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A model for hillock growth in Al thin films controlled by plastic deformation
Acta Materialia 55 (2007) 5297–5301 www.elsevier.com/locate/actamat
A model for hillock growth in Al thin films controlled by plastic deformation Soo-Jung Hwang a, William D. Nix b, Young-Chang Joo a
a,b,*
Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea b Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA Received 17 April 2007; received in revised form 30 May 2007; accepted 31 May 2007 Available online 20 July 2007
Abstract The dependence of total hillock volume (per unit area) on film thickness and annealing temperature has been studied in pure aluminum films. The total hillock volume (per unit area) increases linearly with both the film thickness and annealing temperature. Other characteristics involve a critical temperature and a critical thickness for hillock formation. It is shown that these phenomena can be explained by assuming that hillock growth is controlled by plastic deformation in the surrounding film. A simple analytical model is developed to account for these observations. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminum; Plastic deformation; Creep; Hillock; Thin films
1. Introduction During integrated circuit fabrication processing, interconnect metals are subjected to a number of thermal cycles. The differences in the thermal expansion coefficients between the various materials in these devices give rise to large thermal stresses, which may result in hillock or whisker formation when the stresses are compressive. Hillocks can cause reliability problems, such as dielectric cracks and interconnection shorts. To obtain highly reliable devices, it is important to understand the mechanisms of hillock formation. In metal films, various mechanisms, including interfacial diffusion [1], surface diffusion [2] and grain-boundary diffusion [3,4], have been proposed as the controlling process for hillock growth. Chaudhari proposed that hillock growth is a form of stress relaxation in which atomic species diffuse along the film–substrate interface to the base * Corresponding author. Address: Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea. Tel.: +82 2 880 8986; fax: +82 2 883 8197. E-mail address: [email protected] (Y.-C. Joo).
of the hillock and that the hillock, in turn, extrudes out of the plane of the film [1]. A different view was given by Chang et al. [2], who suggested that hillock growth is a surface diffusion controlled process, based on the observation that during annealing large hillocks became larger while smaller hillocks decreased in size and eventually disappeared, thereby increasing the hillock size and decreasing the hillock density. Iwamura et al. [3] suggested that the atoms diffuse from the film into the hillock, which gives rise to lateral diffusion along the grain boundaries that lie in the plane of the film and around the hillock. They considered that the driving force for lateral diffusion is a gradient of stress, which is established by stress relaxation near the hillock. Ericson et al. [4] observed that hillocks are formed at triple junctions using cross-sectional transmission electron microscopy (TEM) and also suggested that grain-boundary diffusion is the controlling mechanism of hillock growth. Until now, most studies of the controlling mechanism of hillock growth have focused on diffusion. However, even if the hillock grows locally by diffusive processes, the inplane displacements that drive the hillock growth can still be controlled by plastic deformation in the surrounding film.
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.05.046
In an effort to identify the dominant mechanism of hillock growth, Al films of various thicknesses have been prepared and studied. The hillock volume (per unit area) was measured quantitatively for different film thicknesses and annealing temperatures. From these results, a model for hillock formation controlled by plastic deformation was developed. 2. Experiments Pure Al films of different thickness (250, 400 and 600 nm) were deposited by sputtering onto a Mo barrier (50 nm)/glass substrate (700 lm). The Al sputtering process was performed at a sputter power of 16 kW and an Ar pressure of 0.42 Pa without breaking the vacuum. The base pressure prior to deposition was lower than 106 Torr. All of the Al films were annealed in an annealing chamber (<103 Torr) for 400 min at 180, 200, 220, 250 and 280 C with a heating rate of 5 C min1 (five different samples for each of three thicknesses). The samples were observed using a field emission scanning electron microscope (FE-SEM; JEOL JSM 6330F) before and after annealing. The hillocks in the SEM images were marked on transparency papers that were scanned to digitize and transform them to a binary form. Then the individual hillock areas were analyzed by means of an image analysis program (Scion Corp.). In each of the samples, four different locations, with dimensions 30 · 40 lm, were examined. The diameters of the hillocks (Di) were calculated from the measured individual hillock areas (Ai), assuming that the hillocks are circular in shape 4Ai 1=2 Di ¼ p . The total hillock volume in a unit area of the film (Vhillock) was calculated by adding the individual hillock volume in a measured film area (Afilm) assuming that the hillocks are hemispherical in shape. The stress change in the 600 nm thick Al film was measured by means of a wafer curvature system using a twodimensional laser array deflected from the sample (k-Space Associates, Multi-beam Optical System) during heating from room temperature to 200 C and during isothermal annealing for 400 min at 200 C. The biaxial stresses in the films were derived from the measured curvature using Stoney’s formula. 3. Results The total hillock volume in a unit area of film, Vhillock, was calculated for all Al samples annealed at various temperatures and plotted against annealing temperature (Fig. 1a) and film thickness (Fig. 1b). Fig. 1a shows that the Vhillock increases linearly with annealing temperature for all film thicknesses. The slope of Vhillock vs. T increases with increasing film thickness. To further investigate these dependencies, we made use of a linear fit. The fitting parameters, such as the slopes and x-axis intercepts, are summarized in Table 1a. The x-axis intercept indicates that
Hillock volume/area, Vhillock [nm]
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
8 7 6
600 nm 400 nm 250 nm
5 4 3 2 1 0 50
100
150
200
250
300
o
Annealing temperature,T [ C] Hillock volume/area, Vhillock [nm]
5298
8
o
280 C o 250 C o 220 C o 200 C o 180 C
7 6 5 4 3 2 1 0
0
100
200
300
400
500
600
Film thickness, h [nm] Fig. 1. The total hillock volume in a unit area of film assuming the hillock is a hemisphere (scatter) distributed as a function of (a) the annealing temperature and (b) the film thickness. Their linear fit is plotted with a solid line.
