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Residual strains of Pb thin films deposited onto Si substrates
RESIDUAL
STRAINS OF Pb THIN FILMS ONTO Si SUBSTRATES
DEPOSITED
~NIASAIXORIMURAKAMI IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Receiced 28 March 1977; in raisedform 6 July 1977)
investigation has been carried out of the strain and strain-relaxation in (111) oriented Pb films deposited onto oxidized Si substrates at 300 K and then cooled down to temperatures as low as 4.2 K. The strain in the Pb caused by differences in thermal expansion of the Pb and Si was measured using X-ray diffraction techniques. For film thicknesses of _ 1400 A, the magnitude of the observed strain at 4.2 K was found to approximate the strain value calculated from the difference in thermal expansion coefficients using the biaxial strain model. For greater film thicknesses, the value of the strain measured immediately after cooling was much lower, indicating that significant strainrelaxation occurred during the cooling process. It is probable that most of the strain was relaxed by a dislocation-slip mechanism. During isothermal annealing above 50 K a second, slow relaxation process was observed. Nonuniformity of the strain normal to the film-surface was found by analyzing the asymmetric X-ray line broadening that occurred during cooling. A strong dependence of strain on grain orientation was observed, and interpreted in terms of the biaxial strain model.
Abstract-An
Resume-On a ttudit la deformation et la relaxation de la deformation de couches minces de Pb oricntees (11I). deposees sur des supports de Si oxydes a 300 K. et refroidies ensuite a des temperatures pouvant descendre jusqu’a 4.2 K. On a mesure par des techniques de diffraction de rayons X. la diformation du Pb produite par la difference des coefficients de dilatation thermique du Pb et du Si. Pour des couches d’epaisseur 14OOA environ, la deformation observee a 4.2 K etait de I’ordre de grandeur de la deformation calculee a partir de la difference des coefficients de dilatation thermique’ dans un mod&e de deformation biaxiale. Lorsque I’epaisseur des couches est plus grande, la deformation mesuree immidiatement aprts le refroidissement est beaucoup plus petite, ce qui montre qu’une importante relaxation de la deformation s’est produite au tours du refroidissement. La plus grande partie de In dtformation a vraisemblablement ete relaxee par un mecanisme de glissement de dislocations On a observe au tours de recuits isothermes au dessus de 50 K un second processus lent de relaxation. On a observe les heterogeniitis de la deformation perpendiculairement a la surface de la couche mince en analysant un elargissement asymetrique de la raie de rayons X au tours du refroidissement On a observe une influence importante de I’orientation du grain sur la deformation, et on l’a interpritee grace au modtle de la deformation biaxiale. Zusammenfassung--In (I 1 t)-orientierten Pb-Schichten, die bei 300 K auf oxidierte Si-Substrate aufgewachsen und dann bis zu 4,2 K abgekiihlt waren, wurde die Verspannung und deren Relaxation untersucht. Die von den Unterschieden in der thermischen Ausdehnung von Pb und Si herriihrenden Verspannungen wurden mittels Rontgenbeugungstechniken untersucht Bei Schichtdicken von - 1400 A entspricht die beobachtete Verspannung bei 12 K ungefahr derjenigen, die man mit einem zweiachsigen Spannungsmodell errechnet. Bei grogeren Schichtdicken war die unmittelbar nach dem Abktihlen gemessene Verspannung vie1 kleiner; das deutet auf eine betrachtliche Spannungsrelaxation wghrend des Abkiihlprozesses hin. Es ist sehr wahrscheinlich, daB der gri5Bte Teil der Verspannung tiber einen Versetzungsgleitmechanismus relaxiert wird Wahrend des isothermen Ausheilens wird oberhalb 50 K ein zweiter, langsamer Relaxationsprozess beobachtet. Die Analyse der wlhrend des Abkiihlens auftretenden Verbreiterung der asymmetrischen Riintgenlinien ergibt eine UngleichmMigkeit der Verspannung senkrecht zur SchichtoberGche. Eine starke Abhlngigkeit der Verspannung von der Kornorientierung wurde beobachtet und mit dem zweiachsigen Spannungsmodell gedeutet.
1.
INTRODUCXION
If a thin film is grown on a relatively thicker substrate at 300K and then cooled to a lower temperature T, and if the thermal expansion coefficient of the film differs from that of the substrate, a strain in a direction parallel to the film surface is introduced. The maximum value of the strain E,,_ introduced is, 300 K
(1) (rfilm - rrub) dT ir where qilm and rsub are thermal expansion coefficients of film and substrate respectively. Strain-relaxation phenomena in Pb thin films during thermal cycling have been investigated by several %Iax=
175
authors. The general conclusions from their studies are that at lower temperatures the strain is relaxed primarily by a nondiffusive process [I] such as dislocation slip or twinning as witnessed by the observations of the slip bands in Pb films [2,3]; at relatively high temperatures the strains are relaxed mainly by a diffusion controlled mechanism [4,5], Strain relaxation in Pb films was studied by Gangulee[4]. He found that the lattice parameter changes during the thermal cycling agreed well with calculated values obtained using a diffusional creep model [6.7-J. His results suggested that the dominant relaxation mechanism at temperatures above 120 K was diffusion creep.
Ii6
MURAKAMI:
STRAINS M DEPOSITED
The purposes of the present investigation were to use X-ray diffraction to measure nondestructively the strains, introduced upon cooling to temperatures down to 4.2 K in Pb films deposited onto oxidized Si substrates, and to study the relaxation mechanisms of these strains. A special effort was made to detect non-uniform strains along directions parallel or normal to the film surface. If the adhesion between the film substrate is excellent, more strains at the substrateinterface and less strains at the film surface are expected [S]. This strain non-uniformity in a direction normal to the film surface was analyzed from the measured asymmetric X-ray line broadening. In this analysis, the line broadening was simulated using the computer simulation method of Honska et al. [9, lo]. This simulation method was originally developed to obtain the diffused concentration protie of bimetallic couples. Application of this method to the present strain distribution analysis will be described in Section 2.2. Also, as Pb is elastically anisotropic, it is expected that the strain in a polycrystalline film gill vary for grains of differing orientation. eon-unifo~ity of strain parallel to the film surface is experimentally evidenced by the observation of hillocks after thermal cyclings [2,3,11]. In the present experiment the strains in grains with different crystallographic orientations were measured, and the measured values were compared with those calculated using the biaxial strain model. The theory of this model will be reviewed in Section 2.1. For Pb, the following are applicable: (a) it has a relatively large thermal expansion coefficient, resulting in measurably large lattice parameter changes for small temperature changes [12], (b) deposits of Pb on SiOt have a strong (111) fiber structure, which makes it possible to accurately measure the interplanar spacing normal to the lihn surface from (hbh) diffraction peaks, (c) Pb has a large CuKa X-ray absorption coefficient which enables one to analyze the strain distribution in thin films normal to the surface from asymmetric line broadening, and (d) the elastic constants of Pb are known over the desired temperature ranges [13]. 2. REVIEWS OF THEORIES
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to.the stresses by Eij =
sijkl
where S,r are the compliances which are calculated from the reported elastic constants. For an arbitrary (~~~ oriented crystal the relation (3a) also holds,
6ij = S’ijk@t where the transformed
=
V
-
Wdo,
W
compliance S;,%,,is given by
I2419 Si&[= ffi,Ujm UbUtp S-p
(4a)
The rotation matrix aetpis written by, a* =
cos 9 cos 6 sin 9 ( sin 8 sin 9
-sin 9
0 -sin 9 cos 8 1
cos 0 cos 9
sin t? cos 9
E’11 = S;l,,G;l
+
S;122f7;2
E)22 =
+
e;* EL3
= 3
S2211ail S;211~;l S;31141
+ +
S,ldit
(54
S222241+
S1212d12
PW
S;222&2
, h212ai2
(54
S;32242
+
+ +
S;31242.
(54
Also the maximum value of ci3 of an arbitrary oriented crystal is calculated from equation (5) using r;r which is equal to E_ of equation (1). 2.2 Diffraction intensity simulation method Based on an assumed non-uniform strain distribution normal to the film surface, an expected shape of line broadening for intensities diffracted by crystal
‘hkl
(2)
where d is a measured interplanar spacing normal to the f&n surface, and do is a strain-free value. As Pb has a strong anisotropy in elastic modulii, the calculation of strain l ij and stress oij from the measured eiJ must be carried out using a relationship given by Nye [14]. (The subscripts ‘Q”denote the directions of orthogonal axes: ‘11’ and ‘22’ are directions parallel to the film surface and ‘33’ is the one normal to the film surface). For (001) oriented crystal the strains are related
(4b)
where rotation angles 9 and 0 are shown in Fig. 1. The strains in thin films deposited onto substrates were studied theoretically by Vook and Witt [15, 161 for various crystals. They postulated that stresses acting on the film are applied at the interface between the f&n and the substrate. Consequently, all tensile and shear stress components in directions normal to the film surfaces are zero, which means that a;r = 0;2 = a& = 0. Because the film is assumed to adhere to the substrate, E;~ = 0. Using these assumptions adequate equations of equation (3b) to calculate unknown values of E; i( = e&), a\ i, and ai2 from the measured E;~, value are rewritten by,
The strain &a, normaf to a f&n surface is obtained from 43
(34
Gkl,
Fig. 1. Rotation-matrix
coordinate system.
