Atomistic modeling of the electrostatic and transport properties of a simplified nanoscale field effect transistor
A first-principle model is proposed to study the electrostatic properties of a double-gated silicon slab of nano scale in the framework of density functional theory. The applied gate voltage is approximated as a variation of the electrostatic potential on the boundary of the supercell enclosing the system. With the electron density estimated by the real space Green’s functions, efficient multigrid and fast Fourier Poisson solvers are employed to calculate the electrostatic potential from the charge density. In the representation of localized SIESTA linear combination of atomic orbitals, the Kohn-Sham equation is established and solved self-consistently for the wavefunction of the system in the local density approximation. The transmission for ballistic transport across the atomic silicon slab at small bias is calculated. The charge distribution and electrostatic potential profile in the silicon slab versus the gate voltage are then analyzed with the help of the equivalent capacitive model. Quantum confinement and short gate effects are observed and discussed.
The size of Si based metal-insulator-semiconductor (MIS) devices has been shrunk aggressively to nanoscale nowadays (1) in design of very large scale integrated circuits. The quantum-mechanical nature of transport becomes inevitably prominent and there are extensive efforts devoted to modeling quantum behaviors in nano-MIS devices based on the effective mass approximation or the empirical tight-binding models. (9); (2); (3); (4); (5); (6); (7); (8); (10) Besides the traditional top-down technologies, bottom-up techniques has also been developed to fabricate Si nanostructures such as nanowires for future nanodevices. (11) As a result, the first-principle study on electronic and transport properties of atomic Si systems becomes attractive. (12); (13)
Due to the extensive application of Si based field effect transistors, some efforts have been carried out to the first-principle understanding of their behaviors in the past years. Evans et al. (14) have studied the effect of Si-SiO interface roughness on the electron mobility in a Si based MIS structure by describing the roughness by first principles. Based on the above method, Hadjisavvas et al. (15) have proposed an explanation for electron mobility enhancement in strained MIS field effect transistors (FETs). Fonseca et al. (16) and Liu et al. (17) have established a two-terminal system of Si-HfO-Si and Al-SiO-nSi respectively described by a self-consistent density functional theory (DFT) and studied the transport properties through the HfO and SiO slab in the framework of the nonequilibrium Green’s function (NEGF). In this way, they have estimated the leakage current through the ultra-thin oxide barriers of MISFETs as a function of the voltage drop. Landman et al. (18) have studied the quantum transport through short Si nanowires passivated by hydrogen atoms and attached between Al electrodes by solving the eigenchannels of the scattering states in the framework of the density functional method with plan wave basis. Similarly, Ng et al. (19) studied the quantum transport through hydrogenated Si nanowires with Li electrodes employing the NEGF-DFT approach integrated in the Atomistix ToolKit (ATK) package. Fernández-Serra et al. (20) have investigated the effect of doping on electronic quantum transport in Si nanowires by assuming the same material (Si) for electrodes and the device regions. Later on, Markussen et al. (21) have continued the work for longer wires and studied the crossover from ballistic to diffusive transport based on the DFT and a recursive Green’s function approach. Besides the above two-terminal models, three terminal models have also been used for Si systems to take into account the gate effect. Using the NEGF-DFT approach with the help of ATK package, Dai et al. (22) have investigated the transport properties of a Si cluster sandwiched between two Al electrodes. By assuming a constant potential shift in the molecular region (23); (24) when a gate voltage is applied to the third terminal, they have observed charge transfer from the molecular and transconductance oscillation.
The aim of this work is to go further from small three-terminal molecular systems to more realistic multi-terminal MIS devices of nanoscale using the NEGF-DFT approach. However, a rigorous treatment (25) of realistic multiterminal systems is computationally too expensive. Fortunately, particle exchange between the gate and the channel is usually very limited as they are separated by the insulator slab and the gate terminal works as a controller of electrostatic potential in the device region. To assure the simulation feasible with the present computation capability while catching the most important characteristics of nano MISFETs, we use a simplified model and focus ourselves on the gate effects on the electrostatic and transport properties under a small source-drain bias (in the linear region). As we know, in a real MIS device, how the gate voltage is distributed into the channel of the device or the electrostatic property of the device as a MIS capacitor is critical in determining the characteristics of the whole device. (26) For example, double-gate (DG) structure has been considered as the most promising device geometrical structure for nano MIS devices due to its good electrostatic integrity (27); (28). In our model, we will take the effect of gate voltage distribution into account instead of simply adding a constant shift to the electrostatic potential. The gate voltage that controls the boundary condition and the corresponding variation of potential inside the device is self-consistently determined by Poisson equation.
