Real-space Green's tensors for stress and strain in crystals with ...

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Real-space Green's tensors for stress and strain in crystals with ...

033534-2 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005TABLE I. Elastic parameters for common semiconductor materials Ref. 43. The anisotropy coefficient isequal to C 11 −C 12 /2C 44 . and r are expansion coefficients defined by Eqs. 12 and 10, respectively.C 11GPaC 12 =GPaC 44 =GPaAnisotropycoefficient r Si 165.77 63.93 79.62 0.64 0.29 −0.26Ge 124.00 41.30 68.30 0.61 0.30 −0.30AlP 165.00 75.50 39.70 1.13 0.13 0.07AlAs 120.20 57.00 58.90 0.54 0.38 −0.31AlSb 87.69 43.41 40.76 0.54 0.38 −0.30GaP 140.50 62.03 70.33 0.56 0.36 −0.31GaAs 118.80 53.80 59.40 0.55 0.36 −0.31GaSb 88.34 40.23 43.22 0.56 0.36 −0.30InP 101.10 56.10 45.60 0.49 0.43 −0.31InAs 83.29 45.26 39.59 0.48 0.43 −0.33InSb 66.69 36.45 30.20 0.50 0.42 −0.31the physics, revealing trends and simple relations. 12–20 Forexample, within the isotropic approximation, full analyticalexpressions for the strain distribution have been derived forellipsoidal, cuboidal, and truncated-pyramidal dots 13,16 andfor quantum wires of arbitrary shape 12 and near to freesurfaces. 15 The Green’s technique is a powerful method forthe calculation of QD- or QWI-induced strain distributionsand its ease of application makes it attractive to practitionersin the field.Most applications of the Green’s approach assume isotropyof the elastic constants. For some problems an isotropicsolution may be justified because there are much greater uncertaintiessuch as dot size and shape which dominate theaccuracy of the calculation, or anisotropy is not significantfor the phenomena under investigation. Yet most semiconductorsare cubic crystals with an anisotropy coefficient definedin terms of the elastic constants by C 11 −C 12 /2C 44 typically equal to 0.5 compared to the isotropic value of 1see Table I, suggesting that the simple isotropic approachmay miss some key features of the strain field. For example,Holý et al. and Pinczolits et al. demonstrated that the lateralordering of QDs can be explained only if the full cubic anisotropyof the crystal is taken into account, 21,22 Faux andPearson demonstrated that the hydrostatic strain which influencesthe electronic structure is finite in the matrix materialoutside the QD if the anisotropy of the elastic stiffnessesis taken into account but zero if isotropy is assumed, 23 Panand Yang demonstrated significant differences between isotropicand anisotropic strain fields for a QD close to a freesurface, 24 and Tadić et al. showed that the calculated valuesof electron and hole energies are influenced by the anisotropyof the crystal structure. 3 It is reasonable to assume that,of all the approximations pertinent to a strain evaluation,neglect of the cubic anisotropy of the elastic constants ispotentially the most important.Classic work by Burgers, 25 who determined some termsof a series approximation for a linear elastic material withcubic symmetry; Barnett, 26 who determined the elasticGreen’s functions for cubic crystals that require a single numericalintegral, and Mura and Kinoshita, 27 who determineda series equation for the Green’s functions for cubic crystals,all contributed to our knowledge of strain fields in cubicmaterials many years before semiconductor QDs became important.Notable recent work has been performed by Pan andco-workers who have determined the Green’s tensor includingthe effect of electromechanically coupled piezoelectricfields for a QD or a QWI buried in an anisotropic full orhalf-space. 