=Paper=
{{Paper
|id=Vol-2930/paper8
|storemode=property
|title=DFT Analysis of Different Shaped Cu Nanowires for Interconnect Application
|pdfUrl=https://ceur-ws.org/Vol-2930/paper8.pdf
|volume=Vol-2930
|authors=Sonal Agrawal,Anurag Srivastava,Gaurav Kaushal
}}
==DFT Analysis of Different Shaped Cu Nanowires for Interconnect Application==
https://ceur-ws.org/Vol-2930/paper8.pdf
DFT Analysis of Different Shaped Cu Nanowires for Interconnect
Application
Sonal Agrawala, Anurag Srivastavaa and Gaurav Kaushalb
a
Advanced Material Research Group CNT lab, ABV-Indian Institute of Information Technology and
Management Gwalior 474015, India
b
VLSI design lab ABV-Indian Institute of Information Technology and Management Gwalior 474015, India
Abstract
In the present work, Density functional theory (DFT) based comparative analysis has been
implemented to analyze the structural stability, electronic and transport properties of Copper
(Cu) nanowires with varied morphologies and diameters. The calculation of formation energy
shows the stability of Cu nanowires increases with increasing the diameter. Further, from the
transport properties analysis, it confirms that the rectangular morphology of Cu nanowire at
1.8 nm diameter shows linear I-V characteristics and significantly low interconnect
parameters i.e. kinetic inductance and quantum capacitance in comparison to other
morphologies and diameters of Cu nanowires, and which is good for interconnect
perspective. Hence, it can be concluded that Cu nanowire with 1.8 nm diameter of
rectangular morphology may be a possible candidate for interconnect application.
Keywords 1
Copper nanowires, Dynamical parameters, Interconnects, morphology
1. Introduction
Advances in integrated circuit technology require that the chip dimensions decreases in size, to
satisfy the Moore’s law of continuous miniaturization, which creates the need to scale down the size
of the metallic interconnects too [1]. At nanoscale the scaling of interconnect dimensions leads to rise
in Cu resistivity that causes performance degradation. The resistivity of a Cu wire increases with
decreasing area of cross-section due to quantum effects, which become significant as the interconnect
dimensions get closer in magnitude to the electron mean free path (∼40 nm) around room temperature
[1]. Cu faces two critical problems as a nanoscale interconnect material, first, its inability to carry
high current densities, and increased resistivity [2] due to surface scattering of electrons and grain-
boundary issues [3]. The increase in resistivity of the nanoscale Cu interconnects severely impacts the
interconnect performance, in turn affecting adversely the performance of the nanoelectronic circuits
incorporating them.
In spite of these problems, there are qualities that maintain Cu as the material of choice for making
interconnects. First, Cu is reasonably inert under a variety of conditions. Other reason for choosing Cu
is quite abundantly available making it lucrative from a commercial point of view. There are some
experimental reports [3],[4] available which reveals that bulk Cu may be replaced with Cu nanowire
as interconnects in future nanoscale devices due to various challenges in the bulk Cu.
The analysis of electronic, magnetic, thermoelectric, optical and transport properties of various
nanowires like CdS, CdO, CdTe, Fe and Co have been recently reported elsewhere [5], [6], [7], [8]
depending on the shape, geometry and applied pressure. In another report [9] the analysis of Pt, Rh,
Ir nanowire, have been presented and compared their performance with bulk Cu, and concluded that
VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),
September 14–16, 2021, Khabarovsk, Russia
EMAIL: profanurag@gmail.com (Anurag Srivastava)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
70
these nanowires are relatively better than Cu bulk structure for interconnect application. In another
report on Cu nanowires, Ma et al. [10] have analysed the structural and electronic properties of Cu
nanowires with different cross sections and concluded that number of channel increases as nanowires
diameter increases. In [11], the structural and mechanical properties of nanowires have been
compared with the experimental results. In spite of available reports on nanowires, this attempt of
ours is probably for the first time to compute the structural electronic and transport properties of
morphology dependent Cu nanowires for interconnect application. Reason being, the different
morphologies and size of cu nanowires can be formed during the synthesis process.
