© 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

Wire width w

ReliabilityMTF [h]

I = constantT = constant

wMTF_min

w < ∅ Grains(Bamboo wires)

Grain boundaries

we-

wmin …Thermal self-heatingwidth limit

wmin

100

101

102

103

104

Wid

th in

nm

2020 2024Year

2016

Minimum width (metal 1)Maximum bamboo structure SLAMaximum bamboo structure TA

– +

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-

e–

(−) (+)

Segment

e–

Top view

Side view

σcritical

0

Tens

ilest

ress

σ

Lx

t

e−σ(0) σ(L)

Cathode Anode

0L

x

t

σcritical

Tens

ilest

ress

σ

e−

Void σ(L)

Cathode Anode

Lvoid

L

e−

e−

Lvoid

L

100 102 104

100

105

1010

Net lengths as a multiple of the gate pitch

Number of nets

2016 2020 2024Year

Mean segment lengths (metal 1 through 6)Blech lengths

Leng

thin

nm

101

102

103

104

105

Metal n+1

Metal n

Via layer e−

Metal n

Metal n−1

Via layer e−

Via-above

Via-below

e−

Failure

Void

VoidVia-above

Via-below

Metal n+1

Metal n

Via layer

Metal n

Metal n−1

Via layer

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

+e−

SinkSource –

e–

End-of-line reservoirVia

L

Side reservoir Via

L

e–

102

103

104

105

Leng

thin

nm

Blech lengthsBlech lengths with reservoirMean segment lengths (metal 1 through 6)

2016 2020 2024Year

Single via Double via Via array

(a) (b) (c) (d)

3

2

1

0

e− e−

j / jref

e–

j / jref

1.6

1.2

0.8

0.4

0

e−

min

max

j / jrefjref

jref

100 103 1061

10

100

1000

f in Hz109 1012

Self-healing

Skin effect

Current frequencies

MTF(AC)MTF(DC)

Power supply net Clock net Signal netSource Sink Sink SinkSource Source

I I I

Metal(Cu)

Metal liner Dielectric cap

Dielectric (e.g., SiO2)

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]

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

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]

(a) Armchair (b) Zig-zag (c) Chiral

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

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