The insertion of high-k dielectric materials in multilayer graphene nanoribbon interconnects for reducing propagation delay and expanding bandwidth

Journal Pre-proof The insertion of high-k dielectric materials in multilayer graphene nanoribbon interconnects for reducing propagation delay and expanding bandwidth Peng Xu, Zhongliang Pan PII:

S1566-1199(19)30634-2

DOI:

https://doi.org/10.1016/j.orgel.2019.105607

Reference:

ORGELE 105607

To appear in:

Organic Electronics

Received Date: 24 September 2019 Revised Date:

4 December 2019

Accepted Date: 24 December 2019

Please cite this article as: P. Xu, Z. Pan, The insertion of high-k dielectric materials in multilayer graphene nanoribbon interconnects for reducing propagation delay and expanding bandwidth, Organic Electronics (2020), doi: https://doi.org/10.1016/j.orgel.2019.105607. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

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The Insertion of High-k Dielectric Materials in Multilayer Graphene Nanoribbon Interconnects for Reducing Propagation Delay and Expanding Bandwidth Peng Xu, Zhongliang Pan* School of Physics and Telecommunications Engineering, South China Normal University, Guangzhou, 510006, China *Correspondence: [email protected]

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Abstract: The multilayer graphene nanoribbons (MLGNR) by inserting the high-k dielectric material between adjacent graphene nanoribbons (GNR) layers to reduce propagation delay and to expand bandwidth of the conventional pristine (undoped) MLGNR interconnects is presented in this paper. An equivalent distributed circuit model of the proposed MLGNR interconnect with high-k dielectric materials is established to derive the analytical expressions of propagation delay, step response, transfer gain and 3-dB bandwidth for 7.5 nm technology node at global level. The numerical simulation results show that the maximum reduction of propagation delay between the proposed MLGNR by inserting SrTiO3 dielectric material and the pristine MLGNR can reach 12.746 ns for an interconnect length of 4000 µm. The corresponding 3-dB bandwidth for them can be expanded over 4 times for the interconnect length. It is demonstrated that the proposed MLGNR by inserting the high-k dielectric material can greatly reduce propagation delay and enhance transfer gain and 3-dB bandwidth compared with the conventional pristine MLGNR interconnects. Moreover, it is found that the results obtained by the proposed model have close agreement with Synopsys HSPICE simulation and the maximum relative error for them is less than 5%, and the average efficiency of CPU runtime is improved 95% by using the proposed model, as compared to HSPICE simulation. The proposed design method has many potential applications in improving performance of MLGNR interconnect and providing guidelines for the next generation on-chip interconnect system.

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Keywords: MLGNR interconnect; high-k dielectric material; propagation delay; step response; transfer gain; 3-dB bandwidth; on-chip interconnect system

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1. Introduction

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With the feature size of semiconductor integrated circuits (IC) shrinking into the nanometer order, a series of performance degradation and reliability problems on the traditional copper (Cu) based interconnects have been reported, such as the high resistivity, electro-migration effect and small current capacity issues [1,2]. Owing to these limitations, it is crucial that the IC designer ought to explore a new emerging material for next generation on-chip interconnect. Carbon nanomaterials, including the carbon nanotubes (CNT) and the graphene nanoribbons (GNR), have received considerable attention from many researchers because of their excellent electrical, mechanical and thermal properties [3,4]. In general, GNR is more suitable for the design of on-chip interconnects over CNT in view of the latter's complex fabrication process [5,6]. In addition, compared with the CNT, GNR is much easier to use high-resolution lithography for establishing the horizontal interconnect due to their planar geometry structure [7-9]. GNR has several micrometer long mean free path (MFP) far over Cu material. This can lead to a lower resistivity and realize the ballistic transport for the shorter interconnect [10]. Moreover, the current carrying capacity of GNR can exceed 109 A/cm2, which will greatly alleviate the electro-migration and skin effect of Cu interconnects at the high operating frequency [11,12]. In the light of the stacked number of layers, GNR can be categorized into the multilayer GNR (MLGNR) and the single-layer GNR (SLGNR). SLGNR is not applicable to on-chip interconnect in terms of its larger intrinsic resistance [7-9]. Based on their connection types with other devices or interconnects, MLGNR can be further classified into top contact MLGNR (TC-MLGNR) and side contact MLGNR (SC-MLGNR) [10-12]. TC-MLGNR is only the top most layer coupled with surrounding contacts, whereas all layers of SC-MLGNR are connected to the other contacts [13-15]. The unique connection of SC-MLGNR can enable it to have a lower distributed scattering resistance when compared with TC-MLGNR. Therefore the SC-MLGNR is selected as the candidate material of interconnects in this work.

