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Nomenclature

List of Figures

1.1 Fabrication process of a aluminium metallization layer. By repeating the steps a multi-layer structure is realized. [146].

1.2 Cut through a single and dual-damascene interconnect structure [61].

1.3 Schematic of a chip-to-wafer bonded structure. Stapled dies hold together by solder bumps.

1.4 Schematic of the TSV [83]. The orange regions are aluminium, the red are tungsten, the dark blue are the galvanic isolation layer from the metal to the silicon bulk, and the green are the passivation layers. For bonding the bottom wafer is prepared with a silicon oxide (in light blue) and the top wafer is planarized silicon.

1.5 Schematic of the Bosch process.

1.6 The two forces of EM.

2.1 The dependence of the effective valence on the distance of an adatom to a semi-infinite metal surface for two different location above the crystal lattice shown on the right [15]. $\tau$ is the time scale of relaxation of the electronic charge density.

3.1 Segment with an interface domain and a neighboring bulk domain for the diffusion coefficient calculation.

3.2 Schematics for the derivation of the segregation model according to [90].

3.3 Schematics for the derivation of the segregation model as an infinite thin representation of an interface with $\delta$ thickness.

3.4 Local coordinate system.

3.5 Two shapes proposed for the bulk free energy function $f(\phi )$ . The quadratic double well potential in black and the double obstacle potential in red.

4.1 Example meshing elements for 2D and 3D domains.

4.2 An example mesh for an $\mathrm {L}$ shaped structure with a fillet.

4.3 Mapping between an arbitrary triangle and the reference triangle for 2D triangular meshing with the transformation law.

4.4 The hat test function for a single node located in the middle.

4.5 Flowchart of the EM simulation in FEDOS.

4.6 Flowchart of the EM simulation in COMSOL.

4.7 Flowchart of the EM induced void simulation in COMSOL.

5.1 Schematics of the current crowding calculation at a corner.

5.2 The current density at corners for different angles versus the radial distance. The current densities are normalized to $J_{0}$ at the radius $r_{0}$ . The lower the angle of the corner, the higher the current density rises in the vicinity of the corner. In the last plot the three curves are put together to facilitate comparison.

5.3 The current density is presented by the red arrows at the interface between the planar connecting structure and the via, for different ratios of the via width and the via height. The lines represent the electric flow lines.

5.4 Profile view of the TSV structure: Aluminium in yellow, tungsten in green, and substrate and seal layer in red. The tungsten cylinder is shortened to 10% of the real length.

5.5 The structure of a segment of the overlapping area of the tungsten cylinder and the aluminium cylinder. Green represents titanium nitride, red silicon oxide, dark gray tungsten, and light gray aluminium.

5.6 The current density is shown by the red arrows at the interface between the aluminium layer on the top and the tungsten layer at the bottom for different ratios of the thicknesses of the two layers. The lines represent the electric flow lines.

5.7 The current density is shown by the red arrows at the interface between the planar connecting structure and the via for different ratios of the resistances of the upper and the lower layer. The lines represent the electric flow lines.

5.8 The current density is shown by the red arrows at the interface between the planar connecting structure and the via. The lines represent the electric flow lines and show that the transfer of current between the two metals is mainly occurring closed to the right interface end.

5.9 Component of the current density vector normal to the interface.

5.10 Segregation model: Variation of the transport coefficient $h$ . Vacancy concentration in aluminium and in tungsten versus the location of the interface.

5.11 Simulation of a highly blocking interface $(h=10^{-5}\mathrm {m}\mathrm {m}/\mathrm {s})$ for different time steps in arbitrary $t_{0}$ units.

5.12 Variation of the equilibrium concentration in the interface region.

5.13 Rosenberg-Ohring model: Results at different time steps for the lowest possible equilibrium concentration $(C_{inter}=0)$ .

5.14 The maximum concentration of vacancies piled up at the interface.

5.15 Maximum Von Mises stress build-up at the interface.

5.16 Segment of the TSV closest to the current introducing planar interconnect line. The inset shows the half of the upper open TSV structures aluminium cylinder at the top and tungsten cylinder at the bottom. The blue line indicates the cross section for which the following simulation results are plotted. In this zone the highest current densities are foreseen and therefore EM has the biggest influence.

5.17 Current density in $10^{3}\mathrm {A}/\mathrm {c}\mathrm {m}^{2}$ in the simulated segment of the open TSV. At the bottom of the aluminium cylinder the current crowding is pronounced.

5.18 Relative vacancy concentration deviation $(C_{\mathrm {v}}/C_{\mathrm {v},\mathrm {e}\mathrm {q}}-\ 1)$ piled up in the structure for three different simulated time periods. The cross section is the cut of the blue line in Figure 5.16.

