ilayer GNR Mobility Model in allistic Transport Limit S. Mahdi Mousavi, M.Taghi Ahmadi, Hatef Sadeghi, and Razali Ismail Computational Nanoelectronics (CoNE) Research Group, Electrical Engineering Faculty, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor Darul Takzim, MALAYSIA Correspondence address: taghi@fke.utm.my Abstract ilayer graphene nanoribbon (GN) as a future nanoelectronic material with interesting physical properties leading to controllable band gap semiconductor by perpendicular electric field is focused in our study. A stable structure in comparison with unstable AA stack of GN is considered in a FET channel. Carrier density and temperature dependent mobility of carriers in GNs is reported. Carrier mobility model is explained based on quantum confinement effect which indicates that carriers behave like traveling wave only in channel direction, and their behavior in other two directions can be approximated by standing wave. We prove that carrier mobility in GNs is a function of temperature and carrier density which is in good agreement by experimental data. Keywords: Carrier Mobility, Transport, Conductance, ilayer graphene (GN) 1. Introduction ilayer graphene nanoribbon (GN) as a new material with interesting physical properties catches the attention of many research groups [1-]. A band gap in GNs can be created by applying a perpendicular electric field which is consequence of inversion symmetry breaking between double layers in the atomic structure. Two different stacking shapes of the GN, (AA, A) in layers result from interlayer coupling effects in low energy, which shows different band structure [3]. A stack is stable and used in graphite, whereas AA stack of GN s being unstable kept it from being investigated experimentally. The different stacking structures of GN in inter-
coupling between top and bottom layers are weaker than those of intra-coupling within layers leading to electrical conduction. The electrical conduction through the layers is shown by the carrier hopping between π orbits and, Van der Waals forces contribute in the coupling interaction between the layers structures [4-5]. Then, the transport conduction channels are essentially done through layers of GNR rather than between them [3-4]. The different configuration of A stacking in comparison to AA is due to a shift in distance of a lattice in the armchair edges [5]. Figure 1 expresses the position of the different atoms on top of each other in honeycomb lattice. In GN with two Cartesian directions less than De-roglie wavelength (λ D ) with typical value of 10 nm where carriers in y and z direction are confined while can move only in x direction [6]. Figure 1: Structure of bilayer graphene nanoribon in honeycomb lattice with width (W<<λ D ) and length (L>> λ D ) The carrier transport physics is essentially different in single and bilayer graphenes [7]. They move ballistically where the distances are very low (submicron) and can be produced by applying under an external electric field [7-9]. Carriers transport on quantum mechanical tunneling between two layers have important role of improved parabolic band dispersion with effective mass in conduction and valence bands [7]. They can be measured in field effect transistors with ernal stacking indicating a gate-induced insulating state to find a large band gap of GN [10]. In theory, carrier transports of GN have a lower amount of the minimum conductivity around the Fermi level (Dirac point), in comparison with single layer graphene nanoribbon (SGN) [7]. Hence, it showed lower amount of mobility in GN compared the SGN [7, 11]. The minimum conductance of GN is function of temperature and carrier density n. In spite of the zero carrier density close to the Dirac point, graphene reveals a minimum
conductivity and integer quantum effect on the order of e²/h [1]. This conductance value can change roughly with increasing gate voltage (Vg) linearly in momentum space. In this study, we express a computational model of mobility in GNs based on systematic exposition which is used for calculation of mobility by conductance limit effect. The Landauer s formula on the definition of conductance normalized of GN through the central sample at finite temperature is [5]: q df G( E) de. M ( E). T ( E) (1) h de Where T (E) is the transmission probability of carriers or injected electron at one channel side to the other end which is equal to one in ballistic limit, and q is the electron charge in ballistic channel, and plank constant is h. The energy band structure GN (E-k) close the Fermi level versus wave vector with tight binding method can be explained as [8] E( k) k k4 () 4 where we assume ( V t ) v, (1 V. t ) v, V and v f =10 6 m/s is the Fermi velocity and f f t + ~0.35 ev is effective interlayer hopping energy, nearest neighbor carbon-carbon atoms distance is a c-c = 1.4 o A [8]. y taking the derivatives of electron wave vector (k) over the energy E (dk/de) [8, 13-14] the number of the modes (sub bands) M(E) that are above the cut-off point at energy E in the transmission channel can be obtained as 1 de k( 4 k ) M( E) (3) L dk L The amount of parameters M(E) and de, is used in equation (1),thus conductance function of GN is obtained as 3 1 4. xk T 4 xk. T q 1 G. d GN h. L 1 exp ( E E ) k. T f Where k is oltzmann constant, this equation explains conductance with the normalized Fermi energy function f(e)=1/1+exp(e-e f )/k T, in mathematical symbols. Where E f is 1D Fermi energy when probability of occupation is half and T is the ambient temperature. The off state conductivity can be decreased by increasing channel length. Also minimum conductivity value (4)
