Transient Modeling of TSV-Wire Electromigration andLifetime Analysis of Power Distribution Network for 3D ICs

Xin Zhao1, Yang Wan2, Michael Scheuermann3, and Sung Kyu Lim2

1IBM Systems and Technology Group, Hopewell Junction, NY, USA.2School of ECE, Georgia Institute of Technology, Atlanta, GA, USA.3IBM T. J. Watson Research Center, Yorktown Heights, NY, USA.

xzhao@us.ibm.com, yangwan@gatech.edu, mikeman@us.ibm.com, limsk@ece.gatech.edu

Abstract— In this paper, we present a transient modeling of electromi-gration (EM) in TSV and TSV-to-wire interfaces in the power deliverynetwork (PDN) of 3D ICs. In particular, we model atomic depletion andaccumulation, effective resistance degradation, and full chip-scale PDNlifetime degradation due to EM. Our major focuses are on: (1) time-dependent multi-physics EM modeling approach to model TSVs andconnecting wires under the influence of coupled physical phenomenonincluding electric field, temperature, and stress; (2) time-dependent EM-aware power integrity analysis methodology, which is integrated withthe TSV modeling approach to predict long-term IR-drop degradationin full-chip 3D power delivery networks. Our studies show that voids andhillocks grow at various TSV-to-wire interfaces and degrade the effectiveresistance of TSVs significantly. In addition, our full-chip PDN lifetimeanalysis shows significant increase in maximum IR drop during lifetimedue to EM effects.

I. INTRODUCTION

Through-silicon-via (TSV)-based 3D integration has recently re-ceived a lot of interest due to its potential to overcome conventionalCMOS scaling limitations and its potential to enable heterogeneousintegration. Reliability of TSV-based 3D ICs is an important issuefor main stream acceptance. Electromigration (EM) can reduce ICreliability and potentially cause short or open circuit failures. TSVs,especially the power/ground (P/G) TSVs in 3D power deliverynetworks (PDNs), carry large currents and typically have higher localcurrent densities due to current crowding [1]. Regions of high localcurrent density are more susceptible to EM degradation. Therefore,long-term EM reliability issue for TSV-based 3D PDNs needs to beaddressed.

EM modeling of interconnects has been extensively studied formany decades [2], [3]. However, very little work has been done tomodel the reliability of TSVs. A recent paper investigated DC currentcrowding inside TSVs and at the connections between TSVs andpower wires [1]. Current crowding increases the effective resistanceof the TSV and voltage drop in the 3D PDNs. The impact ofTSV stress on back-end-of-line (BEOL) interconnects and its EMlifetime were modeled [4], [5]. Another modeling approach for TSVEM reliability was proposed in [6], where EM measurements ona TSV connection indicates that voids may occur at the TSV-to-wire interfaces [7]. However, none of these works discuss time-

This research is supported by the National Science Foundation (CCF-0917000) and the Semiconductor Research Corporation (CADTS-2239).

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dependent EM modeling for TSV-based 3D connection or study long-term reliability of chip-scale PDNs.

Since atomic diffusion is dominated by the fast diffusion alonggrain boundaries compared to grains, any realistic EM modelingneeds to include grain and grain boundary geometry. Most existingworks on TSV EM modeling do not consider grain structure [4],[5], [6], but instead model the entire TSV as a single perfectcrystal. The grain size distribution measured from actual TSVs canvary significantly, from 2.3um [8] in mean, 2.0-5.5um [9], or 1.0-10.0um [10] depending on the process technology.

The focus of this work is to model and investigate EM andreliability of TSV-based 3D connections and to analyze the impacton the lifetime of 3D PDNs. Our contributions are as follows: (1)multi-physics transient modeling is used to model and simulate EMin TSV-based 3D connections. This approach enables simultaneousconsideration of the atomic transport in the structure, evolution ofelectric current, stress distribution, atomic concentration, and TSVeffective resistance. (2) We investigate how different physical effectsand parameters impact EM in TSVs and ultimately the PDN itselfincluding the effects of temperature, electric current, back flow andthermal expansion stress, activation energy and grain structure. (3)We present a methodology for EM reliability analysis of full-chip-scale 3D PDNs. This methodology integrates our TSV EM modelingwith time-dependent full-chip PDN analysis. Simulations show thatthe degradation of PDN IR-drop due to current crowding and EMcan be modeled and quantified. Related experiments show that themaximum IR drop can increase 5% to 9% due to current crowdingand later increase approximately by 11% to 26% during life time dueto current and stress-driven EM effects.

