Graphene Nanoribbon Fermi Energy Model in Parabolic Band Structure

Mohammad Taghi Ahmadi Faculty of Electrical Engineering

Universiti Teknologi Malaysia Skudai, Malaysia

ahmadiph@gmail.com

Razali Ismail Faculty of Electrical Engineering

Universiti Teknologi Malaysia Skudai, Malaysia

razali.ismail@gmail.com

Abstract—Graphene nanoribbon (GNR) has possibility to overcome the carbon nanotube chirality challenge as a nanoscale device channel. Because of one dimensional behavior of GNR, carrier statistic study is attractive. More research work has been done on carrier statistic study of GNR especially in Boltzmann approximation (Nondegenerate regime). Based on quantum confinement effect to improve the fundamental studies in degenerate regime we focused on, parabolic part of GNR band energy. Our method demonstrates that the band energy of GNR near to the minimum band energy is parabolic. In this part of the band structure, Fermi-Dirac integrals are sufficient for carrier concentration study. Similar to any other one dimensional device in nondegenerate regime Fermi energy shows temperature dependent behavior, on the other hand normalized Fermi energy with respect to the band edge is function of carrier concentration in the degenerate regime. However band structure is not parabolic in other parts of the band energy and numerical solution of GNR Femi-Dirac integrals are needed.

Keywords-component; Graphene Nanoribbon; carrier statistics; parabolic band structure; Fermi energy.

I. INTRODUCTION Single-layer of graphite called Graphene has discovered

as a material with attractive low-dimensional physics, and possible applications in electronics [1–6]. Graphene nanoribbons (GNRs), with quasi-one dimensional structures and narrow widths (<~10 nm) ,are predicted to use as a channel of field effect transistors with high switching speed and excellent carrier mobility specially for ballistic transport behavior [7–13].A single wall carbon nanotube is a rolled-up piece of a graphene, on the contrary a nanoribbon is an unrolled nanotube. Based on quantum confinement effect, the semiconducting and metallic behavior in the nanotube and nanoribbon configurations are similar together. In both cases, the wave vectors are confined to form a standing waves along the chiral vector Cr= ma1 + na2. Chiral vector shows the rolling up direction for single wall CNT, where m and n are integers and a1 and a2 are the unit cell vectors of graphene lattice. In a GNR, two node of the wave vector must be at the ribbon edge; on the other hand in a single wall carbon nanotube (SWNT), they can be anywhere. There are two different ways to unfold SWNTs that result in two different classes of GNRs by unzipping the SWNT along the axial direction through a row of atoms and then opening the atom line onto both edges of the resulting GNR. Thus any given (m, m) armchair SWNT can spread out into either a (m, m) or (m-1/2, m-1/2) armchair (see figure 1) GNR, and any

given (m, 0) zigzag SWNT can spread out into either a (m-1, 0) or (m-1/2, 0) zigzag (see figure 2) GNR [14].

Figure 1 Armchair (m, m) SWNT can spread out into either a (m, m) or (m-1/2, m-1/2) armchair GNR, Because of the symmetric or asymmetric structure we can label all the GNRs in a combined series of (m/2, n/2), where m and n are integers. [15-20]. Armchair and zigzag GNRs show metallic or semiconducting electronic properties depends on the width of the nanoribbon. Armchair ribbon with N = 3p + 2, where p is an integer shows semimetallic behavior. The semiconducting property in armchair CNRs occurs when N = 3p or 3p + 1, for integer p [21].

Figure 2.zigzag (m, 0) SWNT can spread out into either a (m-1, 0) or (m-1/2, 0) zigzag GNR.

2010 International Conference on Intelligent Systems, Modelling and Simulation

978-0-7695-3973-7/10 $26.00 © 2010 IEEEDOI 10.1109/ISMS.2010.78

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2010 International Conference on Intelligent Systems, Modelling and Simulation

978-0-7695-3973-7/10 $26.00 © 2010 IEEEDOI 10.1109/ISMS.2010.78

401

II. GRAPHENE NANORIBBON BAND STRUCTURE The band energy throughout the entire Brillouin zone of

graphene is [22].