The rest of the paper is organized as follows. In Sec. II, we define the device model and briefly describe the NEGF-DFT formalism and the computation method. In Sec. III, the result for electrostatic properties, charge and potential, are reported and analyzed with the help of density of states (DOS) and transmission spectra. Finally, a brief summary is given in Sec. IV.
Ii model and method
ii.1 Device Model
We consider a prototype DG MISFET as schematically shown in Fig. 1(a). The device is made from a Si slab with thickness of nm order in the direction. The channel () direction is chosen along Si crystal orientation and the interfaces along which is the preferred direction due to lower interface defect density and higher mobility. H atoms are used to passivate the dangling Si bonds on the surfaces. The H-Si bond length is relaxed and has a value of 1.49 Å while the Si-Si bond length is 2.35 Å. Further on either side of the Si slab a metal gate of variable length along direction is located a few angstroms away from the H atom sheet, as indicated by the thick solid bars in Fig. 1(a). Note that here we insert the vacuum slabs, instead of oxide slabs, between the gates and the Si slab as the insulator. This is one of the simplifications introduced in this model in order to reduce the computation load. Since the dielectric constant of vacuum () is much lower than that of oxide (e.g. for SiO), we expect a lower insulator capacitance (between the gate and the channel) in our model and, hence, a larger gate threshold than that for devices with oxide insulator slabs.
Four different model structures of total length 21.72 Å are studied in this paper and they are denoted as for models with short gate (length Å) and for the model with long gate (in the full central region of length Å). The index () is used to distinguish models with different thickness of the Si slab (vacuum slab). The Si slab in the models is unit-cells thick ( and Å for and respectively). The corresponding vacuum slab thickness is and Å for and respectively.
Another simplification is to approximate the gate effect with a boundary condition based on the following physical consideration. When we apply a small voltage drop between the Si slab and the metal gates we shift the Fermi energy in the latter by letting the former grounded. Due to the high electron density in metal, the electrostatic potential in the gates will shift in parallel with the gate Fermi energy or the voltage drop. As a result, the role of the metal gates on the channel is to control the electrostatic boundary of the supercell as enclosed by the solid lines in Fig. 1(a). Understanding this gating mechanism, we can exclude the gate materials from our model system (inside the supercell) while taking its most important effect into account by shifting the boundary condition as further explained in next section.
A third simplification is to use the same material for the source/drain electrodes and the channel. In real MISFET devices, the drain and source electrodes are doped and the contact between the electrodes and the channel adds to the complexity of the system and makes the result difficult to analyze. In this work, we focus on the gating effect in the channel region and it would be more practical to handle a simplified system. This simplification is expected not to affect the conclusion when the bias is low and has been used by other authors. (20)
After the above simplifications, in our model, the double-gated MISFET is a Si slab with known electrostatic boundary and can be studied by the NEGF-DFT method. The system is composed of a hydrogen passivated Si slab and two vacuum slabs. Along axis, as separated by the vertical dash lines in Fig. 1(a), the system is divided into three regions: the left electrode (L) the central region, or the channel region (C), and the right electrode (R). In direction, the electrostatic potential of a Si slab without gate is used as the boundary condition of the electrostatic potential on the left and right boundaries of the supercell. This implies that the left and right electrodes are assumed semi-infinite. In direction, the periodic boundary condition is applied to assume an extending system in this direction. In direction, the boundary is variable depending on the system size and the gate voltage and will be specified in the corresponding context. Furthermore, we assume a uniform Fermi level inside the supercell. This approximation is justified in case of small bias between the source and drain electrodes because the gate voltage induced variation of Fermi energy happens mainly in the vacuum slab and the density of states there is negligible.