II. TRANSIENT EM MODELING

A. Fundamental EM PDEs

A set of well-known partial differential equations (PDEs) isused to model the time-dependent evolution of atomic concentrationN(x, y, z, t) at location (x, y, z) at time t.

∂N

∂t+ � · J = 0 (1)

J = Jc + JT + Js + JN (2)

JN = −D �N (3)

Jc =N

kTeZ∗ρj D (4)

JT = −NQ∗

kT 2D �T (5)

Js =NΩ

kTD �σH (6)

363978-1-4799-1071-7/13/$31.00 ©2013 IEEE

TABLE INOTATIONS, MEANINGS, AND TYPICAL VALUE IN OUR EM PDES.

Term Meaning Typical valueN Atomic concentration N0=1.53e28 Atoms/um3

EA Activation energy EA(g)=2.1eV; EA(gb)=0.8eVk Boltzmann constant k=1.38e-23 J/Ke Electric charge e=1.6e-19CZ∗ Effective valence charge Z∗=-4[13]ρ Electrical resistivity ρ∗0=1.8e-8Ω·mD0 Self-diffusion coefficient D0=1e-7 m2/sQ∗ Heat of transport Q∗=1.387e-20Ω Atomic volume Ω=1.182e-29m3

B Back flow stress modular 5e7Paj Current density in mA/um2 from simulationT Absolute temperature in K from simulationσH Hydrostatic stress in Pa from simulationσbf Back flow stress in Pa from simulation

σbf = BN −N0

N0(7)

D = D0 exp(−EA

kT) (8)

The continuity equation (Equation (1)) governs the atomic fluxdivergence (� · J) over spatial dimensions and determines howthe atomic concentration evolves over time. The atomic flux J isdetermined by the combined mechanisms described in Equations (3)through (6), where JN is governed by atomic concentration gradient�N ; Jc by current density j; JT by thermal gradient �T ; and Js

by stress gradient �σH. The hydrostatic stress σH primarily involvesthe back flow stress σbf in Equation (7). The stress modeling andsimulation will be discussed in detail in Sections II-C, IV-C, and IV-D. COMSOL multiphysics, a commercial finite element simulationtool [11], is used to build and solve these PDEs and conduct electric,thermal, and stress simulation. The notations are defined in Table I.

B. Grain and Grain Boundary Modeling

The generic TSV model used in this study is illustrated in Figure 1and consists of a copper TSV with 5um diameter and 25um heightand landing pads with 6um×10um and 1um thick. This test caseconstrains the current flow into the structure from the top-left cornerand out of the bottom-right corner. A simplified grain structure isused for the TSV and nearby wire connections. This grain structurewas chosen to approximate the grain structure described in [8].

Electromigration in a metal depends strongly on its grain structure,which in turn depends on the fabrication process and subsequentthermal and electrical history [8]. In our TSV model, we construct asimplified regular grain structure on the TSV and wires. The grainsare 1.9um cubes separated by 0.1um-thick grain boundaries, bothdefined interior to the bounding surfaces of the TSV as illustratedin Figure 1. These dimension are consistent with the measuredmean value of TSV grain size presented in [8]. In principle, themodel could represent grains of any shape and size distribution.We believe this simplified structure is sufficiently detailed to reflectthe essential basic physics intrinsic to the actual grain structure.Different activation energies are used for the grains and the grainboundaries. For grain boundaries, the activation energy, EA(gb), asmall value reflects high diffusion in grain boundaries; and for grains,the activation energy, EA(g), a large value reflects low diffusion ingrains. Since no specific activation energies have been reported forTSVs to date, we use values reported for wires: 0.9eV for EA(gb)and 2.1eV for EA(g) [12].

Current

Current

wire-to-TSVwire-to-TSVinterfaceinterface

wire-to-TSVwire-to-TSVinterfaceinterface

Grains(1.9um)

GB(0.1um)

Fig. 1. Our TSV model with uniform grains and grain boundary (GB).

C. Stress Modeling

The EM atomic flux (Js in Equation (6)) is affected by the gradientof hydrostatic stress. A TSV-based 3D connection can be influencedby the following three types of stress:

• Back flow stress: Back flow stress is generated by EM [14]. Theatom flux leads to a distribution of atomic concentration alongthe 3D connection, where the atomic depletion generates tensilestress, and the atomic accumulation generates compression. Thisstress gradient tends to move the atoms back toward theiroriginal locations, thereby forming a “back flow” of atoms. Thisis the opposite to the atomic flux caused by the electric currentdensity. Equation (7) is a linear equation which relates the backflow stress to the atomic concentration taken from the Korhonenmodel [15]. Our simulations described in Section IV-D show thesignificance of back flow stress, which also increases over timedue to the EM-induced atomic concentration evolution.

