Metal Width Variation (Summary)

Summary of Metal Width Variation

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Now, I think this much is sufficient for the width variation. Let’s revise few of them.

Below figure has

• Ideal shape (which we draw on layout or say theoretical shape)
• Practical Shape (Which will come after different manufacturing variation effect).
• Here, I have flipped the shape of “practical shape” which we have discussed till now to show you that if you understand this then you can model any type of shape.

Below figure explain about the modeling as per variation Type2:

• Red color dotted shape is Ideal shape / Layout shape or say Theoretical Shape.
• Light brown color shape is Desired Shape.
• Green color shape is modeled shape (as per variation type 2)
• In variation type 2, top and bottom delta are same.
• Since modeling is with respect to theoretical shape’s edges, it’s difficult to model exact delta either on Top side or bottom side. But this one is more closure to the desired shape.

Below figure explain about the modeling as per variation Type3:

• Red color dotted shape is Ideal shape / Layout shape or say Theoretical Shape.
• Light brown color shape is Desired Shape.
• Green color shape is modeled shape (as per variation type 3)
• In variation type 3, top and bottom delta can be different and that’s the reason we are able to match top and bottom delta width properly (except right hand side bottom delta).
• This is because Variation type 3 has condition that
• top width delta on either side should be same
• Bottom width delta on either side should be same.

Below figure explain about the modeling as per variation Type4 along with the variation Type3:

• Red color dotted shape is Ideal shape / Layout shape or say Theoretical Shape.
• Light brown color shape is Desired Shape.
• Green color shape is modeled shape (as per variation Type3 and Type4)
• In variation type 3, top and bottom delta can be different and that’s the reason we are able to match top and bottom delta width properly and in type 4 right and left side variation can be different and that the reason we are able to match right hand side bottom delta (which were missing in the type 3).

In the above figure related to non-linear top-to-bottom variation can be modeled with the help of Variation type 5.

Below figure explain about the modeling as per variation Type6 (basically it will explain now from the Top view – which also include the length of the metal wire)

• Red color dotted shape is Desired Shape (which will come after manufacturing). (Note: here just changed the color scheme)
• Light green shapes are as per modeling (after applying variation type6)
• Dark green (in reference to above point) shapes are Ideal shape / Layout shape or say Theoretical Shape.

For variation type 7 below figure is self-explanatory. I think it’s not required to give much more details of that. Just remember that these 2 wires are oriented differently (one in vertical and other in horizontal).

For variation type 8 again below figure is self-explanatory. Just point to noted that it is because of n number of non-uniformities. It effects differently for Resistance and Capacitance. This variation type can be combined to any other variation type and accordingly can be modeled.

Metal Width Variation (Type 8)

Metal Width Variation (Part 6)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Width Variation Type 8:

There are few data provided by the foundry which is only effecting the Resistance or Capacitance of the design. Means there are different width variation values for effecting the Resistance and Capacitance. These type of info usually provided by 2 different tables (for any type of the variation), one specific to Resistance and other specific to Capacitance.
I know you might be thinking that how it can be possible.

Before I will explain that, it’s my duty :) wanted to remind you following thing. In design wire is long and thick (like showed in the 3D view). Length part which I am referring to is 2D view (Top View in below picture) and Thick part which I am referring to is 2D view (Front view in below picture).

In previous articles (where I have discussed about the different type of Width Variations), I have explained the width variation Type 1 to 5 using the Front-View and Variation type 6 -7 using the Top View.
Actually, in design it’s the combination of both views.

Everywhere, we are talking about the variation linearly But what about the below picture?

Yes, now we are talking about the non-uniformity in the wire itself. These non-uniformities can be introduced as part of Etching, Deposition or Lithography process. And remember, these are different from what we have discussed till now (Means variation type).

Now, if I will ask you how to model these non-uniformities, then maybe you can say that it’s very simple. Let’s take the average of this and model it as part of Width variation type. Means the calculation of the values of Bias which we are going to apply on the width should be as average value of these non-uniformities.
Your approach is 100% correct but there is catch.:0 :)You have mentioned “Average of these non-uniformities”, that’s the part I want to stress here … Average of non-uniformities in terms of which parameter? These non-uniformities have different effect on Resistance and Capacitance calculation. How? :) :)

C is more determined by outer edge of interconnect, R by average resistivity.