ii.2 Theoretical Formalism
To analyze the electrostatic and transport properties of the atomic scale MISFET, we utilize density functional theory (DFT), (29); (30) which converts the many-body system into a single-particle system, and the two-terminal nonequilibrium Green’s function (NEGF) technique (31); (25), which takes care of the coherent transport through the system between the source and the drain electrodes in Landauer picture (32); (25); (33); (34); (35). With standard norm-conserving pseudopotentials (36) to describe the effective interaction of the valence and core electrons, we construct a localized SIESTA linear combination of atomic orbitals (LCAO) basis set (37) as the representation for expanding the electronic wavefunctions. According to the Hohenberg-Kohn theorem, (29) in the Born-Oppenheimer approximation, the fully interacting electron problem for the ground state can be mapped into a variational problem in terms of a single-particle density . The Kohn-Sham (KS) equation (30) is then established for the atomic system in the representation with the corresponding KS Hamiltonian operator expressed in matrix form as . In extending systems, we use the supercell technique with the periodic boundary condition to define and solve the KS equation in the whole real space. The dimension of the supercell is chosen larger than the screening length of the KS potential in any direction so the real system is well mimicked by that inside the supercell. In open systems, the wavefunction differs from that in closed system and this is taken care of by using the Green’s function of a semi-infinite lead (38) as the boundary condition for the Green’s function inside the supercell. With the KS Hamiltonian we calculate the retarded (advanced) Green’s function (GF) , Keldysh GF and retarded (advanced) self-energy . The density matrix then reads (31); (25); (39)
The final density matrix is self-consistently reached from an initial guess, for which we use the neutral atom density matrix obtained by assuming no interaction between atoms. In the linear response region where the bias between the electrodes is small, the Fisher-Lee relation (40) connects the Green’s functions with the transmission coefficients. With the density matrix and the Green’s functions known, the transmission for the quantum transport between the electrodes is then given by (25)
where , are line-width functions of left and right electrodes which indicate the corresponding coupling strength between the electrodes and central region. The linear zero-temperature conductance is determined by the transmission at Fermi energy
ii.3 Computational Method
Atomistix ToolKit (ATK) (41) for two-probe systems together with a multigrid Poisson solver (42); (43) is used to carry out the numerical calculation. We at first apply the periodic boundary condition in the and directions and solve the corresponding electron density and electrostatic potential . on the boundary is denoted as the boundary condition at . The boundary condition at finite is obtained by shifting with on the gate but keep the same as on the electrodes. On each boundary region between a electrode and a gate the potential shift decreases linearly from to zero. The modified boundary condition is then applied to the system to solve and at arbitrary . In the calculation we use the SZP, single valence and orbitals ( orbitals) plus single -polarization () orbitals for Si (H), real space SIESTA LCAO basis set (41); (37); (13) and the standard nonlocal norm-conserving pseudopotential (36) from the database in ATK. The reason why we use the SZP basis set instead of a simpler one is because the result presented in this paper converges for SZP and more complete basis sets. The local density approximation (LDA) with the Perdew-Zunger parametrization (44) of the correlation energy for a non spin-polarized homogeneous electron gas (45) is used for the exchange-correlation functional. The mesh cutoff is set to eV corresponding to a grid size of ÅÅÅ. The MonckHorst-Pack k-point grid is along directions.
Iii Results and Discussions
To express a physical property in our microscopic model on the same footing as its counterpart in traditional macroscopic model, we measure it from its value at and use its average over the - range inside the central unit cell ( Å Å), , in the following analysis. The total net charge is the total charge transferred into the supercell when is applied. Note that there may be a shift between in our model and the observed in experiments because i) the real boundary condition at may differ from the periodic (bulk) boundary condition used in the calculation, ii) the boundary potential shift when a finite being applied is determined selfconsistently and can vary in different environments, and iii) no workfunction is specified for the gate metal.