In the present work, the performance of interconnects has been measured in terms of the structural
stability, bandstructure, density of states, conductance, quantum capacitance and kinetic inductance.
The quantum capacitance and kinetic inductance are calculated following the approach reported in
[12, 13]. Further, analysing the interconnect parameters with different diameters, the relatively best
morphology of Cu nanowires has been considered as a candidate for interconnect application.
2. Computational methods
The structures of the nanowires are created using the FCC crystal structure with bulk lattice
constant of a = 3.61 Å for Cu. The wire axis is taken along the [100] lattice direction. Further, the
electronic structure calculations are based on first principles density functional theory (DFT) using the
Atomisticx-tool kit virtual nanolab (ATK-VNL) [14,15]. DFT calculations with generalized gradient
approximation (GGA) [16] in form of Perdrew-Burke-Erenzhorf (PBE) functional are considered for
ground state properties calculations. Mesh cut-off energy of 75 Hartee. Double zeta double polarized
(DZDP)) basis set and brillouin zone integration with k-point Monkhorst-Pack sampling of 1 x 1 x 20
are selected for geometry optimization and electronic properties evaluation. Whereas 1 x 1 x 100 k
points are selected for the computation of transport properties. Structures are relaxed until forces on
each atom reaches to 0.05 eV/Å. To confine the dimension of nanowires along transverse direction
with respect to longitudinal one (nanowire periodically growth) large cross section of super cell has
been chosen to avoid mirror image interaction and to ensure negligible interaction between the wire
and its replica, vacuum space of 15 Å is used.
To compute the transport properties, we have designed a two probe setup has been modelled by
extending the channel regions in electrodes, which can be explained through NEGF approach and
computes the current using the Launder–Büttiker formula reported elsewhere[17]
2𝑒 μR
𝐼 = ℎ ∫μL (𝑇(𝐸)[ FL (E) − FR (E)])dE (1)
R A
T(E, V) = Tr(ГL (E, V)G (E, V)ГR (E, V)G (E, V)) (2)
Hereµ𝑅 and µ𝐿 represents the chemical potential of left and right electrode, respectively, with Fermi
distribution functions FL(E) and FR(E). T(E) is total transmission probability of all the channels at
applied voltage V and energy E. GR and GA are the retarded and advanced Green’s function with ГL
and ГR as coupling functions of left and right electrode self-energies.
3. Result and Discussion
The unit cell of Cu nanowires with three different diameters (1 nm, 1.4 nm and 1.8 nm) of
tetragonal morphology have been optimized and shown in fig. 1(a), 2(a), 3(a) respectively. Further
different morphologies of nanowires like rectangular, square and triangular of nanowires are created
by removing the number of atoms with varied diameter of nanowires (shown in fig. 1(b-d), 2(b-d),
3(b-d)) and their structural, electronic and transport properties have been investigated. All the
theoretical characterizations are done in terms of formation energy per atom, bandstructure, density of
states, conductance, kinetic inductance, quantum capacitance and I-V characteristics.
71
3.1. Stability analysis
As the stability of any structure at nanoscale is a big challenge, especially for certain device
application. The stability of structure has been analysed by examining the formation energy (Eform) per
atom for each morphology using equation (3) and reported in table 1.
𝐸total − 𝑛𝐸𝐶𝑢
Eform = 𝑛
(3)
Where Etotal is the total energy of the unit cell of nanowire, ECu are the energy of isolated Cu atom,
n represent the total number of atoms in periodic unit cell of the nanowire.
From the Table 1, it can be observed that as the diameter of nanowire increases, the stability also
increases evaluated in terms of formation energy per atom. It is also observed that, analysis with
varied diameters (1 nm, 1.4 nm and 1.8 nm), the tetragonal morphology has higher stability in
comparison to its other counterpart. The modelled Cu nanowire geometry follows the similar trend i.e.
tetragonal > rectangular> square > triangular, with an interesting fact that the triangular morphology
at 1.4 nm diameter is unstable due to bond breaking.
