State Space Model and Numerical Simulation of

2017 2nd International Conference on Image, Vision and Computing

State Space Model and Numerical Simulation of Overlay Error for Multilayer Overlay Lithography Processes He Fuyun, Zhang Zhisheng

*

School of Mechanical Engineering Southeast University Nanjing, China e-mail: [email protected]

Ab stract-Considering

multiple

error

sources

that

between process parameters and product and quality characteristics by vector tolerance[5-7].

affect

overlay deviation in multilayer overlay lithography processes, according to the error source is applied to exposure field or

II.

wafer, the deviation sources can be divided into two aspects: intrafield error and interfiled error. And based on the concept

In order to form the circuit pattern on the wafer, it is necessary to correctly align the pattern on the wafer with the pattern on the mask. The accuracy of matching between each successive pattern and adjacent-layer pattern is known as the alignment deviation, which is generally presented as the vector difference between the geometric position vector of substrate and the geometric position vector of the corresponding point in the next circuit pattern [8, 9].

of virtual location point, these error forms can be expressed uniformly. Then, the state space model of overlay deviation for multilayer overlay lithography processes are established based on stream of variation. The propagation characteristics of multilayer overlay deviation is analyzed by applying state space model, and compared with the result from Monte Carlo simulation

to

completely

parameter

the

physical

overlay

set, the

deviation

relative

model

with

error between two

methods is less than 2%, so the feasibility of the proposed

A. Problem Description

method is verified.

Keywords-multilayer overlay error; lithography process; state space; stream of variation; semiconductor manufacturing

I.

INTRODUCTION

Multilayer overlay lithography process is one of the most important steps in semiconductor manufacturing. Integrated circuit chip manufacturing needs dozens or more independent lithography steps. If circuit patterns of each layer of circuit can't well align with the above layer of patterns in a circuit, it will result in the failure of the whole circuit [I, 2]. Therefore, overlay alignment on each layer of circuit patterns is an important index that is difficult to guarantee. The problem will be more obvious for multilayer lithography process due to the accumulation and propagation of overlay error. Assembly modeling and error control in multi-step manufacturing process are emerging areas of research on the basis of engineering and statistical studies, which made rapid development in the 1990s [3-5]. Jin and Shi et al presented a method of quality control based on state-space model, by which, variation propagation in the multi-step manufacturing process was expressed [3]. Ceglarek et al presented an analytical model for stream of variation based on key product characteristics to represent all restrictions by unified generalized virtual fixture model. The model described error relations between key product characteristics and key control characteristics and laid a solid foundation for the multi-step sophisticated assembly modeling [5]. Jin, Cegkarek, Huang et al presented modeling multi-step manufacturing process by model of state-space equation and described errors

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PROBLEM DESCRIPTION AND ASSUMPTION

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Many factors, such as lens distortion, alignment error and wafer deformation in semiconductor lithography process lead to alignment deviation [10]. Deviation is usually nanometer in order of magnitude, and is strongly affected by the frequently changing process conditions. Thus, alignment deviation usually exists in the patterns that form on wafer. As integrated circuit manufacturing needs more layers and the increase of wafer size, how to effectively identify the causes of alignment deviation and control them has been one of the most difficult problems in the process of semiconductor manufacturing [1I]. Overlay error is a key factor for restricting functionality and reliability of integrated circuit. At present, studies concerned are primarily aimed at inferring accurate superposition models for overlay errors resulting from all kinds of process-level error sources concerned at any given layer in a chip. The models are applied to automatic control of overlay errors at each lithography layer by modulation of controllable process parameters [12]. The lithography process is a typical multi-step manufacturing process, in which, overlay errors are introduced in each step and variation in the previous sediment reference layer influences overlay errors in the subsequent sediment layer. Figure I shows this situation, in which, overlay errors in neighboring layers are the same in case 1 and 2 but superposed overlay errors between non-neighboring layers are clearly better in case I than case 2. The control of multi-layer superposed overlay errors, as shown in Figure 1, are in urgent demand for a multi-layer superposition model that describes the generation of one-layer overlay errors and characteristics of multi-layer propagation.

B. Model Parameters

Case2:

Case1:

The most basic physical model of overlay errors can be divided into two parts based on the different error sources [14]. One part of the model is focused on the objective error caused by the matching of light source glass filter and mask in the same graphic field. Another part of the model is related to the grid errors caused by the change of relative position between mask and wafer. The schematic diagram of alignment error source is shown in Figure 3 [15] . Ol1hogonality ... .--_....