In case of C, when I am talking about the outer edge,

• These non-uniformities can change the distance between 2 adjacent metal wires at different point (along the length of wire), so while you are averaging the effect – may be you will consider this factor and then come up with an average biasing value which suppose to apply on width of wire.
• C is related to deposition of Charge in a plate (outer edge of plate) (we have studied this in school. There will be a lot of irregularity or say non-uniformities in the density of charge at each and every point of layer (sharp edges Vs flat edge) and if that’s the case – we have to consider that part also while doing averaging and decide a final bias value for entire edge.

In case of R

• We are more concerned about the flow of current (Means flow of electron). So mobility, drift current, effective current – these concepts comes into the picture. So while averaging these non-uniformities for R, we will consider these factor more in compare to outer distance from other metal.

You can now imagine that the same non-uniformities effects Resistance and Capacitance differently and if we want to model this – we have to divide our variation into 2 categories.

1. Width variation for Resistance.
2. Width variation for Capacitance.

Now let’s talk about one more reason of this. :) Below picture is self-explanatory (I think so). Color shades represents the metal density.

Now, may be you are thinking, why it will happen and how it will effect Resistance and Capacitance.

As, we have already discussed that interconnects are fabricated using 2 major steps

• Etching a portion of dielectric and
• then fill it up with the interconnect material.

This filling is done in different steps and somehow final material does not have uniform resistivity in cross-section (horizontal or vertical). Just because of this we have to apply different biases for R and C as a function of width.

Etching is done by bombarding loaded particles out of a plasma on the wafer surface under guidance of an electrical field. This field is not uniform but is (slightly) deformed by longer range density of the pattern and influencing etch speed. Additionally the etched away material may also influence local etching performance.
Reference: www.semiwiki.com

Variation info can be in the following way.
Note: I just took the example from variation type 1, but same concepts can be applied for other tables or other type of variation parameter also.

Table 10: Different Bias Value for Resistance and Capacitance effect
Metal Layer	Width (um)	Variation in % (+/-)(Resistive Only)	Variation in % (+/-)(Capacitive Only)
Metal 1	0.2	8	9
Metal 2	0.4	8	9
Metal 3	0.4	9	10
Metal 4	0.4	10	11

Metal Width Variation (Type 7)

Metal Width Variation (Part 5)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Width Variation Type 7:

Now what?? What is remaining now?:) :)Anyways – still few things are remaining. Have we discussed anything about the wire orientation? Wire width variation based on wire orientation.
Please refer the below figure and think how can you model such scenario.

Don’t think that I am creating variation type unnecessary :) it’s the scenario for lower nodes (like 28nm and below). Soon you will get the all the explanation of this. But right now, modeling is important.:). So what’s the solution? Yes you are thinking in the right direction that we will add one more parameter “Direction” along with the normal wire variation parameter. So as per you, Horizontal and Vertical are 2 different direction and we can specify 2 different biasing tables for these. But I would like to generalize this.

Usually, foundry provides the info with respect to the some reference orientation. Actually this direction concept come from the manufacturing instrument (just explained in layman language). Every machine has different accuracy margin between left-right (horizontal) and top-bottom (vertical). So orientation thing comes into the picture.

Your deign will have structure only in 2 direction (45degree structure or any other degree orientated structure are now obsolete in lower nodes) and once you decide that your reference direction (may be based on maximum structure in a particular direction or with respect to any other reason), you will apply those rules. Once your reference direction is decided, machine will be orientated for that direction.

Structure in that direction will have rules corresponding to that and for other structure (which are in other orientation) machine will behave differently (so different rules). So this reference direction is important.

Foundry will provide the rules as per Reference direction.

• Parallel_to_Reference
• Perpendicular_to_Reference

Variation info can be in the following way.

Note: I just took the example from variation type 1, but same concepts can be applied for other tables or other type of variation parameter also.

Table 9: In the form of absolute variation number
Metal Layer	Width (um)	Variation in % (+/-)(Parallel_to_Reference)	Variation in % (+/-)(Perpendicular_to_Reference)
Metal 1	0.2	8	9
Metal 2	0.4	8	9
Metal 3	0.4	9	10
Metal 4	0.4	10	11

Metal Width Variation (Type 6)

Metal Width Variation (Part 4)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Width Variation Type 6:

After the previous variation type (especially Type4), where I made a sentence that “width variation also depends on the surrounding environment (like metal density or say space with respect to neighboring metal layer)”, you may have a lot of questions or questions came in your mind. Like what will happen in the below scenario.

