Requirement for accurate interconnect temperature measurement for electromigration test

Abstract

In this work, we show accurate temperature measurement is important in electromigration (EM) test as it can affect the measured activation energy EA and current density exponent n. The deviation of measured EA and n due to the interconnect temperature variations are derived analytically and illustrated in Cu/low-k interconnects using finite element analysis (FEA) under typical experimental conditions. The derived formulations are verified by our previous experimental works.

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This document is downloaded from DR-NTU, Nanyang TechnologicalThis document is downloaded from DR-NTU, Nanyang Technological
University Library, Singapore.University Library, Singapore.
Title
Requirement for accurate interconnect temperature
measurement for electromigration test.
Author(s)Hou, Yuejin.; Tan, Cher Ming.
Citation
Hou, Y., & Tan, C. M. (2009). Requirement for accurate
interconnect temperature measurement for
electromigration test. In proceedings of the 12th
International Symposium on Integrated Circuits:
Singapore, (pp.522-525).
Date2009
URLhttp://hdl.handle.net/10220/6394
Rights
© 2009 IEEE. Personal use of this material is permitted.
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ISIC 2009
Requirement for Accurate Interconnect Temperature
Measurement for Electromigration Test

Yuejin Hou, Student Member, IEEE and Cher Ming Tan, Senior Member, IEEE
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639 798
Email: ecmtan@ntu.edu.sg


Abstract—In this work, we show accurate temperature
measurement is important in electromigration (EM) test as it can
affect the measured activation energy EA and current density
exponent n. The deviation of measured EA and n due to the
interconnect temperature variations are derived analytically and
illustrated in Cu/low-k interconnects using finite element analysis
(FEA) under typical experimental conditions. The derived
formulations are verified by our previous experimental works.
Index Terms—Electromigration, Activation energy, Current
density exponent
I.
INTRODUCTION
scaling With continuous of device dimensions,
electromigration (EM) remains to be a severe reliability
concern for integrated circuit (IC) interconnects [1]. The
influence of the current density j and metal line temperature T
on the median time to failure (MTF) t50 is described by the
Black’s equation [2],
/
50
AB
EK Tn
tA j
⋅
e
−
=⋅
(1)
where EA is the activation energy, n is the current density
exponent, A is a parameter sensitive to the fabrication process
and the geometry of the interconnect, and
meaning. It is critical to measure the metal line temperature
accurately as it can affect the experimentally determined EA
and n as shown in (1).
B
K T has the usual
The interconnect temperature increment due to Joule
heating is usually estimated by utilizing the temperature
coefficient of resistivity (TCR) of the metallization in EM test
[3]. In practice, the actual interconnect temperature can deviate
from the estimated interconnect temperature due to the oven
temperature variations, individual interconnect resistance
variations, the accuracy of TCR, etc. With more severe Joule
heating expected in future Cu/low-k interconnects, the sample
resistance variations are becoming more important since Joule
heating is dependent on the sample resistance, resulting in
higher temperature variations in interconnects.
In this work, we show the importance of accurate
interconnect temperature measurement in EM test as the
variations of the interconnect temperatures can lead to the
variations of experimentally determined EA and n although the
actual EA and n does not change. The Joule heating effect and
the associated temperature variation are estimated using finite
element analysis (FEA) under typical experimental conditions
for Cu/low-k interconnects. The variations of EA and n due to
interconnect temperature variations are derived analytically so
that they can be re-estimated once the temperature variations
are known. The variations of EA and n are found to be
dependent on the temperature variation, EM test temperature,
and actual activation energy EA.
II.
TEMPERATURE VARIATIONS IN
INTERCONNECTS
Interconnect temperature in oven EM test can be expressed
as
oven
T
joule
TT
=+ Δ
(2)
where
temperature increment due to Joule heating. For typical high
temperature oven used in EM test,
oven
T
is the oven temperature and
joule
T
Δ
is the
oven
T
setoven
TT
=+Δ
(3)
where
temperature variation of the oven. Typical oven shows 1%
uniformity in the spatial temperature distribution as in the
specification for Blue
0.01
ovenset
TT
Δ= ±⋅
.
set
T is the oven temperature setting and
oven
T
Δ
is the
MTM oven. Therefore,
Temperature rise due to Joule heating is formulated as [3]
0
joule
R
⋅
T
TCR R
Δ
Δ=
(4)
In (4), R
sample
resistance
also referenced at 0 °C. It is noted that R
stress current, non-Joule heated sample resistance and TCR.
Depending on the manufacturing process, the initial sample
resistance can have a variations up to 10% [4], leading to
different
joule
T
Δ
among different samples. Besides that, TCR is
also different for different samples as it is dependent on the
microstructure of the sample [5]. For Cu interconnect, constant
TCR cannot be used because Cu resistivity is no longer linearly
dependent on temperature above 200 °C [6], and EM test for
Cu interconnect is usually performed around 300 °C. All the
above-mentioned facts result in the fluctuations of
especially in the presence of severe Joule heating.
Δ is the resistance difference between the Joule heated
resistance
J
R
and non-Joule
R,
0
heated sample
N
R is the sample resistance at 0 °C and TCR is
Δ
is dependent on
joule
T
Δ
,
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To simplify the analysis, we only consider the fluctuation of
resistance among the samples and temperature variation in the
oven in this work. In fact, this is an optimistic assumption
because the TCR variations shall further increase the
interconnect temperature variations. We denote the variation of
interconnect temperature T in (2) as ε . From (2) and (3),
T
TRT
R
δ
joule
ovenN ovenjoule
N
δ
εε
Δ
= Δ+×Δ = Δ+
(5)
joule
ε
under Joule heating effect which is due to the variation of
sample resistance and it is estimated by FEA as shown below.
in (5) is the variation of the temperature increment (
joule
T
Δ
)
Joule heating effect is estimated by the coupled field
electrical-thermal analysis using finite element software
ANSYSTM for a typical Cu/low-k interconnect. The finite
element model is shown in Fig. 1 with Ta diffusion barrier and
SiN cap layer included. The dimensions of line width and line
thickness are 0.28 and 0.35 μm respectively. Via height is 0.7
μm. Low-k material carbon-doped oxide (CDO) is chosen as
the dielectrics in this work. The material properties are taken
from [7]. In the analysis, the substrate bottom surface is kept at
constant EM test temperature with a pre-determined current
density applied to the interconnect. The simulation results are
summarized in Table I.
Table I.
TEMPERATURES AND CURRENT DENSITIES FOR CU/CDO
INTERCONNECTS
Current
density
(MA/cm2)
100
0.01 100
0.5 100.55
1 102.23
2 109.08
4 139.61
6 191.07

