Subsections

• 6.4.1 Introduction
• 6.4.2 Experiment
• 6.4.3 Simulation and Inverse Modeling
  • 6.4.3.1 Mathematical Models
  • 6.4.3.2 Simulation Setup
  • 6.4.3.3 Inverse Modeling
  
  
• 6.4.4 Results and Discussion
• 6.4.5 Conclusions

6.4 Inverse Modeling of Polycrystalline Fuses

6.4.1 Introduction

For deep-sub-micron semiconductor process technology, the use of Polysilicon fuses, as one-time-programmable devices providing memories up to several kilobits offers a cheap, efficient, and area-saving alternative to small non-volatile memories for System-on-a-Chip solutions. Approaches to increase the memory density by using 3-state fuses of layered materials are also reported [185]. Another important application is the use in simple field programmable gate arrays or for trimming CMOS circuits for specific analog performance [186]. Furthermore, the fuses are used to provide variable elements as trimmable resistor or capacitor arrays [187]. Finally, the fuses may act as the classical protective elements for improved protection and replacement of critical components before actual failures [188]. Programming is performed by sending a broad current pulse through the fuse, resulting in an open-circuit after transition to a second-breakdown state. The transition occurs when parts of the Polysilicon layer reach the melting point, and the molten Silicon is transported from the negative end through drift of ions in the applied field [189]. Fuses implemented in deep sub-micron technologies become more and more attractive in terms of power and area consumption, and hybrid approaches using other materials are getting less important [190]. Nevertheless, going to smaller ground rules below 350 nm, implies decreasing supply voltages to 1.5 V and below [191]. This constraint requires a careful optimization of the fuse layout, ensuring an efficient and reliable programming mechanism [192] and minimizing the necessary power consumption of the fusing process. As the fusing process takes place in a short time interval (between a couple of nanoseconds up to the microsecond range), direct thermal measurements of this process are quite hard to obtain. Previous work [193] already shed some light on the physics behind the fusing mechanism, but the optimization of the fuse structure for reliable and fast fusing was only possible via expensive experimental work by using test chips.

This work focuses on gaining better insight into the materials characteristics used in the structure, to enable a layout optimization through simulation. Since the electrical and thermal properties of Polysilicon are a complex function of Polysilicon film doping, grain size, and grain morphology[194], the average electrical and thermal properties as a function of temperature were obtained by experimentally measuring the transient resistivity response of the fuse through Joule self-heating, and subsequent inverse modeling the measured data, to fit the observed behavior. The electro-thermal self-heating simulations were performed with the Smart-Analysis-Package (SAP) for three-dimensional interconnect simulation [195] in combination with SIESTA, a TCAD optimization framework combining gradient based and genetic optimizers[143]. This approach enabled the optimization of the fuse layout by significantly saving costs normally spent in design and production of layout test chips. Furthermore, a better insight into the transient electro-thermal effects occurring in the first couple of microseconds was gained.

6.4.2 Experiment

The fuse devices were fabricated in an industry-standard deep sub-micron polycided gate CMOS process. On a specialized test chip multiple layout variations were placed to find the optimum layout for fast and reliable fusing. A more complicated example of a fuse structure is shown in Figure 6.25. The first experiments were performed with rectangular pulses. Nevertheless, through the steep slope of the fuse terminal voltage the initial time regime of the fuse heating is not well resolved. Furthermore, the initial transient behavior of the measurement circuit yields high errors in the measured current data. To overcome these problems a voltage ramp was applied and the resulting fusing resistance was calculated by assuming ohmic behavior. The Polysilicon layer in the fuse is doped to solid solubility, and therefore its conductivity may considered to be approximately ohmic. Since all materials in the fuse except the Polysilicon layer are metallic, this assumption shall give a reasonable estimate for the fuse resistivity. The devices were stressed with different triangular voltage ramps for a few microseconds. A pulse generator was used to define the length of the pulse. As the generator has a typical output impedance of 50 $\mathrm{\Omega}$ and the resistor of the Polysilicon fuse is lower than that, the source has to be buffered by an operational amplifier with a high slew rate to get a stable voltage. In order to avoid an additional voltage drop on a shunt resistor a current probe was used. In addition, the voltage on the fuse was monitored by an oscilloscope to calculate the right resistor value. The measurement principle can be seen in Figure 6.23.

