Research Article Influence of the Molecular Adhesion Force ...Considering molecular adhesion force...
Transcript of Research Article Influence of the Molecular Adhesion Force ...Considering molecular adhesion force...
Research ArticleInfluence of the Molecular Adhesion Force on the IndentationDepth of a Particle into the Wafer Surface in the CMP Process
Zhou Jianhua,1 Jiang Jianzhong,1 and He Xueming2
1 School of Mechanical Engineering, Jiangnan University, Wuxi 214122, China2 Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology, Jiangnan University, Wuxi 214122, China
Correspondence should be addressed to Zhou Jianhua; [email protected]
Received 8 July 2014; Revised 2 November 2014; Accepted 15 November 2014; Published 25 November 2014
Academic Editor: Rui Zhang
Copyright Β© 2014 Zhou Jianhua et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By theoretical calculation, the external force on the particle conveyed by pad asperities and the molecular adhesion force betweenparticle and wafer are compared and analyzed quantitatively. It is confirmed that the molecular adhesion force between particleand wafer has a great influence on the chemical mechanical polishing (CMP) material removal process. Considering the molecularadhesion force between particle and wafer, a more precise model for the indentation of a particle into the wafer surface is developedin this paper, and the new model is compared with the former model which neglected the molecular adhesion force. Throughtheoretical analyses, an approach and corresponding critical values are applied to estimate whether the molecular adhesion forcein CMP can be neglected. These methods can improve the precision of the material removal model of CMP.
1. Introduction
During the course of chemical mechanical polishing (CMP)process and as the particleβs diameter gets close to the nmscale and the particleβs indentation depth into waferβs surfaceis around 0.01βΌ1 nm, the attractive intermolecular forcesacting at the particle/wafer interface and the external forceapplied on the particle may be close in magnitude. Ignoringthe molecular forces will influence the accuracy of the modelof CMP process and leads to errors. Therefore, in this paperin-depth study was conducted on the influential law of themolecular forces between particle and wafer surface on CMPprocess, and a more complete particle indentation depthmodel considering the effect of the molecular forces betweenparticle and wafer surface is established. Several importantmodeling assumptions are as follows.
In theCMPprocess, the abrasive particles are in the liquidenvironment. According to DLVO theory [1], the molecularforces between particle and wafer surface are mainly theelectric double-layer repulsion and van der Waals attraction;other intermolecular forces are smaller than them in the orderof magnitude and therefore can be ignored.
DLVO theory also shows that double-layer repulsion ismajor, and van der Waals attraction can be ignored at largeparticle/wafer surface approaching distance. van der Waalsattraction is dominant and electrical double-layer repulsioncan be neglected, when the approaching distance is less thanthe critical value of 3 nm. In theCMPprocess, the indentationdepth of particle into wafer surface is about 0.01βΌ1 nm, so justvan der Waals attraction is considered.
There will be molecular force between particle and pol-ishing pad surface. Igor Sokolov found that in liquid environ-ment the magnitude of the Hamaker constant of the particle/pad system is about 0.1 Γ 10
β20 J, while the magnitude of theHamaker constant of the particle/wafer system is about 1 Γ
10
β20 J [2]. van der Waals attraction is proportional toHamaker constant, so the van der Waals attraction betweenparticle and wafer is ten times the magnitude of that betweenparticle and pad.Hence, the van derWaals attraction betweenparticle and pad can be neglected [3].
It is assumed that the particle and wafer surface is com-pletely smooth; therefore, the molecular force between theparticle and the asperities of wafer surface is negligible.
Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014, Article ID 696893, 6 pageshttp://dx.doi.org/10.1155/2014/696893
2 Advances in Materials Science and Engineering
Pad
Particle
Wafer
P
DπΏp
β³SπΏW
Figure 1: Schematic of pad/wafer/particle three-body contact.
2. Applied Forces Exerted on a Particle
2.1. Force Balance Equation of a Particle. In the CMP process,the abrasive particle is pressed into the contact area ofpad/wafer under the external force, which forms a three-bodycontact. Pad is so soft that after wafer is pressed to contact padasperities, pad asperities will wrap abrasive particles. In thiscase, the work pressure π is sustained by the pad asperitiesand the abrasive particle together [4].The schematic diagramof a three-body contact of pad/wafer/particle is shown inFigure 1.