JOULE HEATING EFFECT AT DIFFERENT TEST
Temperature (° °C)
200 300 400
200
200.72
202.90
211.81
251.51
318.43
300
300.88
303.57
314.54
363.42
445.79
400
400.97
403.91
415.94
469.48
559.75

Fig 1: Finite element model for Joule heating effect simulation.

From Table I, at a test temperature of 300 °C and current
density of 4 MA/cm2, the temperature increment is 63.42 °C,
close to the 65 °C as reported by Wu et al. [8] for the same
stress conditions for low-k interconnect. For the same test
temperatures, the temperature increment is proportional to the
square of current density, which shows the temperature
increment is indeed due to the Joule heating effect. It is also
noted that under the same current density, the temperature
increment is higher for higher test temperature.
For ε in (5),
The variation of temperature increment under Joule heating
ε
is estimated by FEA. For illustration, we choose a typical
stress current of 4 MA/cm2 and a typical 10% variation in
interconnect resistance [4]. The change of interconnect
resistance is computed by changing Cu resistivity values in the
material library in the FEA simulations.
combination of different oven temperatures and resistivity is
simulated and the results are summarized in Table II. It is noted
that
joule
ε
is dependent on the resistivity variation and the Joule
heating effect. Copper resistivity variation due to non-uniform
temperature is taken into account using TCR in the simulation.
The temperature variation will be much lower if a smaller
current density is used in the EM test.
oven
T
Δ
is dependent on the oven temperature.
joule
joule
ε
in (5) under a
As shown in Table II, lower Cu resistivity leads to a
negative temperature variation since the generated Joule heat is
linearly dependent on the resistivity. Table II, together
with
oven
T
Δ
, will be used for the numerical calculations in the
following sections.
III.
ACTIVATION ENERGY VARIATIONS
Activation energy EA of EM is often estimated as the slope
of t50 versus 1/kBT in the lifetime plot. If the line fitting is based
on the inaccurate interconnect temperature, the estimated EA
can be different from the actual EA. It remains unknown how
the variation of interconnect temperature affects the estimated
EA.
The derived correction function is shown in (6),
ε
⎛⎞
=+
⎜⎟
⎝
'
/ 1
AA
EE
T
⎠ (6)
In (6),
on test conditions, and
activation energy. For Cu/low-k interconnect with an
eV for interfacial diffusion [9], with the estimated ε and T in
A
E is the actual activation energy which is independent
E is the experimentally determine
'
A
A
E of 0.9
Table II, the experimentally determined
oven temperature and Cu resistivity is shown in Fig. 2.
'
A
E under different
For interconnect resistivity at 0.9ρ0, the temperature
variation is negative as shown in Table II. In other words, the
estimated interconnect temperature is lower than the actual
interconnect temperature and this results in slightly higher
estimated activation energy than the actual EA of 0.9 eV for
interfacial diffusion. With higher interconnect resistivity, the
estimated EA decreases. The variation of estimated EA on oven
temperature is small due to the low variation of oven
temperature in EM test. For the same interconnect resistivity,
higher oven temperature is associated with higher EA variation
as shown in Fig. 2, but its effect is negligible.





