JOURNAL OF APPLIED PHYSICS
VOLUME 96, NUMBER 11
1 DECEMBER 2004
Nanomechanical properties studied by atomic force microscopy in combination with an inverse methodology Win-Jin Changa) Department of Mechanical Engineering, Kun Shan University of Technology, Tainan 710, Taiwan
Te-Hua Fang Department of Mechanical Engineering, Southern Taiwan University of Technology, Tainan 710, Taiwan
(Received 4 May 2004; accepted 7 September 2004) This study presents a method for calculating the applied force during the nanoindentation process using atomic force microscope (AFM). The determination of the applied force in the nanoindentation system is regarded as an inverse vibration problem. The conjugate gradient method is applied to treat the inverse problem using available displacement measurements. Initially, the nanoindentation force should be applied and then the nanomechanical properties of the thin film, including the elastic modulus and the hardness, can be obtained using the geometric relationships between the indenter, the applied force, and the penetration depth. Without using the inverse methodology the nanomechanical properties of ultrathin coatings or films are very difficult to be obtained with precision using the nanoindentation equipment or solely using AFM. This proposed method is useful for designing a surface measurement system for nanostructured materials. © 2004 American Institute of Physics. [DOI: 10.1063/1.1811386] I. INTRODUCTION
In this study, a method is presented using an AFM cantilever to press into a thin film in order to determine its mechanical properties. This method considers the nanoindentation system using an AFM cantilever as an inverse vibration problem using available displacement measurements including the unknown applied force. The conjugate gradient method, is a function estimation approach and is used to solve the inverse problem. Generally, an inverse problem implies that one of boundary conditions, initial conditions, source terms, and material parameters in the differential equations is unknown. It is used to determine the cause from the results obtained and is an ill posed problem. The analysis process and the optimization process are necessary for solving the inverse problem. The combination of the estimated results obtained from an initial guess and the measured values, forms a set of nonlinear functions in the analysis process. Then the optimization algorithm is used to systematically search and obtain a set of optimum values. The method can quickly approach the target function and is very powerful. It has been used to solve the function estimation problem by many researchers.13–16 Once the applied force is obtained from the inverse method, the mechanical properties of the thin films can be obtained in terms of the relationships between the penetration depth, tip geometry, and applied force.
In the past decade, the physical properties of thin film and surface coatings have become an increasingly important research topic due to the continuing miniaturization of electronic components and the development of other nanostructured materials. Measurement of the nanostructured materials properties, including the hardness and the elastic modulus, is significant for understanding the tribological and adhesive behaviors and the fracture mechanisms. The nanoindentation technique is often used by many researchers to obtain the nanomechanical properties of a material.1–7 Recently, DiCarlo and Yang8 presented a semiinverse method for determining the stress-strain relationship of a material by an indentation test. Hochstetter et al.9 studied the contact behavior of polymers for small indentation depths (below 3 m) using nanoindentation. Generally, the penetration depth is usually smaller than 1 / 5 of the thickness of the thin film to avoid the substrate effect. However, the penetration depth cannot be too small in order to obtain the actual surface properties. In order to study the mechanical properties of ultrathin films, the combination of a nanoindenter and atomic force microscope (AFM) has been used.10 Recently, Riedo and Brune11 utilized an AFM study of nanoscopic sliding friction on thin films and found that the friction coefficient is related to the elastic modulus. In addition the AFM cantilever has also been used to perform nanolithography. Fang and Chang12 studied the surface analysis of nanomachining Al material using AFM. However, the contact stiffness of the tip-sample interface in the operating process varies over time and is difficult to precisely measure and the result could be that the applied force is measured incorrectly.
