Paolo Marconcini, Alessandro Cresti, Francois Triozon, Blanca Biel, Yann-Michel Niquet, Demetrio Logoteta, Stephan Roche, "Effect of Boron Doping on the Characteristics of Graphene FETs", Latest Trends in Circuits, Automatic Control and Signal Processing. Proceedings of the 3rd International Conference on Circuits, Systems, Control, Signals (CSCS '12), Barcelona, Spain, 17-19 October 2012, Recent Advances in Electrical Engineering Series 8, 254 (2012).

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Effect of boron doping on the characteristics of graphene FETs PAOLO MARCONCINI Dipartimento di Ingegneria dell’Informazione Universit`a di Pisa Via Caruso 16, I-56122 Pisa ITALY [email protected]

ALESSANDRO CRESTI IMEP-LAHC, UMR 5130, Grenoble INP/UJF/CNRS Universit´e de Savoie, Minatec 3 Parvis Louis N´eel, F-38016 Grenoble FRANCE [email protected]

FRANC ¸ OIS TRIOZON CEA LETI-MINATEC 17 rue des Martyrs, F-38054 Grenoble FRANCE [email protected]

BLANCA BIEL Dpto. Electr´onica y Tecnolog´ıa de Computadores Facultad de Ciencias, Universidad de Granada Campus de Fuente Nueva E-18002 Granada SPAIN [email protected]

YANN-MICHEL NIQUET CEA, INAC, SP2M, L Sim 17 avenue des Martyrs, F-38054 Grenoble FRANCE [email protected]

DEMETRIO LOGOTETA Dipartimento di Ingegneria dell’Informazione Universit`a di Pisa Via Caruso 16, I-56122 Pisa ITALY [email protected]

STEPHAN ROCHE Catalan Institute of Nanotechnology Universitat Aut´onoma de Barcelona Campus UAB, E-08193 Bellaterra (Barcelona) SPAIN [email protected] Abstract: We describe a numerical method which allows to self-consistently simulate the effect of boron doping in graphene-based field-effect-transistors, using a tight-binding description with a proper distribution of fixed charges. Using this simulation technique, we show, in the case of field-effect transistors based on narrow graphene nanoribbons, that low boron doping concentrations generate a clear electron-hole asymmetry in the transfer characteristics of the device. Key–Words: graphene FETS, boron doping, unipolar characteristics, tight-binding

1

Introduction

the other hand, graphene has properties, as high optical transparency, mechanical flexibility and thermal conductivity, which make it a very promising candidate for various practical applications [8]. From an electronic point of view, it represents a very interesting material, due to its high mobility and single-atom thickness (which allows a better control of the gate voltage on the channel and thus a scaling of the device dimensions, if a proper high-κ material is used as a dielectric, in order to limit the gate leakage current). However, unconfined graphene has a null energy gap (since its π energy bands touch in correspondence of the so-called Dirac points) and has an am-

After the quite recent experimental isolation, by mechanical exfoliation [1], of graphene (a single layer of carbon atoms arranged in a hexagonal lattice [2, 3]), a large research activity has developed around this new material and its very interesting properties [4]. On one hand, due to its particular lattice structure, electrons in graphene obey the Dirac-Weyl equation [5], that represents also the relativistic wave equation of massless fermions. Therefore, graphene represents an ideal material to test relativistic effects (as Klein tunnelling or Zitterbewegung) at non-relativistic speeds [6, 7]. On

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bipolar behavior. These two aspects make it difficult to use graphene in digital electronics. Indeed, high ratios Ion /Iof f between the current flowing through the device in the conducting and in the nonconducting state are possible only if an energy gap exists, which limits the interband tunneling between the conduction and the valence bands in the off state. Moreover, in order to develop complementary logic circuits it is necessary to have an unipolar electrical behavior. Several techniques have been proposed to introduce an energy gap into graphene [9]. A possibility is to transversally confine the structure, obtaining a graphene nanoribbon, the energy gap of which increases, on the average, when the ribbon width decreases. However, very narrow ribbons would be necessary to obtain an energy gap sufficient for logic applications. Moreover, the value of the gap strongly depends on the atomic detail of the edges. For example, in the case (that we will consider in this paper) of armchair nanoribbons (ribbons characterized by armchairshaped edges), the value of the gap strongly depends on whether the number of dimer lines across its width is a multiple of 3, a multiple of 3 plus 1, or a multiple of 3 minus 1 (in this last case the nanoribbon is quasimetallic and would be exactly metallic if edge distortion was disregarded). Since present lithographic techniques do not allow a so accurate control of the ribbon dimensions, lateral confinement is not sufficient to obtain the desidered values for the energy gap. One of the methods that have been proposed to increase the Ion /Iof f ratio and to introduce an asymmetry in the transfer characteristics of graphene-based field-effect transistors (FETs) is to dope the nanoribbon with boron or nitrogen. The few experimental results reported in the literature [10–13] do not give a definitive answer about the potentiality of the method, due to their discrepancy. In this paper, we describe the numerical method that we have adopted to simulate the electrical properties of transistors based on boron-doped graphene nanoribbons and we report the results that we have obtained for low boron concentrations (up to 0.6%). In particular we show that for these low values of doping, while obtaining only a small increase in the Ion /Iof f ratio, a clear asymmetry appears in the transfer characteristics, due to the effect of the localized states introduced by the boron impurities.