5.19 Von Mises stress in $\mathrm {N}/\mathrm {m}^{2}$ build-up in the structure for three different simulated time periods. The cross section is the cut of the blue line in Figure 5.16.

5.20 Maximum Von Mises stress build-up at the interface for different currents.

5.21 TTF for different currents with a fitting to Black’s equation. The axis are logarithmically scaled.

5.22 Void evolution in time. The void is placed into the structure and starts to move in the current density direction. After a certain time it has reached the end of the aluminium close to the tungsten. The circle shape deforms to a half circle to reduce the surface energy which is proportional to the surface itself. After this phase the vacancies accumulate at the void surface whereby it grows. Due to the increase of the current density along the surface the growth process gets accelerated.

5.23 Resistance growth due to void movement and growth in the open TSV structure for different electrical currents.

5.24 Fitting of the TTF to Black’s equation.

A.1 Figure A.1.: Illustration of the partitioning of the void metal region into interface, void, and metal domains for the inner expansion [9].

B.1 Figure B.1.: Crystal lattice with atoms migrated to the surface leaving back Schottky defects [51].

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2D	Two-Dimensional
3D	Three-Dimensional
CMOS	Complementary Metal-Oxide-Semiconductor
CVD	Chemical Vapor Deposition
DRIE	Deep Reactive Ion Etch
EM	Electromigration
FEDOS	Finite Element Diffusion and Oxidation Simulator
FEM	Finite Element Method
GB	Grain Boundary
IC	Integrated Circuit
ILD	Inter-Level Dielectric
IVD	Inter-Via Dielectric
KKR	Korringa-Kohn-Rostoker
LKKR	Layer KKR
PDE	Partial Differential Equations
PECVD	Plasma Enhanced CVD
TCAD	Technology Computer Aided Design
TSV	Through Silicon Via
TTF	Time to Failure
UV	Ultraviolet
$A$	Area
$B$	Applicable Mechanical Modulus
$C_{\mathrm {a}}$	Atom Concentration
$c_{\mathrm {T}}$	Specific Heat Capacity
$C_{\mathrm {v}}$	Vacancy Concentration
$C_{\mathrm {v}}^{\mathrm {e}\mathrm {q}}$	Equilibrium Vacancy Concentration
$d$	Grain Size
$D_{\mathrm {a}}$	Atom Diffusion Coefficient
$D_{\mathrm {e}\mathrm {f}\mathrm {f}}$	Effective Diffusion Coefficient
$D_{\mathrm {G}\mathrm {B}}$	Grain Boundary Diffusion Coefficient
$D_{\mathrm {s}}$	Surface Diffusion Coefficient
$D_{\mathrm {v}}$	Vacancy Diffusion Coefficient
$E$	Electric Field
$e$	Unit Charge
$E_{a}$	Activation Energy
$F$	Mechanical Force
$f$	Relaxation Factor
$j$	Current Density
$J_{\mathrm {a}}$	Atom Flux
$J_{\mathrm {v}}$	Vacancy Flux
$k_{\mathrm {B}}$	Boltzmann Constant
$Q^{*}$	Heat of Transport
$\dot {q}$	Joule Heat Generated per Time
$R_{\mathrm {p}}$	Flaw Radius
$T$	Temperature
$T_{\mathrm {r}\mathrm {e}\mathrm {f}}$	Reference Temperature
$\mathrm {u}$	Displacement Field
$V$	Volume
$W$	Elastic Strain Energy
$Z^{*}$	Effective Valence
C	Elastic Tensor
$\delta$	Grain Boundary Width
$\delta _{i,j}$	Kronecker Delta
$\epsilon$	Mechanical Strain
$\epsilon _{\mathrm {p}\mathrm {f}}$	Phase Field Interface Thickness Parameter
$\epsilon _{\mathrm {T}}$	Temperature Induced Mechanical Strain
$\epsilon _{\mathrm {v}}$	Vacancy Related Mechanical Strain
$\phi _{\mathrm {E}}$	Electric Potential
$\gamma _{\mathrm {S}}$	Surface Energy
$\kappa$	Surface Curvature
$\lambda$	Lamè Parameter
$\lambda _{\mathrm {T}}$	Thermal Conductivity
$\mu$	Chemical Potential
$\rho$	Specific Electric Resistance
$\rho _{\mathrm {M}}$	Material Density
$\sigma$	Isotropic Stress
$\sigma _{\mathrm {E}}$	Electric Conductivity
$\sigma _{\mathrm {t}\mathrm {h}}$	Threshold Stress for Void Nucleation
$\tau$	Characteristic Relaxation Time
$\Omega$	Atomic Volume