Layer 0 Figure I. The schematic diagram of multilayer overlay error

Y translation

B. Basic Assumption

X translation

The analysis of alignment deviation based on the state-space model should meet the following assumptions: 1) The offset of exposure field and wafer can be described by the offset of any reference point; 2) The offset of exposure field and wafer is the deviation from its nominal position; 3) Only the objective lens error and grid error are considered; 4) Small error assumption, objective lens error and grid error are 2-4 orders of magnitude smaller than the exposure field and wafer. III.

STATE SPACE MODEL FOR OVERLAY ERROR

A. Basic Concept of State Space Model The alignment process of N layers of lithography is shown in Figure 2:

X Scale

"l

Reticle rotation

Magnifiurion

D D

YScale

V Wafer rotation

Figure 3. Illustration of overlay error sources

Primarily related to multiplying power errors, distortions, masking rotations, and inclinations and so on, objective errors generally refer to position errors of all points in a field. Their existence may result in patterns transformation of all exposed fields and as exposed circumstances change, errors may differ in magnitude. Objective errors are represented as follows in the physical model under Cartesian coordinate system [16],

5e� = Tfi +(M, +MJxr- (R, +R,,)Yr +o ( x�,Y� ) (3) 5ej=Tfj. +(M, - Ma)Yr +(R, - R,,)xj +o ( x�'Y� ) (4) where, I) Coordinate ( xr 'Yr ) represents the position of

�

-n- . .

sampling point in the exposed field coordinate system; 2) T/y and Tfy represents shifting errors of graph of exposed field in Direction x and y respectively; 3) M, represents magnification error of mask and M a represents asymmetrical magnification error of mask; 4) R, represents rotation error of mask and R" represents asymmetrical

til

Figure 2. The schematic diagram of multilayer lithography processes

General linear state-space model [13] is applied to describe the multi-layer lithography process shown in Figure 2.

rotation error of mask;

x(k)= A(k-1 )x(k-1 )+B(k ) u(k)+w(k ),k= 1,2,.··,N (I) (2) Y(k)=C(k)x(k)+v(k),k = 1,2,.··,N where, x( k) represents one state vector of process; u(k) re-presents the control vector of process; A(k) represents the state-transition matrix of process; B( k) represents the input matrix of process that determines the effect of control vector u(k) on state vector x(k) ; Y(k) represents the key quality characteristic value; represents the measurement matrix;

DO

w( k)

and

represents second or

high order error. The grid error is mainly related to the moving parts of lithography machine, such as the positioning accuracy of work piece table, orthogonally of work piece table movement, which mainly refers to the position deviation of central point of each image field. It may lead to the rotation or offset of all exposure fields on wafer. Similarly, the physical model of grid error in Cartesian coordinate system is presented as [16]:

C( k) v( k)

represent the noise and measurement noise of process, respectively.

5) o ( x�'Y� )

where,

5e:= Tw+ x S,x,{ R;t ) Rn Y{ l 4 oe;; = T". y+SyYj- R gXt.( 20) I) Coordinate (xw'y w) represents

(5) (6) the

central

position of exposure field in the wafer coordinate system; 2)

1124

T"x and T"v represent the shifting deviation of wafer in x

and y directions; 3)

Sx

and

Syrepresent

deviation in x and y directions, respectively; Rg represents the vertical rotation error of grid, and represents the non-vertical rotation error of grid; 5) and

0

(y�,)

o

(19) f J' They represent coordinate information for objective and grid errors in each layer. Further, system input u (k) and

4)

Rn ( x);',

y(k) are defmed as follows, u(k)= [c5e;:,(k) c5e;(k) c5e;(k) c5e;(k)J

measurement output

represent the second or higher order errors.

The overlay error of each layer will result in the distortion of image field in the corresponding layer, and the error is passed to the next layer along the multilayer cascade operation. Therefore, the final quality deviation is composed by the objective errors and grid errors of all the middle layers. In this paper, based on the concept of Virtual Fixture, it was thought that two Virtual anchor points were imposed on the each exposure field, so any bias error of the field exposure could be equivalent to according deviation of Virtual anchor point and the unified expression of the error sources was realized. Expressions (3-6), though nonlinear models, are accessible to linear processing in a way similar to Taylor's series. Therefore, models are approximated to linear models in two parts given that their second-order or high-order form is neglected. Part I and II are objective error model and grid error model, respectively. The objective error model is represented as

c5e;= Try +(Mr +Ma)xr -(R" +Ra)Yj oej = Tty +(M,. - M,,)Yr +(Rr - R,,)xf

(7) (8)

