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Voltage-Dependent Parameter Extraction for Graphene Nanoribbon Intercon-nect Model through Ab Initio Approach

Serhan Yamacli

PII: S0040-6090(14)00529-XDOI: doi: 10.1016/j.tsf.2014.04.095Reference: TSF 33443

To appear in: Thin Solid Films

Received date: 15 November 2013Revised date: 25 April 2014Accepted date: 30 April 2014

Please cite this article as: Serhan Yamacli, Voltage-Dependent Parameter Extraction forGraphene Nanoribbon Interconnect Model through Ab Initio Approach, Thin Solid Films(2014), doi: 10.1016/j.tsf.2014.04.095

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Voltage-Dependent Parameter Extraction for Graphene Nanoribbon

Interconnect Model through Ab Initio Approach

Serhan YAMACLI

Nuh Naci Yazgan University, Faculty of Engineering, Department of Electrical-Electronics

Engineering, 38400, Kocasinan, Kayseri, Turkey.

Tel: +90 352 32 40 000, Fax: +90 352 32 40 003

syamacli@nny.edu.tr

ABSTRACT

This paper presents electrical parameter extraction for metallic graphene nanoribbon (GNR)

interconnects utilizing ab initio approach. Unlike the studies taking the kinetic inductance,

quantum capacitance and Fermi velocity as constant values, voltage-dependencies of these

parameters are obtained for GNR transmission line model. The variations of the kinetic

energy and the current by the applied voltage are taken as bases for voltage-dependent kinetic

inductance calculation. Quantum capacitance and the Fermi velocity are also computed from

the kinetic inductance variation. It is concluded that voltage-dependencies of the kinetic

inductance and the quantum capacitance have to be taken into account for accurate GNR

modelling in nanoelectronics design.

Keywords: Kinetic Inductance; Quantum Capacitance; Fermi Velocity; Graphene Nanoribbons;

Interconnects; Ab Initio Calculations

1. INTRODUCTION

Power and ground lines, clock pulse and other signals are transported among circuit

components via interconnects inside the integrated circuits (ICs). Copper, gold and aluminium

interconnects are conventional structures for the transmission of these signals. However, as

the scaling down of the IC technology node continues, the resistances of conventional bulk

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interconnects increase due to grain-boundary and surface scattering mechanisms because the

interconnect dimensions become comparable to electron wavelength [1-3]. These

disadvantages of bulk interconnects lead the research to various feasible alternatives. Nano

scale interconnects such as metallic carbon nanotubes (CNTs), graphene nano ribbons

(GNRs) and gold nanowires are promising alternatives according to theoretical and

experimental studies [4-6]. From the fabrication viewpoint, GNRs are advantageous [5].

However in order to utilize metallic GNRs as interconnects in the future IC technology, their

accurate models have to be developed.

There are various circuit models for GNR interconnects in the literature [7-11] and the

transmission line model shown in Figure 1 is the backbone of these studies. In the GNR

transmission line model of Figure 1, the δ symbol denotes the infinitesimal of the

corresponding parameter. Rq is the contact resistance that is generally taken as the resistance

quantum value, 12.9 k, for good quality contacts [7]. Lm is the magnetic inductance that is

obtained from the classical electromagnetic analyses [7, 8]. Ce is the electrostatic capacitance

which is also calculated using classical electrostatic arguments. Lm and Ce are material and

geometry dependent parameters therefore they are calculated using the classical methods in

the same way as the calculation utilized for the conventional bulk interconnects [11]. Hence,

Lm and Ce are well-defined. On the other hand, there are two other parameters in the GNR

transmission line model, namely the kinetic inductance, Lk, and the quantum capacitance, Cq.

These two parameters arise from quantum effects during the electron transport in nano scale

interconnects [12-14] and they do not have classical counterparts. Therefore Lk and Cq have to

be obtained using quantum mechanical arguments.