increases when the temperature is rising which indicates the characteristic of an insulating system. The conductance model is considered to point out the mobility of GN [15-16] which relates mobility with the measurable conductance, and the carrier density.. Mobility Model of ilayer Graphene The channel length of graphene nanoribbons in quasi-1d structures with narrow widths (<~10 nm) are predicted to display a high carrier mobility in ballistic transport [17-18]. In oltzmann point the intrinsic mobility of a graphene layer calculated where the low-field Coulomb scattering time ( ) between collisions is considered for a charge and also the effective mass (m*) approximation of the carriers in the room temperature is used [19]. q. (5) m * The quasi one dimensional (1D) intrinsic mobility at low field and short channel is a function of electron conductivity and carrier density [9]. (6) n. q Where is the electrical conductivity as a function of conductance and channel length GL. which is given in units of 1.cm in one dimensional systems and the unit of µ is Cm /V.s particularly [9, 0-]. The high carrier density n=7.1*10 1 cm - per unit length is one of the most important advantages of a bilayer graphene device that can be controlled effectively through a gate voltage by applying an external electric field between the layers [3-4]. 3 1 4. xk T 4 xk. T q 1. GN d hn. x 1 e To simplify the mobility equation the derivation of Fermi Dirac distribution function is employed also the limit of integral is changed accordingly thus the mobility is: (7)
q vg 1 3 4 x k T GN n. h vg V q 4 vg 1 exp( x ) 4x k T 4 x k T k T 8 4 x k T (8) Where x= (E-Δ)/k T, although this integral can be assessed numerically when the normalize Fermi energy is defined as η= (E f -Δ)/k.T. General mathematical model of GN carrier mobility is depends on carrier density which can be solved numerically as shown in Fig where interlayer baised voltage is V, vg is gate voltage. dx Figure : mobilility model for GN with T=300K in Dirac point. Fig. explains the values of mobility magnitude which will be increased by increasing the gate voltage. At low field effect, because of filling the empty bands reduces the phase space for the scattering therefore mobility will be increased. Mobility on first band at low field vanishes with increasing the gate voltage independent on the occupation of the second band. Thus, field effect mobility according to Fig. indicates a peak shifts linearly for low voltage similar to the conventional MOSFET. This values decrease rapidly when the band filling emerge at the second band with higher charge density. The gate voltage is changeable in device operation but the effect of voltage variation on mobility in nanoribbon transistors especially at high fields need to be explored.
Figure 3: Comparison of carrier ballistic mobilities of a GN model (red spots) with experimental data (blue spots) at T=300 o K for different voltage gates [5]. The experimental data extracted from conductance that presented by Oostinga, et al 008 and Zou et al 011, [6][7]. Temperature effect on carrier mobility of bilayer graphene is presented by the Coulomb impurity scattering within the random phase approximation method [6]. As shown in figure 4 this model explains the relationship between temperature and the magnitude of mobility in corporation with voltage that indicates incremental effect on mobility in different temperatures (88K, 60K and 340K).
Figure 4: Carrier mobility of GNs as a function of gate voltage plotted at various temperatures at the charge neutrality point. Significant shift on the mobility model at Dirac point illustrates effect of carrier mobility reduction in GN [9, 7]. 3. Conclusion ilayer graphene nanoribon with stable A structure can be used in a FET channel. Carrier mobility model based on the quantum confinement effect indicates that GNs can be assumed as a one dimensional device in a FET channel. In this study the carrier density and temperature effect on mobility of carriers in GNs is explored. Also variation of gate voltage during the device operation and its effect on mobility is discussed and numerical mobility model is presented based on channel conductance. The comparison between presented model and published experimental data is reported which points out acceptable agreement between them. Acknowledgment The authors would like to acknowledge the financial support from Research University grant of the Ministry of Higher Education (MOHE), Malaysia. Also thanks to the Research Management Centre (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.