MURAKAMI:
STRAIXS IN DEPOSITED
planes parallel to the substrate surface can be calcuIated by using the method of Houska et al. [9,10,17]. They determined the solute concentration profiles in diffusion couples by analyzing the shape of the X-ray diffraction lines. The line broadenings were due to the range of lattice parameters encountered by direct beam as it traversed regions of varying concentrations. In the present study, their method was used to obtain the strain distribution profile. A brief review of the theory is given below. A monotonically varying d-variation normal to the film surface is schematically shown in the left figure of Fig, 2 where xf is a film thickness. To pursue the analysis, it is convenient to subdivide mentally the total film thickness into thin slabs of constant thickness Allx.As illustrated in the right figure of Fig. 2, the X-ray intensity P, diffracted from a volume element AV, centered on x, will be given by. P, = ~,QmAVmA.llm,
where k, = 2,‘sin 8,. The parameter p is the linear absorption coefficient of Pb. It is apparent that a knowIedge of the d(x) profile suffices to synthesize an intensity band P,(d,) by combining equations (6)-(g) to yield. P, = floAoQ,g,&
k,exp[ - k,,&x,
- x,)].
(9)
The peak shapes P,,, co~~ponding to various assumed d(x) profiles can be calculated and compared with the P, obtained experimentally to infer the strain distribution normal to the surface. This procedure, ‘Houska’s Computer Simulation hfethod’, [9,10] will be adopted here to analyze the strain distribution in the thin film. 3. EXPERIlIENTAL
PROCEDURES
Oxidized (111) Si wafers of 0.2 mm thickness were
(6) used as substrates. They were precleaned in siru using
where I0 is the intensity of the incident beam, Qm is the reflectivity per unit volume and Aplnr is the absorption correction factor. The reflectivity Q, per unit volume is given by, Q, =
177
LEAD FILMS
!f$(’ lIfn”z?)P: exp(-2M,).
The parameter r, is the classical radius of the electron, ,l is the X-ray wavelength, V, is the volume of unit cell, F, is the structure facfor, exp( - 2M,) is the Debye-Waller factor, and 0, is the Bragg angle for element m. All parameters with suffix m vary with the distance x,,, of the diffracting element. In equation (6) the effective volume of material AV, is given by
an O2 glow discharge before film deposition. Lead source material of 99.9990/:,nominal purity was evaporated at 4 30 &‘s in a vacuum of lo- ’ Torr. The substrate was kept at room temperature. In order to minimize the non-uniformity of film thickness across the substrate the Pb source and the substrate were kept 50cm apart. Films with thicknesses from 0.0s5pm were prepared; the thicknesses were measured using a calibrated quartz crystal monitor placed near the substrate. 3.2 Heat treatments
Slight lattice parameter changes were observed in as-deposited films at room temperature that conAV,,, = %,A&, (7) tinued for a 4 h after deposition. In order to relax the strain introduced during the deposition, all where A, is the cross-sectional area of the section samples were kept in a vacuum desiccator at least exposed to an incident beam and is given by A~Js~~~~ for one day before further thermal treatments_ Ther(Ao: cross-sectional area of the incident beam), and mal treatments of specimens were carried out using g,,, is a factor which has been introduced to allow a liquid helium cooled X-ray diffraction stage. To imfor crystallite misorientations. Here, gn designates the prove thermal conductivity between the specimen and fraction of volume with crystal planes oriented within the specimen holder, a Cu grease (Cry-con grease supmeasurement range. A fraction of the incident and plied by Air Product and Chemicals) was used. The diffracted beam intensity is absorbed in reaching to environment of a specimen was evacuated to and leaving from the diffracting elements. The absorp4 x 10m6Torr before cooling by using 8 L’s iontion correction factor Ap,,, is given by pump. The typical cooling time down to liquid He temperature was approx~ately jmin when the transfer tube was sufficiently pre-cooled and the reheating time to 300 K was about 20 min. Temperatures were kept within + 1 K of the desired temperature by controlling flow rate of the coolant (liquid He or NJ and by adjusting the power to a heater incorporated in the sample mount. 3.3 X-ray measurements
Fig. 2. Cross-section of a thin tilm deposited on the substrate (right-hand side) and the corresponding interplanar spacing profile (left-hand side).
X-ray diffraction intensity measurements were made through a Beryllium window in the refrigerator housing. A computer-controhed GE XR-5 diffractometer was used with CuK* radiation (operated at 40 KV and 2OmA) and divergent (2”), receiving
175
MURAKAMI:
STRAINS TN DEPOSITED
f0.005’) and soiler slits. Angles ctfand x were adjusted to obtain the maximum intensity of the (333) reflection from the Si-substrate. Using a proportional counter the intensities were step-scanned for sufficient times (2 5 40s) to obtain enough measurable intensity in an interval of 26 = 0.005” or 0.02” for predetermined diffraction angle ranges; typical times required to take a Pb (333) reflection profile were 15 min. The data were directly punched out to computer cards to allow convenient subsequent analyses. The deconvolution of Kzi-K,Z peaks was carried out by modified Rachinger correction method [20] where the (2fla,-20cx(2) separation was continuously incremented with increasing diffraction angle within broad intensity band. The interplanar spacing was determined at the middle point in 28 position of the half width (A28) of K,, peak. In cases where an accurate measure of the line broadening due only to strains was required (as will be seen in Fig. 5b), analysis was carried out by Stokes’ correction method [2l] using the peak profile at room temperature as an instrumental line broadening. Dislocation densities were determined from (111)and (222) difraction-peaks using Warren and Averbach’s method [22]. This method was originally developed for poly~stalline samples and recently was shown to be applicable to thin films by Gangulee [23] and Sen et al. [24]. For Houska’s computer simulation analysis [9, lo] the linearly averaged Gaussian and Cauchy’s distribution [25] was used as an i~t~mental broadening. 4. EXPERIMEhTAL RESULTS AND DIScussIONS 4.1 Preliminary results (a) Stability of liquid helium refrigerator. An excellent mechanical stability of the refrigerator during heat-treatments was confirmed by the laser reflection and by X-ray diffraction technique before and after the refrigerator was set up on the X-ray diffractometer. As an example, the X-ray diffraction results for Si are shown in Fig. 3: the (333) reflection from the Si-substrate on which the 5 pm-Pb was deposited was measured at each temperature ranging between 300 and 4.2 K. Experimental results are shown by open circles in Fig. 3(a), the closed circles are the values calculated using the reported thermal expansion coefficients [12]. An agreement between the calculated and measured values is excellent, the maximum deviation of d is 0.00018 A. Also, the negative thermal expansion coefficients below about 120K were confirmed by the present experiment. In Fig. 3(b) the half-width values for the (333) reflection are shown; the values remain constant throughout the temperature range studied, which indicates that the Si-substrate was not significantfy strained by the SiGl or the Pb film. (b) Fiber structure of Pb f;lm. Pb thin films have the strong (Ill) fiber structures if deposited onto oxidized Si wafers. Info~atio~ about the fiber structure
LEAD FILMS I
3
I
(al DW1)
I
I
I
2 b
0 EXPERIMENT . CALCULATION
2 a ki
3,13200
.z a
I 150
8 200
I 250
300
TEWERATURE ?K,
’ (b)A2@ 0.07 ‘-
j$ 0.06 :
I 100
so
0
--o y =
0
0.05 rj
I 0
I
50
I
100
I
I
150 200
I
250
1
300
TEMPERATURE PK)
Fig. 3. Interplanar spacings (a) and X-ray peak half-widths (b) for the Si substrate, as measured at various
temperatures.
in films of various thicknesses were obtained by measuring different X-ray (hkl) reflections. It was found that the ratios of peak intensities of 1(311)/1(222) were 0.03 for 0.1 pm thick films and 0.002 for I pm thick film. It is seen that the (111) fiber structure monotonic~ly improves for thicker films. However, even for the 0.1 pm thick film, the film had the strong (111) fiber structure as judged by the ratio of 1(311)/1(222) of the powder sample which is 3.6. 4.2 Strains nnd stresses at 77 or 4.2 K Strains E& normal to the film-surfaces were measured at 77 or 4.2 K for films with various thicknesses. The (333) diffraction peaks of the films with various thicknesses of 0.1 _ 5.0~ which were cooled down to and measured at 77 K are shown in Fig. 4. The dashed curves represent the K,, and K,a deconvoluted peaks. The interplanar spacings d and angles (628) which subtend the half-maximum intensities were obtained from the deconvoluted &i peaks. In Fig. 5(a) the (111) interplanar spacings d are plotted versus film thickness; the points are the d-values measured immediately after cooling; the dashed line is the strain-free (or bulk) d-value calculated using the reported thermal expansion coef?% cients [12]. The arrows indicate the changes in d observed after one day of isothermal annealing, which means strains were relaxing at these temperatures over a period of one day. (This secondary strainrelaxation will be described elsewhere). It is noted that smaller d-values are observed for thinner f&as. For the 5 pm thick film the d-value almost reaches at the calculated strain-free value. The peak widths A26 are plotted in Fig. 5(b); A26 values are also
STRAISS
MuRAK.4blI:
IN DEPOSITED
o,5
179
LEAD FILMS
’
1\n
_2%___---__ l
-x___-
(al
T=77’K
\ I
\
!