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Although the SC-MLGNR has a multitude of outstanding electrical properties, there are still some limitations in the practical application of on-chip interconnect. The SC-MLGNR is easy to convert into graphite because of the interlayer electron hopping, thus reduces electron mean free path (MFP) and the total number of GNR stacked layers [16]. The reason for forming the interlayer electron hopping is due to modulation of the carbon-carbon bond lengths induced by elastic strain of stacked multiple layers [16,17]. As a consequence, it is essential to find an alternative interconnect structure for improving MFP of SC-MLGNR. In the past few years, there are some reports that MFP could be enhanced by inserting a thin dielectric layer between successive GNR layers, which can avoid the SC-MLGNR turning into graphite and overcome the inherent performance limitations of SC-MLGNR interconnect [16,18]. This is due to that the carrier mobility of GNR layers can be increased owing to existing the thin dielectric layer [16]. Moreover, inserting the thin dielectric layers in SC-MLGNR can also reduce the electron scattering rate of GNR and thus decrease the distributed scattering resistance of SC-MLGNR interconnect. With the emergence of atomic layer deposition (ALD) technique, a series of dielectric materials could be grown on the graphene layer [19]. In reference [20], Dean et al. have processed the hexagonal boron nitride h-BN ( Ɛh-BN =4) dielectric layer on the top of graphene layer and observed that the carrier mobility of graphene is much larger than that the SiO2 dielectric case. In reference [21], Liao et al. have fabricated graphene-based field effect transistors on aluminium oxide Al2O3 (ƐAl2O3 =10) substrate to improve the transconductance of the back-gate GNRFETs. In reference [22], Zou et al. have deposited the dielectric hafnium oxide HfO2 (ƐHfO2 =25) on graphene layer, which can achieve an extremely high carrier mobility of 20000 cm2/Vs at low-temperatures environment. Other potential dielectric candidates, including of tantalum pentoxide Ta2O5 (ƐTa2O5 =25) , lanthanum oxide La2O3 (ƐLa2O3 =30), titanium dioxide TiO2 (ƐTiO2 =95) and strontium titanate SrTiO3 (ƐSrTiO3 = 200), have been reported in reference [16,23]. However, the works mentioned above investigate the impacts of dielectric on transmission performance of single or two-layer graphene structures. To the best of our knowledge, so far only few literatures have studied SC-MLGNR interconnect in which the high-k dielectric materials are inserted between adjacent layers. In reference [16], Nishad et al. presented an analytical model to analyze propagation delay, energy-delay-product of SC-MLGNR interconnect by inserting the hafnium oxide HfO2 (ƐHfO2 =25) dielectric between successive GNR layers. In reference [24], Mekala et al. proposed the mathematical method of finite-difference time-domain (USFDTD) to investigate propagation delay, step response, crosstalk noise and power dissipation of SC-MLGNR interconnect, which is also based on inserting the HfO2 (ƐHfO2 =25) dielectric between two adjacent GNR layers. However, it was found that these literatures focused on the performance parameters of SC-MLGNR interconnect in time domain. Certainly, it is clearly that the performance parameters of SC-MLGNR interconnect in frequency domain are also the paramount consideration factor for the design of on-chip interconnect system. Bandwidth determines the capability of data transmitting for a system and plays a significant role for an interconnect system [25,26]. For an interconnect system, it is no doubt that a greater bandwidth can effectively reduce the total time to convey a certain number of data [27,28]. At present, the study with respect to the transfer gain and 3-dB bandwidth of the SC-MLGNR interconnect is still rare. Therefore, in this paper, an analytical model for transfer gain and 3-dB bandwidth based on the SC-MLGNR interconnect by substituting the traditional pristine SC-MLGNR structure with inserting the HfO2 dielectric material is proposed. Besides the HfO2 dielectric (ƐHfO2 =25), another potential ultra-high-k dielectric candidates including of TiO2 (ƐTiO2 =95) and SrTiO3 (ƐSrTiO3 = 200) as the interlayer dielectric material of SC-MLGNR interconnect are also discussed in this work.

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2. Interconnect model

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Fig. 1 shows that a single-line MLGNR interconnect placed above the ground plane, where the high-k dielectric is inserted between successive GNR layers to replace the vacuum layer. It is observed that the two adjacent GNR layers are separated by a thin dielectric layer (Ɛ2). However, the two adjacent layers of MLGNR interconnect structure separated by a vacuum layer is widely defined as the pristine MLGNR interconnect. W, Tgnr, Tox and Ɛ1 are line width, line height, thickness of SiO2 dielectric medium and relative dielectric constant of the SiO2 dielectric material, respectively. The total number of layers for MLGNR interconnect is determined by the line thickness Tgnr and can be written as Nlayer =1+Integer[Tgnr /δ] [10]. Here the operator Integer[.] denotes that only the integer part is considered. The thicknesses of each GNR layer and each dielectric layer are all δ (=0.34nm) [3].

