FFT-based Analysis

The objective in performing a spectral analysis on the clock period (treated as a signal) is to check if there are some particular spatial frequencies at which signal power is observed. An answer to this question is provided in Fig. 5.21. The FFI' is performed on the 100 period observations of Fig. 5.13a along the X direction, as well as in the Y direc- tion. Note that the DC component was not shown to avoid masking the phenornena of interest. This analysis reveals that the signal power is shared between:

- A DC component corresponding to the average clock period.

-

Several spectral components, out of which three appear at the following angular spatial frequencies:

period of 100 cells, while the second component,

(a5

=

-

^), confirms the "periodic- 20

ity" visually noticed in Fig. 5.13. In other words, the value of the clock period dong the X direction follows a periodic pattern that more or less repeats itself every 20 observations (which corresponds to returning back to the same ce11 but in a different die). The third component, (aio =

%),

2 R which is actually an h m o n i c of H5 (Fig.

5.2 1 a). Note that Hl 5 (Fig. 5.2 1 b) can also be considered an harmonic of H5. Har- monics H l 0 and H l 5 are, in fact, located at multiple frequencies compared to that of H5.

It is remarkable that harmonics are shifted by MATLAB one position to right. The DC component (corresponding to an angular spatial frequency of zero) is in fact reported as the first frequency component. Also, note that Fourier analysis produces symmetric harmonics.

- A noise component corresponding to the rest of FFT points. The elements of the noise magnitude are distributed around a mean value of approximately 1.3 ps. This noise reflects probably the spatial "white" noise introduced by Pelgrom in [ 6 ] , which describes short-distance process variations. If it is the case, the standard deviation of this noise should tend toward zero as more period measurements are added, aIong the X direction, for example. According to Pelgrom, this kind of noise is related to distri- butions of ion-implanted, diffused, or substrate ions; local mobility fluctuations; oxide

granularity and oxide charges.

A similar spectral analysis was performed for the two directions in Fig. 5.13b. The power (in picosecond square) is shared arnong the various components of the period signal as shown in Table 5.3. The power of an harmonie is calculated as the square of its magni- tude. The power in the large-scaIe component is the sum of powers in component H l and its imrnediate neighbors, which are clearly well above the noise floor (Fig. 5.21 a and Fig.

5.21b). The same applies to the environmental component composed of H5, H l 0 and H15, and the respective immediate components. The squared magnitudes of the remaining FFI' points are summed to form the noise cornponent. Sinee some noise contribution is also included in components H l through H15, and in order not to count noise more than once, an amount of power corresponding to the noise mean-value is substracted from the power of spectrum components associated with the large-scale and environmental components.

Table 5.3 shows that, in most cases, the contribution of noise to the total period variation is I l to 15%. However, it can be as large as 20% in the particular case of direction X in Fig. 5.13b, where the environmental component was almost buried into noise. Also, depending on the wafer considered (Fig. 5.13a or Fig. 5.13b) and which direction along the wafer, the environmental and large-scale components have a variable significance. The power carried by the DC component remains much Iarger than al1 components of the period variation.

Table 5.3: Power in various components of the dock signal (in ^p2)

In the following, we are interested in the spatial correlations (Fig. 5.22) between:

1- immediate neighbors (physical neighbors on the same die),

2- cells with the sarne environment in different dies (can be considered as neighbors at WSI scale, since there is only one ce11 with an identical environment on each die),

3- and cells which share neither the same die nor the same eIectrical environment (non- neighbors).

For each type, a number of pairs were constructed from Fig. 5.13a along the X direction.

For each pair of cells, the absolute value of the difference in clock period is considered.

Table 5.4 surnmarizes the mean value of the distribution of these differences for the vari- ous types of neighbors.

Note that along 5 die sides, there are 20 groups of 4 pairs of same-environment neigh-

Table 5.4:

Means of absolute period differences

Immediate Neighbors 17 Same-Environment 70

Neighbors

Non-Neighbors 50

bors, such that ce11 pairs of different groups have different environments. Table 5.4 shows that the period differences are 1.4 to 4.2 times more important between same-envi- ronrnent neighbors than for other types of ce11 pairs. Despite the fact that a certain "perio- dicity" seems to be respected for same-environment neighbors in Fig. 5.13, the gradient of changes in the magnitude of clock period seems to be more conserved from die to die than the magnitude itself. Also, over long distances (5 dies in Our case), the Iarge-scale compo- nent modulates the environmental component, thus affecting the magnitude levels to which same-environment neighbors are set in different dies.

