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Modeling of lightly-doped drain and source contact with boron and nitrogen in graphene nanoribbon
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Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon Kien Liong Wong , Beng Rui Tan , Mu Wen Chuan , Afiq Hamzah , Shahrizal Rusli , Nurul Ezaila Alias , Suhana Mohamed Sultan , Cheng Siong Lim , Michael Loong Peng Tan PII: DOI: Reference:
S0577-9073(19)30932-3 https://doi.org/10.1016/j.cjph.2019.09.026 CJPH 955
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Chinese Journal of Physics
Received date: Revised date: Accepted date:
7 August 2019 9 September 2019 19 September 2019
Please cite this article as: Kien Liong Wong , Beng Rui Tan , Mu Wen Chuan , Afiq Hamzah , Shahrizal Rusli , Nurul Ezaila Alias , Suhana Mohamed Sultan , Cheng Siong Lim , Michael Loong Peng Tan , Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.09.026
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Highlights
Theoretical tight binding model of semiconducting armchair graphene nanoribbons. Lightly doped semi-metallic zigzag nanoribbons used as source and drain contact. Exploration on the band structure and density of states. Semimetallic contact does not vary the channel semiconducting properties.
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Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon Kien Liong Wonga, Beng Rui Tana , Mu Wen Chuana, Afiq Hamzaha, Shahrizal Ruslia, Nurul Ezaila Aliasa, Suhana Mohamed Sultan, Cheng Siong Lima and Michael Loong Peng Tana a
School of Electrical Engineering, Faculty of Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia
E-mail: [email protected] Received xxxxxx Accepted for publication xxxxxx Published xxxxxx
ABSTRACT Graphene, a monolayer carbon atoms arranged in hexagonal honeycomb lattice possesses impressive electronic properties. It is utilized as channel, source and drain contact in graphene nanoribbon field-effecttransistor (GNR-FET). Zigzag graphene nanoribbon (ZGNR) is used as semi-metallic drain and source terminal to a pristine armchair graphene nanoribbon (AGNR) that acts as a semiconducting channel. In addition, a single dopant, either nitrogen or boron is added to create lightly doped drain and source contact. The electronic properties of graphene nanoribbon (GNR) with lightly doped drain and source contact is obtained from tight binding approach. With self-energy matrices, the lightly doped contacts Hamiltonian operator is combined with the pristine channel Hamiltonian operator. The density of states (DOS) are simulated based on the non-equilibrium Green’s Function (NEGF) formalism. Our findings are then compared with other research work. Furthermore, it is demonstrated that the DOS of the overall GNR structure still has a small band gap and possess semiconducting properties although the channel is connected to semi-metallic contact at the drain and source terminal. Keywords: Graphene Nanoribbon (GNR); Lightly Doped Contact; Non-Equilibrium Green’s Function (NEGF); Tight-Binding; Band Structure; Density of States (DOS)
1.0 Introduction Recent research has been comprehensively focusing on two-dimensional graphene in semiconducting field due to its unique electronic properties, such as zero mass Dirac fermions behaviour, high carrier mobility and tuneable band gap [1, 2]. According to Moore’s Law, the number of transistor in an integrated circuit would be doubles every 18 months [3]. However before the limit of metal-oxide-semiconductor field-effect-transistor (MOSFET) technology scaling is reached, many explorations have been carried out on alternative material that is compatible with silicon process [4]. Graphene nanoribbon field-effect-transistor (GNR-FET) is
2
identified as one of the nanomaterial that can enhance silicon and overcome solve severe short channel effects [5]. In this case, the production of channel with lightly doped drain and source in GNR-FET fulfils the needs as novel semiconductor where the electronic properties are able to be done through modulation of their structural parameters, such as width, channel ribbon of length, contact length and doping location. Doping process is very important to change to resistivity of the material, create traps in the material that speed up recombination of electron and holes, and produce junction. In optoelectronic devices, the choice of the electrode material and its work function is vital to enhance conductivity [6]. In semi-classical physics, Schrödinger’s equation is used to describe the quantum mechanical behaviour. It act as the analogue Newton’s second law in classical physics [7]. Schrödinger’s equation deals with probability in quantum mechanics world, thus, it usually in linear partial differential equation. Therefore, in the whole system, integration is involved in solving each energy states. Once the system goes larger, the mathematical analysis would be further complicated and it takes time for the researcher to study the model, as well as calculation part. Schrödinger’s equation also been widely applied by many researchers to study the electronic properties of channel and contacts in GNR. However, most atomistic simulation work considered pristine GNR as channel and contact. Less emphasis are given on non-pristine GNRs research that work on dopants along the contacts since it is largely concluded that doping varied the electronic properties of the structure [8]. Doping in the contacts is relatively important as it change the resistivity of the material, create traps in the material that speed up recombination of electron and holes, and produce junction. As such, semiconducting channel with lightly doped contact is investigated in this work. This paper mainly focused on the modelling of GNR channel drain and source contact nanostructure that can be defined based on the input parameters such as width, channel ribbon of length, contact length and doping location. However, only ideal AGNR channel and lightly doped drain and source contact without any vacancies or line-edge roughness are constructed. Furthermore, the numerical computation study on the electronic properties of AGNR channel, lightly doped drain and source contact, and the combination of AGNR channel with lightly doped ZGNR drain and source are performed separately. All the modelling and computational work is done using MATLAB. 2.0 Related Research Tight-binding method with NEGF formalism is useful for nanomaterial modeling [9]. Initially, carbon nanotube had become the interest of most of the researcher. Thus, Chico et. al employed the tight-binding method to research on the carbon nanotube with defects [10]. After some years, the attention had shifted from carbon nanotube to GNRs as the latter possess some advantages compared with carbon nanotubes. In 2008, Li et. al had studied the GNRs with edge defects via tight-binding method [11]. Huang et. al had studied the edge doping of Zigzag edged GNRs with Nitrogen and Boron dopants but they only focus on the channel [12]. Besides that, Choudhary et. al [13] has also explored the effect of n-type and p-type doping in the channel of GNRs. In 2017, lightly doped drain region is proposed by Naderi [14] for tunnelling GNR field effect transistor. However, only armchair edge GNR is utilized at the contacts. The development of the modelling framework is based on Ref. [15], where an ideal GNR model is introduced using
3
numeric computational approach via MATLAB. By modification on the model, few conditions such as doping, unit cell for drain and source shift downwards and upwards respectively are added. The model is established based on NEGF-based, tight-binding model under the device modeling assumptions [16, 17] that are discussed in simplifying assumptions section. It is demonstrated that edge defects and effects due to passivating hydrogen atoms at edges of GNR have to be considered in the Hamiltonian operator to obtain better accuracies in describing the device [18]. In addition, vacancy incorporated GNRs had been studied [19, 20]. Some reported work have shown that doping in GNR eventually change its electronic properties. Therefore, tight- binding parameters of dopants, namely nitrite and boron are utilized in this modeling work [21]. It is observed that AGNR channel can be connected to ZGNR contacts as both chirality show promising and size invariant performance due to the semi-metallic properties and semiconducting properties respectively [22]. The width also remains the same from the source, channel and to the drain end, to ensure the transmission rate are the same throughout the model. 2.1 Schrödinger’s Equation and Hamiltonian Schrödinger’s Equation took the Bohr’s Model and postulated into a more solid mathematical expression [23] which is a linear partial differential equation that is used for the prediction of behaviour of the particular system’s wave function over time. Equation 1 shows the general form of time-independent Schrödinger’s equation.
i
r, t Hˆ r , t t
(1)
Ψ represent the wave function of the system and describe the wave-like behaviour of electrons in the system. The Hamiltonian operator, H in the equation indicates the energy values of the respective wave functions. The Schrödinger’s equation complements Bohr’s model that there are variety of wave function corresponding to the individual degenerate orbital shells, for a given energy value. When the Hamiltonian operator is applied on a particular wave function Ψ, resulting in a proportional wave function Ψ as the former, then Ψ is said to be in a stationary state, at where the proportionality constant, E is to be the energy of the state Ψ. Equation 2 shows the time–independent Schrödinger’s equation.