24,28–30 For the case of a QD, Pan’s Green’s tensoris nearly analytic, requiring the numerical solution to aneighth-order polynomial equation followed by a numericaldifferentiation to evaluate the stress or strain. However, inmost III-V systems piezoelectric fields are not important IIInitridesystems and InAs/GaAs with the 111 orientation areexceptions.In 2000, 13 we published simple expressions for the strainGreen’s tensor that enabled the strain to be calculated in QDsystems including the cubic anisotropy of the elastic stiffnesses.The Green’s tensor was expressed as a polynomialseries with the zeroth-, first-, and second-order terms presented.Although an approximate solution, truncation at secondorder appeared to provide a very good approximation tothe exact solution for the specific case of a spherical QD.In this paper we present the theory leading to explicitreal-space Green’s tensors which enable the stress and straindistributions in and around QDs and QWIs to be calculatedrapidly and accurately including the cubic anisotropy of theelastic stiffnesses. The Green’s tensor is compared directly tothe formalism of Pan and Yang 24 and we show that the polynomialexpression for the Green’s tensor is indeed veryaccurate.There are many advantages to using the real-spaceGreen’s tensor expansion presented here compared to thePan-Yang formalism. First, the implementation is verystraightforward as the determination of either the stress orstrain involves a volume integral surface integral for wiresof a simple polynomial function. The Pan-Yang approach ismuch more complex to implement as several stages are requiredto determine the Green’s function for displacementfollowed by numerical differentiation to obtain the strainGreen’s function, then a surface integral to calculate thestrain. The computation of the strain at a field point, forexample, is about 50 times faster using the Green’s tensorexpansion than the Pan-Yang method. One example of a resultthat can only be computed using the current method, andDownloaded 31 Oct 2005 to 131.227.244.81. 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033534-3 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005not with the Pan-Yang approach, is the estimation of theenergy of interaction of two point defects. 31 Here the differenceof two energies is required, each energy computed byintegrating the product of two strains over a large volume ofspace. The computational resource required for the Pan-Yangapproach is too demanding to be practical at present to obtainaccurate energies. Even if computation time was not aproblem, we found that numerical computations intrinsic tothe Pan-Yang approach lead to results that are only accurateto three or four significant figures. Normally, this would beperfectly acceptable, but when two similar energies are subtracted,the numerical errors become significant.This paper is organized as follows. The theory is presentedin Sec. II for both QDs and QWIs, based on the workof Mura and Kinoshita. 27 Comparison to the Pan-Yang formalismand to exact results in certain limits is presented inSec. III. We draw our conclusions in Sec. IV.II. THEORYA. Quantum dotThe Green’s tensor expressions determined in this sectionare derived from the work of Mura and Kinoshita 27 whodescribed, from first principles, how the Green’s tensor foranisotropic elasticity can be determined in a series formbased on the work of Lifshits and Rosentsveig. 32 Their expressions,which are mathematically complex in the generalcase, can be simplified for anisotropic cubic crystals. Muraand Kinoshita did not evaluate explicitly the Green’s tensors,instead, they provided a series expression involving a largenumber of differentials with respect to the Cartesian spatialcoordinates. Modern computer algebra packages enable thedifferentials to be executed and the full Green’s tensors obtained.Below, the results of Mura and Kinoshita’s theory foranisotropic cubic crystals are summarized using, as far aspractical, their notation. This theory is then used to producethe Green’s tensors for stress and strain for both quantumdots and wires.The theory that follows is developed for the case of aQD inclusion. It is assumed that the elastic properties arelinear, that the same elastic stiffnesses pertain to the QD andthe surroundings, and that the QD is embedded in an infinitematrix. The use of the elastic stiffnesses of the matrix materialfor both matrix and QD is the most significant approximationand this has been discussed elsewhere. 