpha }{F}V_{\rm gs} , \end{align}$$

(3)

where $q$ is the electron charge,$d =$ 1.42 {\AA} is the length of carbon-carbon (C-C) bond,$t =$ 2.7 eV is the nearest neighbour C-C tight binding overlap energy,$k_{\rm B}$ is the Boltzmann's constant and $L$ shows the length of the conducting channel. Furthermore,$j-0.5(\eta)$ and $j-0.5(-\eta)$ are the Fermi-Dirac integral of orders $-0.5$ whose value can numerically be solved by employing the partial integration method. $\alpha$ is the DNA sensing factor and different concentrations of DNA molecules are expressed in the form of the $F$ parameter. The relationship between $\alpha$ and $F$ is a quadratic function as:

$$\begin{align} \alpha =AF^2+BF+C, \end{align}$$

(4)

where $A =$ 13,$B =$ 50 and $C =$ 4070 are the correlation coefficient between $\alpha$ and $F$. Thus,based on the electrical mechanism of DNA molecules and Equation (3),the current-voltage model of GFET for DNA detection can be simplified as:

$$\begin{align} I_{\rm d} =M\frac{\alpha }{F}V_{\rm gs} , \end{align}$$

(5)

where $M$ is a parameter as the representation of GFET properties and its specific form is equal to the different part between Equations (3) and (5). This model is reliable for graphene-based DNA sensors,since the values of the parameters $A,B$ and $C$ in Equation (4) are calculated based on trial and error.

3. Results and discussion

In this section,the simulation for GFET was executed and compared with experimental data,firstly. Then,GFET responding for ssDNA immobilization and hybridization with complimentary ssDNA was simulated in the next two subsections.

3.1 Simulation of blank GFET

First,volt-ampere characteristic curves under different $V_{\rm gs}$ and the transfer characteristic curves under different $V_{\rm ds}$ were shown in Figures 2(a) and 2(b),respectively. It was exhibited that the slopes of the $I$-$V$ characteristic curves in Figure2(a) were varied by $V_{\rm gs}$. When $V_{\rm gs}$ was varied negatively from 0.75 to $-0.5$,the curves stood up gradually. It means when the gate is more negatively powered more holes will be accumulated in the conducting channel,then the $I$-$V$ curves' slopes will be increased. While being positively varied $V_{\rm gs}$ from 0.75 to 1.5 will make more electrons accumulated and holes depleted,which means $I_{\rm ds}$ will be increased at the same $V_{\rm ds}$,as given in Figure2(a). That is to say,at the constant $V_{\rm ds}$,increased $I_{\rm ds}$ could be got when $V_{\rm gs}$ is varied in two directions,as shown in Figure2(b) which is one of the typical bipolar characteristic curves of GFET.

The other of the GFET's bipolar feature could be expressed in the channel's conductance (G) versus $V_{\rm gs}$. The value of G is defined in Equation (6):

$$\begin{align} G=\frac{\partial I_{\rm ds} }{\partial V_{\rm ds} }\Big\vert _{V_{\rm gs}}. \end{align}$$

(6)

Herein the values of G were calculated and plotted in Figure2(c). The operating mechanism of the field-effect transistor given in Section 2.2 implies that major carriers can be tuned continuously between holes and electrons by $V_{\rm gs}$. So there should be a $V_{\rm gs}$-depended conductance (G),as given in Figure2(c). From the G-$V_{\rm gs}$ curves it could be identified clearly that the turning point exists in both experimental data^[29] and our simulated result. This point is called the charge neutrality point (CNP). The appearance of the CNP is in accordance with the one of the working state,which is given in Section 2.2,Figure1(e). At this point,the channel conductance of the GFET is the lowest. The simulated results were compared with the experimental results from reference^[29]. There is a good agreement between the simulated results and the experimental data in the reference. So it is believed that the proposed model is reasonable.

3.2 Simulation of ssDNA immobilized GFET

DNA is an electron-rich molecule. When the ssDNA molecules are covalently grafted on graphene,the number of carriers will be changed in graphene so as to induce the variation of its conductance. For GFET,this chain reaction is expressed as the fluctuation of $I_{\rm ds}$. The simulation of ssDNA modified GFET provides an opportunity to examine GFET's responding and gives assistance to have an optimal working condition for the detection of target ssDNA.

The simulations of GFET's responding for ssDNA immobilization and hybridization were executed according to Equations (3),(4) and (5). The effects of ssDNA concentration on the transferring characteristic curves could be observed in Figures 3(a) to 3(c) for $V_{\rm ds}$ $=$ 0.2,0.6,1.0,respectively. The moving of CNP accompanying the increasing of ssDNA concentrations is obvious for the three plots in Figure3. The changing tendency of CNP and its relation with ssDNA concentration is extracted and plotted in Figure4(a).