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Fig. 1. Geometry of single-line MLGNR interconnect with inserting the high-k dielectric material.

As shown in Fig. 2, an equivalent distributed circuit model for MLGNR interconnect is established. Cd, Rd and Cl are the effective driver capacitance, driver resistance and load capacitance of this driver-MLGNR interconnect-load system, respectively.

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Fig. 2. The equivalent distributed transmission line model of single-line MLGNR interconnect.

The equivalent resistances of SC-MLGNR interconnect can be modeled as the lumped and distributed elements. The lumped resistance Rlu includes quantum resistance Rc and imperfect contact resistance Rq, which are equally arranged at two symmetrical terminals of the equivalent circuit. The distributed scattering resistance exists only when the SC-MLGNR interconnect length Lgnr is greater than the mean free path (MFP) λeff. The lumped resistance Rlu and the per unit length (p.u.l.) distributed scattering resistance Rds are expressed as below [13], Rqm Rcm Rlu = + , (1) Nlayer Nch Nlayer Nch

Rqm   Rds =  N layer N ch λeff  0 

Lgnr > λeff

.

(2)

Lgnr < λeff

Here, the value of monolayer imperfect contact resistance Rcm ranges from 1 KΩ to 20 KΩ as reported in [4]. The monolayer quantum resistance Rqm can be given as Rqm = h/2e2 (herein h is Plank's constant and e is electron charge). Due to the total number of conducting channels of zigzag MLGNR (zz-MLGNR) over armchair MLGNR (ac-MLGNR), thus zz-MLGNR interconnect has lesser distributed scattering resistance than that of ac-MLGNR interconnect. Accordingly, only the zz-MLGNR as interconnect material is analyzed in this paper. Nch is the total number of conducting channels in each GNR layer and can be calculated as [25,29], (3) N ch = b0 + b1W + b2W 2 + b3 E f + b4 E f W + b5 E f 2 . Where b0 to b5 represent the parameters for zz-MLGNR at room temperature (300 K) when the Fermi energy Ef exceed 0 [29]. The analytical expression of effective mean free path (MFP) λeff of MLGNR interconnect with inserting the high-k dielectric material is given as below [16,24],

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λeff = τ v f .

(4)

Here τ and vf represent the scattering time and Fermi velocity of electrons in graphene (=8×10 m/s). The analytical expression of scattering rate τ -1 of GNR with inserting the dielectric materials between successive GNR layers is derived as [24], 5

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2    n π 2π  k BT    τ −1 = 2v f k F i ∫  sin(θ )  π + 2hv f ε 2ε 0 e2 sin(0.5θ ) + 4 − (5)    dθ .  3ne  hv f    ne 0     0.5 Where kF =(πne) is the Fermi momentum. ni and ne are the impurity concentration and electron concentration of graphene, respectively [16]. ħ, kB and T represent the reduced Planck’s constant, Boltzmann’s constant and reference temperature. Ɛ0 is the vacuum dielectric constant (=8.854·10-12 F/m). In order to analyze the effects of high-k dielectric materials on performance of on-chip MLGNR interconnect, the relative dielectric constant Ɛ2 is used to distinguish different high-k dielectric materials in this paper. As shown in Fig. 2, the total distributed capacitance CT of MLGNR interconnect consists of the electrostatic capacitance Cel and the equivalent quantum capacitance Ceq, and their relationship are described as Equation (6). The p.u.l. electrostatic capacitance Cel is depended on the interconnect geometrical dimension and relative dielectric constant Ɛ1 of insulator medium material, and can be deduced as [30],

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CT = Ceq −1 + Cel −1

131

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(

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−1

,

(6) (7)

Herein M [.] is expressed as [31], 2π 1  0≤k ≤   4 4 2 2  2  ln  (2 + 2 × 1 − k ) (1 − 1 − k )    M [k ] =  . (8)  2 2+2 k  1 ≤ k ≤1  ln   2  π  1 − k  The p.u.l. equivalent quantum capacitance Ceq is derived by applying a recursive method as follows [7,13,32],

C1 = Cqm = rec Ci

rec

4e2 Nch , hv f

  1 1  = +  C i −1 Cm   rec  Ceq = C

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)

  πW   Cel = ε1ε 0 M  tanh   .  4Tox   

147

148

)

(

(9)

−1

Nlayer

+ Cqm , .