• Thermal stress: Thermal stress is caused by the coefficients ofthermal expansion (CTE) mismatch among different materials.Mechanical expansion can occur due to a temperature change,which may be caused by joule heating or the heat flux throughthe TSV. The motivation to study thermal expansion stress wasdriven by a concern that joule heating due to the large TSVcurrent may cause self heating. In this work, we have ruled outself-heating as a significant effect in TSV modeling. Details willbe discussed in Section IV-C.

• Residual stress: Residual stress is generated during the manu-facturing process and “frozen” into the structure. For instance,thermo-mechanical stress [16] occurs during fabrication andthermal cycling of TSVs and is caused by the mismatch ofthermal expansions coefficients among different materials. Theresidual stress can be released over time [17]. Thus, our EMmodeling in this study ignores the residual stress.

D. Effective Resistance Modeling

One objective of modeling a single TSV structure in detail is tocorrelate simulations with experimental results. This is achieved if themodeling results can be compared with measurements from a TSVtest site. There are typically many test sites which can be stressedin various ways and characterized using a simple 4-point probe tomeasure the effective resistance (Reff). To make such a comparisonit is necessary to be able to calculate Reff from the simulation. We

364

Atomic conc. Nt

Resistivity ρt(Nt)

Current density Jt (ρt)

COMSOL DC current simulation

COMSOL PDE solver

Atomic flux due to

�Atomic conc. gradient JN(Nt)�Stress gradient JS(Nt , σt)�Current density JC(Nt, ρt , Jt)

Transient EM simulation in COMSOL

Evolution of

� Atomic concentration distribution N(t)� Resistance degradation R(t)� Current density distribution J(t)� Stress distribution σ(t)

For each time step Δtt = t + Δt

Back flow stress σt(Nt)

COMSOL stress simulation

Geometry and mesh generation

Fig. 2. Our transient EM simulation flow based on COMSOL Multi-physicstoolset.

can do this if we know the resistivity ρ as a function of x, y, z, andt.

In this work, the resistivity is modeled as a function of atomicconcentration N . When the local atomic concentration decreases,the corresponding local resistivity increases; and likewise, whenthe local atomic concentration increases, the corresponding localresistivity decreases. Examples of resistivity function are depicted inFigure 9(a). The form and accuracy of our resistivity function needfurther investigation. Several variations of this function yield similarresults, which will be discussed in Section V-B. We believe thisfunction is a reasonable approximation to the relationship between theatomic concentration and resistivity. Regardless of the actual func-tional form, our modeling approach is able to easily accommodateany well-defined resistivity function.

III. SIMULATION FLOW AND ASSUMPTIONS

We use COMSOL Multiphysics [11] for customized EM modelingand simulation, which enables us to combine user-specified EMPDEs, stress/resistivity modeling with multiple physics (e.g., electriccurrent, joule heating, mechanical thermal expansion stress, etc). Anoverview of our transient EM simulation flow is illustrated in Fig-ure 2. This flow starts by defining and creating the geometries of theTSV, connection wires, grains, and grain boundaries. This is followedby the mesh generation. EM transient simulation is then performedas an iteration loop shown with the red arrows in Figure 2. At eachtime step Δt, atomic concentration Nt at time t and the resistivitydistribution ρt(Nt) are calculated based on the resistivity function.COMSOL is used to calculate the current density distribution Jtusing the resistivity distribution ρt. In addition, COMSOL performsstress simulation based on user-specified equations. Next, the atomicfluxes are updated: (1) Flux Jc due to current density Jt, atomicconcentration Nt, and resistivity ρt; (2) Flux Js due to gradientof hydrostatic stress; and (3) Flux JN due to atomic concentrationgradient, which tends to balance Jc. Finally, COMSOL solves the EMPDEs and calculates the atomic concentration at time t +Δt. In thistransient analysis, COMSOL determines the iteration times and stepfor convergence automatically. Using this flow, we are able to derivethe evolution of each physics phenomenon due to EM over time,e.g., atomic concentration gradient, effective resistance degradation,electric current, stress, etc.

For simplicity, a number of assumptions were made. We assumeuniform grain and grain boundary geometry. We do not consider

Z=1um

Z=0um

Z=1.05um

Time = 1e9s = 31.7yrs

Current

Current

Fig. 3. Current density distribution in a TSV-wire structure, where TSV andwires contain grains. We show a 3D view and XY plane views.

the grain orientation and grain/grain boundary propagation. Nu-cleation sites for void and hillock formation were not included.We assume that grain and grain boundaries have the same initialatomic concentration and resistance. Our simulation shows negligiblethermal gradient from joule heating and thermal stress due to highthermal conductivity of copper TSVs. Quantum effects such as atomictunneling through grain boundaries were ignored. In principle, all ofthese effects which were simplified or ignored can be included in themodel later.