there is a critical temperature, Tcr, needed for hillock growth [4,5]. Another characteristic of Fig. 1a is that the critical temperature increases as the film thickness decreases. Fig. 1b shows that Vhillock increases linearly with the film thickness at temperatures between 180 and 280 C. From the linear fit, the slopes and x-axis intercepts were obtained and are summarized in Table 1b. The x-axis intercepts of Fig. 1b indicate that there is a critical thickness, hcr, needed for hillock growth. The critical thicknesses increase with decreasing annealing temperature. A similar relationship Table 1 Summary of (a) slopes of Vhillock vs. T and critical temperatures, Tcr and (b) slopes of Vhillock vs. h and critical thicknesses, hcr (a) Film thickness (nm)
250
400
600
Slope (nm C1) Tcr (C)
0.016 158
0.024 90
0.036 82
(b) Annealing temperature (C)
180
200
220
250
280
Slope hcr (nm)
0.008 155
0.012 189
0.010 135
0.014 128
0.013 89
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
100
Drrelax ¼ M 600 nm Al films
Change in stress [MPa]
50 0 -50 -100 -150 -200
To
T yield
-250 -350 0
20 40 60 80 100 120 140 160 180 200 220 o
Temperature [ C] Fig. 2. Change in stress during thermal cycling and isothermal annealing was measured by the wafer curvature method in 600 nm Al films. The calculated T0 is shown by the dotted line. The solid line shows the thermoelastic phase, during which thermal strain is accommodated predominantly by elastic strain.
2h Drrelax : M
4. Discussion and model development The total hillock volume increases linearly as the film thickness increases from 250 to 600 nm and as the annealing temperature increases from 180 to 280 C. Furthermore, there is a critical temperature and critical film thickness for hillock growth. These results can be understood using a model based on stress relaxation, assuming that hillock formation is the only plastic deformation mechanism to play a role in the relaxation of the compressive stresses and also that the hillocks are fully grown during the lengthy annealing time at the highest temperature [5]. We describe a model for the dependence of hillock volume (per unit area) on film thickness and annealing temperature, which also accounts for the film thickness dependence of the critical temperature and the temperature dependence of the critical thickness. Hillocking is taken to be the relaxation mechanism for the compressive stresses in the metal film. Thus, if the small change in films thickness due to the stress is neglected, the amount of stress relaxed due to the hillock formation (Drrelax) can be expressed as follows:
ð2Þ
We note that Drrelax is positive for positive hillock growth; that is, compressive stresses in the film are relieved by hillock growth and become more tensile. We consider now complete relaxation of the stress in the film by hillock growth. Chaudhari [1] suggested that if hillock growth is the only mechanism for relaxing the compressive stresss then Drrelax for complete relaxation would be: Drrelax ¼ MDaðT T 0 Þ;
between the film thickness and critical temperature was reported by Ericson et al. [4]. They found that, as the film thickness increases from 0.25 to 2.2 lm, the critical temperature decreases from 150 to 90 C. Fig. 2 shows the variation of biaxial stress in the 600 nm Al film as a function of temperature between 20 and 200 C. Note that the y-axis is the change in stress, not the absolute stress. With increasing temperature to 120 C, the biaxial stress decreases and becomes more compressive because of a thermal expansion mismatch and then remains nearly constant with further increasing temperature. During isothermal annealing at 200 C, the stress decreases by relaxing the compressive stress by approximately 75 MPa.