E H
(2)
The wave function in equation 2 is said to be eigenvalue of Hamiltonian operator. It contains the detailed information of the physical system including the position of constituent particles and the interaction among each other [24]. Once the Hamiltonian being paired up with respective wave function, the available energy spectrum that exists for the particular system can be obtained [25]. 2.2 Simplifying Assumptions for Device Modeling As Schrödinger’s equation involves the solution of integrals for each energy states existing in the whole system, therefore the numerical analysis for a large system is extremely complicated and time consuming. Datta had suggested simplifying assumptions for device modelling [26]. The first assumption is applying basis function that adapted from Hartree’s
4
assumption. In this assumption, a single electron wave function is equivalent to the number of electrons in the entire system. This characteristic function simplifies the many electrons system to many-one electron system. This represents the wave functions of electrons of the molecules that are only involved in their respective transport, regardless of the interaction with the rest of the valence band electrons. Considering one-dimensional (1D) system that consists of three hydrogen atoms in Figure 1.
Figure 1. 1D system with three hydrogen atoms with 1s orbital wave function A free hydrogen atom has a valence electron and thus, the wave function of the of the hydrogen 1s electron orbital is served as the basis function as the modelling of the system. The extension of this concept to a larger system, says in a multi – level orbital system, for example, a 2D GNR. Although it has 2 shells, but only the π-electron of the 2pz orbital and its wave function are involved as the basis function of the system. The second assumption is plane wave approximation, also known as the single-neighbour tight binding model, putting a limitation on the consideration of variety forces acting on particles is well illustrated by Figure 1, at which the wave functions are only affected by its nearest neighbour [27]. One notes that from the Figure 1, the overlapping of wave functions of 1s only occur among the atoms which are directly next to each other. The last assumption is that the discretization of the Hamiltonian operator and its basis functions into matrix equations. This can be done by the concept in which the structure of lattice itself consists of discrete atoms at atomic level. By applying all the simplifying assumptions, the Hamiltonian matrix of three hydrogen atoms in Figure 1 is constructed by using equation 3.
1 u11 u12 0 1 E 2 u21 u22 u23 2 0 u u 32 33 3 3
(3)
where u1, u2 and u3 representing the 1s orbital wave function. For the term u11, u22 and u33, also known as Alpha matrix are the self – interacting energies within the unit cell, whereas u12, u21, u23 and u32, also known as Beta matrix are the interacting energies between respective nearest neighbour. First and second assumptions can be seen from this equation: One should note that the system is effectively zero because 1s orbital electron of hydrogen atoms does not interact within itself. Meanwhile, u13 and u31 do not exist due to the assumption that the first and third hydrogen atoms have no interaction with each other. The concept is used in solving the Schrödinger’s equation semi-empirically to work on the 2D GNR problem.
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2.3 Dispersion Relation Dispersion relation of a system is a function that relates the energy of its electron to its wave vector [17]. It also describes as the effect of dispersion in the system over the travel of wave within the system. By applying the basis function as constraint, the relationship between band energies, E and the wave number, k can be described by the dispersion relation. At quantum level, some of the energies might be unavailable at certain momentum as the energy levels are no longer continuous. The summation of all allowed energy levels at k-points for a given momentum, the band structure of the system can be obtained. From the visualization of band structure, one can induce the intrinsic characteristics of the system from the minimum band – gap between minimum conduction band (CB) and maximum valence band (VB). Range of k values is usually taken from the first Brillouin Zone (BZ) which is from -π to π. Thus, corresponding to this, the Hamiltonian matrix can be refined with the derivation of dispersion relation in a general form equation 4, with the visual aids as shown in Figure 2.
Figure 1. 1D N – atom chain
1 t 2 0 E N 1 N t
t
0 t
t
0 t
0
0
t
t 1 2 N 1 N
(4)
6
By definition, ε represents the self-interacting energies of the atoms, represented by a red cross, whereas t represents the interaction energies of atoms among each other. A periodic boundary condition is added in the chain as bolded t, which simply shows that the first atom and Nth are adjacent to each other. The periodic condition allows the expression of overall matrix equation collapsed into a single linear equation form as shown in equation 5.
En tn1 n tn1
(5)
Due to the coefficients of linear matrix equation stays static, the approximation of wave function can be done with a plane wave equation 6, effectively transforming it into a function of wavenumber, k. Then, dividing the nth minus one wave function with the nth wave function, Equation 7 is obtained [17].
n 0 e inka n1 ei ( n1) ka i nka ei ka n e
(6) (7)
Thus, the final dispersion relation is formed as equation 8, and it is ready to be used for the energy spectra calculation.
E te ika te ika 2t cos ka
(8)
7
2.4 Contact effect in GNR The performance of Schottky barrier GNR is affected by the types and the shape of contact. Many concerns such as transmission rate, band gap energy, size, structure, and material are taken in consideration before it is chosen as contact. There are various types and shapes of contact had been proposed by most researchers are shown in Figure 3 [22].
Figure 2. (a) The schematic diagram of dual-gate graphene nanoribbon SBFETs. (b) Semiinfinite metal is used as the contacts (L and W ∞) (c) Semi-infinite graphene is used as the contacts (L and W ∞). (d) Finite rectangular graphene is used as contacts (e) ZGNR with wedges shapes is used as the contacts (f) ZGNR with equal width as channel is used as contact [22]. The semi-infinite metal shows the promising performance whereas the semi-infinite graphene have the worst performance among all the methods [22]. For finite rectangular graphene and ZGNR with wedges shapes contacts imply improvement on the ION/ OFF ratio in comparison with the graphene case contacts. However, with the strong semi-metallic properties of ZGNR, ZGNR with wedges shapes and ZGNR with equal width as channel contacts show promising performance.
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For the device that used finite rectangular graphene as contact, its transmission coefficient is relatively low due to its size mismatch with the channel and small band gap within the contact. In addition, devices that used ZGNR with wedges shapes contacts has the worst ION/IOFF ratio in comparison with methods in Figure 3 (d), (e), and (f). This phenomenon happened due to the shape of contacts which narrow down from bigger width to the width of channel, causing the larger transmission probability in the band gap region. Through the contribution of quantum tunnelling currents, this eventually increase the OFF-state current and causing the worst ION/IOFF ratio. Lastly, the ZGNR with equal width as channel contacts have the best performance compared with other methods in term of ION/IOFF . Its ION/IOFF remain the same even the length of the contact is varied as ZGNR is a material with semi-metallic properties and the width is same from the beginning until the end of the device. Although other types of contact able to made ION/IOFF ratio improved significantly, ZGNR is still the priority to be used as contact due to its promising and size invariant performance, that possess by the semi-metallic properties [22, 28]. Therefore, ZGNR with wedges shapes is used as the contacts as shown in Figure 3 (f). 2.5 Doping and On-Site Energy The doping of impurities into an intrinsic semiconductor to alter electrical and structural properties [29, 30] is utilized. Doping in semiconductor showed obvious impacts on the electronic properties of GNR [31]. The heavily doped semiconductor increases significantly on the electron mobility [32] whereas lightly doped in semiconductor reduce hot electron effect [33]. Lightly doped is applied to the drain and source contact as to demonstrate less leakage current, large ION/IOFF ratio, improved sub-threshold swing and switching parameters [34]. Besides, the optimal concentration for lightly doped is one dopant in every thousand carbon atoms [34]. Further doping will cause the semiconductor to be degenerate. Furthermore, edge doping has stronger dopant effect in these structures [35]. The tight-binding energy between carbon atoms and the dopants are depicted in Table 1. Table 1. Tight-Binding Parameter and On-Site Energy of the Nearest Hooping Parameters (All in eV) [21, 36] εC
εB
εN
γoCC
γoCB
γoCN
0.00
-6.64
-11.47
-2.70
-2.25
-1.70
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2.6 Density-of-States (DOS) The number of states available to be occupied at each energy level also known as DOS. When there are no states at the particular energy level is available to occupied, it simply known as zero DOS. In semi-classical physics, DOS of semiconductor represents the energy of the number of solutions to the Schrödinger’s equation. DOS calculation can be done in various methods. One of the methods is that DOS calculation is done mathematically through the delta δ function [37]. Equation 9 represents the total DOS of a particular system with N bands. The value of N is twice the number of electrons that involved in the transport due to conduction-valence sub-band pair by each electron. N 1 (9) DOS ( E ) Ei k E dk i 1 2 Solving the delta function within the integral gives equation 10, that is applied in the work as one of the DOS that are calculated numerically. N
DOS ( E ) i 1
1 2
G 2 all k E k E i
(10)
2
G
DOS can also be calculated using Green’s function [11]. The relation between Green’s Function and DOS is expressed as
DOS ( E ) where
1
Im Trace G retarded
(11)
is the retarded Green’s Function as shown in equation 12,
G retarded E i * I H
1
(12)
where I is an identity matrix, H is the Hamiltonian operator matrix and η is a very small imaginary value preventing the inverse matrix from diverging. The value 10-3 is chosen so that the peaks in the DOS are clearer and more dominant. One should note that the calculation of DOS via band structure do not contains any length information of the system unlike Green’s Function method, also known as localised density of states (LDOS), due to the existence of Hamiltonian operator. Both methods are used in the computational for comparison purposes [22]. Moreover, the equation is further modified to calculate the DOS of the overall model where channel is link to the contact as shown in equation 13.