IV. MULTIPHYSICS SIMULATIONS

A. Simulation of Electrical Current

An existing study [1] has been conducted on detailed currentdensity simulation of TSV-wire 3D connection and has shown alarge amount of current concentration at the TSV-wire interfaces(called current crowding). However, this work did not consider thetime-dependent atomic movement due to EM and grain structures,nor the resulting resistance degradation due to atomic depletion.Our EM modeling approach is able to simulate the time-dependentdetailed current density distribution due to EM-induced atomicdepletion/accumulation in grains and grain boundaries: the atomicdepletion increases the electric resistance, which then reduces thelocal current density and slows down the atomic depletion. Inaddition, since grain boundaries have lower activation energy thangrains, atoms flux is faster in grain boundaries than in grains. Thisresults in higher resistance and thus lower current density in grainboundaries than in grains.

An example of how EM alters the current density is plotted inFigure 3. First, the bottom and top TSV-to-wire interfaces are subjectto higher current density due to current crowding. In particular,current is significantly crowded at the bottom-right corner for Z=1um.Second, current concentrates in the grains rather than grain bound-aries due to the resistivity distribution (high in grain boundaries andlow in grains).

B. Simulation of Atomic Transport

Our EM modeling approach is able to simulate the time-dependentatomic accumulation and depletion due to multi-physics EM phe-nomenon. The atomic concentration at the top and bottom TSV-wireinterfaces at time=1e5s and 1e7s are plotted in Figure 4. Here 60mAcurrent enters at the top-left and exits from the bottom-right of the 3D

365

+2.2%

-1.5% -4.1%

+4.2%

Time = 27.8hrs Time = 116 days

(a) (b)

N/N0 N/N0

0 0

(c)

je- flow(Atom flow)

Atomic depletion

Atomic accumulation

Current j

j

Top interface

Bottom interface

Fig. 4. Atomic concentration on TSV-based 3D connection. (a) electron/atomflow directions, (b) concentration at time=1e5s, (c) time=1e7s.

connection. The simulation results explicitly demonstrate the impactof electric current on atomic flux, in particular the current crowdingeffects. Since electrons flow from the bottom right to top left,copper atoms move from bottom right to top left due to momentumexchange after electron collision. Because of a large amount ofcurrent crowding at top-left and bottom-right TSV-wire interfaces [1],the bottom-right interface has depleted atoms (ΔN/N<0), and thetop-left interface has accumulated atoms (ΔN/N>0). In addition,since atoms move faster along grain boundaries than grains, atomicdepletion and accumulation occur along grain boundaries. Over along period, more atoms will accumulate or deplete at the interfacesas shown in Figure 4(b).

C. Simulation of Thermal Expansion Stress

Due to the high thermal conductivity of copper, the thermalgradient is very small inside the copper TSV. To verify this, wesimulate the joule heating in a TSV with 60mA current. The structureconsists of three silicon layers (each is 25um thick), two inter-layerdielectric (ILD) layers (each is 4um thick), a TSV liner (SiO2 with0.2um thick), and a copper TSV with two landing wires. The heatsink is assigned at the top surface with the heat transfer coefficientof 25e3W/(m2·K). Related experiments showed that a small thermalgradient is observed in the ILD layers, landing wires, and theTSV. Across this structure, the temperature varies from 349.84K to349.90K. The resulting thermal gradient across the TSV between thetop and bottom landing pads is only 0.04K.

The thermal expansion stress and its gradient impact on EMatomic flux are simulated for TSV-based 3D connection, where thecopper TSV and wires are buried inside the silicon substrate, andthe insulation layer (SiO2) and barrier layer (TiN) surround the3D connection. The thermal expansion stress is generated by jouleheating effect, where the current flows from the top-left wire, throughthe TSV, and out of the bottom-right wire. In this case, the thermalexpansion stress presents an initial compressive stress at the 3Dconnection. However, the stress gradient from thermal expansionprimarily occurs at the TSV-wire interfaces, not inside the entire TSVconnection. In addition, the stress gradient does not vary significantlyover time. Therefore, thermal expansion stress presents minor impacton the overall EM atomic transport. The simulation result is plotted inFigures 5(a), where only the thermal expansion is considered. Notethat in this stress distribution, the tensile stress has positive value,and the compressive stress has negative value.