2

the hopping and onsite energy parameters of a nearestneighbor TB model were fitted in such a way to recover the energy levels of the fully occupied π molecular orbitals of boron-doped graphene clusters of various sizes, computed through density functional theory (DFT) simulations. Instead, in Ref. [15] the results of self-consistent DFT transport calculations on isolated doped nanoribbons were reproduced with a TB model in which the onsite energy was directly extracted from the ab-initio calculations and the only fitting parameter was the boron onsite energy. However, in both cases, since the TB model was developed to reproduce the DFT results, it already included the DFT self-consistency. Therefore those TB schemes can not be directly used inside a selfconsistent Poisson-Schr¨odinger calculation (which is needed for the study of the complete field-effect transistor), otherwise overscreened results are obtained. Here we have solved this problem using a different approach [16, 17]. We have used a nearest neighbor TB model including the 2 pz atomic orbitals, where the onsite energy of carbon atoms is considered equal to 0 and the hopping parameter between nearest neighbor atoms equal to -2.7 eV (apart from for the edge dimers, where it is multiplied by a factor 1.12 to include the effect of edge distortion). These represent the values commonly used for undoped graphene [18]. The boron energy is left as a fitting parameter. Moreover we include in our schematization a proper distribution of fixed charges. In detail, we consider as fixed charge in correspondence of each atom the total charge of the nucleus, of the core electrons, and of the three sp2 hybridized valence orbitals. The charge of the nucleus and of the core electrons is equal to +4 e for carbon atoms and to +3 e for boron atoms. If we add the charge −3 e of the sp2 hybridized valence orbitals, we obtain that the fixed charge to consider is +e for carbon atoms and 0 for boron atoms. This TB schematization and this distribution of fixed charged are used inside the NanoTCAD ViDES code, which solves self-consistently the Poisson and Schr¨odinger equation (with open boundary conditions), and therefore gives a proper description of screening effects. Inside the calculation, the electrostatic effect of the fixed charge will sum with that of the electrons in the 2 pz atomic orbitals (which are directly considered by the TB model). In order to fit the value of the boron onsite energy and to validate this schematization, we have compared the impurity potential and the transmission obtained, for isolated graphene sheets and graphene nanoribbons with a single boron atom in a substitutional position, with the method we have just described and with a DFT calculation (performed through the SIESTA software). We have found that, considering a null

Tight-binding description

We have first developed a tight-binding (TB) description able to reproduce ab initio results in the case of isolated boron-doped graphene nanoribbons. A previous attempt was made in Ref. [14], where

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3

5

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2 1.5 1

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−0.4

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Figure 2: Potential introduced by a boron impurity in an isolated graphene sheet, obtained with a DFT calculation (empty squares) and with the self-consistent TB model (solid dots). In the DFT simulation a double-ζ basis set has been used.

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2 1.5 1 0.5 0 −0.6

gate −0.4

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Figure 1: Transmission of an isolated armchair nanoribbon (with a number of dimer lines across its width equal to 35 in panel (a) and to 32 in panel (b)) and with a single boron atom in substitutional posi˚ (bold lines) and at 19.68 A ˚ (thin tion located at 1.23 A lines) from the ribbon edge. Dashed lines have been obtained with a DFT simulation, while solid lines with the self-consistent TB calculation.

source

t ox =1 nm

x gate y

z

Figure 3: Sketch of the simulated graphene-based FET. bon. In particular, the considered channel is a 20 nm long armchair nanoribbon with 32 dimer lines across its width, with Schottky contacts at the source and drain ends. The two gates are separated from the ribbon by a 1 nm thick region of silicon dioxide (see Fig. 3). Since 32 is a multiple of 3 minus 1 the undoped nanoribbon is a quasi-metallic one. In Fig. 4 we show the transfer characteristics obtained for an undoped ribbon and for a 0.3% and a 0.6% atomic concentration of randomly distributed boron dopants (at the moment, numerical convergence problems have made impossible for us to consider higher boron doping concentrations). The considered drain-source voltage is 0.1 V. The calculations have been repeated for about 20 spatial distributions of the dopants for each value of the doping concentration. The curves in the figure are obtained averaging these results, while the error bars represent their standard deviation. Despite the large dispersion (due to the small number of dopants), two main results can be observed. The low boron doping concentrations that we have considered determine a small increase in the