G retarded E i * I H S D
1
(13)
where ΣS and ΣD are the self-energies for the right (source) and left (drain) respectively and the formula is shown in equation 14 S , D S , D g S S , D S , D (14) where τ S, D is the interaction between the device and contacts and gS S, D is the surface Green’s Function of the drain contact and source given by
g S S E i * I H source_contact S
g S D E i * I H drain _ contact D
1
1
(15) (16)
10
3. Research Design and Implementation The research methodology of this work is classified into 4 different stages. First, an ideal AGNR model for the channel is constructed [15], followed by the simulation and verification of electronic properties for the channel GNR with other research work [38]. Next, the drain and source ZGNR will be constructed through the modification of the AGNR channel Hamiltonian. Based on the Hamiltonian of the drain and source ZGNR, the generic script for single doped ZGNR is developed which allow the user to define single dopant type and location in the contact Hamiltonian. Besides, the electronic properties of the single doping contact Hamiltonian are simulated and compared with the pristine contact Hamiltonian. Furthermore, the self-energies matrices are developed to allow the connection between the channel and the contact. Last but not least, the density-of-states (DOS) of the overall structure is simulated and compared with the pristine DOS of the overall structure. 3.1 Design of Channel Hamiltonian By applying the three simplifying assumptions [16], namely basis function that adapted from Hartree’s assumption, plane wave approximation, and discretization of Hamiltonian operator, the model of channel AGNR is constructed. The basis function is the wave function for the 2pz electron that forms free moving delocalized π-orbital. Besides, as the AGNR Hamiltonian is generated with open boundary condition, it is assumed that the hydrogen atoms are passivated with all the dangling bonds at the carbon edge. Due to plan wave approximation, the overlapping of wave functions will only occur among the atoms which are directly next to it. In conjunction with it, the width and length of AGNR are defined in unit cells and the numbering sequence of carbon atoms that indicates the position of carbon atom in the Hamiltonian is strictly followed throughout the research, as shown in Figure 4.
Figure 4. Pristine 5-AGNR with length of 3 and the numbering of the carbon atoms.
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The quasi – 2D AGNR structure is decomposed into 1D structure through the collapsing of the unit cell into single matrix form, defined as alpha matrix, α. In other words, alpha matrix, α also indicate the carbon atoms interaction within the unit cell while the beta matrix, β is defined as the interaction between the unit cell. The number of rows in alpha matrix is always equal to the number of in beta matrix and it indicates the number of carbon atoms in the unit cell. After the alpha matrix and beta matrix are defined, both matrixes are combined to generate the channel Hamiltonian matrix. All the AGNR Hamiltonian will have the same general structure for same widths, but different contents in alpha and beta matrices. 3.2 Design of Contact Hamiltonian with Single Dopant For pristine drain and source Hamiltonian, it could be developed through the modification of the channel Hamiltonian as it applied exactly the same concept as the channel Hamiltonian. For a perfect connection of contact Hamiltonian to the channel Hamiltonian, the width of contact has to be greater than the width of channel by one unit cell through adding two carbon atoms. Besides, the unit cell of the drain and source will be shifted downwards and upwards respectively as shown in Figure 5. An important step must be considered is the removal of excess carbon atom that causes unwanted dangling bond. For instance, the carbon atom 11 and 12 in L1 and L3 region on both source and drain contact respectively as illustrated in Figure 5.
(a)
(b)
Figure 5. Modified 6-AGNR with length of 3, single dopant and the numbering of the carbon atoms for (a) drain contact Hamiltonian and (b) source contact Hamiltonian.
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With a single dopant applied, the contact Hamiltonian is further modified in term of the tight-binding energy. In graphene, each π-orbital interacts with three of its nearest neighbour πorbital electrons with tight-binding energy of -2.7eV [18]. Single dopant applied means that the tight-binding energy between the impurities with the nearest neighbour π-orbital electrons will be changed according to the type of dopant. In Figure 5, the blue dot indicates the N-type doping (Nitrogen) whereas the red dot indicates the P-type doping (Boron). This indicator is applied to any doping location within the width and length that is predetermined by users. The algorithm for the doping process has been constructed to process the input given by the users on the doping location and doping type. The complete Hamiltonian for the contact is simulated to obtain the electronic properties. One should take notes that besides the dopant location, the location of remaining carbon atoms are assumed to remain unchanged although the tight-binding energy changed, as this would involves complicated and countless probability of geometrical structure changes in consideration. 3.3 Design of Self-Energy Matrix By applying the assumptions, self-energy matrices for source and drain contacts, ΣS and ΣD respectively are developed through equation 17 [22]. The interaction between channel and contact is defined as τ and the surface Green’s function of contact is defined as gS. The method of developing matrix for τ is same as the beta matrix, β in channel Hamiltonian while the dimensions for the self-energy matrix must be equivalence with channel Hamiltonian. For the location of the τ matrix will be located at top-left corner for drain contact and bottom-right corner for source contact as shown in equation 18. Therefore, the algorithm for the self-energy matrix has been developed to process the width input given by the users. (17)
0 0 D 0 0 0 0 0 0
(18)
0 0 0 S 0 0 0 0 0
(19)
13
3.4 Electronic Properties By practicing basis functions assumptions as the restriction, the dispersion relation is the relationship between band energies, E, and the wavenumber, k. The band structure of channel’s AGNR and contact’s ZGNR are obtained through the summation of all the allowed energy levels, at k-points at given momentum. In order to compute the band energy spectrum, the E against k graph is plotted within the first Brillouin Zone where the wavenumbers, k = -π to π. By applying the periodic boundary condition along with the simplified assumptions [16], the overall matrix is transformed into equation 8, that is used to compute energy spectrum. In equation 8, ε indicates the self-interacting energies of the atoms and t indicates the interaction energies between atoms, which are alpha and beta matrix respectively. In this research, two different methods are used to calculate the DOS of channel and contact band structure. These methods are the delta δ function in equation 10 [26] and Green’s Function in equation 11 [9, 11] and both are implemented for comparison purposes. Note that the doped contact Hamiltonian is computed using Green’s Function as it capture the full dimension of the modelled structured to compute energy spectrum. Finally, the DOS of the overall structure is calculated based on the Green’s Function method on equation 13 [16] with the addition of selfenergy matrix. The DOS of doped structure is compared with the un-doped structure.
14
4. Results, Analysis and Discussion 4.1 Electronic Properties for Channel AGNR The electronic properties for channel with carbon dimmer lines, n= 5 to 20 are simulated. The simulation results for 5-AGNR, 7-AGNR, and 15-AGNR with the channel ribbon of length, L=35 unit cells are shown in Figure 6. The band gap energies for the specific carbon dimmer lines are calculated in Figure 6, which are 0ev (n=5), 1.267eV (n=7), and 0.6eV (n=15).
(a)
(b)
(c)
Figure 6. Simulated sub-band structures for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively.
15
Besides, the numerical DOS for the specific width is simulated. Figure 7 shows the simulated numerical DOS for 5-AGNR, 7-AGNR, and 15-AGNR.
(a)
(b)
(c)
Figure 7. Numerical DOS plots for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively. Furthermore, the Green’s Function approach is used to simulate the DOS for the specific width. Figure 8 shows the simulated Green Function DOS for carbon dimmer lines, n=5, 7, and 15.
(a)
(b)
(c)
Figure 8. Green’s Function DOS plots for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively, with channel ribbon of length, L=35 unit cells.