ð1Þ
where h is the film thickness, M is the biaxial elastic modulus, and V hillock is the hillock volume on a surface area of Afilm [6,7]. This result is valid for both diffusive and plastic hillock growth. We can rewrite this equation as a hillock volume (per unit area), Vhillock, as follows: V hillock ¼
-300
V hillock ; 2hAfilm
5299
ð3Þ
where T is the annealing temperature, T0 is the temperature at which the stress changes from tension to compression. If we put Eq. (3) into (2), Vhillock can be expressed as: V hillock ¼ 2hDaðT T 0 Þ:
ð4Þ
From Eq. (4), we can explain the linear dependence of Vhillock on the film thickness and annealing temperature. However, this cannot explain the existence of critical thickness, hcr, for hillock formation or the dependence of the critical thickness on the annealing temperature. Furthermore, the critical temperature Tcr should be the same as T0, the temperature at which the stress changes from tension to compression. In order to explain the dependence of the critical thickness on the annealing temperature and critical temperature on the film thickness, we assume that hillock growth is controlled by plastic deformation of the surrounding film. According to this picture hillock growth occurs only if plastic flow occurs in the surrounding film; having a net compressive stress in the film is an insufficient condition for plastic flow and hillock growth. In this case, Drrelax would be expressed as follows: Drrelax ¼ MDaðT T yield Þ þ Driso ;
ð5Þ
where Tyield is the temperature at which the compressive yielding is initiated and Driso is the additional stress relaxation during the isothermal annealing at high temperature (Fig. 2). Because the compressive yield stress can be expressed as the thermal stress that develops from T0 to Tyield and the yield stress is thickness dependent [8,9], we may assume that the Tyield of the Al film is thickness dependent, as follows: T yield T 0 ¼
A ; h
where A is a proportionality constant.
ð6Þ
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
0.04
o
If we put Eqs. (5) and (6) into (2) and define B as riso B ¼ T 0 MDa , then Vhillock can be expressed as follows:
Slope of Vhillock vs T [nm/ C]
5300
ð7Þ
Now we may obtain the slopes and x-axis intercepts of Vhillock vs. T and Vhillock vs. h, respectively, by arranging Eq. (7) to reveal the temperature (T) dependence and film thickness (h) dependence. Those parameters are summarized in Table 2. From Table 2, we can expect the slopes of Vhillock vs. T and Vhillock vs. h to increase linearly with the film thickness and the annealing temperature, respectively, and Tcr and hcr to be inversely proportional to the thickness and the annealing temperature, respectively. In order to determine if this model for hillock growth matches the experimental results, the slopes and x-axis intercepts obtained experimentally can be compared with the model predictions. First, the slopes of Vhillock vs. T and Vhillock vs. h were plotted against film thickness (Fig. 3a) and annealing temperature (Fig. 3b), respectively. The slopes of the lines in Fig. 3a and b should be 2Da according to the model. The expected difference of thermal expansion coefficient, Da, can then be calculated from these hillocking data; we find 29 and 24 ppm C1, respectively. These values of Da are somewhat larger than the known value of Da, 20 ppm C1 (aAl = 23.8 ppm C1, aglass = 3.8 ppm C1). One possible explanation for this difference is that there is a discrepancy between the calculated and the actual hillock volumes, because of the assumption that the hillocks have a hemispherical shape [5,10]. If the hillock shape is not hemispherical, but has a spherical cap with a contact angle between the film and the hillock under 90, the value of total hillock volume will decrease. Then the Da values measured from slopes of Vhillock vs. T and Vhillock vs. h will also decrease. In this case, if the hillocks have a contact angle of 78, the measured value Da from the slope of Vhillock vs. T will fit well with the known difference in thermal expansion coefficients. From Table 2, the x-axis intercept Tcr of Vhillock vs. T should be inversely proportional to the film thickness. From a plot of ln Tcr vs. ln h, we obtained A = 34,000 nm C1 and B = 17 C. Using these values of A and B, the critical temperature can be predicted as a function of film thickness. Fig. 4a shows the measured values of critical temperature (square) along with the expected values based on the present analysis (line). Likewise, hcr should be inversely proportional to the temperature. From reciprocal fitting, we obtained A = 25,000 nm C1 and B = 25 C. Fig. 4b shows the measured values of hcr (square) along
Table 2 Summary of slopes and x-axis intercepts of Vhillock vs. T and Vhillock vs. h from Eq. (7)
Temperature dependence (Vhillock vs. T) Thickness dependence (Vhillock vs. h) riso Here, B ¼ T o MDa .
