NANO: Brief Reports and Reviews Vol. 9, No. 3 (2014) 1450032 (8 pages) © World Scienti¯c Publishing Company DOI: 10.1142/S1793292014500325
GRAPHYNE-BASED SINGLE ELECTRON TRANSISTOR: AB INITIO ANALYSIS J. V. N. SARMA* Center of Nanotechnology Indian Institute of Technology Roorkee, Roorkee 247667, India MEMS & Microsensors Group, Solid State Devices Division CSIR-Central Electronics Engineering Research Institute Pilani 333031, India NANO 2014.09. Downloaded from www.worldscientific.com by 191.103.40.69 on 06/22/14. For personal use only.
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RAJIB CHOWDHURY Department of Civil Engineering, and Center of Nanotechnology Indian Institute of Technology Roorkee, Roorkee 247667, India
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R. JAYAGANTHAN Department of Metallurgical and Materials Engineering and Center of Nanotechnology Indian Institute of Technology Roorkee Roorkee 247667, India
[email protected]
Received 14 October 2013 Accepted 10 December 2013 Published 29 January 2014
The application of graphyne for a single-electron transistor (SET) that is operating in the Coulomb blockade regime is investigated in the ¯rst principles framework. Density functional theory modeling for graphyne has been used and the device environment has been described by a continuum model. The interaction between graphyne and the SET environment is treated with self-consistent Poisson equations. The charging energy as a function of gate voltage thus calculated has been used to obtain the charge stability diagram for the present system. The e®ect of electrode separation and the position of the molecule with respect to the dielectric on the gate coupling have been studied further. As compared with the previously studied systems on this line, graphyne has been observed to provide the gate coupling that is nearly close to that of benzene and graphene, but signi¯cantly greater than fullerene-based systems. Keywords: Ab initio; Coulomb blockade; graphyne; single electron transistor.
*Corresponding
author. 1450032-1
J. V. N. Sarma, R. Chowdhury & R. Jayaganthan
1. Introduction
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1
Graphyne was predicted by Baughman et al. as one of the new carbon allotropes consisting of layered planar sheets of equally occupied sp and sp 2 carbon atoms. The layered structures can be constructed by replacing one-third of the carbon–carbon bonds in graphite with acetylenic (C C) linkages. This material was shown to be a semiconductor with an energy bandgap of 1.2 eV with high-temperature stability and graphite-like mechanical properties. Challenged by the ever increasing demand of miniaturization in the electronics industry, molecular electronics, on the other hand, addresses a promising answer, mainly because of the precise reproducibility and versatile chemical tailoring of its basic components.2 With the advent of highly reproducible and stable fabrication techniques such as electromigration and break junction for the fabrication of metal electrodes with a nanogap separation,3,4 electron transport through single molecules was achieved in the framework of three terminal devices. The nature of transport in such metal– molecule–metal junctions has been examined by employing the molecules p-benzenedithiol, xylenedithiol,5,6 and thiolated alkanes,7–11 as the prototypical cases. Recently, graphene has been evolved as a new candidate in this context12–15 for its electron conduction and magnetic properties. Zhou et al.,16 using ¯rst-principles calculations, have studied the electronic structure of graphyne sheet and observed that its bandgap can be tuned by changing the size of hexagonal ring and the length of carbon chain. This recently synthesized17,18 new allotrope of carbon, graphyne, is also predicted to be one of the competitors to graphene in nanoelectronics applications.19 The studies employing graphyne molecule in the single electron transistor (SET) operation were found to be rare, in spite of its useful electronic applications. The acetylenic linkages present in the structure of graphyne possesses various geometries that result in various forms of graphyne i.e., alpha-graphyne, betagraphyne, gamma-graphyne, and ð6; 6; 12Þ-graphyne. In this work, gamma-graphyne molecule has been investigated and modeled in a SET environment and the corresponding total energy and charge stability diagrams are predicted for this system. The quantum con¯nement e®ect leads to tunneling of electrons through thin potential barrier in a device con¯guration that is referred to as SET.20,21
Although the use of nonequilibrium Greens functions (NEGF) together with density functional theory (DFT) or semiempirical models were highly successful in modeling coherent transport in various types of molecular junctions, the transport in SETs is incoherent. The existence of theoretical or experimental studies employing graphyne for molecular SET are very limited, although some studies do exist on graphene-based SETs. We introduce here the ab initio framework of graphyne molecule that is weakly coupled to gold electrodes forming an SET con¯guration operating in the Coulomb blockade regime.22 The molecular SET geometry model has been used to calculate the charging energy of graphyne in an electrostatic environment. The charging energy as a function of gate potential has been calculated and from this the charge stability diagram for graphyne that maps out the di®erential conductance as a function of gate and source–drain voltage has been obtained. The signi¯cance of including renormalization23 of the molecular charge states due to the polarization of the SET environment has been demonstrated, in particular.