23 33( ) 1 4 cos( ) cos( ) 4 cos ( )2 2 2

y C C y C Cx C Ck a k ak a

E k t

(1)

Where 1.42C Ca A is Carbon-Carbon(C-C) bond length, t=2.7 (eV) is the nearest neighbor C-C tight binding overlap energy and , ,x y zk is wave vector component [23, 24]. In low energy limit [25] due to the approximation for the graphene band structure near the Fermi point, the E (k) relation of the GNR in the low energy limit is

2 23( )2

C Cx

t aE k k (2)

Whit

2 2( )1 33

pna

(3)

Where p is the sub band index and xk is the wave vector along the length of the nanoribbon.

3g ccE ta (4)

By using equation 4 as a band gap we can rewrite band energy as

2

2(1 )2

g xE k

E (5)

According to equation (2) relationship between energy and wave vector is not parabolic. In semiconducting GNR which is our focus, Square root approximation can be employed to formulate parabolic relation between energy and wave vector, therefore band energy near the minimum band (k=0) is

22( )

2 4g g

x

E EE k k (6)

2 2

*2 2g x

E kE

m (7)

Where *m is effective mass of GNR, using this approximation leads to Fermi Dirac integrals in the parabolic band structure for carrier concentration, velocity and current similar to the one-dimensional devices [26]. However comparison by a parabola near the minimum band energy shows parabolic relationship for E-k, as shown in figure 3.

Fig. 3 The band structure of GNR near the minimum energy is parabolic. In parabolic part of one dimensional GNR band energy, using gradient of k from equation (7), definition of density of state incorporate with effect of electron spin parameter leads to density of state (DOS) in quasi one dimensional semi-conducting GNR as shown below [27].

12

1* 2

2

1 22 2

gEmDOS E (8)

By substituting 2gE with shifted conduction band energy

CE this equation is able to predict the DOS of other one dimensional material.

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III. CARRIER STATISTICS IN PARABOLIC BAND STRUCTURE m

The number of electrons/cm3 and holes/cm3 with energies between E and E+dE has been established to be

( ) ( )Dos E f E dE and ( )[1 ( )]Dos E f E dE ,therefore the total carrier concentration in a band is obtain simply integrating the Fermi-Dirac distribution function over energy band that is[28].

( ) ( )top

C

E

E

n Dos E f E dE (9)

( )[1 ( )]V

bottom

E

E

p Dos E f E dE (10)

And substituting the density of state ( )Dos E and Fermi-Dirac distribution function ( )f E expressions into carrier concentration, Fermi-Dirac integral appears in our interpretation

22

D Cd (D ) Dn N (11)

Here D is dimensionality D=3(3D), 2(2D), 1(1D), (D 2) 2 is

the Fermi-Dirac integral of order D-2 2 and normalized Fermi energy is ( )D F CD BE E k T . The Fermi-Dirac integral of order i is defined as

0

11 1D

i

i D (x )

x( ) dx

(i ) e (12)

Under nondegenerate condition one can be neglected from the denominator thus Fermi integral reduces into.

( )i eF (13)

In degenerate regime exponential part of equation (13) is very small and all the levels are occupied by electrons up to Fermi level thus occupation probability is 1 and Fermi- Dirac integral can be solve analytically.

111 1

iD

i D( )(i ) i

(14)

The inequalities adjacent to equation.(12) are simultaneously satisfied if the Fermi level lies in the band gap more than 3KBT from either band edge (recall KB is Boltzmann constant). For the cited positioning of the Fermi level, the semiconductor is said to be nondegenerate and equation (13) are referred to as nondegenerate relationships conversely, if the Fermi level is within 3KBT of either band edge or lies inside a band, the semiconductor is said to be degenerate .The simplified form of the occupancy factors is a Maxwell-Boltzmann type function. The simplified occupancy factors lead directly to the nondegenerate relationships. Although in closed form relationships find limited usage in device analyses. Since the nondegenerate relationship are obviously valid for GNR. Quasi one dimensional GNR in Fig. 4 has two axis directions that are less than De–Broglie wave length (one layer atomic thickness with width les than10 nm). On the other hand, GNR has length that is more than De-Broglie wave length,

DL and width less than De-Broglie wave length DW .

