Physics Letters A 367 (2007) 373–378

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Tilt growth of MnAs on the GaAs(001) substrate

Li Wan a,∗, Jinsi Shangguan a, Haijun Luo a, Yunmi Huang a, Ning-Hua Tong b

a Department of Physics, Wenzhou University, Wenzhou 325027, People’s Republic of Chinab Department of Physics, Renmin University, Beijing 100872, People’s Republic of China

Received 12 December 2006; received in revised form 11 March 2007; accepted 13 March 2007

Available online 19 March 2007

Communicated by R. Wu

Abstract

The complete strain tensor in the tilted-grown epilayer is studied based on the linear elastic theory. With the boundary constrain conditions,the tilt angle of an epilayer can be obtained from the minimization of the strain energy. In this way we can also describe the distortion of crystalcells in the epilayer. In this Letter, as an application of our theory, we focus on the growth of the MnAs epilayer on the GaAs(001) substrate. It isshown that the lattice mismatch strain between the MnAs epilayer and the GaAs substrate can be relaxed by two mechanisms. On the one hand,by forming the lattice coincidence construction at the interface, the type A growth can be realized. On the other hand, by tilting the epilayer byabout 30◦ with respect to the substrate, the type B growth is favored. The competition between the two mechanisms is near an equilibrium for thisspecific system, and depending on the growth conditions, it may lead to either an A type growth or a B type growth. Our theoretical results agreewell with the reported experimental observations.© 2007 Elsevier B.V. All rights reserved.

PACS: 68.55.Jk; 68.60.-p

Keywords: Tilt; MnAs; Elastic theory

1. Introduction

In the hetero-epitaxy growth experiments, the tilt growthis a common phenomenon for various material systems, suchas MnAs/GaAs(001), GaN/GaAs{11n}, MnAs/GaAs(113) andCu/GaAs [1–3]. The tilt growth is considered as consequenceof certain mechanisms to relax the mismatch strain between theepilayer and the substrate. Through tilting of the epilayer by acertain angle, the strain energy stored in the epilayer can be de-creased dramatically compared to the coherent growth. Basedon this idea, several models have been proposed to describe thetilted growth of the epilayers [2,4,5]. The final growth orienta-tion of the epilayer is dominated by the mechanism which canrelax more strain energy than the others. However, in Ref. [4],the model needs a vicinal substrate. In Ref. [2], the author con-siders that no shear strain exists in the plans parallel to the tilt

* Corresponding author.E-mail address: liwan_china@yahoo.com.cn (L. Wan).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.03.013

axis and obtains the tilt angle from the minimization of strainenergy. Although the tilt angle was predicted precisely, the dis-tortion of the crystal cell is not described. We will show belowthat in the case of a large tilt angle, a large shear strain existsand the crystal cell cannot be treated as in Ref. [2], without con-sidering the shear strain. For large lattice mismatch epitaxy, aninteresting model proposed in Ref. [5] is based on reciprocalspheres to find the growth orientation. The idea is that the strainenergy will be minimized if the reciprocal spheres of the epi-taxy and the substrate coincide most. However, by this methodthe crystal distortion still cannot be obtained.

For the large lattice mismatch growth, normally the unit cellnumbers of the epilayer and the substrate are different for theirlattice matching. Misfit dislocations should be introduced intothe tilted epilayer or the substrate in order to get a better latticematching [2]. The difference in the unit cell number plays animportant role in determining the tilt angle. Therefore, a gen-eral theory for the tilt angle must take into account the numberdifference. In this Letter, we present such a theory to describethe complete strain tensor. As an example, here we focus on the

374 L. Wan et al. / Physics Letters A 367 (2007) 373–378

tilted growth of MnAs/GaAs(001). This theory is quite generaland can be applied to other tilted-grown epilayers. The applica-tion of our theory to other material systems will be discussed inthe forthcoming papers.