(10) (11)

rec

Herein, Cqm denotes the p.u.l. quantum capacitance of monolayer GNR. Cm represents the p.u.l. coupling capacitance between two adjacent GNR layers and can be expressed as Cm= Ɛ2Ɛ0W/δ. The total distributed inductance LT of MLGNR interconnect comprises of the equivalent kinetic inductance Leq and the magnetic inductance Lma. Similarly, the p.u.l. equivalent kinetic inductance Leq also can be solved by applying a recursive scheme as [7,32],

L1 = Lk = h / 4e2v f Nch ,

(12)

rec

i

L

rec

  1 1   = +  Li −1 + L Lk  m  rec  N layer

Leq = L

.

−1

,

(13) (14)

rec

Herein, Lk denotes the p.u.l. kinetic inductance of monolayer GNR. Lm is the p.u.l. coupling inductance between two adjacent GNR layers and can be written as Lm=µ 0δ/W (herein µ 0 represents the vacuum magnetic permeability). Therefore, the total distributed inductance LT of MLGNR interconnect is calculated as,

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µT LT = Leq + Lma = Leq + 0 ox . W

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(15)

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3. Interconnect delay model

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Based on our previous paper as shown in [10], a 50% propagation delay model for MLGNR driverinterconnect- load system is expressed as,

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herein the parameter ξ is derived as below,

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Tdelay = (1.48ξ + e−2.9ξ

Cl 1 ξ = 1 +  2  CT Lgnr

   

1.35

(

)

) LT Lgnr CT Lgnr + Cl ,

(16)

−0.5  ( 1

  2 Rds Lgnr + 2 Rlu + Rd ) CT / LT +   . ( R L + 2 R + R ) C 2 /  L C L 2   lu d l  T T gnr    ds gnr

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4. Interconnect bandwidth model

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The ABCD parameter matrix for the MLGNR transmission line excluding the driver and load terminals can be deduced as [10], cosh(θ Lgnr ) Z sinh(θ Lgnr )  A B   (17) C D  =  sinh(θ Lgnr ) .    cosh(θ Lgnr )    Z Where θ and Z are propagation constant and characteristic impedance of the MLGNR interconnect, which are given by, (18) θ = sCT ( Rds + sLT ) ,

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Z=

Rds + sLT . sCT

(19)

In order to investigate the step response, frequency response and bandwidth of MLGNR interconnect, it is essential to obtain the transfer function of input–output interconnect system by means of ABCD parameter matrix approach. Taking the impacts of the driver terminals and lumped elements into consideration, the total ABCD parameter matrix of the MLGNR interconnect is derived as, Rlu  cosh(θ Lgnr ) Z sinh(θ Lgnr )   Rlu  0 1  At Bt  1 Rd   1   1  = (20) 2 2 . sinh(θ Lgnr ) C D  0 1   sC  cosh(θ Lgnr )  0 1  t    d 1 0 1    t    Z  Where all parameters of the total ABCD parameter matrix are obtained by matrix computation as below, 1   2 ( sCd Rd + 1) Rlu + Rd  sinh(θ Lgnr ) + ( sCd Rd + 1) cosh(θ Lgnr ) , At = (20a) Z  1     2 ( sCd Rd + 1) Rlu + Rd  Rlu   Bt =   + ( sCd Rd + 1) Z  sinh(θ Lgnr ) 2Z , (20b)    

+ ( ( sCd Rd + 1) Rlu + Rd ) cosh(θ Lgnr )

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1   sCd Rlu +1 sinh(θ Lgnr ) 2  Ct =  + sCd cosh(θ Lgnr ) , (20c) Z  1     2 sCd Rlu + 1 Rlu   Dt =   + sCd Z  sinh(θ Lgnr )+ ( sCd Rlu + 1) cosh(θ Lgnr ) . (20d) 2Z     Combining of the theory of transmission line, the relationship between the voltage and current of input-output terminals for the MLGNR interconnect displayed in Fig. 2 can be solved as,

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Vin ( s )   At  I ( s )  = C  in   t

Bt  Vout ( s )  . Dt   Iout ( s ) 

(21)

Substituting the expression of voltage and current output of load capacitance Iout(s)=sClVout(s) into Equation (21), the transfer function of driver-MLGNR interconnect-load system is obtained as follows, −1