The applied forces exerted on abrasive particle includethe external force πΉ
π
conveyed by pad asperities, antiforce πΉβ
by wafer surface, the molecular adhesion force πΉ
π
betweenparticle and wafer surface, and abrasive particle gravity. Asthe abrasive particle diameter is in nanometer scale, takingAl2
O3
abrasive whose density is larger in CMP as an exampleto calculate its gravity gives 1.88 Γ 10
β9 nN, while the magni-tude of the external force acting on abrasive particle generallycan reach tens to hundreds of nN [5], and the abrasivegravity can be ignored. Small deformation takes place inthe contact area between wafer surface and particle, as itsindentation depth into wafer surface πΏ
π
is far less than theparticle diameter π·; the whole process is considered as acompletely plastic deformation [4], and the total plasticcontact stress between wafer and particle, that is, antiforce bywafer surface πΉ
β
, is given as
πΉ
β
= π»
π
ππ·πΏ
π
. (1)
When a particle is in equilibrium state, the force balanceequation is shown as follows:
πΉ
β
= πΉ
π
+ πΉ
π
. (2)
The schematic diagramof the force applied on the particleis shown in Figure 2.
2.2. External ForceπΉπ
. Fu andChandra used a creative equiv-alent beam bending model to calculate the external forceπΉ
π
conveyed by pad asperities to particle during CMP [10].In his model, the particles are assumed to be distributeduniformly in the contact area between pad and wafer and
D
Fa
Fh
FZ
Figure 2: Applied forces on the particle.
the part of the pad between two particles is reduced to a fixed-fixed bending beam. According to beam theory of mechanicsof materials, πΉ
π
can be expressed as
πΉ
π
= (
4
5
3
3
)
1/4
[πΈ
π
π‘
π
3
π·
2
(
π΄
π
π
)
2
π
3
]
1/4
,(3)
whereπΈπ
is Youngβsmodulus of pad, π‘π
is the equivalent beamthickness, π΄
π
is the effect contact areas between pad andwafer, and π is the number of effective particles embeddedinto the contact areas between pad and wafer.
According to Zhao andChang [4], the number of effectiveparticles embedded into the contact areas between pad andwaferπ can be calculated by
π = π΄
π
(
6π
ππ·
3
)
2/3
= π΄
π
(
6πΆ
π
π
π
ππ·
3
π
π
)
2/3
,(4)
where π is abrasive particle volume concentration,πΆπ
is abra-sive particle mass concentration, π
π
is slurry density, and π
π
is abrasive particle density.Factors affecting the thickness of equivalent beam π‘
π
include particle diameterπ·, Youngβs modulus of pad πΈ
π
, andthe working pressureπ. Based on the theory of dimensionlessunit analysis of the four variables, we have
(
π‘
π
π·
)
3/4
= π
2
(
πΈ
π
π
)
π
1
,(5)
where π2
, π1
are dimensionless constants.Substituting (4), (5) into (3), we obtain
πΉ
π
= (
4
5
3
3
)
1/4
π
2
(πΈ
π
)
1/4
(
πΈ
π
π
)
π
1
(π)
3/4
π·
2
(
6πΆ
π
π
π
ππ
π
)
β1/3
.
(6)
According to Qinβs analysis [11], in the range of theworking pressure π in a typical CMP, the average stress of theactual contact area between pad and wafer π
π
approximatelychanges very slowly with the increase of the working pressureπ. It is reasonable to assume that, with a wafer being pressedonto a polishing pad, π
π
is independent of π, the incrementof which is determined by the increase of abrasive particlesbeing embedded into the contact area between pad asperities
Advances in Materials Science and Engineering 3
D
Z0
Figure 3: Schematic of adhesion between a particle and a surface.