Equivalent capacitive circuits are widely used in qualitative analysis of electrostatic properties of MISFETs in the literature. Our system is macroscopically symmetric about the - (top-down) and - (left-right) planes and three capacitors, , and may be used for this purpose. As shown in Fig. 1(b), left circuit, accounts qualitatively for the average voltage drop across the insulator slab while for the one across the Si slab in direction and in direction. For long channel as usually exists in traditional DG MISFETs, one-dimensional approximation is valid and only is important for the electrostatic analysis. For short devices, however, the boundary at the ends of the channel also plays an important role in determining the potential profile in the center of the channel and the effect of should be taken into account. Because only the linear region is concerned in this paper, we can also use a single capacitor to describe the electrostatic properties of the Si slab with , as illustrated in the right circuit of Fig. 1(b). The total net charge in the Si slab reads
with the total capacitance . Here is the surface potential and the central potential of the channel as further specified in Sec.III.B.
iii.1 Charge and Potential Distributions in Real Space
The charge density distribution along the gate () direction is shown in Fig. 2(a) for model . Similar distribution is observed for other models and is not plotted here. Attracted by the charge in the metal gates, most of the net charges appear near the surface of the Si slab. As a result, as illustrated in Figs. 2(b) and (c), the electric field concentrates mainly in the vacuum slab where varies rapidly and is weak inside the Si slab where becomes flat. The FET is in its off region at small and shifts with until reaches the threshold voltage where saturates in the center of the Si slab.
An interesting short gate effect is observed when comparing the potential distribution for the short gate model and that for the long gate model . For the short gate model , as shown in Fig. 2(b), versus in the vacuum regions (Å or Å Å) is nonlinear. This is a result of the fact that the length of the gate in model is comparable to the thickness of the vacuum slabs in model . The electric field in the vacuum slabs deviates greatly from along the direction and the parallel-plate capacitor model fails. For longer gate model , the fringe effect on the channel region is reduced and the versus curves in the vacuum slabs become straight as shown in Fig. 2(c). The short gate effect described above is unique in nanoscale MISFET and does not appear in conventional MISFET since there the insulator slab’s thickness is usually much smaller than its length. We will further discuss its effect on the FET’s performance in Fig. 3(c) and (d).
iii.2 Total Transferred Charge and Surface Potential
We use the surface just enclosing the H and Si atoms as indicated by the horizontal dashed lines in Fig. 1(a) or the vertical dashed lines at Å and Å for model in Fig. 2(a) as the surfaces separating the channel and the insulator slab. The averaged value on these surfaces is used as the surface potential . The central potential is approximated as the value in the middle of the two gates, i.e. at Å in Fig. 2(a) for model .
With and known at each , we can estimate the capacitance from Eq. (4). For traditional MISFETs, usually we have in the off region and in the on region. However, the Si slab here is extremely thin and in both regions. In addition, the span of the device in the direction is larger than that in the direction. This fact leads to in the off-region and may increase with the Si slab thickness because is proportional to the thickness while is proportional to the inverse of it. In the on region, when a large amount of free charges are created on the surface, is almost pinned due to the screening effect of these free charges.
The total net charge versus the gate voltage for model is shown by the dashed curve in Fig. 3(a). At , the Fermi energy is located in the energy gap of the Si slab. As varies, the relative position of the Fermi level in the energy band shifts accordingly. remains almost constant when the Fermi level is in the energy gap but increases quickly as the Fermi level enters the conduction or the valence band. This sudden increase of happens at the positive and negative threshold points, i.e., near V and V, respectively. Near these two points, due to the shift of the ratio, the versus curve also changes its slope as shown by the dashed curve in Fig. 3(b).
To show the effect of the vacuum-slab thickness on the electrostatic characteristics, we use model with a thinner vacuum slab and redo the calculation. and have similar profiles between the gates as that for model but with a larger amplitude at the same gate voltage. In the equivalent capacitive circuit, increases with the shrinking of the vacuum slab, which results in the shift of and as shown by the solid curve in Fig. 3(a) and (b). The versus curve of thicker insulator slab model has a wider subthreshold region in the axis. One interesting observation is that the threshold surface potentials, the surface potentials at the threshold points, for model and are very close as indicated by the short bars beside the curves in Fig. 3(b). This is because the surface potential reflects directly the Fermi level in the energy band of the Si slab. For a system of thicker insulator or smaller it takes a bigger gate voltage difference to shift the Fermi energy from the valence band to the conduction band.