MnAs, as one of the important ferromagnetic materialswith hexagonal structure, has been grown on semiconduc-tor substrates to integrate its ferromagnetic properties withthe semiconductor techniques [7,16]. Such integration pro-vides opportunities to realize spin filter in semiconductor de-vices. These electron spins could then be used as a new de-grees of freedom for information processing [12,17]. Varioussubstrates have been used for such sample growth, such asGaAs(001), GaAs(113) [6,7,13,18]. It has been reported that onthe GaAs(001) substrate, the MnAs epilayer has two main crys-tallgraphic orientations, namely, the type A and the type B [6,18]. For the type A orientation, MnAs[112̄0] and MnAs[0001],the a axis and c axis, respectively, are both in plane and paral-lel to the GaAs〈110〉. Lattice coincidence construction (LCC)is formed at the interface to relax the lattice mismatch betweenthe epilayer and the substrate in this type of growth [14]. Thetype B growth has similar crystal orientation with the type A,being different only in that the c axis is out of the plane andtilted around the MnAs[112̄0] by ∼ 30◦ [8,9]. The two growthorientations can be controlled by varying beam equivalent pres-sure (BEP) ratios of As4 and Mn in the growth chamber [8,9]. For the As-rich condition, the type A with nontilt growth isachieved while for the As-poor growth condition the tilt growthof type B is obtained [8,9]. Physical properties of the two kindsepilayers have been studied, showing that the tilt epilayer hasa higher phase transition temperature than the nontilt one [10].It is explained that in the tilted MnAs epilayer, the c/a ratiois smaller and the charge transfer from Mn to As is enhanced,which increases the phase transition temperature in the tiltedepilayer [11]. However, a detailed explanation is still lacking.To fully understand the growth of MnAs on the GaAs(001) sub-strate, it is necessary to obtain the complete strain tensor of theepilayer. Here, we will get the strain tensor by using linear elas-tic theory. Combined with the boundary constrain conditions,the tilt angle are obtained and compared with the reported ex-perimental results. To be specific, we study the epilayer systemMnAs/GaAs(001) with the α phase MnAs as the epilayer. Forsimplicity, the substrate is considered to be rigid and the mate-rials isotropic.

2. Method

The geometrical construction of the tilted epilayer boundedon substrate is illustrated in Fig. 1(a). Two sets of orthogonalcoordinates are shown in the figure by (x, y, z) and (x′, y′, z′)with coincident in-plane x and x′ axes parallel to the tilt axis,which is MnAs[112̄0] for MnAs/GaAs(001). Here, the y′ axis isperpendicular to the interface and y is perpendicular to the tiltedcrystal cell surface. Let εij be the components of strain tensorand σpq the components of stress tensor. Subscripts of all thesymbols throughout this Letter, such as the i, j , p, q of εij andσpq , refer to the axes described above. The stress tensor canthen be obtained from the stress-strain equation with Einstein

Fig. 1. (a) Illustration of the geometrical construction of a tilted epilayerbounded on the substrate. Here, we only show the cross section of the struc-ture construction, which is perpendicular to the tilt axis. Parameters m, n andk represent the numbers of lattice units in one tilt period. (b) Distortion of thetilt period. Dashed lines comprise the tilt period without considering the shearstrain while the solid lines are for the distorted one. (c) The structure construc-tion of the interface for the interface energy calculation. The dotted line parallelto the interface represents the neutral line without normal strain component.

notation: σpq = cpqij εij , where the cpqij are the componentsof elastic constant tensor. To get the strain components, twoboundary constrain conditions are used. The first one is thatthe components of strain tensor εxx , εzz and εxz in the film arefixed for a given material system, since the film is constrained tothe substrate. In most cases, the lattice meshes at the interfaceare rectangular with εxz = 0. The second boundary constraincondition is that the normal stress σy′y′ along the y′ axis is equalto zero since the whole sample is free along this direction. Touse the second boundary condition, it is necessary to transformthe stress tensor from the (x, y, z) coordinate to (x′, y′, z′) andthen let σy′y′ be zero. The coordinate transformation tensor is afunction of the tilt angle ϕ. After some algebra, we obtain thecomplete strain tensor in the tilted thin film of the followingform:

εyy = −ν Cos(2ϕ)εxx + (Sin2(ϕ) − ν)εzz

Cos2(ϕ) − ν,

εyz = −ν Sin(2ϕ)εxx − Sin(2ϕ)εzz

Cos2(ϕ) − ν,

(1)εxy = εxz = εyz = 0.