Slope
x-axis intercept
2Dah 2Da(T B)
A h
þB
A T B
0.03
2Δα 0.02
0.01
200
400
600
Film thickness [nm] 0.016
Slope of Vhillock vs h
V hillock ¼ 2DahT 2DaA 2DahB:
0.014 0.012
2Δα
0.010 0.008 0.006 160
180
200
220
240
260
280
300
o
Annealing temperature [ C] Fig. 3. (a) Slope of Vhillock vs. T distributed as a function of film thickness (scatter) and its linear fit (line). (b) Slope of Vhillock vs. h distributed as a function of annealing temperature (scatter) and its linear fit (line).
with the expected values (line). Fig. 4a and b may have some practical importance because we can describe the dependence of critical thickness on annealing temperature and the dependence of critical temperature on film thickness, respectively. When the film is thin, a high critical temperature, Tcr, is needed for hillock growth, but when the film is thick, Tcr decreases. Likewise, when the annealing temperature is high, even thin flms can exhibit hillocking, but when the annealing temperature is low, only thick films can be expected to hillock. Using the values of A and B obtained from the hillocking experiments, we can calculate T0 and Tyield. From riso , T0 can be determined using the expected B ¼ T 0 MDa value of riso/MDa = 38 C; this is found by taking riso to be 75 MPa (measured from Fig. 2), M to be 100 GPa, and Da to be 20 ppm C1. Then T0 is about 55 and 63 C when B is 17 and 25 C, respectively. The calculated value of T0 is shown to be around 60 C in Fig 2. From the calculated value of T0, Tyield can be calculated using Eq. (6). When T0 is 55 or 63 C, Tyield is calculated as 112 or 107 C, respectively. Fig. 2 also shows that a yield temperature, in this case about 120 C, is needed to initiate plastic flow in the film. Therefore, the calculated value of Tyield matches with experimental result well. It should be noted that, while the present model assumes that hillock growth is controlled by compressive plastic or creep flow in the surrounding film, local diffusive processes
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
measured value our model
350
o
Critical temperature, Tcr [ C]
400
300 250
Hillock
200 150 100 50 0
No hillock 200
400
600
800
1000 1200 1400
Film thickness[nm] measured value our model
Critical thickness, hcr [nm]
350 300
hillock
150 100 50 0 100
No hillock 200
300
total hillock volume was measured quantitatively using an image analysis program. The total hillock volume in a unit area of film was found to be linearly proportional both to the film thickness and the annealing temperature. A critical thickness and critical temperature were observed below which no hillocks were formed; the critical thickness and critical temperature decrease with annealing temperature and film thickness, respectively. A simple analytical model for hillocking was proposed based on plastic deformation of the surrounding film. The model can explain the observed linear dependences of hillock volume (per unit area) on annealing temperature and film thickness as well as the existence of a critical temperature and critical thickness for hillock formation. Acknowledgements
250 200
5301
400 o
Annealing temperature[ C] Fig. 4. (a) The measured value of critical temperature (square) due to film thickness and its expected value (line). (b) The measured value of critical thickness (square) due to annealing temperature and its expected value (line).
are still needed at the hillock to accommodate those plastic or creep displacements. We consider the rate-limiting process to involve plastic or creep displacements in the surrounding film, even if diffusion occurs locally. A comparison of purely diffusion-controlled hillock growth with hillock growth controlled by power-law creep in the surrounding film is currently being made [11]. 5. Summary Aluminum films of thickness 250–600 nm were annealed in the temperature range 180–280 C. After annealing, the
This work was supported by the Ministry of Education and Human Resources Development (MOE), the Ministry of Commerce, Industry and Energy (MOCIE) and the Ministry of Labor (MOLAB) through the fostering project of the Lab of Excellency. Y.-C. Joo acknowledges a 2005 LG Yonam Foundation Fellowship for the partial support of his stay at Stanford University. The support of one of the authors (WDN) by a Grant from the US Department of Energy (DE-FG02-04-ER46163) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Chaudhari P. J Appl Phys 1974;45:4339. Chang CY, Vook RW. J Mater Res 1989;4:1172. Iwamura E, Ohnishi T, Yoshikawa K. Thin Solid Films 1995;270:450. ˚ . J Vac Sci Technol B Ericson F, Kristensen N, Schweitz J-A 1991;9:58. Hwang S-J, Lee J-H, Jeong C-O, Joo Y-C. Scripta Mater 2007;56:17. Kim D-K, Heiland B, Nix WD, Arzt E, Deal MD, Plummer JD. Thin Solid Films 2000;371:278. Kim D-K, Ph.D. Dissertation, Stanford University, CA, USA; 2001. Nix WD. Met Trans A 1989;20 A:2217. Thompson CV. J Mater Res 1993;8:237. Hwang S-J, Lee Y-D, Park Y-B, Lee J-H, Jeong C-O, Joo Y-C. Scripta Mater 2006;54:1841. Berla L, Joo Y-C, Nix WD, Unpublished research, Stanford University.