Table 1
Formation energy/atom (eV) of nanowires with different morphologies
Morphology Diameter 1 nm Diameter 1.4 nm Diameter 1.8 nm
Tetragonal -2.02477 -2.2195 -2.37347
Rectangular -1.87984 -2.15021 -2.33543
Triangular -1.6571 -1.80474 NA
Square -1.65708 -2.07072 -2.2951
(a) (b) (c) (d)
Figure 1: Optimized geometries of Cu nanowire with different morphologies at 1 nm diameter
(a)Tetragonal (b) Triangular (c) Rectangular (d) Square
(a) (b) (c) (d)
Figure 2: Optimized geometries of Cu nanowire with different morphologies at 1.4 nm diameter (a)
Tetragonal (b) Triangular (c) Rectangular (d) Square
72
(a) (b) (c) (d)
Figure 3: Optimized geometries of Cu nanowire with different morphologies at 1.8 nm diameter (a)
Tetragonal (b) Rectangular (c) Square (d) Triangular
3.2. Electronic Properties
To analyze the electronic properties, the bandstructure and density of states profiles have been
computed and shown in fig S1(a-d), S2(a-d), and S3(a-d) for 1 nm, 1.4 nm and 1.8 nm diameter
respectively, in the supplementary information. From the bandstructure and density of states profile, it
is clear that all the nanowires with different morphologies and different diameters shows metallic
behaviour. As the diameter increases the metallicity also increases, number of bands at the Fermi level
increases resulting in dense bandstructure with increase in diameter. From the bandstructure (E-K
diagram) the Fermi velocity of each morphology has been calculated using equation (4) reported
elsewhere [18]
1 𝑑𝐸
Vf= ℎ 𝑑𝐾 (4)
Where ‘h’ is the Planck constant and ‘dE/dK’ is the slope of the E-K diagram. From the equation
(4) [18] it can be seen that Fermi velocity is proportional to the slope of E-K diagram. The Fermi
velocity has been calculated at the Z point of brillouin zone. The table 2 confirms that Fermi velocity
follows the order,(abbreviations:-Cu_radius in Å_Morphology) Cu_5_squ > Cu_7_rec > Cu_5_tetra >
Cu_9_squ >Cu_9_rec > Cu_9 >Cu_7_tetra > Cu_7_squ > Cu_5_rec > Cu_7_tri > Cu_5_tri, which
conclude that the Cu nanowires at 1nm diameter have slightly higher fermi velocity in comparison to
that of 1.4 and 1.8 nm diameters, due to the higher slope of bandstructure profile of Cu nanowires.
The calculated Fermi velocity will be used in the analysis of dynamical parameters.
Table 2
Fermi Velocity (m/s) of nanowires with different morphologies
Morphology Diameter 1 nm Diameter 1.4 nm Diameter 1.8 nm
Tetragonal 1.86x10^4 8.38x10^3 8.48X10^3
Rectangular 5.78x10^3 1.87x10^4 9.83X10^3
Triangular 1.34x10^4 3.86x10^2 NA
Square 2.37x10^4 6.97x10^3 1.2X10^4
3.3. Transport properties
Transport properties of nanowires has been analysed using two probe approach. A two-probe
model has been designed by repeating the unit cell of nanowire by 3 times and extending the channel
region into the electrode region.
73
3.3.1. Conductance analysis
The conductance of all the varied diameter and morphologies of Cu nanowires computed by
calculating the transmission spectrum. From the bandstructure analysis, it has been confirmed that as
the diameter increases metallicity increases, hence conductance also increases with increasing
diameter as shown in Fig. 4. From the fig. 4, it is observed that the conductance increases with
increase in diameter and at each diameter 1 nm, 1.4 nm and 1.8 nm diameters, tetragonal
morphologies show higher conductance in comparison to other considered morphologies. Other
morphologies (rectangular and square) also show good conductance with a smaller number of atoms
as compared to tetragonal morphology.