In Fig. 2 we have shown that the electric field in the insulator slab can deviate from normal to the slab due to the fringe effect in short gate cases. This short gate effect in nano FET reduces the per area capacitance of the insulator slab and results in a larger threshold gate voltage than estimated from parallel capacitance approximation. In Fig. 3(c) and (d) we plot the versus curves and the versus curves respectively for model (dashed) and longer gate model (dash-dotted). The smaller total capacitance in model , due to its half gate length plus the short gate effect in it, results in a less than half net charge in the system at any and a much gentler slope of its versus curve in Fig. 3(c). In Fig. 3(d), we also observe a gentler slope of the versus curve for model in the subthreshold region. Combining the relation as expressed in Eq. (4), this means a smaller ratio of and confirms the conclusion drawn from Fig. 2 that the short gate effect mainly reduces the insulator capacitance.
Up to now, we have seen that both making the insulator slab thinner and suppressing the short gate effect can enhance the insulator capacitance and reduce the threshold gate voltage as illustrated in Fig. 3(b) and (d). On the other hand, a close comparison of the curves shows that the two changes to the system have opposite effects on the threshold surface potential. The threshold surface potential has a tendency to increase in the former case (the solid bar on the positive side in Fig. 3(b) is slightly higher than the dashed one) while the threshold surface potential decreases in the latter case. A detailed study shows that the variation of the threshold surface potential is related to the potential distribution inside the Si slab. Given the same surface potential in the subthreshold region, the potential distribution along changes little when varying the insulator slab thickness but a potential drop appears and broadens in the center of the versus curves when shortening the gate.
In Fig. 3(e) and (f), we present the dependence of electrostatic characteristics on the Si slab thickness exemplified by model and . Similar to the case of thicker insulator slab as shown in Fig. 3(a) and (b), the and versus curves for a system of thicker Si slab also have a wider subthreshold region in the axis. However, this apparent common feature in the two cases has different origins and it is interesting to compare the details of their electrostatic characteristics.
At first, as pointed out earlier in this section, because the Si slab inside the supercell has a dimension in the direction bigger than that in the direction, and increases with the Si slab thickness in the off-region. In this case, the versus curve of thicker Si slab model (dotted) in Fig. 3(e) has a sharper slope in the off-region due to an increased total capacitance , in contrast to the gentler slope of the curve for thicker insulator slab model (dashed) in Fig. 3(a). In addition, the bigger in the thicker Si slab case, in stead of the smaller as in the thicker insulator slab case, results in the gentler slope of the dashed versus curve in Fig. 3(f).
Secondly, it is well known (46); (47); (9) that the energy gap of semiconductor materials in nanostructures is enlarged from their bulk value as a result of the quantum confinement effect. The change of the gap depends on the confining potential profile characterized by parameters such as the confining dimension and the potential height. This effect is automatically taken into account by our NEGF-DFT model. As will be illustrated later in Sec.III.C, the thicker Si slab in model has a narrower and shifted energy gap compared to that in model . As a result, the subthreshold region in the axis, i.e. the difference between the surface potentials at the positive and negative threshold points, of model ( V for the dotted curve in Fig. 3(f)) is narrower than that of model ( V for the solid curve). Furthermore, we also observe a shift of the subthreshold region in the and axes when the Si slab thickness varies.
As discussed above and illustrated in Fig. 3(e) and (f), the thicker Si slab in model , compared to that in model , narrows the subthreshold region via the quantum confinement effect on the one hand and reduces ratio in the off-region due to the specific length-thickness ratio of the Si slabs in the models on the other hand. The observed wider subthreshold region is a result of the competition between these two independent but opposite factors.