Y(k)=

(9)

[�::�:�] = [�:��:�] +[�� �:�]

where, the input variable

(21)

u (k) represents the objective error

each wafer. Assuming that

u(k) in each layer is independent,

and the system noise is included in the random impact of u (k), the propagation model of overlay error in the N layer can be simplified as follows:

x(k)= A(k-1)x(k-1)+B(k) u(k),k = 1,2 ,.. ·,N (22) (23) y(k)=C(k)x(k),k=1,2 ,... ,N Furthermore, y(k) can be presented as, (k) (k) (k) Y(k) = = 5e,:(k) + xw(k) (24) sv M + +Ma a+ (k) x -(R R R R g' ( r r a) f ) C(k)x(k) J S,, Yw(k) = M M lRR R, R Yf(k)

] [de,5e;(k) ',; ]

[5eX, ] [5e:,' 5eY(k) a

"

]

-

a

Because data normalization processing, translation form was ignored here. Further, status updates expression is as follows,

(10)

The offset point in the corresponding coordinate system can be presented as,

x;, (k-1) oe;:(k) x�(k 1) c5eX(k) x ()k = , (k-1) + f (k) = x(k-l) +u(k) c5e,:. Yw y�(k-1) c5ej(k)

(11) (12)

(25)

Therefore, in the above multi-layer overlay error state-space model, there are be,

(13)

A

]

a+ R,,) (R + R,) (k)_- B (k)_- [M'C (k)_- Sx, M,, M -(RS,' M M R R R ,

[

,

+

-

(14)

IV.

a

a

-

a

,-

.

a

NUMERICAL SIMULATION

A three-layer lithography process is stimulated to validate the model. In the three-layer overlay simulation process, it is assumed that the error source comes from the objective error and grid error. At the same time, in order to simplity the calculation, it is assumed that the coordinate axes of the exposure field coordinate system are parallel to the coordinate axes of wafer coordinate system. The physical model parameters used in the simulation come from the literature [14], as shown in Table I. Therefore, the state-transition matrix A(k) , control matrix B(k) and

Then, Expressions (7-10) can be presented as

c5e� = (M" +Ma)x �-(R" +Ra)y'r c5e;= (M" - Ma)Y �+(Rr - Ra)x� c5e;:,= SyX;,-( Rg +R,,) Y;, c5e;;' = S,Y;, +Rg x;,

(20)

and grid error corresponding in each layer, and the output variable y(k) represents the total measurement deviation of

-

The grid error model is represented as

w g +Rn) Yw r5e�:" = Twx+Sxx-(R oe;= T"y +S,Yw+Rg xW

}

X(k) = [l

the grid scale

(15) (16) (17) (18)

In each layer of lithography process, the state variables is defined as,

x (k)

output matrix

11 52

C(k)

of the model are

A(k)= B(k)=144, C(k)= "

[

1

-

I

82 4 -1 -1 I I 1 .53 .

].

linearized intact physical models. Results show that relative errors drawn from state-space model and Monte-Carlo simulation are less than 2%.

THE PHYSICAL MODEL PARAMETERS OF SIMULATION MULTILAYER LITHOGRAPHY PROCESSES

TABLE I.

Twx

Twy

Grid

27. 39

34.26

error

Sy

Rn

Tfx

Try

M,

0

0

-40.345

Ma

R,

Ra

V. CONCLUSION

S, Rg

0

Objective error

-41.965

0

The virtual anchor point of each layer and the measurement points of the third layer are given in Table II and Table III, respectively. TABLE II.

COORDINATE SYSTEM, UNIT (MM»

TABLE 111.

Co ordinate

Ml1

(5. 4,5.4)

M"

(-5.4,5.4)

M1J

(-5.4,-5.4)

MJ4

(5.4,-5.4)

ACKNOWLEDGMENT

This work was funded by the National Natural Science Foundation of China (No.51275090), and the Fundamental Research Funds for the Central Universities and Jiangsu Postgraduate Innovation Program (No.KYLX15-0208).

THE COORDINATES OF MEASUREMENT POINT (FIELD

Anchor point

In view of the alignment deviation of multilayer lithography process, based on the linear processing for single-layer alignment deviation physical model, the state-space model of the multi-layer lithography alignment process is established, and the different forms of errors are described as the offset of positioning point according to virtual positioning point. Compared to the physical model with complete parameters, the advantage of linear state space model lies in increasing the calculation efficiency within the appropriate error tolerance, which is very advantageous in realizing the online monitoring and adjustment of process parameters in large and complicated industrial processes.