I am sure you can say that as per the diagram and as per foundry data, small segments of wire (enclosed by dotted red box) see a very large spacing in the horizontal direction. However the next section (enclosed by dotted green box) of the same wire shows a much closure spacing. As per the modeling (whatever we have discussed till now), if large and small spacing concept are used, width (silicon width) changes drastically which we know very well that it’s practically impossible.
What does it mean???
It means … Our list of variation type is not complete. :)

Practically, the above structure will be in somewhere like below shapes after manufacturing. You can notice that the place where spacing is drastically change (Common boundary of red and green dotted box), manufactured shape is not drastically changed. There is a curve/slope or say gradient in the width variation from low value to high value.
Since our goal is to model the variation as close as possible, foundry provide these info also. So that EDA vendors can model such portion also (which can impact RES and CAP values – especially in lower technology nodes).

Info will be somewhere in the following form.

W = width of the wire,
S =space between 2 wires,
L = minimum parallel distance between 2 wires,
dD = delta distance after L length.
dW = Delta width variation = X

data/info will be as per following function
dW(W, S, L, dD) = X ;

e.g
dW(0.5, 0.5, 0.5, 0.0) = 0.01 ;
dW(0.5, 0.5, 0.5, 0.05) = 0.02 ;
dW(0.5, 0.5, 0.5, 0.1) = 0.03 ;
dW(0.5, 1.0, 1.0, 0.0) = 0.04 ;

Again this type of info can be provided in “n” no of ways and I can’t discuss all those (because that may create an issue from confidential point of view). I have just provided the concepts here which can help you to understand. If you have worked on such cases anyhow, you can easily correlate and if you haven’t (till now), then such concept is sufficient for you.:)

After modeling as per variation type 6, above wire structures will be converted into following structures (which is very closure to the actual one).

Metal Width Variation (Type 4 and Type 5)

Metal Width Variation (Part 3)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Width Variation Type 4:

Variation type 3 has condition that

• Top width delta on either side should be same
• Bottom width delta on either side should be same.

I know you can say that it’s easy and let’s divide the modeling parameter top and bottom into left and right side. Like bottom_left, bottom_right, top_right and top_left. But it’s not as easy as you are thinking. Because these modeling has some background. We have to think, what are different reasons which can output different left and right delta? Even I am trying to figure out the exact reason of that (if you find, please let me know ). Point is, I will update you later but till then you have to search out. 

Just a hint that width variation also depends on the surrounding environment (like metal density or say space with respect to neighboring metal layer). So in this variation type, we have to consider the fact that width variation also depends on the space with respect to neighboring metal wire. In such case foundry provide below type of info.

Table 7: Metal effective Silicon Width based on Drawn Width and Spacing pattern
Metal	Drawn Width (um)	Drawn Space (um)	Silicon Width (um)
Metal 1	0.5	0.5	0.409
Metal 1	0.5	0.75	0.429
Metal 1	0.5	1.0	0.439
Metal 1	0.5	1.5	0.448
Metal 1	0.75	0.5	0.682
Metal 1	0.75	0.75	0.701
Metal 1	1.0	0.5	0.930
Metal 1	1.0	1.0	0.960
Metal 1	1.0	1.5	0.969
Metal 1	1.5	0.5	1.420
Metal 1	1.5	1.0	1.449
Metal 1	1.5	1.5	1.459

Similar type of table you can get for other metal layers with more spacing and width points. These are just for the reference level. Also remember – in place of Silicon Width, info can be in the form of delta value or delta %. It depends on foundry to foundry. 

Sometime same info can be provided into following way (final silicon width with respect to different drawn width and spacing combination)

Table 8: Metal effective Silicon Width based on Drawn Width and Spacing pattern
w/s	0.1000	0.1300	0.1500	0.2000	...
0.1000	0.1025	0.1125	0.1220	0.1470	...
0.1300	0.1265	0.1285	0.1340	0.1525	...
0.1500	0.1459	0.1459	0.1459	0.1599	...
0.2000	0.1783	0.1783	0.1783	0.1783	...
...	...	...	...	...	...