Quantum capacitance and kinetic inductance of metallic GNR interconnects are

generally taken as having constant and voltage-independent values in the literature. For

example in [11], Lk and Cq of GNR interconnects are taken as 8 nH/µm and 200 aF/µm,

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respectively. Similarly another study takes Lk and Cq as having the values of 16.1 mH/m and

96.8 pF/m, respectively [9]. In [15], an analysis on the kinetic inductance is given and

assuming that the Fermi velocity, vF, is 8.105 m/s in GNRs, kinetic inductance is calculated as

4 nH/µm. It is worth noting that in [15], the obtained Lk value is valid only under the

assumption that the Fermi velocity has the mentioned value, which is not an exact value. In

other studies, the average value for vF is also taken around 106

m/s for GNR interconnects

[11, 16, 17]. [8] and [16] also give expressions for Lk and Cq dependent on an average value

for vF, however it does not discuss the accurate quantum mechanical calculation of Lk, Cq or

vF. The relation of vF and Lk are also discussed in [17] and a kinetic inductance of 16 nH/µm

is calculated using an average value for vF. In [7], Lk and Cq are discussed in the case of

parallel GNR interconnects and again taking vF as having an average constant value, Lk and

Cq are found to be 8 nH/µm and 200 aF/µm, respectively. As summarized above, in the

studies considering the kinetic inductance and quantum capacitance of GNR interconnects in

the literature, various Lk and Cq values are calculated under the assumption of average and

constant Fermi velocity values. However, Fermi velocity of electrons which are being

transported in a nano scale interconnect is expected to have a voltage-dependent or

bias-dependent characteristics due to the modulation of the transmission spectrum in nano

scale conductance channels, which is analysed in detail in various studies in the literature

[1, 13, 18, 19]. Hence, in order to obtain accurate voltage-dependent Lk and Cq characteristics,

quantum mechanical simulations need to be performed which take the transmission spectrum

modulation and kinetic energy variations into consideration. There is only one study in the

literature for the accurate calculation of the quantum capacitance value in this manner [20]

however it considers only metallic CNTs and not GNRs. Moreover, it is limited to the

quantum capacitance computation and does not consider the calculation of the kinetic

inductance.

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In this study, accurate voltage-dependent variations of the kinetic inductance and the

quantum capacitance of a GNR interconnect sample are obtained in the quasi-static regime

using ab initio quantum mechanical simulations. The variation of the Fermi velocity is also

discussed. It is concluded that the voltage-dependencies of the kinetic inductance and

quantum capacitance have to be taken into account for accurate GNR interconnect

transmission line modelling.

2. THE RELATIONS AMONG KINETIC ENERGY, CURRENT, KINETIC

INDUCTANCE, QUANTUM CAPACITANCE AND THE FERMI VELOCITY

The relation between the kinetic energy Ek at voltage V, and the current I at voltage V,

passing through a nano scale interconnect is given by (1) in the quasi-static regime [14, 21].

(1)

where Lk(V) denotes the voltage-dependent kinetic inductance. Taking the derivatives of both

sides and arranging the equation, the expression for the accurate voltage-dependent kinetic

inductance is given by (2).

(2)

In (2), dEk(V) is the change of the kinetic energy and dI(V) is the change in the current

passing through the interconnect. Hence, accurate voltage-dependent kinetic inductance can

be computed if the change in the kinetic energy and the current by the voltage applied to the

interconnect is obtained by quantum mechanical simulations. In this study, ab initio

simulations of a sample GNR interconnect is performed using Quantumwise ATK® software

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package in order to compute dEk and dI for use in the calculation of the kinetic inductance. In

addition, the relation of vF-Lk and vF-Cq can be given as in (3) and (4), respectively [10].

(3)

(4)

where q is the electron charge, h is Planck’s constant and Nch is the number of transmission

channels. Using (3) and (4), the relation between Cq and Lk can be found as in (5).

(5)

Hence, voltage-dependent quantum capacitance can also be computed when voltage-

dependent kinetic inductance is obtained. Voltage-dependent vF can also be obtained from (3)

as follows.