We first observe that compressive stress on both TSV top andbottom is observed along the 3D connection initially. Since the

(a) Thermal stress @ t=0s (b) Back-flow stress @ t=1e7s

-3

-2

-1

0

1

2

x106

3

Tensile

stress

Compressive

stress

-1.75

-1.7

-1.65

-1.6

-1.55

x108

Compression

HighTension low high

Compression high low

Compression

Low

Fig. 5. Stress simulation results. (a) thermal expansion stress only attime=0s, where compressive stress gradient occurs at both interfaces. (b)back flow stress only at time=1e7s, where tensile stress occurs at the topand compressive at the bottom. Note that the tensile stress has positive value,and the compressive stress has negative value.

Fig. 6. Stress impact on atomic concentration evolution (ΔN/N0) at t=1e7s,5e7s, and 1e8s. Back flow stress slows down the concentration degradation,but thermal stress has minor impact.

connecting wires are asymmetric on top and bottom, the top-rightand bottom-left TSV-wire interfaces experience larger compressivestress (with short connecting wires) than the top-left and bottom-right interfaces (with long connection wires). In addition, due to theSiO2 layer, the TSV-silicon interfaces experience less compressivestress on the top and bottom. Thermal expansion stress does not varysignificantly over time. Second, we note that the stress gradient dueto thermal expansion presents minor impact on atomic degradation.The stress impact on atomic concentration degradation (ΔN/N0) isplotted in Figure 6 at time=1e7s, 5e7s, and 1e8s. At each time,we compare the range of ΔN/N0 with (1) no stress, (2) thermalexpansion stress only, (3) back flow stress only, and (4) both thermalexpansion and back flow stress. From the comparison between “NoStress” and “Thermal Stress” bars, we note that ΔN/N0 does notchange significantly by thermal stress.

D. Simulation of Back Flow Stress

The back flow stress and its gradient impact on EM atomic flux aresimulated for TSV-based 3D connection, where we keep the same 3Dconnection structure as the thermal expansion stress, and apply the

366

(a) (b) (c)

Fig. 8. (a) MTTF vs. average current density, T=350K. (b) MTTF vs. temperature, jTSV=3.1mA/um2, (c) MTTF vs. EA(gb).

(b) t=1e8s(a) t=1e7s

Compression

High

-1.75

-1.7

-1.65

-1.6

-1.55

-1.5x108

Compression

Low

Compression reduces

Compression increases

Fig. 7. Stress simulation with both thermal expansion and back flow stressconsidered. (a) at t=1e7s, (b) at t=1e8s. Back flow stress changes over time,resulting in compressive stress increased on the top and reduced on the bottom.

atomic flux equations due to back flow stress, current crowding, andconcentration gradient. This back flow stress is originated from thecompression at atomic depletion and tension at atomic accumulation,and presents long-time effect on atomic flux due to EM.

The simulation results are plotted in Figure 5(b) with t=1e7s.We first observe that compressive stress occurs at the top TSV-wireinterface and tensile stress at the bottom TSV-wire interface. The backflow stress generates a considerable stress gradient through the entireTSV connection, especially along the grain boundaries where theatomic accumulation and depletion are likely to take place. Second,the TSV top-left and bottom-right corners suffer larger compressionand tension than the top-right and bottom-left corners, respectively,which is mainly due to the current crowding impact on the atomicdegradation. Third, from the comparison between “No Stress” and“Backflow Stress” bars, we note that ΔN/N0 changes significantlyby back flow stress. This clearly shows that the atomic movement isreversed by the back flow stress.

We perform another EM simulation, where we consider boththermal expansion stress and back flow stress. The simulation resultsare plotted in Figure 7 at t=1e7s and t=1e8s. The thermal expansionstress creates compressive stress along the TSV initially. When EMeffects last longer, however, back flow stress dominates the stressgradient, which in turn causes the top TSV-wire connection to bemore compressive and the bottom connection less compressive. In

addition, referring to Figure 6 and by comparing back flow stress onlyvs combined stress, we note that thermal expansion stress does notadd any significant contribution. This confirms our observation thatit is reasonable to ignore thermal expansion stress in EM simulation.

V. TSV FAILURE ANALYSIS

In this section, we present failure analysis discussion that fallsinto two categories: one is the mean-time-to-failure (MTTF), whichis defined as the 10% deviation of atomic concentration over theinitial concentration N0; the other is resistance degradation due toEM effects. Again, transient EM modeling is applied to the structureshown in Figure 1. More general TSV connection structure for full-chip power integrity analysis will be discussed in Section VI. Theimpact of various design and modeling factors on EM reliability willbe investigated, including the current density, temperature, activationenergy, resistivity function, and grain structure on the wire.