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4   1 (22) H ( s) = =  1 + ∑ ( ki s i )  .   At + sCl Bt  i =1  Herein, in order to ensure the signal integrity characteristics at the output terminal of interconnect system, the transfer function is adopted a fourth-order pade's expansion. The coefficients in the Equation (22) can be listed as below, 1 1 k1 = Lgnr 2 Rds CT + Cd Rd + Cl ( Rlu + Rd ) + Lgnr Rd CT + Lgnr Rlu CT + Lgnr Cl Rds , (22a) 2 2 1 1 k2 = Lgnr 4 Rds 2CT 2 + (Cd Rd + Cl ( Rlu + Rd )) Lgnr 2CT Rds + Cd Rd RluCl +LgnrCl LT 24 2 1 1 1 1 + Lgnr3CT 2 Rds Rd + Lgnr 3CT 2 Rds Rlu + Lgnr 3CT Cl Rds 2 + Lgnr Cl Rlu 2CT , (22b) 6 12 6 4 1 1 1 + LgnrCl Rd RluCT + LgnrCd Rd RluCT + LgnrCl Cd Rds Rd + Lgnr 2CT LT 2 2 2 1 1 1 k3 = Lgnr 6 Rds3CT 3 + (Cd Rd + Cl ( Rlu + Rd ))( Lgnr 2 CT LT + Lgnr 4 Rds 2 CT 2 ) 720 2 24 1 1 + Rd Rlu Cl Lgnr 2 Rds (6Cd CT + Lgnr CT 2 + Lgnr CT 2 Cd ) + Lgnr 4 Rds LT CT 2 12 12 , (22c) 1 1 + Lgnr 5CT 2 Rds 2 (2CT Rd + CT Rlu + 2Cl Rds ) + Lgnr Cl LT ( Lgnr 2 CT Rds + Cd Rd ) 240 3 1 1 + Lgnr 3CT 2 (4 LT Rd + 2 LT Rlu + Cl Rds Rlu 2 ) + Lgnr Cl Cd Rd Rlu 2 CT 24 4 1 1 k4 = Lgnr 4 LT CT 2 (10 LT + Lgnr 2 Rds 2 CT ) + Lgnr 5CT 2 LT Rds (CT Rlu + 3Cl Rds ) 240 120 1 1 +(Cd Rd + Cl ( Rlu + Rd )( Lgnr 4 Rds LT CT 2 + Lgnr 6 Rds3CT 6 ) 12 720 1 1 1 +Cd Rd Rlu Cl ( Lgnr 2 LT CT + Lgnr 4 Rds 2 CT 2 ) + Lgnr 3CT 2 LT Rd Rlu (Cl + Cd ) 2 24 12 1 1 . (22d) + Lgnr 7 CT 3 Rds 3 (2CT Rd + CT Rlu + 2Cl Rds )+ Lgnr 3CT Cl LT 2 10080 6 1 1 + Lgnr 5CT 3 Rds (Cl Rds Rlu 2 + 8 LT Rd ) + Lgnr 8 Rds 4 CT 4 480 40320 1 3 2 + Lgnr CT Cl (CT LT Rlu + CT Cd Rds Rd Rlu 2 + 8Cd LT Rds Rd ) 24 1 + Lgnr 5CT 2 Rds 2 Rd (CT Cl Rlu + CT Cd Rlu + 2Cl Cd Rds ) 240 Step response is defined as transient response at the output terminal for an interconnect system, which is triggered by a unit step signal at the input terminal. It represents the time domain performance of the output voltage signal for an interconnect system when the input voltage signal switches from logic "0"to logic "1" in a very short time. Therefore, step response can be applied to evaluate the stability of an interconnect system. Here, the input voltage signal Vin(s) =1/s is set as the ideal step-signal in this paper. And the step response of MLGNR interconnect in the Laplace domain is derived as Equation (23). Subsequently, the step response Vout(t) at the time domain can be obtained by using the inverse Laplace transform for the Equation (23) , −1

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4   Vout ( s ) = Vin ( s ) 1 + ∑ (ki s i )  ,    i =1 

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−1   4  i    Vout (t ) = L Vin ( s ) 1 + ∑ (ki s )       i =1   

(23) −1

.