13,33,34 Currently,even the most sophisticated Green’s analyses assumeconstant elastic stiffnesses for all materials. 24,28The calculation begins with the evaluation of a 33tensor V k,6n+5 whose elements are polynomials of degree6n+5 and where = 1 , 2 , 3 is a unit vector. There isone tensor for each spatial direction represented by k=1,2,3. The x 1 , x 2 , and x 3 axes correspond to the 100,010, and 001 crystallographic directions, respectively.V k,6n+5 is given byV k,6n+5 = 6n+8Ĝ n, 1 k=in which Ĝ n is a rank 3 tensor and = 1 , 2 , 3 is avector. The term 6n+8 is equal to 1 because is a unitvector and is included to ensure that this expression remainsdimensionally correct and homogeneous of order 6n+5 in. Readers seeking the full mathematical treatment leadingto Eq. 1 are referred to Mura and Kinoshita. 27For crystals with cubic anisotropy, the function Ĝ n may be written as 27Ĝ n = −1 n n N 6n+6 , = b 2 2 2 3 2 + 3 2 1 2 + 1 2 2 2 + c 1 2 2 2 3 2 ,a = 2 +2 + ,b = a −1 2 +2 + ,c = a −1 2 3 +3 + ,where =C 44 and =C 12 are the usual Lamé constants and=C 11 −C 12 −2C 44 is the anisotropy index. Cubic crystalsare isotropic when =0 in which case b=c==0 and onlythe n=0 term of Eq. 2 remains. N is 2723456= 2 4 + 2 2 2 + 2 3 + 2 22 3 − + 1 2 2 + 2 3 ¯ ¯ − + 1 3 2 + 2 2 2N − + 2 1 2 + 2 3 2 4 + 2 2 3 + 2 1 + 2 23 1 ¯ ¯ − + 2 3 2 + 2 1 − + 3 1 2 + 2 2 − + 3 2 2 + 2 1 ¯ ¯ 2 4 + 2 2 1 + 2 2 + 2 1 27wherer = maxb/4,b/3 + c/27.10 = + + , = 2 +2 + .The expansion of Eq. 2 is valid if r1, where89Mura and Kinoshita included a table of some common materials,mainly metals, and showed that typical values of rrange from 0.5 to 0.7. The values of r for common semiconductorcompounds are listed in Table I and typically rangefrom 0.3 to 0.45. Therefore the series expansion will con-Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-4 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005TABLE II. The coefficients K ij are of the form fA 2 +B+C 2 where and are the Lamé constants.i j f A B C3 0 2 22 41 572 0,1 −3 34 97 171 0,2 3 18 49 331 −18 4 12 90 0,3 −1 10 5 −31,2 −36 0 0 1verge more rapidly for semiconductor materials than formost metals.The real-space Green’s tensor can be obtained by summationover k and n, thus,G x =− 1 4a n=0136n +6! k=1x kx 2V k,6n+5x 6n+5 ,11where x=x 1 ,x 2 ,x 3 are the usual Cartesian coordinates,x 2 =x 1 2 +x 2 2 +x 3 2 , and where the superscripts “” and “”have been added to indicate that the Green’s tensor is forstress and for the case of QDs, respectively.In Eq. 11 the components of the unit vector in Eq.1 have been replaced by the differential operator so that,for instance, a term in the matrix V k,6n+5 involving i j isreplaced by the differential operator j /x i acting on x 6n+5 .As terms in the matrix V k,6n+5 are homogeneous of order6n+5 in 1 , 2 , 3 , V k,6n+5 involves terms in which 1 , 2 ,and 3 may each be raised to a power between 0 and 6n+5, but where the sum of the powers is equal to 6n+5. Thismeans that V k,6n+5 contains terms with operators of theform i /x 1 j /x 2 6n+5−i−j /x 3 with each i and j takingvalues between 0 and 6n+5. If we take the n=2 term, forexample, x 6n+5 has to be differentiated a total of 17 times ineach of 1818 different ways. This would be a formidabletask for even the most ardent theorist!Mura and Kinoshita proceeded no further than Eq. 11but the calculation can be performed using an algebraicpackage such as XMAPLE. 