A. MTTF Analysis

The impact of TSV current density (jTSV) on MTTF is shown inFigure 8(a), where jTSV increases from 1.5mA/um2 to 6mA/um2,and temperature is 350K. As a result, the EM lifetime of the TSVdramatically reduces from 2.6e9s to 1.0e7s. This high current densityaccelerates the atomic depletion and accumulation and decreases theEM lifetime. For P/G TSVs, which can carry the current densitylarger than 5mA/um2, the EM reliability may become critical.

Temperature also plays an important role in atomic concentrationand EM reliability. Diffusivity D exponentially depends on tem-perature (Equation (8)). In addition, atomic fluxes are affected bytemperature based on Equations (4) to (6) and thermal gradient inEquation (5). To analyze the impact of temperature on EM, thecurrent value is kept constant, and the temperature is increased from300K to 400K. This temperature range is affected by both the powerdensity from neighboring devices and the joule heating of TSV. Theimpact of temperature on EM lifetime is shown in Figure 8(b). As thetemperature increases from 300K to 400K, the MTTF dramaticallyreduces from 5.9e9s to 8.7e6s.

The diffusivity D exponentially depends on −EA (Equation (8)).Small activation energy of grain boundary (EA(gb)) results in fastatomic diffusion, thus degrading the EM lifetime. The impact ofEA(gb) on MTTF is depicted in Figure 8(c), where EA(gb) re-duces from 0.9eV to 0.7eV. When EA(gb) decreases, the MTTFdramatically reduces from 3.55e9s to 5.2e6s. This curve shows theexponential impact of activation energy on atomic flux.

B. Evolution of TSV Resistance

The resistivity evolution model deals with how TSV resistancedegrades over time during EM lifetime. In our transient EM modeling

367

Fig. 9. Impact of resistivity function on resistance degradation. (a) fourresistivity curves (Func 300ρ0, Func 200ρ0, Func 100ρ0 , and Func 32ρ0),(b) corresponding resistance degradation.

approach, the resistivity is updated after each time step according tothe local atomic concentration. These latest resistivity values changethe current density distribution, which in turn changes the atomictransport and atomic concentration distribution. In this section wecompare four possible resistivity evolution functions and show theimpact on TSV resistance degradation. Most existing studies assumethat a void occurs when the atomic concentration drops down to aspecific percentage. However, there is no clear answer on how low theconcentration threshold should be to declare a void.1 Therefore, wedescribe a continuous resistivity increase when atomic concentrationdecreases, but vary the “infinite” resistivity values from 32ρ0 to300ρ0 (see Func 32ρ0 to Func 300ρ0 in Figure 9(a)). The simulationresults of ΔR/R0 shown in Figure 9(b) reveal a consistent resistancedegradation over time among the 4 models (ΔR can reach to morethan 20% R0 due to EM). In addition, we observe that higher ρ0values lead to more resistance degradation. We note that the resistivityfunction with 32ρ0 shows reasonable accuracy and thus is used inthis study.

C. Adding Grains in Wires

The TSV resistance evolution for a test case shown in Figure 1that has no grains in the wires is plotted in Figure 10. This testcase corresponds to the bamboo wire structure, where wires have nograins. We observe from Figure 10 (= red line) that at the early timeperiod, TSV effective resistance increases rapidly. After 3e8 seconds,however, the resistance reaches to a saturation value. This is due to anequilibrium of atomic flux divergence between the opposing currentstress and atomic concentration gradient. Up to 12% TSV resistancedegradation is observed in this case.

1Our proposed modeling approach is able to integrate more complicatedresistivity function if, in the future, more comprehensive study andmeasurement data on resistivity function of atomic concentration areavailable.

Fig. 10. Evolution of TSV effective resistance (RTSV) when wires have ordo not have grains.

This bamboo structure is extended to include grains in the wiresto investigate a non-bamboo wire structure. Wires and the TSV havesimilar grain structure. In the grained-wire case, atoms can nowtransport between wires and the TSV. Two curves of TSV resistancedegradation are plotted in Figure 10 (= black line). When wires arenon-bamboo (with grains), TSV resistance increases by more than15% compared with the initial value. This larger TSV degradationis due to the high local current density in the wires when wireshave grains. Whereas, for bamboo wires (no grains in wires), theTSV resistance degradation is approximately 10% due to the uniformcurrent density in the wires.

VI. LONG-TERM RELIABILITY OF FULL-CHIP 3D PDNS

In this section, a transient analysis methodology for chip-scale 3DPDN EM reliability is implemented.