(24)

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In the frequency domain, the transfer gain and bandwidth are considered as the significant frequency parameters to investigate the performance of MLGNR transmission line. We take the s = j2πf into Equation (22) to obtain the analytical expression about the magnitude of transfer gain, which is solved as, 1 Ttran ( f ) = . (25) (1 − k2 (2π f )2 + k4 (2π f )4 )2 + (k1 2π f − k3 (2π f )3 )2

The analytical expression of Equation (25) can be used for obtaining the bandwidth, where the input signal can be transmitted to output terminal without significant loss. Meanwhile, it is no doubt that the driver-MLGNR interconnect-load system can be considered as the RC low pass filter, which can results in a cut-off frequency. In general, the bandwidth of interconnect system can be expressed by the 3-dB bandwidth f3dB. According to the definition of 3-dB bandwidth, the 3-dB bandwidth is equivalent to the cut-off frequency in which the magnitude of transfer gain drops 1 2 . Based on the aforementioned discussion, we propose an analytical expression to calculate accurately the 3-dB bandwidth f3dB as,

a4 (2π f3dB )8 + a3 (2π f3dB )6 + a2 (2π f3dB )4 + a1 (2π f3dB )2 − 1 = 0 .

(26)

Where all coefficients of Equation (26) are exhibited as follows,

a4 = k42 ,

(26a)

222

a3 = −2k2 k4 + k32 ,

(26b)

223

a2 = 2k4 − 2k1k3 + k22 ,

(26c)

224

a1 = −2k2 + k12 .

(26d)

225

5. Results and discussions

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This section will analyze the effects of different high-k dielectric materials on step response, propagation delay, transfer gain and 3-dB bandwidth of the proposed MLGNR interconnect structure at global level. All geometrical and physical electrical parameters at 7 nm technology node are listed as follows [2,33], W=11.5 nm, Tgnr=26.91 nm, Tox=17.25 nm, Ef =0.3 eV, Ɛ0=1.95·10-11 F/m, Ɛ1=3.9Ɛ0, µ 0=4π·10-7 H/m, ni=2×1015 m-2, ne=8×1016 m-2, Rd=20.51 kΩ, Cd= 0.063 fF, Cl= 0.2 fF. Herein Rd, Cd and Cl are the equivalent resistance and capacitance of minimum-sized gate. For the global level (100µm≤Lgnr≤10mm) interconnect [34], the sizes of driver and load terminals are 100 times larger than that of the minimum-size gate [10], then their equivalent values can be redefined as Rd' = Rd /100, Cd' = Cd·100 and Cl' = Cl·100. All the numerical simulation results of the proposed model are solved by MATLAB R2013a, which are validated with Synopsys HSPICE simulation. In HSPICE simulation setup, the long MLGNR interconnect is divided into equal 5 µm length spatial segments in order to maintain high accuracy. For instance, the number of spatial segments is set to 200 for an interconnect length of 1000 µm. Here the MATLAB simulation and HSPICE simulation are implemented by using a PC with Intel Core I7 CPU (3.9GHz). The impacts of different high-k dielectric materials on step response and propagation delay of the proposed MLGNR interconnect are compared and analyzed as shown in Fig. 3, which are obtained by the Equation (24) and Equation (16), respectively. The term of “pristine MLGNR” occurring in the next all figure means that the gap between two adjacent GNR layers is not filled with the dielectric material and can be considered as a vacuum layer. Hence, the relative dielectric constant Ɛ2 for pristine MLGNR case is 1.

244

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(a )

(b)

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Fig. 3. The impacts of different high-k dielectric materials on (a) step response and (b) propagation delay of MLGNR interconnect. Here the interconnect length of Fig. 3(a) is set as Lgnr=1000 µm.