V. Conclusion

In this paper, we have presented the first large scale experimental characterization of spatial process variations on a parameter that is directly involved in timing issues: the MOS transistor time constant. This is achieved by measuring the oscillation period of

0

high-speed (500

MHz)

CMOS ring oscillators that are implemented at different locations on individual dies and over wafers.

An environmental phenomenon that modulates the clock period was discovered. That phenomenon manifests itself as a strong periodical influence due to variations in surround- ing electrical devices, and which can be viewed as a generalization of the "edge" effect observed and reported in previous work. Also, a large-scale process component induces shifts into the environmental part of the period variations. This component is characterized by a much lower spatial frequency. Based on their measurements, Gneiting et al. assumed in [13] that on-wafer variations are negligible. Due to new phenornena that we observed, their assumption appears to be very questionable, not only at wafer scale, but particularly at die level. Sirnilarities can be drawn between the "striation" effect observed by Pavaso- vic et al. 11 11 and Andreou et al. 1121 with the environmental component (in ternis of

"periodicity") as well as between the "gradient" effect and the process component (in ternis of low frequency). Also, important "edge" effects were observed on the wafer bor- ders. The efiects of the power-supply distribution network on the clock period variations was also analyzed and characterized.

Acknowledgment

We would like to acknowledge Prof. John Currie for making available the facilities of LISA laboratory, Nor- mand Grave1 for his helpful experience in wafer probing and Ali RahaI from POLYGRAM laboratory for helpful discussions on microwave issues, al1 at Ecole Polytechnique of Montreal. The authors also acknowI- edge the Canadian Microelectronics Corporation and NORTEL for manufacturing our experimental device and for providing full wafers for testing. This work was also partly supported by grant OGP0006574 from the National Sciences and Engineering Research Council of Canada and by a grant of the Synergie program of the province of Quebec.

Fig. 5.1. Ring oscillator (schematic)

Fig. 5.2. Ring oscillator and inverter's output (photograph)

Fig. 5.3. HSPICE clccmcal simulation

Fig. 5.4. Oscillation cell (photognpb)

Fig. 5.5. Our design (photograph)

Fig. 5.6. Vdd & Gnd pads (photopph)

Fig. 5.8. Wder (1

-Fig. 5.9. Equipment @hocograph)

Fig. 5.10. Two signal outputs (photograph)

Y- 0 - - 0 x-a) first mdomlychosea die

Y- 0 - 0

X e k œ b m

b) second randomly-chosen die

C) die from ^a non-xribed w a f u

Fig. 5.11. Four-si& npresentation of clock pcriod ai the die levcl.

2300 2400 2500 2600 2700 Period (ps)

Fig. 5.12. Histogram of pcriod distribution in die b) of Fig. 5.11.

(a) fint non-rn&d w&r

(b) second non-scribcd w a f w

Fig. 5.13. Cortelauon dong X & Y directions

(a) horizoncal pair of die sides

(b) vertical pair of die sides.

Fig. 5.14. Cornlalions bciwœn neighbon

(a) fint non-xriùed aider

O O Y direction

(b) second non.scribcd wafcr

Fig. 5.15. Spatial charactcrizauan of clock period oci WSI scale

Fig. 5.16. Numerical values of the process cornponcni (Note: Sincc the wafer is not nctangular, incxisicnr dies have been rcplaccd by the minimal p e n d value Le., 1938 ps)

Gnd rail

Fig. 5.17. Schematic of the power-supply distribution octwork

c) Voluge diffcrcnce Pi-Gi

Fig. 5.18. Voltage fluctuations Joog pow-wpply Nh.

(a) in X direction of Fig. 5.13a

(b) in Y direction of Tig. 5.13a

Fig. 5.21. Che-dimensional FFT of pcnod distribution

Chapitre 6.

Dans le document SYNTHÈSE DE MOHAMED NEKILI (Page 136-158)