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The AGNR channel that developed in this paper only limited to an odd number carbon dimmer lines, which indicate the width in the form of carbon numbers. The purpose of using odd number carbon dimmer lines is to avoid extra dangling bond that create a mismatch between channel and contact. From the results, the electronic properties for channel with carbon dimmer lines, n=3p and 3p+1, the AGNR channel possess semiconducting properties whereas n=3p+2, it possesses semi-metallic properties, which equivalent to the benchmarks given and tallies with the expected results. Besides, the numerical DOS for the specific width is simulated. However, this approach is length independent and it only varies with the width. From the simulation result, it is observed that it converge to approximately the same shape although the width is varying. Furthermore, the Green’s Function approach is used to simulate the DOS for the specific width. Unlike the numerical approach, this method takes the length of the structure into consideration. Prominent peaks start to appear as the length increases as shown in Figure 10 where ribbon of length, L=35 unit cells. In previous work, it is demonstrated that channel ribbon of length less than 10 unit cells has less van Hove singularities peaks [15]. For both numerical DOS and Green’s Function DOS, the simulation results show the peaks at ±2.7eV which is the tight-binding energies for the interaction between carbon atoms that contribute to the 2pz π-orbital.
17
4.2 Electronic Properties for Contact ZGNR The following work consider GNR with carbon dimmer lines, n=7 and 15 as it matches with band gap of silicon and germanium respectively. Hence, it is useful to simulate the electronic properties with these widths. The simulation results for 7-ZGNR, and 15-ZGNR with contact length, L=35 unit cells are shown in Figure 9.
(a)
(b)
(c)
(d)
Figure 9. Simulated sub-band structures for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15ZGNR drain, and (d) 15-ZGNR source. Besides, both the numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for both drain and source of 7-ZGNR, and 15-ZGNR. The simulated DOS is shown in Figure 10 and Figure 11 respectively.
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(a) (b) (c) (d) Figure 10. Numerical DOS plots for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source.
(a)
(b)
(c)
(d)
Figure 11. Green’s Function DOS plots for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source.
19
The paper is followed by applying single dopant into drain and source contact. Figure 12 shows the comparison of sub-band structure for N-type doping, using Nitrite as a dopant in drain and source contact with the pristine electronic properties, for 7-ZGNR, and 15-ZGNR.
(a)
(b)
(c)
(d)
Figure 12. Comparison of sub-band structures for N-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure. Furthermore, both the numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for lightly doped drain and source contact, 7-ZGNR, and 15ZGNR and compare with the pristine results. The simulated DOS is shown in Figure 13 and Figure 14 respectively. The simulation is repeated with P-type doping, using Boron as the dopant and the results are shown in Figure 15, Figure 16 and Figure 17.
(a)
(b)
(c)
(d)
Figure 13. Comparison of Numerical DOS plots for N-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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(a)
(b)
(c)
(d)
Figure 14. Comparison of Green’s Function DOS plots for N-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
(a)
(b)
(c)
(d)
Figure 15. Comparison of sub-band structures for P-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
21
(a)
(b)
(c)
(d)
Figure 16. Comparison of Numerical DOS plots for P-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
(a)
(b)
(c)
(d)
Figure 17. Comparison of Green’s Function DOS plots for P-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
22
The ZGNR contact that developed in this work is the modification from pristine channel AGNR, by shifting the unit cell upwards or downwards for source and drain respectively. The contact width must be equivalent with the channel width, yet in the Hamiltonian, it has two more rows and columns to fill up the extra carbon atoms that required for modification purpose. According to the simulated results, the contact always possesses semi-metallic properties and similar sub-band structure as semi-metallic form AGNR as it is the modification of AGNR structure. Regardless of the width, the contacts are semi-metallic state with zero band gap energy. In addition, it is noticeable that the drain and the source have similar sub-band structures. Therefore, it can concluded that the shifting of the unit cell, either shifting upwards or downwards do not affect the electronic properties Both numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for both drain and source of 7-ZGNR, and 15-ZGNR. For the numerical DOS, it is observed that the results have the same trend as the channel DOS, where it converge to approximately the same shape although the width is varying. Furthermore, the simulation results using Green’s Function approach differ from the numerical approach, as this method take the length of the structure into consideration. From the simulation result, it also shows prominent peaks at ±2.7eV. After the simulation of pristine drain and source ZGNR contact, a single dopant is added to observe the effect of lightly doped towards the electronic properties of the contact. In this research scope, edge doping concept is applied to both drain and source. The dopant in the drain and source interacts with three and two carbon atoms respectively. It is observed that the dopant has major effects on the sub-band structure and DOS that is computed numerically. This is due to the calculation of the sub-band structure and numerical DOS do not consider the length. Thus, doping has a larger impact. On the other hand, Green’s Function DOS considers the length of contact. As a result, the effect of doping on Green’s Function DOS becomes less impactful with increasing length. In our investigation, a single dopant is utilized and thereby the Fermi energy of the structure can be preserved [34]. Generally, Green’s Function DOS is more accurate than numerically calculated DOS. Furthermore, the simulation data such as band structure and numerically calculated DOS for drain and source contact with doping show much more sensitivity as clear variation is evident when dopant are placed at different position respectively. It can be concluded that the doping location also plays a very important role in electronic properties. Dopant that has two carbon atoms directly near to it shows different electronic properties with the dopant that has three carbon atoms directly near to it. Besides, the type of doping also plays an important role in this context. It is also found that only minimal variation for P-type structure are observed as compared to the pristine structure but stronger variation for N-type structure. The reasons are because P-type dopant has tight-binding energy of -2.25eV with carbon atoms whereas N-type dopant has tight-binding energy of -1.70eV with carbon atoms. A small difference in tight-binding energy causes minor changes.
23
4.3 Electronic Properties for Overall Structure Green’s Function DOS is simulated for the overall structure. The comparison of Green’s Function DOS N-type structure with the pristine structure of 7-GNR and 15-GNR are simulated as shown in Figure 18 (a) and (b). The process is repeated by the P-type structure as shown in Figure 18 (c) and (d).
(a)
(b)
(c)
(d)
Figure 18. Comparison of Green’s Function DOS plots for N-type overall structure (a) 7-GNR, (b) 15-GNR, and for P-type overall structure (c) 7-GNR, and (d) 15-GNR. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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Self-energy matrices are developed to combine the lightly doped drain and source contact with the pristine channel AGNR. For the overall structure, there is no clear alpha and beta matrix as all the channel and contact are combined together. Therefore, only DOS based on Green’s function can be utilized whereas numerical DOS only detect the alpha and beta matrix from the overall structure. From the simulation result, the overall structure shows prominent peaks at ±2.7eV which indicates the tight-binding energy for the carbon atoms interaction. Besides, the light-doped showed fewer discrepancies with the pristine structure. Therefore, it can be concluded that the light-doped have a low impact on the simulated DOS. Furthermore, it is observed that the Green’s Function DOS of the overall structure still has a band gap that indicates it is still a semiconductor. In short, the channel properties are still preserved and not affected by drain and source contact. 5. CONCLUSION A tight-binding model for AGNR with ZGNR drain and source contact is presented. By applying appropriate simplifications and assumption, doping effects which alter the composition of Hamiltonian are implemented by replacing on-site self-energies and hopping integral or tightbinding energy of carbon atoms. In addition, the simulated electronic properties of channel have good agreement with reported data from other researcher [15]. DOS that is based on Green’s function able to provide the states availability at each energy level when the full dimension of the structure is considered. The DOS simulation result for all case showed a prominent peak at ±2.7eV which are the tight-binding energy for carbon atoms interaction. Furthermore, the dopant showed an insignificant shift for all the electronic properties. Finally, the semiconducting properties for the overall structure remain unchanged as the drain and source contact are semimetallic does not vary the channel semiconducting properties. Acknowledgements Kien Liong would like to express his appreciation for the award of Zamalah PhD Scholarship from the School of Graduate Studies UTM. Michael Tan acknowledges the financial support from a FRGS Research Grant (Vote no. R.J130000.7851.5F043) that allowed the research to proceed smoothly.