0
I
I
I
I
0.2
0.4
0.6
0.8
FILM
DiFFRACTION
ANGLE
THICKNESS
Xf
I 1.0
(pm)
28tDEG)
Fig. 4. (333) diffraction peaks of Pb films with various thicknesses measured at 77 K.
observed to change with annealing time as shown by arrows. A peak at x/ = 0.5 pm was observed in this plot, which will be interpreted in Section 4.4. Strains&, calculated from equation (2) were plotted versus film thickness in Figs. 6(a) and (b) at 77 and at 4.2 K respectively. Each dashed line indicates the theoretical maximum value (E&) of the strain at these temperatures. It is found that the films of approximately 0.1 pm thickness almost reach the maximum strain value at each temperature. For films thinner than 0.1 pm the strains drop off from the eLMIas seen in Fig. 6(b). This is believed to be caused by the discontinuity of films in such thinner films, since many small holes of about 0.5 pm diameter were observed
I
0
0.2
1
I
I
0.4
0.6
0.8
FILM
THICKNESS
I
Xf
1.0 (pm1
Fig. 6. Residual strains l‘s3 for Pb films of various thicknesses measured at 77 K (a) and 4.2 K (b) respectively.
in 0.1 pm thick film by the scanning electron microscopy. It is noted that the log E vs film thickness plots are linear at both temperatures in the thickness region of 0.1-1.0~. This behavior is described by, exp
(b)A28 Ta77.K
9
O.Oo
I
2
3
4
5
X, (pm1
FLM THICKNESS
Fig. 5. Interplanar spacings (a) and half-widths (b) for Pb films of various thicknesses measured at 77 K.
(10)
where do and d, and d-values at X, = x and x, = 0, respectively, and - l/rj is a proportionality constant. Residual stresses a;i were calculated from the measured strains c’~~according to the procedures described in Section 2.1 using equation (5). The results at 77 and 4.2 K are shown in Figs. 7(a) and (b), respectively. The data are plotted on the normal scales instead of semi-logarithm scales in order to compare with the other data. Stresses are found to have a very
180
~~~~MI:
STRAINS IN DEPOSITED LEAD FILMS
strong film thickness dependence; thinner films have more residual stresses. The maximum stress a;,, shown by the dashed lines in these figures, are calculated using the expected maximum strains e;,, of Pb on Si given in equation (1). The dashed curves reproduce Caswell er al.‘s data [2]. They prepared Pb films of various thicknesses by evaporation onto Ni-substrates and measured the residual stresses by a cantilever beam technique, immersing the measurement unit into liquid nitrogen or helium. At 77 K the agreement between the present results and their data is within experimental errors for films thinner than OSpm, but the stresses of the thicker films were smaller than Casweli et al’s resuhs. At 4.2 K there is a factor of two difference in the stresses. Good agreement of the stresses would be expected for films of thickness between 0.2 and O.Spm if the films were of similar structure and impurity level. In this thickness range, significant stress relaxation is occurring in the Pb films and the effect of having substrates with different thermal expansion coefficients would be unimportant. For thicker films the stress inferred from the X-ray me~urements would be lower than those determined by the cantilever beam technique because the film thickness is becoming comparable to the X-ray absorption length ( * 1 pm for CuKr radiation). For thinner films where stress relaxation is not occurring both z,ub,lrate differences and measurement method differences should result in larger stress values for the present work by a factor of about 1.3. The factor of two higher stresses observed by Caswell et al. at 4.2 K (Fig. 7b) is surprising in light of the agreement at 77 K and the antici) for their experipated lower value of (xpi, - CL,,,,~,.~+ ment. The reason for this difference is not understood. The primary mechanism of the strain~eI~ation observed at these temperatures for thicker films is considered. to be dislocation-slip. High densities of dislocation slip lines were observed on the film surfaces by multi-thermal cyclings of Pb films [2,3]. In addition, a dislocation slip model has been used successfully to explain the grain orientation dependence of strain relaxation in Pb films [26]. The dislocation density can be calculated using Warren and Averbath’s analysis [22] if the dislocations are uniformly distributed through the fihn. Reasonably symmetric (Ill) and (222) reflection peaks were obtained at 77 K (after one day) and at 4.2 K from which the disiocation densities were caIculated to be 0.7 x lO”/cm at 77 K and 1.8 x 10’O/cm” at 4.2K respectively. Similarly low dislocation densities were also obtained by Gangulee for thin films [23]. 4.3 Strain-relaxation
during isothermal annealing
Strain-relaxation during isothermal annealing was studied by following the d-changes. The films were cooled from 300 K to lower temperatures and isothermally annealed at these temperatures. In Fig. 8 the d-values normal to the (1 II) planes of Pb films of various film thicknesses are shown. The closed circles
_____-__
crmax _____
--
____
-___
40(a) T=77OK X---X
CASWEU
-
FILM THICKNESS
cl al
PRESENT RESULTS
X, (pm)
(b) T= 4.2*K
x---GGWEU PRESENT
flLM
THICKNESS
et al RESULTS
X, (pm,
Fig. 7. Residual stresses for Pb films of various thicknesses at 77 K (a) and 4.2 K (b) respectively. are the strain-free d-values of buik Pb wtrich were calculated using the reported thermal expansion coefficients of Pb. The closed squares are the calculated d-values using the biaxial strain model (Section 2) and assuming no strain relaxation occurred. The experimental values measured immediately after cooling are indicated by the specific symbols. The vertical arrows indicate the d-value changes that occur during isothermal annealing over a period of about one day. It is noted that for relatively thick films most of the
MURAKAMI:
STRAINS IN DEPOSITED
2.860 1
ric peak-shapes changed to symmetric shapes. with narrowing A20’ as shown by arrows. The endpoints of arrows indicate the final values of AZ at each temperature. The A/0 value for the specimen annealed at 260 K comes back to the value observed at 300 K which is presumed to be the strain-free value and is shown by a dashed line. Specimens annealed below 260 K never reach the strain-free value. It is seen that the amount of relaxation in A.28 during the isothermal annealing is maximum at 77 K. At 4.2 K no change in the A28’ was observed during the annealing, which is consistent with no change in the d-values at this temperature as seen in Fig. 8. These decreases in line broadening will be interpreted in terms of a non-uniform strain distribution normal to the film surface in Section 4.4.
2.850 z
1x1
LEAD FILMS
1
4.4 Non-uniform strain distributions
0
I
I
50
100
I
I
I
150 200
I
250
300
TEMPERATURE(OK) Fig. 8. Summary of the interplanar spacings for Pb films of various thickness measured at each temperature. strain was relaxed concurrently
with cooling. The kinetics of this primary relaxation process could not be studied by the present experimental technique, because it took at least 15 min to obtain the first data. Changes in the half-widths are also observed during isothermal annealings. Half-widths (A20’) at various temperatures obtained from the deconvoluted (333) K,, peaks are shown in Fig. 9. (Note A.20 are not corrected by Stokes’ method[21]). The 620’ values became larger with lowering temperatures and reach the maximum value at approximately 77 K. (It should be noted that no line broadenings were observed for Si peaks). During isothermal annealing the asymmetPb (333)
T
(a) Non-uniformity normal to film surface. Non formity of the strain normal to the film surface was analyzed from the measured X-ray line broadenings on the basis of the exponential strain distribution which was experimentally observed as illustrated in Fig. 6. The (333) K,,-diffraction profiles for the 1 ,umthick films measured shortly after cooling to 77 or 200 K, are shown in Fig. 10. The profile at T = 77 K has a noticeable asymmetric line broadening. These profiles were analyzed with Houska’s computer simulation technique (Section 2.1) using the following procedure: First, the trial values of the unknown parameters, I, do and dl in equation (10) were chosen to obtain a first iteration for the strain distribution. Based on this strain distribution the peak profiles were calculated by equation (?), and they were compared with the experimental values. Changing q, do, and d, values, iteration was continued until the mini‘mum rms. value of the difference in the calculated and measured intensities was obtained. Peak profiles thus calculated are shown by the dashed curves in Fig. 10. Excellent fits between the calculated and the measured intensities are observed at both temperatures. The strain distributions corresponding to the peak profiles obtained during the isothermal annealings were shown in Fig. 11. As expected, the strain distribution is widest immediately after cooling and (olT=77*K
(b)T*200”K
/
T
II
L
.___-_I
o.lot , 0
50
(
,
,
-I-,
,f
100 150 200 250 300 TEMPERATURE(OK1
Fig. 9. Half-widths of the 1 w thick Pb films measured at each temperature.