(6)

3. AB INITIO QUANTUM MECHANCAL SIMULATIONS OF THE GNR

INTERCONNECT SAMPLE

Ab initio simulations of nano scale devices provide accurate results for the electronic

behaviours of various structures such as CNTs [20], metal nanowires [22] and GNRs [23].

First principles quantum mechanical simulations have several types that are used for different

aims. The voltage-dependent variations of Ek and I are needed in this study hence, a quantum

mechanical simulator capable of calculating voltage-dependent variations is required.

Quantumwise ATK® is an ab initio simulator utilizing density functional theory (DFT) to

solve Kohn-Sham equations in conjunction with non-equilibrium Green’s function formalism

(NEGF) to provide voltage-dependent variations of electronic parameters [24]. ATK® can

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also be used for accurate computations of GNR structures in consistent with experimental

results [25]. Therefore this software package is used in this work.

The metallic 5 atoms wide zigzag GNR (5-ZZGNR) interconnect shown in Figure 2 is

simulated in ATK® in order to obtain Ek-I variations for computing Lk, Cq and vF. The

simulated GNR interconnect sample is 4.79 nm long which is a realistic length for nanometre

scale ICs. In addition, the sample GNR is chosen as 5 atoms wide which is asymmetric since

asymmetric GNRs have linear I-V characteristics without a transmission pinch off by the

applied voltage permitting to be utilized as conductors [26]. The GNR sample provides also a

trade-off between the width and the simulation cost. The GNR interconnect considered in our

study is consistent with the GNRs utilized in the previous studies by having hydrogen

terminated edges and asymmetric structure. Parallel ferromagnetic state is taken in

simulations as discussed in the literature [26, 27]. The DFT simulation parameters are selected

as follows: the basis set is double-zeta polarized with 0.001 Bohr of radial sampling, mesh

cut-off energy is 250 Ry, the pseudopotentials are chosen as norm conserving and

exchange-correlation functional is Perdew-Zunger type local density approximation that gives

accurate results for carbon-based materials [28]. Electron temperature is 300 K and Brillouin

zone integration path is taken as (1,1, 400) in order to provide sufficient momentum space

(k-space) sampling. The geometry is relaxed until the forces are below 0.02 eV/Å before the

actual NEGF simulations. Dirichlet boundary conditions are used in the transport direction.

The applied voltage is swept from 0 V to 1 V with 0.1 V of sampling. This voltage

range is selected considering the International Technology Roadmap for Semiconductors

estimations for the future interconnect technology [29]. The variation of the kinetic energy by

the current passing through the GNR interconnect is obtained with ATK® simulations as

shown in Figure 3.

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4. VOLTAGE-DEPENDENT KINETIC INDUCTANCE AND QUANTUM

CAPACITANCE

The variation of the kinetic inductance is computed from the Ek-I variation according to (2)

and given in Figure 4. Note that kinetic inductance is obtained from ab initio quantum

mechanical simulations using the kinetic energy and the current. In [11] and [17], Lk per unit

length are given as 8 nH/µm/Nch where Nch is the number of transmission channels having the

value of 2 for GNRs [30]. Similarly, a value of 16.1 nH/µm for the total Lk per unit length is

reported in [9]. Hence the value range of the kinetic inductance values obtained in our study

is in consistent with the average values existing in the literature. However, our analysis gives

the variation of the kinetic inductance with the applied voltage to provide an accurate GNR

interconnect model. From Figure 4, it can be seen that the value of Lk changes from

21 nH/µm to 13.8 nH/µm. Considering that the change is 34% in the voltage range of 0.1 V to

1 V, it is clear that the voltage-dependency of Lk has to be considered for accurate modelling.

Quantum capacitance is a circuit parameter arising from the transmission of electrons

as mentioned in Section 1. The voltage-dependent variation of the quantum capacitance is

also obtained from (5) and given in Figure 5. The constant Cq values existing in the literature

are taken around 2 pF/cm [11, 17]. Therefore, the quantum capacitance values computed in

this study are also compatible with the average Cq values in the literature. Cq changes from

2.05 pF/cm to 1.35 pF/cm as seen from Figure 5. In other words, there is a 34% change also in

the quantum capacitance as the voltage increases from 0.1 V to 1 V implying that taking an

average value for Cq can lead to inaccurate results hence the voltage-dependency of Cq has to

be taken into account. It is worth emphasizing that the voltage-dependency of the quantum

capacitance is obtained through the variation of the kinetic inductance.

