SPIE Proceedings [SPIE Photonics West '96 - San Jose, CA (Saturday 27 January 1996)] Oxide...

Invited Paper

Stress relaxation and critical layer thickness of high temperature superconductor thin films,heterostructures and superlattices.

J.P. Contoura, A. Abta,b d A. Défossez

a C.N.R.S. I Thomson CSF -LCR, F 91404 Orsay, Franceb Fakultat für Physik, UniversitAt Konstanz, D 78434 Konstanz, Deutschland

C Physique du Solide, ESPCI, 10 rue Vauquelin, F 75231 Paris, France.

ABSTRACT

The heteroepitaxial growth of high temperature superconductor thin films on single crystal substrates producesstrained heterostmctures when the mismatch is small and the thickness ofthe epilayer is not large. Although a large numberof studies have been carried out in the case of semiconductor epitaxy, only a few papers report models or experimentalresults dealing with the relaxation ofthe elastic strain in cuprate heterostructures. We apply here the calculations performedfor semiconductor epitaxial layers and pseudomorphic superlattices to estimate the critical layer thickness of cuprate thinfilms, heterostructures and superlattices. The result of these calculations is discussed with respect to the previously reporteddata and also to our resultsin the case ofYBaCuO heterostructures.

Keywords: HTSC thin film, superlattice, strained epitaxial layer, critical thickness.

1. INTRODUCTION

Since 1986, a large amount of work had been performed to produce high quality high temperature superconductorthin films and heterostructures on more or less well matched single crystal substrates1'2. The resulting epilayer correspondsto an heteroepitaxial structure which is strained if the mismatch between the substrate and the film is small and thethickness not large, this phenomenon being possibly enhanced by the cuprate anisotropy3'4. Although strain and relaxationhave been studied for more than 20 years in the case of the semiconductor epitaxy5, this is not true for cuprate growth,although the possible effect of strain have been mentionned in several previous papers68. Moreover, future HTSC activedevices could be founded on heteroepitaxial structures, such as ultra-thin cuprate or artificially layered films, hence anunderstanding ofthe strain effect is required9.

In this paper, we shall first summarize the previous studies carried out on strained semiconductor epitaxial filmsand superlattices, then apply them to HTSC thin films and heterostructures and finally discuss this approach with respect tothe experimental results

2. STRESS AND RELAXATION IN SEMICONDUCTOR HETEROSTRUCTURES

Silicon-germanium alloy thin films are the strained layer systems which have been the most comprehensivlystudied over the past 20 years so they will be used here to introduce the effect of strain, dislocations and critical thickness inthe HTSC thin films. Since 1955, a great amount ofwork has been performed to study the properties ofbulk GexSiix, butthe first high quality pseudomorphic layers on silicon were only grown in 1975 by Kasper et al. .During the eighties,structures based on GexSiix strained layers started to play an important role in developing new devices11'12. However theusefulness of the strained thin films and superlattices is determined by their morphology and cristallinity. One of the mostimportant parameters governing these properties is the critical layer thickness below which a good quality heterostructurecan be grown without introducing misfit dislocations. An extensive review article on this subject has been already publishedin 1991 by Jam et al.5'13.

2.1. Critical thickness of a single epitaxial layer

The theories of critical layer thickness stem from the work of Frank and van der Merwe on bicrystals which hasbeen extended and compared with experiment for 40 years1418. It was demonstrated that if the lattice mismatch is smalland the thickness not large, the mismatch is accommodated by an elastic strain in the epilayer: the growth is then said

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pseudomorphic (coherent or commensurate). As the layer thickness increases the homogeneous strain energy becomes solarge that a thickness is reached, the critical thickness hc, where it becomes energetically favorable for misfit dislocations tobe introduced: the epitaxial layer is then relaxed by introduction ofthese dislocations16'19'20.

The total energy E,which is minimum at the equilibrium, is given by:

E=Eh+nEd (2.1)where Eh is the energy of the homogeneous strain and Ed the one of the misfit dislocation, n being the number ofdislocation sets.