DIFFRIICTKJN
ANGLE
ZBVJEGI
DIFFRACTION
ANGLE
28 (DEG)
Fig. 10. (333) K,,-diffraction peak profiles of 1pm thick Pb films measured at 77 K (a) and 200 K (b) respectively.
152
MURAKAMI:
STRAINS IN DEPOSITED
0.0
(alTq77.K
% 2.850 t r-
7
-‘-t
SZmin
il
--0.2
i btt*2OO*K
2B440
0.0 a !z
___*__________________
I 0.2
I 04
/ 0.8
I 0.6
I.0
DISTANCE FROM SUBSTRATE INTERFACE (pm)
Fig. 11. Calculated interplanar spacings distributions normal to the film-surface for the 1 pm thick Pb films measured at 77 K (a) and 200 K (b) respectively.
it becomes narrower with the annealing time. The dashed lines show the calculated equilib~um d-values obtained using the thermai expansion coefficients of bulk Pb. The d-value indicated by the open circle on each curve represents the one calculated from the 28 position which corresponds to the middle point of the half-width (A2fY). Additional experimental evidence for the existence of the non-~ifo~ strain distribution normal to the film surface was obtained by the thickness dependence of A28 in Fig. 5(b), in which the maximum value of the A28 was observed for the 0.5pm thick film. This is explained nicely as follows: The interplanar spacing difference (Ad) between the surface and the substrate-interface is proportional to the peak width A28 of the diffracted intensities. For a thin fihn Ad is very small, thus a smaif A28 value is expected. For a thick film, although Ad is very large, the intensities diffracted from the volume elements close to the interface are very weak because of the strong absorption of intensities by the Pb film. The measured intensities are mainly from the volume elements close to the film-surface in which the effective Ad is small, thus producing a small A2@-value. For an inter-
LEAD FILMS
mediate thickness film, Ad is relatively large and the intensities from the volume elements at the interface are measurable, which produces a very large A28 value. This dependence of A28 on film thickness would not be expected if the Iine broadening was caused by strain variations in the plane of the film. Evidence for large amounts of strain relaxation in l.Opm thick films at the substrate interface upon cooling is seen in the simulated d-distributions of Fig. ii(a). The strain at the interface of the film. in the as-cooled state is 0.18%. This strain is about 3.2 times smaller than the maximum strain of O.jT<, which was obtained for the 0.1 pm thick film at 77 K (Fig. 6a). The difference in strains (0.4%) may be relaxed at the interface upon cooling and/or at the very early stages of the isothermaf annealing. It was confirmed by computer calculations that the intensities diffracted from the volume elements at the substrate interface of the 1.0~ thick film should be measurable and should be observable in the peak profiles. (b) Non-uniformity parallel to the film sur$ze. Nonuniformity of the strain parallel to the film surface was investigated by measuring the strains e& in the grains whose surface planes (hkl) are different. As thinner films do not have perfect fiber structures, the (hkl) reflection peaks other than (hhh) reflections are measurable by the present X-ray technique. In Table 1 the aia values measured at 4.2 K for 0.1 and 0.2 pm thick films are listed for grains with different crystal orientations. Since the (200) reflection peak partially overlaps with the reflection of the substrate, an accurate me~urem~t for this plane was not possible. The e& values were found to be strongly dependent on crystallographic orientation: the E& for the (311) plane is about two times larger than the & for the (111) plane. Using the biaxial strain model the theoretical values of the & were calculated for each crystal orientation. In this calculation the elastic constants of Cl1 = 5.554, Cz2 = 4.542, and C,, = 1.942 x IO” dyn/cm’ were used (131. The calculated values of the ratio E;,/E;~ are listed in the fourth column of Table 1. In the final column the values of ~5~ are given that were calculated using c’;“;’ = 0.706% as obtained from equation (1) for Pb on Si at 4.2 K. Reasonable agreements between the calculated and measured E; s values are seen. From this calculation
Table 1. Residual strains E;, normal to the film surface in the different grains whose (hkl) planes are parallel to the film-surface
$3 (“/,I 2oooA
tw
1OOOA
Gm
-0186
(220) (222) I:::; (333)
e3
f&/f; t
-0:66 --0.71 1.14
- 0.72 - 0.63 --0.13 1.18
-
-0.65
- 0.63
- 0.959
* Measurements
l
were not carried out.
1.635 1.094 0.959 1.065 1.457
(%I
(eI I = 0.?0570/,) - 1.15 -0.71 -0.68 --0.75 1.03 -0.68
MURAKAMI:
STRAINS IN DEPOSITED LEAD FILMS
result it is expected that the c’s3 for the (100) plane would be largest. 5. SUMMARY The experimental results obtained in the present investigation are summarized below: (1) Residual strains at low temperatures of Pb films deposited onto oxidized Si substrates display a strong thickness dependence. For a 0.1 pm-thick film, the strain at 4.2 K was found to reach the maximum value calculated for the thermal expansion coefficient difference between Pb and Si. This indicates that no strain-relaxation has occurred on cooling to 4.2 K. However, for 0.07 and 0.05 pm thick films relaxation was observed which may be associated with holes in the films. (2) Reasonably good agreement was obtained between values of residual stress a;r calculated from the measured residual strains and values of stress measured by Caswell et al. [Z]. (3) For thicker Pb films, most of the strains were relaxed by plastic deformation during cooling. The dislocation densities introduced by the deformation were 0.7 x 10’“/cm’ at 77 K and 1.8 x 10’“/cm2 at 4.2 K. (-I) Residual elastic strains li3 in the plane of the films were found to have a strong crystal orientation dependence, which agreed with theoretical results calculated using the biaxial strain model. The ratio of e’,&rr is maximum and minimum for grains whose planes are parallel to the film surface are (100) and (11 I), respectively. (5) Non-uniform strains normal to the surfaces of the films were observed by analyzing the measured X-ray diffraction profiles. The regions of the films near the substrate interface were found to be more strained than those near the film surfaces.
Acknowledgemenrs-The author would like to express his deep gratitude to C. J. Kircher, J. W. Matthews, P. Chaud-
133
hari and I. Ames for stimulating discussions and for manuscript reviewing, to J. An@ello and A. Segmuller for assistance with X-ray diffraction techniques. to S. K. Lahiri for many informative discussions on stresses in thin films, to P. S. Ho and F. M. D’heurle for discussion on diffusion in thin films and to V. Tom for sample preparation.
REFERENCES 1. M. F. Ashby, Acta Met. 20, 887 (1972). 2. H. L. Caswell, J. R. Priest and Y. Bado, J. appl. Phys. 34, 3261 (1963). 3. S. K. Lahiri, J. appl. Phys. 46, 2791 (1975). 4. A. Gangulee, Acta Met. 22, 177 (1974). 5. P. Chaudhari, J. appl. Phys. 45, 4339 (1974). 6. F. R. N. Nabarro, Proc. Conf on the Strengths ofSolids, p. 75. Physical Society, London (1948). 7. C. Herrine. J. aool. Phvs. 21. 137 (1950). 8. P. ChaudKari, l&f JI’Res. bev. 13, 197 (1969). 9. C. R. Houska, J. appl. Phys. 41, 69 (1970). 10. J. Unnam. J. A. Carnenter and C. R. Houska J. a&. .. Phys. 44, 1957 (197;.
II. S. K. Lahiri and 0. C. Wells. Aaol. . . Phvs. _ Lert. 15, 234 (1969). 12. D. E. Gray, American Institute of Physics Handbook. McGraw-Hill, New York (1972). 13. D. L. Waldorf and G. A. Alers. >. norrl. . . Phvs. _ 33. 3266 (1962). 14. J. F. Nye, Physicaf Properties of Crystals, p. 131. University Press. Oxford (1972). 15. R. W.-Vook and F. Witt, J.‘appl. Phys. 36, 2169 (1965). 16. F. Witt and R. W. Vook, J. appl. Phys. 39, 2773 (1968); 40, 709 (1969).
17. M. Murakami, D. deFontaine and J. Fodor. J. appl. Phys. 47, 2850, 2857 (1976). 18. G. B. Gibbs, Phil. Msg. 13, 589 (1966). 19. R. L. Coble, J. app[. Phys. 34, 1679 (1963). 20. W. A. Rachinger, J. Sci. Instrum. 25, 254 (194S). 21. A. R. Stokes, Proc. Phys. Sot. London 61. 382 (1948). 22. B. E. Warren and B. L. Averbach, J. ap&. Phys. 2i, 595 (1950). 23. A. Ganguiee, J. appl. Phys. 43, 867 (1972). 24. S. Sen, S. K. Halder and S. P. S. Gupta, J. Phys. D: Appl. Phys. 8, 1709 (1975). 25. B. E. Warren, X-ray Diflraction. Addison-Wesley, Reading, p. 258 (1969). 26. M. Murakami and P. Chaudhari. Thin Solid Films. 46. 109 (1977).