A. Full-chip PDN Reliability Analysis Methodology

The analysis flow for our TSV-based 3D PDN long-term reliabilitysimulation is depicted in Figure 11. The EM modeling approach dis-cussed in Section II is applied to calculate TSV effective resistance. Alibrary of TSV model which computes and returns the effective TSVresistance based on currents into and out of the TSV is generated.This methodology allows us to analyze the IR drop degradation of3D PDNs. The entire flow consists of the following steps. First,an initial netlist for 3D PDN is generated from SPICE simulation.The initial TSV resistance is calculated using ρ × l/s. Second, theinitial current density of the PDN is calculated. Third, transient PDNreliability analysis is performed over multiple iterations. For eachiteration at time t, SPICE simulation is performed using the updatedTSV resistance value. The current density distribution and voltagedrop are analyzed. Finally, the TSV resistance increase is calculatedby using the TSV resistance library.

The simple TSV modeling structure shown in Figure 1 is extendedto consider neighboring wires connecting to all four terminals ofthe TSV as well as the relative P/G C4 location to the TSV. TheTSV resistance degradation due to EM is mainly affected by twofactors: current density through the TSV, and the current distributedto neighboring wires, which dictates how the current is crowded atthe TSV-to-wire interfaces. Two sample cases are shown in Figure 12.In Figure 12(a), current is provided from the C4 directly below theTSV and is distributed to the four neighboring wires. In Figure 12(b),TSV and C4 are offset, and current can either come into or flow outfrom the TSV neighboring wires.

We build and utilize a library to efficiently calculate the effectiveTSV resistance degradation for full-chip PDN analysis. The input

368

Power integrity analysis using SPICE simulation

*.out

Analyze simulation results

initial netlist

*.sp

initial TSV resistance

tsvRes.0

sum.voltsum.cur

sum.curTSV

TSV resistance update

TSV R library from COMSOL

tsvRes.t

t=0

t →t+∆t

Step1 Step2

Step3

Step4

Step5

Fig. 11. Transient reliability analysis methodology for long-term powerintegrity of 3D PDNs.

IBL IBRIBU

ITL ITR

TSV

IC4

TSV

IBL IBRIBU

ITL ITR

IC4

(a) (b)

Fig. 12. Two sample cases for multi-terminal current flows in our P/G TSVstructure.

to the library is the current density and direction at each cornerconnecting to the TSV, and the output is the TSV effective resistancecurve with respect to the time. A large number of simulations areperformed, where each case contains the combination of currentdensity and direction for the wires (e.g. IBL, IBR, ITL, ITR, or IC4).Curve fitting is applied on these data points to derive analyticalequations for the TSV resistance. By querying the library with thein-situ TSV currents, the corresponding TSV model is chosen tocalculate the effective resistance of the TSV at a given time andthe currents through the neighboring wires and TSVs.

The TSV effective resistance degradation under the multi-terminalcurrent flow is shown in Figure 13, where the TSV and the C4are aligned, and temperature is set to 380K. To study the TSVcurrent density effect, current densities in neighboring wires are keptbalanced, where IBL=IBR and ITL=ITR in this experiment. The TSVresistance evolution curves are plotted in Figure 13 under differentTSV current values. We observe that with the TSV current increasingfrom 96mA to 240mA, the TSV resistance degradation becomesmuch larger. In particular, the TSV resistance value increases from33mΩ to 53mΩ with IBU of 240mA.

B. Transient Power Integrity Analysis

Our 3D power delivery network is represented as a 3D resistivemesh. Each TSV is represented as a single resistor having an effectiveresistance Reff. Our time-dependent TSV library described earlier isable to capture the impact of both current crowding and EM on TSVresistance degradation over time. The power wire is 6um wide and1um thick. The TSV has 5um diameter and 25um height. Grains inboth TSVs and wires are 1.9um large with 0.1um grain boundary.

Fig. 13. Impact of TSV current density on EM in multi-terminal structure.TSV and C4 are aligned.

Temperature is set to 380K. Current density is obtained by SPICEsimulation.

Our transient power integrity analysis is applied to a 3D PDN,which is 2-die stack, each die having 1.4mm x 1.4mm footprint area.The global PDN contains 16x16 power wires (yellow and blue linesin Figure 14) with 4x4 TSVs and C4s (white boxes). C4s provide1.0V power supply to the bottom die (die-1), and P/G TSVs deliverthe supply to the top die (die-2). IR drop maps for die-2 at t=0 andt=1e9s are shown in Figures 14(a) and 14(b), respectively. Due to theEM-induced TSV resistance degradation, IR drop increases over time.Especially at the current hot spot (top right corner of the design), theTSV resistance increases faster over time than the current cool spot.As a result, the maximum IR increases from 23.0mV to 43.2mV.