The energy Eh be calculated in an elastic continuum theoiy and is given by:.

(2.2)1—v

where .t is the shear modulus, v is the Poisson's ratio, h the thickness and c is the elastic strain which depends on the misfitparameterfm Lfld the average number ofdislocations at the interface; s is given by:

IEtfmb/P (2.3)where b is the active component of the Burgers's vector and p the average distance between the dislocations. In the case ofGexSiix,fm is defined by:

fm(')a(x)—a(Si) 0.042x (2.4)

For a diamond structure the energy E ofa single dislocation per unit length is given by21'22:

E7=__ 11+ln—--') (2.5)4it(1—v)L q')

where q' is the inner cut-off radius ofthe dislocation. E means that the distance between 2 dislocations is so large that theydo not interact. The total energy due to i/p dislocations per unit length is:

E7= (2.6)pBy using the inner cut-off radius calculated by Kasper eta!. q' 0.609 nm22, the critical thickness can be

calculated from (2.2) and (2.5)2123:

h = b [1+1J1 (2.7)87t(1V)fmL q' )]

or by substituting the numerical values:1 175x 1O_2h= ln(8.9h) (2.8)

Some other equations have been proposed byBfl and People20 and by Matthews and Blackeslee24, but in factWillis et a!. have shown that they give identical results, the difference in their form being due to the assumed expressionfor the dislocation energy25.

2.2. Critical thickness of a superlattice

A strained superlattice is an artificial periodic structure in which each period consists of 2 pseudomorphic strainedelementary layers of different materials. An isolated superlattice shows an in-plane lattice constant (aav) which can becalculated from the unstrained lattice constants of the two elementary layers (a1, a2) and their respective thickness (d1, d2),d1 and d2 being less than h. Assuming that the elastic constants are the same and the substrate not used to force the latticeconstants of the elementary layers to that of the substrate, both layers of the superlattice will be constrained to the in-plane

lattice constant aav. Furthermore equation (2.2) shows that Eh stored in the two layers is proportional to s?di and sd2respectively. The minimun energy in a period is obtained when26:

= —s2d2 (2.9)The strain in the two layers can be then calculated from equations (2.3) and (2.4):

.aiaav (i=1,2) (2.10)aav

Substituting equation (2.10) in equation (2.9), the average parameter is given by:

340/SPIE Vol. 2697


a1d1+a2d2 211aav—d1+d2

( . )

In fact the superlattice has to be supported on a substrate the lattice constant of which forces the elementary layers

ofthe superlattice to acquire its own in-plane lattice constant (a5). The critical thickness of the superlattice (hi) can be now

calculated using equation (2.8), with a misfit parameter defined as20:

jiaav—as (2.12). a

This result implies that, in terms of stability, the superlattice can be taken as one single layer of GeSii alloywith a value of x equal to an average value Xav. Ifl a equally strained superlattice, the composition and thickness of the twoelementary layers are adjusted so that they are equally and oppositely strained. If the thickness of each elementary layer isless than h, there is no limit to the thickness of the coherent superstructure when it grown on a buffer layer having thelattice constant equal to aay, by equation (2. 1 1)27.

3. CALCULATION OF THE CRITICAL THICKNESS OF CUPRATES HETEROSTRUCTURES

3.1. Critical thickness of the YBaCuO single layer

The calculations reported in section 2. 1 have been used to determine the dislocation energy and critical thicknessaccording to Willis's calculations, i.e. in a energy model ofFrank and van der Merwe to which one adds a periodic networkof dislocations1618'2530. They have been first carried out using an isotropic approach, the elastic constants being equal toan average value calculated from the experimental ones ofYBaCuO (Table 1), according to the following hypothesis:

1) Mechanical equilibrium between the substrate and the epitaxial layer, the total energy being set to its minimumvalue.

2) 2D growth mechanism, flat interface parallel to YBaCuO {OO1} and SrTiO3 {OO1} the <010> axes beingparallel, growth defects are neglected.

