The Investigation of SAWs propagation in Layered Anisotropic Media

for Mechanical Properties Measurement of Porous Low–k Dielectrics Zhi-Guo Li, Su-Ying Yao, Xia Xiao*, Mao-Sen Bai, Xin-Hui Zhang

School of Electronic and Information Engineering, Tianjin University, Tianjin 300072, China * Email: XiaXiao@tju.edu.cn

Abstract Porous low-k materials are required for the advanced 45nm node and beyond ULSI interconnect system. The deteriorative mechanical properties of the porous materials must be evaluated accurately. Surface acoustic wave (SAW) technique can measure the Young’s modulus of thin films accurately and nondestructively by analyzing the dispersive characters of SAW waves’ propagation. In this paper, the theoretical investigation of SAW waves propagating in the transversely isotropic porous low-k film/silicon substrate layered structure is expressed. Several numerical examples are gived.These computation results can be applied to the SAW experiments to determinant the anisotropic Young’s modulus of periodic porous low-k film.

1. Introduction Porous low dielectric constant (k) dielectric materials are required to minimize RC delay, crosstalk noise and power dissipation for the 45nm node and beyond advanced ULSI interconnect systems [1]. But the large porosity decreases the mechanical properties of dielectrics enormously. That becomes the great challenge for the manufacture process of Cu/low-k integration, such as Chemical Mechanical Polishing (CMP) [2-5]. Thus, the accurate evaluation of mechanical properties of the low-k films is necessary. The surface acoustic wave (SAW) technique is an accurate and nondestructive method available for determination of elastic properties for thin fragile materials. Experimental results indicate that the Young’s modulus of porous low-k films has been measured successfully [6-8]. The SAW measurement bases on both theoretical and experimental analyses of dispersive characters of surface acoustic waves propagation in the layered media. For the computation simplicity, all the existent theoretical models treat the low-k films as isotropic structure. However, because the introduction of nano-pores into the films, the dielectrics are not isotropic actually. In this paper, the real symmetry of porous films has been taken into account in the transversely isotropic model. The theoretical deduction for SAW waves propagating in the transversely isotropic dielectric film / Si substrate layered media is presented in detail. Several numerical examples are calculated and a series of dispersive curves has been made.

2. Theoretical Deduction SAW waves are dispersive in the layered structure [9]. The frequency ( f ) dependent SAW velocity ( ) is

determined by the propagation directions ( l )and the physical properties of the film and substrate:

v

1 2 3( , , )l l l

,ˆ( , , ( , , ) , ( ) )ijkl film ijkl substratev f l c h c (1)The schematic diagram of SAW waves propagating in the porous low-k film on Si structure shows in Fig. 1.

z(z´)

Figure 1. schematic diagram of SAW waves propagatingin the porous low-k film on Si structure.

An orthogonal Cartesian system x-y-z has establishedwith its origin o located at the interface between the filmand the substrate. The film is modeled as the compositeof bulk material and identical cylindrical pores with anangle anticlockwise from x axis. Assuming the poreshave uniform distribution in bulk material, the porousfilm can be regard as transversely isotropic structure. By rotating the coordinate system x-y-z to new coordinatesystem x´-y´-z´ in which the x´ axis is parallel to thenano-pores’ direction, the elastic constants of the filmcan be expressed as the following form:

11 12 12

12 22 23

12 23 22

22 23

55

55

ˆ ˆ ˆ 0 0 0ˆ ˆ ˆ 0 0 0ˆ ˆ ˆ 0 0 0

ˆ ˆ ˆ0 0 0 0 02ˆ0 0 0 0 0

ˆ0 0 0 0 0

c c cc c cc c c

c c c

cc

(2)

Si (100) substrate

hf

x

yx´y´nano-pore

o(o´)

1-4244-0161-5/06/$20.00 ©2006 IEEE

The elastic constants of the film in coordinate system x-y-z can be obtained by the transformation of caccording to:

0 50 100 150 200 250 300 350 400 450 5004.90

4.92

4.94

4.96

4.98

5.00

5.02

5.04

5.06

5.08

5.10

Velo

city

(km

/s)

Frequency (MHz)

=45°=30°, 60° =0°, 90°, 180° =120°, 150°

ijkl im jn kp lq mnpqc c (3)

ij is the cosine of the angle between the two coordinate systems. The particle displacement of surface acoustic waves is assumed to be the following form:

3 1 1 2 2 3 3exp( )exp[ ( )]j j lu ikbx ik l x x l x vt

3ˆˆ

T

(4)The b gives the variation with depth of the amplitude and phase of the partial wave. The gives the relative amplitudes of the different components of partial wave. To satisfy boundary conditions, the elastic waves in the film and substrate must be written by linear combinations with weight factors Cm and Cn:In the substrate:

In the film:

( ) ( )3 1 1 2 2 3 3exp( )exp[ ( )]m m

j m jm

u C ikb x ik l x l x l x vt

( ) ( )

3 1 1 2 2 3 3ˆ exp( )exp[ ( )]n nj n j

nu C ikb x ik l x l x l x vt

For this transversely isotropic film/ cubic Si substrate structure, the 9 boundary conditions include continuity of the displacement ( ) and the

stress ( ) at the interface (x

1 1 2 2 3ˆ ˆ, , u u u u u u

31 31 32 32 33 33ˆ ˆ, , T T T T T T

3=0), and vanishing of the stress: ( T T ) at the free surface (x31 32 33

ˆ ˆ ˆ0, 0, 0 3=hf),

where 3 3i

j jill

uT c x

A 9×9 matrix is built by these boundary conditions. Solving the secular equation of this matrix gives the solution of , then the dispersion relation according to

k/(2 )f kv .

3. Numerical examples The SAW waves are assumed to propagate along the

[110] direction whose direction cosines are l1= 22 ,

l2= 22 and l3=0.

The density and elastic constants of Si (100) substrate are s =2.330 g/cm3, c11 =165.7 GPa, c12=63.9 GPa, c44=79.6 GPa, respectively. The density ( f) and thickness (hf) of the film are set to be 0.85 g/cm3 and 500nm respectively. The independent elastic constants of transversely isotropic film are decided by the Young’s modulus (E), Poisson’s ratio ( ) in transversely isotropic plane, and Young’ modulus (E ), Poisson’s ratio ( ) and shear

modulus (G ) along the direction of pores according to: 2 2

11 12 132

2

22 442

1 ' '(1 ) 'ˆ ˆ ˆ, ,' '

1ˆ ˆ, '

c c cEE EEE

c c GE

(5)

Here2

2(1 )(1 2 ' )

E EThe dispersive curves when the angle changes are shown in Fig.2. It is obvious that, with the increase of frequency, the SAW velocity reduces continually. Comparing with the relationship between the propagating direction and the nano-pores’ direction illustrated in the Fig.3, it is found that the curvature changes with the angle between the propagating direction and the pores’ direction. And the bigger the angle is, the larger the curvatures are.

Figure 2. The dispersive curves when SAW propagating along the [110] direction, changes from 0ºto 180º, E=8GPa , E´=16GPa. .

x 1(0°)

x 2(90°) Propagating direction: [110]

120°

30°

60°

150°

180°

Figure 3.The relationship between the propagating direction and the nano-pores’ direction

When the angle =90º, the three dimension (3D) dispersive curves and their projections on the f-v plane

with different Young’s moduli E and E’ of the porous film are shown in the Figs.4 and 5 respectively.

Figure 4.The dispersive curves and their projections on the frequency-SAW velocity plane along [110] direction, =90º, Young’s modulus E changes from 4GPa to 12GPa

and Young’s modulus E´ maintain 16GPa.

Figure 5.The dispersive curves and their projections on the frequency-SAW velocity plane along [110] direction, =90º, Young’s modulus E´ changes from 12GPa to

20GPa, Young’s modulus E maintain 8GPa.

Fig.4 and Fig.5 show that the curvatures of the dispersive curves will become larger in the high frequency area when Young’s modulus E or E’ decreases. And each dispersive curve includes the information of a

pair of E and E’. These dispersive characters indicate that both the Young’s moduli E and E’ can be measure in experiment when the SAW waves propagating along [110] direction. And raising the experimental detecting frequency higher than 300MHz will remarkably improve the measurement accuracy.

E=12GPa

E=10GPa 4. Summary The dispersive characters of SAW waves propagating in the layered structure of transversely isotropic porous film deposited on Si(100) substrate has been expressed.

E=8GPa

From numerical examples, the effects of the nano-pores’ direction and Young’s moduli E and E´ on the dispersive curves were found. Numerical examples also demonstrate that both the Young’s modulus E and E’ can be measured. These model and results can be used for the mechanical properties determination of porous low-k film by SAW, such as periodic porous materials.

E=6GPa

E=4GPa

Acknowledgments The Authors are grateful to the support by the National Natural Science Foundation of China (60406003) and the SRF for ROCS, State Education Ministry of China.

References

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