In this section, we analyze the electrostatic characteristics of our model nano FET in a way widely employed in the analysis of conventional MISFET. (26) Here we want to emphasize that our system (of size 2 nm) is much smaller than any of the conventional MISFETs even the smallest predicted by ITRS in its latest report. (1) As a result, our model device shows characteristics between molecular electronics and conventional electronics. For example, we observe a large slope of the versus curve in the off region in all models which is similar to the result of the Si cluster (22) but different from that in conventional MISFET where a flat - dependence is usually found in the off region. In addition, the threshold gate voltage in our systems ( V) is much higher than the value usually observed in conventional MISFET ( V). This is because the Si slab in our systems is very thin and a vacuum instead of SiO slab is used as the insulator slab. The ratio then becomes much larger in the off region.
iii.3 Transmission and Density of States
To explore further the origin of the electrostatic properties of nano FETs and the gate effect on transport, we calculate the transmission, which gives the conductance in the linear region via Eq. (3), and the density of states (DOS). In Fig. 4(a), the transmission of an electron propagating along the direction is calculated at for quantum transport between the source and drain electrodes in model (solid) and (dashed). Because the electron sees a perfect crystal in the direction, its transmission is for each channel and the transmission versus energy curve tells the number of propagating channels for an electron of the energy. The one-dimensional DOS in direction for model (solid) and (dashed) is plotted in Fig. 4(b). The band gap of model is wider than that of model due to the quantum-confinement effects in nano semiconductor structure (9). In addition, most probably due to the change of the surface effect or the ratio between the numbers of the H and Si atoms in the system, a shift of the energy gap is observed when varying the Si slab thickness.
The transmission versus electron energy at different gate voltages is plotted in Fig. 5 for model . When a negative gate voltage is applied, the energy band in the gate region shifts to higher energy. This shift forms an energy barrier for electrons and a well for holes. As a result, we observe a significant decrease of transmission for electrons near the bottom of the conduction band as shown in Fig. 5(a). On the contrary, if a positive gate voltage is applied, an energy barrier is formed for the holes in the middle of the device and the propagation of holes near the top of the valence band becomes unfavored. The transmission profile changes quickly with the gate voltage in the subthreshold region and then becomes less sensitive when and saturate at higher than the threshold gate voltage.
In summary, we have proposed an atomistic multi-terminal model for electrostatic simulation of nano MISFETs in the framework of the NEGF-DFT approach and studied the behaviour of devices with Si slabs of one and two unit cells in thickness, with variable insulator (vacuum in this model) thickness, and with different gate length. The subthreshold region in terms of the surface potential reflects the band gap of the Si slabs and the surface potential at the threshold points remains little changed in devices of different insulator thickness. For a device with thicker Si slab, the subthreshold region in surface potential becomes narrower due to the quantum confinement effect while the subthreshold region in gate voltage becomes wider as a result of the competition between the confinement effect and the geometry effect on capacitance. In nano FETs where the gate length is comparable to the insulator thickness, the short gate effect results in a much wider subthreshold region than that observed in conventional MISFETs. This short gate effect can be suppressed by using longer gates to reduce the fringe effect on the electric field in the insulator slab.
In addition, we have calculated the band structure of Si slabs with different thickness. The density of states and the transmission probability of electrons in our nano FET system are then estimated to demonstrate the quantum confinement effect and the gating effect on transport in the linear region. The energy gap widens and shifts to higher energy when the Si slab varies from one to two unit-cells in thickness. The application of a gate voltage introduces energy barriers or wells to the carriers in the transport channel and modifies the performance the nano FETs. A negative gate voltage favors the hole transport between the electrodes while a positive one favors the electron transport.
Limited to the computation capability, our model system is still not fully realistic and is smaller than even the latest feasible MISFETs (1). Nevertheless, our model may provide qualitative information about realistic MISFETs and serve as a prototype framework for nano MISFET simulations.
Acknowledgements.This work has been supported in part by the Economic Development Board (EDB), Singapore, and in part under the joint Research Collaboration Agreement between Nanyang Technological University (NTU) and Atomistix Asia Pacific Pte Ltd (AAP). Support from the Nanocluster and Microelectronics Center of NTU is also acknowledged.
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