Figure 6.23: Schematic of the Polysilicon fuse measurement

The resulting measurement data for three different source voltages as a function of time are shown in Figure 6.24.

Figure 6.24: Measured current through fuse and voltage at the fuse terminals as a function of time

The resistance difference between the three voltages is caused by self heating of the whole structure (including the contact barrier layers) in the first microsecond of the applied pulse. The negative temperature coefficient of the resistance in all three curves occurs through the combined Joule self-heating of the Polysilicon/Polycide layer sandwich (see Figure 6.25). The high noise in the data during the first $10 \mathrm{{\mu}s}$ originates from the low voltage level in this time regime and the resulting low signal-to-noise ratio.

Figure 6.25: Fuse device structure showing the variety of included materials

6.4.3 Simulation and Inverse Modeling

6.4.3.1 Mathematical Models

For the numerical calculation of Joule self-heating effects two partial differential equations have to be solved. POISSON's equation

$$\nabla \cdot (\gamma_\mathrm{E} \nabla \varphi)=0$$ (6.5)

gives the electric potential $\varphi$ where $\gamma_\mathrm{E}$ denotes the electric conductivity. The power loss density $p$ is obtained by computing $p=\gamma_\mathrm{E}(\nabla \varphi)^2$. The heat conduction equation

$$c_\mathrm{p} \rho_\mathrm{m} \frac{\partial{T}}{\partial{t}} - \nabla \cdot (\gamma_\mathrm{T}\nabla T)=-p$$ (6.6)

is solved to obtain the temperature distribution where $\gamma_\mathrm{T}$ represents the thermal conductivity, $c_\mathrm{p}$ the specific heat, and $\rho_\mathrm{m}$ the mass density. The temperature dependence of the conductivities is modeled with

$$\gamma(T)=\frac{\gamma_0}{1+\alpha(T-T_0)+\beta{(T-T0)}^2}$$ (6.7)

where $\gamma_0$ is the thermal or electrical conductivity at the temperature $\mathrm{T}_0$, $\alpha$ and $\beta$ are the linear and quadratic temperature coefficients of the specified materials [162].

6.4.3.2 Simulation Setup

The layout of the fuse was transformed into a three-dimensional representation of the device using a detailed process description of the interconnect forming deposition and etch steps. With the well known electrical conductivity of the interconnect and barrier layers the structural setup was checked by calculating the overall resistance of the structure excluding self-heating effects. The Polysilicon conductivity was matched to the observed overall resistance and the resulting value was compared to independently measured sheet resistances of the polycrystalline layer in fabrication, resulting in an excellent agreement between the ohmic simulation and the measurements. The subsequent transient simulations were set up including the thermal coefficients of the electrical conductivity, the thermal conductivity, and the heat capacity of all layers in the structure. The starting values of these parameters were taken from literature data [196], [197], [198].

6.4.3.3 Inverse Modeling

The simulation framework SIESTA provides a wide range of optimizers that can be chosen to fit best for the current problems. Reference data for this optimization are measurements of the resistance calculated from Figure 6.24. At start time SIESTA provides the initial values of the free parameters for the three-dimensional interconnect simulator STAP of the SAP package, as introduced in [195]. The output of the simulation is parsed by SIESTA in order to compare it with the reference data. It produces a score value that indicates how good these two data sets match. This value is submitted to the optimizer which generates corresponding to the score value the next n-tuple of free parameters to improve the next score value that will be evaluated after the next simulation run with the currently produced values.

Wide interval ranges of free parameters can result in convergence problems because of non-physical parameter values which would cause negative resistance or negative doping. To avoid this, the simulation framework SIESTA provides a kind of divergence detection where SIESTA is signaled when the simulator has problems to converge. This feature allows the user to expand the intervals of the free parameters in a larger range as before.

6.4.4 Results and Discussion

The simulation framework SIESTA has to fit the thermal parameters of the electrical and thermal conductivities in order to minimize the difference between the reference and the simulation. To check the consistency of the setup, all thermal and electrical parameters were used for the automated simulation run, resulting in a total of 10 parameters. The best fit to the measured reference data is given in Table 6.3. The electrical and thermal conductivities ${\gamma}_\mathrm{0,E}$ and ${\gamma}_\mathrm{0,T}$ as well as the linear temperature coefficient of the thermal conductivity ${\alpha}_\mathrm{T}$ for Polysilicon are in excellent agreement with data reported in  [194]. The electrical conductivity of the Polysilicon/Tungsten Silicide sandwich as a function of temperature is comparable to data measured electrically by external heating of the layers.

