Modeling of Moisture Diffusion and Moisture-Induced Stresses in Semiconductor and MEMS Packages

Abstract

This chapter describes recent progresses in modeling of moisture diffusion and moisture-induced stresses in semiconductor and MEMS packages. It addresses both fundamental and practical aspects including (1) the theoretical background of moisture diffusion and hygroscopic swelling, (2) modeling schemes to solve moisture diffusion and combined hygro-thermo-mechanical stresses, and (3) validation of those modeling schemes. Three specific modeling schemes are presented: (v1) thermal–moisture analogies that enable a heat transfer module in commercial finite element analysis software to solve moisture diffusion equations, (2) an effective-volume scheme that can effectively handle moisture diffusion into and out of a cavity surrounded by polymeric materials, and (3) hygro-thermo-mechanical stress analysis scheme that combines hygroscopic swelling-induced stresses with conventional thermo-mechanical stresses through a sequential procedure.

Appendix

8.1.1 A.1 FDM Schemes for Mass Diffusion Equations

8.1.1.1 A.1.1 Anisothermal 1-D Problem

For a 1-D mass diffusion problem, the diffusion equation of equation (8.9) can be reduced to

$$\phi \frac{{\partial S}}{{\partial t}} + S\frac{{\partial \phi }}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {D\frac{{\partial \left( {S\phi } \right)}}{{\partial x}}} \right].$$

((8.54))

The explicit finite difference form of equation (8.54) can be expressed as

$$\begin{array}{l} \displaystyle S_x^t \frac{{\phi _x^{t + \Delta t} - \phi _x^t }}{{\Delta t}} + \phi _x^t \frac{{S_x^{t + \Delta t} - S_x^t }}{{\Delta t}} \\ \displaystyle\quad = \frac{{\left( {\frac{{D_{x + \Delta x}^t + D_x^t }}{2}} \right)\frac{{S_{x + \Delta x}^t \phi _{x + \Delta x}^t - S_x^t \phi _x^t }}{{\Delta x}} - \left( {\frac{{D_x^t + D_{x - \Delta x}^t }}{2}} \right)\frac{{S_x^t \phi _x^t - S_{x - \Delta x}^t \phi _{x - \Delta x}^t }}{{\Delta x}}}}{{\Delta x}} \\ \displaystyle\quad = \frac{{\left( {D_x^t + D_{x - \Delta x}^t } \right)S_{x - \Delta x}^t \phi _{x - \Delta x}^t - \left( {D_{x + \Delta x}^t + 2D_x^t + D_{x - \Delta x}^t } \right)S_x^t \phi _x^t + \left( {D_{x + \Delta x}^t + D_x^t } \right)S_{x + \Delta x}^t \phi _{x + \Delta x}^t }}{{2\left( {\Delta x} \right)^2 }} \\ \end{array}.$$

((8.55))

The superscript t and the subscript x denote the time and spatial dependency of φ, respectively. The values of S ^t, $S^{t + \Delta t}$, and D ^t can be calculated directly from equation (8.4). For an isothermal problem, equation (8.55) is reduced to

$$\frac{{\phi _x^{t + \Delta t} - \phi _x^t }}{{\Delta t}} = \frac{{D\phi _{x - \Delta x}^t - 2D\phi _x^t + D\phi _{x + \Delta x}^t }}{{\left( {\Delta x} \right)^2 }}.$$

((8.56))

For multi-material systems, the normalized concentration at the interface must satisfy a certain condition to ensure a continuous mass flux. Assuming that the materials I and II are located on the left and right sides of the interface, respectively, the continuity requires

$$D_{\textrm{I}} S_{\textrm{I}} \left. {\frac{{\partial \phi }}{{\partial x}}} \right|_{x_i ^ - } = D_{{\textrm{II}}} S_{{\textrm{II}}} \left. {\frac{{\partial \phi }}{{\partial x}}} \right|_{x_i ^ + } $$

((8.57))

where $x_i-$ and $x_i+$ denote the left side and right side at the interface x _i . The discretization of equation (8.57) yields

$$D_{\textrm{I}}^t S_{\textrm{I}}^t \frac{{\phi _{x_i }^t - \phi _{x_i - \Delta x}^t }}{{\Delta x}} = D_{{\textrm{II}}}^t S_{{\textrm{II}}}^t \frac{{\phi _{x_i + \Delta x}^t - \phi _{x_i }^t }}{{\Delta x}}.$$

((8.58))