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Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon Kien Liong Wong , Beng Rui Tan , Mu Wen Chuan , Afiq Hamzah , Shahrizal Rusli , Nurul Ezaila Alias , Suhana Mohamed Sultan , Cheng Siong Lim , Michael Loong Peng Tan PII: DOI: Reference:
S0577-9073(19)30932-3 https://doi.org/10.1016/j.cjph.2019.09.026 CJPH 955
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
7 August 2019 9 September 2019 19 September 2019
Please cite this article as: Kien Liong Wong , Beng Rui Tan , Mu Wen Chuan , Afiq Hamzah , Shahrizal Rusli , Nurul Ezaila Alias , Suhana Mohamed Sultan , Cheng Siong Lim , Michael Loong Peng Tan , Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.09.026
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Highlights
Theoretical tight binding model of semiconducting armchair graphene nanoribbons. Lightly doped semi-metallic zigzag nanoribbons used as source and drain contact. Exploration on the band structure and density of states. Semimetallic contact does not vary the channel semiconducting properties.
1
Modeling of Lightly Doped Drain and Source Contact with Boron and Nitrogen in Graphene Nanoribbon Kien Liong Wonga, Beng Rui Tana , Mu Wen Chuana, Afiq Hamzaha, Shahrizal Ruslia, Nurul Ezaila Aliasa, Suhana Mohamed Sultan, Cheng Siong Lima and Michael Loong Peng Tana a
School of Electrical Engineering, Faculty of Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia
E-mail: [email protected] Received xxxxxx Accepted for publication xxxxxx Published xxxxxx
ABSTRACT Graphene, a monolayer carbon atoms arranged in hexagonal honeycomb lattice possesses impressive electronic properties. It is utilized as channel, source and drain contact in graphene nanoribbon field-effecttransistor (GNR-FET). Zigzag graphene nanoribbon (ZGNR) is used as semi-metallic drain and source terminal to a pristine armchair graphene nanoribbon (AGNR) that acts as a semiconducting channel. In addition, a single dopant, either nitrogen or boron is added to create lightly doped drain and source contact. The electronic properties of graphene nanoribbon (GNR) with lightly doped drain and source contact is obtained from tight binding approach. With self-energy matrices, the lightly doped contacts Hamiltonian operator is combined with the pristine channel Hamiltonian operator. The density of states (DOS) are simulated based on the non-equilibrium Green’s Function (NEGF) formalism. Our findings are then compared with other research work. Furthermore, it is demonstrated that the DOS of the overall GNR structure still has a small band gap and possess semiconducting properties although the channel is connected to semi-metallic contact at the drain and source terminal. Keywords: Graphene Nanoribbon (GNR); Lightly Doped Contact; Non-Equilibrium Green’s Function (NEGF); Tight-Binding; Band Structure; Density of States (DOS)
1.0 Introduction Recent research has been comprehensively focusing on two-dimensional graphene in semiconducting field due to its unique electronic properties, such as zero mass Dirac fermions behaviour, high carrier mobility and tuneable band gap [1, 2]. According to Moore’s Law, the number of transistor in an integrated circuit would be doubles every 18 months [3]. However before the limit of metal-oxide-semiconductor field-effect-transistor (MOSFET) technology scaling is reached, many explorations have been carried out on alternative material that is compatible with silicon process [4]. Graphene nanoribbon field-effect-transistor (GNR-FET) is
2
identified as one of the nanomaterial that can enhance silicon and overcome solve severe short channel effects [5]. In this case, the production of channel with lightly doped drain and source in GNR-FET fulfils the needs as novel semiconductor where the electronic properties are able to be done through modulation of their structural parameters, such as width, channel ribbon of length, contact length and doping location. Doping process is very important to change to resistivity of the material, create traps in the material that speed up recombination of electron and holes, and produce junction. In optoelectronic devices, the choice of the electrode material and its work function is vital to enhance conductivity [6]. In semi-classical physics, Schrödinger’s equation is used to describe the quantum mechanical behaviour. It act as the analogue Newton’s second law in classical physics [7]. Schrödinger’s equation deals with probability in quantum mechanics world, thus, it usually in linear partial differential equation. Therefore, in the whole system, integration is involved in solving each energy states. Once the system goes larger, the mathematical analysis would be further complicated and it takes time for the researcher to study the model, as well as calculation part. Schrödinger’s equation also been widely applied by many researchers to study the electronic properties of channel and contacts in GNR. However, most atomistic simulation work considered pristine GNR as channel and contact. Less emphasis are given on non-pristine GNRs research that work on dopants along the contacts since it is largely concluded that doping varied the electronic properties of the structure [8]. Doping in the contacts is relatively important as it change the resistivity of the material, create traps in the material that speed up recombination of electron and holes, and produce junction. As such, semiconducting channel with lightly doped contact is investigated in this work. This paper mainly focused on the modelling of GNR channel drain and source contact nanostructure that can be defined based on the input parameters such as width, channel ribbon of length, contact length and doping location. However, only ideal AGNR channel and lightly doped drain and source contact without any vacancies or line-edge roughness are constructed. Furthermore, the numerical computation study on the electronic properties of AGNR channel, lightly doped drain and source contact, and the combination of AGNR channel with lightly doped ZGNR drain and source are performed separately. All the modelling and computational work is done using MATLAB. 2.0 Related Research Tight-binding method with NEGF formalism is useful for nanomaterial modeling [9]. Initially, carbon nanotube had become the interest of most of the researcher. Thus, Chico et. al employed the tight-binding method to research on the carbon nanotube with defects [10]. After some years, the attention had shifted from carbon nanotube to GNRs as the latter possess some advantages compared with carbon nanotubes. In 2008, Li et. al had studied the GNRs with edge defects via tight-binding method [11]. Huang et. al had studied the edge doping of Zigzag edged GNRs with Nitrogen and Boron dopants but they only focus on the channel [12]. Besides that, Choudhary et. al [13] has also explored the effect of n-type and p-type doping in the channel of GNRs. In 2017, lightly doped drain region is proposed by Naderi [14] for tunnelling GNR field effect transistor. However, only armchair edge GNR is utilized at the contacts. The development of the modelling framework is based on Ref. [15], where an ideal GNR model is introduced using
3
numeric computational approach via MATLAB. By modification on the model, few conditions such as doping, unit cell for drain and source shift downwards and upwards respectively are added. The model is established based on NEGF-based, tight-binding model under the device modeling assumptions [16, 17] that are discussed in simplifying assumptions section. It is demonstrated that edge defects and effects due to passivating hydrogen atoms at edges of GNR have to be considered in the Hamiltonian operator to obtain better accuracies in describing the device [18]. In addition, vacancy incorporated GNRs had been studied [19, 20]. Some reported work have shown that doping in GNR eventually change its electronic properties. Therefore, tight- binding parameters of dopants, namely nitrite and boron are utilized in this modeling work [21]. It is observed that AGNR channel can be connected to ZGNR contacts as both chirality show promising and size invariant performance due to the semi-metallic properties and semiconducting properties respectively [22]. The width also remains the same from the source, channel and to the drain end, to ensure the transmission rate are the same throughout the model. 2.1 Schrödinger’s Equation and Hamiltonian Schrödinger’s Equation took the Bohr’s Model and postulated into a more solid mathematical expression [23] which is a linear partial differential equation that is used for the prediction of behaviour of the particular system’s wave function over time. Equation 1 shows the general form of time-independent Schrödinger’s equation.
i
r, t Hˆ r , t t
(1)
Ψ represent the wave function of the system and describe the wave-like behaviour of electrons in the system. The Hamiltonian operator, H in the equation indicates the energy values of the respective wave functions. The Schrödinger’s equation complements Bohr’s model that there are variety of wave function corresponding to the individual degenerate orbital shells, for a given energy value. When the Hamiltonian operator is applied on a particular wave function Ψ, resulting in a proportional wave function Ψ as the former, then Ψ is said to be in a stationary state, at where the proportionality constant, E is to be the energy of the state Ψ. Equation 2 shows the time–independent Schrödinger’s equation.
