PHYSICAL REVIEW B 66, 045305 ~2002!

Strain-mediated phase coexistence in MnAs heteroepitaxial films on GaAs:An x-ray diffraction study

V. M. Kaganer, B. Jenichen, F. Schippan, W. Braun, L. Da¨weritz, and K. H. PloogPaul-Drude-Institut fu¨r Festkorperelektronik, Hausvogteiplatz 5-7, D-10117 Berlin, Germany

~Received 6 September 2001; published 10 July 2002!

The temperature-dependent phase coexistence between crystalline phases in heteroepitaxial films of MnAson GaAs is studied. The epitaxial constraints on the film expansion are analyzed. The x-ray-diffraction data arefitted to a model of periodic elastic domains. The temperature dependencies of phase fractions, the domainsizes, and the misfits are simultaneously obtained. The domain sizes correspond to the minimum of elasticenergy, which proves the equilibrium state of the heteroepitaxial system at each temperature.

DOI: 10.1103/PhysRevB.66.045305 PACS number~s!: 68.35.Rh, 64.70.Kb, 61.50.Ks, 05.70.Np

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I. INTRODUCTION

Recently we found1 that a first-order structural phase trasition in MnAs heteroepitaxial films on GaAs proceeds inway qualitatively different from the same transition in buMnAs crystals. Instead of an abrupt transition with a teperature hysteresis inherent to the first-order transitionbulk crystals, we have observed a phase coexistencetemperature interval of more than 20 °C, with the fractionthe low-temperature phase linearly increasing on coolinglinearly decreasing on heating. We explained this phaseexistence by the restriction on lateral expansion of the fiimposed by the substrate. The coexistence is a result obalance between the free energy released at the phaseformation and the emerging elastic energy. In the preswork, we perform a detailed x-ray-diffraction study of thtemperature-dependent phase coexistence. We develmodel of periodic elastic domains in the layer, and find teperature dependencies of the phase fraction, misfit, andmain size by fitting the x-ray data to the model. We demostrate, by comparing the observed domain structure withenergy-minimizing one, that the film is close to the equilrium.

The formation of equilibrium polydomain structures asway to reduce elastic energy at structural phase transfortions in epitaxial films was first proposed theoreticallyRoytburd.2 The energetics of polydomain phases and eqlibrium structures were considered further by Bruinsma aZangwill,3 Kwak and co-workers,4,5 Speck andco-workers,6–9 Sridhar and co-workers,10,11 Roytburd andco-workers,12–15and Bratkovsky and Levanyuk.16,17 Most ofthe cited papers concentrate on the phase transformationsisting in a tetragonal distortion of a cubic crystal~inter-changeably referred to as martensitic, ferroelectric, orroelastic phase transformation!. At this transition, the elasticenergy is minimized by a coexistence of domains~twins!with three orthogonal directions of the tetragonal distortioThe orientations of domain boundaries are determinedcrystallographic orientations of neutral planes which allone to fit phases without strain at the boundary.2 A possiblecoexistence between the parent and a derivative phaseconsidered theoretically by Roytburd and co-workers12–15as-suming that the period of the domain structure is sma

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than the thickness of the film. This assumption is not apcable for the experimental system studied in the preswork. We therefore develop a theory which is not limitedthe domain widths.

MnAs on GaAs is a promising heteroepitaxial systewhich integrates magnetic and semiconductor propertOne of the aims in combining such materials is to injespin-polarized electrons into a semiconductor.18 The struc-ture of bulk MnAs crystals,19–22the epitaxial relationships oMnAs on GaAs,23–27 and the structure of the interface28,29

are known. Figure 1 sketches the epitaxy of MnAs onGaAs~001! surface. The hexagonal prism of the MnAs uncell is attached to the GaAs~001! surface by a side facetEpitaxial growth proceeds despite a large mismatch alongc axis of the prism which amounts to 33%. Transmissielectron microscopy studies28,29 showed that every sixthGaAs$220% plane fits into every fourth MnAs$0002% plane,which reduces the actual mismatch to 5%. This mismatchwell as the mismatch along the perpendicular direct~7.7%!, are released by regular arrays of misfit dislocatioThe MnAs films are grown28,29 with a unique epitaxial ori-entation with respect to the polar GaAs~001! surface, namely,(1100) MnAs i ~001! GaAs and @0001# MnAs i@110#GaAs, which was checked in the present study byin situreflection high-energy electron diffraction. It is essential fthe considerations below that the orientation of the film wrespect to the substrate is unique. There are no rotationequivalent domains~twins!, albeit translational domains cabe present.