II. ANALYSIS
AFM is used to indent the specimens and the cantilever is a rectangular elastic beam as shown in Fig. 1. The cantilever has a length L, thickness b, width a, and tip length h. When the indentation is in progress, the tip contacts with the specimen and results in the vertical reaction force F共t兲, which is a function of time t. The indentation system can be
a)
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© 2004 American Institute of Physics
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J. Appl. Phys., Vol. 96, No. 11, 1 December 2004
W. Chang and T. Fang
J关F共t兲兴 =
冕
tf
关y共L,t兲 − Y共L,t兲兴2dt,
6713
共2兲
0
FIG. 1. Nanoindentation using atomic force microscope.
modeled as the flexural vibration motion of the cantilever. The motion is a partial differential equation and its transverse displacement y is dependent on time t and the spatial coordinate x.16,17 When the forces are unknown, the system becomes an inverse vibration problem. In this paper, the conjugate gradient method is used to deal with the inverse problem. The calculation process of the method includes the following problems: the direct problem, the sensitivity problem, and the adjoint problem, which are discussed as follows.
A. Direct problem
The linear differential equation of motion and the corresponding boundary and initial conditions for the free vibration of the cantilever beam are
4y A 2y + = 0, x4 EI t2
共1a兲
y共0,t兲 = 0,
共1b兲
y共0,t兲 = 0, x
共1c兲
where y共L , t兲 is the computed displacement at x = L and time t and Y共L , t兲 is the measured displacement at the same location and time as y共L , t兲. t f is the final time of the measurement. To derive the sensitivity problem, it is assumed that when F共t兲 undergoes a variation ⌬F共t兲, the displacement y共L , t兲 changes by a corresponding amount ⌬y. By replacing F with F + ⌬F and y with y + ⌬y in the direct problem and subtracting it from its original problem expressed by Eq. (1). The sensitivity problem is defined as follows:
4⌬y A 2⌬y + = 0, x4 EI t2
共3a兲
⌬y共0,t兲 = 0,
共3b兲
⌬y共0,t兲 = 0, x
共3c兲
2⌬y共L,t兲 = 0, x2
共3d兲
3⌬y共L,t兲 ⌬F共t兲 , = x3 EI
共3e兲
⌬y共x,0兲 =
2y共L,t兲 = 0, x2
共1d兲
3y共L,t兲 F共t兲 , = x3 EI
共1e兲
⌬y共x,0兲 = 0. t
C. Adjoint problem and gradient equation
To derive the adjoint problem, Eq. (1) is multiplied by the Lagrange multiplier 共x , t兲, and the resulting expressions are integrated over time. Then the result is added to the righthand side of Eq. (2) and the following form is obtained:
冕 冕冕 冋 tf
J关F共t兲兴 =
关y共L,t兲 − Y共L,t兲兴2dt
0
y共x,0兲 =
y共x,0兲 = 0, t
共1f兲
where E is the modulus of elasticity, I is the area momentum of inertia, is the volume density, and A is the uniform cross-sectional area of the cantilever. When the applied forces are given, Eq. (1) can be solved for the single unknown F共t兲 using the technique for treating the timedependent boundary condition.18,19
L
tf
+
0
0
⌬J关F共t兲兴 = 2
冕 冕冕 冋 tf
共4兲
关y共L,t兲 − Y共L,t兲兴⌬y共L,t兲dt
0
tf
The solution of the direct problem with the unknown applied force F共t兲 is now regarded as a problem of optimum control, which has the control function F共t兲, and is intended to minimize the functional J共F兲 defined by
册
4y A 2y + dxdt. x4 EI t2
The variation of ⌬J is obtained by perturbing y共L , t兲 with ⌬y共L , t兲 in Eq. (3) and then by subtracting it from Eq. (4), the following is obtained:
L
+ B. Sensitivity problem
共3f兲
0
0
册
4⌬y A 2⌬y + dxdt. x4 EI t2
共5兲
