We investigate the effect of electron-electron interaction on voltage distribution, charge distribution and current-voltage curve of two dimensional nano-MOSFETs with dimension equal to 1 × 1 nm^{2}, 3 × 3 nm^{2}, and 6 × 6 nm^{2} by using non-equilibrium Green function method. It is shown that the turn on voltage increases by decreasing the size of sample because of size quantization. Also we show that for a critical drain-source voltage a negative resistance is seen at current-voltage curve of 1 × 1 nm^{2} sample because of electron-electron interaction, and in consequence it can tolerate lower gate voltage in real practical applications.

## I. INTRODUCTION

The number of transistors that can be placed on an integrated circuit (IC) would double approximately every two years.^{1} The ongoing trend toward the miniaturization of electronic circuits is using a single molecule as functional devices.^{2} Nowadays, there are three obstacles in this way: the rising costs of fabrication, the limitations of lithography and the size of fundamental elements such as diode and transistor.^{3} Many researchers investigate new ways for manufacturing ICs and they believe that the fabrication of nanoelectronic devices is suitable for solving the above obstacles. By decreasing the size of the device, the electron-electron and electron phonon interactions are important phenomena and should be considered when one studies the electron transport. A p-channel MOSFET includes n^{+}-doped source and drain. By applying positive (negative) bias voltage to gate the holes deplete (accumulate) under the gate. When the positive voltage is greater than a critical value the electrons accumulate under the gate and the channel is inverted.^{4,5} Generally a bias voltage is applied along the length of device i.e., between source and drain and the conductivity of channel is controlled by applying the voltage of gate along the width of device.^{4,5} Therefore by decreasing the dimension of device, we should explain the electron transport based on the non-equilibrium nano-physics theory.^{6} Non-Equilibrium Green Function method (NEGF) is an approach to address the quantum transport under non-equilibrium condition.^{7–10} For calculating the Green function and finding the current through the device in self-consistent method, we should calculate the Hamiltonian of the system. The Hamiltonian matrix can be calculated by Hartree-Fock (HF),^{11} Density Functional Theory (DFT),^{12} Finite Element Method (FEM)^{13} or Finite Difference Method (FDM).^{13} We used DFT-NEGF method and investigated the electron transport through a benzene molecule connected to zigzag and armchair leads at zero bias regime.^{14} Jiang *et al.*, studied the charge transport through two dimensional MOSFET by using FEM and FDM without considering the electron-electron and electron-phonon interactions.^{13} Also Shao *et al.*,^{15} studied the accurate calculation of Green function of the Schrödinger equation in a block layered potential and Cheng *et al.*,^{16} studied the three dimensional transport solver based on the perfectly matched layer. Non-studied electron-electron interaction by Ref. 13 and non-studied non-zero bias regime by Ref. 14 motivated us to study the electron transport through nano-MOSFETs in presence of electron-electron interaction by using NEGF method based on FDM.

In this paper we consider the two dimensional nano-MOSFETs with dimension equal to 1 × 1 nm^{2}, 3 × 3 nm^{2}, and 6 × 6 nm^{2} including one layer of left and right lead atoms. The width of devices is sandwiched between two layers of SiO_{2} and the gate voltage is applied in width-direction. A bias voltage is applied along the length of the device i.e., between the source and drain. The Local Density Approximation (LDA) is assumed for electron-electron interaction potential.^{17,18} By using FDM, the self-consistent NEGF equation is solved for finding the potential and charge distribution over the nano-MOSFET devices and their current-voltage curve. It is shown that the turn on voltage increases by decreasing the size of sample i.e., because of size quantization. Also we show that for a critical source-drain voltage a negative resistance is seen at current-voltage curve of 1 × 1 nm^{2} sample and in consequence the device fails. Therefore the current-voltage curve of two dimensional nano-MOSFETs is affected not only by size quantization but also by electron-electron interaction and the device can tolerate lower gate voltage in real practical applications if its dimension decreases. The paper is organized as follows: In Sec. II, the calculation method is given. In Sec. III, the results are presented and discussed. Finally, the summary is given in Sec. IV.