The crystal structure of the bulk MnAs phases andphase diagram of MnAs were reviewed in Refs. 30–32. Blow 40 °C, the bulk MnAs crystal is ferromagnetic and form

FIG. 1. Scheme of the epitaxy of MnAs on GaAs~001!.

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KAGANER et al. PHYSICAL REVIEW B 66, 045305 ~2002!

the hexagonalaMnAs phase. At approximately 40 °C, thbulk aMnAs undergoes a first-order phase transition toparamagnetic orthorhombic phasebMnAs. Its unit cell isshown in Fig. 1 by the dashed line: the hexagon anisotrocally shrinks in both directions, while the height of the prisdoes not change. The orthorhombic distortion of thebMnAsunit cell is small20 (,0.2%). The transition is accompanieby a large ('1.2%) lattice parameter discontinuity in thhexagonal plane and a large ('15 °C) temperature hysteresis. The discontinuous ferromagnetic transition~while mag-netic phase transitions are commonly continuous at Ctemperature! is explained by strong magnetostrictioeffects.33 A further phase transition in bulk MnAs takes plaat 125 °C. This transition is continuous and results in a pamagneticgMnAs phase, which is again hexagonal. The scessive phase transformations in MnAs can be explaiwith the Landau theory involving two coupled order paraeters responsible for the orthorhombic distortion amagnetization.34 The magnetostriction suppresses distortat the ferromagnetic transition.

II. THEORY

A. Phase transition under mean-strain constraint

An epitaxial film experiencing a structural phase transfmation cannot freely change its size and shape, as is posfor a bulk single crystal, but is the subject of several costraints. One of them is an integral constraint imposed onwhole film: the film cannot change its lateral size and hethe mean lateral lattice spacing in the film is constant. Trequirement gives rise to elastic strain in the film and resin the phase coexistence at the first-order phtransformation,1 but does not restrict the domain sizes of tcoexisting phases. This is considered in the present subtion.

Further constraints are local and follow from the latticontinuity at the interfaces between the film and the substand between the domains. The latter requirement determthe orientation of the boundaries between domains of tetonal phase at the martensitic~ferroelectric! transition asplanes of zero misfit.2 The present paper deals with the unform dilatation in the hexagonal plane of the film, and straat the domain boundaries is unavoidable. The calculationthe displacement field, the minimization of the elastic enerand the calculation of the x-ray-diffraction pattern are psented in the subsequent subsections.

We adopt the notation to theaMnAs-bMnAs transition,which is studied experimentally in the present paper, arefer to the high-temperature phase as theb phase and thelow-temperature phase asa phase. We take the substrate ucell as a reference. First consider the film in a single-phstate~either thea or b phase! detached from the substratso that the film is free to expand. Then the relative diffences between lattice spacings of the substrate and thothe corresponding phase in afreeMnAs single crystal can bedescribed by the tensorsha andhb , commonly called inter-nal strain tensors. The differenceh5ha2hb consists of the

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phase transformation strain and the difference between tmal expansions of the two phases.

The latter contribution cannot be neglected for taMnAs-bMnAs transition, since theaMnAs phase experi-ences thermal contraction, whilebMnAs thermally expands.The values of the corresponding thermal coefficients19–22areof the order of 1024K21, i.e., very large compared to ththermal expansion coefficient of the GaAs substrate whis35 631026K21. The shear components of the misfit aabsent in the problem under consideration, and hence alldiagonal components ofha and hb are zero. We take intoaccount that the lattice parameters at theaMnAs-bMnAstransition are discontinuous only in the hexagonal planethathzz50. Orientation of the axes is shown in Fig. 1: theyaxis is normal to the film, andx and z axes are in the filmplane. We neglect the small orthorhombic distortion of tbMnAs phase compared to the change of the lattice pareter at the transition and takehaxx2hbxx5hayy2hbyy[h.These simplifications allow us to present the results incompact form.