Strictly speaking, equation (8.57) is valid for an infinitesimal interface volume whose moisture content is negligible. Once the analysis domain is discretized, however, the interface volume takes finite values and retains a certain amount of moisture. A more rigorous formulation for the interface can thus be derived by considering the moisture accumulation at the interface volume (interface length, dx, after normalized by the area in the 1-D case). It is mathematically expressed as

$$S_{\textrm{I}} \frac{{\partial \phi }}{{\partial t}}\left( {\frac{{dx}}{2}} \right) + S_{{\textrm{II}}} \frac{{\partial \phi }}{{\partial t}}\left( {\frac{{dx}}{2}} \right) = - J^ + + J^ - = D_{{\textrm{II}}} S_{{\textrm{II}}} \left. {\frac{{\partial \phi }}{{\partial x}}} \right|_{x_i + } - D_{\textrm{I}} S_{\textrm{I}} \left. {\frac{{\partial \phi }}{{\partial x}}} \right|_{x_i - }.$$

((8.59))

The corresponding finite difference form is obtained as

$$\frac{{\Delta x}}{2}\left( {S_{\textrm{I}} + S_{{\textrm{II}}} } \right)\frac{{\phi _{x_i }^{t + \Delta t} - \phi _{x_i }^t }}{{\Delta t}} = D_{{\textrm{II}}}^t S_{{\textrm{II}}}^t \frac{{\phi _{x_i + \Delta x}^t - \phi _{x_i }^t }}{{\Delta t}} - D_{\textrm{I}}^t S_{\textrm{I}}^t \frac{{\phi _{x_i }^t - \phi _{x_i - \Delta x}^t }}{{\Delta x}}$$

((8.60))

When node spacing is sufficiently small, equations (8.58) and (8.60) yield practically the same result.

It would be worth noting that the implicit scheme can be obtained simply by replacing all superscript t in the right-hand sides of equations (8.55) and (8.60) with t + Δt.

8.1.1.2 A.1.2 Isothermal Axisymmetric Problem

The explicit finite difference form of equation (8.38) can be expressed as

$$\frac{{\phi _r^{t + \Delta t} - \phi _r^t }}{{\Delta t}} = \frac{{D\left(\phi _{r + \Delta r}^t - 2\phi _r^t + \phi _{r - \Delta r}^t \right)}}{{\left( {\Delta r} \right)^2 }} + \frac{D}{r}\frac{{\phi _{r + \Delta r}^t - \phi _{r - \Delta r}^t }}{{2\left( {\Delta r} \right)}}.$$

((8.61))

Mass continuity at the volume about the center node $(r = 0 \sim \Delta r/2)$ yields

$$S\frac{{\partial \phi }}{{\partial t}}dV = - JA = DSA\frac{{\partial \phi }}{{\partial r}}.$$

((8.62))

In the finite difference form,

$$\frac{{\phi _0^{t + \Delta t} - \phi _0^t }}{{\Delta t}} \cdot \left( {\frac{{\Delta r}}{2}} \right) = D\frac{{\phi _{\Delta r}^{t} - \phi _{0}^{t} }}{{\Delta r}}.$$

((8.63))

Since the area is not constant along the r axis, the mass continuity at the interface (r = r _i ) depicted in equation (8.59) should be modified as

$$S_{\textrm{I}} \frac{{\partial \phi }}{{\partial t}}{\textrm{d}}V^ - + S_{{\textrm{II}}} \frac{{\partial \phi }}{{\partial t}}{\textrm{d}}V^ + = - J^ + A^ + + J^ - A^ - = D_{{\textrm{II}}} S_{{\textrm{II}}} A^ + \left. {\frac{{\partial \phi }}{{\partial r}}} \right|_{r_{\textrm{i}} + } - D_{\textrm{I}} S_{\textrm{I}} A^ - \left. {\frac{{\partial \phi }}{{\partial r}}} \right|_{r_{\textrm{i}} - }.$$

((8.64))

The finite difference form is then obtained as

$$\begin{array}{l} S_{\textrm{I}} \frac{{\phi _{r_{\textit{i}} }^{t + \Delta t} - \phi _{r_{\textit{i}} }^t }}{{\Delta t}} \cdot \pi \left[ {r_{\textit{i}} ^2 - \left( {r_{\textit{i}} - \frac{{\Delta r}}{2}} \right)^2 } \right] + S_{{\textrm{II}}} \frac{{\phi _{r_{\textit{i}} }^{t + \Delta t} - \phi _{r_{\textit{i}} }^t }}{{\Delta t}} \cdot \pi \left[ {\left( {r_{\textit{i}} + \frac{{\Delta r}}{2}} \right)^2 - r_{\textit{i}} ^2 } \right] \\ \quad = D_{{\textrm{II}}} S_{{\textrm{II}}} \cdot 2\pi \left( {r_{\textit{i}} + \frac{{\Delta r}}{2}} \right) \cdot \frac{{\phi _{r_{\textit{i}} + \Delta r}^{t} - \phi _{r_{\textit{i}} }^{t} }}{{\Delta r}} - D_{\textrm{I}} S_{\textrm{I}} \cdot 2\pi \left( {r_{\textit{i}} - \frac{{\Delta r}}{2}} \right) \cdot \frac{{\phi _{r_{\textit{i}} }^{t} - \phi _{r_{\textit{i}} - \Delta r}^{t} }}{{\Delta r}} \\ \end{array}.$$

