Tips & Tricks
6. Variability Analysis Using Impedance Field Method

6.1 What is the impedance field method?
6.2 Does IFM analysis require calibration?
6.3 How does IFM analysis of RDF effects compare to atomistic RDF analysis?
6.4 Can IFM describe non-Gaussian distributions?
6.5 How do I extract σ(V_t) and σ(I_d) from IFM simulation results?
6.6 Which parameters are necessary for oxide thickness variation analysis?
6.7 How do I simulate variation of the gate length of MOSFETs?
6.8 How are 2D and 3D simulations handled in the IFM?
6.9 Relevant Application Notes
6.10 References

Objectives

• To present how to perform variability analysis using the impedance field method.

6.1 What is the impedance field method?

The random variability of leading-edge transistors is an issue of growing concern as transistor scaling continues. With each new technology generation, new materials and process steps are introduced to chip manufacturing that affect transistor variability. The impedance field method (IFM) is an important tool for variability analysis and is available in Sentaurus Device.

The basic concept behind the IFM is to treat randomness as a perturbation of a reference device. Rather than solving the full nonlinear Poisson and drift-diffusion equations for a large number of random device realizations, you obtain the 3D TCAD solution only once for the reference device. Then, you compute the current fluctuations at the device terminals caused by these random perturbations. This computation is based on linear response theory using a Green's function technique.

For modeling statistical variability, Sentaurus Device supports variants of the IFM: noise-like IFM, statistical IFM, and deterministic IFM.

6.1.1 Noise-Like Impedance Field Method

The noise-like IFM performs noise analysis of semiconductor devices. Its basic concept is to establish a model of the noise sources of the device and, for each local fluctuation, to evaluate its effect on the device terminals where the output noises are measured. The total noise observed on the device terminals is calculated by summing the contribution of all the local fluctuations. There are two main steps in performing noise-like IFM simulations.

The first step identifies the appropriate microscopic models of the noise sources, that is, the local fluctuations inside the device. Depending on the problem and the kind of noise of interest, the noise sources could be, for example, fluctuations of the device geometry or the dopants inside the device.

The second step determines the impact of the noise sources on the device terminal outputs. It is assumed that the terminal voltages and the local fluctuations follow linear relationships, which are obtained by computing the Green's function of the terminal voltages. When the Green's functions are computed, the total noise on device terminals can be readily calculated from the noise sources.

Noise source models related to variability analysis are available for random dopant fluctuations, random geometric fluctuations, random trap concentration fluctuations (bulk and interface traps), and random workfunction fluctuations.

6.1.2 Statistical Impedance Field Method

Within the statistical impedance field method (sIFM), you use the Green's function to compute the linear current response to actual random perturbation. To obtain statistical samples, for example, for random dopant fluctuations, you assume that doping is spatially uncorrelated and that the number of dopants in a given volume follows a Poisson distribution, with an average given by the average number of dopants in the volume.

Sentaurus Device can generate actual random perturbations for any of the following quantities under investigation: dopant fluctuations, geometric fluctuations (for example, oxide thickness variation and line edge roughness), trap concentration fluctuations (bulk and interface traps), workfunction fluctuations, band-edge fluctuations, metal conductivity fluctuations, and dielectric constant fluctuations.

You also can include the effects of a random contact resistor attached to the drain contact of a MOSFET. The effect of this resistor on the I_d–V_g characteristics can be computed in a postprocessing step with Sentaurus Visual.

For details, see the Sentaurus™ Visual User Guide, Appendix G. Specifically, the documentation for the ifm::GetsIFMStdDev and ifm::ReadsIFM procedures.

From the linear current response, you compute the full I–V curves of the randomized devices. The IFM approximates the generally nonlinear system by a linear, small-signal, equivalent one. You can improve the accuracy of this inherent approximation by leveraging all otherwise available information about the system.

The IFM provides a certain degree of freedom in selecting which linearized quantities to consider. For example, for a MOS transistor I_d–V_g simulation, you can consider either the linear gate-voltage response, or the linear drain-current response of the system, or a combination of both. For a transistor in saturation, the gate voltage only weakly controls the drain current and, therefore, considering the linear drain-current response is more appropriate. In the subthreshold regime, on the other hand, random fluctuation effects are very well described by threshold voltage fluctuations, which are intuitively linked to the linear gate-voltage response. This additional information is incorporated into the I–V computation algorithm by formulating a set of boundary conditions specific to the devices and biasing schemes at hand.

An application note [App. 1] includes examples of performing sIFM simulations.

6.1.3 Deterministic Impedance Field Method

For the deterministic IFM, you specify the variations directly. Instead of using doping profile randomizations, you use your own modified doping profile. Sentaurus Device computes the effect of the variations on the observation terminal voltages and currents. Compared to random fluctuations, deterministic variations give you more control over the variation and are easier to understand, because no statistical interpretation is required. This method is particularly useful for screening and corner analysis.