The free film can be coherently attached to the substraan external stress is applied to the film, to make the film asubstrate lattices match along the interface. After the filmattached and the external stress is removed, the film becoelastically strained. The coexistence of thea andb phases isa way to minimize the elastic energy under the constrainthe constant lateral size of the film imposed by epitaxy. Lea and eb be the elastic strain tensors in the correspondphases. Then the relative change of the lattice spacing ofilm in the corresponding phase with respect to the substis given by the sum of the internal and the elastic strain«a

5ha1ea (a5a or b), which is called the total strain.Let j be the fraction of thea phase in the film. The latera

size of the epitaxial film is restricted by the substrate, ahence the lateral components of the mean total strain infilm are equal to zero:

j«axx1~12j!«bxx50,~1!

j«azz1~12j!«bzz50.

In this subsection, we do not take into considerationinterfaces between the substrate and the film and betweedomains of the film. Then the free-energy density of the ficonsists of the free-energy densities of the phases in thestrained state,f a and f b , and the elastic energy densitiesEaandEb :

f 5j~ f a1Ea!1~12j!~ f b1Eb!. ~2!

The elastic energy densities of the phases can bepressed through the elastic strainsea andeb . The absence ofthe stress normal to the filmsyy50 relates the strain components:eayy52@n/(12n)#(eaxx1eazz), wherea5a or bandn is the Poisson ratio. The film is assumed to be elacally isotropic. Then the elastic energy density of each phis

Ea5Y~eaxx2 12neaxxeazz1eazz

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STRAIN-MEDIATED PHASE COEXISTENCE IN MnAs . . . PHYSICAL REVIEW B 66, 045305 ~2002!

whereY5Y0/2(12n2), andY0 is the Young modulus.We substitute the elastic strainsea5 «a2ha (a5a,b)

into Eqs.~3! and, using Eqs.~1!, find the minimum of thefree energy density@Eq. ~2!# with respect to the total strainat a given fractionj of the a phase. The result is

f 5 f b1jD f 1Yj2h2, ~4!

where

D f 5 f a2 f b12Yh~hbxx1nhbzz!. ~5!

The bulk phase transition temperatureTc is determined bythe condition f a5 f b . The difference between the freeenergy densities in the two phasesf a2 f b is a linear functionof the temperature close to the transition,f a2 f b5Q(T2Tc)/Tc , whereQ is the latent heat. The last term of Eq.~5!shifts the transition temperature in the film with respectthe bulk transition temperature. We denote the transition tperature in the film byTc* .

The minimum of the free energy density@Eq. ~4!# isachieved by the phase coexistence with a fraction of thaphase:

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The temperature range of the phase coexistence is limitethe condition 0,j,1. Therefore, the low-temperatureaphase appears at the temperatureTc* and its fraction almostlinearly increases when the temperature is decreased bTc* . Some nonlinearity is introduced by the temperaturependence ofh which arises from the thermal expansionthe phases.1 We can estimate the temperature interval ofphase coexistenceDT52Yh2Tc* /Q using the values of la-tent heat36 Q51.8 cal/g, Young modulus37 Y053.231011 dyn/cm2, the internal strainh50.01, and the transition temperatureTc* 5320 K asDT'20 K, which is in agood agreement with the experimental results presebelow.

The mean-strain constraint@Eq. ~1!# gives rise to the finitetemperature interval of the phase coexistence, but doesrestrict the domain sizes of the coexisting phases. We nturn to this more elaborate problem, which requires a coplete treatment of the compatibility between the domainsbetween the film and the substrate.

B. Elastic domains

We consider an epitaxial film with a periodic arrayelastic domains of two alternating phases, Fig. 2. The th

FIG. 2. Scheme of a periodic domain structure with alternatdomains of two phases. The period of the domain structure isl.

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nesst of the film is taken as the length unit. The period of tdomain structure is 2l ~in units of t) and the domain boundaries are normal to the film. The elastic problem for thdomain geometry with arbitrary internal strain tensors wsolved by Sridhar, Rickman, and Srolovitz.10,11 In brief, thesolution consists of a Fourier expansion of the periodicternal strain and solution of the elastic equilibrium problefor each Fourier component. The elastic equilibrium equat]s i j /]xj50 is solved in the film where the stresss iscaused by the elastic strain«2h and in the substrate wherthe internal strainh is zero. The boundary conditions argiven at the free surface~tractions are absent!, at the film-substrate interface~displacements and tractions are continous!, and in the substrate far away from the interface~stressvanishes!. We repeat the calculations since we need the dplacement field in the filmuy(x,y) while only the elasticenergy is given in the cited papers. We calculate the elaenergy for the diagonal components of the internal strtensors and found only one minor error in the final formuof Ref. 10: in the coefficienta7 in Eq. ~B1! of Ref. 10 read2d11d22 instead of 2nd11d22. The final expression for thenormal displacementuy reads

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Figure 3 shows the normal displacementsuy(x,y) whichare calculated by Eqs.~7! and~8! for different periods of thedomain structure 2l. When the domains are narrow (l

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FIG. 3. The normal displacements in the film calculated by E~7! and~8! for different periods of domain structure 2l. The phasefraction isj50.3 for all plots. The misfit between phasesh50.01,the displacementuy is magnified by a factor of 10.