35 The aim is to obtain the stressGreen’s tensor expansion where the first term is the isotropicstress Green’s tensor and subsequent terms are successiveorders of correction for the anisotropic case. In order to convenientlyexpress the tensor expansion, a quantity, , isdefinedwhereTABLE III. The coefficients K ij are of the form fA 2 +B+C 2 where and are the Lamé constants.i j f A B C3 0 2 14 19 92 0,1 −9 14 35 171 0,2 9 6 19 151 −36 2 3 00 0,3 −1 2 1 91,2 6 4 12 3TABLE IV. The coefficients L ki are of the form fA 2 +B+C 2 where and are the Lamé constants.k i f A B C2 0 3 22 51 391 0,1 −3 26 83 420 0,2 3 22 41 241 −3 26 93 57 = +2 .12The elements contained in the n=0,1,..., tensors of Eq. 11can be expanded in terms of , provided 1, and termsinvolving increasing powers of can be collected to yield anexpansion of the formG = G 0 + G 1 + 2 G 2 ¯ = m G m .m=013This expression applies to both stress and strain, in otherwords, the superscript can be replaced by to representstrain or by to represent stress. Equation 13 for stress isequivalent to Eq. 11 where the expansion is expressed inincreasing powers of .It is now convenient to revert to nontensorial Voigt notationas the preferred way to express stress and strain components.In this case, Eq. 13 represents an expansion interms of six-element column matrices. Using Voigt notation,the strain components are obtained from the stress componentsin the usual way through Hooke’s Law, namely,G = C −1 G ,where C −1 is the inverse of the matrix1411 0 0 0 C 11 0 0 0 C 11 0 0 0C =C 150 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2,where we note the presence of 2 in the matrix necessary toconvert the stresses defined by Mura and Kinoshita into theusual form for the strains. G 0 is obtained from the n=0term of Eq. 11 only, G 1 is obtained from the n=0 andn=1 terms, G 2 from the n=0, n=1, and n=2 terms, and soon. If we take the isotropic limit, =0, all terms disappearexcept for the zeroth which therefore contains the wellknownstress or strain Green’s tensor for isotropic elasticity.The dimensionless quantity is presented in Table I for arange of semiconductors and is typically equal to −1/3.We have found that the mth Green’s six-element matrixof the series, G m ,inEq.13 may be written in terms oftwo functions,Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-5 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005TABLE V. The coefficients M ij are of the form fA 3 +B 2 +C 2+D 3 where and are the Lamé constants.i j f A B C D5 0 −2 118 177 78 574 0,1 1 1910 5229 5292 19593 0,2 −2 1180 5712 7221 27121 −4 310 2547 2691 1922 0,3 1 1850 6379 7572 26491,2 −3 130 2031 2308 −1291 0,4 −2 280 422 111 1321,3 2 950 3547 3531 13622 −6 230 1449 2822 12090 0,5 1 14 25 36 −151,4 −5 46 5 −90 −1112,3 −5 26 103 522 219G m x 1 ,x 2 ,x 3 =4m+5G mP m G mx G m x 1 ,x 2 ,x 3 G mx 2 ,x 3 ,x 1 x 3 ,x 1 ,x 2 16x 2 ,x 3 ,x 1 x 1 ,x 3 ,x 2 x 1 ,x 2 ,x 3 ,G mG mwhere the two distinct functions are given the symbols and to indicate plane and shear components. In general, theplane components represented by G mdepend on whetherthey relate to stress or strain but the shearcomponents differ only through the constant P m .The constants P m areP m = 0m!4 m+1 m 2 + , 17P m =2P m .18where 0 is the uniform misfit strain in an unrelaxed QDusing the convention that 0 is negative for a compressivelystrained QD. The Green’s functions for m=0 areG 0x 1 ,x 2 ,x 3 =−2 +3x 2 −3x 2 1 ,19TABLE VI. The coefficients M ij are of the form fA 3 +B 2 +C 2+D 3 where and are the Lamé constants.i j f A B C D5 0 −2 −6 35 44 34 0,1 −1 1350 2477 2060 7413 0,2 2 1860 7336 7405 23731 4 1350 5611 4975 11582 0,3 −1 1690 7307 11080 43711,2 3 1170 3223 340 −6211 0,4 2 80 −74 −25 2731,3 −2 1270 5291 5755 11582 6 150 197 380 6210 0,5 1 2 −17 92 31,4 5 14 −11 122 392,3 −5 86 457 230 75TABLE VII. The coefficients N ki are of the form fA 3 +B 2 +C 2+D 3 where and are the Lamé constants.k i f A B C D4 0 3 38 37 8 1173 0,1 −6 374 1221 1524 5312 0,2 3 818 3687 4248 16771 8 176 1094 666 −811 0,3 −6 294 871 929 2761,2 2 784 5566 5499 9810 0,4 3 118 177 78 571,3 −2 1042 3103 3312 12332 1 2054 10 781 14 004 5121G 0 x 1 ,x 2 ,x 3 =32 +3x 1 x 2 .20Equations 19 and 20 combined with Eq. 16 with m=0yield the well-known stress or strain Green’s matrix for isotropiccrystals. 