STRAINS OF Pb THIN FILMS ONTO Si SUBSTRATES
DEPOSITED
~NIASAIXORIMURAKAMI IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Receiced 28 March 1977; in raisedform 6 July 1977)
investigation has been carried out of the strain and strain-relaxation in (111) oriented Pb films deposited onto oxidized Si substrates at 300 K and then cooled down to temperatures as low as 4.2 K. The strain in the Pb caused by differences in thermal expansion of the Pb and Si was measured using X-ray diffraction techniques. For film thicknesses of _ 1400 A, the magnitude of the observed strain at 4.2 K was found to approximate the strain value calculated from the difference in thermal expansion coefficients using the biaxial strain model. For greater film thicknesses, the value of the strain measured immediately after cooling was much lower, indicating that significant strainrelaxation occurred during the cooling process. It is probable that most of the strain was relaxed by a dislocation-slip mechanism. During isothermal annealing above 50 K a second, slow relaxation process was observed. Nonuniformity of the strain normal to the film-surface was found by analyzing the asymmetric X-ray line broadening that occurred during cooling. A strong dependence of strain on grain orientation was observed, and interpreted in terms of the biaxial strain model.
Abstract-An
Resume-On a ttudit la deformation et la relaxation de la deformation de couches minces de Pb oricntees (11I). deposees sur des supports de Si oxydes a 300 K. et refroidies ensuite a des temperatures pouvant descendre jusqu’a 4.2 K. On a mesure par des techniques de diffraction de rayons X. la diformation du Pb produite par la difference des coefficients de dilatation thermique du Pb et du Si. Pour des couches d’epaisseur 14OOA environ, la deformation observee a 4.2 K etait de I’ordre de grandeur de la deformation calculee a partir de la difference des coefficients de dilatation thermique’ dans un mod&e de deformation biaxiale. Lorsque I’epaisseur des couches est plus grande, la deformation mesuree immidiatement aprts le refroidissement est beaucoup plus petite, ce qui montre qu’une importante relaxation de la deformation s’est produite au tours du refroidissement. La plus grande partie de In dtformation a vraisemblablement ete relaxee par un mecanisme de glissement de dislocations On a observe au tours de recuits isothermes au dessus de 50 K un second processus lent de relaxation. On a observe les heterogeniitis de la deformation perpendiculairement a la surface de la couche mince en analysant un elargissement asymetrique de la raie de rayons X au tours du refroidissement On a observe une influence importante de I’orientation du grain sur la deformation, et on l’a interpritee grace au modtle de la deformation biaxiale. Zusammenfassung--In (I 1 t)-orientierten Pb-Schichten, die bei 300 K auf oxidierte Si-Substrate aufgewachsen und dann bis zu 4,2 K abgekiihlt waren, wurde die Verspannung und deren Relaxation untersucht. Die von den Unterschieden in der thermischen Ausdehnung von Pb und Si herriihrenden Verspannungen wurden mittels Rontgenbeugungstechniken untersucht Bei Schichtdicken von - 1400 A entspricht die beobachtete Verspannung bei 12 K ungefahr derjenigen, die man mit einem zweiachsigen Spannungsmodell errechnet. Bei grogeren Schichtdicken war die unmittelbar nach dem Abktihlen gemessene Verspannung vie1 kleiner; das deutet auf eine betrachtliche Spannungsrelaxation wghrend des Abkiihlprozesses hin. Es ist sehr wahrscheinlich, daB der gri5Bte Teil der Verspannung tiber einen Versetzungsgleitmechanismus relaxiert wird Wahrend des isothermen Ausheilens wird oberhalb 50 K ein zweiter, langsamer Relaxationsprozess beobachtet. Die Analyse der wlhrend des Abkiihlens auftretenden Verbreiterung der asymmetrischen Riintgenlinien ergibt eine UngleichmMigkeit der Verspannung senkrecht zur SchichtoberGche. Eine starke Abhlngigkeit der Verspannung von der Kornorientierung wurde beobachtet und mit dem zweiachsigen Spannungsmodell gedeutet.
1.
INTRODUCXION
If a thin film is grown on a relatively thicker substrate at 300K and then cooled to a lower temperature T, and if the thermal expansion coefficient of the film differs from that of the substrate, a strain in a direction parallel to the film surface is introduced. The maximum value of the strain E,,_ introduced is, 300 K
(1) (rfilm - rrub) dT ir where qilm and rsub are thermal expansion coefficients of film and substrate respectively. Strain-relaxation phenomena in Pb thin films during thermal cycling have been investigated by several %Iax=
175
authors. The general conclusions from their studies are that at lower temperatures the strain is relaxed primarily by a nondiffusive process [I] such as dislocation slip or twinning as witnessed by the observations of the slip bands in Pb films [2,3]; at relatively high temperatures the strains are relaxed mainly by a diffusion controlled mechanism [4,5], Strain relaxation in Pb films was studied by Gangulee[4]. He found that the lattice parameter changes during the thermal cycling agreed well with calculated values obtained using a diffusional creep model [6.7-J. His results suggested that the dominant relaxation mechanism at temperatures above 120 K was diffusion creep.
Ii6
MURAKAMI:
STRAINS M DEPOSITED
The purposes of the present investigation were to use X-ray diffraction to measure nondestructively the strains, introduced upon cooling to temperatures down to 4.2 K in Pb films deposited onto oxidized Si substrates, and to study the relaxation mechanisms of these strains. A special effort was made to detect non-uniform strains along directions parallel or normal to the film surface. If the adhesion between the film substrate is excellent, more strains at the substrateinterface and less strains at the film surface are expected [S]. This strain non-uniformity in a direction normal to the film surface was analyzed from the measured asymmetric X-ray line broadening. In this analysis, the line broadening was simulated using the computer simulation method of Honska et al. [9, lo]. This simulation method was originally developed to obtain the diffused concentration protie of bimetallic couples. Application of this method to the present strain distribution analysis will be described in Section 2.2. Also, as Pb is elastically anisotropic, it is expected that the strain in a polycrystalline film gill vary for grains of differing orientation. eon-unifo~ity of strain parallel to the film surface is experimentally evidenced by the observation of hillocks after thermal cyclings [2,3,11]. In the present experiment the strains in grains with different crystallographic orientations were measured, and the measured values were compared with those calculated using the biaxial strain model. The theory of this model will be reviewed in Section 2.1. For Pb, the following are applicable: (a) it has a relatively large thermal expansion coefficient, resulting in measurably large lattice parameter changes for small temperature changes [12], (b) deposits of Pb on SiOt have a strong (111) fiber structure, which makes it possible to accurately measure the interplanar spacing normal to the lihn surface from (hbh) diffraction peaks, (c) Pb has a large CuKa X-ray absorption coefficient which enables one to analyze the strain distribution in thin films normal to the surface from asymmetric line broadening, and (d) the elastic constants of Pb are known over the desired temperature ranges [13]. 2. REVIEWS OF THEORIES
LEAD FILMS
to.the stresses by Eij =
sijkl
where S,r are the compliances which are calculated from the reported elastic constants. For an arbitrary (~~~ oriented crystal the relation (3a) also holds,
6ij = S’ijk@t where the transformed
=
V
-
Wdo,
W
compliance S;,%,,is given by
I2419 Si&[= ffi,Ujm UbUtp S-p
(4a)
The rotation matrix aetpis written by, a* =
cos 9 cos 6 sin 9 ( sin 8 sin 9
-sin 9
0 -sin 9 cos 8 1
cos 0 cos 9
sin t? cos 9
E’11 = S;l,,G;l
+
S;122f7;2
E)22 =
+
e;* EL3
= 3
S2211ail S;211~;l S;31141
+ +
S,ldit
(54
S222241+
S1212d12
PW
S;222&2
, h212ai2
(54
S;32242
+
+ +
S;31242.
(54
Also the maximum value of ci3 of an arbitrary oriented crystal is calculated from equation (5) using r;r which is equal to E_ of equation (1). 2.2 Diffraction intensity simulation method Based on an assumed non-uniform strain distribution normal to the film surface, an expected shape of line broadening for intensities diffracted by crystal
‘hkl
(2)
where d is a measured interplanar spacing normal to the f&n surface, and do is a strain-free value. As Pb has a strong anisotropy in elastic modulii, the calculation of strain l ij and stress oij from the measured eiJ must be carried out using a relationship given by Nye [14]. (The subscripts ‘Q”denote the directions of orthogonal axes: ‘11’ and ‘22’ are directions parallel to the film surface and ‘33’ is the one normal to the film surface). For (001) oriented crystal the strains are related
(4b)
where rotation angles 9 and 0 are shown in Fig. 1. The strains in thin films deposited onto substrates were studied theoretically by Vook and Witt [15, 161 for various crystals. They postulated that stresses acting on the film are applied at the interface between the f&n and the substrate. Consequently, all tensile and shear stress components in directions normal to the film surfaces are zero, which means that a;r = 0;2 = a& = 0. Because the film is assumed to adhere to the substrate, E;~ = 0. Using these assumptions adequate equations of equation (3b) to calculate unknown values of E; i( = e&), a\ i, and ai2 from the measured E;~, value are rewritten by,
The strain &a, normaf to a f&n surface is obtained from 43
(34
Gkl,
Fig. 1. Rotation-matrix
coordinate system.