REFERENCES [1]

Narita, Hiroaki. "Semiconductor device manufacturing method and semiconductor device." Nursing & Information Journal of the Japan Nursing Library Association 45.7(2016):1347-1350.

[2]

Nara, Kei, and T. Hamada. "Method for manufacturing display element, manufactur ing apparatus of display element and display device." (2016).

[3]

Jin, Jionghua, and 1. Shi. "State Space Modeling of Sheet Metal Assembly for Dimensional Control." Journal of Manufacturing Science & Engineering 121.4(1999):756-762.

[4]

Cai, Na, L. Qiao, and N. Anwer. "Unified variation modeling of sheet metal assembly considering rigid and compliant variations." Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture 229. 3(2014):495-507.

[5]

.lin, Jionghua, and J. Shi. "State Space Modeling of Sheet Metal Assembly for Dimensional Control." Journal of Manufacturing Science & Engineering 12l.4(1999):756-762.

[6]

Shi, Jianjun. "Stream of Variation Modeling and Analysis for Multistage Manufacturing Processes." Crc Press (2006).

[7]

Ding, Yu, D. Ceglarek, and 1. Shi. "Design evaluation of multi-station assembly processes by using state space approach." ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference American Society of Mechanical Engineers, 2002:408-418.

[8]

Jiao, Yibo, and D. Djurdjanovic. "Stochastic Control of Multilayer Overlay in Lithography Processes." IEEE Transactions on Semiconductor Manufacturing 24. 3(2011):404-417.

[9]

He, Fuyun, and Z. Zhang. "An empirical study-based state space model for multilayer overlay errors in the step-scan lithography process." Rsc Advances 5. 126(2015): I 03901-103906.

THE COORDINATES OF ANCHOR POINT IN EACH LAYER (FIELD COORDINATE SYSTEM, UNIT (MM»

Anchor point

Coordinate

Lll

(-8.6,0)

Anchor point L22

Coordinate

(-3.8,6.5)

L12

(8.6,0)

L3)

(8.6,8. 6)

L"

(3. 8,6.5)

L"

(-6.5,6. 5)

THE ERROR OF MEASUREMENT POINT OF STATE SPACE MODEL AND MONTE CARLO SIMULATION, UNIT (NM)

TABLE IY.

Anchor point

State space

Monte carlo

model

simulation

Relative error (%)

L'.x

t,y

L'.x

t,y

M3)

-50.04

2.52

-49.23

2. 45

l.65

l.86

M12

48.84

l.32

49.75

l.36

l.83

l.94

M33

50.04

-0.52

50.27

-0.53

0.46

M14

-48.84

0.68

-47.93

0.69

1.90

-1. 8 1. 45

[10] Schmidt, Dennis, and G. Charache. "Wafer process-induced distortion study for x ray technology. "Journal of Vacuum Science & Technology B Microelectronics & Nanometer Structures 9.6(1991):3237-3240. -

Table IV shows results drawn from Monte-Carlo simulations based on state-space model and physical model for overlay errors, in which, L'.x and ll.y refer to overlay errors in the Axis x and y respectively. Monte-Carlo simulation makes 10000 computations based on not

[11] Jiao, Yibo, and D. Djurdjanovic. "Compensability of errors in product quality in multistage manufacturing processes." Journal of Manufacturing Systems 30. 4(2011):204-213.

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[12] ChenFu Chien, KuoHao Chang, and ChihPing Chen. "MODELING OVERLAY ERRORS AND SAMPLING STRATEGIES TO IMPROVE YIELD." Journal of the Chinese Institute of Industrial Engineers 18. 3(2001):95-103. [13] Bazdar, Aliasghar, R. B. Kazemzadeh, and S. T. A Niaki. "Variation source identification of multistage manufacturing processes through discriminant analysis and stream of variation methodology: A case study in automotive industry." Journal of Engineering Research 3.2(2015).

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[14] Brink, M. A Van Den, C. G. D. Mol, and R. A George. "Matching Performance for Multiple Wafer Steppers Using an Advanced Metrology Procedure." Proc Spie 921(1988):180-197. [15] Bode, C. A, B. S. KO, and T. F. Edgar. "Run-to-run control and performance monitoring of overlay in semiconductor manufacturing." Control Engineering Practice 12.7(2004):893-900. [16] Lin, Zone Ching, and W. J. Wu. "Multiple linear regression analysis of the overlay accuracy model." IEEE Transactions on Semiconductor Manufacturing 12.2(1999):229-237.