6.2 Does IFM analysis require calibration?

The IFM analysis of random dopant fluctuation (RDF) effects does not require any calibration. The amplitude of the RDF effect is proportional to the local doping level and is determined automatically. Each doping species contributes to RDF effects individually. For example, in an area of a device with compensated doping, the contributions from donors and acceptors add rather than subtract.

Comparisons to experimental data and to atomistic simulations show that this approach works accurately without the need to calibrate any IFM parameters.

After the traps themselves have been calibrated, no additional calibration is needed for the IFM analysis of random trap fluctuation. Similar to RDF, the amplitude of the random trap fluctuation effect is determined automatically by the local trap concentration.

The models for random geometric fluctuations and random workfunction fluctuations offer several use-adjustable parameters that require calibration.

6.3 How does IFM analysis of RDF effects compare to atomistic RDF analysis?

Accurate atomistic RDF analysis requires 3D simulations because, in a 2D structure, each dopant atom becomes a charged rod that runs across the entire channel width, which significantly exaggerates the effect. A 3D simulation uses a large number of randomized samples. To obtain statistically relevant results, you must use a minimum of 200 samples. Sometimes 100 000 samples are used [Ref. 1], which constitute a huge computational effort.

In the IFM approach, which can be used in two and three dimensions, the simulation is performed only once and takes about twice as long as DC analysis. Therefore, IFM analysis can be performed 100 to 50 000 times faster than the atomistic approach.

In terms of accuracy, direct comparisons of the IFM and atomistic approaches have shown that they result in almost identical threshold variabilities (see, for example, [Ref. 2]) if the comparison is performed correctly.

However, not all of the well-accepted TCAD transport models and parameters are directly applicable to atomistic RDF analysis. Many TCAD transport models are specifically designed to simulate an "average" device structure with a continuous "average" doping profile. When randomizing a doping profile, this implicit assumption of smooth doping profiles is violated. For certain TCAD transport models, this results in spurious – and at times very strong – nonlinearities. Therefore, you must carefully select the TCAD transport models you want to use for your atomistic simulations [Ref. 3].

Furthermore, atomistic simulations do not automatically reproduce the correct average (see, for example, [Ref. 4]). If you use the same TCAD transport models and parameters, the threshold voltage of the unperturbed "average" device will not correspond to the average of the threshold voltages of the randomized devices. Now, you typically have calibrated your TCAD transport models and parameters such that you obtain the correct results for the unperturbed "average" device. Therefore, you will need to redo this calibration for the atomistic simulations to recover these correct results.

With the IFM, first you compute the TCAD solution, then you determine the linear response of the TCAD solution to the given perturbation. Consequently, all well-established TCAD transport models and all calibrated transport model parameters can be used directly for the IFM. In addition, the linearization guarantees by design that, for example, the threshold voltage of the unperturbed "average" device will be the same as the average of the threshold voltages of the randomized devices.

6.4 Can IFM describe non-Gaussian distributions?

The noise-like IFM is fast, easy to use, and efficient when it is sufficient to know the fluctuation-induced standard deviations of terminal currents and voltages.

The noise-like IFM does not make any assumptions about the actual shape of the distribution functions, so a standard deviation can be uniquely defined for any distribution function. However, if all you know is the standard deviation, you cannot draw any conclusions about possible non-Gaussian distribution tails.

Experimental data published in [Ref. 5] [Ref. 6] exhibits purely Gaussian threshold voltage and driving current distributions for a million transistors, which correspond to the range of 10 standard deviations – which raises at least the possibility that non-Gaussian behavior observed in certain atomistic simulations might be artifacts of the method (see Section 6.3 How does IFM analysis of RDF effects compare to atomistic RDF analysis?).

Using the statistical IFM, you can describe the non-Gaussian behavior of derived quantities. An example of such a derived quantity is the static noise margin of an SRAM cell (see the application note [App. 2]).

Variability due to surface traps can also show pronounced non-Gaussian distributions when the average number of traps is low, for example, two. In this case, it is statistically possible for a given device to have 5 or 6 traps, but it is impossible to have less than zero traps. Consequently, the distribution must be very asymmetric. Such a case is illustrated and discussed in an application note [App. 1].