MURAKAMI:
STRAIXS IN DEPOSITED
planes parallel to the substrate surface can be calcuIated by using the method of Houska et al. [9,10,17]. They determined the solute concentration profiles in diffusion couples by analyzing the shape of the X-ray diffraction lines. The line broadenings were due to the range of lattice parameters encountered by direct beam as it traversed regions of varying concentrations. In the present study, their method was used to obtain the strain distribution profile. A brief review of the theory is given below. A monotonically varying d-variation normal to the film surface is schematically shown in the left figure of Fig, 2 where xf is a film thickness. To pursue the analysis, it is convenient to subdivide mentally the total film thickness into thin slabs of constant thickness Allx.As illustrated in the right figure of Fig. 2, the X-ray intensity P, diffracted from a volume element AV, centered on x, will be given by. P, = ~,QmAVmA.llm,
where k, = 2,‘sin 8,. The parameter p is the linear absorption coefficient of Pb. It is apparent that a knowIedge of the d(x) profile suffices to synthesize an intensity band P,(d,) by combining equations (6)-(g) to yield. P, = floAoQ,g,&
k,exp[ - k,,&x,
- x,)].
(9)
The peak shapes P,,, co~~ponding to various assumed d(x) profiles can be calculated and compared with the P, obtained experimentally to infer the strain distribution normal to the surface. This procedure, ‘Houska’s Computer Simulation hfethod’, [9,10] will be adopted here to analyze the strain distribution in the thin film. 3. EXPERIlIENTAL
PROCEDURES
Oxidized (111) Si wafers of 0.2 mm thickness were
(6) used as substrates. They were precleaned in siru using
where I0 is the intensity of the incident beam, Qm is the reflectivity per unit volume and Aplnr is the absorption correction factor. The reflectivity Q, per unit volume is given by, Q, =
177
LEAD FILMS
!f$(’ lIfn”z?)P: exp(-2M,).
The parameter r, is the classical radius of the electron, ,l is the X-ray wavelength, V, is the volume of unit cell, F, is the structure facfor, exp( - 2M,) is the Debye-Waller factor, and 0, is the Bragg angle for element m. All parameters with suffix m vary with the distance x,,, of the diffracting element. In equation (6) the effective volume of material AV, is given by
an O2 glow discharge before film deposition. Lead source material of 99.9990/:,nominal purity was evaporated at 4 30 &‘s in a vacuum of lo- ’ Torr. The substrate was kept at room temperature. In order to minimize the non-uniformity of film thickness across the substrate the Pb source and the substrate were kept 50cm apart. Films with thicknesses from 0.0s5pm were prepared; the thicknesses were measured using a calibrated quartz crystal monitor placed near the substrate. 3.2 Heat treatments
Slight lattice parameter changes were observed in as-deposited films at room temperature that conAV,,, = %,A&, (7) tinued for a 4 h after deposition. In order to relax the strain introduced during the deposition, all where A, is the cross-sectional area of the section samples were kept in a vacuum desiccator at least exposed to an incident beam and is given by A~Js~~~~ for one day before further thermal treatments_ Ther(Ao: cross-sectional area of the incident beam), and mal treatments of specimens were carried out using g,,, is a factor which has been introduced to allow a liquid helium cooled X-ray diffraction stage. To imfor crystallite misorientations. Here, gn designates the prove thermal conductivity between the specimen and fraction of volume with crystal planes oriented within the specimen holder, a Cu grease (Cry-con grease supmeasurement range. A fraction of the incident and plied by Air Product and Chemicals) was used. The diffracted beam intensity is absorbed in reaching to environment of a specimen was evacuated to and leaving from the diffracting elements. The absorp4 x 10m6Torr before cooling by using 8 L’s iontion correction factor Ap,,, is given by pump. The typical cooling time down to liquid He temperature was approx~ately jmin when the transfer tube was sufficiently pre-cooled and the reheating time to 300 K was about 20 min. Temperatures were kept within + 1 K of the desired temperature by controlling flow rate of the coolant (liquid He or NJ and by adjusting the power to a heater incorporated in the sample mount. 3.3 X-ray measurements
Fig. 2. Cross-section of a thin tilm deposited on the substrate (right-hand side) and the corresponding interplanar spacing profile (left-hand side).
X-ray diffraction intensity measurements were made through a Beryllium window in the refrigerator housing. A computer-controhed GE XR-5 diffractometer was used with CuK* radiation (operated at 40 KV and 2OmA) and divergent (2”), receiving
175
MURAKAMI:
STRAINS TN DEPOSITED
f0.005’) and soiler slits. Angles ctfand x were adjusted to obtain the maximum intensity of the (333) reflection from the Si-substrate. Using a proportional counter the intensities were step-scanned for sufficient times (2 5 40s) to obtain enough measurable intensity in an interval of 26 = 0.005” or 0.02” for predetermined diffraction angle ranges; typical times required to take a Pb (333) reflection profile were 15 min. The data were directly punched out to computer cards to allow convenient subsequent analyses. The deconvolution of Kzi-K,Z peaks was carried out by modified Rachinger correction method [20] where the (2fla,-20cx(2) separation was continuously incremented with increasing diffraction angle within broad intensity band. The interplanar spacing was determined at the middle point in 28 position of the half width (A28) of K,, peak. In cases where an accurate measure of the line broadening due only to strains was required (as will be seen in Fig. 5b), analysis was carried out by Stokes’ correction method [2l] using the peak profile at room temperature as an instrumental line broadening. Dislocation densities were determined from (111)and (222) difraction-peaks using Warren and Averbach’s method [22]. This method was originally developed for poly~stalline samples and recently was shown to be applicable to thin films by Gangulee [23] and Sen et al. [24]. For Houska’s computer simulation analysis [9, lo] the linearly averaged Gaussian and Cauchy’s distribution [25] was used as an i~t~mental broadening. 4. EXPERIMEhTAL RESULTS AND DIScussIONS 4.1 Preliminary results (a) Stability of liquid helium refrigerator. An excellent mechanical stability of the refrigerator during heat-treatments was confirmed by the laser reflection and by X-ray diffraction technique before and after the refrigerator was set up on the X-ray diffractometer. As an example, the X-ray diffraction results for Si are shown in Fig. 3: the (333) reflection from the Si-substrate on which the 5 pm-Pb was deposited was measured at each temperature ranging between 300 and 4.2 K. Experimental results are shown by open circles in Fig. 3(a), the closed circles are the values calculated using the reported thermal expansion coefficients [12]. An agreement between the calculated and measured values is excellent, the maximum deviation of d is 0.00018 A. Also, the negative thermal expansion coefficients below about 120K were confirmed by the present experiment. In Fig. 3(b) the half-width values for the (333) reflection are shown; the values remain constant throughout the temperature range studied, which indicates that the Si-substrate was not significantfy strained by the SiGl or the Pb film. (b) Fiber structure of Pb f;lm. Pb thin films have the strong (Ill) fiber structures if deposited onto oxidized Si wafers. Info~atio~ about the fiber structure
LEAD FILMS I
3
I
(al DW1)
I
I
I
2 b
0 EXPERIMENT . CALCULATION
2 a ki
3,13200
.z a
I 150
8 200
I 250
300
TEWERATURE ?K,
’ (b)A2@ 0.07 ‘-
j$ 0.06 :
I 100
so
0
--o y =
0
0.05 rj
I 0
I
50
I
100
I
I
150 200
I
250
1
300
TEMPERATURE PK)
Fig. 3. Interplanar spacings (a) and X-ray peak half-widths (b) for the Si substrate, as measured at various
temperatures.
in films of various thicknesses were obtained by measuring different X-ray (hkl) reflections. It was found that the ratios of peak intensities of 1(311)/1(222) were 0.03 for 0.1 pm thick films and 0.002 for I pm thick film. It is seen that the (111) fiber structure monotonic~ly improves for thicker films. However, even for the 0.1 pm thick film, the film had the strong (111) fiber structure as judged by the ratio of 1(311)/1(222) of the powder sample which is 3.6. 4.2 Strains nnd stresses at 77 or 4.2 K Strains E& normal to the film-surfaces were measured at 77 or 4.2 K for films with various thicknesses. The (333) diffraction peaks of the films with various thicknesses of 0.1 _ 5.0~ which were cooled down to and measured at 77 K are shown in Fig. 4. The dashed curves represent the K,, and K,a deconvoluted peaks. The interplanar spacings d and angles (628) which subtend the half-maximum intensities were obtained from the deconvoluted &i peaks. In Fig. 5(a) the (111) interplanar spacings d are plotted versus film thickness; the points are the d-values measured immediately after cooling; the dashed line is the strain-free (or bulk) d-value calculated using the reported thermal expansion coef?% cients [12]. The arrows indicate the changes in d observed after one day of isothermal annealing, which means strains were relaxing at these temperatures over a period of one day. (This secondary strainrelaxation will be described elsewhere). It is noted that smaller d-values are observed for thinner f&as. For the 5 pm thick film the d-value almost reaches at the calculated strain-free value. The peak widths A26 are plotted in Fig. 5(b); A26 values are also
STRAISS
MuRAK.4blI:
IN DEPOSITED
o,5
179
LEAD FILMS
’
1\n
_2%___---__ l
-x___-
(al
T=77’K
\ I
\
!