References [1] K. S. Novoselov, et al., "Electronic properties of graphene," Physica Status Solidi (), vol. 44, pp. 4106-4111, 007. [] K. S. Novoselov, et al., "Unconventional quantum Hall effect and erry s phase of," Nature Physics, vol., pp. 177-180, 006. [3] Y. Xu, "Infrared and Raman spectra of AA-stacking bilayer graphene," Nanotechnology vol. 1, p. 6pp, 010. [4] S. J. Sun and C. P. Chang, "allistic transport in bilayer nano-graphite ribbons under gate and magnetic fields," European Physical Journal, vol. 64, pp. 49-55, 008. [5] N. X. a. J. W. Ding, "Conductance growth in metallic bilayer graphene nanoribbons with disorder and contact scattering," Journal of Physics D-Applied Physics, vol. 0, pp. 48513 doi: 10.1088/0953-8984/0/48/48513, 008 [6] M. T. Ahmadi, "Graphene Nanoribbon Conductance Model in Parabolic and Structure," Journal of Nanomaterials, vol. doi:10.1155/010/753738, p. 4, 010. [7] e. a. Das Sarma, "Theory of carrier transport in bilayer graphene," Physical Review, vol. 81, p. 161407, 010. [8] Castro, "Electronic properties of a biased graphene bilayer," Journal of Physics-Condensed Matter, vol., p. 175503, 010. [9] S. V. Morozov, et al., "Giant intrinsic carrier mobilities in graphene and its bilayer," Physical Review Letters, vol. 100, p. 01660, 008. [10] F. N. Xia, et al., "Graphene Field-Effect Transistors with High On/Off Current Ratio and Large Transport and Gap at Room Temperature," Nano Letters, vol. 10, pp. 715-718, 010. [11] K. Nagashio, et al., "Mobility Variations in Mono- and Multi-Layer Graphene Films," Applied Physics Express, vol., p. 05003, 009. [1] D. Dragoman, "Low-energy conductivity of single- and double-layer graphene from the uncertainty principle," Physica Scripta, vol. 81, 010. [13] S. Russo, "Electronic transport properties of few-layer graphene materials," Graphene Times vol. arxiv:1105.1479v1 [cond-mat.mes-hall] 011. [14] S. M. Mousavi, "ilayer Graphene Nanoribbon Carrier Statistic in Degenerate and Non Degenerate Limit," Journal of Computational and Theoretical Nanosceince, p. In Press, 011. [15] Sadeghi, "allistic Conductance Model of ilayer Graphene Nanoribbon (GN) " Journal of Computational and Theoretical Nanoscience, p. In Press, 011. [16] Gnani et al., ""Effective Mobility in Nanowire FETs Under Quasi-allistic Conditions."," IEEE Transactions on Electron Devices, vol. 57(1), pp. 336-344 010. [17] X. L. Li, et al., "Chemically derived, ultrasmooth graphene nanoribbon semiconductors," Science, vol. 319, pp. 19-13, 008. [18] A. etti, et al., "Physical insights on graphene nanoribbon mobility through atomistic simulations," 009 Ieee International Electron Devices Meeting, vol. 10.1109/ IEDM.009. 54476, pp. 837-840, 009. [19] Kechao Tang, "Electric Field Induced Energy Gap in Few Layer Graphene," The Journal of PHYSICAL CHEMISTRY C, vol. dx.doi.org/10.101/jp01761p,115, pp. 9458-9464, 011. [0] M. A. Aziziah Amin, Johari, Seyed Mahdi Mousavi, and Razali Ismail, "Effective mobility model of graphene nanoribbon in parabolic band energy," Modern Physics Letters,In press, 011. [1] X. Du, "Suspended graphene: a bridge to the Dirac point," Nature Nanotechnology, vol. 3, pp. 491-495, (008). [] Z.-M. Liao, "Current regulation of universal conductance fluctuations in bilayer graphene," New Journal of Physics vol. 1, p. 083016 (9pp), 010. [3] H. M. Dong, et al., "Optical conductance and transmission in bilayer graphene," Journal of Applied Physics, vol. 106, pp. 043103-043103-6 009. [4] Y. L. Mao, et al., "First-principles study of the doping effects in bilayer graphene," New Journal of Physics, vol. 1, 010. [5] J.. Oostinga, et al., "Gate-induced insulating state in bilayer graphene devices," Nature Materials, vol. 7, pp. 151-157, 008. [6] M. Lv, "Screening-Induced Transport at Finite Temperature in ilayer Graphene," Phys. Rev., vol. 81, p. 195409, 009. [7] X. H. K. Zou, "Electron-electron interaction and electron-hole asymmetry in bilayer graphene," arxiv:1103.1663v1 [cond-mat.str-el] 011.