L. Wan et al. / Physics Letters A 367 (2007) 373–378 375

We take the Poisson ratio to be ν = 0.37 in this Letter. In pre-vious work, the shear strain component εyz is neglected whenthe epilayer is tilted with very small angle and the crystal cellis reasonably assumed to be still rectangle after tilting [12,13].However, Eq. (1) shows that the strain component εyz will bevery large, if a large tilt angle is considered and it will play animportant role in determining the tilt angle.

To be general, we illustrate the distortion of the two tiltperiods in Fig. 1(b). Coordination (x′′, y′′, z′′) deviates from(x, y, z) due to the crystal cell distortion. The dashed lines rep-resent the tilt period without considering the shear strain whilethe solid lines are for the distorted one. Bulk lattice parameterof the epilayer along the z axis is denoted by dz and the onealong y is by dy . The lattice parameter of the substrate along z′axis is dz′ . As indicated in Fig. 1(a), parameters m, n and k rep-resent the unit cell number of dz′ , dz and dy in one tilt period,respectively. We note that m and n may take the values of halfinteger since the epilayer and the substrate may have atomicplanes at the positions of a half integer lattice parameter, if thestep height for tilt is fixed to be a certain value. Since the atomicplanes of the epilayer are constrained by the substrate, the straincomponents εzz and εy′′y′′ can then be written as

εzz = mdz′ Cos(ϕ)

ndz

− 1,

(2)εy′′y′′ = mdz′ Cos(β)

kdz

− 1.

By transforming the (x′′, y′′, z′′) coordination to (x, y, z), εy′′y′′can then be expressed by εzz and εxx , which is to be solved forthe strain components. From the equation |AO|2 = |AB|2 +|BO|2 − 2|AB||BO|Cos(ϕ + β) for the tilt periods, we have

(mdz′)2 = (ndz(1 + εx′′x′′)

)2 + (kdy(1 + εyy)

)2

(3)− 2nkdydz(1 + εx′′x′′)(1 + εyy)Cos(ϕ + β).

From Eq. (3), we solve m and n as functions of k. The strainenergy stored in the tilted epilayer can in general be expressedas

Estrain ∼ ν

2(1 − 2ν)(εxx + εyy + εzz)

2 + 1

2

(ε2xx + ε2

yy + ε2zz

)

(4)+ 1

4ε2yz.

In this Letter, the energies always refer to the energies perunit volume. Substituting all the solved strain components intoEq. (4) and replacing β by ω + π

2 − ϕ, we obtain the strainenergy as a function of ϕ, ω, k and εxx . Here, the angle ω rep-resents the distortion of the crystal cell, as shown in Fig. 1(b).Finally the tilt angles ϕ and ω can be found at the minimum ofthe strain energy. We resort to the numerical calculation to solvethe minimization problem. In the calculation, the influence ofthe numerical error is treated carefully. In our calculation for theMnAs/GaAs(001), k = 1 is used as one bilayer height for tilt-ing and the tilt angle ϕ is restricted in the range 0◦ < ϕ < 50◦.For larger tilt angle, k should take the value 2 or even larger.

The interface energy between the tilted epilayer and the sub-strate also plays an important role in determining the tilt an-gle. The structure construction of the interface is illustrated in

Fig. 1(c). Here, we only show the cross section of the struc-ture construction, which is perpendicular to the tilt axis. Theinterface energy is confined in the space formed by the bottomlayer of the thin film and the top layer of the substrate. Sincethe interface structure construction is periodic, the interface en-ergy obtained in one tilt period is enough for our calculation.In the figure, one typical tilt period is represented by the trian-gle ABO with point D the projection of point B on the lineAO . The length of the lines of the triangle has also been indi-cated in the figure with AO = a, AD = c and BD = b. Theangle � ABO is not always a right angle if the existence of celldistortion in the epilayer is considered.

