ICSE2012 Proc. 2012, Kuala Lumpur, Malaysia

Temperature Effect on Quantum CapacitanceZig-Zag Graphene Nanoscrolls (ZGNS) (16,0)Afiq Hamzah1, M.T.Ahmadi*1, Mohammad Javad Kiani1,2, Fatimah. K. A. Hamid1, Azlin Bahador1

,

Razali Ismail1

1Department of Electronic Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Johor Bahru, 81310 Johor, Malaysia

2Department of Electrical Engineering, Islamic Azad University, Yasooj branch, Yasooj, Iran.Tel No.: +6075535246, Fax No.: +607-5566272,

*Email address: taghi@fke.utm.my

Abstract - Device scaling of the electronic devices has brings the dominancy of quantum effect in nano-size device characterization. This paper presented the first band analytical model of the quantum capacitance for (16,0) zig-zag graphene nanoscroll (ZGNS). The derivation of quantum capacitance is based on the differentiation of carrier density towards the Fermi energy. The Taylor’s series expansion is employed on parabolic energy band structure so that it can be modified in the form of Fermi Intergal. Owing to its unique geometry structure that provides high intercalationarea, it is expected that ZGNS exhibit high quantum capacitance.

I. INTRODUCTION

A Graphene nanoscroll (GNS) is a novel carbon-based structure which is made by rolling a layer of graphene sheet in the form of Archimedean type spiral [1-3]. It is also known asthe “Swiss roll” graphene due to its spiral shape structure [4].In distinct to the carbon nanotube (CNT), GNS has an open edge along the translational axis. Since GNS is a scroll graphene layer, the chirality can be describe as ⃗ = ⃗ + ⃗ where the nomenclature can be describe as zigzag for = 0, armchair for = 90°, and chiral for 0 < < 90° [1] as shown in Figure 1(B). GNS exhibit fascinating properties due to its novel structure [4-7] and it cannot be determined uniquely by its chirality (n,m) [4] since the tuneable core size that alters its properties that signify the GNS properties dependence on its geometry structure [6, 7]. Moreover, GNSinterlayer galleries can be used to intercalate with dopants and the diameter can be expand to accommodate the volume of the dopant-layer interactions [8-11]. Because of its novel structure, researchers speculate the variety of promising applications to be implemented using the GNS especially as energy storage devices [11-14]. Despite rapid advances research in carbon-based material [15] transport properties, the carrier statistics and electrostatic temperature dependence properties such as quantum capacitance remain unexplored for GNS.

x

y

C

Ta1

a2

Fig. 1. Shows a graphene layer as the precursor to the formation of GNS where is the scroll angle with respect to the axes, and is the scroll

vector and translational vector respectively.

Owing to the GNS geometry and temperature dependence properties, the quantum capacitance of a semiconducting zigzag GNS (ZGNS) at (16,0) chirality in a semi-classical regime is reported. The quantum capacitance is derived based on the parabolic energy dispersion approximation. In this paper, the ZGNS of chirality (16,0) is used and the energy dispersion is equated as [4]

2cc3 2 2( ) 1 4 42 2 2Zx

xGNSk a v vE k t cos cos cos

n n (1)

where the circumferential direction is around the y-axis, thus √3 2⁄ = (2 − ) 2⁄ is replaced into the graphene general equation to form Eq. (1) and is the only parameter that control the geometry of the GNS which also modulate the energy gap [6] and is the nearest neighbour C-C overlap energy which is between 2.5eV to 3.2eV. As a consequence to Chen’s et al., = 1.9242for ZGNS (16,0). is the wave vector along the x-axis and is the length of carbon-carbon atom and n is the chirality of ZGNS (n,m). The derivation of

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ICSE2012 Proc. 2012, Kuala Lumpur, Malaysia

the quantum capacitance is based on the approximation of Maxwell-Boltzmann within 5% agreement with Fermi-Dirac probability distribution law in parabolic function is employed for estimating Fermi level for classical regime of 1D ZGNS and high carrier density effect with probability of occupation equal to one in band energy is approximated for degenerate regime.

II. QUANTUM CAPACITANCE

Capacitance is an essential parameter that helps to gain physical insights in determining the MOSFET characteristics and the electrical properties such as carrier statistic, conductance and mobility [16, 17]. There have been numbers of research on GNS characteristic through molecular dynamic simulation and fabrication process [5, 18-20]. Numerical simulation does give accurate result but very time consuming for fast circuit simulation [17]. Thus by analytically model the quantum capacitance of GNS, could provide the accessibility on its device physics and the device characteristic performance can be measured [16, 21, 22]. Because of its interlayer galleries which able to intercalated with dopant atoms, signify the concept of generating electricity from solar energy [8, 23]. As the interconnect capacitance increased due to the quantum confinement and CMOS scaling, it can also store the harvested photon energy without the battery materials to be integrated on-chip. is the quantum capacitance which was significantly affected by the quantum confinement into a nanoscale size device. Thus, the application of the quantum capacitance has been considered for nanoscale devices modelling especially in the carbon-based material [16, 21, 24]. The quantum capacitance limit is dominant in one dimensional (1D) device compare to the conventional two-dimensional (2D) and three-dimensional (3D) structures. As for the case of GNS, research shows that it exhibit 1D properties [1, 4]. A general expression for quantum capacitance is = = , (2)

where = ∙ is the change in charge per unit length.= / is the voltage differential applied to the device and is the magnitude of an electron charge. For 1D devices, the numbers of electrons/cm with energies between E and E+dE is established as ( ) ( , ) . Thus the total of carrier concentration within a band can be obtained by integrating the Fermi-Dirac distribution function against energy that equated as [17] = ∫ ( ) ( , ) (3)

whereby ( ) is the density of states (DOS) for the ZGNS. Since ZGNS is a confined 1D structure, hence the carrier density is defined as [25]

1 1 1/2( ) ( )D F D Fn E N F . (4)

is the 1D effective DOS and / ( ) is Fermi integral of order minus half and = ( − )/ is the normalized Fermi energy, The effective DOS for ZGNS (16,0) is = 2 /(3 √8 ). In order to fulfil the constraints for the calculations and physical interpretation, the first unfilled energy level (conduction band) at = 0 is being considered where the probability for electrons to occupy is high. Then, by employing the Taylor’s series expansion, Eq. (1) is approximated to be

2 2cc

(16,0)90.5

8x

zz xk a

E k t . (5)

The quantum capacitance only considered within a parabolic band. Eq. (5) indicate that the parabolic energy dispersion for (16,0) ZGNS as illustrated in Figure 2.