A model for hillock growth in Al thin films controlled by plastic deformation Soo-Jung Hwang a, William D. Nix b, Young-Chang Joo a
a,b,*
Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea b Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA Received 17 April 2007; received in revised form 30 May 2007; accepted 31 May 2007 Available online 20 July 2007
Abstract The dependence of total hillock volume (per unit area) on film thickness and annealing temperature has been studied in pure aluminum films. The total hillock volume (per unit area) increases linearly with both the film thickness and annealing temperature. Other characteristics involve a critical temperature and a critical thickness for hillock formation. It is shown that these phenomena can be explained by assuming that hillock growth is controlled by plastic deformation in the surrounding film. A simple analytical model is developed to account for these observations. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminum; Plastic deformation; Creep; Hillock; Thin films
1. Introduction During integrated circuit fabrication processing, interconnect metals are subjected to a number of thermal cycles. The differences in the thermal expansion coefficients between the various materials in these devices give rise to large thermal stresses, which may result in hillock or whisker formation when the stresses are compressive. Hillocks can cause reliability problems, such as dielectric cracks and interconnection shorts. To obtain highly reliable devices, it is important to understand the mechanisms of hillock formation. In metal films, various mechanisms, including interfacial diffusion [1], surface diffusion [2] and grain-boundary diffusion [3,4], have been proposed as the controlling process for hillock growth. Chaudhari proposed that hillock growth is a form of stress relaxation in which atomic species diffuse along the film–substrate interface to the base * Corresponding author. Address: Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea. Tel.: +82 2 880 8986; fax: +82 2 883 8197. E-mail address: [email protected] (Y.-C. Joo).
of the hillock and that the hillock, in turn, extrudes out of the plane of the film [1]. A different view was given by Chang et al. [2], who suggested that hillock growth is a surface diffusion controlled process, based on the observation that during annealing large hillocks became larger while smaller hillocks decreased in size and eventually disappeared, thereby increasing the hillock size and decreasing the hillock density. Iwamura et al. [3] suggested that the atoms diffuse from the film into the hillock, which gives rise to lateral diffusion along the grain boundaries that lie in the plane of the film and around the hillock. They considered that the driving force for lateral diffusion is a gradient of stress, which is established by stress relaxation near the hillock. Ericson et al. [4] observed that hillocks are formed at triple junctions using cross-sectional transmission electron microscopy (TEM) and also suggested that grain-boundary diffusion is the controlling mechanism of hillock growth. Until now, most studies of the controlling mechanism of hillock growth have focused on diffusion. However, even if the hillock grows locally by diffusive processes, the inplane displacements that drive the hillock growth can still be controlled by plastic deformation in the surrounding film.
1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.05.046
In an effort to identify the dominant mechanism of hillock growth, Al films of various thicknesses have been prepared and studied. The hillock volume (per unit area) was measured quantitatively for different film thicknesses and annealing temperatures. From these results, a model for hillock formation controlled by plastic deformation was developed. 2. Experiments Pure Al films of different thickness (250, 400 and 600 nm) were deposited by sputtering onto a Mo barrier (50 nm)/glass substrate (700 lm). The Al sputtering process was performed at a sputter power of 16 kW and an Ar pressure of 0.42 Pa without breaking the vacuum. The base pressure prior to deposition was lower than 106 Torr. All of the Al films were annealed in an annealing chamber (<103 Torr) for 400 min at 180, 200, 220, 250 and 280 C with a heating rate of 5 C min1 (five different samples for each of three thicknesses). The samples were observed using a field emission scanning electron microscope (FE-SEM; JEOL JSM 6330F) before and after annealing. The hillocks in the SEM images were marked on transparency papers that were scanned to digitize and transform them to a binary form. Then the individual hillock areas were analyzed by means of an image analysis program (Scion Corp.). In each of the samples, four different locations, with dimensions 30 · 40 lm, were examined. The diameters of the hillocks (Di) were calculated from the measured individual hillock areas (Ai), assuming that the hillocks are circular in shape 4Ai 1=2 Di ¼ p . The total hillock volume in a unit area of the film (Vhillock) was calculated by adding the individual hillock volume in a measured film area (Afilm) assuming that the hillocks are hemispherical in shape. The stress change in the 600 nm thick Al film was measured by means of a wafer curvature system using a twodimensional laser array deflected from the sample (k-Space Associates, Multi-beam Optical System) during heating from room temperature to 200 C and during isothermal annealing for 400 min at 200 C. The biaxial stresses in the films were derived from the measured curvature using Stoney’s formula. 3. Results The total hillock volume in a unit area of film, Vhillock, was calculated for all Al samples annealed at various temperatures and plotted against annealing temperature (Fig. 1a) and film thickness (Fig. 1b). Fig. 1a shows that the Vhillock increases linearly with annealing temperature for all film thicknesses. The slope of Vhillock vs. T increases with increasing film thickness. To further investigate these dependencies, we made use of a linear fit. The fitting parameters, such as the slopes and x-axis intercepts, are summarized in Table 1a. The x-axis intercept indicates that
Hillock volume/area, Vhillock [nm]
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
8 7 6
600 nm 400 nm 250 nm
5 4 3 2 1 0 50
100
150
200
250
300
o
Annealing temperature,T [ C] Hillock volume/area, Vhillock [nm]
5298
8
o
280 C o 250 C o 220 C o 200 C o 180 C
7 6 5 4 3 2 1 0
0
100
200
300
400
500
600
Film thickness, h [nm] Fig. 1. The total hillock volume in a unit area of film assuming the hillock is a hemisphere (scatter) distributed as a function of (a) the annealing temperature and (b) the film thickness. Their linear fit is plotted with a solid line.