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KAGANER et al. PHYSICAL REVIEW B 66, 045305 ~2002!

,1), the coherence at the interfaces between the domresults in a common lattice spacing for the two phases infilm’s interior. In contrast, when the domains are widel@1), the spacings differ from each other and are close tofree spacings in the corresponding phases. We show inII C that this dependence of the spacing on the domain pemakes the x-ray-diffraction pattern sensitive to the domwidths.

The expression for the elastic energy10,11 is significantlysimplified for the case under consideration (hxx5hyy[h,hzz50). We find that the free energy density@Eq. ~4!# issupplemented with an additional term which depends onperiod of the domain structure:

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~pn!2. ~9!

The energyEd(l,j) for a fixed phase fractionj has a mini-mum at some finite value of the period 2lmin(j). This periodis shown in Fig. 4~a! as function of the phase fraction.diverges atj50 and 1 and reaches the minimum of 5.65j51/2. The width of the smaller domain, equal to 2jlmin forj,1/2 and 2(12j)lmin for j.1/2, remains finite and variefrom 1.4 atj50 and 1 to 2.8 forj51/2.

The energy-minimizing domain period 2lmin is a result ofa competition between the coherence requirements atfilm—substrate interface and at the interfaces betweendomains in the film. The coherence at the film-substrateterface favors narrow domains, since they localize stranear the interface in a layer with a thickness comparablethe domain width.12–15 On the other hand, for narrow do

FIG. 4. ~a! Period of the domain structure minimizing the elasenergy 2lmin ~full line! and the width of a smaller domain@equal to2jlmin for j,1/2 and 2(12j)lmin for j.1/2, broken lines#. ~b!Elastic energy for the energy-minimizing period,Ed(lmin ,j) ~fullline! and its approximation by a parabola~broken line!.

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mains the coherence at the interfaces between domainquires equal spacings in the direction normal to the filwhile for wide domains the elastic strains in that directi~i.e., eyy) relax independently in the two phases. The opmum is realized with domain widths comparable with tfilm thickness.

Figure 4~b! shows the elastic energy density for thenergy-minimizing period,Ed(lmin ,j). The broken line is anapproximation of this energy by a parabola

Ed~lmin ,j!'0.28Yh2j~12j!. ~10!

Adding this energy to Eq.~4!, we find that the domain energincreases the temperature range of phase coexistencefactor (120.28)21'1.4 and also causes an additional shof the transition temperature with respect to the bulk trantion.

C. X-ray scattering from elastic domains

The x-ray diffraction from the film can be calculated busing the kinematic scattering formula

I~qx ,qy!5U E exp@ iQ•u~x,y!1 i ~qxx1qyy!#dxdyU2

,

~11!

whereqx and qy are the deviations of the scattering vectfrom the reciprocal-lattice vectorQ and the integration isperformed over the film thickness and one period of its leral structure. We consider only the symmetric Bragg refltions ~vector Q is parallel to they axis!, and the verticaldisplacementuy(x,y) given by Eqs.~7! and ~8! is sufficientfor the calculations. The sharp surfaces of the film give rto thickness intensity oscillations while the domain periodity leads to satellite peaks. These features are not seen inexperiments presented below because of film thickness vations, misorientation of different parts of the film~mosaic-ity!, nonideal periodicity of the domains, and the finite agular resolution of the experiment. Therefore, we do nattempt to calculate the satellite peaks and average the thness intensity oscillations by taking into account the fimosaicity and the angular resolution of the detector.

Let f be the angular deviation of the diffracted beafrom the reference geometry andc the deviation of the filmnormal from the reference orientation. The correspondingviation of the wave vectors aredqx5k(c1f sinu) anddqy5kf cosu, where k is the wave vector andu is theBragg angle. We consider thev22u scan in the experimenand takeqx50 in the reference geometry. Then the averaof the intensity~11! with the detector resolution functionRd(f) and the film mosaicity distributionRm(c) is

I ~qy!5E I~qx8 ,qy8!RdS qy82qy

k cosu D3RmS qx82~qy82qy!tanu

k Ddqx8dqy8 . ~12!