3) Frenkel-Kontorova's model, the interfacial atoms being forced into a periodic potential.4) Substrate and film are governed by Hooke's law, their elastic constants are supposed to be the same.5) The growing system does not contain any impurity.

Table 1: Elastic constants ofYBaCuO and SrTiO3, units are GPa.

c11 c22 C33 c44 c55 c66 c12 c13 c23

YBaCuOCalcul.31

182 185 111 42 21.5 101 105 50 55

YBaCuOExp.32

231 268 186 49 37 95 132 71 95

SrTiO3

Exp.33

318 123 103

Furthermore, it has been assumed that 2 orthogonal networks ofedge dislocations are parallel to <100> and <010>,the Burgers's vector b1 being in the ab plane with a modulus equal to the lattice parameter. According Gosling et a!. thecalculation leads to the dislocation energy ofthe isotropic model:

E - [b? +b +(1_v)b]ln]_b? +b

(3.1)4it(1—v) q 2 j

where v is the Poisson's ratio, the shear modulus and q the core-cut off radius. Substituting numerical values gives thecritical thickness from:

4!= 1.55lnh+3.1 (32)a h

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In order to take into account the YBaCuO anisotropy31'32, hc has then been estimated from an anisotropic approachfrom Gosling and Willis29 where Ed is expressed as a function of the critical thickness and the elastic constants, howeverthe elastic constants of the subtrate and the film are still supposed to be the same and equal to those of the film32 (Table 1).The total energy ofthe epitaxial layer is calculated as a function of h,fand p. then it is minimized versus p keeping h andfconstant as above. The critical thickness hc corresponds to the energy minimum when p —+ oo ,or f 0:

h = 2(E+E) (33)C

where mark 1,2 or 3 denotes the 3 reference axes parallel to <100>, <001> or <010> and b the j th component of theBurgers vector along the x1 axis; is the tensor of the mean elastic stress without dislocation which is given by:

cli = CjklSkl (3.4)where CjjU S the elastic constant and c the bending tensor. Ec, the hard core energy of the dislocation, which iscalculated from Bacon et al.34, is small with respect to E; it can mostly be neglected in the calculation.

4)

I

Fig. 1. Calculated critical thickness of an YBaCuO (123) single layer as a function of the substrate mismatch: a) isotropichypothesis, b) anisotropic hypothesis. The plotted data are the experimental critical thicknesses reported in table 2.

The results of the numerical calculations are presented as a function of the parameter mismatch in Fig. 1 togetherwith the experimental critical thicknesses of YBaCuO thin films and superlattices deposited on { 100} SrTiO3 reported inprevious studies (Table 2). The results are discussed in section 4.

3.2. Critical thickness of YBa2Cu307 based superlattices

Two critical lengths have to be taken into account in the case of the superlattices: i) the critical thickness of theelementary layers h(el), ii) the critical thickness of the superstructure hc(sl)5"7'29. The first is calculated as in section 3.1.The second one has been estimated by treating it as a single layer and assuming that the superstructure shows mean latticeparameters calculated from (2. 1 1):

— a1d1 +a2d2 b — b1d1 +b2d2 _ c1d1 +c2d2a1 —d1 + d2

' sid1 + d2

' ci —

d1 + d2(3.5)

where d1 and d2 are the respective thickness of the elementary layer of the superlattice materials A1 and A2. As anillustration, the lattice constants of YBaCuOIPrBaCuO superlattices as a function of the relative thickness of the PrBaCuOto YBaCuO elementary layers are given in Fig. 2. For short period YBaCuO based superlattices grown on SrTiO3, the meanparameters which are calculated from the relative thickness leads to a smaller mismatch than that of an YBaCuO singlelayer, i.e. a larger critical thickness. As in the case of GeSi equally strained superlattices an appropriate choice of the

342 / SPIE Vol. 2697

Mismatch aJa [%}


substrate, A1 and A2 can give oppositely strained structures showing extremely large critical thickness, however due to theorthorhombic structure ofYBaCuO the limit to the thickness of the coherent stucture is lowered.