there is a critical temperature, Tcr, needed for hillock growth [4,5]. Another characteristic of Fig. 1a is that the critical temperature increases as the film thickness decreases. Fig. 1b shows that Vhillock increases linearly with the film thickness at temperatures between 180 and 280 C. From the linear fit, the slopes and x-axis intercepts were obtained and are summarized in Table 1b. The x-axis intercepts of Fig. 1b indicate that there is a critical thickness, hcr, needed for hillock growth. The critical thicknesses increase with decreasing annealing temperature. A similar relationship Table 1 Summary of (a) slopes of Vhillock vs. T and critical temperatures, Tcr and (b) slopes of Vhillock vs. h and critical thicknesses, hcr (a) Film thickness (nm)
250
400
600
Slope (nm C1) Tcr (C)
0.016 158
0.024 90
0.036 82
(b) Annealing temperature (C)
180
200
220
250
280
Slope hcr (nm)
0.008 155
0.012 189
0.010 135
0.014 128
0.013 89
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
100
Drrelax ¼ M 600 nm Al films
Change in stress [MPa]
50 0 -50 -100 -150 -200
To
T yield
-250 -350 0
20 40 60 80 100 120 140 160 180 200 220 o
Temperature [ C] Fig. 2. Change in stress during thermal cycling and isothermal annealing was measured by the wafer curvature method in 600 nm Al films. The calculated T0 is shown by the dotted line. The solid line shows the thermoelastic phase, during which thermal strain is accommodated predominantly by elastic strain.
2h Drrelax : M
4. Discussion and model development The total hillock volume increases linearly as the film thickness increases from 250 to 600 nm and as the annealing temperature increases from 180 to 280 C. Furthermore, there is a critical temperature and critical film thickness for hillock growth. These results can be understood using a model based on stress relaxation, assuming that hillock formation is the only plastic deformation mechanism to play a role in the relaxation of the compressive stresses and also that the hillocks are fully grown during the lengthy annealing time at the highest temperature [5]. We describe a model for the dependence of hillock volume (per unit area) on film thickness and annealing temperature, which also accounts for the film thickness dependence of the critical temperature and the temperature dependence of the critical thickness. Hillocking is taken to be the relaxation mechanism for the compressive stresses in the metal film. Thus, if the small change in films thickness due to the stress is neglected, the amount of stress relaxed due to the hillock formation (Drrelax) can be expressed as follows:
ð2Þ
We note that Drrelax is positive for positive hillock growth; that is, compressive stresses in the film are relieved by hillock growth and become more tensile. We consider now complete relaxation of the stress in the film by hillock growth. Chaudhari [1] suggested that if hillock growth is the only mechanism for relaxing the compressive stresss then Drrelax for complete relaxation would be: Drrelax ¼ MDaðT T 0 Þ;
between the film thickness and critical temperature was reported by Ericson et al. [4]. They found that, as the film thickness increases from 0.25 to 2.2 lm, the critical temperature decreases from 150 to 90 C. Fig. 2 shows the variation of biaxial stress in the 600 nm Al film as a function of temperature between 20 and 200 C. Note that the y-axis is the change in stress, not the absolute stress. With increasing temperature to 120 C, the biaxial stress decreases and becomes more compressive because of a thermal expansion mismatch and then remains nearly constant with further increasing temperature. During isothermal annealing at 200 C, the stress decreases by relaxing the compressive stress by approximately 75 MPa.