To calculate the interface energy, it is reasonable to assumethe chemical bonding energy in the interface is same to thechemical bonding energy in the epilayer, even though the mate-rial in the epilayer is difference to the material in the substrate.Based on the assumption, the interface energy is dominated bythe strain energy in the interface. And the strain energy in theinterface is mainly contributed by the normal strain εy′y′ , whichis perpendicular to the interface. The strain component εy′y′ isnegative at the point A but positive at point B . Thus, there ex-ists a neutral line parallel to the interface without any normalstrain εy′y′ . Due to the symmetry of the tilt period, the neutralline is located at the center of BD and has been indicated in thefigure as the dotted line. The neutral line meets the line AB atpoint E. In the following calculation of the interface energy, wetake the point E as origin. The distance between the point E

and a random point in the neutral line is noted as x. Based onthe elastic theory, the strain energy per unit volume can be ex-pressed as E = 1

2kε2 with k the elastic constant and ε the straincomponent. Thus, the interface energy can be integrated in thevolume ABO to be:

E = 2 ×c/2∫0

1

2k

(b

cx

)2

+ 2 ×a/2∫

c/2

1

2k

[b

a − c×

(x − a

2

)]2

,

which can be simplified to be:

E = kab2

24.

The total strain energy of the hetero-structure is comprised withtwo part energies. One part energy is the strain energy stored inthe epilayer and the other part energy is the interface energy,which has been derived as the above formula. The interface en-ergy is independent to the film thickness. However, while thetotal energy has been referred to the energy per unit volume,say the strain energy in the epilayer is divided by the whole vol-ume of the thin film, the interface energy also should be treatedin this way. Thus, the interface energy used for our tilt anglecalculation is obtained as:

Einterface = E/(a × Hdy) = kb2

24Hdy

.

In this formula, k is proportional to 4ν(1−2ν) after the Young’smodulus are cancelled as be done in Eq. (4). And b is derivedto be b = a

Tan(ϕ)+Tan(β)with a = mdz′ . Parameter H is the bi-

layer number. Thus we obtain the interface energy referred to

376 L. Wan et al. / Physics Letters A 367 (2007) 373–378

the energy per unit volume as the following formation:

(5)Einterface ∼ ν(1 − 2ν)(mdz′)2

6Hdy(Tan(ϕ) + Tan(β))2.

This part of energy is large at the initial stage of growth but de-creases rapidly with the increase of bilayer number H . Fig. 5(b)is a plot of the interface energy versus H for MnAs/GaAs(001).As shown in the figure, the tilted growth occurs only after a pe-riod of growth time, if the tilted growth can happen at all. Thewhole energy Etilt is the sum of the interface energy Einterfaceand the strain energy Estrain. Since the strain energy in the epi-layer is independent of the film thickness, it will dominate thewhole energy, when the film is thick enough. In the followingcalculations, we take H = 50 for a static analysis.

3. Discussions

Fig. 2 plots the Etilt versus the tilt angles ϕ and ω of theMnAs/GaAs(001) system with lateral strain component εxx

zero for simplicity. We use the parameters dz = 5.71 Å, dy =6.409 Å and dz′ = 3.998 Å for the plot. There exist two groupsof points in Fig. 2(a). Each point has its energy and tilt angle.

Fig. 2. (a) Energy stored in the tilted MnAs epilayer on GaAs(001) as a functionof the tilts angles ϕ and ω. (b) Project image of (a) on ϕ and ω planes.

The tilt angles for the MnAs epilayer should be obtained at thepoints with minimum energy. To get the tilt angles, we projectthe points on ϕ and ω planes, as shown in Fig. 2(b). Corre-sponding to the two group points in the figure, we find two spotswhich are located at the centers of the points, i.e., (ϕ = 33.5◦,ω = −6.5◦) and (ϕ = 45.2◦, ω = −7◦), denoted as P1 and P2,respectively. The tilt angle of P1 is consistent with the experi-mentally reported result, which is about 30◦ [10]. The existenceof a finite angle ω shows that the distortion of the crystal cellsmust be taken into account in the theory, as shown by the solidlines in Fig. 1(b), instead of the dashed lines in Fig. 1(b) withoutconsidering the distortion. The shear strain component εyz isobtained as 5.4%. The influence of such a crystal cell distortionwas neglected in previous works. Parameters m and n are foundto be m = 2.5 and n = 1.5 at point P1, which means that threeMnAs unit cells of dz are required to match five GaAs ones, andthat two bilayers are required for the tilting. Mismatch strain inthe epilayer along z axis εzz is obtained as −1.8%, which iscompressive, favoring the reduction of the c/a ratio. At pointP2, the growth surface of the epilayer is exactly the atom plane{11̄01} with m = 2 and n = 1. To our knowledge, experimentalresult consistent with this solution is still lacking.