Fig. 2. The energy dispersion of ZGNS near the minimum energy is parabolic which will be useful for order determination of known Fermi integral form in

calculating the quantum capacitance.

By differentiating energy with respect to the wave vector, the DOS (normalized per unit length) for infinite length of ZGNS is obtained as in Eq. (6)

122 ( 0.5 )

3 8x cc

nD E E tEL a t

. (6)

The total summation of the product of DOS and Fermi-Dirac distribution function is obtained as in Eq. (7) by resolving in terms of where = ( − 0.5 )/ and = ( −0.5 )/ . The carrier concentration to be the total numbers of electron resides within the conduction band. is the

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ICSE2012 Proc. 2012, Kuala Lumpur, Malaysia

Boltzmann constant and is a temperature in Kelvin. The expression for the total DOS is the same for the conduction band and valence band and exhibits Van Hove singularities at energies is depicted as the bottom of the sub-band (assumed one) [25].

12

e2

xp( )

13 8

top

C

EB

Ecc

K T xn dxa xt

. (7)

Hence, the quantum capacitance at a Dirac point or charge neutrality is obtained by the differentiation of carrier concentration against the states of energy level which is

12 22 (

e3 xp 1)

8Qcc

e xx

Ca t

(8)

In the non-degenerate regime, “1” from the denominator is neglected. At room temperature (T=300K), the states within the conduction band (allowed band) are partially filled, thusthe exponential part of Eq. (8) is big enough for “1” to be neglect. Thus in non-degenerate regime, the quantum capacitance is obtained as

12 2

e p2 (3 8

x)QND

cc

e xCa t

x . (9)

While for degenerate regime, the exponential part is very small because of the probability for electron to fill all the available states up to Fermi level one is “1”, signifying that there are no available states within the conduction band for − < 3 . Therefore the quantum capacitance within degenerate regime is

12 22 ( )

3 8QDcc

e xCa t

. (10)

From the result obtained in Figure 3, at low concentration the non-degenerate regime can be approximate using Maxwell-Boltzmann approximation within 5 per cent of each other, which is valid for approximately < −1. It is also shows that quantum capacitance operating in the degenerate regime where it reach degenerate limit at > 9 for ZGNS (16,0) at = 1.9242. At this particular condition, it can be seen that for ZGNS (16,0) at ground state, the quantum capacitance has reach its limit at ≅ 1.7 × 10 F/m @ 170 pF/m. The dominancy of the quantum capacitance can be describe at different concentration level. In degenerate regime, quantum capacitance has more influence for the fact that carrier concentration within the conduction band exceeds the density of states. Therefore the Fermi level resides in the conduction band.

Fig. 3. The validity of the approximation on both classical and degenerate regime based on the general quantum capacitance curve.

The temperature dependence on the quantum capacitance for the ZGNS (16,0) is shown in Figure 4 where the non-degenerate regime has cause the normalized Fermi level, to decreased (shifted to the left) by increasing the temperature.The distance between Fermi energy to the conductance band is decreased. By intuition, the electron can be easily elevated to the conduction band by the energy provided by the temperature so that the electron moves freely within the 1D GNS structure to conduct current. Degenerate regime is theoretically not affected by the temperature due to the assumption of the high carrier density effects that cause the probability of the occupation for conduction band is one. Thereby, the quantum capacitance in degenerate regime is temperature independent and according to Kliros et al. that the temperature independent is due to its Fermi energies level higher than the broadening parameter [26]. Furthermore, as the temperature increased the quantum capacitance reach its limit at low . Since the quantum capacitance is a function of the total charge, the electron concentration is vastly increased by the temperature increment causing the quantum capacitance reach its limit faster.

Fig. 6. The quantum capacitance temperature dependence for ZGNS (16,0).

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ICSE2012 Proc. 2012, Kuala Lumpur, Malaysia

III. CONCLUSION

As device is being shrunk into nanoscale topology, quantum mechanics play important role in determining the carrier transport. In 1D structure, it is very important to consider the quantum transport in analytical model. Quantum capacitance is one of the parameter that needs to be considered. In this paper we model the quantum capacitance by applying the first derivative of Taylor’s expansion approximation on energy dispersion relation. The Maxwell-Boltzmann’s distribution law is employed at low concentration where Fermi level within a range of 3 for a validity of < −1 where the Maxwell-Boltzmann probability function assumption is in accordance with the Fermi-Dirac distribution function. Asimple expression is used as the quantum capacitance reach degeneracy at 170 pF/m for < 3 . By having the quantum capacitance model, one can predict the efficiency of its storage capability at different temperature condition.

ACKNOWLEDGEMENT

Authors would like to acknowledge the financial support from Research University grant of the Ministry of Higher Education (MOHE), Malaysia under Projects Q.J130000.7123.02H24 and Q.J130000.7123.02H04. Also thanks to the Research Management Center (RMC) of Universiti Teknologi Malaysia (UTM) for providing excellent research environment in which to complete this work.

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