ð1Þ
where h is the film thickness, M is the biaxial elastic modulus, and V hillock is the hillock volume on a surface area of Afilm [6,7]. This result is valid for both diffusive and plastic hillock growth. We can rewrite this equation as a hillock volume (per unit area), Vhillock, as follows: V hillock ¼
-300
V hillock ; 2hAfilm
5299
ð3Þ
where T is the annealing temperature, T0 is the temperature at which the stress changes from tension to compression. If we put Eq. (3) into (2), Vhillock can be expressed as: V hillock ¼ 2hDaðT T 0 Þ:
ð4Þ
From Eq. (4), we can explain the linear dependence of Vhillock on the film thickness and annealing temperature. However, this cannot explain the existence of critical thickness, hcr, for hillock formation or the dependence of the critical thickness on the annealing temperature. Furthermore, the critical temperature Tcr should be the same as T0, the temperature at which the stress changes from tension to compression. In order to explain the dependence of the critical thickness on the annealing temperature and critical temperature on the film thickness, we assume that hillock growth is controlled by plastic deformation of the surrounding film. According to this picture hillock growth occurs only if plastic flow occurs in the surrounding film; having a net compressive stress in the film is an insufficient condition for plastic flow and hillock growth. In this case, Drrelax would be expressed as follows: Drrelax ¼ MDaðT T yield Þ þ Driso ;
ð5Þ
where Tyield is the temperature at which the compressive yielding is initiated and Driso is the additional stress relaxation during the isothermal annealing at high temperature (Fig. 2). Because the compressive yield stress can be expressed as the thermal stress that develops from T0 to Tyield and the yield stress is thickness dependent [8,9], we may assume that the Tyield of the Al film is thickness dependent, as follows: T yield T 0 ¼
A ; h
where A is a proportionality constant.
ð6Þ
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
0.04
o
If we put Eqs. (5) and (6) into (2) and define B as riso B ¼ T 0 MDa , then Vhillock can be expressed as follows:
Slope of Vhillock vs T [nm/ C]
5300
ð7Þ
Now we may obtain the slopes and x-axis intercepts of Vhillock vs. T and Vhillock vs. h, respectively, by arranging Eq. (7) to reveal the temperature (T) dependence and film thickness (h) dependence. Those parameters are summarized in Table 2. From Table 2, we can expect the slopes of Vhillock vs. T and Vhillock vs. h to increase linearly with the film thickness and the annealing temperature, respectively, and Tcr and hcr to be inversely proportional to the thickness and the annealing temperature, respectively. In order to determine if this model for hillock growth matches the experimental results, the slopes and x-axis intercepts obtained experimentally can be compared with the model predictions. First, the slopes of Vhillock vs. T and Vhillock vs. h were plotted against film thickness (Fig. 3a) and annealing temperature (Fig. 3b), respectively. The slopes of the lines in Fig. 3a and b should be 2Da according to the model. The expected difference of thermal expansion coefficient, Da, can then be calculated from these hillocking data; we find 29 and 24 ppm C1, respectively. These values of Da are somewhat larger than the known value of Da, 20 ppm C1 (aAl = 23.8 ppm C1, aglass = 3.8 ppm C1). One possible explanation for this difference is that there is a discrepancy between the calculated and the actual hillock volumes, because of the assumption that the hillocks have a hemispherical shape [5,10]. If the hillock shape is not hemispherical, but has a spherical cap with a contact angle between the film and the hillock under 90, the value of total hillock volume will decrease. Then the Da values measured from slopes of Vhillock vs. T and Vhillock vs. h will also decrease. In this case, if the hillocks have a contact angle of 78, the measured value Da from the slope of Vhillock vs. T will fit well with the known difference in thermal expansion coefficients. From Table 2, the x-axis intercept Tcr of Vhillock vs. T should be inversely proportional to the film thickness. From a plot of ln Tcr vs. ln h, we obtained A = 34,000 nm C1 and B = 17 C. Using these values of A and B, the critical temperature can be predicted as a function of film thickness. Fig. 4a shows the measured values of critical temperature (square) along with the expected values based on the present analysis (line). Likewise, hcr should be inversely proportional to the temperature. From reciprocal fitting, we obtained A = 25,000 nm C1 and B = 25 C. Fig. 4b shows the measured values of hcr (square) along
Table 2 Summary of slopes and x-axis intercepts of Vhillock vs. T and Vhillock vs. h from Eq. (7)
Temperature dependence (Vhillock vs. T) Thickness dependence (Vhillock vs. h) riso Here, B ¼ T o MDa .