17 We note that the only difference betweenthe stress and strain Green’s matrices for the m=0 case is afactor of 2 that appears in the constant given by Eq. 18. Inother words, Eq. 19 is identical for both stress and strainfor the isotropic case m=0. The m0 terms approximate thecubic anisotropy of the crystal. We evaluate these for m=1and m=2 only.The Green’s matrix for m=1 may be written asG 1 x 1 ,x 2 ,x 3 = G 13 3−ij=0i=0 x 1 ,x 2 ,x 3 = x 1 x 2k=0K ij x 1 2i x 2 2j x 3 6−2i−2j ,2 2−ki=0L ki x 2k 3 x 2i 1 x 4−2i−2k 2 ,2122and are polynomials of order 6. The constants K , K , andL are presented in Tables II–IV.The functions forming the m=2 Green’s tensor containpolynomials of order 10,G 2 x 1 ,x 2 ,x 3 = G 25 5−ij=0i=0 x 1 ,x 2 ,x 3 = x 1 x 2k=0M ij x 1 2i x 2 2j x 3 10−2i−2j ,4 4−ki=0N ki x 2k 3 x 2i 1 x 8−2i−2k 2 ,2324where the constants M , M , and N are presented inTables V–VII. Files containing the coefficients presented inthe tables can be obtained by emailing the author.The full Green’s matrix for either stress or strain, tosecond order in , can therefore be obtained from Eqs.16–24. The stress or strain matrix for a QD is obtained byintegrating the stress or strain Green’s matrix over the volumeof the dot. Thus, for stress, x 1 ,x 2 ,x 3 =VG x 1 ,x 2 ,x 3 dVx 1 ,x 2 ,x 3 , 25where x i =x i −x i are Cartesian coordinates where x i refersto points within the volume of the QD and V therefore representsthe volume of the dot.Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-6 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005TABLE VIII. The coefficients J ij for the calculation of the QD hydrostaticstrain are of the form fA 2 +B+C 2 where and are the Laméconstants.ij f A B C40, 00, 04 1 4 −26 2430, 31, 10, 13, 01, 03 −1 284 644 39920, 22, 02 6 114 319 17421, 11, 12 6 108 268 3TABLE X. The coefficients K i for a quantum wire are of the formfA 2 +B+C 2 where and are the Lamé constants.i f A B C3 6 2 1 12 −6 22 51 231 2 26 93 690 2 2 9 3The analytic expressions for the components of theGreen’s matrix can be extremely useful. For example, thehydrostatic strain is defined as h = 11 + 22 + 33 and so themth hydrostatic strain Green’s function, G m h , is easily obtained.We find that these Green’s functions are of a particularlysimple formG 0 h =0,G 1 h = 122 +3P 1 x 7 x 1 4 + x 2 4 + x 3 4 −3x 1 2 x 2 2 + x 2 2 x 3226+ x 3 2 x 1 2 , 27G h 2 = 4P 2 x 114 4−ij=0i=0J ij x 1 2i x 2 2j x 3 8−2i−2j ,28where the constants J are listed in Table VIII. The hydrostaticstrain shifts the conduction-band- and average valenceband-edgeenergies 18,36 and is an important quantity in electronicstructure analysis. Equation 26 indicates that theGreen’s function is zero in the isotropic approximation andso the direct integration over the volume of a QD will alsoyield a zero result. It is always necessary to add the initialconditions to obtain the correct hydrostatic strain so that4 0 /+2 is added to the volume integrals obtainedfrom Eqs. 27 and 28 for a field point inside the QD.Equation 27 was presented by Faux and Pearson. 23The stress or strain can therefore be computed for anyQD shape by computing the Green’s matrix through Eqs.13–24 and then integrating over the volume of the dot.The principle of superposition allows the strain in systemscontaining large numbers of QDs to be determined and compositionvariation within the QD is also easily catered for bymaking 0 a function of position and including it inside theintegral.These polynomial functions constitute very simple expressionswhich are straightforward to compute. No numericalcomputation is required to generate the Green’s tensorsTABLE IX. The coefficients K i for a quantum wire are of the formfA 2 +B+C 2 where and are the Lamé constants.i f A B C3 2 14 23 152 −2 58 157 931 6 6 19 110 −6 2 1 1but the truncated series is not exact. Nevertheless, we showin Sec. III that they form an excellent approximation to theexact results for cubic anisotropy.B. Quantum wiresOne of the significant advantages of developing analyticforms for the QD Green’s tensors is that they may be integratedwith respect to one coordinate, x 2 say, to obtain immediatelythe Green’s matrix for a line source. Thus,G x 1 ,x 3 G =− x 1 ,x 2 ,x 3 dx 2 .29where the symbol is used to indicate