and wafer.Therefore, ππ
can be considered as a constant. As asingle particle is concerned, the applied force of pad on it πΉ
π
keeps unchanged, so the indentation depth of the particle intothe wafer πΏ
π
is unchanged.This point of view is substantiatedby Zhao and Chang [4], Qin et al. [11], and Jeng and Huang[12]. If the working pressure π does not affect the indentationdepth πΏ
π
, we can obtain π
1
= 3/4 in (6). And thenrearranging (6) yields
πΉ
π
= 2.48π
2
πΈ
π
π·
2
(
6πΆ
π
π
π
ππ
π
)
β1/3
.(7)
In (7) π
2
is a dimensionless constant and has no relationto a variety of the pad/wafer/particle and operating parame-ters during the course of CMP. Based on the study ofQin et al.[11] and Jiang et al. [9], π
2
is 0.078 in a typical CMP, so (7) isrearranged to (8). Equation (8) can be applied to every caseof CMP:
πΉ
π
= 0.19πΈ
π
π·
2
(
6πΆ
π
π
π
ππ
π
)
β1/3
.(8)
2.3. Molecular Adhesion Force between Particle and WaferπΉ
π
. As shown in Figure 3, during the course of a particleapproaching a surface, when the approaching distance π
0
is less than 3 nm, the van der Waals attraction between theparticle and the surface is dominant. When the approachingdistance is π
0
, the van der Waals attraction πΉ
0
between aparticle with the diameter of π· and a plane can be expressedas [3]
πΉ
0
=
π΄π·
12π
2
0
, (9)
where π΄ is the Hamaker constant of the adhesion system.An expression of the molecular adhesion force πΉ
π
bet-ween a particle and a plane is given by Bowling [13] as
πΉ
π
=
π΄π·
12π
2
0
(1 +
2π
2
π·π
0
) , (10)
where π is the diameter of the particle and π is the contactradius between particle and plane, as shown in Figure 4.
The geometric relationship equation in Figure 4 is givenas
π
2
= (
π·
2
)
2
β ((
π·
2
) β πΏ
π
)
2
= π·πΏ
π
β πΏ
2
π
.(11)
D
a
Fa
Figure 4: Adhesion force considering particle deformation.
Table 1: Hamaker constant of Cu-Water-Al2O3.
Materials Water Vacuum Unit ReferenceCopper 30 40 10β20 J [6]Water β 4.0 10β20 J [6]Al2O3 4.44 16.75 10β20 J [6]Polyurethane-water-siliconnitride 0.1β1 10β20 J [6]
Substituting (11) into (10) gives
πΉ
π
=
π΄π·
12π
2
0
(1 +
2 (π·πΏ
π
β πΏ
π
2
)
π·π
0
) . (12)
Because the indentation depth of a particle into wafersurface πΏ
π
is far less than the particle diameter π·, (12) canbe reduced as
πΉ
π
=
π΄π·
12π
2
0
(1 +
2πΏ
π
π
0
) . (13)
Assuming the particle and the wafer surface are com-pletely smooth, π
0
can be taken as 0.4 nm [14]. π΄ is theHamaker constant related to the material properties of theadhesion system of the particle/polishing slurry/wafer sys-tem, and the formula is shown as (14) to calculate theHamaker constant of a three-component adhesion system[6]:
π΄ = (βπ΄
11
β π΄
33
)(βπ΄
22
β π΄
33
) , (14)
whereπ΄11
is the Hamaker constant for a particle in a vacuumenvironment, π΄
22
is the Hamaker constant for wafer in avacuum environment, and π΄
33
is the Hamaker constant forpolishing slurry in a vacuum environment. As the abrasiveparticle and oxidant concentration in the slurry are not high,polishing slurry of CMP usually can be approximated as thewater.The Hamaker constants of Cu-Water-Al
2
O3
system arelisted in Table 1.
2.4. Comparison of πΉπ
and πΉ
π
. To further study the relation-ship between the external forces on a single particleπΉ
π
by pad
4 Advances in Materials Science and Engineering
Table 2: Parameters calculating forces between wafer and abrasive.