Figure 4: Zero bias conductance of different morphologies of Cu nanowires at 1 nm, 1.4 nm and 1.8
nm diameters (abbreviations: - Cu_radius in Å_morphology)
3.3.2. I-V Characteristics
Further the I-V Characteristics of the different morphologies of nanowires at different diameters
have also been computed and shown in fig. 5 (a-d). The I-V characteristics of simplest natural
structures i.e. Linear Atomic Chains (LAC) of all the popular metals are also computed for the
comparison of results with the Cu nanowires and shown in Fig. 5(e).
At 1 nm diameter with different morphologies of Cu nanowire, current is highest in tetragonal
morphology up to 0.5 V, however, beyond 0.5 V it shows decrease in current. Whereas, in case of
rectangular morphology the current is highest in the voltage range 0.5 V to 1 V. The triangular and
square morphologies show monotonically increasing I-V characteristics in the bias range 0-1 V.
At 1.4 nm diameter, current is lowest in triangular morphology and highest in tetragonal
morphology with monotonously increasing trend. Whereas, in the rectangular morphology, current is
linear with (constant slope) in the applied voltage range of 0-1 V, hence, this morphology can be a
good candidate for interconnect application. Further, increasing the diameter to 1.8 nm, all the stable
morphologies show monotonically increasing I-V characteristics and rectangular morphology shows
linear current with almost constant slope i.e. constant resistance, hence, these can be a potential
candidate for interconnect application.
When the I-V characteristics of all the morphologies with varied diameters are being compared,
then the morphologies at 1.8 nm diameters shows relatively better conductance. The I-V
characteristics of these nanowires are also compared with the natural linear atomic chains of metals
i.e. gold, silver and copper (as shown in fig. 5(d)), From the results it can be observed that these
metallic chains show maximum current of 80 µA, whereas the tetragonal morphology of Cu
nanowires have maximum current up to 1200 µA at 1.8 nm diameter, which is almost 15 times higher
than the metallic LACs.
From the above observations and comparison of I-V characteristics at different diameter and
different morphologies, it is observed that although the current is highest with monotonously
increasing trend in tetragonal morphology of Cu nanowire at 1.8 nm diameter, suffers its linear
behaviour. Rectangular morphology gives linear current with increases in the bias voltage range 0-1 V
74
which is desired for interconnect application. For much better understanding of the properties of
nanowires, transmission eigen values or transmission channel (Nch) of each nanowires has also been
computed and observed that number of channel increases with increase in diameter, which is in
agreement with reports [10].
Further, the tensile stress in the periodic direction has also been computed and given in table 3.
From the stress analysis, it has been observed that the proposed Cu nanowires of 1 nm and 1.4 nm
have higher stress in the optimized geometries. However, the rectangular, square morphology at 1 nm
diameter and rectangular, square and triangular morphology at 1.4 nm diameter, retain almost linear
current voltage characteristics under stressed condition up to certain bias voltages.
(a) (b) (c)
(d) (e)
Figure 5: I-V characteristics of Different nanowire with different morphologies (a) At diameter of 1
nm (b) 1.4 nm diameter (c) 1.8 nm diameter (d) Comparison of both 1 nm, 1.4 nm and 1.8 nm
diameter (e) I-V characteristics of metallic atomic chains
Table 3
Stress (eV/Å3) in different geometries of nanowires at different diameters
Morphology Diameter 1 nm Diameter 1.4 nm Diameter 1.8 nm
Tetragonal 0.01206921 4.63018388e-3 -4.08972570e-03
Rectangular 0.08139395 1.47867e-3 -0.00114492
Triangular 0.02471341 3.83135199e-3 NA
Square 0.0043406 2.62136173e-3 1.27405730e-03
3.4. Dynamical parameter analysis
For the interconnect point of view, the dynamical parameters like Resistance, kinetic inductance
and quantum capacitance also play important role, especially the challenges of interconnects at
nanoscale, [1, 19, 20, 21], as these parameters are responsible for interconnects delay. Interconnect
delays are proportional to kinetic inductance and quantum capacitance, for lower interconnect delays
these quantities should be low. Further these parameters have been computed by using the equations
given by (5) and (6) for analysing the possible interconnects performance and shown in fig. 6.