6.5 How do I extract σ(V_t) and σ(I_d) from IFM simulation results?

In typical noise-like IFM simulations for MOSFETs, the noise simulation is run at a series of bias points. The noise spectrum of the terminal currents of interest is calculated for each bias point. The $V_t$ variation $σ_{Vt}$ can be calculated from the noise spectrum in a postprocessing step. For example, when the noise-like IFM is applied to an I_d–V_g simulation, the drain current variation $σ_{Id}$ is calculated from the drain noise current spectral density $S_{Id}$, which is a function of $V_g$:

$$σ_{Id} = √{S_{Id}(V_g) ⋅ 1 \text"Hz"}$$

The calculation of the current variations assumes that the contact voltages remain fixed. Assuming fixed drain and gate currents, the gate voltage variation $σ_{Vg}$ can be calculated from the drain noise current spectral density $S_{Id}$ and the drain-gate admittance $A_{dg}$:

$$σ_{Vg} = σ_{Id}/A_{dg} = {√{S_{Id}(V_g) ⋅ 1 \text"Hz"}}/A_{dg}$$

The gate-voltage variation is also a function of the gate voltage $V_g$. The threshold voltage variation $σ_{Vt}$ is $σ_{Vg}$ at $V_g = V_t$, where $V_t$ is the user-defined threshold voltage, typically the linear or saturation $V_t$ based on a constant drain-current level. The variation of the drain-current $S_{Id}$ is directly available from AC analysis results. The relative drain-current variation is:

$$σ_{Id}/I_d = √{S_{Id} ⋅ 1 \text"Hz"}/I_d$$

The drain-current variation is also a function of the gate and drain voltages. A special point commonly referred to is the saturation drain-current $I_{d,Sat}$ variation, located at $V_g = V_{ds} = V_{dd}$ on this curve. Sometimes, 2D simulation structures are used to approximate quasi-3D structures, assuming that the device is homogeneous along the third (width) dimension. The AreaFactor is set to the device width. With 2D simulations, the standard deviations of the 3D devices can be calculated from those of the 2D device by the following relations:

$$\table σ_{Vg}^{3\text"D"},=,σ_{Vg}^{2\text"D"}√{1\: \text"μm"} / √w ; σ_{Id}^{3\text"D"},=,σ_{Id}^{2\text"D"}√w / √{1\: \text"μm"} ; $$

When the AreaFactor ($w$) is set correctly, the standard deviations calculated by the IFM are scaled properly according to the above relations.

6.6 Which parameters are necessary for oxide thickness variation analysis?

Two parameters must be provided for oxide thickness variation analysis: the amplitude and the correlation length of the oxide thickness fluctuations. The amplitude is usually set to the atomic step at the silicon–oxide interface, which is approximately 0.2 nm, as can be seen in a TEM in [Ref. 7].

For high-k gate dielectrics, usually a SiO₂ interlayer is used. If the simulation structure explicitly contains such an interlayer, the amplitude should be set to the atomic step. However, if the equivalent oxide thickness (EOT) is used instead, the amplitude must be scaled according to the ratio of high-k permittivity to SiO₂ permittivity.

The correlation length determines how long are the flat parts of the interface in between the atomic steps. A typical correlation length for the silicon–oxide interface is approximately 4 nm as can be seen in experimental data shown in [Ref. 7].

6.7 How do I simulate variation of the gate length of MOSFETs?

In an actual manufacturing process, when the gate length of a MOSFET varies, the liner oxide, the spacers, the source/drain extensions (SDEs), and the deep source/drain (SD) regions all vary accordingly, as they are formed in a self-aligned process.

In an IFM simulation, however, this does not happen automatically as the IFM usually treats all variability sources as statistically independent. It is possible, however, to correlate different variability sources using the RandomField option. A random field is an abstract, dimensionless, and normalized amplitude field. It can be used in the form of a multiplication factor in one or more statistical IFM (sIFM) variability sources.

For gate-length variability or line edge roughness (LER), for example, you would need to correlate the fluctuations of the side gate edges using the Geometric option of the sIFM and the fluctuations of the doping profiles using the DopingVariation option of the sIFM.

For a spacer-patterning process of the gate stack, the spatial fluctuations of the two gate edges are correlated. For the IFM modeling of this situation, it is sufficient to define a vectorial shift of both the source-side and drain-side gate edges, and the corresponding doping profile. Note, however, that in this case, the LER is typically small and negligible compared to other variability sources.

For direct patterning of the gate stack, the spatial fluctuations of the two gate edges are uncorrelated. For the IFM modeling of this situation, it is necessary to separate the source side and the drain side by limiting the application of the IFM variations to box-shaped regions using the SpatialShape, SpaceMid, and SpaceSig options. Then, you define separate random fields for the source side and the drain side.

As a worse-case scenario, it is also of interest to anticorrelate the spatial fluctuations of the two gate edges. In this case, you can use a single random field for the source-side and the drain-side areas, and correlate them by defining them together in a single RandomizedVariation statement, using equal but opposite shift vectors.