2. Modeling and Simulation Within the Trans-SIESTA framework24–27 that is implemented in the Atomistix Toolkit,28 the DFTbased NEGF formalism has been used for the calculation. The DFT model applied here is based on the pseudopotentials with numerical localized basis ðrÞ is functions, where a compensation charge comp i introduced for each atom site. The compensation charge has the same charge Zi as the pseudopotential, and helps in screening the electrostatic interactions. The total energy functional has been extended to include interactions with the dielectric and metallic regions present in the system. Represented in Fig. 1 is a typical SET con¯guration, where graphyne molecule is positioned at a distance of 6 A o on top of a 3.7 A o thick dielectric material with a metal back-gate and surrounded by metallic source-drain electrodes that are 19 A o apart from each other, with graphyne being at the center. The potential is ¯xed to the applied voltage on each of the electrodes and Neumann boundary conditions are applied on the faces of the supercell to nullify the perpendicular component of the electric ¯eld. To calculate the energies, the charging energies of the isolated graphyne molecule were
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Graphyne-Based SET: Ab Initio Analysis
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(a)
(b)
Fig. 1. (a) Illustration of graphyne molecule in the presence of SET environment, where the top view of the graphyne molecule and the charge transfer mechanism could also be seen; (b) SET con¯guration with graphyne molecule that is 6 A from the bottom and above the 3.7 A thick dielectric. Metallic electrodes are 19 A apart from each other, graphyne being at the center. The contour plot shows the induced electrostatic potential for a gate voltage of 2 V and zero source-drain bias (color online).
obtained ¯rst by performing self-consistent calculations for N and N þ 1 charge states and subtracting their total energies. We employed DFT in the spin polarized local density approximation29 with PZ functional and expanding the wavefunctions in a double- polarized basis set. The k-point sampling used is 12 12 12 with a density mesh cut-o® of 75 Hartree. PulayMixer algorithm with Hamiltonian variables has been used for the iteration control.
3. Results and Discussion The calculations are started with isolated benzene molecule to validate our adopted modeling and methodology. The charging energies are calculated for the isolated case and in the SET environment. We obtained EI EA ¼ 11:46 eV and 7.72 eV, respectively in these cases. These values are in excellent agreement with the available results.30 We then considered the analysis of the graphyne molecule. The ionization energy (EI ) and the a±nity energy (EA ) of the isolated phase of graphyne are obtained to be 7.72 eV and 1.43 eV, respectively. The calculations are repeated for graphyne in the SET environment and the values of EI and EA are found to be 6.65 eV and 4.39 eV, respectively. Note the decrease in these values in the SET environment as compared to the isolated ones, indicate the e®ect of applied electrostatic potential on the molecule. The calculated charging energies for
the system along with the ¯tting parameters are summarized in Table 1 and compared to the calculations on benzene, C60 and graphene-based SETs from the literature. The reductions in the ionization and a±nity energies of the graphyne molecule in the SET environment as that of the isolated molecule are indicative of the e®ect of polarization charge induced by the charges on the molecule in the presence of SET con¯guration. The molecular charge states are shifted typically by an amount of polarization energy.31 This explains the signi¯cance of polarization e®ects on the addition energies due to renormalization of the molecular charge states in the device con¯guration. Graphyne-based system can be seen to be exhibiting gate coupling that is nearly close to benzene and graphene-based systems, whereas signi¯cantly higher than the one of fullerene-based systems. Further, the ground-state ionization energy for the graphyne molecule in the SET environment can be seen from Table 1 to be signi¯cantly higher than those of graphene and fullerenes and also the corresponding a±nity energy is signi¯cantly lower. This di®erence enhances or reduces the sequential tunneling mechanism, whereby the electron localizes at the molecule before subsequent tunneling into or away from each of both the electrodes i.e., source and drain. However, this could be controlled by optimizing the geometry parameters and the bias voltage. Therefore, graphyne is signi¯cant for
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J. V. N. Sarma, R. Chowdhury & R. Jayaganthan Table 1. Ionization ðIÞ and a±nity ðAÞ energies for graphyne molecule along with the ¯tting parameters and its comparison with the calculated values for benzene, C60 and graphene-based systems. System Benzene a Benzene b Fullerene a Fullerene b Graphene a Graphene b Graphyne a Graphyne b
ðeV 1 Þ
E Iþ1 ðeVÞ
EI ðeVÞ
EA ðeVÞ
E A1 ðeVÞ
Refs.