You may ask now, how it will remove the constraint of Type 3 variation. Very simple… left and right side variation in width can be figure out as per the environment around a particular metal layer. Please refer below figure for more understanding. It will help you to understand how left and right variation can be coded differently without using “left” and “right” keywords.

With the help of combination of different type of variation we can model anything. Like different top and bottom delta (using Type variation 3), different left and right delta (using type variation 4). As I have described in type3 how to combine different variation types, same process we can use along with Type 4 and come more closure to actual silicon structure.

Note: In above figure X1, X2, X3 and X4 can be equal or may be different.
Using the variation type 4 (along with other variation type), we are more closure toward the actual shapes after the manufacturing. Please refer the below comparison. (Note: here bottom_delta and top_delta are different along with right and left bottom variation also).

Width Variation Type 5:

Question is what’s now missing. There is one point which we are assuming continuously that variation will be linear in shape from top to bottom but in actual that’s not the case. Like in the below figure.

What’s the solution of that! Very simple – provide the equation or “order of polynomial equation” of variation from top to bottom. 
Fortunately, till now I didn’t come up with any such data provided by Foundry. May be it’s not that important till now or Foundry don’t want to make modeling so complex.(Good for EDA vendor). But this is a proposed solution from me for future. 

Reference: "Including Pattern-Dependent Effects in Electromagnetic Simulations of On-Chip Passive Components" by Sharad Kapur, David Long, Tsun-Lai Hsu, Sean Chen, Chewn-Pu Jou, Sally Liu, Gwan-Sin Chang, Cheng-Hung Yeh, and Hui-Ting Yang (Download)

Metal Width Variation (Type 3)

Metal Width Variation (Part 2)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

Width Variation Type 3:

In this, we are just removing the restriction which we have imposed in the Type 2 . Means in this

“Bottom delta” ≠ “top delta” (top_width_delta is_not_equal_to bottom_width_delta)Concept wise everything else remains (as in the width variation Type 2) same but just provided info will be little bit different.

Table 6: In the form of absolute variation number
Metal	Width (um)	Top_delta/bottom_delta
Metal 1	0.2	+0.01/-0.02
Metal 2	0.4	+0.01/-0.02
Metal 3	0.4	+0.01/-0.02
Metal 4	0.4	+0.01/-0.02

Here I have used different value for Bottom and Top Delta.
Note: Similarly we can have other tables also (similar to table 2 and 4).

Structure wise after applying the variation type 3, we may get below patterns or we can say that we are modeling only below type of variation in the shapes.

Now a simple question…
Can you model this type of variation (Type 3) using tangent values (tan Ɵ) as we have done in the last post (table 5)?
I know, instantly – you can say yes. But remember in case of “tangent”, variation from the vertical line either on top or bottom should be same. Means as shown in figure.

So, now how can you model this type of variation? I am sure something may be clicked in your mind. What about shifting the vertical line little bit outside the actual shape edge. Means, what about applying variation Type 1 first and then variation Type 2.

(Note: Color of original shape changes because of overlapping of other shapes – so please dnt mind or say confused by the color)

We have applied Type 1 variation (discussed in previous post) on the very first shape (Edges AD and BC). New shape is with edges “ad” and “bc”. On this new shape, apply Type 2 variation with some angle (or may be same top and bottom delta width value). This will give you a new shape (trapezoidal) with edges UV and WY.
In the following diagram, you can see that original shape (Ideal one) is in blue color and Final shape (practical one – after variation) is brown in color. Here X1 is not equal to X2 (means top_width_delta is_not_equal_to bottom_width_delta).

This method also shows you the importance of Variation type1 (for those who may be thinking that variation type 1 is not a practical one, so what’s the important –I hope, they should know the importance of that.)

Using the variation type 3, we are more closure toward the actual shapes after the manufacturing. Please refer the below comparison. (Note: here bottom_delta and top_delta are different).

Now, first one are 99% matching (except few curves on the edges) but 2nd figure still have some mismatch. Variation type 4 can help in that.