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As shown in Fig. 3a, it is observed that the time of reaching the steady-state value 1 V will decrease with the increase of the relative dielectric constant Ɛ2. As a consequence, increasing the relative dielectric constant Ɛ2 will lead to the decrease of propagation delay of MLGNR interconnect. The corresponding propagation delay with different interconnect length illustrated in Fig. 3b, it is also obvious that the propagation delay of MLGNR interconnect decreases as the relative dielectric constant Ɛ2 increases. Taking the interconnect length of 2000 µm as an instance, the delay time of MLGNR interconnect without inserting dielectric material (pristine MLGNR case) is 4.568 ns while for inserting the high-k dielectrics HfO2, TiO2 and SrTiO3 cases are 3.381 ns, 1.695 ns and 0.848 ns, respectively. Here the delay time for pristine MLGNR case is 5.387 times greater than that of inserting the SrTiO3 between two adjacent GNR layers scheme. This is due to the fact that the effective mean free path λeff will increase as the relative dielectric constant Ɛ2 increases, thereby, leading to a lesser distributed scattering resistance Rds according to the Equations (2), (4) and (5). Moreover, according to our numerical simulation results, the maximum reduction of propagation delay between pristine MLGNR and inserting the SrTiO3 dielectric material cases can reach 12.746 ns for an interconnect length of 4000 µm. Therefore, inserting the high-k dielectric material between successive GNR layers is an efficient method to reduce propagation delay of MLGNR interconnect. Furthermore, as depicted in Fig. 3, the results of the proposed model are in good matching with HSPICE simulation results for all cases. The maximum relative errors between the proposed model and HSPICE simulation for step response and propagation delay are 3.53% and 3.75%, respectively. In addition, it can be obtained that the proposed model can reduce the CPU runtime by an average of 98% in comparison to HSPICE simulation. For the step response of Fig. 3a in inserting the SrTiO3 case, the CPU runtime using the proposed analytical Equation (24) is 0.318 second with MATLAB programs, in contrast to 16.61 second by HSPICE simulation on the same computer. The corresponding CPU runtime of propagation delay for the proposed analytical Equation (16) and HSPICE simulation are 0.268 second and 18.95 second when the interconnect length is chosen to be 1000 µm. The reason behind this is that modified nodal analysis (MNA) is the core analysis approach applied in HSPICE simulation to formulate the circuit system equations. By using the Kirchhoff's current law and the energy conversion theory of the MNA deduces the set of matrix equations. The order of the matrix is dependent on the number of spatial segments and unknown variables in the circuit system. The unknown variables are solved by performing the matrix inversion calculation, which require more calculation time. However, the proposed model is not containing the matrix inversion calculation, thereby achieving a better computational efficiency when compared to HSPICE simulations. Fig. 4 exhibits the frequency response of MLGNR interconnect for a length Lgnr=1000 µm with different relative dielectric constant Ɛ2 cases. Transfer gain represents the magnitude of frequency response for an interconnect system and is the ratio of voltage amplitude between the output and input signal at different frequencies, which can be obtained by the Equation (25). As illustrated in Fig. 4, at the high frequency region, it is remarkable that the transfer gain increases with the increase of the relative dielectric constant Ɛ2. The reason behind this is that the MLGNR interconnect system can be regarded as a RC low pass filter and its cut-off frequency can be approximately described as: 1/(2πCTRdsLgnr2) [8], meanwhile the distributed scattering resistance Rds decreases as the relative dielectric constant Ɛ2 increases. As a result, the MLGNR

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interconnect for inserting the SrTiO3 has a greater cut-off frequency compared with the pristine MLGNR, inserting the HfO2 and TiO2 cases. Here, it is noteworthy that increasing the relative dielectric constant Ɛ2 can result in the increase of interlayer coupling capacitance Cm. However, based on our numerical simulation data, it is infer that the impact of increasing the interlayer coupling capacitance Cm on the total distributed capacitance CT can be ignored. Apparently, there is no doubt that decreasing the distributed scattering resistance Rds will play a major role for improving the transfer gain of MLGNR interconnect when the high-k dielectric material is inserted between successive GNR layers. Moreover, it can be found from Fig. 4 that the results of the proposed model are in good accordance with HSPICE simulation results with the maximum relative error of 3.61% for any cases.

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Fig. 4. The impacts of different high-k dielectric materials on frequency response of MLGNR interconnect. Here the interconnect length is set as Lgnr=1000 µm.

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Besides the transfer gain, the bandwidth is also regarded as the indispensable parameter in the frequency domain. Bandwidth represents the capability of data transmission, hence the larger bandwidth for an interconnect system can effectively reduce the total time to transmit a certain amount of data. Thus, it is necessary to analyze the impacts of the different high-k dielectric materials on 3-dB bandwidth of the MLGNR interconnect. Based on the Equation (26), the corresponding results of different high-k dielectric materials on 3-dB bandwidth of MLGNR interconnect are shown in Table 1.

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Table 1. The impacts of different high-k dielectric materials on 3-dB bandwidth of MLGNR interconnect with different length. Length (µm) 100 200 500 800 1000 1500 2000 3000 4000

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Pristine MLGNR Our model Hspice 1583.731 1562.242 750.653 738.458 245.688 238.387 128.161 123.986 92.114 88.931 48.805 46.917 30.306 28.986 15.001 14.318 8.937 8.529

3-dB bandwidth (MHz) HfO2 inserted MLGNR TiO2 inserted MLGNR Our model Hspice Our model Hspice 2149.169 2128.258 4102.630 4078.136 1024.817 1012.912 2007.133 1990.246 335.226 327.517 665.822 652.114 174.427 169.526 347.640 338.191 125.187 121.268 249.803 242.720 66.150 63.744 132.225 128.013 41.003 39.313 82.035 79.030 20.250 19.404 40.556 38.833 12.049 11.497 24.145 23.040