0
I
I
I
I
0.2
0.4
0.6
0.8
FILM
DiFFRACTION
ANGLE
THICKNESS
Xf
I 1.0
(pm)
28tDEG)
Fig. 4. (333) diffraction peaks of Pb films with various thicknesses measured at 77 K.
observed to change with annealing time as shown by arrows. A peak at x/ = 0.5 pm was observed in this plot, which will be interpreted in Section 4.4. Strains&, calculated from equation (2) were plotted versus film thickness in Figs. 6(a) and (b) at 77 and at 4.2 K respectively. Each dashed line indicates the theoretical maximum value (E&) of the strain at these temperatures. It is found that the films of approximately 0.1 pm thickness almost reach the maximum strain value at each temperature. For films thinner than 0.1 pm the strains drop off from the eLMIas seen in Fig. 6(b). This is believed to be caused by the discontinuity of films in such thinner films, since many small holes of about 0.5 pm diameter were observed
I
0
0.2
1
I
I
0.4
0.6
0.8
FILM
THICKNESS
I
Xf
1.0 (pm1
Fig. 6. Residual strains l‘s3 for Pb films of various thicknesses measured at 77 K (a) and 4.2 K (b) respectively.
in 0.1 pm thick film by the scanning electron microscopy. It is noted that the log E vs film thickness plots are linear at both temperatures in the thickness region of 0.1-1.0~. This behavior is described by, exp
(b)A28 Ta77.K
9
O.Oo
I
2
3
4
5
X, (pm1
FLM THICKNESS
Fig. 5. Interplanar spacings (a) and half-widths (b) for Pb films of various thicknesses measured at 77 K.
(10)
where do and d, and d-values at X, = x and x, = 0, respectively, and - l/rj is a proportionality constant. Residual stresses a;i were calculated from the measured strains c’~~according to the procedures described in Section 2.1 using equation (5). The results at 77 and 4.2 K are shown in Figs. 7(a) and (b), respectively. The data are plotted on the normal scales instead of semi-logarithm scales in order to compare with the other data. Stresses are found to have a very
180
~~~~MI:
STRAINS IN DEPOSITED LEAD FILMS
strong film thickness dependence; thinner films have more residual stresses. The maximum stress a;,, shown by the dashed lines in these figures, are calculated using the expected maximum strains e;,, of Pb on Si given in equation (1). The dashed curves reproduce Caswell er al.‘s data [2]. They prepared Pb films of various thicknesses by evaporation onto Ni-substrates and measured the residual stresses by a cantilever beam technique, immersing the measurement unit into liquid nitrogen or helium. At 77 K the agreement between the present results and their data is within experimental errors for films thinner than OSpm, but the stresses of the thicker films were smaller than Casweli et al’s resuhs. At 4.2 K there is a factor of two difference in the stresses. Good agreement of the stresses would be expected for films of thickness between 0.2 and O.Spm if the films were of similar structure and impurity level. In this thickness range, significant stress relaxation is occurring in the Pb films and the effect of having substrates with different thermal expansion coefficients would be unimportant. For thicker films the stress inferred from the X-ray me~urements would be lower than those determined by the cantilever beam technique because the film thickness is becoming comparable to the X-ray absorption length ( * 1 pm for CuKr radiation). For thinner films where stress relaxation is not occurring both z,ub,lrate differences and measurement method differences should result in larger stress values for the present work by a factor of about 1.3. The factor of two higher stresses observed by Caswell et al. at 4.2 K (Fig. 7b) is surprising in light of the agreement at 77 K and the antici) for their experipated lower value of (xpi, - CL,,,,~,.~+ ment. The reason for this difference is not understood. The primary mechanism of the strain~eI~ation observed at these temperatures for thicker films is considered. to be dislocation-slip. High densities of dislocation slip lines were observed on the film surfaces by multi-thermal cyclings of Pb films [2,3]. In addition, a dislocation slip model has been used successfully to explain the grain orientation dependence of strain relaxation in Pb films [26]. The dislocation density can be calculated using Warren and Averbath’s analysis [22] if the dislocations are uniformly distributed through the fihn. Reasonably symmetric (Ill) and (222) reflection peaks were obtained at 77 K (after one day) and at 4.2 K from which the disiocation densities were caIculated to be 0.7 x lO”/cm at 77 K and 1.8 x 10’O/cm” at 4.2K respectively. Similarly low dislocation densities were also obtained by Gangulee for thin films [23]. 4.3 Strain-relaxation
during isothermal annealing
Strain-relaxation during isothermal annealing was studied by following the d-changes. The films were cooled from 300 K to lower temperatures and isothermally annealed at these temperatures. In Fig. 8 the d-values normal to the (1 II) planes of Pb films of various film thicknesses are shown. The closed circles
_____-__
crmax _____
--
____
-___
40(a) T=77OK X---X
CASWEU
-
FILM THICKNESS
cl al
PRESENT RESULTS
X, (pm)
(b) T= 4.2*K
x---GGWEU PRESENT
flLM
THICKNESS
et al RESULTS
X, (pm,
Fig. 7. Residual stresses for Pb films of various thicknesses at 77 K (a) and 4.2 K (b) respectively. are the strain-free d-values of buik Pb wtrich were calculated using the reported thermal expansion coefficients of Pb. The closed squares are the calculated d-values using the biaxial strain model (Section 2) and assuming no strain relaxation occurred. The experimental values measured immediately after cooling are indicated by the specific symbols. The vertical arrows indicate the d-value changes that occur during isothermal annealing over a period of about one day. It is noted that for relatively thick films most of the
MURAKAMI:
STRAINS IN DEPOSITED
2.860 1
ric peak-shapes changed to symmetric shapes. with narrowing A20’ as shown by arrows. The endpoints of arrows indicate the final values of AZ at each temperature. The A/0 value for the specimen annealed at 260 K comes back to the value observed at 300 K which is presumed to be the strain-free value and is shown by a dashed line. Specimens annealed below 260 K never reach the strain-free value. It is seen that the amount of relaxation in A.28 during the isothermal annealing is maximum at 77 K. At 4.2 K no change in the A28’ was observed during the annealing, which is consistent with no change in the d-values at this temperature as seen in Fig. 8. These decreases in line broadening will be interpreted in terms of a non-uniform strain distribution normal to the film surface in Section 4.4.
2.850 z
1x1
LEAD FILMS
1
4.4 Non-uniform strain distributions
0
I
I
50
100
I
I
I
150 200
I
250
300
TEMPERATURE(OK) Fig. 8. Summary of the interplanar spacings for Pb films of various thickness measured at each temperature. strain was relaxed concurrently
with cooling. The kinetics of this primary relaxation process could not be studied by the present experimental technique, because it took at least 15 min to obtain the first data. Changes in the half-widths are also observed during isothermal annealings. Half-widths (A20’) at various temperatures obtained from the deconvoluted (333) K,, peaks are shown in Fig. 9. (Note A.20 are not corrected by Stokes’ method[21]). The 620’ values became larger with lowering temperatures and reach the maximum value at approximately 77 K. (It should be noted that no line broadenings were observed for Si peaks). During isothermal annealing the asymmetPb (333)
T
(a) Non-uniformity normal to film surface. Non formity of the strain normal to the film surface was analyzed from the measured X-ray line broadenings on the basis of the exponential strain distribution which was experimentally observed as illustrated in Fig. 6. The (333) K,,-diffraction profiles for the 1 ,umthick films measured shortly after cooling to 77 or 200 K, are shown in Fig. 10. The profile at T = 77 K has a noticeable asymmetric line broadening. These profiles were analyzed with Houska’s computer simulation technique (Section 2.1) using the following procedure: First, the trial values of the unknown parameters, I, do and dl in equation (10) were chosen to obtain a first iteration for the strain distribution. Based on this strain distribution the peak profiles were calculated by equation (?), and they were compared with the experimental values. Changing q, do, and d, values, iteration was continued until the mini‘mum rms. value of the difference in the calculated and measured intensities was obtained. Peak profiles thus calculated are shown by the dashed curves in Fig. 10. Excellent fits between the calculated and the measured intensities are observed at both temperatures. The strain distributions corresponding to the peak profiles obtained during the isothermal annealings were shown in Fig. 11. As expected, the strain distribution is widest immediately after cooling and (olT=77*K
(b)T*200”K
/
T
II
L
.___-_I
o.lot , 0
50
(
,
,
-I-,
,f
100 150 200 250 300 TEMPERATURE(OK1
Fig. 9. Half-widths of the 1 w thick Pb films measured at each temperature.