Besides the tilt growth, the nontilt growth of MnAs/GaAs(001) of type A has also been obtained in the experimentsunder different growth conditions. LCC structure is formed atthe interface to relax the mismatch strain between the epilayerand the substrate in this growth type [14]. Competition betweenthe tilt growth and the LCC growth determines the final growthorientation. Before discussing the competition, we first specifythe geometrical construction of the LCC structure at the inter-face. For the type A orientation with ϕ = 0, Eq. (1) becomes

εyy = −ν(εxx + εzz)

1 − ν,

(6)εyz = εxy = εxz = εyz = 0.

This result can also be obtained from the general solution of thecomplete strain tensor for the nontilted epilayer [15]. For clar-ity, we use l and j to denote the numbers of the crystal cellunits, and define w = l − j as the number of the lattice mis-match. The strain components of Eq. (6) and εzz = ldz′

jdz− 1 are

substituted into Eq. (2) to get the strain energy Entilt in the non-tilt epilayer. Fig. 3 is a plot of Entilt as the function of l forvarious w. For simplicity, εxx = 0 is still used here. In the fig-ure, geometrical construction of the interface can be found atthe energy minimum. As a result, we obtain l : j = 3 : 2 if sec-ondary dislocations are not considered. The numbers here arethe nearest integers to the calculated values. The residual strainεzz is found to be 5% after forming such LCC structure, whichis tensile and increases the ratio c/a. This result is consistentwith the analysis of the transmission electron microscopy datafor the MnAs/GaAs(001) system [14].

At the beginning of the epitaxial growth, the MnAs epilayeris pseudomorphic on the substrate with l : j = 1 : 1. The mis-match strain εzz can be as high as 30%. With the increase of thefilm thickness, the mismatch strain has to be relaxed to avoidtoo high a strain energy. As mentioned before, the two strain re-laxation mechanisms compete with each other to relax the strain

L. Wan et al. / Physics Letters A 367 (2007) 373–378 377

Fig. 3. Strain energy stored in the nontilted MnAs epilayer on GaAs(001) versusthe number of substrate lattice unit l for various w.

energy. In one case, LCC is formed at the interface, leading tothe type A growth. In the other case, the epilayer is tilted byabout 30◦ to realize the type B growth. It is reported that thetwo growth modes are controllable by manipulating the growthconditions. Now there is the question why the two mechanismsboth can be observed for the thin film growth. To answer thisquestion, we should compare their residual strain energies tofind which mechanism can relax more mismatch strain. Here,the energy term of lateral strain εxx has to be taken into accountin the whole strain energy expression. We define the energyratios ρ1 = Etilt

Epsand ρ2 = Entilt

Eps, and plot them in Fig. 4 as func-

tions of the lateral strain εxx . The term Eps represents the strainenergy stored in the pseudomorphic epilayer. As indicated inthe figure, the two energy ratios are both smaller than 1, mean-ing that the two relaxation mechanisms truly effect in the strainrelaxation process. To investigate the competition between thetwo mechanisms, we define the energy ratio ρ = Etilt

Entiltand plot

it in Fig. 5(a) versus the lateral strain εxx with H = 50. As in-dicated in the figure, certain part of the curve is below ρ = 1and there the tilted growth is preferred. However, as shown inFig. 5(b), the interface energy is strongly dependent on the filmthickness. In particular, at the initial stage of the growth, the in-terface energy is high enough to push the curve over the line ofρ = 1. In this case, LCC is preferred for the growth. Fig. 5(c)gives the detail explanation. In Fig. 5(c), only the range of εxx

from 0 to 0.2 is plotted and the critical point (0.08, 1) is indi-cated since the lateral strain component εxx at the initial growthis 0.08. By varying the bilayer number H from 5 to 40, thecurve passes the critical point. That means, if the thickness forthe growth mode transition is lager than the critical thickness,then tilt growth is favored. On the other hand, LCC growth pre-vails.