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Table II: THE TEMPERATURE INCREMENT UNDER JOULE HEATING EFFECT AND ITS VARIATIONS AT DIFFERENT OVEN TEMPERATURE
AND INTERCONNECT RESISTIVITY. THE SIMULATION IS BASED ON CU/CDO INTERCONNECT WITH A CURRENT DENSITY OF 4 MA/CM2.
Oven temperature
(° °C)
joule
(° °C)
T
Δ
in (1)
joule
ε
at
0.9ρ ρ0 (° °C)
-6.43
joule
ε
at
0.95ρ ρ0 (° °C)
-3.16
joule
ε
at
ρ ρ0 (° °C)
0.00
joule
ε
at
1.05ρ ρ0 (° °C)
3.48
joule
ε
at
1.1ρ ρ0 (° °C)
7.08 150.00 45.57
200.00 51.52 -7.19 -3.53 0.00 3.90 7.92
250.00 57.47 -7.95 -3.91 0.00 4.31 8.75
300.00 63.42 -8.71 -4.28 0.00 4.72 9.59
350.00 69.55 -9.47 -4.65 0.00 5.13 10.43
400.00 75.67 -10.23 -5.03 0.00 5.54 11.26

0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10
0.880
0.885
0.890
0.895
0.900
0.905
0.910
0.915
0.920


Activation energy (eV)
ρ/ρ0
Oven at 150
200
250
300
350
400
oC
oC
oC
oC
oC
oC

Fig. 2: The variation of Ea due to the variation of sample resistance and oven
temperature. Actual Ea is assumed to be at 0.9 eV.

The variation of EA in Fig. 2 shall lead to large variations of
t50 if these EA is used in the Black’s equation for lifetime
extrapolation. The variation of t50 due to the variation of EA is
shown in Fig. 3. The variation of t50 is much higher than that of
EA since t50 is exponentially dependent on EA. For the same
interconnect resistivity, higher oven temperature is associated
with lower t50 variation as shown in Fig. 3.
Equation (6) is validated by our previous experimental
work [4]. The reported variation of EA (delta EA) is within
0.01~0.02 eV with ΔT=13.83 °C, T=275 °C, EA is within
0.85~0.88 eV. Using (6), the variation EA is estimated to be
~0.02 eV, close to the experimentally reported value.
0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10
0.6
0.8
1.0
1.2
1.4
1.6


Oven at 150
200
250
300
350
400
oC
oC
oC
oC
oC
oC
t50/t50(ideal)
ρ/ρ0

Fig. 3: The variation of EM lifetime due to the variation in EA.