The resistance of the two-probe system can be modelled by the equation (7) shown in the
following [22]:
𝐶𝑄 =4𝑒 2 𝑁 (5)
ℎ𝑣𝑓 𝑐ℎ
75
LK ℎ 1 (6)
= 2
4𝑒 𝑣𝑓 𝑁𝑐ℎ
1 ℎ 1
RQ = = (7)
𝐺𝑄 2𝑒 2 𝑁𝑐ℎ
Where ‘RQ’ is defined as the quantum of resistance and ‘GQ’ is quantum of conductance defined
for very small lengths. ‘e’ represents electronic charge, ‘h’ represents Planck’s constant, and ‘Nch’
represents the number of transmission channels. From the equation (7) the dependence of resistance
on the number of transmitting channel (Nch) can be observed. Here the transmission
modes/transmission channel are number of half wavelength that can travel in a given width. From the
table 4, it is observed that the dynamical parameters of Cu nanowires are lowest at 1nm diameters,
with poor I-V characteristics and less stability. Whereas, at 1.8 nm diameter Cu nanowire shows
acceptable dynamical parameters with linear as well as monotonically increasing I-V characteristics
with higher stability and hence can be defended as a potential candidate for interconnect application.
Table 4
Number of transmission channel (Nch), quantum capacitances (CQ), kinetic inductance (LK) and
resistance (RQ) of different morphologies of Cu nanowires at different diameter
Diameters Cu nanowire Nch CQ(F/m) LK(H/m) RQ(kΩ)
morphologies
1 nm Cu_Tetragonal 10 8.3087E-08 3.48E-02 1.29
Cu _Rectangular 5 1.3369E-07 2.24E-01 2.58
Cu _Triangular 4 4.6132E-08 1.21E-01 3.225
Cu _Square 5 3.2604E-08 5.46E-02 2.58
1.4 nm Cu_Tetragonal 17 3.1351E-07 4.54E-02 0.758824
Cu _Rectangular 17 1.4049E-07 2.04E-02 0.758824
Cu _Triangular 4 1.6015E-06 4.19E+00 3.225
Cu _Square 13 2.8824E-07 7.14E-02 0.992308
1.8 nm Cu_Tetragonal 46 8.3832E-07 1.66E-02 0.280435
Cu _Rectangular 32 5.0309E-07 2.06E-02 0.403125
Cu _Square 32 4.1211E-07 1.69E-02 0.403125
Figure 6: Dynamical parameters (quantum capacitance and kinetic inductance) of different
morphology of Cu nanowires at different diameters
76
4. Conclusion
On analyzing the electronic and transport properties of Cu nanowires with three different diameters
and four different morphologies. It is observed that the stability of Cu nanowires increases with
increase in diameter. The transport of Cu nanowire with rectangular morphology at 1.8 nm diameter is
better as compared to other counterparts, due to its linear I-V characteristics, with higher current than
LACs as well as higher number of transmission channel. The kinetic inductance and quantum
capacitances values of rectangular morphologies are 503.09 nF/m and .0206 H/m, respectively. From
all the above observations it can be concluded that Cu nanowire with rectangular morphology at 1.4
nm diameter may be a potential candidate for interconnect application.
5. Acknowledgement
The authors are thankful to Computational Nanoscience and Technology Laboratory, ABV-IIITM
Gwalior for providing infrastructure support and one of us S.A. is thankful to Ministry of Education
for the Ph.D. fellowship.
6. References
[1] N. Kharche, S. R. Manjari, Y. Zhou, R. E. Geer, and S. K. Nayak, “A comparative study of
quantum transport properties of silver and copper nanowires using first principles calculations,”
vol. 085501.
[2] D. Josell, S. H. Brongersma, and Z. Tőkei, “Size-Dependent Resistivity in Nanoscale
Interconnects,” Annu. Rev. Mater. Res., vol. 39, no. 1, pp. 231–254, 2009.
[3] W. H. Xu, L. Wang, Z. Guo, X. Chen, J. Liu, and X. J. Huang, “Copper nanowires as nanoscale
interconnects: Their stability, electrical transport, and mechanical properties,” ACS Nano, vol. 9,
no. 1, pp. 241–250, 2015.