0 1 2 4 5

Fig. 2. Lattice constants of YBaCuOIPrBaCuO superlattices as a function of the PrBaCuO to YBaCuO relative thickness ofthe elementary layers.

Fig. 3. Relative critical layer thickness of YBaCuOIPrBaCuO superlattices to YBaCuO single layer hc(sl)Ihc(el) as afunction ofthe PrBaCuO to YBaCuO relative thickness ofthe elementary layers: a) SrTiO3, b) NdGaO3.

This effect is illustrated by the plot in Fig. 3 which gives the relative critical layer thickness of YBaCuOIPrBaCuOsuperlattices to YBaCuO single layer, hc(sl)fhc(el), as a function of the PrBaCuO to YBaCuO relative thickness of theelementary layers for SrTiO3 and NdGaO3 substrates. It is assumed in this calculation that the superlattice misfit resultingfrom the averaged lattice parameters leads to a homogeneous strain energy Ed which is calculated by adding thecontribution of the different lattice parameter misfits. In the case of NdGaO3, which shows a lattice constant between the a

.1

0.384

1.17 1

.11170

1.169

1.168

Ratio dpBco/dyBco

3

Ratio

40

30

4).1o

Ratio /

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and b parameters of YBaCuO, the mean misfit varies from a positive to negative value when the PrBaCuO layer thicknessincreases, explaining the large h foreseen as the mean misfits are close to zero.

Table 2: Experimental critical thickness ofYBaCuO single layers and superlattices deposited on SrTiO3

Source hc (nm) Method ofdetermination Growth technic/Oxygen source

Frey eta!. 36 20.4 AFM Laser MBE IRF plasma

Gau et a!. 7.0 XRD PLD I 02Gupta eta!. (inref. 36) 8.3 RHEED Laser MBE I

RF plasmaTerashima eta!. 6.0 RHEED Reactive MBE I

RF plasmaxi eta!. 8.0 RBS Sputtering I02Zegenhagen eta!. 4.0 - 8.0 XRD MBE I 02 + H20Zheng eta!. ' 9.5 - 19 STM PLD I 02Contour eta!. 42 6.0 - 8.0

4.8 (YBCOIPBCGO S.L.)XRD PLD I 02

Horiuchi et a!. 4.5 XRD Laser MBE I

Gupta et a!. 6(YBCOILSCO SL)

10 - 12

(YBCOINCCO SL)Tc(R=O)

°2 0PLD I 02

Tabata eta!. 5

(LaSrCuO/SmCuO SL)

XRD Laser MBE I

02+03

4. EXPERIMENTAL DETERMINATIONS OF THE CRITICAL LAYER THICKNESSOF CUPRATES HETEROSTRUCTURES

In the case of YBaCuO single layers, the experimental values which are plotted in Fig. 1 are distributed betweenthe curves given by the isotropic and the anisotropic models, they are strongly underestimated by the first one, with thesecond one appearing as an upper limit. This result confirms that the strong anisotropy of the material cannot be neglectedin the study of the stress relaxation of YBaCuO based ultra-thin films and heterostructures35. The scattering of theexperimental data could be attributed to the difference between the true growth conditions and the modeling hypotheses,especially the second one assuming the 2D growth mechanism but also to the laws of the continuous state mechanics whichare extrapolated to the atomic scale in the present calculation. The various measurement techniques and the associatedexperimental uncertainty have also to be taken into account. Although the results do not suggest any clear correlation withthe growth technique, STM and AFM yields to the largest hc whereas RHEED leads to the lowest ones, XRD producingintermediate results (Table 2). In the case of our [(YBa2Cu3O7)M/(PrBa2Cu3xGaxO7)Nlp superlattices, the XRD 0t20scans recorded across the (025)1(205) planes and 4 scans across the (225) ones of four superlattices evidence the tetragonalstress of the YBaCuO elementary layers of the shortest period structures when the YBaCuO thickness is lower than 4.8 nm(4 YBaCuO unit cells) (Fig. 4). The 1x4 (M=1, N4) and 3x5 structures are strongly tetragonaly strained with a = b0.3907 nm and 0.3889 urn respectively. Around M = 4 a transition into the orthorhombic phase is observed, which iscompleted for M = 6. However for M = 4 a strong in-plane stress is still found (a = 0.3868 nm, b = 0.3896 nm) and,similarly, for M = 6 a large residual stress can be detected with a = 0.3850 nm and b = 0.3897 nm. These results suggestthat the structure relaxation is governed by the thickness of the elementary layers rather than by that of the superstructure,since the calculation performed according to section 3. 1 and 3.2 in the range of our M and N values leads to hc(sl) valueswhich are around one order of magnitude larger than the corresponding hc(el). The upper limit of the superlattice criticalthickness is then better estimated by the anisotropic critical thickness ofthe YBaCuO single layer2.