IV. CURRENT DENSITY EXPONENT VARIATIONS
Current density exponent n is an indication of EM failure
process. It is commonly accepted that n=2 is for void
nucleation limited failure and n=1 is for void growth limited
failure [10]. However, n can be larger than 2 under Joule
heating effect [11].
Consider EM test at two different current densities, j1 and j2
(j1<j2) with test temperature T. The temperature increment due
to Joule heating is denoted as ΔT1 and ΔT2.
/
1
50
'
1
AB
EK T
n
tA j
⋅
e
−
=⋅
(7)
'
/
2
50 2
AB
EK T
n
tA j
⋅
e
−
=⋅

(8)
where
exponent excluding Joule heating effect. Combine (7) and (8),
(
50501
lnlnlnlnttnj
−= − ⋅−
'n is the experimentally determined current density
)
12'
2
j

(9)
By considering Joule heating effect,
1
/()
1
501
AB
EKTT
n
tA j
⋅
e
+Δ
−
=⋅
(10)
2
/()
2
502
AB
EKTT
n
tA j
⋅
e
+Δ
−
=⋅
(11)
where n is the Joule heating effect corrected current density
exponent. Combine (10) and (11),
()
1
50
t
2
50
t
12
12
11
lnlnlnln
A
B
E
k
njj
TTTT
⎛
⎜
⎝
⎞
⎟
⎠
−= − ⋅−+−
+Δ+Δ
(12)
Combine (9) and (12), we have
()
() (
⋅
)
12
(ln
'
1212
ln)
A
B
ET
+Δ
T
nnnn
KTTTTjj
Δ−Δ
= += +Δ
+Δ−
(13)
The current density exponent correction factor n
can be further simplified from the intrinsic nature of Joule
heating. The temperature increment can be written as ΔT=K⋅j2.
Therefore, n
Δ can be re-written as
Δ in (13)
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Page 5

()
() (
+
)
−
() ()
2
1
+
2
2
(ln
2
1
2
212
1
T
212
22
2112
ln)
(
⋅
) (
⋅
)
(lnln)
A
B
A
B
E KjKj
n
KTKj
⋅
T Kjjj
E
+
Kjj
Kj
jj
KT Kjjj
−
Δ =
+⋅−
⋅
=
+−
(14)
To obtain n
such that lim(j1-j2)=0, and we have
Δ as a function of j, we take the limit of (j1-j2)
( ) (
E
T
)() ()
12
2222
(0)
2
2 2
)
2
()()
lim
j
−
(ln )
22
(()
A
+
A
+
j
BB
AA
B
In the above derivations, the temperature variations due to
oven and interconnect resistivity are not taken into
consideration. In (15), T is the oven temperature and T
the temperature increment due to Joule heating. T
can be replaced by ε and T shall represent interconnect
temperature including Joule heating effect. Equation (15) can
be rewritten as
ε
ε⋅+
Δ due to temperature variation ε is
shown in (16). The current density exponent variation factor
n
Δ is dependent on the amount of temperature variation ε ,
test temperature T, as well as actual activation energy EA. In
other words, the value of n will be seemingly temperature and
current density dependent experimentally.
B
E
T
K
Kj
j
⋅
j
+
E
T
K
Kj
j
⋅
j
+
dj
nj
dj
KTKj KTKj
E
T
Kj
Kj
T
TKK
=
⋅⋅+⋅⋅+
Δ =⋅=⋅
⋅⋅⋅⋅Δ
+Δ
==
⋅+⋅
(15)
Δ
is
Δ
in (15)
2
2
()
A
B
E
T
n
K
⋅⋅
Δ =
(16)
The correction function n
With the previous determined EA in Fig. 2 and temperature
variation in Table II,
n
Δ
is plotted with the variation of
resistivity as shown in Fig. 4. The estimated n
one for the temperature variation considered in this work.
Δ is less than
Using experimental data from our previous work, EA=0.87
eV, ε≈3 °C, T=300 °C, substitute into (16), Δn is estimated to
be 0.18, close to the experimentally corrected n of 0.1 and 0.2
for narrow and wide interconnects [4]. Therefore, the
estimation from (16) is consistent with the experimental values.
0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0


Δn
ρ/ρ0
Oven at 150
200
250
300
350
400
OC
OC
OC
OC
OC
OC

Fig. 4: Joule heating corrected n due to the variation of sample resistance and
oven temperature.

V.
CONCLUSIONS
In this work, the formulations for the variation of Ea and n
are derived analytically and verified through our recent
experimental work. As the temperature variation in
interconnect is becoming more important in EM test in future
low-k interconnects, the results in this work can provide a
quick and accurate assessment on the variation of EM
parameters in high current EM test.

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