— 0.62 — 0.38 — 0.64 — 0.51
— 0.003 — 0.025 — 0.003 — 0.012
15.73 7.70 10.09 7.24 — 7.70 10.31 9.003
9.15 5.41 6.84 5.89 6.58 5.97 7.72 6.65
2.34 2.26 1.90 2.85 3.71 4.24 1.43 4.39
8.39 4.88 1.32 1.53 — 2.51 1.16 2.15
26 26 26 26 13 13 — —
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Note: a Isolated; b SET.
potential application for SET devices. Figure 1(a) illustrates the top view of the gamma-graphyne molecule that is positioned between the electrodes and a few angstroms below the molecule is the dielectric on a metallic gate. Under the applied voltage of 2 V, the electrostatic potential distribution is as represented in Fig. 1(b). The total energy for the di®erent charge states of the SET system has been calculated as a function of the gate voltage as shown in Fig. 2. The total energy includes the reservoir energy pW, where p being the charge on graphyne molecule, and W (5.28 eV) is the work function for the gold electrode. The state with lowest energy is the stable charge state. It is interesting to observe that, the neutral graphyne has the lowest energy at the zero gate potential and the positive charge states are stabilized at negative gate voltages and the negative charge states at the positive gate voltages, in agreement with the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) levels following eVG . Thus, at positive bias, the LUMO level gets dominated by the electrode Fermi level and attracts an electron, and graphyne becomes negatively charged. At negative bias, the HOMO level gets above the electrode Fermi level and an electron is removed out from graphyne, making it positively charged. The gate voltage dependence is almost linear and the slope is related to the charge states of the graphyne. The data have been ¯tted into a quadratic form to understand the dependence: E ¼ qVG þ ðeVG Þ 2 :
ð1Þ
Here the linear term is assumed to be proportional to the charge q on graphyne; whereas the quadratic
term comes from the polarization of the graphyne and so is independent of q. By ¯tting the data in Fig. 2, we ¯nd for graphyne, ¼ 0:5064 and ¼ 0:012 eV 1 . The gate coupling parameter must be as large as possible in order to access as many charge states as possible. This value typically depends on the geometry and can be increased by the appropriate optimization of the graphyne-electrode separation. Graphyne shows an almost linear relationship between total energy and the gate voltage, as all atoms are shifted equally by the potential. This fact is illustrated by the induced potential diagram in Fig. 1(b). The contour plot shows that, the potential is almost constant at the atomic sites, which is 1:1 eV. Dividing with the gate potential 2 eV, we ¯nd ¼ 0:55, in good agreement with the above quadratic ¯t. The electronic transport from
Fig. 2. The total energy as a function of gate voltage for graphyne in the SET environment. Di®erent curves are for di®erent charge states: blue (2), green (1), red (0), turquoise (1) and violet (2) (color online).
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Graphyne-Based SET: Ab Initio Analysis
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source to molecule and molecule to drain should separately meet the following criterion: E s ðnÞ E s ðn 1Þ E m ðp þ 1Þ E m ðpÞ;
ð2Þ
E m ðp þ 1Þ E m ðpÞ E d ðq þ 1Þ E d ðqÞ;
ð3Þ
where E s ðnÞ, E m ðpÞ and E d ðqÞ represent the total energy functions for source, molecule and drain, respectively. The incoherent transport sets the limit on the energy in the source and the drain. For zero Fermi energy level and taking into account the work function (W ¼ 5:8 eV) for gold electrode, the energy maximum of an electron in source would be W þ eV=2, whereas the energy minimum of an electron in drain is W eV=2. Thus,
Fig. 3. The charge stability diagram at the zero voltage. The colors show the number of charge states in the applied voltage range for a given gate voltage. The color map is blue (2), light blue (1), red (0), green (1) and yellow (2). The onset of transport through excited states is given by the diagonal lines coming from the diamonds, while the edges of diamond-shaped regions give the onset of current (color online).