344 ISPIE Vol. 2697


The critical layer thickness can also be determined by using a procedure of structural refinement of superlatticefrom XRD spectra, such as the refinement programme of E.E. Fullerton et a!. '. The critical thickness of thesesuperlattices, which is estimated to 4.8 nm as shown in Fig. 5, is in good agreement with the direct XRD measurements.The experimental hc-value of 4.5 nm which is reported by Horiuchi et al.7'8 for LaSrCuOIYBaCuO superlattices is also ingood agreement with our measurements although lower than the calculated value reported in Fig. 1, as the 30 urn thickLaSrCuO barriers are supposed to be relaxed, consequently, the mismatch of the elementaiy layers is the one betweenLaSrCuO and YBaCuO, i.e. 26%7,8 The case ofNdCaCuOTYBaCuO superlattices reported by Gupta et a!. 6is different asit could correspond to equally strained superlattices, explaining the large h (> 10 nm) which is deduced from theirexperimental results.

4. 1) 0/29 scans of (025)/(205) reflection. 4.2) 4 scans of the (225) reflection.Fig. 4. XRD patterns of strained and relaxed YBaCuOIPrBaCuGaO superlattices: [(a) (1x4)30, b) (3x5)20, c) (4x14)10, d)

(6x4)20].

However, for [(YBa2Cu3O7)M/(PrBa2Cu3XGaXO7)Np superlattices, as in the case of YBCO/LSCO superlatticesreported by Horiuchi et al. the lattice parameter varies continuously with the YBCO layer thickness from the substrateparameter to the one of single crystal in addition of the occurence of the plastic accomodation, as shown in Fig. 6, implyingthat an elastic accomodation is superimposed to the plastic one. This last observation could suggest that the wholerelaxation is occunng over a thickness which is significantly larger than the so called critical layer thickness. Such longdistance stress in YBCO layers has already been reported from the BBS study of a -axis YBCO layers grown on SrTiO345.This phenomenon was also observed by Ece et a!. who compared the FWHM of the rocking curves of the (005) X-rayreflection of YBCO films grown on SrTiO3 and NdGaO346. They demonstrated that the FWHM of the films grown onNdGaO3, a perfectly matched substrate, is independant of the thickness and very close to that of the substrate (0.04 <Ee <

0.1 deg.), whereas the one of films grown on SrTiO3, &ila 2.3%, decreases continuously from 0.4 to 0.15 when thethickness increases from 40 to 275 nm. Furthermore, if we include our F\VHM measurements carried out using electronsynchrotron radiation over ultra-thin YBCO layers grown on SrTiO3 (7.5, 15 and 30 nm) and reported in Fig. 7, they showthat before plastic accomodation the FWHM is exactly the same as that recorded for a NdGaO3 substrate, and that itincreases drastically across the range of the critical thickness and then overtakes the values observed by Ece et al (Fig. 8).