Simulation of a graphene FET with boron doping

After validating the method, we have used it to obtain the transfer characteristics of a double-gate field effect transistor based on a boron-doped graphene nanorib-

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drain

L

SiO 2

boron onsite energy, our self-consistent TB calculation correctly reproduces the DFT results for all the boron positions and nanoribbon widths that we have examined. In particular, in Fig. 1 we show the comparison between the results obtained with the two methods for the transmission of an isolated armchair nanoribbon with a number of dimer lines across its width equal to 35 in panel (a) (4.18 nm wide ribbon) and to 32 in panel (b) (3.81 nm wide ribbon), with a ˚ and at 19.68 A ˚ from the ribboron atom at 1.23 A bon edge. In Fig. 2 instead the impurity potential in a graphene sheet, computed with the two methods, is shown: as can be noticed, the boron atom, which behaves as an electron acceptor, introduces a repulsive potential.

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SiO 2

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12 V DS =0.1 V

0%

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I D ( μ A)

10 0%

8 6 4

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2 0 −1

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0.5

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−1

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Figure 5: Transmission obtained using an envelopefunction method (including only the electrostatic effect of boron impurities) for a nanoribbon with 32 dimer lines along its width and for 3 boron concentrations.

Figure 4: Transfer characteristics for the considered FET with VDS = 0.1 V, for a 0%, 0.3%, and 0.6% boron doping concentration in the graphene channel. For the last two cases, the error bars represent the standard deviation of the results obtained considering different spatial distributions of the dopant atoms for each value of the doping concentration.

static effect of boron doping replicating the screened potential corresponding to each boron atom along the nanowire. This continuum model includes the electrostatic effect of boron impurities, but not the backscattering effect of the quasi-localized states, which derive from atomistic details not considered in this schematization. In Fig. 5 we show the transmission obtained using this model for different boron concentrations. As we can see, if only the electrostatic effect of boron atoms is considered, the electron branch is suppressed more than the hole one.

Ion /Iof f ratio with respect to that in the absence of doping, still insufficient for logic applications. However a clear asymmetry and a mobility gap appear in the transfer characteristics. In particular, as could be observed also from the DFT and TB results reported in Fig. 1, there is a clear suppression of the hole branch with respect to the electron branch. This is due to the presence of quasi-bound states around the impurities that give rise to anti-resonances in the transmission coefficients in correspondence of negative energies, as previously observed in carbon nanotubes [19]. Our results show the importance, in the explored doping concentration range and for the narrow nanoribbons we have considered, of the hole backscattering introduced by the states localized around the impurities. However, further analysis would be necessary to investigate the transport behavior for different boron concentrations and for larger graphene structures, where quasi-bound states could play a smaller role. In that case also the effect of electrostatic repulsion between the negatively charged boron impurities and the electrons moving inside the ribbon could significantly impact transmission. In order to simulate this last effect on the trasmission of a graphene nanowire, we have used a numerical code based on a envelope-function approach [5] (analogous to the effective mass approximation commonly used for the study of semiconductor nanostructures [20–24]), in which the Dirac equation is solved properly converting the Dirichlet boundary conditions into periodic boundary conditions [25] and working in the reciprocal space [26]. We have considered an isolated graphene nanowire with 32 dimer lines along its width and we have approximately reproduced the electro-

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4

Conclusion

We have developed a numerical method to correctly reproduce the DFT results through a self-consistent simulation based on a TB schematization with a proper distribution of fixed charges. We have adopted this method to study the effect of low boron doping concentrations (up to 0.6%) on the transfer characteristics of a FET based on a graphene nanoribbon. The results show the onset of an unipolar behavior in the transfer characteristics of the device, while the increase of the Ion /Iof f ratio is still insufficient for logic applications. Further numerical work is however necessary to numerically simulate the behavior of ribbons with larger sizes and higher boron concentrations. Acknowledgements: We thank Dr. Gianluca Fiori for his important contribution to the research activity reported in this article Support from the European Union under the contract No. 215752 GRAND (GRAphene-based Nanoelectronic Devices) is gratefully acknowledged.

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