Metal Width Variation (Type 1 and Type 2)

Metal Width Variation (Part 1)

Chapter 3: Manufacturing Effects and Their Modeling
3.1	3.2a	3.2b	3.3a	3.3b	3.3c	3.4
Introduction	Effect Of Etching Process	Effect Of Etching Process	Chemical Mechanical Planarization	Importance Of CMP process	Dishing & Erosion (CMP effects)	Lithography
3.5a	3.5b	3.5c	3.5d	3.5e	3.5f	3.5g
Metal Width Variation (Type:1-2)	Metal Width Variation (Type3)	Metal Width Variation (Type:4-5)	Metal Width Variation (Type6)	Metal Width Variation (Type7)	Metal Width Variation (Type8)	Metal Width Variation (Summary)

In the previous articles, we have discussed a lot about the Etching, CMP, Lithography and their effects. Now it’s the time to know

1. How these variations are modeled in real?
2. How foundry provide the corresponding data?
3. How to read that one and provide the info to different tools?

Let’s start with the Summary figures of last post.



Note: Blue is what we need ideally and brown is what we will get actually/practically.

As we have discussed in last few post that there are basically 3 parameters which are affected a lot.

• Width of metal
• Thickness of Dielectric and
• Thickness of Metal.

From the above figure and also from statement, you can easily conclude that there are 2 mainly type of variation – In width and In height. Let’s start one by one.

Width Of Metal:

There are different ways to model variation information of this parameter. Different EDA vendors code this info in different way (I will summarize this in the last of this Article series). Similarly, Foundries also provide this info in different way.

Width Variation Type 1:

Most common and simpler form of variation is “variation in percentage” or “absolute numbers” in the form of table.

Table 1: In the form of variation %
Metal	Width (um)	variation in % (+/-)
Metal 1	0.02	8
Metal 2	0.04	8
Metal 3	0.04	9
Metal 4	0.04	10

Structure wise after applying the variation type 1, we may get below patterns or we can say that we are modeling only below type of variation in the shapes.

In this type of variation, we assume that width variation is same from top to bottom OR if there is any difference, their effect in Capacitance and Resistance are negligible. I have just used 2 words, CAPACITANCE and RESISTANCE, so it’s my moral duty to ask this question- “How, above type of width variation impact the CAP and RES of the Circuit?” :)

I would say – think and if you forget then please refer Parasitic Interconnect Corner article. It will help you to refresh your concept And don’t worry I will also summarize this later on.

Width Variation Type 2:

In this we will remove the restriction of Type 1 (same bottom and top width variation). It’s now more closure toward the practical shape. And foundry consider this for 180nm and below nodes.

Ideal Width of Metal = W (Rectangle shape)
Because of several fabrication steps (already discussed in last few Articles of this series), final shape of the Metal is not rectangular. It’s trapezoidal, so we have to define 2 widths.

Top_width = W+2A
Bottom_Width = W-2A

Note:

• Here we are considering that “bottom delta” = “top delta”.
• In case, top_width_delta=bottom_width_delta, we can model this by using the angle Ɵ also. Where tan Ɵ = 2B/2A and known as Tangent.

So, in all the above case the table (or say info provided by Foundry) can be any of the following.

Table 2: In the form of absolute Numbers (final width)
Metal	Width (um)	Top_width/bottom_width
Metal 1	0.2	0.21/0.19
Metal 2	0.4	0.41/0.39
Metal 3	0.4	0.41/0.39
Metal 4	0.4	0.41/0.39

Table 3: In the form of absolute variation number
Metal	Width (um)	Top_delta/bottom_delta
Metal 1	0.2	+0.01/-0.01
Metal 2	0.4	+0.01/-0.01
Metal 3	0.4	+0.01/-0.01
Metal 4	0.4	+0.01/-0.01

Table 4: In the form of % delta variation number
Metal	Width (um)	%Top_delta/%bottom_delta
Metal 1	0.2	+5%/-5%
Metal 2	0.4	+10.0%/-10.0%
Metal 3	0.4	+10.0%/-10.0%
Metal 4	0.4	+10.0%/-10.0%

Table 5: In the form of Tangent (angle)
Metal	Width (um)	Thickness (A)	tan Ɵ
Metal 1	0.2	1000	5
Metal 2	0.4	1200	6
Metal 3	0.4	1200	6
Metal 4	0.4	1200	6

Structure wise after applying the variation type 2, we may get below patterns or we can say that we are modeling only below type of variation in the shapes.

I am sure you are able to co-relate these with the real structure or shapes (Snapshot of last few article summary). But if you are still confused, please refer below figure.

After seeing above figure, you may be thinking that it’s not 100% matching. For that variation Type 3 can help you.