SrTiO3 inserted MLGNR Our model Hspice 7660.850 7628.128 3914.607 3893.411 1323.508 1307.438 694.314 683.117 499.890 490.137 265.471 258.339 165.051 159.953 81.808 78.890 48.779 46.824

As described in Table 1, it can be seen that 3-dB bandwidth will be expanded as the relative dielectric constant Ɛ2 increases at the same interconnect length. Giving the interconnect length of 3000 µm as an example, the 3-dB bandwidth for inserting the SrTiO3 scheme is 5.454 times larger than that of pristine MLGNR case. The corresponding values for inserting the TiO2 scheme is 2.704 times larger than that of pristine MLGNR case. The another situation for inserting the HfO2 scheme is 1.350 times larger than that of pristine MLGNR case. Based on the aforementioned discussion, it can be explained that distributed scattering resistance Rds will decrease with the increase of the relative dielectric constant Ɛ2, hence inserting the SrTiO3 has a larger cut-off frequency compared with the pristine MLGNR, inserting the HfO2 and TiO2 cases.

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Moreover, Table 1 shows that 3-dB bandwidth of MLGNR interconnect decreases as the interconnect length increases for all of pristine MLGNR, inserting the HfO2, TiO2 and SrTiO3 cases. This is due to the fact that the increase of interconnect length gives rise to parasitic resistance, inductance and capacitance. In addition, based on the data of Table 1, it is indicate that 3-dB bandwidth of inserting the SrTiO3 scheme is enhanced over 4 times than that of pristine MLGNR case for all interconnect length. Therefore, inserting the high-k dielectric material as a new emerging technology for the global level interconnect has a significant application prospect in improving bandwidth. Furthermore, it can be seen from the Table 1 that the results obtained by the proposed model for all cases are fairly consistent with HSPICE simulation results. The maximum relative error involved in the proposed model and HSPICE simulation is 4.80% in case of inserting the HfO2 scheme with the interconnect length of 4000 µm. Here the simulation runtime of the proposed model and HSPICE is investigated. An average reduction of 95% is achieved by applying the proposed model. As an instance, using the SrTiO3 inserted case of interconnect length of 4000 µm with the 800 spatial segments, the CPU runtime using the proposed model is 0.618 second by carried out the Equation (26) with MATLAB programs, against 29.52 second using HSPICE simulator.

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6. Conclusions

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Based on the transmission line model, an equivalent distributed circuit of MLGNR interconnect is modeled according to the extracted parasitic parameters of resistance, inductance and capacitance for 7.5nm technology node at global level. Using the extracted parasitic parameters, an analytical propagation delay model is presented. Furthermore, the analytical expressions of step response, transfer gain and 3-dB bandwidth of the equivalent MLGNR interconnect circuit can be obtained by applying the ABCD parameter matrix approach. Based on the proposed the analytical model, the impacts of different high-k dielectric materials on propagation delay, step response, transfer gain and 3-dB bandwidth are predicted. The results show that substituting the conventional pristine MLGNR case with inserting the high-k dielectric material for the MLGNR interconnect has a greater performance advantage in terms of the propagation delay, step response, transfer gain and 3-dB bandwidth at the same conditions. The results obtained by the proposed model have great consistency with HSPICE simulation, and the proposed model is highly runtime efficient than HSPICE simulation. According to our simulation results, it can be expected that inserting the high dielectric materials between successive GNR layers can be regarded as a new emerging technology to enhance the performance of delay time, step response, transfer gain and bandwidth of the on-chip MLGNR interconnect.

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Acknowledgements

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References

This work was supported by Guangzhou Science and Technology Project (Grant No. 201904010107), Guangdong Provincial Natural Science Foundation of China (Grant No. 2019A1515010793), Guangdong Province Science and Technology Project (Grant No. 2016B090918071), and National Natural Science Foundation of China (Grant No. 61072028).

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The analytical expressions of propagation delay, transfer gain and 3-dB bandwidth of SC-MLGNR interconnects with inserting high-k dielectric materials is firstly derived. The performance of propagation delay, transfer gain and 3-dB bandwidth of SC-MLGNR interconnects can be improved by inserting high-k dielectric material. The maximum reduction of propagation delay between SC-MLGNR by inserting SrTiO3 dielectric material and conventional SC-MLGNR can reach 12.746 ns for an interconnect length of 4000 µm, and corresponding 3-dB bandwidth for them can be expanded over 4 times.

Declaration of Interest Statement We declare that no conflict of interest exits in the submission of this manuscript. Zhongliang Pan and Peng Xu School of Physics and Telecommunication Engineering, South China Normal University, Panyu District, Guangzhou City, Guangdong Province, China Dec. 4, 2019