Fig.4 A prototype one dimensional GNR with DW and DL for rectangular cross-section

In nondegenerate condition the carrier concentration is

C F

B

E EK T

Cn N e (15)

Here NC is the effective carrier concentration and 0 0 0C C y zE E is shifted conduction band because of

confinement effect. This simplified distribution function is extensively used in determining the transport parameters. This simplification is true for nondegenerately-doped semiconductors also GNR near the minimum band energy. However, most nano electronic devices these days are degenerately doped. Hence any design based on the Maxwellian distribution is not strictly correct and often leads to errors in our interpretation of the experimental results, and for degenerate regime we have:

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1* 2

2 2

8 ( )F Cm E En (16)

In quasi one dimensional GNR semiconductors the Fermi-Dirac integral proportion of exponential of in

nondegenerate approximation and proportion of 1 22 e in degenerate approximation as shown in Fig.5.

Fig.5 Comparison of the Fermi-Dirac integral, with e and 1 22 e one dimensional GNR

Fig. 5 shows Fermi order -1/2 is closely approximated by exponential of when -3 in nondegenerate regime, and

for 6 approximated by 1 22 e in degenerate regime. For three dimensional bulk semiconductors D=3(3D), quasi-two-dimensional nanostructures D=2(2D), and quasi-one-dimensional D=1(1D) GNRs. The Fermi integral with Maxwellian approximation is always an exponential for all values of i and is given by ( )i eF (nondegenerate). In the strongly degenerate regime, the Fermi integral transforms to equation (14). The normalized Fermi energy Fd as a function of normalized carrier concentration for D=3(3D), 2(2D), 1(1D) Fig. 6 as expected in the nondegenerate regime

F c dE E as a function of ( / )dn N is given by

ln( )F c B dd

nE E k T

N(Nondegenerate) (17)

whereF c dE E is a weak (logarithmic) function of carrier

concentration, but varies linearly with temperature in the nondegenerate (ND) regime. However, for strongly degenerate statistics, the Fermi energy is independent of temperature and is a strong function of carrier concentration:

2/2

2 1 / 2* 2

d

F c dd

dE E n

m (18)

The Fermi energy is proportional to 2/33n for bulk (3D)

configuration, 2n for 2D nanostructure, and 21n for 1D

nanostructure.

Fig. 6 Comparison of Fermi-Dirac integral, and Fermi -1/2, 0, 1/2 for Q3D bulk, Q2D and Q1D devices respectively.

IV. CONCLUSION The one dimensional GNR approaches degeneracy at relatively lower values of carrier concentration as compared to 2D and 3D structures. Also Fermi energy with respect to band edge on GNR is function of temperature that independent of the carrier concentration in the nondegenerate regime. In the other strongly degenerate, the Fermi energy is a function of carrier concentration but is independent of temperature. Induced and doped carrier density in most nanoscale devices are now in degenerate regime generating a great interest in degenerate. Because of simplicity in the expressions for nondegenerate statistics, it is not uncommon to base the findings on nondegenerate statistics that sometimes leads to erroneous results.

ACKNOWLEDGMENT The authors would like to acknowledge the financial support from FRGS grant (Vot. No: 78251) of the Ministry of Higher Education (MOHE), Malaysia. The author is thankful to the Research Management Centre (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.

REFERENCES [1] A. K. Geim, K. S. Novoselov, Nat. Mater. 6, 183 (2007). [2] K. S. Novoselov et al., Science 306, 666 (2004). [3] C. Berger et al., Science 312, 1191 published online 12 April (2006). [4] Y. B. Zhang, Y. W. Tan, H. L. Stormer, P. Kim, Nature438, 201

(2005).