The second term in the above equation is integrated by parts; the initial conditions of the sensitivity problem are utilized and then ⌬J is set to zero. After some manipulation, the adjoint problem is defined as follows:
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J. Appl. Phys., Vol. 96, No. 11, 1 December 2004
W. Chang and T. Fang
4 A 2 + = 0, x4 EI t2
共6a兲
共0,t兲 = 0,
共6b兲
共0,t兲 = 0, x
共6c兲
2共L,t兲 = 0, x2
共6d兲
3共L,t兲 = 2关y共L,t兲 − Y共L,t兲兴, x3
共6e兲
共x,t f 兲 = 0. t
共6f兲
共x,t f 兲 =
⌬J =
冕
0
冕
tf
⌬F共t兲 dt. 共L,t兲 EI
共7兲
J⬘⌬F共t兲dt,
共8兲
0
where J⬘ is the gradient of the functional J. A comparison of Eqs. (7) with (8) leads to the following gradient equation 共L,t兲 . EI
J⬘ =
共9兲
D. Conjugate gradient method for minimization
Assuming the function, y共L , t兲, ⌬y共L , t兲, 共L , t兲, and J⬘ are available at the Kth iteration, the iterative process for estimating the unknown function F共t兲 by the conjugate gradient method is performed. The cutting force F共t兲 at the 共K + 1兲th step can be evaluated by 共10兲
K = 0,1,2 . . . ,
where  is the step size of the search and P is the direction of the descent given by K
K
PK = J⬘K + ␥K PK−1,
K = 0,1,2 . . . ,
共11兲
where ␥K is the conjugate coefficient and determined from
␥ = K
兰t0f 关J⬘K共t兲兴2dt 兰t0f 关J⬘K−1共t兲兴2dt
with
␥ = 0. 0
共12兲
The functional J关F共t兲兴 for iteration K + 1 is obtained by rewriting Eq. (2) as J共FK+1兲 =
冕
tf
共14兲
The sensitivity function ⌬y共FK兲 is taken as the solution of Eq. (5) at the measured time by letting ⌬F = PK. The search step size K can be determined by minimizing the function given by Eq. (14) with respect to . After rearrangement, the following expression is obtained:
K =
兰t0f ⌬y共PK兲关y共FK兲 − Y兴dt 兰t0f 关⌬y共PK兲兴2dt
.
关y共FK − K PK兲 − Y共L,t兲兴2dt.
0
Expanding Eq. (13) into a Taylor series yields
共13兲
共15兲
E. Stopping criterion
The convergence condition for the minimization of the criterion is applied to the optimal problem and is 共16兲
where ⑀ is a small specified number. The computational procedures for estimating the applied force of AFM nanoindentation by inverse methodology are described above. F. Determination of the elastic modulus and the hardness of the film
Once the applied force is obtained from the above inverse methodology, the elastic modulus and the hardness of the film can be obtained by the relationships between the indenter geometry, displacement, and applied force. In this paper, a conical indenter is assumed to be used; and then the relationship can be expressed as20 F共t兲 =
FK+1 = FK − K PK,
关y共FK兲 − K⌬y共FK兲 − Y共L,t兲兴2dt.
J共FK+1兲 艋 ⑀ ,
From previously published works,14 the following is derived: ⌬J =
冕
tf
0
Finally, the integral term is left as tf
J共FK+1兲 =
2E * tan ␣ 2 y 共t兲.
共17兲
where ␣ represents a half angle of the cone; y共t兲 denotes the total indentation depth and is a function of time; E* is the combined elastic modulus of the film Ei and the indenter E j and their relation is 共1 − 2i 兲 共1 − 2j 兲 1 = + , E* Ei Ej
共18兲
where i and j are the Poisson’s ratio of the film and the indenter, respectively. The elastic modulus of the film can be obtained from Eqs. (17) and (18). According to the geometry of the conical indenter, the relationships of the applied force F and the radius of circle of contact a, which are a function of time, can be expressed as20 F共t兲 =
E * a2共t兲cot ␣ , 2
共19兲
where a共t兲 is related to the plastic depth y p共t兲, which is a function of time, and given by a共t兲 = y p共t兲tan ␣ .