C. Large-scale 3D IC PDN Analysis

The simulation results of PDN long-term reliability on five large-scale 3D PDNs are shown in Table II. The maximum IR dropdegradation in the top die includes the following results: (1) IR dropat t=0s without EM and current crowding (Column 8), and (2) IRdrop due to EM and current crowding at times t=0s, 1e8s, 2e8s,and > 10yrs (Columns 9-12). The percentage increase of IR drop iscomputed with respect to the IR drop with no current crowding attime t=0s.

We observe that IR drop degradation is affected by both currentcrowding and EM. First, current crowding at t=0s (Column 9)increases the maximum IR drop by 5.5% to 8.5% compared with theinitial IR drop that does not include EM and crowding. Second, foreach design after approximately 3 years (Column 10), maximum IRincreases from 5.9% to 14.1%. This degradation continues increaseover time. After approximately 10 years (Column 12), the PDNssuffer between 10.5% to 25.6% degradation due to IR drop. Finally,the bottom die also shows similar IR drop. The chip-scale PDNanalysis requires tens of iterations for the entire PDN network withTSV resistance update for each iteration. The SPICE simulations arewithin a few seconds.

VII. CONCLUSIONS

In this paper, we studied electromigration (EM) reliability forTSV-based 3D ICs and 3D power delivery networks (PDNs). Weproposed a transient EM modeling approach to simulate time-dependent atomic concentration and effective resistance degradation.This method iteratively updates the resistivity based on the latest

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TABLE IIEM RELIABILITY ANALYSIS OF LARGE-SCALE 3D PDNS INCLUDING THE FOOTPRINT, POWER DENSITY, DEGRADATION OF MAXIMUM IR DROP WITH

PERCENTAGE INCREASE, AND RUNTIME FOR EACH SPICE SIMULATION.

Pwr dens. Power IR top in mV(% increase) RuntimeDesign Footprint top bot grid #TSVs #C4 no crowding crowding+EM IR bot per itr.

mm2 W/mm2 (#×#) t=0 t=0 t=1e8s≈3.2yrs t=2e8s≈6.3yrs t >10yrs mV secPDN1 5×5 0.57 0.57 50×50 144 144 23.7 25.0 ( 5.5 ) 25.1 ( 5.9 ) 25.6 ( 8.0 ) 26.2 ( 10.5 ) 20.8 3PDN2 6×6 0.80 0.75 60×60 225 225 23.1 24.6 ( 6.5 ) 24.9 ( 7.8 ) 25.8 ( 11.7 ) 26.7 ( 15.6 ) 16.2 6PDN3 9×9 0.80 0.80 90×90 484 484 45.8 48.7 ( 6.3 ) 50.1 ( 9.4 ) 52.7 ( 15.1 ) 54.5 ( 19.0 ) 31.0 20PDN4 11×11 0.71 0.91 110×110 729 729 29.3 30.9 ( 5.5 ) 31.2 ( 6.5 ) 32.1 ( 9.6 ) 32.9 ( 12.3 ) 32.9 41PDN5 15×15 0.47 0.49 150×150 1369 1369 57.5 62.4 ( 8.5 ) 65.6 ( 14.1 ) 69.6 ( 21.0 ) 72.2 ( 25.6 ) 44.7 64

(a)

(b)

IR (mV)

Fig. 14. EM-induced IR drop degradation on die-2 for a 3D PDN. (a) t=0(max noise: 23.0mV) and (c) t=1e9s (max noise: 43.2mV).

atomic concentration, simulates the electric current, stress gradient,and calculates the atomic concentration for the next time step. Grainsand grain boundaries were also included in the TSVs and neighboringwires. In addition, we developed a transient power integrity analysisflow for chip-scale 3D PDNs lifetime prediction, which integratedour EM modeling approach, simulated the IR-drop evolution, andanalyzed the maximum IR degradation due to current crowding andEM.

Our observations are as follows: (1) Atomic depletion or accumula-tion occurs at wire-to-TSV interfaces, in regions of high current den-sity and in particular where current crowding occurs; (2) Back flowstress gradient presents non-negligible impact on atomic flux, whereasthe thermal expansion stress has minor impact; (3) High temperature,large current density, small grain size, or low activation energy ofgrain boundaries can accelerate EM, thus shortening the lifetimeof TSVs; (4) EM-induced degradation of TSV effective resistanceis simulated, and the neighboring wires with grains accelerate the

EM degradation; (5) For chip-scale PDN reliability analysis, undersome typical conditions, current crowding and current-driven EM canincrease the maximum IR drop significantly over time. The modelingand PDN reliability analysis flow described in this paper can also beused for other 3D structures subject to high current density and highthermal and mechanical stress.

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