404404

[5] K. S. Novoselov et al., Nature 438, 197 (2005). [6] C. Berger et al., J. Phys. Chem. B 108, 19912 (2004). [7] Y.-W. Son, M. L. Cohen, S. G. Louie, Phys. Rev. Lett.97, 216803

(2006). [8] M. Y. Han, B. Ozyilmaz, Y. B. Zhang, P. Kim, Phys. Rev.Lett. 98,

206805 (2007). [9] D. A. Areshkin, D. Gunlycke, C. T. White, Nano Lett. 7,204 (2007). [10] Z. Chen, Y. M. Lin, M. J. Rooks, P. Avouris,

http://arxiv.org/abs/cond-mat/0701599 (2007). [11] K. Nakada, M. Fujita, G. Dresslhaus, M. S. Dresselhaus,Phys. Rev.

B 54, 17954 (1996). [12] V. Barone,O.Hod, G. E. Scuseria, Nano Lett. 6, 2748 (2006). [13] G. C. Liang, N. Neophytou, D. E. Nikonov, M. S. Lundstrom,IEEE

Trans. Electron. Dev. 54, 677 (2007). [14] Qimin Yan, Bing Huang,Jie Yu, Fawei Zheng, Ji Zang, Jian

Wu,Bing-Lin Gu, Feng Liu,and Wenhui Duan,”Intrinsic Current-Voltage Characteristics of Graphene Nanoribbon Transistors and Effect of Edge Doping” Nano Lett., Vol. 7, No. 6, (2007)

[15] Miyamoto, Y.; Nakada, K.; Fujita, M. Phys. ReV. B, 59, 9858-9861,(1999)

[16] Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B, 54, 17954-17961(1996)

[17] Son, Y.; Cohen, M. L.; Louie, S. G. Nature, 444, 347-349 (2006) [18] Son, Y.; Cohen, M. L.; Louie, S. G. Phys. ReV. Lett. 97, 216803

(2006). [19] Barone, V.; Hod, O.; Scuseria, G. Nano Lett., 6, 2748-2754 (2006) [20] Semiconductor Industry Association, International Roadmap for

Semiconductors2005, [Online]. Available: http://public.itrs.net/(2005)

[21] Gengchiau Liang, Neophytos Neophytou, Dmitri E. Nikonov, , and Mark S. Lundstrom,“Performance Projections for Ballistic Graphene Nanoribbon Field-Effect Transistors” IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 54, NO. 4, APRIL (2007)

[21] Mark Lundstrom, Jing Guo,”NANOSCALE TRANSISTORS , Springer, “(2006)

[22] J. W. G. Wild¨Oer, L. C. Venema, A. G. Rinzler, R. E. Smalley, And C. Dekker, Electronic Structure of Atomically Resolved Carbon Nanotubes,” Nature (London), Vol. 391, No. 6662, Pp. 59–62, (1998)

[23] R. A. Jishi, D. Inomata, K. Nakao, M. S. Dresselhaus, And G. Dresselhaus, “Electronic And Lattice Properties Of Carbon Nanotubes,” J.Phys.Soc.Jap., Vol. 63, No. 6, Pp. 2252–2260, (1994)

[24] Robert F.Pierret” Advanced semiconductor fundamentals” (2003) [25] Huaixiu Zheng, Zhengfei Wang, Tao Luo, Qinwei Shi, and Jie Chen,

“ Analytical Study of Electronic Structure in Armchair Graphene Nanoribbons” Cond-mat , 0612378v2 . 16 Dec (2006)

[26] V. K. Arora, “Quantum enginering of nanoelectronic devices: the role of quantum emission in limiting drift velocity and diffusion coefficient,”Elsevie, Microelectronics Journal, vol. 31, no.11-12, pp 853-859. (2000)

[27] Vijay K. Arora, “Failure of Ohm's Law: Its Implications on the Design of Nanoelectronic Devices and Circuits,” Proceedings of the IEEE International Conference on Microelectronics,May 14-17, ,Belgrade, Serbia and Montenegro, pp.17- 24 (2006)

[28] Supriyo Datta ,”Quantum Transport Atom to transistor”Cambridge University Press ,(2005)

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