Parameter Value ReferenceYoungβs modulus of wafer πΈ
π
(MPa) 130000 [5]Poissonβs ratio of wafer ]
π
0.26 [5]Wafer surface hardnessπ»
π
(MPa) 5000 [7]Youngβs modulus of particle πΈ
π
(MPa) 310000 [4]Poissonβs ratio of particle ]
π
0.26 [4]Particle density π
π
(kg/m3) 3600 [4]Youngβs modulus of pad πΈ
π
(MPa) 10 [8]Poissonβs ratio of pad ]
π
0.5 [8]Mass concentration of particles πΆ
π
0.034 [9]Slurry density π
π
(kg/m3) 1000 [9]Particle diameter π· (nm) 200 [9]
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
Forc
e on
part
icle
(nN
)
Fa
Particle diameter, D (nm)
FZ
Figure 5: Comparison of πΉπ
and πΉ
π
(πΈπ
= 10MPa, πΏπ
= 1 nm).
and the molecular adhesion force between particle and waferπΉ
π
, an example is used by theoretical calculation to compareπΉ
π
and πΉ
π
that assumes an Al2
O3
abrasive is pressed intothe surface of Cu wafer. The Hamaker constant of Cu-Water-Al2
O3
system is calculated as 9 Γ 10
β20 J by (13), the inden-tation depth πΏ
π
is taken as 1 nm, and other parameters arelisted in Table 2.
Using (8), (13) to calculate the external forces on a singleparticle πΉ
π
by pad and the molecular adhesion force betweenparticle and wafer πΉ
π
, the results are shown in Figure 5.As shown in Figure 5, the critical value when the external
force πΉπ
is equal to the molecular adhesion force πΉπ
is 33 nm.When the particle diameter is less than the critical value, πΉ
π
is less than πΉ
π
. When the particle diameter is greater than thecritical value, πΉ
π
is greater than πΉ
π
, and, along with increaseof the particle diameter, the increasing extent of πΉ
π
is greaterthan that of πΉ
π
.The above analysis shows that when the parti-cle diameter is small, the molecular adhesion force πΉ
π
may begreater than or close to the external force applied on a singleparticle πΉ
π
. Therefore, CMP models ignoring the molecularadhesion force between particle and wafer will bring about
large errors. Only when the particle diameter is large enoughand the external force πΉ
π
is much greater than the molecularadhesion force πΉ
π
, πΉπ
can be ignored. The critical values usedto judge whether the molecular adhesion force in CMP canbe neglected or not are related to Youngβs modulus of pol-ishing pad, particle diameter, particle indentation depth, andparticle volume concentration. In order to find the criticalvalues, further study is needed.
3. Particle Indentation Depth consideringthe Molecular Adhesion Force
Substituting (1), (8), and (13) into (2) gives the specific forcebalance equation of the particle shown as
π
π
πΈ
π
π· +
π΄
12π
2
0
(1 +
2πΏ
π
π
0
) = π»
π
ππΏ
π
. (15)
In (15), a dimensionless parameter ππ
is defined as
π
π
= 0.19 (
6πΆ
π
π
π
ππ
π
)
β1/3
.(16)
From (15), we have
πΏ
π
π·
=
(π
π
πΈ
π
/π»
π
) + (π΄/12π
2
0
π»
π
π·)
π + (π΄/6π
3
0
π»
π
)
.(17)
Two dimensionless parameters are defined as
Ξ
1
=
π΄
6π
3
0
π»
π
,
Ξ
2
=
π΄
12π
2
0
π»
π
π·
.
(18)
If theΞ 1
,Ξ 2
items are ignored in (17), the particle inden-tation depth without considering the molecular adhesionforce between particle and wafer surface can be expressed as
πΏ
π
π·
=
π
π
πΈ
π
ππ»
π
. (19)
The result of calculating by using (17), (19) is shown inFigure 6. Relative parameters are taken as follows: πΈ
π
=
10MPa, πΆπ
= 0.01, and the rest are listed in Table 2. Inaccordance with the analysis of Figure 6, when the particlediameter π· is equal to 25 nm or so, the relative particleindentation depths calculated by (17), (19) are equal; whenπ· is less than 25 nm, the relative particle indentation depthconsidering the molecular adhesion force is greater than thatignoring the molecular adhesion force, and with the decreaseof the particle diameter, their deviation will increase; whenπ·
is greater than 25 nm they are in small deviation and can beseen as equal.Therefore, in case of large particle diameter, theinfluence of the molecular adhesion force between particleand wafer on the indentation depth πΏ
π
of particle into waferis negligible. From the analysis of (17), the critical values usedto judgewhether themolecular adhesion force inCMP can beneglected or not still have relationwith theHamaker constantof CMP system, the approaching distance between particleand wafer surface, and wafer surface hardness.