a quantum wire witha cross section in the x 1 ,x 3 plane. The stress or strainGreen’s matrix for a line source, if integrated over the crosssectionalarea of a QWI, yields the stress or strain, respectively,at a position in the x 1 ,x 3 plane due to a strainedQWI. The Green’s tensors for the QWI can be expressed asG m x 1 ,x 3 =4m+4G mP mX x 1 ,x 3 0G mx 3 ,x 1 0G m, 30x 1 ,x 3 0where, for the QWI, x 2 =x 2 1 +x 2 3 . The QWI Green’s tensorfor m=0 may be written asG 0x 1 ,x 3 =22 +3x 2 1 − x 2 3 ,31G 0 x 1 ,x 3 =42 +3x 1 x 3 .32Equations 31 and 32 combined with Eq. 30 with n=0yield the well-known stress or strain Green’s matrix for isotropiccrystals. 14The QWI Green’s tensor for m=1 may be written as x 1 ,x 3 = K i x 2i 1 x 6−2i 3 ,G 13i=033TABLE XI. The coefficients L k for a quantum wire are of the formfA 2 +B+C 2 where and are the Lamé constants.k f A B C2,0 4 14 23 151 −8 10 37 21Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-7 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005TABLE XII. The coefficients M i for a quantum wire are of the formfA 3 +B 2 +C 2 +D 3 where and are the Lamé constants.i f A B C D5 −2 38 37 −24 214 6 234 531 536 2433 −4 470 2445 3048 10052 4 358 1157 2512 5011 −6 58 43 24 270 −2 10 3 24 3TABLE XIII. The coefficients M i for a quantum wire are of the formfA 3 +B 2 +C 2 +D 3 where and are the Lamé constants.i f A B C D5 −2 10 3 24 34 −6 90 11 48 513 20 174 641 584 1772 −4 310 1581 2416 8851 −6 22 237 168 510 −6 2 23 −16 −1 x 1 ,x 3 = x 1 x 3G 12k=0L k x 3 2k x 1 4−2k .34The constants K , K , and L are presented in TablesIX–XI.The functions forming the m=2 QWI Green’s tensor arepolynomials of order 10, x 1 ,x 3 = M i x 2i 1 x 10−2i 3 ,G 2G 25i=0 x 1 ,x 3 = x 1 x 34k=0N k x 3 2k x 1 8−2k ,3536where the constants M , M , and N are listed in TablesXII–XIV.The zeroth-order hydrostatic strain, as for the QD, iszero. The hydrostatic Green’s function G h 1 x 1 ,x 3 , definedas G 1x 1 ,x 3 +G 1x 3 ,x 1 as the first-order correction for aQWI, is found to beG h 1 x 1 ,x 3 = 8P 1 2x 6 H i x 2i 1 x 4−2i 3 ,37i=0where H 0 =H 2 =2+3 and H 1 =−62+3. We notethe especially simple form of this expression. The secondorderhydrostatic Green’s function G h 2 x 1 ,x 3 isG h 2 x 1 ,x 3 = 16P 2 4x 10 J i x 2i 1 x 8−2i 3 ,38i=0whereJ 0 = J 4 = −2 2 −9 +3 2 ,J 1 = J 3 =−410 2 +21 +21 2 ,J 2 =1018 2 +49 +21 2 .394041These Green’s functions enable the hydrostatic strain in andaround a QWI to be computed simply and quickly.III. RESULTSWe undertake a careful assessment of the accuracy of theGreen’s tensors produced from the formalism presented inSec. II. In earlier work, 23 the strain component 11 was calculatedfor a spherical QD, for the region outside the QD inthe 100 direction, and compared to the approximate Fourierapproach of Andreev et al. 33 A very good agreement wasobtained. Here, we use the Pan approach as thestandard. 24,28–30 In principle, Pan’s theory provides an exactGreen’s tensor for crystals with any form of anisotropy andso is extremely powerful. However, as usual with a moregeneral formalism, implementation is less straightforward.Pan’s approach requires significant computation, includingfinding the numerical solution to an eighth-order polynomialequation and the numerical differentiation of the displacementGreen’s tensor to obtain the stress or strain Green’stensor. For the purposes of direct comparison, we use thevolume integral form of the Pan’s Green’s tensor. For simplicity,we refer to the Green’s approach presented in Sec. IIas Faux-Christmas FC.First, the numerical values of the strain FC Green’s tensorfor a QD are compared to the Pan equivalent. Directcomparison of the Green’s functions is preferable to a comparisonof the strain for a specific QD as it avoids the numericaldifficulties associated with integration over a volumeof the QD. Material parameters for GaAs are used see TableI and 0 =0.01. The source is at the origin. The Green’stensor is calculated for each of the three cases; first, theisotropic case where =0 and values