Now we focus our attention on the lateral strain εxx . Asshown in Fig. 5(c), if the lateral strain εxx is relaxed to zerobefore the growth mode transition, only the LCC growth canbe obtained with energy ratio ρ always larger than 1. There-fore, the delay of the strain relaxation of εxx may lead to the tilt

Fig. 4. Dependence of the energy ratios ρ1 and ρ2 on the lateral strain εxx .

growth. Since the strain relaxations along the two directions x

and z axes are relatively independent, the relaxation of the lat-eral strain εxx plays an important role in determining the growthorientation of the epilayer. The determination of the thicknessesfor the growth mode transition and the relaxation of the lat-eral strain εxx are strongly dependent on the growth conditions.By manipulating As2 pressure in the growth chamber, differ-ent As atom coverage on the growth surface can be obtained,which results in various kinds of surface defects, such as steps,kinks and threading dislocations. These defects will produce lo-cal strain and modify the interface energy to some extent, andhence may push the competition away from the equilibrium,leading to a final growth orientation. To model the influence ofthe growth conditions on the lateral strain relaxation and the de-termination of the thicknesses is beyond the scope of this Letter.The main reason for the controllable growth is that the com-petition between the two strain relaxation mechanisms is nearthe equilibrium point and the different growth conditions caneasily push such competition away from the equilibrium point.Additionally, in Fig. 5(a), a peak with ρ = 3.5 and εxx = −3%is found in the range ρ > 1, where the nontilt growth is fa-vored. At this peak position, the tilted growth is less likely tooccur since the residual strain energy in the tilted epilayer is 3.5times larger than that in the nontilted one. In order to suppressthe tilted growth to get a uniformly well-oriented epilayer, wesuggest either to choose suitable substrates with smaller latticeparameters or to bend the substrate around the z′ axis during thesample growth to decrease the lateral strain εxx .

It has been reported that the MnAs epilayer tilted-grownon GaAs(001) substrate has a higher phase transition temper-ature than the nontilted one. Analysis shows that c/a ratio isdecreased more in the tilted epilayer to enhance the chargetransfer from Mn to As [10]. This compressive strain is con-firmed by our calculation. However, besides such compressivestrain, we also find a large shear strain εyz = 5.4% existing inthe tilted epilayer according to Eq. (1), while for the nontiltedone it is zero. It has been suggested that the Mn–As bond islargely ionic [11]. The bond energy is more dependent on the

378 L. Wan et al. / Physics Letters A 367 (2007) 373–378

Fig. 5. (a) The energy ratio ρ versus the lateral strain εxx . (b) Interface energyas a function of the MnAs bilayer number H , showing the energy is very largeat the initial growth stage and decreases rapidly with the increase of H . (c) Theenergy ratio ρ versus the lateral strain εxx with the consideration of the in-terface energy, showing that the competition between the two strain relaxationmechanisms is near the critical point (0.08,1). Different growth conditions orthe relaxation of the lateral strain εxx can push the competition away from thecritical point to realize different growth orientations.

bond length than on the bond angle [11]. Therefore, it seemsthat the shear strain makes less effects on the charge transferfrom Mn to As to increase the phase transition temperature.However, at the same time, the shear strain can influence thegrowth surface reconstruction and interface construction, whichmay influence the magnetization of the epilayer. In this respect,a detailed analysis is still in progress.

4. Conclusion

We solve the complete strain tensor in the tilted epilayer. Thetilt angle for the epilayer can be predicted by using the linearelastic theory with the proper boundary constrain conditions.We apply the theory to the system MnAs/GaAs(001) and showthat the lattice mismatch strain between the MnAs epilayer andthe GaAs substrate can be relaxed by two mechanisms. Oneis by forming LCC at the interface to get type A growth andthe other is by tilting the epilayer by about 30◦ to get type Bgrowth. The competition between the two mechanisms is nearan equilibrium point for this specific system. We suggest thatthe relaxation of the lateral strain εxx plays an important role indetermining the growth modes. The shear strain has also beenfound to be large in the tilted epilayer. However, the role ofa large shear strain in changing the magnetization of epilayeris still not clear. The method proposed in this Letter is quitegeneral and can be applied to the study of other tilted epilayers.

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