Therefore it can be assumed that the strained epitaxial layer shows a perfect crystal structure, which is stronglyperturbed as the plastic accomodation begins and that this perfect crystal structure is only recovered after the deposition of alayer having a thickness 5 to 10 times larger than hc. Consequently in the case of ETC superconductor strained structures,two critical thicknesses can be proposed: i) the plastic critical layer thickness corresponding to the rising of misfit

2000

1500

08

1000

500

0

C)

8

0C)

>aC)a

20 [O} cp [0]

60 61 62 63 64 43 44 45 46 47

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dislocations, ii) the elastic critical thickness corresponding to the end of elastic accomodation and to the recovering of thebulk lattice parameters and the perfect ciystal structure. The first one is determined through RHEED, AFM, STM, XRDmeasurements, the second one, which is around one order of magnitude higher, through RBS or XRD rocking analyses.

0 2 4 6 8

Number of YBCO unit cells

Fig. 5. Interfacial stress as determined from XRDrefinement versus the number of YBaCuO cells in theelementary period of the YBaCuOIPrBaCuGaOsuperlattices.

I

1 2 3 4 5 6

Numberof YBCO cells

Fig. 6. Lattice constants a and b versus the number ofYBaCuO cells in the elementary period of theYBaCuOIPrBaCuGaO superlattices.

0.21

a

0.20

U,.

0.19

IO.18

U

0.17 /U'

0390

0388U,

0.386

0.384

• a,!, b

--. .

bbulkYBCO

a'S,U

abulk YBCO

c

CO [O}

40000

20000

0

CO [O]

Fig. 7: Rocking curves ofthe (005) reflection ofultra-thin YBaCuO layers on SrTiO3: a) 7.5 nm; b) 15 nm; c) 30 nm.

5. CONCLUSION

In summary, we show that a Frank-van der Merwe energy model can be adapted to evaluate the critical layerthickness of YBaCuO thin films and YBaCuO based superlattices. The critical thickness of a single layer on SrTiO3calculated from the anisotropic model (16 nm) is in good agreement with the previously published experimental values

346 I SPIE Vol. 2697

CO [0]

18 19 20 21


which are spread out from 4 to 20 nm. In the case of superlattices, relaxation appears to be governed by the criticalthickness of the elementary sub-layers and is then better evaluated through the calculation performed for YBaCuO singlelayers. In addition of this critical layer thickness which corresponds to the beginning of the plastic accomodation, a longdistance disorder is induced by the lattice mismatch between the film and the substrate, the perfect crystal structure beingrecovered only after the growth of an epitaxial layer which is around one order of magnitude larger than the criticalthickness in the case of a SrTiO3 substrate. This first approach has however to be refined taking into account the stronganisotropy of the cuprates, their growth mechanisms and defects. Furthermore, experiments must first be carried out todetermine the nature of of the dislocation arrays that could be considered in the relaxation mechanisms, then to determinethe experimental critical layer thicknesses by in situ measurements, such as those that have been previously carried out inthe case of semiconductor epitaxy79, but taking into account the temperature range of the growth and the cooling steps50.

Q7

0 0 YBcD/STO, this works—' Q6

g a YBc0/sT0,Fecel a!.. as . YBc0/Ncio,Eceetal1

Q4 0D :Q3. 0

Q2

:: •0 75 100 125 150 175 2O 225 O 275 300

Filmthickrss [nm]

Fig. 8. The FWHM of the rocking curve of the (005) reflection of YIBaCuO films grown on SrTiO3 and NdGaO3, from ourmeasurements and those reported in ref. 46

6. ACKNOWLEDGMENTS

The authors thank N. Devisscher, M. Drouet and D. Ravelosona (C.N.R.S./Thomson-CSF, Orsay, Fr.), W.Schwegle (Diffraction Group of ESRF, Grenoble, Fr.) for their contribution and Pr. J. Bok (LPSIESPCI, Paris, Fr.) forhelpful discussions. The XRD refinement has been performed using Suprex 1.0 (E.E. Fullerton et a!. 44), the authors aregrateful to Dr. K. Temts (University ofLeuven, Be) for his assistance.

7. REFERENCES

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