Table 6.3: Parameters of electrical and thermal conductivity for materials used in the polysilicon fuse structure

	Poly Si	$\mathrm{WSi_2}$
${\gamma}_\mathrm{0,E}$ [ ${1}/\mathrm{{\mu}{\Omega}m}$]	$0.12$	$1.25$
${\alpha}_\mathrm{E}$ [ ${1}/\mathrm{K}$]	$9.1 \times 10^{-4}$	$8.9 \times 10^{-4}$
${\beta}_\mathrm{E}$ [ ${1}/\mathrm{K^2}$]	$7.9 \times 10^{-7}$	$8.1 \times 10^{-7}$
${\gamma}_\mathrm{0,T}$ [ $\mathrm{W}/\mathrm{Km}$]	$45.4$	$119.4$
${\alpha}_\mathrm{T}$ [ ${1}/\mathrm{K}$]	$2 \times 10^{-2}$	$2.98 \times 10^{-2}$

Figure 6.26: Comparison of measured and simulated polysilicon fuse resistance as a function of time

The optimized parameter set results in resistance characteristics as shown in Figure 6.26 where an excellent match with the measurements is obtained. The strong increase of the resistance as a function of time because of self heating can be clearly seen. In addition, after a certain critical temperature is reached, the resistance drops dramatically and the ohmic approximation looses its validity. In order to generalize this result to other fuse geometries, this critical temperature has to be extracted. As expected, the critical temperatures of all the three samples are inside of a small interval about $1150 \mathrm{K}$ (cf. Figure 6.28). This value is much smaller as the single crystal Silicon melting point of $1414 ^\circ\mathrm{C}$ and the Tungsten Silicide (WSi$_2$ phase) melting point of $2015 ^\circ\mathrm{C}$[199]. The maximum temperature of the Polysilicon fuse is observed in the center of the Tungsten Silicide layer as shown in Figure 6.28.

Several mechanisms for this low critical temperature are possible. First, the disordered region between the Tungsten Silicide and the Silicon may have a stoichiometry closer to the eutectic point of the Tungsten-Silicide system and therefore a lower melting point. But since the lowest eutectic temperature of the W-Si system is $1389 ^\circ\mathrm{C}$[199], this is not likely for pure alloys. Second, the high doping concentration of the Polysilicon layer reduces the melting temperature as reported for Silicon glasses with high Boron and Phosphorus contents. Finally, the assumption that all materials show Ohmic behavior over the full temperature range between $300$ and $1200 \mathrm{K}$ does not hold for higher temperatures.

The intended target to optimize fuse layouts for better performance is not affected, since it is obvious from Figure 6.27 that the melting begins always at approximately the same temperature. Therefore, the method should be applicable for other geometries as well. The agreement between experiment and simulation is excellent and provides a reliable base for carrying out predictive simulations of the transient temperature distribution during the initial heating phase of the fusing.

Figure 6.27: Comparison of the simulated temperature and measured polysilicon fuse resistance showing the extracted critical temperature

Figure 6.28: Temperature distribution in the polysilicon fuse interconnect structure at $\textrm {65 }{\mu }\textrm {s}$ and $\textrm {1.7 V}$

6.4.5 Conclusions

A method for obtaining important material parameters by inverse modeling using finite element simulations of complex interconnect structures was presented. This method is capable of describing the electrical behavior of interconnect materials over a significant temperature interval. Furthermore, it uses the transient thermal self-heating effect to separate different materials and their electrical and thermal properties. Nevertheless, the exact conduction mechanism inside the Polysilicon layer is still not well reflected in this analysis. In further work the impact of the grain boundary barriers and their behavior at high temperature has to be addressed by implementing a more accurate model like the model of Mandurah [200]. It was demonstrated that the method is consistent and gives an excellent match to experimental results. With the extracted critical temperature, where the material looses its ohmic properties, the geometry can be optimized in terms of reliability and speed.