共20兲
The plastic depth is the total indentation depth subtracting the elastic depth of penetration. The projected contact area in term of the plastic depth can be written as
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J. Appl. Phys., Vol. 96, No. 11, 1 December 2004
W. Chang and T. Fang
A共y p兲 = y 2ptan2 ␣ .
The hardness of the film can be obtained from Eqs. (19)–(21) and is expressed as H=
Y共L,t兲 = Y exact共L,t兲 + ,
共21兲
F共t兲 . A
共22兲
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共24兲
where Y exact共L , t兲 is the displacement of the direct problem with the exact applied force and is a random variable within −2.576 to 2.576 for a 99% confidence bounds and is the standard deviation of the measurement. C. Comparison of applied force using AFM nanoindentation
III. DISCUSSION A. Computation procedure
The direct problem source terms and material parameters in the governing equation, boundary condition, and initial condition are all well defined, and its solution can be obtained (i.e., displacement) from any textbooks written on vibration mechanics. In this paper, the force in the boundary condition is unknown and the cantilever system forms an inverse vibration problem. To solve the inverse vibration problem, the following procedures are necessary and the details are described below. The computational procedures for estimating the applied force of AFM nanoindentation by inverse methodology are summarized as follows. (i) Step 1: Guess the value for FK共t兲 and suppose that K F 共t兲 is available at iteration K. (ii) Step 2: Solve the direct problem given by Eq. (1) to obtain y共t兲. (iii) Step 3: Check the stopping criterion given by Eq. (16) and continue the iteration if not satisfied. (iv) Step 4: Solve the adjoint problem given by Eq. (6) to obtain 共x , t兲. (v) Step 5: Compute the gradient of the functional from Eq. (9). (vi) Step 6: Compute the conjugation coefficient from Eq. (12) and then the direction of descent from Eq. (11). (vii) Step 7: Set ⌬F = PK and then solve the sensitivity problem given by Eq. (3). (viii) Step 8: Compute the step size of the search K from Eq. (15). (ix) Step 9: Compute the new estimations for FK+1 from Eq. (10) and go back to Step 1. B. Error analysis of the inverse methodology
The error analysis is important in assessing the accuracy of the result obtained using the inverse methodology. The relative error between the exact and estimated value of the applied force can be computed by =
再冕
tf
0
关F*共t兲 − F共t兲兴2dt/
冕
tf
0
F*2共t兲dt
冎
1/2
,
共23兲
where F * 共t兲 is the exact applied force. The measurements of the displacement of the cantilever tip are necessary for estimating the applied force of the nanoindentation. If computer simulation is used, the displacements involve random measurement errors and random noise is added to the exact simulated data to generate the measured displacement Y共L , t兲, which is
The elastic modulus and the hardness of thin films or surface coating are often obtained by using the nanoindentation technique. However, when the films or the coatings are very thin such as 200 nm, the applied force in the technique cannot be precisely controlled and result in the wrong mechanical properties. Therefore, a stiffer AFM cantilever should be used to press into very thin films to obtain the elastic modulus and the hardness. However, the indentation force applied by the cantilever is difficult to precisely calculate during operation. In order to improve this point, the applied force can be obtained using the inverse methodology, which utilizes the available displacement measurements of the cantilever tip. To precisely obtain the applied force during AFM operation, a carbon nanotube is attached to the cantilever tip. IV. CONCLUSION
In this study, a method for determining the nanomechanical properties of very thin films, including the hardness and the elastic modulus, using the atomic force microscope (AFM) in combination with an inverse methodology has been proposed. The nanomechanical properties of very thin films are usually difficult to precisely obtain by the nanoindentation or solely using AFM. This method will contribute to promote the measurement techniques of nanoelectromechanical systems. ACKNOWLEDGMENT
The authors wish to thank the National Science Council of the Republic of China in Taiwan for providing financial support for the present study through Project No. NSC 932212-E-168-020. 1
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