Finally, Fermi velocity is plotted against the applied voltage using (6) in Figure 6. As

mentioned before, the average Fermi velocity is given around 106

m/s for GNR interconnects

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[11, 16, 17], but our work provides the variation of vF with the applied voltage and again the

values obtained in this study are consistent with average vF values given in the previous

studies [31, 32]. Although vF is not a circuit parameter of the GNR interconnect transmission

line model, it provides a verification for the consistency of the transmission line parameters. It

is worth noting that any parameter that affects vF would also have effects on Lk and Cq

according to (3) and (4). Hence, as the structural defects and edge termination variations

decrease vF [33, 34], these would lead to the increment of Lk and Cq. Moreover, a similar

effect is also expected when the length of the GNR increases that also induce scattering states

[35]. However, increment of the width of the GNR affects Lk and Cq by changing the number

of transmission channels. As the width of the GNR increases, Nch would increase [30] leading

to an increment in the quantum capacitance and decrement in the kinetic inductance according

to (3) and (4).

5. CONCLUSIONS

Kinetic inductance and the quantum capacitance are circuit parameters in the GNR

interconnect transmission line model arising from quantum mechanical effects hence their

modelling requires quantum mechanical treatments. In this study, kinetic inductance and

quantum capacitance of a 4.79 nm long GNR interconnect sample are computed using ab

initio quantum mechanical simulations. Kinetic energy and the current variations of the GNR

interconnect with the applied voltage are taken as bases for the calculation of Lk and Cq.

Obtained Lk and Cq value ranges are found to be in consistent with the average values already

existing in the literature. However, the variations of the kinetic inductance and the quantum

capacitance are found to be 34% as the voltage changes from 0.1 V to 1 V. Hence it is

obvious that taking average values for these transmission line parameters leads to inaccurate

results and the voltage-dependent variations of Lk and Cq have to be considered for accurate

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modelling during nanoelectronics design. The variation of the Fermi velocity with the applied

voltage is also obtained to verify with vF values given in the previous studies.

ACKNOWLEDGEMENT

The authors would like to thank QuantumWise AS and Dr. Anders Blom for their valuable

support.

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FIGURE CAPTIONS

Figure 1. The generic graphene nanoribbon interconnect transmission line model

Figure 2. The simulated asymmetric zigzag GNR interconnect sample

Figure 3. Variation of the kinetic energy of the GNR interconnect sample with the current

passing through it

Figure 4. Variation of the kinetic inductance of the GNR interconnect sample with the

applied voltage

Figure 5. Variation of the quantum capacitance of the GNR interconnect sample with the

applied voltage

Figure 6. Variation of the Fermi velocity of the GNR interconnect sample with the applied

voltage

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Figure 1. The generic graphene nanoribbon interconnect transmission line model

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Figure 2. The simulated asymmetric zigzag GNR interconnect sample

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Figure 3. Variation of the kinetic energy of the GNR interconnect sample with the current

passing through it

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Figure 4. Variation of the kinetic inductance of the GNR interconnect sample with the

applied voltage

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Figure 5. Variation of the quantum capacitance of the GNR interconnect sample with the

applied voltage

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Figure 6. Variation of the Fermi velocity of the GNR interconnect sample with the applied

voltage

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Highlights

Metallic graphene nanoribbon interconnects are studied using ab initio approach.

Variations of the kinetic inductance (LK) and the current are obtained.

Voltage-dependency of the kinetic inductance is extracted.

The variations of quantum capacitance (CQ) and Fermi velocity are calculated.

LK and CQ change by 34% in the voltage range of 0-1V.