In particular, for device structures with complicated spacers, it can be challenging to properly define and correlate the various interfaces involved in the gate length or LER variability simulation. Note also that the accuracy of the results of gate-length variability IFM can be sensitive to meshing. Furthermore, when simulating devices with an actually changed gate length, the current as a function of amplitude tends to show some nonlinear behavior, which cannot be captured by linear response theory. Therefore, it is recommended to validate the IFM setup, the mesh, and the applicability by comparing IFM results for a spatially uniform gate-length variation with standard TCAD simulation results for device structures with an actually changed gate length.

An alternative approach to modeling gate-length variation is to create a parameterized simulation structure and to vary the gate length parameter. This method ensures that all of the mentioned variation factors are included automatically. Multiple gate-length splits can be defined and organized using Sentaurus Workbench.

6.8 How are 2D and 3D simulations handled in the IFM?

Even when considering a reference device structure that exhibits 2D symmetry (extruded devices), the randomization will break the 2D symmetry, and any variability analysis must be performed using full 3D TCAD simulations.

Fortunately, when using the IFM, you can use a very efficient hybrid approach. For the IFM, you need the Green's function and the randomization. The Green's function depends only on the reference device and this retains 2D symmetry, while the randomization will almost always exhibit 3D symmetry.

Therefore, if your device structure exhibits 2D symmetry, you can rely on the much faster 2D TCAD simulations to compute the DC results as well as the Green's functions. The current response to the randomization, however, will be computed in three dimensions. For that, the device structure is extruded internally in the third dimension. For some variability sources, namely, RDFs, the convolution integral of the Green's function and the randomization can be performed analytically in the third dimension, and the simulation speed does not depend on the selected device width.

For other variability sources, namely, sources with a finite correlation length, the convolution integral must be performed numerically in the third dimension and, therefore, the evaluation speed of the convolution integral for an extruded 2D structure will be approximately the same as for a real 3D structure and will depend on the selected device width.

The device width is set with the keyword AreaFactor, which defaults to 1 µm. To activate this hybrid approach, use the keyword ExtrudeTo3D. This is necessary to model variations correctly with spatial correlations, but it will increase the runtime for these variations. Without this keyword, for all variability sources with a finite correlation length, the randomization is performed assuming that the variations are correlated perfectly in the third spatial direction. The keyword ExtrudeTo3D has no effect on 3D structures.

6.9 Relevant Application Notes

App. 1

Modeling Random Variability Effects With the Statistical Impedance Field Method, TCAD Sentaurus Applications Library, available from
../Applications_Library/Variability/FinFET_Variability_sIFM.

App. 2

Modeling Statistical Variability of Static Noise Margins of SRAM Cells Using the Statistical Impedance Field Method, available from
../Applications_Library/Variability/SRAM_Variability/SRAM_3d.

Check the List of Available TCAD Sentaurus Application Examples and Notes article, available from the SolvNetPlus support site, for the latest versions of an application example.
Go to https://solvnetplus.synopsys.com/s/article/List-of-Available-TCAD-Sentaurus-Application-Examples-and-Notes-1576165785936.

6.10 References

Ref. 1

D. Reid et al., "Statistical Enhancement of the Evaluation of Combined RDD- and LER-Induced V_T Variability: Lessons From 10⁵ Sample Simulations," IEEE Transactions on Electron Devices, vol. 58, no. 8, pp. 2257–2265, 2011.

Ref. 2

G. Roy et al., "Comparative Simulation Study of the Different Sources of Statistical Variability in Contemporary Floating-Gate Nonvolatile Memory," IEEE Transactions on Electron Devices, vol. 58, no. 12, pp. 4155–4163, 2011.

Ref. 3

K. El Sayed, E. Lyumkis, and A. Wettstein, "Modeling Statistical Variability with the Impedance Field Method," in International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), Denver, CO, USA, pp. 205–208, September 2012.

Ref. 4

H.-S. Wong and Y. Taur, "Three-Dimensional "Atomistic" Simulation of Discrete Random Dopant Distribution Effects in Sub-0.1μm MOSFET's," in IEDM Technical Digest, Washington, DC, USA, pp. 705–708, December 1993.

Ref. 5

T. Tsunomura et al., "Analyses of 5σ V_th Fluctuation in 65nm-MOSFETs Using Takeuchi Plot," in Symposium on VLSI Technology, Honolulu, HI, USA, pp. 156–157, June 2008.

Ref. 6

X. Song et al., "Impact of DIBL Variability on SRAM Static Noise Margin Analyzed by DMA SRAM TEG," in IEDM Technical Digest, Washington, DC, USA, pp. 62–65, December 2011.

Ref. 7

M. Niwa, K. Okada, and R. Sinclair, "Atomically flat, ultra thin-SiO₂/Si(001) interfaces formation by UHV heating," Applied Surface Science, vol. 100–101, pp. 425–430, July 1996.