Figure 5 presents the x-ray-diffraction peaks calculatedusing Eqs.~7!, ~8!, ~11!, and ~12! for the same domains a

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STRAIN-MEDIATED PHASE COEXISTENCE IN MnAs . . . PHYSICAL REVIEW B 66, 045305 ~2002!

shown in Fig. 3. The diffraction pattern qualitatively changas l is changed. Only one diffraction peak is seen forl,1, since the coherence between the domains whose wis essentially smaller than the film thickness results in a comon lattice spacing for both phases. The separation betwthe peaks gradually increases forl>1 and reaches~for l@1) a limiting value given by the mean-strain constraint. Wconclude that the diffraction pattern is sensitive to themain period.

III. EXPERIMENT

The MnAs layers were grown28 by solid sourcemolecular-beam epitaxy on 100-nm-thick GaAs buffer layat 250 °C with a growth rate of 19 nm h21. The GaAs sub-strate wafers were soldered with indium onto the Mo sstrate holder. The x-ray measurements on samples 1, 2, a~the thicknesses of the MnAs film were 60, 120, and 180 nrespectively! were performed on the as-grown samples. Fsamples 4 and 5~the thicknesses of the MnAs films were 10and 180 nm!, the soldering In layer was removed by thinninthe sample from the back side to thicknesses of 330267 mm, respectively.

The x-ray-diffractometric measurements were performin a high-resolution x-ray diffractometer with temperatucontrolled sample stages described below using a symmcally cut four-reflection Du Mond–Bartels type Ge 22monochromator placed at a distance of 30 cm fromsample and CuKa1 radiation. The angular acceptance of tdetector was 0.1 °.

The working temperature at the diffractometer was 30The measurements above that temperature were perfoby using a resistive heating. It was sufficient for the who

FIG. 5. Calculated x-ray-diffraction peaks for different perio

of the domain structure. (1100) reflection from MnAs, the phasfraction j50.3, film thicknesst5180 nm, phase transformatiostrain h50.01, angular resolution of detectorDf50.1° and filmmisorientationDc50.1°.

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set of measurements on samples 1 and 2. Sample 3 wasstudied under cooling with a separate circulating-fluid dvice. Samples 4 and 5 were studied with the aid of a cobined device with resistive heating and cooling by liqunitrogen. The estimated systematic uncertainty in tempeture determination was at most 2 °C. The curvaturessamples 4 and 5, caused by the misfit between the MnAsand the GaAs substrate, were measured in a double crtopographic camera equipped with a plane Si 440 collimacrystal38 by observing the displacement of the diffractiospot ~GaAs 135 or 115! over the sample surface with thchange of the incidence angle.

IV. RESULTS

Figure 6 presents thev22u diffractometric curve re-corded on sample 5 at 30 °C. Besides the strong subspeak~GaAs 002! two closely spaced peaks of the MnAs filmare visible. These peaks are the 1100 peak ofaMnAs and020 peak ofbMnAs. We use the hexagonal notation of threflections foraMnAs and the orthorhombic notation fobMnAs. If the small orthorhombic distortion is neglectethe 020 peak ofbMnAs can also be referred to as the 1100peak. The latter notation is used in Fig. 5, where the peaktwo phases are calculated.

A weak peak on the right from the substrate peak isMnAs 1101 peak originating from a small fraction of thfilm grown in 1101 direction.25,27 From the relative intensi-ties of the peaks and the ratio of their structure fact(u f 1101/ f 1100u2'10.3) we conclude that the fraction of th1101-grown film is about 0.3% and can be ignored in tanalysis below.

The MnAs peaks are shown in Fig. 7 in more detail aon a linear scale. A sequence of thev22u diffractometriccurves of sample 5 under stepwise cooling from 55 to 7and subsequent heating is shown. The temperature wasconstant while recording each curve. The sample rotaangle v was measured with respect to the position of tGaAs 002 substrate peak; cf. Fig. 6. Only one peak of

FIG. 6. Diffraction curve (v22u scan! near the GaAs~002! reflection measured at a temperature of 30 °C on sam

5 (Cu Ka1 radiation!. The aMnAs(1100) and thebMnAs (020)reflections are clearly distinguished.