of and are used,second the FC polynomial solutions obtained from Eq. 16with the substitutions given by Eqs. 17–24, and finally thePan approach. In the implementation of the Pan theory, thestrain is coupled with the piezoelectric field. The computationis much more stable if the full piezoelectric constantsare included. However, for GaAs, piezoelectric effects arevery small and we have determined that they have a negligibleeffect on the current results.Figure 1 presents the absolute values of G x 1 ,x 2 ,x 3 and G x 1 ,x 2 ,x 3 plotted as a function of distance r fromthe source origin in the 100 and 111 directions, respectively.We have chosen to represent the distance in units ofnm which leaves the Green’s functions in units of nm −3 .Ona log-log plot the results are straight lines, as expected, witha gradient of −3. In all cases, the Green’s tensor is of theform 0 A/r 3 where A is a dimensionless tensor independentTABLE XIV. The coefficients N k for a quantum wire are of the formfA 3 +B 2 +C 2 +D 3 where and are the Lamé constants.k f A B C D4,0 4 38 37 −24 213,1 −16 98 247 280 1112 24 70 405 488 165Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-8 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 2005FIG. 1. The absolute values of G x 1 ,x 2 ,x 3 in the 100 direction andG x 1 ,x 2 ,x 3 in the 111 direction are plotted as a function of distancefrom the source origin r. The results for G x 1 ,x 2 ,x 3 have been displacedvertically by two decades for clarity.of r but a function of the angles and . Indeed, the substantialtheoretical effort involved in deriving the Green’stensors for anisotropic elasticity essentially reduces to findingthe form for A. The values of the tensor components of Afor the isotropic, FC, and Pan approaches are presented in z * z *Table XV. 0In all cases, the FC results are much closer to the Panvalues than for the standard isotropic approach. For example,in the 100 direction, the FC approach makes up about 96%of the difference between the isotropic value and the accuratePan result. Even in the worse case, the FC value contributesmore than 80% of the difference except in one case whereall methods give similar values and therefore constitutes asignificant improvement in accuracy with a minimum of additionalcomputation. Dederichs and Leibfried 37 used a perturbationtheory to approximate the strain field due to a pointsource in crystals with cubic anisotropy. They presented“best” isotropic elastic stiffnesses to be used for strain evaluationfor crystals with cubic anisotropy. We found, however,that their expressions for the effective elastic stiffnesses producedlittle improvement on the standard isotropic theory.A stringent test of the FC approximation is to determinethe component of the strain in the direction normal to theplane of a quantum well, call this the z * direction, for aquantum well orientated such that the z * axis points along001, 110, or111. Analytical solutions are known for theconstant strain within the well and are 38TABLE XV. The value of the components of A in the expression 0 A/r 3describing the form of the strain Green’s components for the Pan and YangRef. 24, Faux-Christmas FC, and isotropic approaches for a point sourceof strain in GaAs.Direction Green’s function Pan FC Isotropic100 A 11 0.0757 0.0830 0.2583A 22 =A 33 −0.0993 −0.1019 −0.1292A 12 =A 13 =A 23 0.000 0.0000 0.0000110 A 11 =A 22 0.134 0.1217 0.0646A 33 −0.199 −0.1947 −0.1292A 12 0.193 0.1919 0.1937A 13 =A 23 0.000 0.0000 0.0000111 A 11 =A 22 =A 33 0.0464 0.0392 0.0000A 12 =A 13 =A 23 0.198 0.1863 0.1292FIG. 2. The strain component z * z * in the direction normal to the plane of aquantum well is presented for the 001, 110, and111 directions as afunction of distance in the z * direction for the Faux-Christmas approximationsymbols and for the exact analytic formulas solid lines. The quantumwell is 2 nm thick with a misfit strain of −0.067 and centerd on the origin.GaAs elastic constants are assumed.