E H
(2)
The wave function in equation 2 is said to be eigenvalue of Hamiltonian operator. It contains the detailed information of the physical system including the position of constituent particles and the interaction among each other [24]. Once the Hamiltonian being paired up with respective wave function, the available energy spectrum that exists for the particular system can be obtained [25]. 2.2 Simplifying Assumptions for Device Modeling As Schrödinger’s equation involves the solution of integrals for each energy states existing in the whole system, therefore the numerical analysis for a large system is extremely complicated and time consuming. Datta had suggested simplifying assumptions for device modelling [26]. The first assumption is applying basis function that adapted from Hartree’s
4
assumption. In this assumption, a single electron wave function is equivalent to the number of electrons in the entire system. This characteristic function simplifies the many electrons system to many-one electron system. This represents the wave functions of electrons of the molecules that are only involved in their respective transport, regardless of the interaction with the rest of the valence band electrons. Considering one-dimensional (1D) system that consists of three hydrogen atoms in Figure 1.
Figure 1. 1D system with three hydrogen atoms with 1s orbital wave function A free hydrogen atom has a valence electron and thus, the wave function of the of the hydrogen 1s electron orbital is served as the basis function as the modelling of the system. The extension of this concept to a larger system, says in a multi – level orbital system, for example, a 2D GNR. Although it has 2 shells, but only the π-electron of the 2pz orbital and its wave function are involved as the basis function of the system. The second assumption is plane wave approximation, also known as the single-neighbour tight binding model, putting a limitation on the consideration of variety forces acting on particles is well illustrated by Figure 1, at which the wave functions are only affected by its nearest neighbour [27]. One notes that from the Figure 1, the overlapping of wave functions of 1s only occur among the atoms which are directly next to each other. The last assumption is that the discretization of the Hamiltonian operator and its basis functions into matrix equations. This can be done by the concept in which the structure of lattice itself consists of discrete atoms at atomic level. By applying all the simplifying assumptions, the Hamiltonian matrix of three hydrogen atoms in Figure 1 is constructed by using equation 3.
1 u11 u12 0 1 E 2 u21 u22 u23 2 0 u u 32 33 3 3
(3)
where u1, u2 and u3 representing the 1s orbital wave function. For the term u11, u22 and u33, also known as Alpha matrix are the self – interacting energies within the unit cell, whereas u12, u21, u23 and u32, also known as Beta matrix are the interacting energies between respective nearest neighbour. First and second assumptions can be seen from this equation: One should note that the system is effectively zero because 1s orbital electron of hydrogen atoms does not interact within itself. Meanwhile, u13 and u31 do not exist due to the assumption that the first and third hydrogen atoms have no interaction with each other. The concept is used in solving the Schrödinger’s equation semi-empirically to work on the 2D GNR problem.
5
2.3 Dispersion Relation Dispersion relation of a system is a function that relates the energy of its electron to its wave vector [17]. It also describes as the effect of dispersion in the system over the travel of wave within the system. By applying the basis function as constraint, the relationship between band energies, E and the wave number, k can be described by the dispersion relation. At quantum level, some of the energies might be unavailable at certain momentum as the energy levels are no longer continuous. The summation of all allowed energy levels at k-points for a given momentum, the band structure of the system can be obtained. From the visualization of band structure, one can induce the intrinsic characteristics of the system from the minimum band – gap between minimum conduction band (CB) and maximum valence band (VB). Range of k values is usually taken from the first Brillouin Zone (BZ) which is from -π to π. Thus, corresponding to this, the Hamiltonian matrix can be refined with the derivation of dispersion relation in a general form equation 4, with the visual aids as shown in Figure 2.
Figure 1. 1D N – atom chain
1 t 2 0 E N 1 N t
t
0 t
t
0 t
0
0
t
t 1 2 N 1 N
(4)
6
By definition, ε represents the self-interacting energies of the atoms, represented by a red cross, whereas t represents the interaction energies of atoms among each other. A periodic boundary condition is added in the chain as bolded t, which simply shows that the first atom and Nth are adjacent to each other. The periodic condition allows the expression of overall matrix equation collapsed into a single linear equation form as shown in equation 5.
En tn1 n tn1
(5)
Due to the coefficients of linear matrix equation stays static, the approximation of wave function can be done with a plane wave equation 6, effectively transforming it into a function of wavenumber, k. Then, dividing the nth minus one wave function with the nth wave function, Equation 7 is obtained [17].
n 0 e inka n1 ei ( n1) ka i nka ei ka n e
(6) (7)
Thus, the final dispersion relation is formed as equation 8, and it is ready to be used for the energy spectra calculation.
E te ika te ika 2t cos ka
(8)
7
2.4 Contact effect in GNR The performance of Schottky barrier GNR is affected by the types and the shape of contact. Many concerns such as transmission rate, band gap energy, size, structure, and material are taken in consideration before it is chosen as contact. There are various types and shapes of contact had been proposed by most researchers are shown in Figure 3 [22].
Figure 2. (a) The schematic diagram of dual-gate graphene nanoribbon SBFETs. (b) Semiinfinite metal is used as the contacts (L and W ∞) (c) Semi-infinite graphene is used as the contacts (L and W ∞). (d) Finite rectangular graphene is used as contacts (e) ZGNR with wedges shapes is used as the contacts (f) ZGNR with equal width as channel is used as contact [22]. The semi-infinite metal shows the promising performance whereas the semi-infinite graphene have the worst performance among all the methods [22]. For finite rectangular graphene and ZGNR with wedges shapes contacts imply improvement on the ION/ OFF ratio in comparison with the graphene case contacts. However, with the strong semi-metallic properties of ZGNR, ZGNR with wedges shapes and ZGNR with equal width as channel contacts show promising performance.
8
For the device that used finite rectangular graphene as contact, its transmission coefficient is relatively low due to its size mismatch with the channel and small band gap within the contact. In addition, devices that used ZGNR with wedges shapes contacts has the worst ION/IOFF ratio in comparison with methods in Figure 3 (d), (e), and (f). This phenomenon happened due to the shape of contacts which narrow down from bigger width to the width of channel, causing the larger transmission probability in the band gap region. Through the contribution of quantum tunnelling currents, this eventually increase the OFF-state current and causing the worst ION/IOFF ratio. Lastly, the ZGNR with equal width as channel contacts have the best performance compared with other methods in term of ION/IOFF . Its ION/IOFF remain the same even the length of the contact is varied as ZGNR is a material with semi-metallic properties and the width is same from the beginning until the end of the device. Although other types of contact able to made ION/IOFF ratio improved significantly, ZGNR is still the priority to be used as contact due to its promising and size invariant performance, that possess by the semi-metallic properties [22, 28]. Therefore, ZGNR with wedges shapes is used as the contacts as shown in Figure 3 (f). 2.5 Doping and On-Site Energy The doping of impurities into an intrinsic semiconductor to alter electrical and structural properties [29, 30] is utilized. Doping in semiconductor showed obvious impacts on the electronic properties of GNR [31]. The heavily doped semiconductor increases significantly on the electron mobility [32] whereas lightly doped in semiconductor reduce hot electron effect [33]. Lightly doped is applied to the drain and source contact as to demonstrate less leakage current, large ION/IOFF ratio, improved sub-threshold swing and switching parameters [34]. Besides, the optimal concentration for lightly doped is one dopant in every thousand carbon atoms [34]. Further doping will cause the semiconductor to be degenerate. Furthermore, edge doping has stronger dopant effect in these structures [35]. The tight-binding energy between carbon atoms and the dopants are depicted in Table 1. Table 1. Tight-Binding Parameter and On-Site Energy of the Nearest Hooping Parameters (All in eV) [21, 36] εC
εB
εN
γoCC
γoCB
γoCN
0.00
-6.64
-11.47
-2.70
-2.25
-1.70
9
2.6 Density-of-States (DOS) The number of states available to be occupied at each energy level also known as DOS. When there are no states at the particular energy level is available to occupied, it simply known as zero DOS. In semi-classical physics, DOS of semiconductor represents the energy of the number of solutions to the Schrödinger’s equation. DOS calculation can be done in various methods. One of the methods is that DOS calculation is done mathematically through the delta δ function [37]. Equation 9 represents the total DOS of a particular system with N bands. The value of N is twice the number of electrons that involved in the transport due to conduction-valence sub-band pair by each electron. N 1 (9) DOS ( E ) Ei k E dk i 1 2 Solving the delta function within the integral gives equation 10, that is applied in the work as one of the DOS that are calculated numerically. N
DOS ( E ) i 1
1 2
G 2 all k E k E i
(10)
2
G
DOS can also be calculated using Green’s function [11]. The relation between Green’s Function and DOS is expressed as
DOS ( E ) where
1
Im Trace G retarded
(11)
is the retarded Green’s Function as shown in equation 12,
G retarded E i * I H
1
(12)
where I is an identity matrix, H is the Hamiltonian operator matrix and η is a very small imaginary value preventing the inverse matrix from diverging. The value 10-3 is chosen so that the peaks in the DOS are clearer and more dominant. One should note that the calculation of DOS via band structure do not contains any length information of the system unlike Green’s Function method, also known as localised density of states (LDOS), due to the existence of Hamiltonian operator. Both methods are used in the computational for comparison purposes [22]. Moreover, the equation is further modified to calculate the DOS of the overall model where channel is link to the contact as shown in equation 13.