Slope
x-axis intercept
2Dah 2Da(T B)
A h
þB
A T B
0.03
2Δα 0.02
0.01
200
400
600
Film thickness [nm] 0.016
Slope of Vhillock vs h
V hillock ¼ 2DahT 2DaA 2DahB:
0.014 0.012
2Δα
0.010 0.008 0.006 160
180
200
220
240
260
280
300
o
Annealing temperature [ C] Fig. 3. (a) Slope of Vhillock vs. T distributed as a function of film thickness (scatter) and its linear fit (line). (b) Slope of Vhillock vs. h distributed as a function of annealing temperature (scatter) and its linear fit (line).
with the expected values (line). Fig. 4a and b may have some practical importance because we can describe the dependence of critical thickness on annealing temperature and the dependence of critical temperature on film thickness, respectively. When the film is thin, a high critical temperature, Tcr, is needed for hillock growth, but when the film is thick, Tcr decreases. Likewise, when the annealing temperature is high, even thin flms can exhibit hillocking, but when the annealing temperature is low, only thick films can be expected to hillock. Using the values of A and B obtained from the hillocking experiments, we can calculate T0 and Tyield. From riso , T0 can be determined using the expected B ¼ T 0 MDa value of riso/MDa = 38 C; this is found by taking riso to be 75 MPa (measured from Fig. 2), M to be 100 GPa, and Da to be 20 ppm C1. Then T0 is about 55 and 63 C when B is 17 and 25 C, respectively. The calculated value of T0 is shown to be around 60 C in Fig 2. From the calculated value of T0, Tyield can be calculated using Eq. (6). When T0 is 55 or 63 C, Tyield is calculated as 112 or 107 C, respectively. Fig. 2 also shows that a yield temperature, in this case about 120 C, is needed to initiate plastic flow in the film. Therefore, the calculated value of Tyield matches with experimental result well. It should be noted that, while the present model assumes that hillock growth is controlled by compressive plastic or creep flow in the surrounding film, local diffusive processes
S.-J. Hwang et al. / Acta Materialia 55 (2007) 5297–5301
measured value our model
350
o
Critical temperature, Tcr [ C]
400
300 250
Hillock
200 150 100 50 0
No hillock 200
400
600
800
1000 1200 1400
Film thickness[nm] measured value our model
Critical thickness, hcr [nm]
350 300
hillock
150 100 50 0 100
No hillock 200
300
total hillock volume was measured quantitatively using an image analysis program. The total hillock volume in a unit area of film was found to be linearly proportional both to the film thickness and the annealing temperature. A critical thickness and critical temperature were observed below which no hillocks were formed; the critical thickness and critical temperature decrease with annealing temperature and film thickness, respectively. A simple analytical model for hillocking was proposed based on plastic deformation of the surrounding film. The model can explain the observed linear dependences of hillock volume (per unit area) on annealing temperature and film thickness as well as the existence of a critical temperature and critical thickness for hillock formation. Acknowledgements
250 200
5301
400 o
Annealing temperature[ C] Fig. 4. (a) The measured value of critical temperature (square) due to film thickness and its expected value (line). (b) The measured value of critical thickness (square) due to annealing temperature and its expected value (line).
are still needed at the hillock to accommodate those plastic or creep displacements. We consider the rate-limiting process to involve plastic or creep displacements in the surrounding film, even if diffusion occurs locally. A comparison of purely diffusion-controlled hillock growth with hillock growth controlled by power-law creep in the surrounding film is currently being made [11]. 5. Summary Aluminum films of thickness 250–600 nm were annealed in the temperature range 180–280 C. After annealing, the
This work was supported by the Ministry of Education and Human Resources Development (MOE), the Ministry of Commerce, Industry and Energy (MOCIE) and the Ministry of Labor (MOLAB) through the fostering project of the Lab of Excellency. Y.-C. Joo acknowledges a 2005 LG Yonam Foundation Fellowship for the partial support of his stay at Stanford University. The support of one of the authors (WDN) by a Grant from the US Department of Energy (DE-FG02-04-ER46163) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Chaudhari P. J Appl Phys 1974;45:4339. Chang CY, Vook RW. J Mater Res 1989;4:1172. Iwamura E, Ohnishi T, Yoshikawa K. Thin Solid Films 1995;270:450. ˚ . J Vac Sci Technol B Ericson F, Kristensen N, Schweitz J-A 1991;9:58. Hwang S-J, Lee J-H, Jeong C-O, Joo Y-C. Scripta Mater 2007;56:17. Kim D-K, Heiland B, Nix WD, Arzt E, Deal MD, Plummer JD. Thin Solid Films 2000;371:278. Kim D-K, Ph.D. Dissertation, Stanford University, CA, USA; 2001. Nix WD. Met Trans A 1989;20 A:2217. Thompson CV. J Mater Res 1993;8:237. Hwang S-J, Lee Y-D, Park Y-B, Lee J-H, Jeong C-O, Joo Y-C. Scripta Mater 2006;54:1841. Berla L, Joo Y-C, Nix WD, Unpublished research, Stanford University.