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KAGANER et al. PHYSICAL REVIEW B 66, 045305 ~2002!

bMnAs phase is seen at high temperatures, and onlypeak of theaMnAs phase is present at low temperatures.intermediate temperatures both peaks are observed, wpoints to a phase coexistence in a temperature range of mthan 20 °C.1 The fraction of theaMnAs phase continuouslydecreases upon cooling and continuously increases uheating. The diffractometric curves are reproduced on thmal cycling without any change, which points to the equilrium nature of the observed phase coexistence.

The lines in Fig. 7 are the best fits of the curves tointensity calculated by Eqs.~7!, ~8!, ~11!, and~12!. The pa-rameters of fit are the phase fractionj, the domain period2l, the misfit between the phases of the filmh5haxx2hbxx , the difference of lattice spacings~normal to thefilm! between theaMnAs phase of the film and the substra

FIG. 7. Diffraction curves (v22u scans! of sample 5 obtainedon cooling and heating the sample. The arrows in the right-hpart indicate the sequence of the measurements. The lines arbest fits to the model of periodic elastic domains.

FIG. 8. The misfit betweenaMnAs and bMnAs phases as afunction of temperature.

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ey5(haxx1nhazz)/(12n), the peak intensity which is ascaling factor, and the intensity background. All these paraeters depend on the temperature. Fortunately, different fitrameters are sensitive to different features of the diffractcurve, which allows one to obtain them from one fit. Trelative integral intensities of the two peaks are given byphase fractionj. The distance between the peaks dependsthe misfit between phases of the filmh and the domain pe-riod 2l. The depth of the minimum between the peakssensitive tol. The misfit to the substrateey is given by thepeak positions with respect to the substrate peak.

Figure 8 shows the temperature dependence of the mbetween the MnAs phasesh obtained from the fits of thediffraction curves of all five samples on cooling and heatinThe experimental points follow a linear temperature depdenceh50.02323.8431024T@°C#. The temperature gra

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FIG. 9. Fractions ofaMnAs phase obtained on heating~up tri-angles! and cooling~down triangles! of the samples. The thicknest of the MnAs films is indicated.

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STRAIN-MEDIATED PHASE COEXISTENCE IN MnAs . . . PHYSICAL REVIEW B 66, 045305 ~2002!

dient of h is in a good agreement with the difference btween thermal contraction of theaMnAs phase and thermaexpansion of thebMnAs phase.19,21,22Further fits were per-formed taking this linear dependence ofh and excludinghfrom the list of fit parameters. These did not changeresults essentially but somewhat reduced scattering ofpoints.

Figure 9 presents the temperature dependencies offraction j of the aMnAs phase. This parameter is rathinsensitive to the fit model. It can be found simply from tratio of the integrated intensities of the peaks.1 The fractionof the aMnAs phase increases on cooling almost linearlyall samples in the temperature intervals of 15 to 25 °C. Tphase coexistence is explained by the theory preseabove. The phase coexistence is accompanied by a temture hysteresis in all samples with the exception of samplThe maximum hysteresis of 7 °C is found in sample 3. Itworth to note that the hysteresis in Fig. 9 cannot beplained in the same way as usual temperature hysteresisfirst order phase transition caused by the absence of nuation sites of the new phase.33 In the system under consideation, both phases are present at any temperature in the pcoexistence range in comparable, albeit different, amouThe change of the phase fraction proceeds without nuation by a barrierless motion of the domain walls.

Figure 10 presents the domain period 2l obtained fromthe fit of the diffraction peaks. The period plotted versusphase fractionj does not show a hysteresis which wouldpresent on al(T) plot. The data from all samples fall oncommon curve and well agree with the domain period foufrom the elastic energy minimum in Sec. II B. The agreembetween the domain periods found from the x-ray-diffractdata and by elastic energy minimization proves the equirium nature of the phase coexistence.

Figure 11 shows the temperature dependence of theference of lattice spacings between theaMnAs phase of thefilm and the substrate in the direction of the normal tofilm, ey5(haxx1nhazz)/(12n). The data from all five

FIG. 10. Period of the domain structure 2l obtained by the fit ofthe x-ray data to the model of periodic domains. Up and dotriangles denote the measurements on heating and coolingsamples, respectively. Samples 1–5 are denoted by the samebols as in Figs. 8 and 9. The solid line is the result of eneminimization presented in Fig. 4~a!.