## II. CALCULATION METHOD

We develop our previous calculation method^{19,20} from one dimensional to two dimensional FDM and also use NEGF method which is introduced by Ref. 13. We assume that the density of n^{+}-source, and n^{+}-drain is equal to 10E20 /Cm^{3} and the density of n-channel is equal to 5E19/Cm^{3}. Also it is assumed that the two dimensional n-channel region includes 100 points i.e., 10 nm along x-axis and 10 nm along z-axis. The effective mass of electron and electrical permittivity of channel is equal to 0.25m_{e} and 4ɛ_{0} respectively, which m_{e} and ɛ_{0} are rest mass of electron and electrical permittivity of vacuum respectively. The voltage of source and gate is adjusted to 0V and 0.4V respectively. At first we use Poisson-Schrödinger solver for finding the voltage distribution along the width-direction of devices. Then we use Liebmann's method as FDM and the voltage of leads on the boundaries of the channel is applied as boundary conditions on the finite difference equation.^{21} The drain voltage, V_{d}, changes from −0.25 V to 0.25 V with 0.02 V increment in each step. The electron-electron interaction potential is given by^{12}

where

and rho is the local density of electrons. The fitted parameters for the above parameterization scheme are given by^{18}

It is noted that there are different kinds of functional for electron-electron interaction. *Becke* have introduced modified gradient-corrected functional on semi-empirical grounds.^{25} Lee *et al.* have obtained functional formulas for the correlation energy and correlation potential based on the insertion of gradient expansions for the local kinetic-energy density.^{26} We used B3LYP (i.e., Becke-3,^{25} Lee, Yang and Parr)^{26} functional for calculating the conductance of a benzene molecule connected to graphene leads.^{14} Since we want to show only the effect of electron-electron interaction on current-voltage curve of samples by attention to their length, we use only the formula (1) but other kinds of functional can be considered and their results can be compared with each others in next studies. The self-consistent NEGF calculation method is shown in Fig. 1. The bare Hamiltonian includes only the kinetic energy terms and potential at first and end of length and wide of the device is considered as boundary conditions. For finding the potential distribution along the device the Poisson and Schrödinger equations are solved simultaneously and in self-consistent regime. The charge distribution along device is found by using Green function^{22} at each cycle of iteration. The electron density in the channel per unit area, n_{s}, is obtained by integrating the electron density per Cm^{3} in the channel by attention to the fact that the wave functions should be normalized.^{22}