G retarded E i * I H S D
1
(13)
where ΣS and ΣD are the self-energies for the right (source) and left (drain) respectively and the formula is shown in equation 14 S , D S , D g S S , D S , D (14) where τ S, D is the interaction between the device and contacts and gS S, D is the surface Green’s Function of the drain contact and source given by
g S S E i * I H source_contact S
g S D E i * I H drain _ contact D
1
1
(15) (16)
10
3. Research Design and Implementation The research methodology of this work is classified into 4 different stages. First, an ideal AGNR model for the channel is constructed [15], followed by the simulation and verification of electronic properties for the channel GNR with other research work [38]. Next, the drain and source ZGNR will be constructed through the modification of the AGNR channel Hamiltonian. Based on the Hamiltonian of the drain and source ZGNR, the generic script for single doped ZGNR is developed which allow the user to define single dopant type and location in the contact Hamiltonian. Besides, the electronic properties of the single doping contact Hamiltonian are simulated and compared with the pristine contact Hamiltonian. Furthermore, the self-energies matrices are developed to allow the connection between the channel and the contact. Last but not least, the density-of-states (DOS) of the overall structure is simulated and compared with the pristine DOS of the overall structure. 3.1 Design of Channel Hamiltonian By applying the three simplifying assumptions [16], namely basis function that adapted from Hartree’s assumption, plane wave approximation, and discretization of Hamiltonian operator, the model of channel AGNR is constructed. The basis function is the wave function for the 2pz electron that forms free moving delocalized π-orbital. Besides, as the AGNR Hamiltonian is generated with open boundary condition, it is assumed that the hydrogen atoms are passivated with all the dangling bonds at the carbon edge. Due to plan wave approximation, the overlapping of wave functions will only occur among the atoms which are directly next to it. In conjunction with it, the width and length of AGNR are defined in unit cells and the numbering sequence of carbon atoms that indicates the position of carbon atom in the Hamiltonian is strictly followed throughout the research, as shown in Figure 4.
Figure 4. Pristine 5-AGNR with length of 3 and the numbering of the carbon atoms.
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The quasi – 2D AGNR structure is decomposed into 1D structure through the collapsing of the unit cell into single matrix form, defined as alpha matrix, α. In other words, alpha matrix, α also indicate the carbon atoms interaction within the unit cell while the beta matrix, β is defined as the interaction between the unit cell. The number of rows in alpha matrix is always equal to the number of in beta matrix and it indicates the number of carbon atoms in the unit cell. After the alpha matrix and beta matrix are defined, both matrixes are combined to generate the channel Hamiltonian matrix. All the AGNR Hamiltonian will have the same general structure for same widths, but different contents in alpha and beta matrices. 3.2 Design of Contact Hamiltonian with Single Dopant For pristine drain and source Hamiltonian, it could be developed through the modification of the channel Hamiltonian as it applied exactly the same concept as the channel Hamiltonian. For a perfect connection of contact Hamiltonian to the channel Hamiltonian, the width of contact has to be greater than the width of channel by one unit cell through adding two carbon atoms. Besides, the unit cell of the drain and source will be shifted downwards and upwards respectively as shown in Figure 5. An important step must be considered is the removal of excess carbon atom that causes unwanted dangling bond. For instance, the carbon atom 11 and 12 in L1 and L3 region on both source and drain contact respectively as illustrated in Figure 5.
(a)
(b)
Figure 5. Modified 6-AGNR with length of 3, single dopant and the numbering of the carbon atoms for (a) drain contact Hamiltonian and (b) source contact Hamiltonian.
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With a single dopant applied, the contact Hamiltonian is further modified in term of the tight-binding energy. In graphene, each π-orbital interacts with three of its nearest neighbour πorbital electrons with tight-binding energy of -2.7eV [18]. Single dopant applied means that the tight-binding energy between the impurities with the nearest neighbour π-orbital electrons will be changed according to the type of dopant. In Figure 5, the blue dot indicates the N-type doping (Nitrogen) whereas the red dot indicates the P-type doping (Boron). This indicator is applied to any doping location within the width and length that is predetermined by users. The algorithm for the doping process has been constructed to process the input given by the users on the doping location and doping type. The complete Hamiltonian for the contact is simulated to obtain the electronic properties. One should take notes that besides the dopant location, the location of remaining carbon atoms are assumed to remain unchanged although the tight-binding energy changed, as this would involves complicated and countless probability of geometrical structure changes in consideration. 3.3 Design of Self-Energy Matrix By applying the assumptions, self-energy matrices for source and drain contacts, ΣS and ΣD respectively are developed through equation 17 [22]. The interaction between channel and contact is defined as τ and the surface Green’s function of contact is defined as gS. The method of developing matrix for τ is same as the beta matrix, β in channel Hamiltonian while the dimensions for the self-energy matrix must be equivalence with channel Hamiltonian. For the location of the τ matrix will be located at top-left corner for drain contact and bottom-right corner for source contact as shown in equation 18. Therefore, the algorithm for the self-energy matrix has been developed to process the width input given by the users. (17)
0 0 D 0 0 0 0 0 0
(18)
0 0 0 S 0 0 0 0 0
(19)
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3.4 Electronic Properties By practicing basis functions assumptions as the restriction, the dispersion relation is the relationship between band energies, E, and the wavenumber, k. The band structure of channel’s AGNR and contact’s ZGNR are obtained through the summation of all the allowed energy levels, at k-points at given momentum. In order to compute the band energy spectrum, the E against k graph is plotted within the first Brillouin Zone where the wavenumbers, k = -π to π. By applying the periodic boundary condition along with the simplified assumptions [16], the overall matrix is transformed into equation 8, that is used to compute energy spectrum. In equation 8, ε indicates the self-interacting energies of the atoms and t indicates the interaction energies between atoms, which are alpha and beta matrix respectively. In this research, two different methods are used to calculate the DOS of channel and contact band structure. These methods are the delta δ function in equation 10 [26] and Green’s Function in equation 11 [9, 11] and both are implemented for comparison purposes. Note that the doped contact Hamiltonian is computed using Green’s Function as it capture the full dimension of the modelled structured to compute energy spectrum. Finally, the DOS of the overall structure is calculated based on the Green’s Function method on equation 13 [16] with the addition of selfenergy matrix. The DOS of doped structure is compared with the un-doped structure.
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4. Results, Analysis and Discussion 4.1 Electronic Properties for Channel AGNR The electronic properties for channel with carbon dimmer lines, n= 5 to 20 are simulated. The simulation results for 5-AGNR, 7-AGNR, and 15-AGNR with the channel ribbon of length, L=35 unit cells are shown in Figure 6. The band gap energies for the specific carbon dimmer lines are calculated in Figure 6, which are 0ev (n=5), 1.267eV (n=7), and 0.6eV (n=15).
(a)
(b)
(c)
Figure 6. Simulated sub-band structures for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively.
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Besides, the numerical DOS for the specific width is simulated. Figure 7 shows the simulated numerical DOS for 5-AGNR, 7-AGNR, and 15-AGNR.
(a)
(b)
(c)
Figure 7. Numerical DOS plots for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively. Furthermore, the Green’s Function approach is used to simulate the DOS for the specific width. Figure 8 shows the simulated Green Function DOS for carbon dimmer lines, n=5, 7, and 15.
(a)
(b)
(c)
Figure 8. Green’s Function DOS plots for (a) 5-AGNR, (b) 7-AGNR, and (c) 15-AGNR respectively, with channel ribbon of length, L=35 unit cells.
