length l width w frequency f material technology
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© J. Lienig, M. Thiele, Fundamentals of Electromigation-Aware Integrated Circuit Design. Springer, ISBN 978-3-319-73557-3, 2018.Figures of Chapter 4: Mitigating Electromigration in Physical Design
Length L Width W Frequency f Material Technology
Bamboo effect, Sect. 4.2
Blech effect, Sect. 4.3
Via effects, Sect. 4.4
Reservoir effect, Sect. 4.5
Via configuration, Sect. 4.6
Self-healing, Sect. 4.7
Passivation, Sect. 4.8
Immunity, Sect. 4.9
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Wire width w
ReliabilityMTF [h]
I = constantT = constant
wMTF_min
w < ∅ Grains(Bamboo wires)
Grain boundaries
we-
wmin …Thermal self-heatingwidth limit
wmin
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100
101
102
103
104
Wid
th in
nm
2020 2024Year
2016
Minimum width (metal 1)Maximum bamboo structure SLAMaximum bamboo structure TA
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– +
FN5FN4FN3FN2FN1
Cu+Cu+ Cu+Cu+ Cu+Cu+
Electromigration (EM)
– +
FN5FN4FN3FN2FN1
Cu+Cu+Cu+ Cu+Cu+
Cu+Cu+
Equilibrium between EM and SMif Lwire < “Blech length”
– +
FN5FN4FN3FN2FN1
Cu+Cu+Cu+ Cu+Cu+Cu+
Stress Migration (SM)
e-
e-
e-
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e–
(−) (+)
Segment
e–
Top view
Side view
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σcritical
0
Tens
ilest
ress
σ
Lx
t
e−σ(0) σ(L)
Cathode Anode
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0L
x
t
σcritical
Tens
ilest
ress
σ
e−
Void σ(L)
Cathode Anode
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Lvoid
L
e−
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e−
Lvoid
L
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100 102 104
100
105
1010
Net lengths as a multiple of the gate pitch
Number of nets
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2016 2020 2024Year
Mean segment lengths (metal 1 through 6)Blech lengths
Leng
thin
nm
101
102
103
104
105
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Metal n+1
Metal n
Via layer e−
Metal n
Metal n−1
Via layer e−
Via-above
Via-below
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e−
Failure
Void
VoidVia-above
Via-below
Metal n+1
Metal n
Via layer
Metal n
Metal n−1
Via layer
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Blech lengths for via belowBlech lengths for via aboveMean segment lengths (metal 1 through 6)
2016 2020 2024Year
102
103
104
105
Leng
thin
nm
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+e−
SinkSource –
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e–
End-of-line reservoirVia
L
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Side reservoir Via
L
e–
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102
103
104
105
Leng
thin
nm
Blech lengthsBlech lengths with reservoirMean segment lengths (metal 1 through 6)
2016 2020 2024Year
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Single via Double via Via array
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(a) (b) (c) (d)
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3
2
1
0
e− e−
j / jref
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e–
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j / jref
1.6
1.2
0.8
0.4
0
e−
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min
max
j / jrefjref
jref
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100 103 1061
10
100
1000
f in Hz109 1012
Self-healing
Skin effect
Current frequencies
MTF(AC)MTF(DC)
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Power supply net Clock net Signal netSource Sink Sink SinkSource Source
I I I
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Metal(Cu)
Metal liner Dielectric cap
Dielectric (e.g., SiO2)
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Metal Specific resistance ϱ in µΩ∙cm
Activation energy Ea in eV Source
Aluminum 2.44 0.61 [EJS91]
Silver 1.47 0.66 [EJS91]
Copper 1.54 0.70 [EJS91]
Gold 2.03 0.75 [EJS91]
Tungsten 4.84 1.89 [MIH+90][EJS91]
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Material εr E in GPa
Vacuum or air 1 —Aerogels 1.1–2.2 ≈ 0.001Polyimide (organic) 2.7 3.7 Silicon oxide (SiO2) 3.9 300 Glass-fiber-reinforced epoxy resin 5 20 Silicon nitride (Si3N4) 7.5 297 Aluminum oxide (Al2O3) 9.5 264 Silicon 11.7 99
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Barrier Activation energy Ea in eV Sources
Ta 2.1 [Gla05] Ta/TaN 1.4 [HGR06] SiN or SiCxNyHz 0.7–1.1 [Gla05][HGR06] SiN on Cu(Ti) 1.3 [HGR06] CoWP 1.9–2.4 [HGR06] SiCxHy 0.9 [HGR+03]
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(a) Armchair (b) Zig-zag (c) Chiral
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Cu Single-wall CNTs Multi-wall CNTs Cu-CNT CompositesMaximum current density (A/cm2) < 1∙107 > 1∙109
[YKD00]> 1∙109
[WVA01]> 6∙108
[SSY16]
Thermal conductivity @300K (W/m∙K) 385 3,000-10,000 [MPG13] 3,000
[MPG13]~ 800
[SYK13]
Electrical conductivity (S/cm) 5.8∙105 7∙105
[LYY04]2.7∙105
[KKR12](2.3 - 4.7)∙105
[SSY16]
Electron mean free path @300K (nm) 40 > 1,000
[PHR07]> 25,000[BYW02]
Coefficient of thermal expansion (1/K)
17∙10-6
[BS06](−0.3 - +0.4)∙10-6
[JLH04]*(4 - 5)∙10-6
[ARW11]
Young's modulus (GPa) 129[Bei03]
~ 1,000[Bel05][MR06]
~ 900[MR06]
Tensile strength (GPa) 0.2[BS06]
~ 100[MR06]
*) @ 400 K, axial direction; the specific value depends on temperature, diameter, CNT structure, and direction
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1 10 100
1
10
100
Interconnect line length (µm)
Resistivity(µΩ⋅cm)
10000.1
Cu line W=10 nm
Cu line W=40 nmCu line W=80 nm
SWCNT D=1 nm
MWCNT D=40 nm
MWCNT D=80 nm
MWCNT D=10 nm