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samples fall on a common curve. The temperature gradof ey is much smaller than that of the misfit between tphasesh shown in Fig. 8~note that the abscissas in Figs.and 11 give the same intervals of values!. Such a temperaturedependence results from the compensation of the therexpansion along thec axis (hazz8 .0, where the prime de-notes the temperature derivative! by the thermal contractionin the hexagonal plane (haxx8 ,0) and indicates thathaxx8 '2nhazz8 . This conclusion is in an agreement with measuments of the temperature dependencies of the MnAs latparameters in bulk crystals,21,22 which give the ratiohaxx8 /hazz8 in the range21/3 to 21/2.

The lateral misfit between the film and the substratebe calculated from the sample curvature measuremeTable I presents the results of the curvature measuremensamples 4 and 5 at 30 °C. The curvature radii are maximalong the direction GaAs@110# ~thec axis of the MnAs film!and minimum in the perpendicular direction~the hexagonalplane of the film!. The curvature radii for intermediate directions follow the Euler theorem, 1/R5sin2g/Ra1cos2g/Rc ,where Ra and Rc are the curvature radii in the hexagonplane and in thec direction of MnAs, respectively, andg is

nheym-y

FIG. 11. Relative differenceey between the (1100) aMnAs and~002! GaAs lattice spacings perpendicular to the surface in sam1–5. Note that the scale is chosen the same as in Fig. 8.

TABLE I. Radius of curvatureR and lateral misfite of theMnAs films at 30 °C in different directions.

Angle to CurvatureSample Reflection c axis radiusR Misfit e

~degrees! ~m! ~%!

115 0 6.78 2.7

4 315 26 7.77 2.3

315 74 19.4 0.94

115 90 32.82 0.55

115 0 3.88 1.7

5 315 26 2.9 2.3

315 74 3.62 1.8

115 90 38.8 0.17

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KAGANER et al. PHYSICAL REVIEW B 66, 045305 ~2002!

the angle between the given direction in the surface pland thec axis. The measured curvature radii allow onecalculate the lateral misfite of the film with respect to thesubstrate in the corresponding directions by using Stonequation39,40

1

R5

6t f

ts2

e, ~13!

where t f and ts are the film and substrate thicknessest f!ts). We havet f5100 nm andts5330 mm for sample 4,and t f5180 nm andts5267 mm for sample 5. The valueof the misfit in the hexagonal plane and in thec direction~i.e., in the two perpendicular directions parallel to the sface! are to be compared with the variations of the corsponding lattice parameters of MnAs in the temperatureterval from 250 °C ~the growth temperature! to roomtemperature, which are 0.6% and 1.5%, respectively.19 Weconclude that, at the growth temperature, the misfit is morelieved by networks of misfit dislocations. The thermal cotraction of the GaAs substrate can be neglected, andchange of the lattice parameters of the MnAs film on coolfrom the growth temperature is the dominant source of stand sample bending at room temperature.

V. CONCLUSIONS

We studied the first-order structural phase transformain heteroepitaxial films of MnAs on GaAs. The substra

,

r,

hy

W

ta

04530

e

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---

ly-hegin

n

does not allow the lateral expansion of the film, which resuin the phase coexistence governed by a balance betweefree energy released at the phase transformation andemerging elastic energy. The local constraints on the coence between the film and the substrate and between thephases of the film define the domain sizes of the coexisphases. We fitted the x-ray-diffraction data to the modelperiodic elastic domains and simultaneously obtainedphase fractions, the domain sizes, and the misfits. We fothat the fraction of the low-temperatureaMnAs phase lin-early increases on cooling and linearly decreases on heaThe phase coexistence occurs in a temperature interva15–25 °C and is accompanied by a temperature hysteres0 –7 °C which probably points to imperfections of thsamples. We found that the domain sizes correspond tominimum of elastic energy, which proves that the heteroetaxial system is at equilibrium at each temperature.

Note added in proof.The results of the present paper habeen confirmed recently by direct observationtemperature-dependent periodic surface corrugations duelastic domains of coexisting phases: see T. Plake, M. Rsteiner, V. M. Kaganer, B. Jenichen, M. Ka¨stner, L. Daweritz,and K. H. Ploog, Appl. Phys. Lett.80, 2523~2002!.

ACKNOWLEDGMENTS

V.M.K. thanks M. Fradkin and A. Roytburd for helpfudiscussions.

-P.

ct.

-

das,

.M.

.M.

.

.

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