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The AGNR channel that developed in this paper only limited to an odd number carbon dimmer lines, which indicate the width in the form of carbon numbers. The purpose of using odd number carbon dimmer lines is to avoid extra dangling bond that create a mismatch between channel and contact. From the results, the electronic properties for channel with carbon dimmer lines, n=3p and 3p+1, the AGNR channel possess semiconducting properties whereas n=3p+2, it possesses semi-metallic properties, which equivalent to the benchmarks given and tallies with the expected results. Besides, the numerical DOS for the specific width is simulated. However, this approach is length independent and it only varies with the width. From the simulation result, it is observed that it converge to approximately the same shape although the width is varying. Furthermore, the Green’s Function approach is used to simulate the DOS for the specific width. Unlike the numerical approach, this method takes the length of the structure into consideration. Prominent peaks start to appear as the length increases as shown in Figure 10 where ribbon of length, L=35 unit cells. In previous work, it is demonstrated that channel ribbon of length less than 10 unit cells has less van Hove singularities peaks [15]. For both numerical DOS and Green’s Function DOS, the simulation results show the peaks at ±2.7eV which is the tight-binding energies for the interaction between carbon atoms that contribute to the 2pz π-orbital.
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4.2 Electronic Properties for Contact ZGNR The following work consider GNR with carbon dimmer lines, n=7 and 15 as it matches with band gap of silicon and germanium respectively. Hence, it is useful to simulate the electronic properties with these widths. The simulation results for 7-ZGNR, and 15-ZGNR with contact length, L=35 unit cells are shown in Figure 9.
(a)
(b)
(c)
(d)
Figure 9. Simulated sub-band structures for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15ZGNR drain, and (d) 15-ZGNR source. Besides, both the numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for both drain and source of 7-ZGNR, and 15-ZGNR. The simulated DOS is shown in Figure 10 and Figure 11 respectively.
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(a) (b) (c) (d) Figure 10. Numerical DOS plots for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source.
(a)
(b)
(c)
(d)
Figure 11. Green’s Function DOS plots for (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source.
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The paper is followed by applying single dopant into drain and source contact. Figure 12 shows the comparison of sub-band structure for N-type doping, using Nitrite as a dopant in drain and source contact with the pristine electronic properties, for 7-ZGNR, and 15-ZGNR.
(a)
(b)
(c)
(d)
Figure 12. Comparison of sub-band structures for N-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure. Furthermore, both the numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for lightly doped drain and source contact, 7-ZGNR, and 15ZGNR and compare with the pristine results. The simulated DOS is shown in Figure 13 and Figure 14 respectively. The simulation is repeated with P-type doping, using Boron as the dopant and the results are shown in Figure 15, Figure 16 and Figure 17.
(a)
(b)
(c)
(d)
Figure 13. Comparison of Numerical DOS plots for N-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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(a)
(b)
(c)
(d)
Figure 14. Comparison of Green’s Function DOS plots for N-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
(a)
(b)
(c)
(d)
Figure 15. Comparison of sub-band structures for P-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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(a)
(b)
(c)
(d)
Figure 16. Comparison of Numerical DOS plots for P-type contact (a) 7-ZGNR drain, (b) 7ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
(a)
(b)
(c)
(d)
Figure 17. Comparison of Green’s Function DOS plots for P-type contact (a) 7-ZGNR drain, (b) 7-ZGNR source, (c) 15-ZGNR drain, and (d) 15-ZGNR source. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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The ZGNR contact that developed in this work is the modification from pristine channel AGNR, by shifting the unit cell upwards or downwards for source and drain respectively. The contact width must be equivalent with the channel width, yet in the Hamiltonian, it has two more rows and columns to fill up the extra carbon atoms that required for modification purpose. According to the simulated results, the contact always possesses semi-metallic properties and similar sub-band structure as semi-metallic form AGNR as it is the modification of AGNR structure. Regardless of the width, the contacts are semi-metallic state with zero band gap energy. In addition, it is noticeable that the drain and the source have similar sub-band structures. Therefore, it can concluded that the shifting of the unit cell, either shifting upwards or downwards do not affect the electronic properties Both numerical DOS and Green’s Function DOS approach are used to simulate the electronic properties for both drain and source of 7-ZGNR, and 15-ZGNR. For the numerical DOS, it is observed that the results have the same trend as the channel DOS, where it converge to approximately the same shape although the width is varying. Furthermore, the simulation results using Green’s Function approach differ from the numerical approach, as this method take the length of the structure into consideration. From the simulation result, it also shows prominent peaks at ±2.7eV. After the simulation of pristine drain and source ZGNR contact, a single dopant is added to observe the effect of lightly doped towards the electronic properties of the contact. In this research scope, edge doping concept is applied to both drain and source. The dopant in the drain and source interacts with three and two carbon atoms respectively. It is observed that the dopant has major effects on the sub-band structure and DOS that is computed numerically. This is due to the calculation of the sub-band structure and numerical DOS do not consider the length. Thus, doping has a larger impact. On the other hand, Green’s Function DOS considers the length of contact. As a result, the effect of doping on Green’s Function DOS becomes less impactful with increasing length. In our investigation, a single dopant is utilized and thereby the Fermi energy of the structure can be preserved [34]. Generally, Green’s Function DOS is more accurate than numerically calculated DOS. Furthermore, the simulation data such as band structure and numerically calculated DOS for drain and source contact with doping show much more sensitivity as clear variation is evident when dopant are placed at different position respectively. It can be concluded that the doping location also plays a very important role in electronic properties. Dopant that has two carbon atoms directly near to it shows different electronic properties with the dopant that has three carbon atoms directly near to it. Besides, the type of doping also plays an important role in this context. It is also found that only minimal variation for P-type structure are observed as compared to the pristine structure but stronger variation for N-type structure. The reasons are because P-type dopant has tight-binding energy of -2.25eV with carbon atoms whereas N-type dopant has tight-binding energy of -1.70eV with carbon atoms. A small difference in tight-binding energy causes minor changes.
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4.3 Electronic Properties for Overall Structure Green’s Function DOS is simulated for the overall structure. The comparison of Green’s Function DOS N-type structure with the pristine structure of 7-GNR and 15-GNR are simulated as shown in Figure 18 (a) and (b). The process is repeated by the P-type structure as shown in Figure 18 (c) and (d).
(a)
(b)
(c)
(d)
Figure 18. Comparison of Green’s Function DOS plots for N-type overall structure (a) 7-GNR, (b) 15-GNR, and for P-type overall structure (c) 7-GNR, and (d) 15-GNR. Line (red) indicates structure with dopant whereas line (blue) indicates pristine structure.
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Self-energy matrices are developed to combine the lightly doped drain and source contact with the pristine channel AGNR. For the overall structure, there is no clear alpha and beta matrix as all the channel and contact are combined together. Therefore, only DOS based on Green’s function can be utilized whereas numerical DOS only detect the alpha and beta matrix from the overall structure. From the simulation result, the overall structure shows prominent peaks at ±2.7eV which indicates the tight-binding energy for the carbon atoms interaction. Besides, the light-doped showed fewer discrepancies with the pristine structure. Therefore, it can be concluded that the light-doped have a low impact on the simulated DOS. Furthermore, it is observed that the Green’s Function DOS of the overall structure still has a band gap that indicates it is still a semiconductor. In short, the channel properties are still preserved and not affected by drain and source contact. 5. CONCLUSION A tight-binding model for AGNR with ZGNR drain and source contact is presented. By applying appropriate simplifications and assumption, doping effects which alter the composition of Hamiltonian are implemented by replacing on-site self-energies and hopping integral or tightbinding energy of carbon atoms. In addition, the simulated electronic properties of channel have good agreement with reported data from other researcher [15]. DOS that is based on Green’s function able to provide the states availability at each energy level when the full dimension of the structure is considered. The DOS simulation result for all case showed a prominent peak at ±2.7eV which are the tight-binding energy for carbon atoms interaction. Furthermore, the dopant showed an insignificant shift for all the electronic properties. Finally, the semiconducting properties for the overall structure remain unchanged as the drain and source contact are semimetallic does not vary the channel semiconducting properties. Acknowledgements Kien Liong would like to express his appreciation for the award of Zamalah PhD Scholarship from the School of Graduate Studies UTM. Michael Tan acknowledges the financial support from a FRGS Research Grant (Vote no. R.J130000.7851.5F043) that allowed the research to proceed smoothly.
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