= −2C 11, 001 42= 2 −3 − C 112 + + C 11, 110 43=2 2 −2 − C 11, 111, 444 +2 + C 11where Eq. 42 also corresponds to the isotropic result. Thestrain component z * z * in the direction normal to the plane ofa quantum well is presented in Fig. 2 for the 001, 110,and 111 directions as a function of distance in the z * directionusing Eqs. 42–44 and the FC method. The FC resultsare computed by numerical integration of the Green’s tensoras described in Sec. II as if the quantum well were a cuboidalQD with dimensions of 2002002 nm 3 . GaAs elasticconstants are assumed throughout and the misfit strain istaken to be −0.067. The FC results are in excellent agreementwith the exact results for all orientations of the well.Finally we examine the hydrostatic strain for a cuboidalQD with dimensions of 18186 nm 3 with growth directionalong the z * axis. The chosen dot size is consistent witha range of QDs including the truncated pyramid characteristicof InAs/GaAs systems for example, see Ref. 39, thelens or cylinder, and so these results illustrate the generaltrends to be found in a range of QD systems. Elastic constantsfor GaAs are again used for all materials and the misfitstrain is taken to be −0.067.We focus on three cases with the growth direction along001, 110, and 111. With the z * axis orientated along110, the x * and y * axes are taken to be along 1¯10 and001, respectively, and when the z * axis is orientated along111 the x * and y * axes are in the 11¯0 and 112¯ directions,respectively. The hydrostatic strain is calculated using Eqs.26–28. Figure 3 presents h as a function of distancealong the z * axis through the center of the dot along 001,110, and 111. Inside the QD, h is only a weak function ofDownloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp


033534-10 D. A. Faux and U. M. E. Christmas J. Appl. Phys. 98, 033534 20051997.13 G. S. Pearson and D. A. Faux, J. Appl. Phys. 88, 730 2000.14 D. A. Faux, J. R. Downes, and E. P. O’Reilly, J. Appl. Phys. 80, 25151996.15 F. Glas, Phys. Status Solidi B 237, 5992003.16 J. R. Downes, D. A. Faux, and E. P. O’Reilly, J. Appl. Phys. 81, 67001997.17 J. H. Davies, J. Appl. Phys. 84, 3581998.18 J. H. Davies, Appl. Phys. Lett. 75, 4142 1999.19 T. J. Gosling, Philos. Mag. A 73, 111996.20 N. A. Gippius and S. G. Tikhodeev, J. Exp. Theor. Phys. 88, 1045 1999.21 V. Holý, G. Springholz, M. Pinczolits, and G. Bauer, Phys. Rev. Lett. 83,356 1999.22 M. Pinczolits, G. Springholz, and G. Bauer, Phys. Rev. B 60, 115241999.23 D. A. Faux and G. S. Pearson, Phys. Rev. B 62, R4798 2000.24 E. Pan and B. Yang, J. Appl. Phys. 90, 6190 2001.25 J. M. Burgers, Proc. K. Ned. Akad. Wet. 42, 378 1939.26 D. M. Barnett, Phys. Status Solidi B 49, 7411972.27 T. Mura and N. Kinoshita, Phys. Status Solidi B 47, 6071971.28 E. Pan and F. Tonon, Int. J. Solids Struct. 37, 943 2000.29 E. Pan, J. Appl. Phys. 91, 3785 2002.30 E. Pan, J. Appl. Phys. 91, 6379 2002.31 N. Cowern, U. M. E. Christmas, J. Benson, and D. A. Faux unpublished.32 I. M. Lifshits and L. N. Rosentsveig, Zh. Eksp. Teor. Fiz. 17, 91947.33 A. D. Andreev, J. R. Downes, D. A. Faux, and E. P. O’Reilly, J. Appl.Phys. 86, 297 1999.34 S. W. Ellaway and D. A. Faux, J. Appl. Phys. 92, 3027 2002.35 B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan,and S. M. Watt, Maple Language Reference Manual: Maple V: The Futureof Mathematics Springer, New York, 1991.36 M. P. C. M. Krijn, Semicond. Sci. Technol. 6, 271991.37 P. H. Dederichs and G. Leibfried, Phys. Rev. 188, 11751969.38 L. De Caro and L. Tapfer, Phys. Rev. B 48, 2298 1993.39 P. W. Fry et al., Phys. Rev. Lett. 84, 733 2000.40 R. A. Masumura and G. Sines, J. Appl. Phys. 41, 3930 1970.41 H. E. Schaefer and H. Kronmüller, Phys. Status Solidi B 67, 631975.42 R. Shneck, R. Alter, A. Brokman, and M. P. Dariel, Philos. Mag. A 65,797 1992.43 M. Levinshtein, S. Rumyantsev, and M. Shur, Handbook Series on SemiconductorParameters World Scientific, Singapore, 1996.Downloaded 31 Oct 2005 to 131.227.244.81. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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