In Figure3(a) the decrease of $I_{\rm ds}$ with the increase of ssDNA concentration could be identified clearly on the left side of the CNP at the same $V_{\rm gs}$. This could be explained that ET from DNA to graphene neutralize some of the accumulated-state holes (as shown in Figure2(c)). The amount of major carriers is decreased so that the current along the channel is decreased. On the right side of the CNP,the shifting of $I_{\rm ds}$ is upward when $V_{\rm gs}$ is constant. It is also coincident with ET theory. While there are differences among Figures 3(a),3(b) and 3(c),when $V_{\rm ds}$ is lower (0.2 V in Figure3(a)),the value of $I_{\rm ds}$ at the CNP is decreased directly with the left moving of the CNP; but in the other two plots,the variations of $I_{\rm ds}$ at the CNP are not in a single direction,they increase firstly,then decrease. This turning sight is more obvious when $V_{\rm ds}$ increases from 0.6 to 1.0 V,as shown in Figures 3(b) and 3(c). In general,the voltages of the CNP ($V_{\rm CNP})$ are lowered with the growing of ssDNA concentrations almost linearly and saturated at 20 $\mu$M. In this figure,the influences of $V_{\rm ds}$ are relatively small,that means the CNP could be developed as an indicator for DNA sensing.

At the same time,the channel conductance $G$ (defined in Equation (6)) is extracted by the same method as the one given in Section 3.1 and plotted in Figure4(b) based on the simulation of $I$-$V$ characteristics (which is very similar to Figure2(a) and omitted here). The curves of $G$-$V_{\rm gs}$ in different ssDNA concentrations exhibit a similar bipolar feature as in Figure2(c). The influences of ssDNA immobilization are expressed on the left downward moving of the curves. It is in coincidence with the ET mechanism,which was discussed in the last paragraph.

3.3 Simulation of dsDNA GFET

The dsDNA comes into being when the immobilized ssDNA hybridize with its complimentary ssDNA (cDNA). Since there are more charges in the chain of dsDNA than ssDNA,the electric features of GFET will be modified further. In this section it mainly concerns the effect of hybridized ssDNA's concentrations on the transferring characteristics of hybridized dsDNA-GFET,which is presented in Figures 5(a),5(b) and 5(c) when $V_{\rm ds}$ is constant at 0.2,0.6 and 1.0 V,respectively. The simulation is conducted by choosing GFET's responding of 20 $\mu$M ssDNA as control,i.e. 0 pM in Figure5. The concentrations of cDNA are arranged from 1 pM to 40 nM. Discussions for Figure5 are presented at here.

First,the shapes of the curves in three plots are very similar,the arising of $V_{\rm ds}$ causes the upward shift of the whole curves. That means there is no offset point (defined in Section 2.2) in the three situations. $I_{\rm ds}$ is constructed by the drift of one kind of majorities. When compared with Figure3,it could be found that at the CNP,$I_{\rm ds}$ are lowered without fluctuations as in Figure3. The deduction is also presented as ET theory. Because there are so many electrons injected by DNA molecules which are immobilized or hybridized on the GFET that the increasing of $V_{\rm ds}$ does not cause the appearance of the offset point in the channel. The working states of the dsDNA-GFET are switched between holes' accumulation and electrons' accumulation as depicted in Section 2.2.

Second,there are still points of intersection in Figures 5(a),5(b) and 5(c),respectively. On the left of this point,$I_{\rm ds}$ decreases with the increasing of cDNA concentration when $V_{\rm gs}$ is constant,while on the right of it,$I_{\rm ds}$ varies in an opposite way. It is deduced that when $V_{\rm gs}$ is lower than this points' $x$-coordinate the GFET is gated more negatively,the working state falls into Figure2(c) and the majorities are holes. So,more electrons are injected from cDNA,more holes will be shielded,and $I_{\rm ds}$ will decrease. While the situations of $V_{\rm gs}$ exceed the point of intersection,the electrons are accumulated as depicted in Figure2(d); the injected electrons will enhance the amount of major carriers,which is just in accordance with the increased $I_{\rm ds}$ (on the right of the intersection points) in Figure5.

Finally,the $V_{\rm CNP}$ and channel conductance G are extracted and plotted in Figures 6(a) and 6(b). When compared with Figure4(a),there are relatively small variations of $V_{\rm CNP}$. The reason is deduced as: the injected electrons from ssDNA enhance the amounts of electrons in GFET and make the neutralization of holes easier; so the $V_{\rm CNP}$ for dsDNA-GFET are relatively smaller than ssDNA-GFET. The channel conductance of dsDNA-GFET is extracted in the same way as described in Section 3.1. The ET mechanism between the DNA molecule and graphene is still applicable. Induced electrons screen the accumulated holes when $V_{\rm gs}$ is on the left of the intersecting point so the conductance is lowered with the increasing of cDNA; while for the right of this point,$G$ is increased by the electrons donated by hybridized cDNA.

4. Conclusions

Based on a popular CAD tool for electric devices,GFET is simulated and compared with the reported experimental data; simulation of ssDNA immobilization on GFET and the hybridization with its cDNA are also performed and discussed based on the ET mechanism of the DNA molecule and between DNA and graphene. The agreement between the simulated results and the experimental data indicates the feasibility of the presented simulation method. The simulated results for ssDNA immobilizing and cDNA hybridizing are also coincident with ET mechanisms. It is believed that the presented method for the simulation of GFET and DNA sensing would be helpful for electronic researchers to understand this kind of biosensor behavior before experiments,and it would be a cost-efficient way for bioelectroinc sensing system design.