[4] S. Khan and A. Srivastava, “Geometry dependent structural and electronic properties of CdS
nanowires: An ab-inito study,” Superlattices Microstruct., 2016.
[5] R. Chaurasiya, S. Khan, and A. Srivastava, “Shape dependent structural stability , electronic and
optical properties of CdO nanowire,” pp. 252–256, 2015.
[6] M. Bhatia, S. Khan, and A. Srivastava, “Pressure Induced Phase Transition in CdTe nanowire : A
DFT Study,” vol. 040033, 2018.
[7] S. Kansara, S. K. Gupta, Y. Sonvane, and I. Lukačević, “Modeling of diameter-dependent Fe and
Co ultrathin nanowires from first-principles calculations,” Phys. Chem. Chem. Phys., vol. 19, no.
23, pp. 15412–15423, 2017.
[8] N. A. Lanzillo, “Ab Initio evaluation of electron transport properties of Pt, Rh, Ir, and Pd
nanowires for advanced interconnect applications,” J. Appl. Phys., vol. 121, no. 17, 2017.
[9] L. C. Ma, J. M. Zhang, and K. W. Xu, “Structural and electronic properties of ultrathin copper
nanowires: A density-functional theory study,” Phys. B Condens. Matter, vol. 410, no. 1, pp.
105–111, 2013.
[10] J. Jia, D. Shi, J. Zhao, and B. Wang, “Structural properties of silver nanowires from atomistic
descriptions,” Phys. Rev. B - Condens. Matter Mater. Phys., vol. 76, no. 16, pp. 1–5, 2007.
[11] A. Thakur and R. Kashyap, “Study of transmission function and electronic transport in (DFT),”
AIP Conf. Proc., vol. 1953, 2018.
[12] S. Salahuddin, M. Lundstrom, and S. Datta, “Transport Effects on Signal Propagation in
Quantum Wires,” vol. 52, no. 8, pp. 1734–1742, 2005.
[13] N. Srivastava and K. Banerjee, “Performance analysis of carbon nanotube interconnects for VLSI
applications,” in Proceedings of the 2005 IEEE/ACM International conference on Computer-
aided design, 2005, pp. 383–390.
[14] “https://www.synopsys.com/silicon/quantumatk.html.” .
[15] J. M. Soler et al., “The SIESTA method for ab initio order-N materials simulation,” J. Phys.
Condens. Matter, vol. 14, no. 11, p. 2745, 2002.
77
[16] J. P. Perdew, K. Burke, and Y. Wang, “Generalized gradient approximation for the exchange-
correlation hole of a many-electron system,” Phys. Rev. B, vol. 54, no. 23, p. 16533, 1996.
[17] R. Lake and S. Datta, “Nonequilibrium Green’s-function method applied to double-barrier
resonant-tunneling diodes,” Phys. Rev. B, vol. 45, no. 12, pp. 6670–6685, Mar. 1992.
[18] S. Yamacli, “Investigation and comparison of the large-signal characteristics and dynamical
parameters of silicene and germanene nanoribbon interconnects,” Comput. Mater. Sci., vol. 141,
pp. 353–359, 2018.
[19] S. Yamacli, “Comparison of the electronic transport properties of metallic graphene and silicene
nanoribbons,” J. Nanoparticle Res., vol. 16, no. 8, 2014.
[20] S. Yamacli, “Extraction of the voltage-dependent quantum capacitance and kinetic inductance of
GNRFETs: a first-principles study,” J. Comput. Electron., vol. 14, no. 1, pp. 249–256, 2015.
[21] C. Xu, H. Li, and K. Banerjee, “Modeling , Analysis , and Design of Graphene,” IEEE Trans.
Electron Devices, vol. 56, no. 8, pp. 1567–1578, 2009.
[22] V. Sharma, P. Srivastava, and N. K. Jaiswal, “Edge-Oxidized Germanene Nanoribbons for
Nanoscale Metal Interconnect Applications,” IEEE Trans. Electron Devices, vol. 65, no. 9, pp.
3893–3900, 2018.
78