W þ eV=2 E s ðnÞ E s ðn 1Þ;
ð4Þ
E d ðq þ 1Þ E d ðqÞ W eV=2;
ð5Þ
(2)–(5) taken together yields the desired criterion as: eV=2 E m þ W eV=2;
one electrode to another via molecule is described by the charge stability diagram for SET. Theoretical energy considerations based on the molecular energy level diagrams can provide us the criterion for charge transfer.27 Assume n, p and q as the number of electrons on the source, molecule and drain, respectively. Then, the charge transport from
ð6Þ
where E m ¼ E m ðp þ 1Þ E m ðpÞ gives the charging energy with p electrons in the graphyne molecule. The corresponding charge stability diagram predicted from (6) is represented in Fig. 3, where di®erent colors shows di®erent number of charge states within the applied bias range.
(a) Fig. 4. The charge stability diagram at the zero voltage for the electrode-to-electrode separation of 18 A in (a) and that of 20 A in (b), respectively, along with the total energy curves. The colors show the number of charge states in the applied voltage range for a given gate voltage. The color map is blue (2), light blue (1), red (0), green (1), and yellow (2). It can be seen that, the bias window decreases as the separation is increased (color online). 1450032-5
J. V. N. Sarma, R. Chowdhury & R. Jayaganthan
(b)
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Fig. 4.
Note that the nonlinear dependence of the total energy on the gate voltage is not observed in the charge stability diagram, due to the fact that, the charge stability diagram depends only on the energy di®erence between the charge states as predicted by (6) and thus the second-order term in (1) is independent of the charge state. To further understand the e®ect of position of the graphyne molecule on the transport properties as well as on the gate coupling, the computations are repeated for di®erent locations of the molecule with respect to dielectric and the electrodes. The corresponding total energies and the charge stability diagrams obtained are represented in Figs. 4(a) and 4(b) for the electrode-to-electrode separation of 18 A and 20 A , respectively, graphyne molecule being situated at the center. It can be seen that, the bias window decreases as the electrode separation is increased, as opposed to the case of dielectric separation, where the bias window increases. The charge stability diagram at the zero voltage is given in Fig. 4(a) for the graphyne molecule location with respect to dielectric at 5.5 A and that of 6.5 A in Fig. 4(b), respectively, along with the total energy curves. The gate coupling constant that is obtained by changing the separation of electrodes indicates an almost linearly increasing behavior which is shown in Fig. 5, whereas Fig. 6 indicates the linearly decreasing response as the molecule is farther from the dielectric substance. The increase in gate coupling and the decrease in bias window as the electrode separation is increased are consistent with the
(Continued )
prediction of (1), showing the linear dependence with respect to the total energy. Similar arguments also apply well for the case of variation of separation of the graphyne molecule with respect to the dielectric. These results are in good agreement with the results studied by Kaasbjerg and Flensberg31 for the case of organic molecules placed in the SET environment and the predictions of Dutta et al.32 based on the ¯nite element electrostatic simulation calculations. This suggests that, the electrode separation could be higher while the molecule is close
Fig. 5. The variation of gate coupling constant as a function of the electrode-to-electrode separation, graphyne molecule being at the center. The gate coupling increases almost linearly with the electrode separation (color online).
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Graphyne-Based SET: Ab Initio Analysis
Computer Center at IIT-R, for the computational facilities.
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References
Fig. 6. The variation of gate coupling constant as a function of the molecule-dielectric separation, graphyne molecule being at the center. The gate coupling decreases almost linearly with the electrode separation (color online).
enough to the dielectric, in order that the device to be operated in strong coupling region and vice versa to achieve the weak coupling condition.
4. Conclusions The application of graphyne molecule has been modeled for a SET operating in the Coulomb blockade regime in the ab initio framework. We have calculated the charging energy of graphyne molecule in the SET environment by applying DFT. The charging energy as a function of gate voltage and the corresponding charge stability diagram are obtained, which shows that graphyne has the lowest energy at the zero gate potential and the positive charge states are stabilized at negative gate voltages and vice versa. The signi¯cance of renormalization of the molecular charge states due to the polarization of the SET environment has been demonstrated in particular. The studied system also shows that, graphyne exhibits the gate coupling that is nearly close to the benzene and graphene-based systems and signi¯cantly greater than one with the fullerenes.
Acknowledgments JVN gratefully acknowledges the support from Dr. Chandra Shekhar, Director, CSIR-CEERI, Pilani, India, by providing the study leave to carry out this work at IIT-R, and thankful to the Institute
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