## III. RESULTS AND DISCUSSION

The voltage distribution over nano-MOSFETs without and with LDA effects is shown in Fig. 2 and Fig. 3 respectively. As the figures show the voltage distribution over sample depends on the size of them and electron-electron interaction has not effective effect on it. It is well known that, by decreasing the size of the channel (generally less than five nm), the size quantization causes that the charge density is peaked near the middle of the channel and potential energy is distributed over the channel based on the boundary condition.^{19,22} The electron distribution on samples per Cm^{3} without and with LDA effects is shown in Fig. 4 and Fig. 5 respectively. For 3 × 3 nm^{2} and 6 × 6 nm^{2} samples the effect of size quantization is seen well when LDA is not considered since the electron density has peak inside the channel. But when LDA is considered, for 3 × 3 nm^{2} sample the charge distribution has maximum value inside the channel while for 6 × 6 nm^{2} it is placed on the lead positions and not inside the channel. It has been shown that by increasing the length of device the peak of charge distribution is placed at lead positions and device shows classical behavior.^{22} Therefore our result means that the classical behavior is seen sooner when LDA is considered. For 1 × 1 nm^{2} sample the charge distribution has maximum inside the channel but in spite of the 3 × 3 nm^{2} and 6 × 6 nm^{2} samples, its peak is placed at the edge of sample (i.e., edge states). As the figures show the charge distribution depends not only on the size of them but also on the electron-electron interaction inside the channel. Jiang *et al.*, have studied a 29 nm double gate MOSFET under gate voltage V_{g} = 0.4 eV and drain bias V_{ds} = 0.4 eV and shown that the charge has a semi-classical behavior and potential has a special distribution along the length of the sample.^{13} As Fig. 2 and Fig. 3 show for 3 × 3 nm^{2} and 6 × 6 nm^{2} samples, the maximum of potential is placed near the right (left) side of samples and its minimum is placed at its left (right) side. Therefore the behavior of our potential curves is in agreement with Fig. 12 of Ref. 13 approximately. As Fig. 5(c) shows the electron density is placed at lead positions and not inside the channel for 6 × 6 nm^{2} sample. Fig. 13 of Ref. 13 shows that the electron density is placed at beginning and end of the channel and therefore our results about 6 × 6 nm^{2} sample is in agreement with Fig. 13 of Ref. 13 approximately when LDA is considered i.e., classical behavior is seen sooner. Fig. 6 and Fig. 7 show the current-voltage curve of the samples without and with LDA effects respectively. It has been shown that, the resistivity of device is proportional to the (1 − m_{s}/m_{d})^{1/2}, where m_{s} and m_{d} are effective mass of electron of s-orbital and d-orbital.^{24} Therefore the interaction between electrons, which belong to same orbital do not create resistance but the interaction between electrons belong to two different bands creates resistance. As Fig. 6 shows at 0.05 V and 0.1 V the current-voltage curve of samples are not similar to each other when the LDA effects is not considered i.e., by decreasing the size of the sample the turn on voltage increases. It shows that the size quantization has effect on turn on voltage. As Fig. 7(a) shows not only the turn on voltage increases by decreasing the size of the sample but also for a critical drain-source voltage a negative resistance is seen and therefore the 1 × 1 nm^{2} sample can tolerate smaller bias voltage. Therefore it can be concluded that the effect is caused by electron-electron interaction i.e., electric field screening. In the other word, by decreasing the size of the sample and approaching to the cluster of atoms the electron-electron interaction effects shows itself more seriously and its effect is seen in current-voltage curve apparently. Here we do not consider the electron-phonon interaction. Therefore by applying the voltage, the electrons do not interact with the acoustic and optical phonons but they may interact with little and heavily screened impurities if there are any^{23} which of course here we did not consider it. It is noted that typically the effect of the interactions is important when the temperature is less than 150 K^{23} By comparing the Fig. 6(a) and Fig. 7(a) it can be concluded that for cluster of atoms (i.e., 1 × 1 nm^{2} sample) the electron-electron interaction creates a negative resistance and in consequence the device fails when the voltage is greater than 0.225 V.

## IV. SUMMARY

Electron-electron and electron-phonon interaction is important phenomena from electron transport point of view. Here we did not consider the electron phonon-interaction and electron-little and heavily screened impurities. We assumed a bias voltage is applied between source and drain of the two dimensional nano-MOSFETs with dimension equal to 1 × 1 nm^{2}, 3 × 3 nm^{2}, and 6 × 6 nm^{2} and the conductivity of their channel is controlled by applying a gate voltage. We used the NEGF method based on Liebmann's method as FDM and showed that, by increasing the length of channel the maximum of voltage moves from right (left) side to the middle side of sample and it depends on the length of the sample with and without considering the electron-electron interaction effects. It has been showed that the electron density per Cm^{3} is placed at the edge, middle and lead positions of the 1 × 1 nm^{2}, 3 × 3 nm^{3}, and 6 × 6 nm^{2} samples respectively with considering the electron-electron interaction effects. We showed that the electron-electron interaction effect shows itself seriously when the size of the sample is approached to cluster of atoms (i.e., 1 × 1 nm^{2} sample) and it causes that the sample can tolerate smaller bias voltage because of electric field screening. Therefore by decreasing the size of nano-MOSFETs, the device can tolerate lower gate voltage i.e., two dimensional nano-MOSFETs work only within the low gate voltage regime.

## REFERENCES

*Electron-Phonon Interactions in molecular electronic devices*(