Method for improving accuracy of imprint force application in imprint lithography
A method for identifying a model for modeling cable stress relaxation dynamics of an imprint head is provided. The method includes performing a non-contact imprint test run for a predetermined number of wafers. Data from a force trace and a position trace of the imprint head during the non-contact imprint test run are collected to compute cable stress relaxation forces. A model having a plurality of different model orders is generated. A set of parameters of the model for each model order is identified. Residual sum of squares for wafer average error for each of the model orders is calculated based on the obtained cable stress relaxation forces. The set of parameters of one of the plurality of model orders may be based on a difference in the residual sum of squares for wafer average error between the one of the plurality of model orders and a next higher model.
1 . A method for identifying a model for modeling cable stress relaxation dynamics of an imprint head, comprising:
performing a non-contact imprint test run for a predetermined number of wafers;
collecting data from a force trace and a position trace of the imprint head during the non-contact imprint test run;
computing cable stress relaxation forces based on the collected data;
generating a model having a plurality of different model orders;
identifying a plurality of sets of parameters of the model for the plurality of different model orders, wherein each of the different model orders has a respective set of parameters of the plurality of sets of parameters;
computing residual sum of squares for wafer average error for each of the different model orders based on the computed cable stress relaxation forces; and
selecting the respective set of parameters of one of the plurality of different model orders based on a difference in the residual sum of squares for the wafer average error between the one of the plurality of different model orders and a next higher model order,
wherein the non-contact imprint test run includes a first motion sequence of the imprint head which is idling at an up-plane with respect to an imprint plane before starting the imprint test run,
wherein the non-contact imprint test run includes a second motion sequence of the imprint head before starting imprint on a first wafer of the predetermined number of wafers, the second motion sequence includes a calibration measurement routine and a template mapping routine during which the imprint head moves to different planes,
wherein the model is generated with a state-space representation as:
{dot over (x)} ( t )= Ax ( t )+ Bu ( t )
f t =Cx ( t )+ Du ( t )
with the sets of parameters A, B, C, and D as:
A
=
[
0
1
0
…
0
⋮
⋮
⋱
⋱
0
-
a
n
-
a
n
-
1
…
…
-
a
1
]
;
B
=
[
0
⋮
1
]
;
C
=
[
b
n
…
…
b
0
]
;
D
=
k
0
where x(t) are state variables at time t, u(t) are positions of the imprint head at time t, k 0 , a n , a n−1 , . . . a 1 and b n , b n−1 , . . . , b 1 are scalar parameters to be determined by fitting; f t is force disturbance due to cable stress relaxation, A is a n×n matrix, B is a n×1 matrix, C is a 1×n matrix, and n is an integer, and
wherein the wafer average error is calculated as:
e
i
,
Cureend
=
1
p
∑
j
=
1
p
(
f
i
,
j
,
Cureend
-
f
model
,
i
,
j
,
Cureend
)
,
where e i.Cureend is the wafer average error, f i.j.cureend is an actual force disturbance due to the cable stress relaxation at an end of a curing process, f model.i.j.cureend is a model force disturbance due to the cable stress relaxation at the end of the curing process, j is a particular imprint field, and p is a number of imprint fields.
2 . The method according to claim 1 , further comprising performing a third motion sequence on the first wafer of the predetermined number of wafers, the third motion sequence including an imprint motion sequence and a metrology sequence.
3 . The method according to claim 2 , further comprising performing a fourth motion sequence of the imprint head after the third motion sequence before starting imprint on a second wafer of the predetermined number of wafers, the imprint head idling at the up-plane and then moves along a calibration measurement routine during the fourth motion sequence.
4 . The method according to claim 3 , wherein times spent during each of the first motion sequence to fourth motion sequence are tunable.
5 . The method according to claim 1 , wherein the residual sum of square for the wafer average error is calculated as:
RSS
=
∑
i
=
1
q
e
i
,
Cureend
2
wherein RSS is the residual sum of squares, i is a particular wafer, q is a number of wafers.
6 . A method of manufacturing an article, comprising:
applying a formable material on a substrate;
moving a template held in an imprint head to contact with the formable material to perform imprint on the formable material, wherein final imprint force applied to imprint the formable material is calibrated by a cable stress relaxation force which is estimated by identifying a first model for modeling cable stress relaxation dynamics of an imprint head, comprising:
performing a non-contact imprint test run for a predetermined number of wafers;
collecting data from a force trace and a position trace of the imprint head during the non-contact imprint test run;
computing cable stress relaxation forces based on the collected data;
generating a second model having a plurality of different model orders;
identifying a plurality of sets of parameters of the second model for the plurality of different model orders, wherein each of the model orders has a respective set of parameters of the plurality of sets of parameters;
computing residual sum of squares for wafer average error for each of the different model orders based on the computed cable stress relaxation forces; and
selecting the respective set of parameters of one of the plurality of different model orders based on a difference in the residual sum of squares for the wafer average error between the one of the plurality of different model orders and a next higher model order,
wherein the non-contact imprint test run includes a first motion sequence of the imprint head which is idling at an up-plane with respect to an imprint plane before starting the imprint test run,
wherein the non-contact imprint test run includes a second motion sequence of the imprint head before starting imprint on a first wafer of the predetermined number of wafers, the second motion sequence includes a calibration measurement routine and a template mapping routine during which the imprint head moves to different planes,
wherein the second model is generated with a state-space representation as:
{dot over (x)} ( t )= Ax ( t )+ Bu ( t )
f t =Cx ( t )+ Du ( t )
with the sets of parameters A, B, C, and D as:
A
=
[
0
1
0
…
0
⋮
⋮
⋱
⋱
0
-
a
n
-
a
n
-
1
…
…
-
a
1
]
;
B
=
[
0
⋮
1
]
;
C
=
[
b
n
…
…
b
0
]
;
D
=
k
0
where x(t) are state variables at time t, u(t) are positions of the imprint head at time t, k 0 , a n , a n−1 , . . . a 1 and b n , b n−1 , . . . , b 1 are scalar parameters to be determined by fitting; f t is force disturbance due to cable stress relaxation, A is a n×n matrix, B is a n×1 matrix, C is a 1×n matrix, and n is an integer, and
wherein the wafer average error is calculated as:
e
i
,
Cureend
=
1
p
∑
j
=
1
p
(
f
i
,
j
,
Cureend
-
f
model
,
i
,
j
,
Cureend
)
,
where e i,Cureend is the wafer average error, f i.j.cureend is an actual force disturbance due to the cable stress relaxation at an end of a curing process, f model.i.j.cureend is a model force disturbance due to the cable stress relaxation at the end of the curing process, j is a particular imprint field, and p is a number of imprint fields.