((8.65))

Re-arranging equation (8.65) yields

$$\begin{array}{l} \frac{{\phi _{r_{\textrm{i}} }^{t + \Delta t} - \phi _{r_{\textrm{i}} }^t }}{{\Delta t}}\left[ {S_{\textrm{I}} \left( {r_{\textrm{i}} \Delta r - \frac{{\Delta r^2 }}{4}} \right) + S_{{\textrm{II}}} \left( {r_{\textrm{i}} \Delta r + \frac{{\Delta r^2 }}{4}} \right)} \right] \\ \quad\;\;\; = D_{{\textrm{II}}} S_{{\textrm{II}}} \left( {2r_{\textrm{i}} + \Delta r} \right)\frac{{\phi _{r_{\textrm{i}} + \Delta r}^{t}} - \phi_{r_{\textrm{i}}}^{t}}{{\Delta r}} - D_{\textrm{I}} S_{\textrm{I}} \left( {2r_{\textrm{i}} - \Delta r} \right)\frac{{\phi_{r_{\textrm{i}}}^{t} - \phi_{r_{\textrm{i}} - \Delta r}^{t} }}{{\Delta r}} \\ \end{array}.$$

((8.66))

8.1.2 A.2 ANSYS Input Templates for the Advance Analogy

8.1.2.1 A.2.1 Transient Case $\left( {\nabla T = 0{\textrm{ but }}\dot T \ne 0} \right)$

The macro listed below is the portion for the setup of material properties and solution steps included in the modeling program of the case study. Since the system is spatially isothermal, the properties of each material are homogenous. Therefore, they can be defined as constants at each solution step:! Defining parameters M1=19.74 ! M of material I M2=6.58 ! M of material II R_gas=8.3145 ! Gas constant D01=5e-3 ! D0 of material I S01=6e-10 ! S0 of material I D02=4e-3 ! D0 of material II S02=2e-10e ! S0 of material II Ed1=-5e4e ! D1 = D01 * exp(Ed1/R_gas/T) Es1=4e4 ! S1 = S01 * exp(Es1/R_gas/T) Ed2=-5e4 ! D2 = D02 * exp(Ed2/R_gas/T) Es2=4e4 ! S2 = S02 * exp(Es2/R_gas/T) ! RH = H0*exp(Eh/R_gas/T) for fixed partial vapor pressure (P at 25C/100%RH) ! H0 = S0*Pv/M in Eq. (16), where Pv = 3207 Pa in this case H0=9.742e-8 ! Note: In this program Evp was set 4e4 for consistency with Es ! Its actual value = 4.15e4 Eh=Es1 ! Will be used to calculate phi_BC(T) for fixed Pv T0=298 ! Initial temperature dt=9 ! Time step ! /prep7 et,1,55 ! 2D thermal element c,1,M1 ! c = M ("Modified" solubility) c,2,M2 dens,1,1 ! Densities = 1 dens,2,1 !............................................................ ! Solid modeling, meshing, and grouping for IC/BC setup !............................................................ /sol antype,4 ! Initial condition setup. Node components pre-defined ! Phi of exterior surface nodes -> RH of environment ic,exterior,temp,H0*exp(Eh/T0) ! Phi of interior nodes -> zero (initially fully dried) ic,interior,temp,0 ! *do,ii,1,400 time,ii*dt ! System temperature at the current solution step T1=T0+(1/60)*ii*dt ! Calculation of boundary RH at the current solution step d,exterior,temp,H0*exp(Eh/R_gas/T1) ! Update of conductivity of the analogy kxx,1,D01*exp(Ed1/R_gas/T1)*M1 ! k = DM kxx,2,D02*exp(Ed2/R_gas/T1)*M2 nsub,1,1,1 solv *enddo