Advances in Materials Science and Engineering 5
0 50 100 150 200 250 3000.000
0.005
0.010
0.015
0.020
0.025
0.030
Relat
ive i
nden
tatio
n de
pth
Considering molecular adhesion forceNeglecting molecular adhesion force
Particle diameter, D (nm)
Figure 6: Comparison of relative indentation depth of abrasiveparticle.
4. Two Dimensionless Judicial Criteria
According to (17), when assumingΞ
1
< 0.1π,Ξ 2
< 0.1π
π
πΈ
π
/
π»
π
; that is,
π΄
π
3
0
π»
π
< 0.6π,
π΄ (6πΆ
π
π
π
/ππ
π
)
1/3
πΈ
π
π
2
0
π·
< 0.231.
(20)
The Ξ
1
, Ξ 2
items in (17) can be neglected; that is, theinfluence of the molecular adhesion force between particleand wafer on the indentation depth πΏ
π
of particle into waferis negligible.
Substituting the Hamaker constant π΄ = 9 Γ 10
β20 J, theapproaching distance π
0
= 0.4 nm, into (20) gives
π»
π€
> 746MPa, (21)
πΈ
π
π·
(6πΆ
π
π
π
/ππ
π
)
1/3
> 2.435. (22)
From the above analysis, when aCMP system is under thecondition shown in (21), (22), the molecular adhesion forcebetween particle and wafer can be ignored. Substituting theparameters in a typical CMP πΈ
π
= 10MPa, πΆπ
= 0.034, π· =
200 nm, ππ
= 3600 kg/m3, and π
π
= 1000 kg/m3 into (22)gives
πΈ
π
π·
(6πΆ
π
π
π
/ππ
π
)
1/3
= 7.626 > 2.435. (23)
What is more, π»π
= 500MPa < 746MPa. Hence, Ξ 2
itemcan be neglected while Ξ
1
item cannot be neglected during
the course of this CMP. And then the relative indentationdepth of the Cu-water-Al
2
O3
system can be simplified as
πΏ
π
π·
=
π
π
πΈ
π
/π»
π
π + (π΄/6π
3
0
π»
π
)
. (24)
From the above analysis, (20) are the two dimensionlessjudicial criteria that determine whether the influence of themolecular adhesion force between particle and wafer on theindentation depth πΏ
π
of particle into wafer can be neglectedor not. The two judicial criteria can apply to all CMP pro-cesses. It can be seen from these two criteria that the greaterthewafer surface hardnessπ»
π
, theYoungmodulus of padπΈ
π
,and the particle diameterπ·, the lower the particle concentra-tion and the greater the possibility of neglecting the influenceof the molecular adhesion force. It is by the inverse calcula-tions indicated that, in typical metal CMP processes, if wetake πΈ
π
= 10MPa, πΆπ
= 0.034, ππ
= 3600 kg/m3, and π
π
=
1000 kg/m3, average surface hardness of pure metal is gener-ally less than 746MPa, so Ξ
1
item cannot be ignored; whenthe particle diameter π· is greater than 64 nm, Ξ
2
item canbe neglected. For a typical nonmetallic oxide CMP processes,under the same conditions, the nonmetallic oxide surfacehardness is generally greater than 746MPa, so Ξ
1
item canbe neglected; when the particle diameter π· is greater than64 nm, Ξ
2
item can be neglected.
5. Conclusions
By considering the influence of the molecular adhesion forcebetween particle and wafer on the indentation depth πΏ
π
ofparticle intowafer, the force equilibriumequation of a particlein wafer/particle/pad CMP system is established, and theexternal force on the particle conveyed by pad asperities andthe molecular adhesion force between particle and wafer arecompared and analyzed by quantitative calculation. The twodimensionless judicial criteria that determine whether theinfluence of the molecular adhesion force between particleand wafer on the CMP process can be neglected are educedthrough theoretical deduction. They can be applicable toevery CMP system, which are important to constructmechanical model of CMP. According to the two criteria, thegreater the wafer surface hardness π»
π
, the Young modulusof pad πΈ
π
, and the particle diameterπ·, the lower the particleconcentration and the greater the possibility of neglecting theinfluence of the molecular adhesion force.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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