DIFFRIICTKJN
ANGLE
ZBVJEGI
DIFFRACTION
ANGLE
28 (DEG)
Fig. 10. (333) K,,-diffraction peak profiles of 1pm thick Pb films measured at 77 K (a) and 200 K (b) respectively.
152
MURAKAMI:
STRAINS IN DEPOSITED
0.0
(alTq77.K
% 2.850 t r-
7
-‘-t
SZmin
il
--0.2
i btt*2OO*K
2B440
0.0 a !z
___*__________________
I 0.2
I 04
/ 0.8
I 0.6
I.0
DISTANCE FROM SUBSTRATE INTERFACE (pm)
Fig. 11. Calculated interplanar spacings distributions normal to the film-surface for the 1 pm thick Pb films measured at 77 K (a) and 200 K (b) respectively.
it becomes narrower with the annealing time. The dashed lines show the calculated equilib~um d-values obtained using the thermai expansion coefficients of bulk Pb. The d-value indicated by the open circle on each curve represents the one calculated from the 28 position which corresponds to the middle point of the half-width (A2fY). Additional experimental evidence for the existence of the non-~ifo~ strain distribution normal to the film surface was obtained by the thickness dependence of A28 in Fig. 5(b), in which the maximum value of the A28 was observed for the 0.5pm thick film. This is explained nicely as follows: The interplanar spacing difference (Ad) between the surface and the substrate-interface is proportional to the peak width A28 of the diffracted intensities. For a thin fihn Ad is very small, thus a smaif A28 value is expected. For a thick film, although Ad is very large, the intensities diffracted from the volume elements close to the interface are very weak because of the strong absorption of intensities by the Pb film. The measured intensities are mainly from the volume elements close to the film-surface in which the effective Ad is small, thus producing a small A2@-value. For an inter-
LEAD FILMS
mediate thickness film, Ad is relatively large and the intensities from the volume elements at the interface are measurable, which produces a very large A28 value. This dependence of A28 on film thickness would not be expected if the Iine broadening was caused by strain variations in the plane of the film. Evidence for large amounts of strain relaxation in l.Opm thick films at the substrate interface upon cooling is seen in the simulated d-distributions of Fig. ii(a). The strain at the interface of the film. in the as-cooled state is 0.18%. This strain is about 3.2 times smaller than the maximum strain of O.jT<, which was obtained for the 0.1 pm thick film at 77 K (Fig. 6a). The difference in strains (0.4%) may be relaxed at the interface upon cooling and/or at the very early stages of the isothermaf annealing. It was confirmed by computer calculations that the intensities diffracted from the volume elements at the substrate interface of the 1.0~ thick film should be measurable and should be observable in the peak profiles. (b) Non-uniformity parallel to the film sur$ze. Nonuniformity of the strain parallel to the film surface was investigated by measuring the strains e& in the grains whose surface planes (hkl) are different. As thinner films do not have perfect fiber structures, the (hkl) reflection peaks other than (hhh) reflections are measurable by the present X-ray technique. In Table 1 the aia values measured at 4.2 K for 0.1 and 0.2 pm thick films are listed for grains with different crystal orientations. Since the (200) reflection peak partially overlaps with the reflection of the substrate, an accurate me~urem~t for this plane was not possible. The e& values were found to be strongly dependent on crystallographic orientation: the E& for the (311) plane is about two times larger than the & for the (111) plane. Using the biaxial strain model the theoretical values of the & were calculated for each crystal orientation. In this calculation the elastic constants of Cl1 = 5.554, Cz2 = 4.542, and C,, = 1.942 x IO” dyn/cm’ were used (131. The calculated values of the ratio E;,/E;~ are listed in the fourth column of Table 1. In the final column the values of ~5~ are given that were calculated using c’;“;’ = 0.706% as obtained from equation (1) for Pb on Si at 4.2 K. Reasonable agreements between the calculated and measured E; s values are seen. From this calculation
Table 1. Residual strains E;, normal to the film surface in the different grains whose (hkl) planes are parallel to the film-surface
$3 (“/,I 2oooA
tw
1OOOA
Gm
-0186
(220) (222) I:::; (333)
e3
f&/f; t
-0:66 --0.71 1.14
- 0.72 - 0.63 --0.13 1.18
-
-0.65
- 0.63
- 0.959
* Measurements
l
were not carried out.
1.635 1.094 0.959 1.065 1.457
(%I
(eI I = 0.?0570/,) - 1.15 -0.71 -0.68 --0.75 1.03 -0.68
MURAKAMI:
STRAINS IN DEPOSITED LEAD FILMS
result it is expected that the c’s3 for the (100) plane would be largest. 5. SUMMARY The experimental results obtained in the present investigation are summarized below: (1) Residual strains at low temperatures of Pb films deposited onto oxidized Si substrates display a strong thickness dependence. For a 0.1 pm-thick film, the strain at 4.2 K was found to reach the maximum value calculated for the thermal expansion coefficient difference between Pb and Si. This indicates that no strain-relaxation has occurred on cooling to 4.2 K. However, for 0.07 and 0.05 pm thick films relaxation was observed which may be associated with holes in the films. (2) Reasonably good agreement was obtained between values of residual stress a;r calculated from the measured residual strains and values of stress measured by Caswell et al. [Z]. (3) For thicker Pb films, most of the strains were relaxed by plastic deformation during cooling. The dislocation densities introduced by the deformation were 0.7 x 10’“/cm’ at 77 K and 1.8 x 10’“/cm2 at 4.2 K. (-I) Residual elastic strains li3 in the plane of the films were found to have a strong crystal orientation dependence, which agreed with theoretical results calculated using the biaxial strain model. The ratio of e’,&rr is maximum and minimum for grains whose planes are parallel to the film surface are (100) and (11 I), respectively. (5) Non-uniform strains normal to the surfaces of the films were observed by analyzing the measured X-ray diffraction profiles. The regions of the films near the substrate interface were found to be more strained than those near the film surfaces.
Acknowledgemenrs-The author would like to express his deep gratitude to C. J. Kircher, J. W. Matthews, P. Chaud-
133
hari and I. Ames for stimulating discussions and for manuscript reviewing, to J. An@ello and A. Segmuller for assistance with X-ray diffraction techniques. to S. K. Lahiri for many informative discussions on stresses in thin films, to P. S. Ho and F. M. D’heurle for discussion on diffusion in thin films and to V. Tom for sample preparation.
REFERENCES 1. M. F. Ashby, Acta Met. 20, 887 (1972). 2. H. L. Caswell, J. R. Priest and Y. Bado, J. appl. Phys. 34, 3261 (1963). 3. S. K. Lahiri, J. appl. Phys. 46, 2791 (1975). 4. A. Gangulee, Acta Met. 22, 177 (1974). 5. P. Chaudhari, J. appl. Phys. 45, 4339 (1974). 6. F. R. N. Nabarro, Proc. Conf on the Strengths ofSolids, p. 75. Physical Society, London (1948). 7. C. Herrine. J. aool. Phvs. 21. 137 (1950). 8. P. ChaudKari, l&f JI’Res. bev. 13, 197 (1969). 9. C. R. Houska, J. appl. Phys. 41, 69 (1970). 10. J. Unnam. J. A. Carnenter and C. R. Houska J. a&. .. Phys. 44, 1957 (197;.
II. S. K. Lahiri and 0. C. Wells. Aaol. . . Phvs. _ Lert. 15, 234 (1969). 12. D. E. Gray, American Institute of Physics Handbook. McGraw-Hill, New York (1972). 13. D. L. Waldorf and G. A. Alers. >. norrl. . . Phvs. _ 33. 3266 (1962). 14. J. F. Nye, Physicaf Properties of Crystals, p. 131. University Press. Oxford (1972). 15. R. W.-Vook and F. Witt, J.‘appl. Phys. 36, 2169 (1965). 16. F. Witt and R. W. Vook, J. appl. Phys. 39, 2773 (1968); 40, 709 (1969).
17. M. Murakami, D. deFontaine and J. Fodor. J. appl. Phys. 47, 2850, 2857 (1976). 18. G. B. Gibbs, Phil. Msg. 13, 589 (1966). 19. R. L. Coble, J. app[. Phys. 34, 1679 (1963). 20. W. A. Rachinger, J. Sci. Instrum. 25, 254 (194S). 21. A. R. Stokes, Proc. Phys. Sot. London 61. 382 (1948). 22. B. E. Warren and B. L. Averbach, J. ap&. Phys. 2i, 595 (1950). 23. A. Ganguiee, J. appl. Phys. 43, 867 (1972). 24. S. Sen, S. K. Halder and S. P. S. Gupta, J. Phys. D: Appl. Phys. 8, 1709 (1975). 25. B. E. Warren, X-ray Diflraction. Addison-Wesley, Reading, p. 258 (1969). 26. M. Murakami and P. Chaudhari. Thin Solid Films. 46. 109 (1977).