8.1.2.2 A.2.2 Anisothermal Case $\left( {\nabla T \ne 0{\textrm{ and }}\dot T \ne 0} \right)$

A truly anisothermal loading gives rise to much more complexity in the analogy model. An example of modeling and solving process is illustrated in Fig. 8.18. The key feature is re-assignment of material properties to each element after sorting all elements by materials and temperatures. In ANSYS, the element table feature [57] is useful when this process is to be implemented: ! key part for material property definition and update ! All parameters used in this macro are identical to those in A.1 /prep7 et,1,55 ! 2D thermal element ! Pre-definition of material properties as a function of temperature ! In this program, they are defined from 1C to 100C with one degree step *do,ii,1,100 dens,ii,1 c,ii,M1 kxx,ii,D01*exp(Ed1/(273+ii))*M1 *enddo *do,ii,101,200 dens,ii,1 c,ii,M2 kxx,ii,D02*exp(Ed2/(173+ii))*M2 *enddo !........................................................... ! Solid modeling, meshing, and grouping for IC/BC setup !........................................................... ! Total element count = 8000 (4000 for each material) ! Node numbers must be controlled material-by-material (using NUMCMP) n_total=8000 *dim,t_out,array,n_total /sol antype,4 ! *do,ii,1,400 /post1 ! Reading temperature distribution data from aniso_ht.rth (result file) file,aniso_ht,rth set,ii etable,t_res,temp ! Element table of temperature distribution ! Storing temperature data into array *do,jj,1,n_total *get,tt1,etab,1,elem,jj t_out(jj)=tt1 *enddo *if,ii,gt,1,then file,,rth *endif ! ! Re-defining element-to-element material numbers /prep7 *do,jj,1,n_total/2 emodif,jj,mat,nint(t_out(jj))-273 emodif,jj+4000,mat,nint(t_out(jj+4000))-173 *enddo /sol *if,ii,eq,1,then antype,trans ! Initial condition setup. Node components pre-defined ! Phi of exterior surface nodes -> RH of environment ic,exterior,temp,H0*exp(Eh/T0) ! Phi of interior nodes -> zero (initially fully dried) ic,interior,temp,0 *else antype, ,rest *endif ! Environment temperature was assumed to be controlled as in A.1 ! Note: Temperature inside polymers was input from the heat transfer T1=T0+(1/60)*ii*dt time,ii*dt ! Instead of ambient temperature, actual surface temperature ! has to used to calculate BC because of temperature difference ! between environment and solid surface (n_bc: node number at surface) ! If temperature is not uniform over boundary surfaces, ! BC should be input node-to-node d,exterior,temp,t_out(n_bc) nsub,1,1,1 solv *enddo

Fig. 8.18

General solution process of the anisothermal problem using ANSYS

From a practical point of view, the spatially anisothermal complexity is not needed for most of the problems since spatial temperature gradients during heating or cooling can be ignored (usually less than 1°C). In addition, the time step can be controlled in an automatic manner by implementing a customized macro for time step adjustment while taking the rate of concentration change at each step into consideration.

8.1.3 A.3 Templates for the Combined Analysis

8.1.3.1 A.3.1 Program to Change the Record Key

This program changes the record key of the nodal concentration (221) obtained from a diffusion analysis using ABAQUS so that the concentration has the record key of the temperature (201). The detailed explanation about subroutines and variables used in this program is available in [61]: SUBROUTINE diff2temp INCLUDE 'aba_param.inc' DIMENSION ARRAY(513), JRRAY(NPRECD,513) EQUIVALENCE (ARRAY(1), JRRAY(1,1)) PARAMETER (MXUNIT=21) INTEGER LRUNIT(2,MXUNIT),LUNIT(10) CHARACTER FNAME*80 DATA LUNIT/1,5,6,7,9,11,12,13,20,28/ NRU = 1 LOUTF = 2 LRUNIT(1,1)=8 LRUNIT(2,1)=2 WRITE(6,60) 60 FORMAT(1X,'Enter the name of the input file(s) (w/o extension):') READ(5,'(A)') FNAME CALL INITPF (FNAME, NRU, LRUNIT, LOUTF) JUNIT=8 KEYPRV = 0 DO 100 INRU = 1, NRU JUNIT = LRUNIT(1,INRU) CALL DBRNU (JUNIT) I2001 = 0 DO 80 IXX2 = 1, 100 DO 80 IXX = 1, 99999 CALL DBFILE(0,ARRAY,JRCD) C WRITE(6,*) 'KEY/RECORD LENGTH = ', JRRAY(1,2), JRRAY(1,1) IF (JRCD .NE. 0 .AND. KEYPRV .EQ. 2001) THEN WRITE(6,*) 'END OF FILE #', INRU CLOSE (JUNIT) GOTO 100 ELSE IF (JRCD .NE. 0) THEN WRITE(6,*) 'ERROR READING FILE #', INRU CLOSE (JUNIT) GOTO 110 ENDIF LWRITE=1 IF (INRU.GT.1) THEN IF (JRRAY(1,2).GE.1900 .AND. JRRAY(1,2).LE.1909) LWRITE=0 IF (JRRAY(1,2).GE.1912 .AND. JRRAY(1,2).LT.1922) LWRITE=0 IF (JRRAY(1,2) .EQ. 2001 .AND. I2001 .EQ. 0) THEN I2001 = 1 LWRITE = 0 ENDIF ENDIF IF (INRU .EQ. 1 .OR. LWRITE .EQ. 1) THEN KEY=JRRAY(1,2) IF((KEY.EQ.1900).OR.(KEY.EQ.1901).OR. (KEY.EQ.1902).OR. 1 (KEY.EQ.1910).OR.(KEY.EQ.1911).OR. (KEY.EQ.1921).OR. 2 (KEY.EQ.1922).OR.(KEY.EQ.1980).OR. (KEY.EQ.2000).OR. 3 (KEY.EQ.2001).OR.(KEY.EQ.1)) THEN CALL DBFILW(1,ARRAY,JRCD) IF (JRCD .NE. 0) THEN WRITE(6,*) 'ERROR WRITING FILE' CLOSE (JUNIT) GOTO 110 ENDIF C C If a current RECORD KEY is 221, the key is changed into 201 and the associated data C are written to an output file. C ELSE IF( (KEY.EQ.221)) THEN JRRAY(1,2)=201 CALL DBFILW(1,ARRAY,JRCD) IF (JRCD .NE. 0) THEN WRITE(6,*) 'ERROR WRITING FILE' CLOSE (JUNIT) GOTO 110 ENDIF ENDIF ENDIF KEYPRV = JRRAY(1,2) 80 CONTINUE 100 CONTINUE 110 CONTINUE CLOSE(9) C RETURN END

8.1.3.2 A.3.2 Example Program for UEXPAN

The macro listed below combines hygroscopic swelling and thermal expansion to define a total volumetric strain in ABAQUS: SUBROUTINE UEXPAN(EXPAN,DEXPANDT,TEMP,TIME,DTIME,PREDEF,DPRED, $ STATEV,CMNAME,NSTATV) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME C DIMENSION EXPAN(*),DEXPANDT(*),TEMP(2),TIME(2),PREDEF(*), $ DPRED(*),STATEV(NSTATV) C EXPAN(1) : VOLUMETRIC STRAIN C TEMP(1) : TEMPERATURE AT T+DT C TEMP(2) : TEMPERATURE INCREMENT C PREDEF(1): FIELD VARIABLE AT T+DT (I.E., NORMALIZED CONCENTRATION ) C DPRED(1) : FIELD VARIABLE INCREMENT ALPHA = 1.0D-05 BETA = 1.D-1 SOLU = 1 EXPAN(1) = ALPHA*TEMP(2)+ SOLU*BETA*PREDEF(1) C RETURN END

8.1.3.3 A.3.3 ANSYS Input Template for Hygro-Thermal Loading

This example program illustrates (1) how to define CHS and (2) how to read the distributions of moisture concentration and temperature from separate external result files: ... ! DEFINE MATERIAL PROPERTIES INCLUDING SWELLING COEFFICIENT (N-MM-SEC-K) EX,1,3E3 NUXY,1,0.4 ALPX,1,30E-6 TB,SWELL,1 ! C72=10 FOR ACTIVATION OF USERSW ! EPS=C67*(FLUENCE)^C68 WHERE C67=SWELLING COEFFICIENT AND C68=1 TBDATA,72,10 TBDATA,67,0.1,1 ... ... ! READ MOISTURE CONCENTRATION FIELD FROM RESULT FILE (HYGRO.RTH) ! ELEMENT NUMBER MUST BE COMPRESSED (NUMCMP) IN ADVANCE ALLS LDREAD,TEMP,NO_STEP,NO_SUBSTEP,HYGRO,RTH ! CONVERT DUMPED TEMPERATURE FIELD TO FLUENCE FIELD *DO,II,1,NO_ELEM *GET,BF_MOIST,NODE,II,NTEMP BF,II,FLUE,BF_MOIST *ENDDO ! READ ACTUAL TEMPERATURE FIELD FROM RESULT FILE (TEMPERATURE.RTH) LDREAD,TEMP,NO_STEP,NO_SUBSTEP,TEMPERATURE,RTH ...